Non-Invertible Dynamical Systems: Volume 3 Analytic Endomorphisms of the Riemann Sphere 9783110769876, 9783110769845

Table of contents :
Preface
List of Figures
Introduction to Volume 3
Contents
Volume 3
22 The Riemann–Hurwitz formula
23 Selected tools from complex analysis
24 Dynamics and topology of rational functions: their Fatou and Julia sets
25 Selected technical properties of rational functions
26 Expanding (or hyperbolic), subexpanding, and parabolic rational functions: topological outlook
27 Equilibrium states for rational functions and Hölder continuous potentials with pressure gap
28 Invariant measures: fractal and dynamical properties
29 Sullivan’s conformal measures for rational functions
30 Conformal measures, invariant measures and fractal geometry of expanding rational functions
31 Conformal measures, invariant measures and fractal geometry of parabolic rational functions
32 Conformal measures, invariant measures and fractal geometry of subexpanding rational functions
Appendix A – A selection of classical results
Bibliography
Index

Citation preview

Mariusz Urbański, Mario Roy, Sara Munday Non-Invertible Dynamical Systems

De Gruyter Expositions in Mathematics

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Edited by Lev Birbrair, Fortaleza, Brazil Victor P. Maslov, Moscow, Russia Walter D. Neumann, New York City, New York, USA Markus J. Pflaum, Boulder, Colorado, USA Dierk Schleicher, Bremen, Germany Katrin Wendland, Freiburg, Germany

Volume 69/3

Mariusz Urbański, Mario Roy, Sara Munday

Non-Invertible Dynamical Systems �

Volume 3: Analytic Endomorphisms of the Riemann Sphere

Mathematics Subject Classification 2010 37A05, 37A25, 37A30, 37A35, 37A40, 37B10, 37B25, 37B40, 37B65, 37C05, 37C20, 37C40, 37D20, 37D35, 37E05, 37E10 Authors Prof. Dr. Mariusz Urbański University of North Texas Department of Mathematics 1155 Union Circle #311430 Denton, TX 76203-5017 USA [email protected]

Dr. Sara Munday John Cabot University Via della Lungara 233 00165 Rome Italy [email protected]

Prof. Dr. Mario Roy York University Glendon College 2275 Bayview Avenue Toronto, M4N 3M6 Canada [email protected]

ISBN 978-3-11-076984-5 e-ISBN (PDF) 978-3-11-076987-6 e-ISBN (EPUB) 978-3-11-076989-0 ISSN 0938-6572 Library of Congress Control Number: 2022950737 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2023 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

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Mariusz Urbański dedicates this book to his wife, Irena. À mes parents Thérèse et Jean-Guy, à ma famille et à mes amis, sans qui ce livre n’aurait pu voir la vie... du fond du coeur, merci! Mario

Preface Dynamical systems and ergodic theory is a rapidly evolving field of mathematics with a large variety of subfields, which use advanced methods from virtually all areas of mathematics. These subfields comprise but are by no means limited to: abstract ergodic theory, topological dynamical systems, symbolic dynamical systems, smooth dynamical systems, holomorphic/complex dynamical systems, conformal dynamical systems, onedimensional dynamical systems, hyperbolic dynamical systems, expanding dynamical systems, thermodynamic formalism, geodesic flows, Hamiltonian systems, KAM theory, billiards, algebraic dynamical systems, iterated function systems, group actions, and random dynamical systems. All of these branches of dynamical systems are mutually intertwined in many involved ways. Each of these branches nonetheless also has its own unique methods and techniques, in particular embracing methods which arise from the fields of mathematics the branch is closely related to. For example, complex dynamics borrows advanced methods from complex analysis, both of one and several variables; geodesic flows utilize methods from differential geometry; and abstract ergodic theory and thermodynamic formalism rely heavily on measure theory and functional analysis. Indeed, it is truly fascinating how large the field of dynamical systems is and how many branches of mathematics it overlaps with. In this book, we focus on some selected subfields of dynamical systems, primarily noninvertible ones. In the first volume, we give introductory accounts of topological dynamical systems acting on compact metrizable spaces, of finite-state symbolic dynamical systems, and of abstract ergodic theory of measure-theoretic dynamical systems acting on probability measure spaces, the latter including the metric entropy theory of Kolmogorov and Sinai. More advanced topics include infinite ergodic theory, general thermodynamic formalism, and topological entropy and pressure. This volume also includes a treatment of several classes of dynamical systems, which are interesting on their own and will be studied at greater length in the second volume: we provide a fairly detailed account of distance expanding maps and discuss Shub expanding endomorphisms, expansive maps, and homeomorphisms and diffeomorphisms of the circle. The second volume is somewhat more advanced and specialized. It opens with a systematic account of thermodynamic formalism of Hölder continuous potentials for open transitive distance expanding systems. One chapter comprises no dynamics but rather is a concise account of fractal geometry, treated from the point of view of dynamical systems. Both of these accounts are later used to study conformal expanding repellers. Another topic exposed at length is that of thermodynamic formalism of countable state subshifts of finite type. Relying on this latter, the theory of conformal graph directed Markov systems, with their special subclass of conformal iterated function systems, is described. Here, in a similar way to the treatment of conformal expanding repellers, the main focus is on Bowen’s formula for the Hausdorff dimension of the limit set and https://doi.org/10.1515/9783110769876-201

VIII � Preface multifractal analysis. A rather short examination of Lasota–Yorke maps of an interval is also included in this second volume. The third volume is entirely devoted to the study of the dynamics, ergodic theory, thermodynamic formalism, and fractal geometry of rational functions of the Riemann sphere. We present a fairly complete account of classical as well as more advanced topological theory of Fatou and Julia sets. Nevertheless, primary emphasis is placed on measurable dynamics generated by rational functions and fractal geometry of their Julia sets. These include the thermodynamic formalism of Hölder continuous potentials with pressure gaps, the theory of Sullivan’s conformal measures, invariant measures and their dimensions, entropy, and Lyapunov exponents. We further examine in detail the classes of expanding, subexpanding, and parabolic rational functions. We also provide, with proofs, several of the fundamental tools from complex analysis that are used in complex dynamics. These comprise Montel’s Theorem, Koebe’s Distortion Theorems and Riemann–Hurwitz formulas, with their ramifications. In virtually each chapter of this book, we describe a large number of concrete selected examples illustrating the theory and serving as examples in other chapters. Also, each chapter of the book is supplied with a number of exercises. These vary in difficulty, from very easy ones asking to verify fairly straightforward logical steps to more advanced ones enhancing largely the theory developed in the chapter. This book originated from the graduate lectures Mariusz Urbański delivered at the University of North Texas in the years 2005–2010 and that Sara Munday took notes of. With the involvement of Mario Roy, the book evolved and grew over many years. The last 2 years (2020 and 2021) of its writing were most dramatic and challenging because of the COVID-19 pandemic. Our book borrows widely from many sources including the books [73, 88, 110]. We nevertheless tried to keep it as self-contained as possible, avoiding to refer the reader too often to specific results from special papers or books. Toward this end, an appendix comprising classical results, mostly from measure theory, functional analysis and complex analysis, is included. The book covers quite a many topics treated with various degrees of completeness, none of which are fully exhausted because of their sheer largeness and their continuous dynamical growth.

List of Figures Figure 24.1

Figure 24.2

Figure 24.3

Figure 24.4

Figure 24.5

Figure 24.6

Figure 26.1 Figure 30.1 Figure 31.1

Case p = 3 (with a = −1). There are 3 attracting directions (indicated by rays and vectors pointing toward the parabolic point) and 3 repelling directions (indicated by dashed rays and vectors pointing away from the parabolic point) � 978 Case p = 5 (with a = −1). There are 5 attracting directions (indicated by rays and vectors pointing toward the parabolic point) and 5 repelling directions (indicated by dashed rays and vectors pointing away from the parabolic point) � 978 Three petals, case p = 3 (with a = −1). Some dynamics are depicted by arrows. There are 3 attracting directions (indicated by rays and vectors pointing toward the parabolic point) with respect to which the petals are symmetric and 3 repelling directions (indicated by dashed rays and vectors pointing away from the parabolic point) � 986 Four petals, case p = 4 (with a = −1). There are 4 attracting directions (indicated by rays and vectors pointing toward the parabolic point) with respect to which the petals are symmetric and 4 repelling directions (indicated by dashed rays and vectors pointing away from the parabolic point). Some dynamics in and near one of the petals are depicted by the arrows. Reproduce them in and near the other three petals � 987 Four petals, case p = 4 (with a = −1). There are 4 attracting directions (indicated by rays and vectors pointing towards the parabolic fixed point) with respect to which the petals are symmetric and 4 repelling directions (indicated by dashed rays and vectors pointing away from the point). Some dynamics within one of the petals are depicted by the arrows. Reproduce them in the other three petals � 995 Four petals, case p = 4 (with a = −1). There are 4 attracting directions (indicated by rays and vectors pointing towards the parabolic fixed point) with respect to which the petals are symmetric and 4 repelling directions (indicated by dashed rays and vectors pointing away from the point). Some dynamics within one of the petals are depicted by the arrows. Reproduce them in the other three petals � 996 The Mandelbrot set � 1093 Graph of the pressure function t 󳨃→ P(t) for an expanding rational function � 1234 Graph of t 󳨃→ P(t) for a parabolic rational function � 1251

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Introduction to Volume 3 In the first volume, we developed the general theory of thermodynamic formalism, that is, for general topological dynamical systems (i. e., continuous self-maps of a compact metrizable space) subject to general potentials (i. e., continuous potentials). In the second volume, we carried on with the study of thermodynamic formalism and its dynamical and geometric applications, but this time for specific classes of systems subject to specific classes of potentials, including topologically transitive open and distance expanding maps under Hölder continuous potentials. The third volume concentrates on the dynamical and fractal properties of endomorphisms of the Riemann sphere. We now describe in more detail the content of each chapter of this third volume, including their mutual dependence and interrelations.

Introduction ̂ emerged in the secThe theory of iteration of rational functions of the Riemann sphere ℂ ond half of the 19th century with the works of Böttcher [14–18], Cayley [22–27], Farkas [47], Koenigs [64–69], Korkine [70], Leau [75–77] and Schröder [116–118] on the local behavior of rational functions near their attracting and indifferent fixed points and on Newton’s method. The mature and modern treatment of the iteration of rational functions appeared at the beginning of the 20th century with the long series of works by Pierre Fatou (see [48] and the references in [3, 4]) and Gaston Julia (see [61] and the references in [3, 4]), which were prompted by the French Academy of Science Award competition announcement. The seminal work of Paul Montel (see the references in [3, 4]) on normal families of analytic functions rendered this modern, comprehensive and systematic approach to iteration of rational functions, both locally and globally, possible. Fatou and Julia introduced many concepts and proved several theorems significantly examining the sets that nowadays bear their names. Since then and until the late 1970s, only scattered isolated papers were published. Among others, let us mention the works of Cremer [29] and Siegel [121], who respectively proved the existence of what is now called Cremer points and Siegel disks. These latter are of profound importance in the KAM theory. This early period of development of the theory of iteration of rational functions is thoroughly presented and well documented in the beautiful books [3, 4]. The contemporary era in the theory of iteration of rational functions truly began when Dennis Sullivan proved in his breakthrough paper [131] Fatou’s conjecture from [48] and [49] that rules out the existence of wandering domains. The importance of this paper also lies in the introduction to complex dynamics of the powerful tool of quasiconformal mappings, which have been widely used in rational and transcendental complex dynamics ever since. We mention here only a few other milestones which highly influenced and accelerated the progress and development of complex dynamics. Namely, Mandelbrot’s original definition and numerical studies of his namesake set [81]; Douady https://doi.org/10.1515/9783110769876-203

XII � Introduction to Volume 3 and Hubbard’s pioneering works [43, 44] on the theoretical understanding of the topological and conformal structure of the Mandelbrot set; Sullivan’s works [127, 129] introducing conformal measures to the realm of rational functions; and Rufus Bowen’s work [19] which launched the use of thermodynamic formalism in conformal dynamical systems (more specifically, quasi-Fuchsian groups). Since then, research on complex dynamics has exploded. It has covered a large variety of topics, such as genericity of expanding maps, the topological and fractal structure of the Mandelbrot set and quadratic polynomials, the topological and fractal structure of Julia sets, the structure of Fatou sets and the boundaries of their connected components, Newton’s method and the measurable dynamics and thermodynamic formalism induced by rational functions. Part of this development has been documented in book monographs such as [8, 21, 94, 150] as well as in Lyubich’s expository article [80] and Urbański’s paper [141]. In this volume, we present the core of the theory of iteration of rational functions in its topological, conformal, and fractal aspects. We base it on fundamental and powerful tools and methods from complex analysis, such as the Riemann–Hurwitz formula, normal families of analytic functions, Montel’s theorem and Koebe’s distortion theorems. Because of their continual use, and in fact their indispensability, we prove them all in Chapters 22–23 for the sake of completeness. Normal families, Montel’s theorem and Koebe’s distortion theorems are classic; the Riemann–Hurwitz formula, too, but following Beardon’s approach in [8], we present it in a form tailored for the needs of complex dynamics, that is, to handle the topological structure of images and preimages of open ̂ connected sets in ℂ.

In Chapter 24, we prove all the standard properties and characterizations of the Julia and Fatou sets, including the density of repelling periodic points in the Julia set, the perfectness of the Julia set, the topological exactness of the map restricted to its Julia set, as well as a detailed analysis of the components of the Fatou set and their boundaries, with a complete proof of a classification of their periodic components. We also provide a thorough account of the local behavior of the map around its periodic points. We pay particular attention to rationally indifferent fixed and periodic points. In addition to qualitative properties such as the Fatou–Leau flower theorem, we give a rigorous quantitative description of the local and asymptotic behavior of the iterates of points in attracting and repelling petals. Chapter 25 comprises a number of technical properties of rational functions, in particular, their behavior around their critical points. It also includes Lyubich’s geometric lemma and special types of partitions such as Mañé’s partition. Chapter 26 introduces the three classes of rational functions whose topological, fractal, conformal and thermodynamic properties are studied in this chapter and from Chapter 30 on, namely the expanding, subexpanding and parabolic rational functions. The expanding functions form a subclass of the subexpanding ones, whereas the classes of expanding and parabolic functions are disjoint. The union of these latter two classes constitutes the family of expansive rational functions. We first treat these three classes

Introduction

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from a topological viewpoint, establishing among others their dynamical and topological characterizations and proving the continuity of the Julia sets of expanding and ̂ Using the techsubexpanding maps in the Hausdorff metric of compact subsets of ℂ. nique of quasisymmetric and quasiconformal mappings developed in Chapter 16, we demonstrate the structural Julia stability of expanding maps, and as a consequence show that the Julia sets of the quadratic maps z 󳨃→ z2 + c, where c belongs to the main cardioid of the Mandelbrot set, are quasicircles and in particular Jordan curves. We then move on to topics that originate from the paper [111] but that have not had any book presentation yet (except for one chapter in [110], which partially overlaps with some of them). Chapter 27 pertains to thermodynamic formalism for all rational functions and Hölder continuous potentials with a pressure gap as described in [36, 106], including further advances due to [60, 133, 134]. Our exposition of this topic is improved and simplified via the combination and further development of the methods from all those papers. The main result is the existence and uniqueness of equilibrium states, their particular form and some stochastic properties such as K-mixing. Chapter 28 treats of invariant measures and fractal and dynamical properties of rational functions, including the nonnegativity of Lyapunov exponents due to Przytycki, Ruelle’s inequality, Pesin’s theory, volume lemmas, and conformal Katok’s theory. We also introduce three significant subsets of the Julia set, namely the radial (conical), expanding radial and uniformly expanding radial subsets. We study them at length in all subsequent chapters, especially in Chapter 29. In Chapter 29, we give a general and detailed treatment of Sullivan’s conformal measures. Our approach arises from [34, 35] and comprises the progress made in [110]. By means of a fairly effective construction, we prove as the main result the existence of a Sullivan conformal measure with a minimal exponent and equate this exponent with not only the Hausdorff dimension but also many other fractal characteristics/dimensions of the Julia set, such as its expanding (hyperbolic) dimension, its dynamical dimension, and the Hausdorff dimension of radial, expanding radial and uniformly expanding radial subsets of the Julia set. For the first time ever, the use of the otherwise very powerful technique of K(V ) sets published in [35] is avoided. In the remaining chapters, we focus on the fractal properties and measurable dynamics generated by the three classes of functions examined in Chapter 26. The most transparent ones are found in the class of expanding maps scrutinized in Section 26.1 and especially in Chapter 30: the classical version of Bowen’s formula, a unique Sullivan’s conformal measure (whose exponent is necessarily equal to the Hausdorff dimension h of the Julia set), the finiteness and positivity of the h-dimensional Hausdorff and packing measures, their equality (up to a multiplicative constant) with the conformal measure and their Ahlfors (i. e., geometric) property. As a repercussion, all fractal dimensions (commonly used in complex dynamics) of the Julia set of an expanding map T are equal. The Gibbs/equilibrium state of the Hölder continuous potential −h log |T ′ | is the unique invariant Borel probability measure absolutely continuous with respect to

XIV � Introduction to Volume 3 the unique conformal measure. Furthermore, those two measures are equivalent with log-bounded Radon–Nikodym derivatives and, in particular, the invariant Gibbs measure is geometric. As mentioned earlier, the expanding and parabolic classes are mutually disjoint but their union coincides with the family of expansive maps. Although there are some similarities, the portrait for parabolic rational functions, given in Section 26.2 and especially in Chapter 31, is quite different from that for expanding and subexpanding maps. The potential −h log |T ′ | is well-defined and even Hölder continuous but because of the presence of rationally indifferent periodic points, it does not exhibit a pressure gap. In particular, the results proved in Chapter 27 do not apply. Nevertheless, there is a neat version of Bowen’s formula, which identifies the Hausdorff dimension h of the Julia set as the least zero of the pressure function P(t) but, unlike in the expanding case, P(t) = 0 for all t > h. We also show that h is equal to the minimal exponent for which a conformal measure exists. As in the subexpanding case in Chapter 32, there exists a unique Sullivan’s h-conformal measure and that measure is atomless; it is even ergodic and conservative. Similarly to the subexpanding case, there exist t-conformal measures for all t > h. All of those are purely atomic and are convex combinations of atomic measures supported on the backward orbits of rationally indifferent periodic points. In Section 31.4, we establish the existence and uniqueness of an invariant measure which is absolutely continuous with respect to the h-conformal measure, and we provide a simple criterion for this measure to be either finite or infinite. In particular, this measure is finite for the quadratic polynomial z 󳨃→ z2 + 41 , while it is infinite for parabolic Blaschke products. In the last section of Chapter 31, we provide a complete description of the h-dimensional Hausdorff and packing measures on the Julia sets of parabolic rational functions. These are sometimes finite, sometimes positive, and sometimes infinite, their nature depending exclusively on the value of the Hausdorff dimension h. Chapter 31 combines and develops the results proved in several of Denker and Urbański’s papers [1, 37, 38, 40]. For subexpanding maps, the picture is in some respects quite similar to that of expanding maps, but is considerably different in some other respects. Despite the possible presence of critical points in the Julia set and the classical topological pressure of the potential −h log |T ′ | not necessarily being well-defined, the equality of all commonly used fractal dimensions holds. There also exists a unique Sullivan’s h-conformal measure and that measure is atomless. As in the case of expanding maps, the h-dimensional Hausdorff and packing measures are finite, positive and equal to the h-conformal measure, up to a multiplicative constant. They also share the Ahlfors property. We further show, and this is the most involved proof in Chapter 32, the existence and uniqueness of an invariant Borel probability measure absolutely continuous with respect to the unique h-conformal measure. Moreover, these two measures are equivalent though this time not necessarily with log-bounded Radon–Nikodym derivatives. As in the expanding case in Chapter 30 and the parabolic case in Chapter 31, there are no t-conformal measures for

Euclidean metric/derivative versus spherical metric/derivative

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t < h. The drastic difference between subexpanding and expanding maps is that if critical points are found in the Julia set of a subexpanding map, then there exist t-conformal measures for all t > h. All of those are purely atomic and are convex combinations of atomic measures supported on the backward orbits of critical points lying in the Julia set. Our presentation of subexpanding maps, given in Section 26.3 and especially in Chapter 32, stems partly from the paper [39] but both improves it substantially and goes beyond it. The field of complex dynamics is beautiful, deep and wide. It has many subfields that, to some extent, live on their own, have their respective goals, methods, problems and results. We cover quite a bit of those but still much less than we left aside. We do not deeply deal with genericity of expandingness, the Mandelbrot set, external rays, Yoccoz puzzles, quadratic polynomials, quasiconformal surgeries, issues of local connectedness, the existence of Siegel disks (and rotation numbers for which these exist), the existence of Herman rings, parabolic implosion, Newton’s method or polynomials with a positive plane Lebesgue measure. Nor do we deal with topics quite close to Chapters 28–32 such as harmonic measure, ergodic theory and fractal geometry of nonrecurrent rational functions and topological Collet–Eckmann maps. We do not dive very deep into the topic of geometric pressure, i. e., for potentials of the form −t log |T ′ | and all rational functions, although we get quite close to it in Chapter 29. Many of these topics would deserve an entire book or even several books.

Euclidean metric/derivative versus spherical metric/derivative Whenever possible, we will work with the standard Euclidean metric and the ordinary Euclidean derivative. This is especially so when local topics are considered since one can then always, by means of a Möbius change of coordinates (i. e., via a Möbius conjugacy), ̂ to a map from an open subset of the complex plane ℂ to ℂ. move the problem from ℂ But when dealing with global issues, the use of the spherical metric and of the spherical derivative is actually necessary. Although this use is inconvenient and somewhat awkward, the essentially only alternative is to work with local holomorphic charts, which may be even more cumbersome. In addition to local issues, the other clear cut context in which one can rely upon the Euclidean metric/derivative is when the Julia set of a rational map is not the whole ̂ and one is only interested in the dynamics on a sufficiently small Riemann sphere ℂ open neighborhood of the Julia set. One can then similarly, i. e., by means of a Möbius conjugacy, send the Julia set outside of an open neighborhood of infinity. For instance, this is the case for any polynomial and any expansive rational function. Since the spherical metric and the spherical derivative behave frequently quite well, and in particular since there are dependable distortion theorems about them (see Section 23.1), we will sometimes not explicitly state which metric or derivative is being considered; we will just write |T ′ (z)| for the modulus (scaling factor) of either the

XVI � Introduction to Volume 3 Euclidean or the spherical derivative of a meromorphic map T. But if distinguishing between these two derivatives is truly substantive and choosing the spherical one is actually necessary, we will then clearly indicate that we work with the spherical metric ρ and the spherical derivative |T ′ (z)|ρ .

Contents Volume 1 1 1.1 1.2 1.3 1.4 1.5 1.5.1 1.5.2 1.5.3 1.6 1.6.1 1.6.2 1.7

Dynamical systems Basic definitions Topological conjugacy and structural stability Factors Subsystems Mixing and irreducibility Minimality Transitivity and topological mixing Topological exactness Examples Rotations of compact topological groups Maps of the interval Exercises

2 2.1 2.2 2.2.1 2.3 2.3.1 2.3.2 2.4

Homeomorphisms of the circle Lifts of circle maps Orientation-preserving homeomorphisms of the circle Rotation numbers Minimality for homeomorphisms and diffeomorphisms of the circle Denjoy’s theorem Denjoy’s counterexample Exercises

3 3.1 3.2 3.2.1 3.2.2 3.2.3 3.3 3.4

Symbolic dynamics Full shifts Subshifts of finite type Topological transitivity Topological exactness Asymptotic behavior of periodic points General subshifts of finite type Exercises

4 4.1 4.1.1 4.1.2 4.2 4.3

Distance expanding maps Definition and examples Expanding repellers Hyperbolic Cantor sets Inverse branches Shadowing

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XVIII � Contents 4.4 4.5 4.6

Markov partitions Symbolic representation generated by a Markov partition Exercises

5 5.1 5.2 5.3 5.4 5.5

(Positively) expansive maps Expansiveness Uniform expansiveness Expansive maps are expanding with respect to an equivalent metric Parabolic Cantor sets Exercises

6 6.1 6.2 6.3 6.3.1 6.3.2 6.4 6.4.1 6.4.2 6.4.3 6.5

Shub expanding endomorphisms Shub expanding endomorphisms of the circle Definition, characterization, and properties of general Shub expanding endomorphisms A digression into algebraic topology Deck transformations Lifts Dynamical properties Expanding property Topological exactness and density of periodic points Topological conjugacy and structural stability Exercises

7 7.1 7.1.1 7.2 7.2.1 7.2.2 7.2.3 7.3 7.4 7.5 7.6

Topological entropy Covers of a set Dynamical covers Definition of topological entropy via open covers First stage: entropy of an open cover Second stage: entropy of a system relative to an open cover Third and final stage: entropy of a system Bowen’s definition of topological entropy Topological degree Misiurewicz–Przytycki theorem Exercises

8 8.1 8.1.1 8.1.2 8.1.3 8.2

Ergodic theory Measure-preserving transformations Examples of invariant measures Poincaré’s recurrence theorem Existence of invariant measures Ergodic transformations

Contents �

8.2.1 8.2.2 8.2.3 8.2.4 8.3 8.3.1 8.3.2 8.3.3 8.4 8.5

Birkhoff’s ergodic theorem Existence of ergodic measures Examples of ergodic measures Uniquely ergodic transformations Mixing transformations Weak mixing Mixing K-mixing Rokhlin’s natural extension Exercises

9 9.1 9.2 9.3 9.4 9.4.1 9.4.2 9.4.3 9.5 9.6 9.7

Measure-theoretic entropy An excursion into the origins of entropy Partitions of a measurable space Information and conditional information functions Definition of measure-theoretic entropy First stage: entropy and conditional entropy for partitions Second stage: entropy of a system relative to a partition Third and final stage: entropy of a system Shannon–McMillan–Breiman theorem Brin–Katok local entropy formula Exercises

10 10.1 10.2 10.3 10.4 10.5

Infinite invariant measures Quasi-invariant measures, ergodicity and conservativity Invariant measures and inducing Ergodic theorems Absolutely continuous σ-finite invariant measures Exercises

11 11.1 11.1.1 11.1.2 11.2 11.3 11.4 11.5

Topological pressure Definition of topological pressure via open covers First stage: pressure of a potential relative to an open cover Second stage: the pressure of a potential Bowen’s definition of topological pressure Basic properties of topological pressure Examples Exercises

12 12.1 12.1.1

The variational principle and equilibrium states The variational principle Consequences of the variational principle

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XX � Contents 12.2 12.3 12.4

Equilibrium states Examples of equilibrium states Exercises

Volume 2 13 13.1 13.2 13.2.1 13.2.2 13.3 13.4 13.5 13.6 13.6.1 13.6.2 13.6.3 13.6.4 13.6.5 13.7 13.7.1 13.7.2 13.7.3 13.7.4 13.7.5 13.7.6 13.8 13.8.1 13.8.2 13.8.3 13.9 13.9.1 13.9.2 13.9.3 13.9.4 13.9.5 13.10 13.11

Gibbs states and transfer operators for open, distance expanding systems Hölder continuous potentials Gibbs measures Definition and properties Invariant Gibbs states are equilibrium states Jacobians and changes of variables Construction of invariant measures from quasi-invariant ones Transfer operators Nonnecessarily-invariant Gibbs states Existence of eigenmeasures for the dual of the transfer operator Eigenmeasures are conformal measures Eigenmeasures are Gibbs states for transitive systems Ergodicity of the eigenmeasures for transitive systems Metric exactness of the eigenmeasures for topologically exact systems Invariant Gibbs states Almost periodicity of normalized transfer operators Existence, uniqueness and ergodicity of invariant Gibbs states for transitive systems Invariant Gibbs states and equilibrium states coincide and are unique Hölder continuous potentials with the same Gibbs states Invariant Gibbs states have positive entropy; pressure gap Absolutely continuous invariant measures for Shub expanding maps Finer properties of transfer operators and Gibbs states Iterates of transfer operators Ionescu-Tulcea and Marinescu inequality and spectral gap Continuity of Gibbs states Stochastic laws Exponential decay of correlations Asymptotic variance Central limit theorem Law of the iterated logarithm Metric exactness, K -mixing and weak Bernoulli property Real analyticity of topological pressure Exercises

Contents �

14 14.1 14.2 14.3 14.4 14.5

Lasota–Yorke maps Definition Transfer operator Existence of absolutely continuous invariant probability measures Exponential decay of correlations Exercises

15 15.1 15.2 15.2.1 15.2.2 15.2.3 15.2.4 15.3 15.3.1 15.3.2 15.3.3 15.3.4 15.3.5 15.3.6 15.3.7 15.4 15.5 15.6 15.7

Fractal measures and dimensions Outer measures Geometric (Hausdorff and packing) outer measures and dimensions Gauge functions Hausdorff measures Packing measures Packing versus Hausdorff measures Dimensions of sets Hausdorff dimension Packing dimensions Packing versus Hausdorff dimensions Box-counting dimensions Alternate definitions of box dimensions Hausdorff versus packing versus box dimensions Hausdorff, packing and box dimensions under (bi-)Lipschitz mappings A digression into geometric measure theory Volume lemmas—Frostman converse theorems Dimensions of measures Exercises

16 16.1 16.2 16.3 16.3.1 16.4 16.5 16.6 16.7

Conformal expanding repellers Conformal maps Conformal expanding repellers Bowen’s formula Special case of Hutchinson’s formula Real-analytic dependence of Hausdorff dimension of repellers in ℂ Dimensions of measures, Lyapunov exponents and measure-theoretic entropy Multifractal analysis of Gibbs states Exercises

17 17.1 17.1.1 17.2

Countable state thermodynamic formalism Finitely irreducible subshifts Finitely primitive subshifts Topological pressure

XXI

XXII � Contents 17.2.1 17.2.2 17.2.3 17.3 17.4 17.5 17.6 17.7 17.8 17.9

Potentials: acceptability and Hölder continuity on cylinders Partition functions The pressure function Variational principles and equilibrium states Gibbs states Gibbs states versus equilibrium states Transfer operator Existence and uniqueness of eigenmeasures of the dual transfer operator, of Gibbs states and of equilibrium states The invariant Gibbs state has positive entropy; pressure gap Exercises

18 18.1 18.2 18.3 18.3.1 18.3.2 18.3.3 18.3.4 18.4 18.5

Countable state thermodynamic formalism: finer properties Ionescu-Tulcea and Marinescu inequality Continuity of Gibbs states Stochastic laws Exponential decay of correlations Asymptotic variance Central limit theorem Law of the iterated logarithm Potentials with the same Gibbs states Exercises

19 19.1 19.1.1 19.1.2 19.1.3 19.2 19.3 19.4

Conformal graph directed Markov systems Graph directed Markov systems The underlying multigraph 𝒢 The underlying matrix A The system itself Properties of conformal maps in ℝd , d ≥ 2 Conformal graph directed Markov systems Topological pressure, finiteness parameter, and Bowen parameter for CGDMSs Classification of CGDMSs Bowen’s formula for CGDMSs The finite case The general case Hutchinson’s formula Other separation conditions and cone condition The strong open set condition The boundary separation condition The strong separation condition The cone condition

19.5 19.6 19.6.1 19.6.2 19.6.3 19.7 19.7.1 19.7.2 19.7.3 19.7.4

Contents

19.7.5 19.8 19.8.1 19.8.2 19.8.3 19.8.4 19.9 19.10

� XXIII

Conformal-likeness Hölder families of functions and conformal measures Basic definitions and properties Conformal measures for summable Hölder families of functions Conformal measures for CGDMSs Dimensions of measures for CGDMSs Examples Exercises

20 20.1 20.2 20.3 20.4 20.5

Real analyticity of topological pressure and Hausdorff dimension Real analyticity of pressure: part I Real analyticity of pressure: part II Real analyticity of pressure: part III Real analyticity of Hausdorff dimension for CGDMSs in ℂ Exercises

21 21.1 21.2 21.3 21.3.1 21.3.2 21.4 21.5

Multifractal analysis for conformal graph directed Markov systems Pressure and temperature Multifractal analysis of the conformal measure mF over a subset of J Multifractal analysis over J Under the boundary separation condition Under other conditions Multifractal analysis over another subset of J Exercises

Volume 3 Preface � VII List of Figures � IX Introduction to Volume 3 � XI 22 22.1 22.2 22.3 22.4 22.5

The Riemann–Hurwitz formula � 925 Proper analytic maps and their degree � 925 The Euler characteristic of plane bordered surfaces � 928 ̂ � 930 The Riemann–Hurwitz formula for bordered surfaces in ℂ ̂ case � 937 Euler characteristic: the general ℂ ̂ case � 940 Riemann–Hurwitz formula: the general ℂ

23

Selected tools from complex analysis � 945

XXIV � Contents 23.1 23.2 24

Koebe distortion theorems � 945 Normal families and Montel’s theorem � 951

24.4 24.4.1 24.4.2 24.4.3 24.5 24.6 24.7 24.8

Dynamics and topology of rational functions: their Fatou and Julia sets � 955 Fatou set, Julia set, and periodic points � 955 Attracting periodic points � 962 Nonattracting periodic points � 971 Rationally indifferent periodic points � 976 Local and asymptotic behavior of holomorphic functions around rationally indifferent periodic points: part I � 976 Leau–Fatou flower petals � 998 Local and asymptotic behavior of rational functions around rationally indifferent periodic points: part II – Fatou’s flower theorem and fundamental domains � 1004 Nonattracting periodic points revisited: total number and denseness of repelling periodic points � 1010 The structure of the Fatou set � 1017 Forward invariant components of the Fatou set � 1017 Periodic components of the Fatou set � 1026 The postcritical set � 1028 Cremer points, boundary of Siegel disks and Herman rings � 1029 Continuity of Julia sets � 1031 Polynomials � 1034 Exercises � 1039

25 25.1 25.2 25.3 25.4 25.5 25.5.1 25.5.2 25.5.3 25.6 25.6.1 25.6.2

Selected technical properties of rational functions � 1045 Passing near critical points: results and applications � 1045 Two rules for critical points � 1061 Expanding subsets of Julia sets � 1068 Lyubich’s geometric lemma � 1079 Two auxiliary partitions � 1081 Boundary partition � 1081 Exponentially large partition � 1082 Mañé’s partition � 1082 Miscellaneous facts � 1084 Miscellaneous general facts � 1084 Miscellaneous facts about rational functions � 1085

26

Expanding (or hyperbolic), subexpanding, and parabolic rational functions: topological outlook � 1087 Expanding rational functions � 1087

24.1 24.1.1 24.1.2 24.2 24.2.1 24.2.2 24.2.3

24.3

26.1

Contents

26.1.1 26.2 26.3 26.4 27 27.1 27.2 27.2.1 27.2.2 27.2.3 27.3 27.4 27.5 27.5.1 27.6 27.7 27.8 27.9 27.10 28 28.1 28.2 28.3 28.4

� XXV

The Mandelbrot set � 1092 Expansive and parabolic rational functions � 1093 Subexpanding rational functions � 1097 Exercises � 1108 Equilibrium states for rational functions and Hölder continuous potentials with pressure gap � 1109 Bad and good inverse branches � 1109 The transfer operator ℒφ : C(𝒥 (T )) → C(𝒥 (T )): its lower and upper bounds � 1120 First lower bounds on ℒφ � 1120 Auxiliary operators Lβ � 1125 ̂φ ; the eigenmeasure mφ � 1128 Final bounds on ℒφ and its normalization ℒ ̂φ ; “Gibbs states” mφ and μφ � 1133 Equicontinuity of iterates of ℒ ̂φ and their dynamical Spectral properties of the transfer operator ℒ consequences � 1137 Equilibrium states for Hölder potentials φ : 𝒥 (T ) → ℝ � 1143 The “Gibbs state” μφ is an equilibrium state for φ � 1143 Continuous dependence on φ of the “Gibbs states” mφ , μφ and of the density ρφ � 1146 Differentiability of topological pressure � 1150 Uniqueness of equilibrium states for Hölder continuous potentials � 1153 Assorted remarks � 1157 Exercises � 1160

28.6.1 28.6.2 28.7

Invariant measures: fractal and dynamical properties � 1161 Lyapunov exponents are nonnegative � 1161 Ruelle’s inequality � 1166 Pesin’s theory in a conformal setting � 1168 Volume lemmas; Hausdorff and packing dimensions of invariant measures � 1172 HD(𝒥 (T )) > 0 and radial Julia sets 𝒥r (T ), 𝒥er (T ), 𝒥uer (T ) � 1178 Conformal Katok’s theory of expanding sets and topological pressure � 1184 Pressure-like definition of the functional hμ (T ) + ∫ φ dμ � 1184 Conformal Katok’s theory � 1188 Exercises � 1196

29 29.1 29.1.1 29.1.2

Sullivan’s conformal measures for rational functions � 1197 General concept of conformal measures � 1198 Motivation for and definition of general conformal measures � 1198 Selected properties of general conformal measures � 1201

28.5 28.6

XXVI � Contents 29.1.3 29.1.4 29.2 29.3 29.4 30 30.1 30.2 30.3 31 31.1 31.2 31.3 31.3.1 31.3.2 31.3.3

31.4 31.5 31.6 32 32.1 32.1.1 32.1.2 32.1.3

32.2 32.3

The limit construction and PS limit measures � 1205 Conformality properties of PS limit measures � 1208 Sullivan’s conformal measures � 1211 Pesin’s formula � 1227 Exercises � 1230 Conformal measures, invariant measures and fractal geometry of expanding rational functions � 1233 Fundamental fractal geometry: Bowen’s formula � 1233 Geometric rigidity � 1238 Exercises � 1241 Conformal measures, invariant measures and fractal geometry of parabolic rational functions � 1243 General conformal measures for expansive topological dynamical systems � 1243 Geometric topological pressure and generalized geometric conformal measures for parabolic rational functions � 1248 Sullivan conformal measures for parabolic rational functions � 1251 Technical preparations � 1252 The atomless hT -conformal measure mT : existence, uniqueness, ergodicity and conservativity � 1259 The complete structure of Sullivan’s conformal measures for parabolic rational functions: the atomless measure mT and purely atomic measures � 1267 Invariant measures equivalent to mT : existence, uniqueness, ergodicity and the finite–infinite dichotomy � 1270 Hausdorff and packing measures � 1275 Exercises � 1283 Conformal measures, invariant measures and fractal geometry of subexpanding rational functions � 1285 Sullivan conformal measures for subexpanding rational functions � 1286 Technical preparations and Bowen’s formula � 1286 The atomless hR -conformal measure mR : existence, uniqueness, ergodicity and conservativity � 1291 The complete structure of Sullivan’s conformal measures for subexpanding rational functions: the atomless measure mR and purely atomic measures � 1308 Invariant probability measure equivalent to mR : existence, uniqueness and ergodicity � 1310 Hausdorff and packing measures � 1312

Contents

32.4 A A.1 A.1.1 A.1.2 A.1.3 A.2 A.3

Exercises � 1313 A selection of classical results � 1263 Measure theory � 1315 Convergence theorems � 1315 Mutual singularity, absolute continuity and equivalence of measures � 1319 The space C(X), its dual C(X)∗ and the subspace M(X) � 1319 Functional analysis � 1322 Complex analysis in one variable � 1323

Bibliography � 1327 Index � 1333

�

XXVII

22 The Riemann–Hurwitz formula In this chapter, we present a relation between holomorphic maps, their degree, their critical points and the topological structure of their images and preimages. This relation is known as the Riemann–Hurwitz formula. This formula has a long history and is treated in many textbooks on Riemann surfaces and algebraic geometry. It is usually formulated for compact Riemann surfaces. However, as we need greater generality, we provide here a complete, self-contained exposition which fulfills our needs. The above-mentioned topological structure enters our considerations in terms of the Euler characteristic. We briefly introduce this latter, in a restricted scope and in the most elementary way; in particular, we only speak of triangulations rather than dealing with the more general and flexible concept of CW-complexes. The results of this chapter, which are various versions of the Riemann–Hurwitz formula and their repercussions, will be instrumental throughout this volume. These formulas most likely constitute the best tool to handle the topological structure of connected components of preimages of open connected sets under holomorphic maps, especially to make sure that such connected components are simply connected. Our exposition stems from that of Beardon [8]. We begin with a preparatory section devoted to proper maps and their topological degree.

22.1 Proper analytic maps and their degree In this short section, we examine the concept of proper holomorphic maps and their degree. We closely follow Section 4.2 of Forster’s book [51]. Let X and Y be topological spaces. A continuous map f : X → Y is said to be proper if f −1 (K) is compact for every compact set K ⊆ Y . Proposition 22.1.1. Every proper map between locally compact Hausdorff topological spaces is closed, i. e., the image of every closed set is closed. Proof. Let X and Y be locally compact Hausdorff spaces and let f : X → Y be a proper map. Let F be a closed subset of X and w ∈ f (F). As Y is a locally compact T2 space, there exists an open neighborhood W of w in Y such that W is compact. Since f is proper, the preimage f −1 (W ) is a compact subset of X. As F is closed in X, the set F ∩ f −1 (W ) is thus compact. The continuity of f then ensures that the set f (F ∩ f −1 (W )) is compact, and hence closed in the Hausdorff space Y . If V is any open neighborhood of w, then so is W ∩ V . As w ∈ f (F), we have (f (F) ∩ W ) ∩ V = f (F) ∩ (W ∩ V ) ≠ 0. Therefore, w ∈ f (F) ∩ W ⊆ f (F) ∩ W . https://doi.org/10.1515/9783110769876-022

926 � 22 The Riemann–Hurwitz formula Since f (F) ∩ W = f (F ∩ f −1 (W )) is a closed set, we deduce that w ∈ f (F ∩ f −1 (W )) = f (F ∩ f −1 (W )) ⊆ f (F). So, f (F) ⊆ f (F), i. e., the set f (F) is closed. A continuous map f : X → Y is called discrete if for every y ∈ Y the preimage f −1 (y) is discrete in X. More precisely, this means that f −1 (y), when viewed as a topological subspace of X endowed with its relative topology, is a discrete topological space. Equivalently, for every point x ∈ f −1 (y) there exists an open set U ⊆ X such that U ∩f −1 (y) = {x}. Lemma 22.1.2. Suppose that X, Y are locally compact Hausdorff spaces and f : X → Y is a proper, discrete map. The following statements hold: (a) For every point y ∈ Y , the preimage f −1 (y) is a finite set. (b) If y ∈ Y and U is a neighborhood of f −1 (y), then there exists a neighborhood V of y such that f −1 (V ) ⊆ U. (c) If D ⊆ X is a closed discrete set, then so is f (D). Proof. Item (a) follows from the fact that f −1 (y) is a compact discrete subset of X. For item (b), we may assume that U is open. As the set X \ U is closed, its image f (X \ U) is closed according to Proposition 22.1.1. Observe also that y ∉ f (X \ U). Thus, V := Y \ f (X \ U) is an open neighborhood of y such that f −1 (V ) ⊆ U. To prove (c), fix x ∈ D. By (a), the set f −1 (f (x)) is finite. Since D is closed and discrete and #f −1 (f (x)) < ∞, there exists an open neighborhood U of f −1 (f (x)) such that U ∩ D = f −1 (f (x)) ∩ D. Then f (x) ∉ f (X \ U), and f (X \ U) is a closed set per Proposition 22.1.1. So, Y \ f (X \ U) is an open neighborhood of f (x) and f (x) ∈ (Y \ f (X \ U)) ∩ f (D) ⊆ f (U ∩ D) ⊆ {f (x)} ∩ f (D) = {f (x)}. Thus, (Y \ f (X \ U)) ∩ f (D) = {f (x)}, hence proving that f (D) is discrete. The closedness of f (D) follows from Proposition 22.1.1. We now turn our attention to holomorphic maps between Riemann surfaces. Recall that given a nonconstant holomorphic map f : X → Y between two Riemann surfaces X and Y , the local degree of f at z, denoted by degz (f ), is the unique k ∈ ℕ such that (using any pair of charts around z and f (z)) f (ζ ) = f (z) + a(ζ − z)k + O((ζ − z)k+1 )

(22.1)

for some a ∈ ℂ \ {0}. Observe that degz (f ) = 1 if and only if z is not a critical point of f .

22.1 Proper analytic maps and their degree

� 927

Theorem 22.1.3. Suppose that X and Y are Riemann surfaces and f : X → Y is a nonconstant proper holomorphic map. Then there exists n ∈ ℕ such that f takes every value w ∈ Y exactly n times (counting multiplicities), i. e., ∑ z∈f −1 (w)

degz (f ) = n.

In particular, the map f is surjective. The number n is called the (topological) degree of f and is denoted by deg(f ). Proof. It is a simple consequence of Taylor’s theorem that the zeros of a nonzero holomorphic map are isolated. Consequently, the map f is discrete. It also ensues from this and the continuity of f ′ that the set Crit(f ) of critical (branch) points of f is closed and discrete. It is called the critical set of f . Since f is proper and discrete, it follows from Lemma 22.1.2 that the set of critical values f (Crit(f )) of f is also closed and discrete. Let X ′ := X \ f −1 (f (Crit(f ))) ⊆ X \ Crit(f ) and

Y ′ := Y \ f (Crit(f )).

Since f |X ′ : X ′ → Y ′ is a proper discrete covering map (as a proper local homeomorphism between locally compact Hausdorff spaces) and since the space X ′ is pathwise connected, it follows from Lemma 22.1.2(a) that there exists n ∈ ℕ such that #(f |X ′ )−1 (y) = n,

∀y ∈ Y ′ .

(22.2)

It only remains to deal with the critical values b ∈ f (Crit(f )). There then are mutually disjoint neighborhoods Ux of the preimages x ∈ f −1 (b) as well as neighborhoods Vx of b such that Vx \ {b} ⊆ Y ′ and for which there exist qx (f ) ∈ ℕ such that #(f −1 (w) ∩ Ux ) = qx (f ),

∀w ∈ Vx \ {b}.

It is clear from Taylor’s theorem that qx (f ) = degx (f ). By Lemma 22.1.2(b), there is a neighborhood V⊆

⋂ Vx

x∈f −1 (b)

of b such that f −1 (V ) ⊆

⋃ Ux .

x∈f −1 (b)

For every point w ∈ V \ {b} ⊆ Y ′ , we then have #(f −1 (w)) =

∑ qx (f ).

x∈f −1 (b)

928 � 22 The Riemann–Hurwitz formula Combining this with (22.2), we conclude that ∑ qx (f ) = n.

x∈f −1 (b)

22.2 The Euler characteristic of plane bordered surfaces We devote this section to a brief, restricted in scope, introduction to the Euler characteristic. More about this concept can be found in virtually any book on algebraic topology. ̂ is said to be an A domain (i. e., an open connected set) in the Riemann sphere ℂ open bordered surface if its boundary consists of a finite number (possibly 0) of mutually disjoint simple closed (i. e. Jordan) curves. Note that the complex plane ℂ is not an ̂ is. An open bordered surface is called open bordered surface but the Riemann sphere ℂ ̂ proper if it is different from ℂ. ̂ is said to be a closed bordered surface if it is the closure of an open A set S ⊆ ℂ bordered surface. Observe again that the boundary (border) of S can be empty and then ̂ In fact, ℂ ̂ is the only set which is simultaneously an open and closed bordered S = ℂ. surface. A triangulation T of S is a finite partition of S into subsets called vertices, edges and faces, denoted by V , E and F respectively, with the following properties: (1) each vertex v ∈ V is a point of S; (2) for each edge e ∈ E, there are a compact interval [a, b] ⊂ ℝ and a homeomorphism φ : [a, b] → e such that φ((a, b)) = e

and

φ({a, b}) ⊆ V ;

(3) for each face f ∈ F, there are a closed triangle Δ ⊆ ℂ and a homeomorphism φ : Δ → f which maps the edges and vertices of Δ (in the usual sense) into E and V , respectively, and such that φ(Int(Δ)) = f . Of course, if (2) holds for some compact interval (resp., if (3) holds for some triangle) then (2) holds for any compact interval (resp., (3) holds for any triangle). We stress once again that T partitions S into mutually disjoint subsets of S. Each such subset is either a vertex, an edge or a face, and we call each of these a simplex of T of dimension 0, 1 and 2, respectively. For any simplex s of dimension m, the Euler characteristic χ(s) is defined to be (−1)m . More generally, if S ′ is any subset of S comprising a union of simplices, say s1 , . . . , sk , where sj has dimension mj , then S ′ is called a subcomplex of S relative to T, and we define

22.2 The Euler characteristic of plane bordered surfaces k

k

j=1

j=1

χ(S ′ , T) := ∑ χ(sj ) = ∑(−1)mj .

� 929

(22.3)

In particular, if a triangulation T of S consists of faces F, edges E and vertices V , then the Euler characteristic χ(S) of S is, by definition, χ(S) = χ(S, T) := #F − #E + #V . The crucial fact that is well known in algebraic topology (which we accept here without proof) is that χ(S, T) is independent of the triangulation T. In particular, it is a topological invariant, meaning that two homeomorphic closed bordered surfaces have the same Euler characteristic. So, we can compute χ(S) using the triangulation of our choice. For this, we need to know, and we indeed do by obvious geometry, that every such surface admits a triangulation. For each edge e ∈ E, the closure e is a subcomplex of S relative to T with χ(e, T) = 1.

(22.4)

The boundary 𝜕S of S is a subcomplex of S relative to any triangulation T and it is again evident that χ(𝜕S, T) = 0, as every connected component of 𝜕S is a Jordan curve, which is a subcomplex of any triangulation of S and whose Euler characteristic is clearly equal to zero. The independence from the triangulation T permits us to write χ(𝜕S) = 0.

(22.5)

We say that the Euler characteristic of 𝜕S is zero. Calculations of χ can often be simplified by making use of the above and the following simple idea extending it. If T is a triangulation of S, and S1 and S2 are two subcomplexes of S relative to T, then S1 ∪ S2 and S1 ∩ S2 are subcomplexes of S relative to T as well, and we obtain from (22.3) that χ(S1 ∪ S2 , T) + χ(S1 ∩ S2 , T) = χ(S1 , T) + χ(S2 , T).

(22.6)

Let us illustrate this idea with examples. The interior Int(S) = S \ 𝜕S of S is a subcomplex of S relative to any triangulation T of S. By (22.6)–(22.5), χ(S) = χ(Int(S), T) + χ(𝜕S) = χ(Int(S), T). Hence, χ(Int(S), T) is independent of T and we can speak of χ(Int(S)), calling it the Euler characteristic of Int(S). We thus have

930 � 22 The Riemann–Hurwitz formula χ(Int(S)) = χ(S),

(22.7)

̂ we define and for open bordered surfaces V ⊆ ℂ χ(V ) := χ(V ).

(22.8)

However, we must issue a warning. We have not yet established that χ(Int(S)) is a topô = χ(Int(S)) for every logical invariant. Indeed, we do not know yet whether χ(Int(S)) ̂ ̂ bordered surface S such that Int(S) is homeomorphic to Int(S). We know this, though, when Ŝ and S are homeomorphic. The more general case will follow from Proposition 22.4.4. ̂ = 2 and χ(D) = 1 By constructing explicit triangulations, it is immediate that χ(ℂ) for any open topological disk D (i. e., a set homeomorphic to 𝔻). ̂ of a By its very definition, each closed bordered surface S is the complement in ℂ finite number, say k ≥ 0, of mutually disjoint open topological disks D1 , . . . , Dk , whose boundaries are Jordan curves. So, S is of connectivity k, meaning that its complement has k connected components. We can triangulate the sphere so that each of the sets S, ̂ Then (22.6) yields D1 , . . . , Dk is a subcomplex of ℂ. k

̂ = χ(S) + ∑ χ(Dj ) = χ(S) + k. 2 = χ(ℂ) j=1

Hence, χ(S) = 2 − k.

(22.9)

̂ we have that: So, for any such closed bordered surface S of the sphere ℂ, ̂ i. e., k = 0; (a) χ(S) = 2 if and only if S is the sphere ℂ, ̂ i. e., S is a closed topological (b) χ(S) = 1 if and only if S is simply connected but is not ℂ, disk (i. e., S is homeomorphic to 𝔻), or equivalently k = 1; (c) χ(S) = 0 if and only if S is doubly connected, i. e., k = 2; (d) χ(S) < 0 in all other cases, i. e., if and only if k ≥ 3.

̂ 22.3 The Riemann–Hurwitz formula for bordered surfaces in ℂ Our first main goal is to prove a formula commonly called Riemann–Hurwitz formula, ̂ and in the following setting. The ambient space will always be the Riemann sphere ℂ ̂ will always be with respect to the topological space the boundary 𝜕A of a subset A of ℂ ̂ i. e., 𝜕A = A ∩ ℂ ̂ \ A. If A ⊆ B ⊆ ℂ, ̂ the boundary of A with respect to the topological ℂ, space B will be denoted by 𝜕B A. Note that 𝜕B A ⊆ 𝜕A.

̂ 22.3 The Riemann–Hurwitz formula for bordered surfaces in ℂ

�

931

̂ is (by definition) an open connected subset of ℂ. ̂ Recall that a domain in ℂ ̂ and K is a compact subset of S, then Observation 22.3.1. If S is a domain in ℂ dist(K, 𝜕S) > 0. ̂ and let Let S1 and S2 be domains in ℂ R : S1 → S2 be an analytic map which has a continuous extension from S1 to S2 . That extension is unique and will generally be denoted by the same symbol R. ̂ If an analytic map R : S1 → S2 admits a Theorem 22.3.2. Let S1 and S2 be domains in ℂ. (unique) continuous extension to S1 , then R is proper if and only if R(𝜕S1 ) ⊆ 𝜕S2 .

(22.10)

Proof. For this proof, we reserve the symbol R for the map from S1 to S2 and denote by R the continuous extension of R to S1 . By way of contradiction, suppose that the map R : S1 → S2 is proper but that (22.10) does not hold. This means that there exists a point z ∈ 𝜕S1 such that R(z) ∈ S2 . As S2 is open, there is r > 0 such that B(R(z), r) ⊆ S2 . Since R is proper, the preimage R−1 (B(R(z), r)) is a compact subset of S1 . As z ∈ 𝜕S1 , Observation 22.3.1 guarantees that there exists ε > 0 for which B(z, ε) ∩ R−1 (B(R(z), r)) = 0.

(22.11)

Given that R : S1 → S2 is continuous, there is δ ∈ (0, ε) such that R(B(z, δ) ∩ S1 ) ⊆ B(R(z), r).

(22.12)

But since z ∈ S1 , there exists a point w ∈ B(z, δ) ∩ S1 . It then follows from (22.11) that R(w) ∉ B(R(z), r) while it ensues from (22.12) that R(w) ∈ B(R(z), r). This contradiction finishes the proof of the direct implication. Now suppose that (22.10) holds but that the map R is not proper. Then there exists

a compact set K ⊆ S2 such that R−1 (K) is not compact. But since R (K) is compact (by −1

the continuity of R and the compactness of S1 ), we get that R (K)\R−1 (K) ≠ 0. However, −1

R (K)\R−1 (K) ⊆ 𝜕S1 , whence 𝜕S2 ⊇ R(𝜕S1 ) ⊇ R(R (K)\R−1 (K)) ⊆ K ⊆ S2 . This means −1

−1

that R(R (K)\R−1 (K)) = 0, and hence R (K)\R−1 (K) = 0. This contradiction completes the proof of the converse implication. −1

−1

932 � 22 The Riemann–Hurwitz formula ̂ and R : S1 → S2 be a proper analytic Theorem 22.3.3. Let S1 and S2 be domains in ℂ map which has a (unique) continuous extension to S1 . If V is a subdomain of S2 and U is a connected component of R−1 (V ), then the restriction R|U : U → V is a proper analytic map satisfying R(U) = V ,

R(𝜕U) = 𝜕V

and U ∩ R−1 (𝜕V ) = 𝜕U.

(22.13)

In particular, R(𝜕S1 ) = 𝜕S2 . Proof. We first prove that the map R|U : U → V is proper. Let K ⊆ V be a compact set. Let 𝒞 be the collection of all connected components of R−1 (V ). Then −1 ̂ R|−1 U (K) = R (K) ∩ [ℂ \

⋃

C∈𝒞\{U}

C].

As the set R−1 (K) is compact (given that R : S1 → S2 is proper) and each C ∈ 𝒞 is an open ̂ we conclude that the set R|−1 subset of ℂ, U (K) is compact. Thus, the map R|U : U → V is proper. Moreover, it follows from Theorem 22.3.2 that R(𝜕U) ⊆ 𝜕V and from Theorem 22.1.3 that R(U) = V . This immediately implies that U ∩ R−1 (𝜕V ) = 0. Hence, U ∩ R−1 (𝜕V ) = 𝜕U.

(22.14)

In order to complete the proof, it remains to show that 𝜕V ⊆ R(𝜕U).

(22.15)

Let w ∈ 𝜕V . There exists a sequence (wn )∞ n=1 of points in V converging to w. For every −1 n ∈ ℕ, choose a point zn ∈ U ∩ R (wn ). Passing to a subsequence if necessary, we may ̂ assume that the sequence (zn )∞ n=1 converges in ℂ. Denote its limit by z. Then z ∈ U and R(z) = lim R(zn ) = lim wn = w. n→∞

n→∞

So, z ∈ U ∩R−1 (𝜕V ), and thus z ∈ 𝜕U by (22.14). Consequently, w = R(z) ∈ R(𝜕U) and (22.15) holds.

̂ 22.3 The Riemann–Hurwitz formula for bordered surfaces in ℂ

� 933

From now on, we assume throughout the rest of this section that S1 and S2 are dô and R : S1 → S2 is a proper analytic map having a (unique) continuous mains in ℂ extension to S1 . We also assume that V is a subdomain of S2 such that V ⊆ S2 and 𝜕V ∩ R(Crit(R)) = 0,

(22.16)

where Crit(R) is the set of critical points of R. So, R(Crit(R)) is the set of critical values of R. Furthermore, we let U be a connected component of R−1 (V ). We will establish some properties of the map R in relation to V and U that will turn out to be useful in the proof of the Riemann–Hurwitz formula. In light of Theorems 22.1.3 and 22.3.3, we may speak of the degree of the map R|U : U → V and state that deg(R|U ) ≥ 1. ̂ and let R : S1 → S2 be a proper analytic Lemma 22.3.4. Let S1 and S2 be domains in ℂ map which admits a continuous extension to S1 . If V ⊆ S2 is an open bordered surface, then any connected component U of R−1 (V ) is also an open bordered surface. (In particular, U ̂ and Int(U) = U and Int(V ) = V .) In addition, the and V are closed bordered surfaces in ℂ restriction R|𝜕U : 𝜕U → 𝜕V is a proper covering map of finite degree. Proof. Because of Theorem 22.3.3 and relation (22.16), the map R|𝜕U : 𝜕U → 𝜕V is a locally homeomorphic surjection. As 𝜕U is compact, the map R|𝜕U : 𝜕U → 𝜕V is thereby a covering map of finite degree, and we write q := deg(R|𝜕U ). If Γ is a connected component of 𝜕V , then Γ is a Jordan curve, and R : R−1 (Γ) → Γ is also a covering map of degree at most q. Let L be a connected component of R−1 (Γ). We claim that R(L) = Γ. Otherwise, there would exist a point x ∈ L such that R(x) ∈ 𝜕Γ R(L). But, because of (22.16), there exists an open topological arc α containing x such that R(α) ⊆ Γ and R(α) is an open topological arc containing R(x). But then L∪α is connected and R(L∪α) ⊆ Γ. Hence, L ∪ α = L and, therefore, R(x) ∈ R(α) ⊆ IntΓ R(L). This contradiction gives that R(L) = Γ. Thus, the map R : L → Γ is a covering surjection. In consequence, it is at most q–to–1 and R−1 (Γ) has at most q connected components. We are thus left to show that L is a Jordan curve. But L is compact and R : L → Γ is a local homeomorphism, so L is a compact connected 1-dimensional topological manifold. It is well known that L is then a Jordan curve. Another argument, less topological and more analytic, goes as follows. Fix a point ξ ∈ Γ. Then #(L∩R−1 (ξ)) ≤ q and each point z ∈ L∩R−1 (ξ) gives rise to a local inverse map R−1 z from a sufficiently small neighborhood of ξ onto a neighborhood of z; in particular, −1 R−1 z (ξ) = z. Extending Rz analytically from ξ to ξ along Γ, we traverse a path Γz in L whose other endpoint ̂z belongs to R−1 (ξ). The path Γz is either a closed topological arc or a Jordan curve depending on whether ̂z ≠ z or ̂z = z, respectively. Moreover, the map

934 � 22 The Riemann–Hurwitz formula R−1 (ξ) ∋ z 󳨃󳨀→ ̂z ∈ R−1 (ξ) is a bijection. We repeatedly extend analytically, successively getting points z, ̂z, ̂̂z, . . ., until we reach z again, through at most q iterations. The consecutive closed topological arcs Γz , Γ̂z , Γ̂̂z , . . ., have exactly one common endpoint, ̂z, ̂̂z, . . ., respectively. Their union, up to the second-last one of them, is a closed topological arc as well. The union of this latter with the last closed topological arc is a Jordan curve since these two arcs have two common endpoints and no other common points. Lemma 22.3.5. Under the hypotheses of Lemma 22.3.4, we have deg(R|𝜕U ) = deg(R|U ). Proof. Fix ξ ∈ 𝜕V . As 𝜕V contains no critical value of R, there exists ε > 0 such that all analytic branches of R−1 are defined on B(ξ, ε). Since 𝜕U consists of finitely many disjoint Jordan curves (by Lemma 22.3.4) and since for every z ∈ 𝜕U ∩ R−1 (ξ), the map R : R−1 z (B(ξ, ε)) → B(ξ, ε) is a homeomorphism, there exists an open (relative to U) neighborhood of z in U, denoted by Γz ⊆ R−1 z (B(ξ, ε)) ∩ U, such that R(Γz ) is an open (relative to V ) neighborhood of ξ in V . Hence, Vξ :=

⋂ z∈𝜕U∩R−1 (ξ)

R(Γz )

is also an open (relative to V ) neighborhood of ξ in V . Fix y ∈ Vξ ∩ V . As each family 󵄨

ℱξ := {Rz (B(ξ, ε)) 󵄨󵄨󵄨 z ∈ 𝜕U ∩ R (ξ)} −1

−1

and

󵄨

ℱy := {Rx (B(ξ, ε)) 󵄨󵄨󵄨 x ∈ U ∩ R (y)} −1

−1

consists of mutually disjoint sets, we know that deg(R|𝜕U ) = #ℱξ

and

deg(R|U ) = #ℱy .

Thus, in order to complete the proof, it suffices to show that ℱξ = ℱy .

(22.17)

If z ∈ 𝜕U ∩ R−1 (ξ), then there exists a (unique) point x ∈ Γz such that R(x) = y. So, −1 x ∈ U ∩ R−1 (V ) = U. Hence, x ∈ U ∩ R−1 (y) and R−1 x (y) = x = Rz (y). Consequently, −1 −1 Rz (B(ξ, ε)) = Rx (B(ξ, ε)) ∈ ℱy , and the inclusion ℱξ ⊆ ℱy

(22.18)

is proved. For the opposite inclusion, suppose that x ∈ U ∩ R−1 (y). Then R−1 x (Vξ ∩ V ) ⊆ U, whence

̂ 22.3 The Riemann–Hurwitz formula for bordered surfaces in ℂ

� 935

−1 −1 z := R−1 x (ξ) ∈ Rx (Vξ ∩ V ) = Rx (Vξ ∩ V ) ⊆ U. −1 −1 −1 Therefore, z ∈ 𝜕U ∩ R−1 (ξ) and R−1 z (ξ) = z = Rx (ξ). Thus, Rz (B(ξ, ε)) = Rx (B(ξ, ε)). So, −1 Rx (B(ξ, ε)) ∈ ℱξ and the inclusion ℱy ⊆ ℱξ is proved. Along with (22.18), this completes the proof of formula (22.17), and, simultaneously, the proof of Lemma 22.3.5.

We now introduce the deficiency of R at a point z ∈ S1 as δR (z) := degz (R) − 1 ≥ 0,

(22.19)

where degz (R) is the local degree of R at z (see (22.1)). Note that δR (z) > 0 if and only if z is a critical point of R. For a set A ⊆ S1 , the total deficiency of R over A is naturally defined as δR (A) := ∑ δR (z) ≥ 0. z∈A

The function A 󳨃→ δR (A) is nondecreasing and additive, i. e., δR (A) ≤ δR (B) whenever A ⊆ B ⊆ S1 while for disjoint sets A and B in S1 , δR (A ∪ B) = δR (A) + δR (B). We are now ready to relate the quantities χ(U), χ(V ), deg(R|U ) and δR (U). ̂ Let S1 and S2 be Theorem 22.3.6 (Riemann–Hurwitz formula for bordered surfaces in ℂ). ̂ and R : S1 → S2 be a proper analytic map which admits a continuous domains in ℂ extension to S1 . Let V be a subdomain of S2 such that V ⊆ S2 and such that V is a bordered ̂ for which surface in ℂ 𝜕V ∩ R(Crit(R)) = 0.

(22.20)

If U is a connected component of R−1 (V ), then χ(U) + δR (U) = deg(R|U ) ⋅ χ(V ).

(22.21)

Proof. We can triangulate V in such a way that all critical values of R in V are vertices of the triangulation. Indeed, since each connected component of 𝜕V is a Jordan curve, given any triangulation, we can connect via a closed topological arc each critical value of R with at least two distinct vertices of the triangulation, bounding a 1-dimensional simplex on one connected component of 𝜕V . This way we form a larger triangulation with the required property. Denote such a triangulation by T. Its sets of vertices, edges and faces are respectively denoted by VT , ET and FT . We construct a triangulation TU of 𝜕U in the following way. Let m = deg(R|U ). Let Δ ∈ FT . Since Δ is an open topological disk, there exists an open topological disk Δ′ ⊆ S2 containing Δ \ V and disjoint from the union of the critical values of R and the vertices

936 � 22 The Riemann–Hurwitz formula of Δ. Then, according to the monodromy theorem (Theorem A.3.4), there are exactly m ′ −1 −1 distinct analytic branches R−1 j : Δ → S1 of R such that Rj (Δ) ⊆ U. Define 󵄨󵄨 FTU := {R−1 j (Δ) 󵄨󵄨 1 ≤ j ≤ m, Δ ∈ FT }. The elements of FTU will be the faces of the triangulation TU . The edges of every face −1 R−1 j (Δ) correspond with the sets Rj (ei ), i = 1, 2, 3, where the ei ’s are the three edges of Δ. So, define 󵄨󵄨 ETU := {R−1 j (e) 󵄨󵄨 1 ≤ j ≤ m, Δ ∈ FT , e ∈ ET and e ⊆ 𝜕Δ}. Finally, the set VTU consists of the endpoints of the closures of all elements of ETU . Note that the endpoints of the closure of each e ∈ ETU are distinct since they are mapped by R onto two different points, namely the endpoints of R(e) ∈ ET . It readily follows from this construction that the sets FTU labeled as faces, ETU labeled as edges and VTU labeled as vertices, form a triangulation of U. We denote that triangulation by TU . It is immediate from this construction that #FTU = m ⋅ #FT ,

#ETU = m ⋅ #ET

and #VTU = m ⋅ #VT −

∑

(degc (R) − 1) = m ⋅ #VT − δR (U).

c∈VTU ∩Crit(R)

Therefore, χ(U) = χ(U, TU ) = #FTU − #ETU + #VTU = m(#FT − #ET + #VT ) − δR (U)

= mχ(V ) − δR (U) = mχ(V ) − δR (U). We now derive a series of consequences of this theorem. Corollary 22.3.7. Under the hypotheses of Theorem 22.3.6 and the additional assumption ̂ (i. e., V is conformally equivalent to the that V is simply connected but different from ℂ unit disk 𝔻 due to Koebe’s uniformization theorem (Theorem 23.2.2)), it turns out that ̂ and hence χ(U) ≤ 1 and U ≠ ℂ, δR (U) ≥ deg(R|U ) − 1, with equality holding if and only if U is simply connected (i. e., is conformally equivalent to 𝔻). In this latter case, #Crit(R|U ) ≤ deg(R|U ) − 1.

(22.22)

̂ case 22.4 Euler characteristic: the general ℂ

�

937

Corollary 22.3.8. Under the hypotheses of Theorem 22.3.6 and the additional assumptions ̂ or conformally equivalent to the (1) that V is simply connected (i. e., is either equal to ℂ unit disk 𝔻); and (2) that U contains no critical point of R, the map R|U : U → V is a conformal homeomorphism. In consequence, U has the same ̂ or U and V are both conformally equivalent to 𝔻). nature as V (i. e., either U = V = ℂ, ̂ then χ(V ) = 2. By (2), δR (U) = 0. As χ(U) ≤ 2, we deduce from TheoProof. If V = ℂ, rem 22.3.6 that deg(R|U ) ≤ 1. Thus, deg(R|U ) = 1, whence R|U is a conformal homeomor̂ phism. Therefore, U = ℂ. ̂ then U ≠ ℂ, ̂ for otherwise V would be compact, and thus equal to ℂ. ̂ If V ≠ ℂ, Hence, χ(U) ≤ 1. We therefore infer from Theorem 22.3.6 that deg(R|U ) ≤ 1. Consequently, deg(R|U ) = 1, whence R|U is a conformal homeomorphism. This immediately yields that U, like V , is conformally equivalent to 𝔻. Corollary 22.3.9. Under the hypotheses of Theorem 22.3.6 and the additional assumptions (1) that V is conformally equivalent to the unit disk 𝔻; and (2) that the map R|U : U → V has a unique critical point c, it holds that: (a) U is conformally equivalent to 𝔻; and (b) deg(R|U ) = degc (R). ̂ (as otherwise V would be compact Proof. Obviously, deg(R|U ) ≥ degc (R). Since U ≠ ℂ ̂ and thus equal to ℂ), we know that χ(U) ≤ 1. It then follows from Theorem 22.3.6 that deg(R|U ) ≤ degc (R). Hence, deg(R|U ) = degc (R), proving (b). Applying Theorem 22.3.6 once more, we conclude that χ(U) = 1 and (a) holds.

̂ case 22.4 Euler characteristic: the general ℂ ̂ has been defined in Section 22.2 whenever its The Euler characteristic of a domain D ⊆ ℂ boundary 𝜕D consists of finitely many Jordan curves. In general, however, the boundary of a domain D is much more complicated. We may not be able to triangulate the closure of D and in such circumstances, χ(D) has yet to be defined. As this is likely to be so in the case of main interest to us (when D is a component of the Fatou set), this presents us with a problem which we will now tackle. We propose to show that given any domain D, we can define χ(D) as the limiting value of the Euler characteristic of smooth subdomains which exhaust D. Once this is done, we can use the Euler characteristic as a tool to study the way in which a rational function maps one component of the Fatou set onto another. The following exposition is closely related to the construction of the ideal boundary components of a Riemann surface. ̂ be a domain, i. e., an open connected set. A subdomain Δ of D is said to be Let D ⊆ ℂ a regular subdomain of D if:

938 � 22 The Riemann–Hurwitz formula (1) Δ is an open bordered surface whose closure Δ is contained in D, i. e. Δ ⊆ D, and 𝜕Δ is a finite (possibly empty) union of mutually disjoint Jordan curves γ1 , . . . , γn , and ̂ \ Δ enumerated so that their (2) if Γ1 , . . . , Γn denote all the connected components of ℂ ̂ boundaries are respectively γ1 , . . . , γn , then Γj ∩ (ℂ \ D) ≠ 0 for every 1 ≤ j ≤ n.

For example, the open unit disk 𝔻 = {z ∈ ℂ : |z| < 1} is a regular subdomain of ℂ, whereas the annulus {z ∈ ℂ : 1 < |z| < 2} is not. ̂ is ℂ ̂ itself. Observation 22.4.1. The only regular subdomain of the Riemann sphere ℂ Per the previous section, the Euler characteristic χ(Δ) is defined for each regular subdomain Δ of D. If a nonregular subdomain Δ of D satisfies condition (1) but not (2) (for example, an annulus), we can form a regular subdomain Δ∗ of D by adjoining to Δ those sets Γj which ̂ \ D. Obviously, χ(Δ∗ ) ≥ χ(Δ); in fact, χ(Δ∗ ) = χ(Δ) + k, where k ≥ 0 is do not intersect ℂ the number of adjoined sets Γj . We want to represent D as a limit of regular subdomains. Given that no canonical sequence of regular subdomains of D presents itself, it is best to reject the idea of a sequential limit and to consider instead convergence with respect to the directed set (or net) of regular subdomains. There is no need for great generality here and the details are quite simple and explicit. First, we prove the following. ̂ then Lemma 22.4.2. If D is a domain in ℂ, (a) any compact subset of D lies in a regular subdomain of D

and (b) if Δ1 and Δ2 are regular subdomains of D, then there is a regular subdomain Δ of D which contains Δ1 ∪ Δ2 . ̂ the lemma is immediate by virtue of Observation 22.4.1. So suppose that Proof. If D = ℂ, ̂ If ξ is an arbitrary point in D, then the Möbius map ℂ ̂ ∋ z 󳨃→ D is a proper subset of ℂ. −1 ̂ (z − ξ) ∈ ℂ sends ξ to ∞ and we may assume without loss of generality that ∞ ∈ D. Let n ∈ ℕ. Cover the plane ℂ with squares whose sides of length 1/2n are parallel to the axes and whose corners have coordinates that are integer multiples of 1/2n . Let Kn be ̂ \ D. If n ∈ ℕ is large enough, say the union of those squares in the cover that intersect ℂ ̂ \ Kn that contains ∞. n ≥ N, then ∞ ∉ Kn and let Dn be the connected component of ℂ It is easy to see that (Dn )∞ is an ascending sequence of regular subdomains of D such n=N ∞ that ∪n=N Dn = D. Having this, the rest of the proof is straightforward. Given any compact set K ⊆ D, the family {Dn }∞ n=N is an open cover of K and so admits a finite subcover. As the sequence ∞ (Dn )n=N is ascending, this subcover encompasses a largest domain Dm which contains K. This proves (a).

̂ case 22.4 Euler characteristic: the general ℂ

� 939

Regarding (b), if Δ1 and Δ2 are regular subdomains of D, then Δ1 ∪ Δ2 is a compact subset of D and so lies, by (a), in a regular subdomain of D. Lemma 22.4.2 states that the class ℛ(D) of all regular subdomains of D is a net under the order determined by the (direct) inclusion relation. Our next task is to show that the χ function is monotone decreasing on ℛ(D) and so tends to a limit, which may be −∞. ̂ the Euler characteristic function χ : ℛ(D) → Lemma 22.4.3. For every domain D ⊆ ℂ, {2, 1, 0, −1, −2, . . . , −∞} is decreasing. That is, Δ1 , Δ2 ∈ ℛ(D), Δ1 ⊆ Δ2

󳨐⇒

χ(Δ2 ) ≤ χ(Δ1 ).

̂ \ Δ1 , and let W1 , . . . , Wn be the Proof. Let V1 , . . . , Vm be the connected components of ℂ ̂ \ Δ2 . Since Δ1 ⊆ Δ2 , we have connected components of ℂ W1 ∪ . . . ∪ Wn ⊆ V1 ∪ . . . ∪ Vm . For each j ∈ {1, . . . , m}, pick a point zj ∈ Vj \ D. As zj is not in D, it lies in some set Wk and so Wk (being connected) is contained in Vj . It follows that each set Vj , j ∈ {1, . . . , m}, contains some set Wk with k ∈ {1, . . . , n}. Hence, m ≤ n. Using (22.9), we deduce that χ(Δ2 ) = 2 − n ≤ 2 − m = χ(Δ1 ). An immediate repercussion of Lemma 22.4.3 is the following. ̂ the limit Proposition 22.4.4. For any subdomain D of ℂ, χ(D) := lim χ(Δ) Δ∈ℛ(D)

exists and χ(D) = inf χ(Δ). Δ∈ℛ(D)

More precisely: (1) Either χ(D) = −∞, and there are regular subdomains Δn of D with χ(Δn ) → −∞ (equivalently, the connectivity of the Δn ’s diverges to ∞) (2) or else there is a regular subdomain D∗ of D such that χ(D∗ ) = χ(D) > −∞, and χ(Δ) = χ(D) for every regular subdomain Δ of D which contains D∗ .

940 � 22 The Riemann–Hurwitz formula If D is a simply connected domain, then 𝜕D is connected and each regular subdomain Δ of D has at most one complementary connected component, regardless of the nature ̂ and exactly one if D ≠ ℂ. ̂ Using the of 𝜕D: it has no complementary component if D = ℂ Riemann mapping theorem (Theorem A.3.14), we obtain the following. ̂ is simply connected if and only if either χ(D) = 2 or Theorem 22.4.5. A domain D ⊆ ℂ ̂ In the latter case, D is either ℂ or is conformally χ(D) = 1. In the former case, D = ℂ. equivalent to the unit disk 𝔻. More generally, if D has connectivity k, then χ(D) = 2 − k for all sufficiently large regular subdomains Δ of D, and so χ(D) = 2 − k,

(22.23)

again irrespective of the complexity of 𝜕D. ̂ Furthermore, the current definition of the Euler characteristic of a domain in ℂ coincides with the one in (22.9) in the case of a bordered surface D. Also, we now know that the Euler characteristic of domains is a topological invariant.

̂ case 22.5 Riemann–Hurwitz formula: the general ℂ The main theorem of this section and of the entire Chapter 22 is a generalization of Theorem 22.3.6. ̂ (i. e., Theorem 22.5.1 (General Riemann–Hurwitz formula). Let S1 and S2 be domains in ℂ ̂ open connected subsets of ℂ). Let R : S1 → S2 be a proper analytic map which admits a (unique) continuous extension to S1 . Let V be a subdomain of S2 . If U is a connected component of R−1 (V ), then χ(U) + δR (U) = deg(R|U ) ⋅ χ(V ).

(22.24)

Proof. Fix a point w ∈ V \ R(Crit(R)). Because of Lemma 22.4.2(a) there exists a regular subdomain Δ0 of U such that: (1) Crit(R) ∩ U ⊆ Δ0 and (2) 0 ≠ U ∩ R−1 (w) ⊆ U ∩ R−1 (w) ⊆ Δ0 .

Again due to Lemma 22.4.2(a) and the compactness of R(Δ0 ) ⊆ V , there exists a regular subdomain Δ1 of V such that R(Δ0 ) ⊆ Δ1 . We shall prove the following.

(22.25)

̂ case 22.5 Riemann–Hurwitz formula: the general ℂ

�

941

Claim. The set Δ2 := U ∩ R−1 (Δ1 ) is a regular subdomain of U containing Δ0 . Proof. Since Δ0 is connected and since R(Δ0 ) ⊆ Δ1 , there exists a unique connected component Δ′2 of R−1 (Δ1 ) such that Δ0 ⊆ Δ′2 .

(22.26)

As every connected component of R−1 (Δ1 ) is either disjoint from U or is contained in U, −1 ′′ and as any component Δ′′ 2 contained in U intersects R (w) (because R(Δ2 ) = Δ1 by The′′ ′ orem 22.3.3), it follows from property (2) of Δ0 that Δ2 ∩ Δ0 ≠ 0. But then Δ′′ 2 ∩ Δ2 ≠ 0 ′′ ′ by (22.26). Consequently, Δ2 = Δ2 . We have thus proved that Δ′2 = U ∩ R−1 (Δ1 ) =: Δ2 . So, Δ2 is a subdomain of U, and Δ0 ⊆ Δ2 by (22.26). Moreover, Δ2 = Δ′2 is a bordered surface by Lemma 22.3.4. We are thus left to show that the subdomain Δ2 of U is regular. ̂ \ Δ2 . Seeking a contradiction, suppose For this, let W be a connected component of ℂ ̂ \ U) = 0. This means that that W ∩ (ℂ ̂ \ Δ2 ) = U \ Δ2 = U \ (U ∩ R−1 (Δ1 )) = U ∩ R−1 (ℂ ̂ \ Δ1 ). W ⊆ U ∩ (ℂ

(22.27)

We know that R(W ) is connected and ̂ \ Δ1 )) ⊆ R(U) ∩ R(R−1 (ℂ ̂ \ Δ1 )) ⊆ V ∩ (ℂ ̂ \ Δ1 ) = V \ Δ1 . R(W ) ⊆ R(U ∩ R−1 (ℂ Let W1 be the connected component of V \ Δ1 containing R(W ) and let W2 be the con̂ \ Δ1 containing W1 . In particular, W2 is a connected component nected component of ℂ ̂ of ℂ \ Δ1 and, therefore, since Δ1 is a regular subdomain of V , we deduce that ̂ \ V ) ≠ 0. W2 ∩ (ℂ

(22.28)

As W1 ⊆ V ∩ W2 , we know that 𝜕W1 ⊆ 𝜕V ∪ 𝜕W2 . We will show that 𝜕W1 ∩ 𝜕V ≠ 0. By way of contradiction, suppose that 𝜕W1 ∩ 𝜕V = 0. Then 𝜕W1 ⊆ 𝜕W2 . Hence, W2 = W1 ∪ (W2 \ W1 ). Since W2 is connected and W1 ≠ 0, this implies that W2 \ W 1 = 0. Equivalently, W2 ⊆ W 1 . As both W1 and W2 are open, this yields W2 ⊆ W1 . Thus, W2 = W1 , whence W2 ⊆ V , contrary to (22.28). We have thereby proved that

942 � 22 The Riemann–Hurwitz formula

𝜕W1 ∩ 𝜕V ≠ 0.

(22.29)

We now show that W is a connected component of R−1 (W1 ) contained in U. Indeed, by (22.27), ̂ \ Δ2 . W ⊆ U ∩ R−1 (W1 ) ⊆ U ∩ R−1 (V \ Δ1 ) = U \ Δ2 ⊆ ℂ ̂ \ Δ2 , it is also a connected component Therefore, since W is a connected component of ℂ −1 of U ∩ R (W1 ). As each connected component of R−1 (W1 ) is either contained in U or is disjoint from U, it follows that W is a connected component of R−1 (W1 ) contained in U. It ensues from this and Theorem 22.3.3 that R(𝜕W ) = 𝜕W1 . Invoking (22.29), we infer that R(𝜕W ) ∩ 𝜕V ≠ 0. Hence, ̂ \ V ) ≠ 0. R(𝜕W ) ∩ (ℂ

(22.30)

But 𝜕W ⊆ 𝜕Δ2 ⊆ Δ2 ⊆ U ∩ R−1 (Δ1 ) ⊆ U ∩ R−1 (V ) ⊆ U. So, R(𝜕W ) ⊆ R(U) = V , contrary to (22.30). The proof of the claim is complete. ◼ Now, since Δ0 ⊆ Δ2 ⊆ U by the claim, it follows from Lemma 22.4.3 that χ(Δ0 ) ≥ χ(Δ2 ) ≥ χ(U).

(22.31)

From property (1) of the subdomain Δ0 , we have δR (Δ0 ) = δR (Δ2 ) = δR (U).

(22.32)

According to the claim, Theorem 22.3.6 applies and yields, with the use of properties (1)–(2) of Δ0 along with (22.31)–(22.32), that deg(R|U )⋅χ(Δ1 ) = deg(R|Δ2 )⋅χ(Δ1 ) = χ(Δ2 )+δR (Δ2 ) = χ(Δ2 )+δR (U) ≥ χ(U)+δR (U). (22.33) Recall that the only requirement on Δ1 is (22.25). Thanks to Lemmas 22.4.2–22.4.3, the infimum of χ(Δ1 ) over all regular subdomains Δ1 of V is χ(V ). Therefore, (22.33) yields deg(R|U ) ⋅ χ(V ) ≥ χ(U) + δR (U).

(22.34)

On the other hand, (22.33) and (22.31), combined with Lemma 22.4.3, give deg(R|U ) ⋅ χ(V ) ≤ deg(R|U ) ⋅ χ(Δ1 ) = χ(Δ2 ) + δR (U) ≤ χ(Δ0 ) + δR (U).

(22.35)

Note that the sole requirements on Δ0 are (1)–(2). In light of Lemmas 22.4.2–22.4.3, the supremum of χ(Δ0 ) over all such domains Δ0 is χ(U). Consequently, (22.35) results in deg(R|U ) ⋅ χ(V ) ≤ χ(U) + δR (U). Along with (22.34), this completes the proof of Theorem 22.5.1.

̂ case 22.5 Riemann–Hurwitz formula: the general ℂ

� 943

Remark 22.5.2. It is important to observe that Corollaries 22.3.7–22.3.9 hold under the weaker hypotheses of Theorem 22.5.1, rather than those of Theorem 22.3.6. In particular, it is not necessary that the boundary consists of Jordan curves or that the critical values lie off the boundary. The proofs are essentially the same. We list these new corollaries below for the convenience of the reader and ease of reference. Corollary 22.5.3. Under the hypotheses of Theorem 22.5.1 and the additional assumption ̂ is simply connected (i. e., V is conformally equivalent to 𝔻), we have that that V ≠ ℂ δR (U) ≥ deg(R|U ) − 1, with equality holding if and only if U is simply connected (i. e., is conformally equivalent to 𝔻). In this latter case, #Crit(R|U ) ≤ deg(R|U ) − 1.

(22.36)

Corollary 22.5.4. Under the hypotheses of Theorem 22.5.1 and the additional assumptions (1) that V is simply connected and (2) that U contains no critical point of R, the map R|U : U → V is a conformal homeomorphism. In particular, U is simply connected. Corollary 22.5.5. Under the hypotheses of Theorem 22.5.1 and the additional assumptions ̂ is simply connected and (2) that the map R|U : U → V has a unique critical (1) that V ≠ ℂ point c, we know that: (a) U is simply connected. (a′ ) If V is conformally equivalent to the unit disk 𝔻, then so is U. (b) deg(R|U ) = degc (R). Item (a′ ) readily follows from item (a) since the possibility that U be ℂ is ruled out by ̂ is excluded by the non-compactness Liouville’s theorem while the possibility that U be ℂ of V . We will frequently use, often without explicitly invoking it, the following reformulation of Corollary 22.5.4. Corollary 22.5.6. Under the hypotheses of Theorem 22.5.1, if V is simply connected, ξ ∈ S1 , R(ξ) ∈ V and the (unique) connected component of R−1 (V ) containing ξ contains no critical point of R, then there exists a unique holomorphic map ψ : V → S1 such that R ∘ ψ = IdV

and

ψ(R(ξ)) = ξ.

Furthermore, ψ(V ) is the connected component of R−1 (V ) containing ξ, and both maps ψ : V → ψ(V ) and R|ψ(V ) : ψ(V ) → V are conformal homeomorphisms. In the sequel, the map ψ : V → ψ(V ) will be referred to as the unique holomorphic branch of R−1 defined on V that sends R(ξ) to ξ. It will be denoted by R−1 ξ .

23 Selected tools from complex analysis 23.1 Koebe distortion theorems This section is devoted to the formulation and the proof of various versions of the celebrated Koebe distortion theorem. They are truly amazing features of univalent holomorphic functions and constitute one of the most indispensable tools when dealing with the dynamical and especially geometric aspects of meromorphic maps in the complex plane. Koebe distortion theorems will be among the most frequently invoked theorems in this volume, particularly from Chapter 27 onward. Koebe’s 41 –theorem (Theorem 23.1.3) was conjectured by Paul Koebe in 1907 and was proved by Ludwig Bieberbach in 1916. The proof is a fairly easy consequence of Bieberbach’s coefficient inequality (Theorem 23.1.2) obtained in [10]. Likewise, analytic Koebe distortion theorems, such as Theorems 23.1.5–23.1.6 follow from Bieberbach’s coefficient inequality. Our proofs are standard and closely follow the exposition in [59]. We start by demonstrating Theorems 23.1.1–23.1.2. The latter will form a crucial ingredient in the proof of a first version of Koebe’s distortion theorem (Theorem 23.1.3). ̂ be a univalent meromorphic function Theorem 23.1.1 (Area theorem). Let g : B(0, 1) → ℂ with a simple pole at 0. Assume that the residue of g at 0 is equal to 1, so that the function g can be represented in the form of a Laurent series g(z) =

1 ∞ + ∑ b zn . z n=0 n

Then ∞

∑ n|bn |2 ≤ 1.

n=1

Proof. For every 0 < r < 1, let Dr = ℂ \ g(B(0, r)). By Green’s theorem, the area of Dr satisfies S(Dr ) = ∬ dx dy = Dr

1 1 ∫ z dz = − 2i 2i 𝜕Dr

∫ g dg.

(23.1)

𝜕B(0,r)

Recall that 1 2πi

∫ zk zl dz = δkl . 𝜕B(0,r)

Substituting the power series expansions of g and g ′ into (23.1), and performing the integration, we obtain https://doi.org/10.1515/9783110769876-023

946 � 23 Selected tools from complex analysis

S(Dr ) = π(

∞ 1 − n|bn |2 r 2n ). ∑ r 2 n=1

Since S(Dr ) ≥ 0, taking the limit as r ↗ 1 yields the desired result. that

Let 𝒮 denote the class of all univalent holomorphic functions f : B(0, 1) → ℂ such f (0) = 0

and

f ′ (0) = 1.

n Theorem 23.1.2 (Bieberbach’s coefficient inequality). If f (z) = z + ∑∞ n=2 an z ∈ 𝒮 , then |a2 | ≤ 2.

Proof. Observe that the formula h(z) :=

∞ ∞ f (z2 ) 2n−2 = 1 + a z = 1 + an z2(n−1) ∑ ∑ n z2 n=2 n=2

defines a holomorphic function from B(0, 1) to ℂ \ {0} such that h(0) = 1. Let √h : B(0, 1) → ℂ \ {0} be the square root of h uniquely determined by the requirement that √h(0) = 1. Let g(z) :=

1

z√h(z)

=

1 1 − a z + ⋅⋅⋅. z 2 2

This series contains only odd powers of z as the function g is odd. If g(z1 ) = g(z2 ), then (g(z1 ))2 = (g(z2 ))2 , whence f (z21 ) = f (z22 ). We infer from the univalence of f that z21 = z22 , i. e., z1 = ±z2 . But since g is odd, we deduce that z1 = z2 . Thus, g is univalent. Theorem 23.1.1 thereby gives that |a2 | ≤ 2. Theorem 23.1.3 (Koebe’s 41 –theorem). If w ∈ ℂ, r > 0 and H : B(w, r) → ℂ is a univalent analytic function, then 󵄨 󵄨 H(B(w, r)) ⊇ B(H(w), 4−1 󵄨󵄨󵄨H ′ (w)󵄨󵄨󵄨r). Proof. Precomposing H with the scaled (by a factor 1/r) translation moving 0 to w and B(0, r) onto B(w, r), and postcomposing H with a scaled (by a factor r) translation moving H(w) to 0, we may assume without loss of generality that H ∈ 𝒮 . Fix c ∈ ℂ \ H(B(0, 1)). An inspection shows that the function B(0, 1) ∋ z 󳨃󳨀→

cH(z) ∈ℂ c − H(z)

n belongs to 𝒮 . Writing H(z) = z + ∑∞ n=2 an z , this function takes on the form

23.1 Koebe distortion theorems � 947

1 B(0, 1) ∋ z 󳨃󳨀→ z + (a2 + )z2 + ⋅ ⋅ ⋅ , c

(23.2)

where a2 is the coefficient of z2 in the Taylor series expansion of H about 0. Applying Theorem 23.1.2 twice (once to the function in (23.2) and once to H), we obtain 󵄨󵄨 1 󵄨 ≤ |a2 | + 󵄨󵄨󵄨a2 + 󵄨󵄨 |c|

1 󵄨󵄨󵄨󵄨 󵄨 ≤ 2 + 2 = 4. c 󵄨󵄨󵄨

In the above theorem, the number 1/4 is optimal as the Koebe function B(0, 1) ∋ z 󳨃󳨀→

∞ z = nzn ∈ ℂ ∑ (1 − z)2 n=1

maps univalently the unit ball B(0, 1) to the slit plane ℂ \ (−∞, −1/4]. We now prove a series of Koebe distortion theorems. First, we establish Lemma 23.1.4, which will form a crucial ingredient in the proof of Theorem 23.1.5. Lemma 23.1.4. If f ∈ 𝒮 , then ′′ 󵄨󵄨 1 󵄨󵄨 󵄨󵄨 󵄨 2 f (z) − z󵄨󵄨󵄨 ≤ 2, 󵄨󵄨 (1 − |z| ) ′ 󵄨󵄨 2 󵄨󵄨 f (z)

∀z ∈ B(0, 1).

(23.3)

Proof. Fix z ∈ B(0, 1). The Möbius transformation w 󳨃󳨀→

w+z 1 + zw

maps B(0, 1) onto itself, sending 0 to z. It follows that for any choice of constants C1 ∈ ℂ and C2 ∈ ℂ \ {0} the function g(w) = C1 + C2 f (

w+z ) 1 + zw

is univalent on B(0, 1). Choose C1 and C2 so that g(0) = 0 and g ′ (0) = 1, i. e., so that g ∈ 𝒮 . These constraints yield C1 = −

f (z) − |z|2 )

f ′ (z)(1

and

C2 =

1 . − |z|2 )

f ′ (z)(1

Direct calculations give g ′ (w) =

1 1 w+z f ′( ) f ′ (z) (1 + zw)2 1 + zw

and g ′′ (w) = −

1

f ′ (z)

2z w+z 1 1 − |z|2 ′′ w + z f ′( )+ ′ f ( ). 3 1 + zw f (z) (1 + zw)4 1 + zw (1 + zw)

948 � 23 Selected tools from complex analysis It ensues that f ′′ (z) 1 g(w) = w + [ (1 − |z|2 ) ′ − z]w2 + ⋅ ⋅ ⋅ . 2 f (z) As g ∈ 𝒮 , the coefficient of w2 has modulus at most 2 by Theorem 23.1.2. We now formulate and prove the central theorem among Koebe distortion theorems. Theorem 23.1.5 (Koebe’s distortion theorem, analytic version I). Let 0 < r < 1. If f ∈ 𝒮 , then 1−r 1+r 󵄨 󵄨 ≤ 󵄨󵄨f ′ (z)󵄨󵄨󵄨 ≤ , (1 + r)3 󵄨 (1 − r)3

∀z ∈ B(0, r)

(23.4)

and there is a choice of argument for f ′ (z) such that 1+r 󵄨󵄨 󵄨 ′ , 󵄨󵄨arg f (z)󵄨󵄨󵄨 ≤ 2 log 1−r

∀z ∈ B(0, r).

(23.5)

Proof. By Lemma 23.1.4, there exists a continuous function η : B(0, 1) → B(0, 1) such that f ′′ (t) η(t) 2t =4 , − f ′ (t) 1 − |t|2 1 − |t|2

∀t ∈ B(0, 1).

Integrating this expression along the line segment from 0 to z in B(0, 1) yields ′

2

z

log f (z) + log(1 − |z| ) = 4 ∫ 0

η(t) dt, 1 − |t|2

∀z ∈ B(0, 1),

(23.6)

∀z ∈ B(0, r).

(23.7)

where 󵄨󵄨 z 󵄨󵄨 r 󵄨󵄨 󵄨󵄨 η(t) 1+r ds 󵄨󵄨 󵄨󵄨 dt 󵄨󵄨 ≤ 4 ∫ = 2 log , 󵄨󵄨4 ∫ 2 2 󵄨󵄨 󵄨 1−r 1 − |t| 󵄨󵄨 1−s 󵄨󵄨 0 󵄨 0

Equating the real parts of both sides of (23.6) and using (23.7), we get the inequality 2 log

1−r 1+r 󵄨 󵄨 ≤ log[(1 − |z|2 )󵄨󵄨󵄨f ′ (z)󵄨󵄨󵄨] ≤ 2 log , 1+r 1−r

∀z ∈ B(0, r).

Since the function B(0, r) ∋ z 󳨃→ log[(1−|z|2 )|f ′ (z)|] is harmonic, it assumes its maximum and minimum values on 𝜕B(0, r). This yields (23.4). Likewise, formula (23.5) results from equating the imaginary parts in (23.6) and using (23.7).

23.1 Koebe distortion theorems

� 949

Relation (23.5) is called the rotation theorem. It was discovered by Bieberbach in 1919. The estimates in (23.4) are the best possible; equality is reached for the Koebe functions fβ (z) =

z

2

(1 + eiβ z)

(23.8)

,

where β is any real number. Let 𝒰 denote the class of all univalent holomorphic functions from the open unit disk B(0, 1) into ℂ. With obvious translations and rescalings, we deduce the following consequence of Theorem 23.1.5. Theorem 23.1.6 (Koebe’s distortion theorem, analytic version II). Let 0 < r < 1. If f ∈ 𝒰 , then |f ′ (z)| 1+r 1−r ≤ ≤ , ′ 3 |f (0)| (1 + r) (1 − r)3

∀z ∈ B(0, r)

(23.9)

and there is a holomorphic branch of argument for f ′ (z) such that 󵄨󵄨 f ′ (z) 󵄨󵄨󵄨 1+r 󵄨󵄨 , 󵄨󵄨arg ′ 󵄨󵄨󵄨 ≤ 2 log 󵄨󵄨 f (0) 󵄨󵄨 1−r

∀z ∈ B(0, r).

(23.10)

As an immediate consequence of Theorem 23.1.6, we obtain the following two facts. Theorem 23.1.7 (Koebe’s distortion theorem, analytic version III). There exists an increasing continuous function K : [0, 1) → [1, ∞) such that K(0) = 1 and with the following property: if w ∈ ℂ, R > 0, and H : B(w, R) → ℂ is any univalent analytic function, then 󵄨󵄨 |H ′ (z)| 󵄨󵄨 󵄨 󵄨󵄨 − 1󵄨󵄨󵄨 ≤ K(r/R)|z − w| 󵄨󵄨 ′ 󵄨󵄨 󵄨󵄨 |H (w)|

(23.11)

for every r ∈ [0, R) and for all z ∈ B(w, r). Theorem 23.1.8 (Koebe’s distortion theorem, Euclidean version). There exists an increasing continuous function k : [0, 1) → [1, ∞) such that k(0) = 1 and with the following property: if w ∈ ℂ, r > 0, and H : B(w, r) → ℂ is any univalent analytic function, then for every t ∈ [0, 1) we have 󵄨 󵄨 󵄨 󵄨 sup{󵄨󵄨󵄨H ′ (z)󵄨󵄨󵄨 : z ∈ B(w, tr)} ≤ k(t) inf{󵄨󵄨󵄨H ′ (z)󵄨󵄨󵄨 : z ∈ B(w, tr)}.

950 � 23 Selected tools from complex analysis ̂ Denote by ρ the spherical metric on the Riemann sphere ℂ. Theorem 23.1.9 (Koebe’s distortion theorem, spherical version). Given R, s ∈ (0, π), there exists an increasing continuous function kR,s : [0, 1) → [1, ∞) with the following property: ̂ r ∈ (0, R], and H : Bρ (w, r) → ℂ ̂ is any univalent analytic function whose range’s if w ∈ ℂ, ̂ complement ℂ \ H(Bρ (w, r)) contains a spherical ball of radius s, then for every t ∈ [0, 1) we have 󵄨 󵄨 󵄨 󵄨 sup{󵄨󵄨󵄨H ′ (z)󵄨󵄨󵄨ρ : z ∈ Bρ (w, tr)} ≤ kR,s (t) inf{󵄨󵄨󵄨H ′ (z)󵄨󵄨󵄨ρ : z ∈ Bρ (w, tr)}, where |H ′ (⋅)|ρ refers to the scaling factor of the spherical derivative. ̂ → ℂ ̂ be the Möbius transformation corresponding via the stereoProof. Let Mw : ℂ graphic projection to the rotation of the sphere 𝕊2 := {x ∈ ℝ3 : ‖x‖ = 1} that moves the point corresponding to 0 to the point corresponding to w. Then Mw−1 (Bρ (w, R)) = Bρ (0, R) = B(0, R1 )

and

Mw−1 (Bρ (w, r)) = Bρ (0, r) = B(0, r1 )

for some appropriate radius R1 ∈ (0, ∞) depending only on R, and some radius r1 ∈ (0, R1 ] depending only on r. Note that for every t ∈ [0, 1) there exists t ′ ∈ (0, 1) such that Mw−1 (Bρ (w, tr)) ⊆ B(0, t ′ r1 ).

(23.12)

̂→ℂ ̂ be the Möbius Let Bρ (ξ, s) be a spherical ball disjoint from H(Bρ (w, r)). Let Mξ : ℂ transformation corresponding via the stereographic projection to the rotation of the sphere 𝕊2 that moves the point corresponding to ξ to the point corresponding to ∞. Then ̂ \ Bρ (ξ, s)) = ℂ ̂ \ Bρ (∞, s) = B(0, s2 ) Mξ (ℂ for some appropriate radius s2 ∈ (0, ∞) depending only on s. Then Mξ ∘ H ∘ Mw : B(0, r1 ) 󳨀→ B(0, s2 ) is a univalent analytic function. Note that the moduli of the Euclidean and spherical derivatives |(Mξ ∘ H ∘ Mw )|′ and |(Mξ ∘ H ∘ Mw )′ |ρ are uniformly comparable on B(0, r1 ) to (multiplicative) comparability constants depending only on R and s. Observe also that the moduli of the spherical derivatives of Mw and Mξ are everywhere equal to 1 as these ̂ Then the result ensues from maps are isometries with respect to spherical metric on ℂ. Theorem 23.1.8 by taking into account (23.12). Employing the mean value inequality, the next two lemmas are respective consequences of the latter two distortion theorems and Koebe’s 41 –theorem (Theorem 23.1.3).

23.2 Normal families and Montel’s theorem

� 951

Lemma 23.1.10. Suppose that D ⊆ ℂ is an open set, z ∈ D and H : D → ℂ is an analytic function which has an analytic inverse Hz−1 defined on B(H(z), 2R) for some R > 0. Then for every 0 ≤ r ≤ R we have 1 󵄨 󵄨−1 󵄨 󵄨−1 B(z, r 󵄨󵄨󵄨H ′ (z)󵄨󵄨󵄨 ) ⊆ Hz−1 (B(H(z), r)) ⊆ B(z, Kr 󵄨󵄨󵄨H ′ (z)󵄨󵄨󵄨 ). 4 ̂ is an open set, z ∈ D and H : D → ℂ ̂ is an analytic Lemma 23.1.11. Suppose that D ⊆ ℂ −1 map which has an analytic inverse Hz defined on Bρ (H(z), 2R) for some R > 0 such that Hz−1 (Bρ (H(z), 2R)) avoids a spherical ball of some radius s. Then for every 0 ≤ r ≤ R we have −1 󵄨 󵄨−1 󵄨 󵄨−1 Bρ (z, [k2R,s (1/2)] r 󵄨󵄨󵄨H ′ (z)󵄨󵄨󵄨ρ ) ⊆ Hz−1 (Bρ (H(z), r)) ⊆ Bρ (z, k2R,s (1/2)r 󵄨󵄨󵄨H ′ (z)󵄨󵄨󵄨ρ ).

23.2 Normal families and Montel’s theorem Introduced at the beginning of the 20th century, the beautiful concept of normal families is powerful. Its treatment can be found in many textbooks on complex analysis. The book by Schiff [115] is devoted to its systematic exposition. Definition 23.2.1. Let ℱ be a family of meromorphic functions, all defined on some open ̂ The family ℱ is called normal on U if it is relatively compact on U. subset U of ℂ. An immediate reformulation of normality is that from any sequence of functions in ℱ one can extract a subsequence which converges uniformly on compact subsets of U. The limit of such a subsequence is a meromorphic function, possibly a constant function ̂ (potentially ∞). identically equal to an element of ℂ Shortly, we will state and prove Montel’s theorem, which is an extremely helpful tool for establishing normality, and which makes normality so important and so useful in complex analysis. It is also absolutely indispensable and frequently used in the theory of iteration of rational functions. We first formulate and use several times, including in the proof of Montel’s theorem, another great theorem of complex analysis. Theorem 23.2.2 (Koebe’s uniformization theorem). Every simply connected Riemann surface (i. e., two-dimensional holomorphic manifold) is conformally equivalent to either the ̂ open unit disk 𝔻 = {z ∈ ℂ : |z| < 1}, the complex plane ℂ or the Riemann sphere ℂ. This theorem was proved by Henri Poincaré and Paul Koebe in 1907. Its proof can be found in several books on complex analysis and Riemann surfaces; for instance, see Theorem 7.4 in Kodaira [63] or Theorem 27.9 (the Riemann mapping theorem) in Forster [51]. As a consequence of Koebe’s uniformization theorem and of the great Picard’s theorem, we prove the following intermediate result.

952 � 23 Selected tools from complex analysis ̂ then the open unit disk 𝔻 is a Theorem 23.2.3. If a, b, c are three distinct points in ℂ, ̂ \ {a, b, c} and there exists a meromorphic covering map universal covering space of ℂ ̂ \ {a, b, c}. π:𝔻→ℂ ̂ \ {a, b, c} and let Proof. Let D be a universal, i. e., simply connected, covering space of ℂ ̂ P : D → ℂ \ {a, b, c} be a covering map. This map uniquely induces a two-dimensional holomorphic manifold structure on D (a Riemann surface). In particular, the map P bê since then comes meromorphic. The surface D cannot be conformally equivalent to ℂ ̂ \ {a, b, c} would be compact, and it cannot be conformally equivalent to ℂ because ℂ of the great Picard’s theorem. According to Koebe’s uniformization theorem, D is conformally equivalent to 𝔻. This means that there exists a conformal homeomorphism ̂ \ {a, b, c} is the sought holomorphic map. h : 𝔻 → D. Then π := P ∘ h : 𝔻 → ℂ We can now prove the ultimate theorem of this section. Theorem 23.2.4 (Montel’s theorem). If ℱ is a family of meromorphic functions, all defined ̂ that omits three points in ℂ, ̂ i. e., if ℂ ̂ \ ⋃f ∈ℱ f (U) contains on some open subset U of ℂ, at least three points, then the family ℱ is normal on U. Proof. We first look at the following weaker version of Montel’s theorem, whose demonstration is a fairly straightforward consequence of Cauchy’s integral formula (Theorem A.3.2) and Arzelà–Ascoli’s theorem (Theorem A.2.3). Claim. If U ⊆ ℂ and the family ℱ is bounded on U, i. e., if 󵄨 󵄨 M := sup{󵄨󵄨󵄨f (z)󵄨󵄨󵄨 : f ∈ ℱ , z ∈ U} < ∞, then ℱ is normal on U. Proof. By Arzelà–Ascoli, it suffices to show that the family ℱw := {f |B(w,R) : f ∈ ℱ } is equicontinuous (with respect to the Euclidean metric on ℂ) for any w ∈ U and R > 0 such that B(w, 4R) ⊆ U. Let γ : [0, 1] → B(w, 4R) be the circle (a simple, closed, smooth and rectifiable curve) γ(t) := w + 3Re2πit . For any function f ∈ ℱ and for every z ∈ B(w, 2R), Cauchy’s integral formula affirms that 󵄨󵄨 󵄨 f (z) |f (z)| M 3M 1 󵄨󵄨 ′ 󵄨󵄨 󵄨󵄨󵄨 1 󵄨󵄨 dz |dz| ≤ < ∞. ∫ ∫ ∫ |dz| = 󵄨󵄨f (z)󵄨󵄨 = 󵄨󵄨 󵄨≤ 󵄨󵄨 2πi (z − w)2 󵄨󵄨󵄨 2π |z − w|2 R 2πR2 γ γ γ Thus, all functions f |B(w,2R) , f ∈ ℱ , are Lipschitz continuous with Lipschitz constant 3M/R. So, the family {f |B(w,2R) : f ∈ ℱ } is equicontinuous. Since it is uniformly bounded (by M) and since the set B(w, R) is compact, this family is relatively compact by the Arzelà–Ascoli theorem and the proof of the claim is complete. ◼

23.2 Normal families and Montel’s theorem

� 953

Passing to a more general case, keep U ⊆ ℂ but assume only that ℱ avoids three ̂ say a, b, c. By Theorem 23.2.3, there exists a meromorphic covering map points in ℂ, ̂ π : 𝔻 → ℂ\{a, b, c}. As 𝔻 is simply connected, for every f ∈ ℱ there exists a continuous ̃ lift f : U → 𝔻 such that f = π ∘ ̃f , i. e. the following diagram commutes: 𝔻 ̃f

U

? π

? ? ℂ\{a, ̂ b, c} f

Since the meromorphic covering map π is locally one-to-one, we deduce that the map ̃f is holomorphic. As the set 𝔻 is bounded, it follows from the claim that the family ̂ {̃f : U → 𝔻 | f ∈ ℱ } is normal. Hence, the family ℱ = {π ∘ ̃f : U → ℂ\{a, b, c} | f ∈ ℱ } is

normal, too. The general case ensues from the fact that normality is a local property. If ∞ ∉ U, then the family ℱ is normal on U\{∞} according to the previous case. If ∞ ∈ U, then ̂ ̂→ℂ ̂ is the reciprocal map ℐ (z) = 1/z, there exists R > 0 such that ℂ\B(0, R) ⊆ U. If ℐ : ℂ then the family {f ∘ ℐ |B(0,1/R) : f ∈ ℱ } avoids three points and is hence normal per the previous case. Then ℱ is normal, too. Invoking again the fact that normality is a local property, we obtain a direct yet effective consequence of Theorem 23.2.4. ̂ is an open set and ℱ is a family of meromorphic functions Corollary 23.2.5. If U ⊆ ℂ ̂ such that for every w ∈ U, there exists a ball B(w, r) ⊆ U for which the family from U to ℂ ̂ then the family ℱ is normal. {f |B(w,r) : f ∈ ℱ } omits three points in ℂ,

24 Dynamics and topology of rational functions: their Fatou and Julia sets 24.1 Fatou set, Julia set, and periodic points ̂→ℂ ̂ be an analytic endomorphism of the Riemann sphere ℂ, ̂ i. e. an holomorLet T : ℂ phic map from the Riemann sphere, considered as a Riemann surface, into itself. After looking at the zeros and the poles of T, it turns out that T is a rational function, i. e., the ratio of two polynomials T=

P , Q

where we may assume that P and Q are relatively prime, i. e., have no common factors. ̂ where the dynamics of T are stable We would like to distinguish the region of ℂ (or tame) from that where the dynamics of T are chaotic (or random). These regions are respectively called the Fatou and Julia sets of T. Recall from Chapter 23 the concept of normality of a family of analytic functions and Montel’s theorem. The Fatou set of a rational function T is the set of all points that admit a neighborhood on which the iterates of T form a normal family while the Julia set is its complement. ̂→ℂ ̂ be a rational function. The Fatou set ℱ (T) of T is the Definition 24.1.1. Let T : ℂ ̂ for each of which there exists an open neighborhood Uz ⊆ ℂ ̂ set of all points z ∈ ℂ n of z such that the family of functions {T |Uz : n ∈ ℕ} is normal on Uz . Its complement ̂ \ ℱ (T) is called the Julia set of T. 𝒥 (T) = ℂ From its very definition, the Fatou set is open, and hence the Julia set is a closed, ̂ Moreover, both sets are fully invariant. and thereby compact, subset of ℂ. Theorem 24.1.2. Both the Julia set 𝒥 (T) and the Fatou set ℱ (T) are completely T-invariant, i. e., T −1 (𝒥 (T)) = 𝒥 (T) = T(𝒥 (T)) and

T −1 (ℱ (T)) = ℱ (T) = T(ℱ (T)).

Proof. It is enough to show that the Fatou set ℱ (T) is completely invariant. Let z ∈ T −1 (ℱ (T)). Then T(z) ∈ ℱ (T), and thus the sequence of iterates (T n )∞ n=1 is ̂ of T(z). Let (T nk |T −1 (U ) )∞ be an arbitrary normal on an open neighborhood UT(z) ⊆ ℂ T(z) k=1 sequence of iterates of T restricted to the open neighborhood T −1 (UT(z) ) of z. Then there

∞ exists a subsequence (nkj )∞ j=1 of (nk )k=1 such that the sequence (T

formly on compact subsets of UT(z) . But then the sequence (T

nkj

nkj −1 ∞ )j=1 nkj −1

=T

converges uni-

∘T)∞ j=1 converges

uniformly on compact subsets of T −1 (UT(z) ). Hence, z ∈ ℱ (T). So T −1 (ℱ (T)) ⊆ ℱ (T). To prove the opposite inclusion, fix z ∈ ℱ (T). Let Uz be an open neighborhood of z upon which the iterates of T form a normal family. Let (T nk )∞ k=1 be an arbitrary sequence https://doi.org/10.1515/9783110769876-024

956 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets ∞ of iterates of T. There then exists a subsequence (nkj )∞ j=1 of (nk )k=1 such that the sequence nkj +1 ∞ )j=1 converges uniformly on compact subsets of Uz . This nkj ∞ sequence (T )j=1 converges uniformly on compact subsets of T(Uz ), −1

of iterates (T

implies that

the

which is an

open neighborhood of T(z). In consequence, T(z) ∈ ℱ (T). Thus z ∈ T (ℱ (T)), and the inclusion ℱ (T) ⊆ T −1 (ℱ (T)) holds. In summary, we showed that T −1 (ℱ (T)) = ℱ (T). As T is surjective, this immediately yields that T(ℱ (T)) = ℱ (T). ̂ → ℂ, ̂ the singleton ∞ is completely Example 24.1.3. For any polynomial P : ℂ −1 P-invariant, i. e., P ({∞}) = {∞}. This property is a characterization of polynomials within the class of rational functions of the Riemann sphere. ̂ \ B(0, r(T))) ⊆ ℂ ̂ \ B(0, r(T)) for any r(T) > 0 If P is of degree at least two, then T(ℂ n ̂ ̂ large enough. So, T (ℂ\B(0, r(T))) ⊆ ℂ\B(0, r(T)) for all n ∈ ℕ and by Montel’s theorem ̂ \ B(0, r(T)). This means that (Theorem 23.2.4) the family {T n : n ∈ ℕ} is normal on ℂ ̂ \ B(0, r(T)) ⊆ ℱ (T). In particular, ∞ ∈ ℱ (T). ℂ According to Theorem 22.1.3, we can speak of the topological degree deg(T) of a ratiô→ℂ ̂ as being the number of preimages of any regular point of T, i. e., nal function T : ℂ of any point not belonging to the set of critical values T(Crit(T)). It is then obvious that deg(T) is the maximum of the algebraic degrees of the polynomials P and Q. The concept of degree permits us to prove the following fundamental theorem (cf. Exercise 24.8.2). ̂→ℂ ̂ is a rational function with deg(T) ≥ 2, then 𝒥 (T) ≠ 0. Theorem 24.1.4. If T : ℂ Proof. Assume, by way of contradiction, that 𝒥 (T) = 0. Then the iterates {T n : n ∈ ℕ} ̂ and hence there is a subsequence (nk )∞ of (n)∞ form a normal family on ℂ, n=1 such that k=1 nk ∞ ̂ to some analytic function S : ℂ ̂ → ℂ. ̂ the sequence (T )k=1 converges uniformly on ℂ Note that S cannot be constant since T is surjective. Moreover, as the map R 󳨃→ deg(R) is continuous on the space of analytic maps of the Riemann sphere endowed with the topology of uniform convergence, it follows that deg(S) = lim deg(T nk ) = lim (deg(T)) n→∞

n→∞

nk

= ∞.

This is impossible since deg(S) < ∞ per Theorem 22.1.3. From this point on, we will always assume that deg(T) ≥ 2, so that the Julia set of T is always nonempty. For a while, we will repeat this assumption but eventually we may omit it. In light of the above two theorems, it makes sense to study the topological dynamical systems

24.1 Fatou set, Julia set, and periodic points

T : ℱ (T) → ℱ (T)

and

� 957

T : 𝒥 (T) → 𝒥 (T).

We begin our investigation with a slew of fundamental properties. ̂→ℂ ̂ is a rational function with deg(T) ≥ 2, then Theorem 24.1.5. If T : ℂ k

ℱ (T ) = ℱ (T)

and 𝒥 (T k ) = 𝒥 (T),

∀k ∈ ℕ.

Proof. Fix k ∈ ℕ. If the sequence of iterates of T is normal on some open set, then so is the sequence of iterates of T k . Therefore, ℱ (T) ⊆ ℱ (T k ). On the other hand, let z ∈ ℱ (T k ) and Uz a neighborhood of z on which the sequence (T kn )∞ n=1 is normal. Fix an arbitrary strictly increasing sequence of positive integers (nj )∞ . For each j ∈ ℕ, write nj = knj′ + rj , where nj′ and rj are nonnegative integers j=1 and rj < k. There then exist 0 ≤ r < k and a strictly increasing sequence of positive kn ∞ integers (ji )∞ i=1 such that rji = r for all i ∈ ℕ. Due to the normality of (T )n=1 on Uz , ∞ there is a strictly increasing sequence (ih )h=1 of positive integers such that the sequence knj′

nj

knj′

ih (T ih )∞ = T r ∘ T ih )∞ h=1 converges uniformly on compact subsets of Uz . Hence, (T h=1 also converges uniformly on compact subsets of Uz . Thus, z ∈ ℱ (T), and the inclusion ℱ (T k ) ⊆ ℱ (T) holds. ̂ \ ℱ (T k ) = ℂ ̂ \ ℱ (T) = 𝒥 (T). In summary, ℱ (T k ) = ℱ (T). So 𝒥 (T k ) = ℂ

The proof of the following result is left to the reader as an exercise. ̂ → ℂ ̂ is conjugate, via a Möbius transforLemma 24.1.6. If a rational function T : ℂ ̃ ̂ ̂ ̂ ̂ i. e., if T = M −1 ∘ T̃ ∘ M, then mation M : ℂ → ℂ, to a rational function T : ℂ → ℂ, ̃ and M(𝒥 (T)) = 𝒥 (T). ̃ M(ℱ (T)) = ℱ (T) ̂→ℂ ̂ is proper, the following fact is an immediate Since each rational function T : ℂ consequence of Theorem 22.3.3. ̂→ℂ ̂ be a rational function with deg(T) ≥ 2. If V is a domain in Lemma 24.1.7. Let T : ℂ ̂ ℂ and U is any connected component of T −1 (V ), then the map T|U : U → V is proper and T(U) = V ,

T(𝜕U) = 𝜕V ,

and

U ∩ T −1 (𝜕V ) = 𝜕U.

(24.1)

Having this, we can easily prove the following. ̂→ℂ ̂ is a rational function with deg(T) ≥ 2 and V is a connected Theorem 24.1.8. If T : ℂ component of the Fatou set ℱ (T), then: (a) T(V ) is a connected component of ℱ (T). (b) Each connected component U of T −1 (V ) is a connected component of ℱ (T). (c) For any connected component U of T −1 (V ), it holds that T(U) = V ,

T(𝜕U) = 𝜕V ,

and U ∩ T −1 (𝜕V ) = 𝜕U.

(24.2)

958 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets ̂ every connected component of ℱ (T) is open and so Proof. Because ℱ (T) is open in ℂ, ̂ Therefore, part (c) is a special case of (24.1) in Lemma 24.1.7. is a domain in ℂ. Regarding (a), note that T(V ) is open since T is an open map and V is open; it is also ̂ By Theoconnected as T is continuous and V is connected. So, T(V ) is a domain in ℂ. ̃ rem 24.1.2, there is a unique connected component V of ℱ (T) that contains the domain ̃) ⊆ ℱ (T). Then V is a connected compoT(V ). By that same theorem, we know that T −1 (V −1 ̃ ̃ nent of T (V ) since it is a connected component of ℱ (T). By (c), it follows that T(V ) = V is a connected component of ℱ (T). Concerning (b), we know thanks to (c) that T(U) = V . By Theorem 24.1.2, the set ̃ of U is a connected subset of ℱ (T). So, there exists a unique connected component U −1 ̃ ℱ (T) containing U. As U ⊆ T (V ), the set T(U) is a connected subset of ℱ (T) intersecting V and, therefore, is contained in the connected component V . Hence, V = T(U) ⊆ ̃ ⊆ V . Thus, T(U) ̃ = V . Consequently, U ̃ is contained in a connected component of T(U) −1 ̃ T (V ). But since U intersects U (it even contains it) and since U is a connected compõ = U. Hence, U is a connected component of ℱ (T). nent of T −1 (V ), we conclude that U Due to Montel’s theorem (Theorem 23.2.4), for every point z ∈ 𝒥 (T) the set ∞

̂ \ ⋃ T n (Uz ). Ez (T) = ℂ n=0

can consist of no more than two points for every neighborhood Uz of z, and is actually the same set for every such z and every small enough Uz . The following theorem gives a complete description of that exceptional set. ̂→ℂ ̂ is a rational function with deg(T) ≥ 2, then: Theorem 24.1.9. If T : ℂ (a) The set Ez (T) is independent of z ∈ 𝒥 (T) and shall henceforth be denoted by E(T). The set E(T) is called the exceptional set of T. ̂ → ℂ, ̂ (b) If E(T) is a singleton, then T is conjugate, via a Möbius transformation M : ℂ d −1 to a polynomial P different from z 󳨃→ az , d ≥ 2, i. e. T = M ∘ P ∘ M, and then E(P) = {∞} and E(T) = M −1 ({∞}). (c) If E(T) consists of exactly two points, then T is conjugate, via a Möbius transformation M, to a rational map R of the form z 󳨃→ azd , |d| ≥ 2, i. e. T = M −1 ∘ R ∘ M, and then E(R) = {0, ∞} and E(T) = M −1 ({0, ∞}). (d) E(T) ⊆ ℱ (T). Proof. Fix z ∈ 𝒥 (T). By definition, the set Ez (T) is backward T-invariant, i. e. T −1 (Ez (T)) ⊆ Ez (T).

(24.3)

Assume that Ez (T) is a singleton, say Ez (T) = {b}. The above inclusion and the surjectivity ̂ → ℂ ̂ be a Möbius transformation such that of T imply that T −1 ({b}) = {b}. Let M : ℂ −1 ̃ = M(b) = ∞. Setting T̃ := M ∘ T ∘ M , we deduce that T̃ −1 ({∞}) = {∞} and EM(z) (T)

24.1 Fatou set, Julia set, and periodic points

� 959

̂ → ℂ ̂ is a polynomial M(Ez (T)) = {∞}. Recalling Example 24.1.3, we infer that T̃ : ℂ d ̃ which differs from z 󳨃→ az , d ≥ 2 per Exercise 24.8.4. Since ∞ is completely T-invariant ̃ ̃ ̃ and ∞ ∈ ℱ (T), it immediately follows that Ew (T) = {∞} for every w ∈ 𝒥 (T). Therefore, ̃ = {b} for all ζ ∈ 𝒥 (T) per Lemma 24.1.6. Thus, (b) is proved and Eζ (T) = M −1 (EM(ζ ) (T)) (a) is true when (b) holds. Moreover, ̃ = ℱ (T). E(T) = {b} = M −1 ({∞}) ⊆ M −1 (ℱ (T)) Thus, (d) is true when (b) holds. ̂→ℂ ̂ be a Möbius Now assume that Ez (T) consists of exactly two points. Let M : ℂ −1 transformation such that M(Ez (T)) = {0, ∞}. Setting T̃ := M ∘T ∘M and invoking (24.3), ̃ = {0, ∞} and T̃ −1 ({0, ∞}) = {0, ∞}. Either we deduce that Ez (T) ̃ T(0) =0

and

̃ T(∞) = ∞,

(24.4)

̃ T(∞) = 0.

(24.5)

or ̃ T(0) =∞

and

In the case of (24.4), the map T̃ must be a polynomial since T̃ −1 ({∞}) = {∞}, and must be of the form z 󳨃→ azd , d ≥ 2, since T̃ −1 ({0}) = {0}. In the case of (24.5), the only zero of the rational function T̃ is ∞, so T̃ = 1/Q, where Q is a polynomial with deg(Q) ≥ 2. But since T̃ −1 ({∞}) = {0}, the polynomial Q must be of the form azd , with d ≥ 2. In both ̃ = {0, ∞} for every w ∈ 𝒥 (T), ̃ and hence Eζ (T) consists of the same two cases, Ew (T) −1 points M ({0, ∞}) for all ζ ∈ 𝒥 (T). Thus, (c) is proved and (a) is true when (c) holds. As in (b), it is easy to show that (d) is true when (c) holds. According to the preceding theorem, the exceptional set is empty unless, up to a conjugation by a Möbius transformation, T(z) = azd for some |d| ≥ 2. Since the exceptional set is contained in the Fatou set and since the Julia set is compact and completely invariant, we deduce the following result. ̂ → ℂ ̂ is a rational function with deg(T) ≥ 2, then for every Corollary 24.1.10. If T : ℂ ̂ intersecting the Julia set 𝒥 (T), there exists a nonnegative integer n(U) open set U ⊆ ℂ such that n(U)

⋃ T k (U ∩ 𝒥 (T)) = 𝒥 (T).

k=0

In other words, T : 𝒥 (T) → 𝒥 (T) is very strongly transitive. We now classify the periodic points of T.

960 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets ̂→ℂ ̂ be a rational function with deg(T) ≥ 2. Let w ∈ ℂ be Definition 24.1.11. Let T : ℂ a periodic point of T with prime period ℓ ∈ ℕ. The periodic point w is said to be (a) attracting if |(T ℓ )′ (w)| < 1; if (T ℓ )′ (w) = 0, then w is also commonly referred to as a superattracting periodic point. (b) repelling if |(T ℓ )′ (w)| > 1. n 󵄨 󵄨 (c) rationally indifferent if 󵄨󵄨󵄨(T ℓ )′ (w)󵄨󵄨󵄨 = 1 and ((T ℓ )′ (w)) = 1 for some n ∈ ℕ. n 󵄨 󵄨 (d) irrationally indifferent if 󵄨󵄨󵄨(T ℓ )′ (w)󵄨󵄨󵄨 = 1 but ((T ℓ )′ (w)) ≠ 1 for all n ∈ ℕ. Indifferent periodic points (i. e., (c) and (d) together) are sometimes called neutral. ̂→ℂ ̂ If ∞ is a periodic point of T, we conjugate T by a Möbius transformation M : ℂ which sends ∞ to a complex number ξ ∈ ℂ and classify the periodic point ∞ of T as ξ is classified for the conjugate map M ∘ T ∘ M −1 . The number (T ℓ )′ (w) is called the multiplier of w with respect to T. ̂ is said to be preperiodic if there is k ∈ ℕ such that T k (z) is a Finally, a point z ∈ ℂ periodic point of T. All the points in the orbit (T j (w))ℓ−1 j=0 of w are periodic points of T with the same prime period ℓ and the same multiplier as w. Hence, they all have the same nature and it makes sense to talk about attracting, repelling and neutral (rationally or irrationally indifferent) orbits. In Theorem 24.3.4, we will see that the set of periodic points of T is dense in 𝒥 (T), and thus is infinite. At this stage, it suffices to establish that this set is infinite. ̂→ℂ ̂ with d := deg(T) ≥ 2 admits infinitely Lemma 24.1.12. Every rational function T : ℂ many periodic points whose prime periods are prime numbers. Proof. Suppose that T has no point of period p, where p is a prime number. This means that every solution z′ to the equation T p (z) = z is of period k ′ for some factor k ′ of p. That is, k ′ = 1 and every z′ is actually a fixed point of T. As T and T p have d + 1 and d p + 1 fixed points respectively (counting multiplicities), we deduce that there is a fixed point z+ of T such that T p has more fixed points at z+ than T does. Comparing the Taylor expansions of T and T p around z+ reveals that this can only happen if T ′ (z+ ) ≠ 1 but [T ′ (z+ )]p = 1 (see Exercise 24.8.5). Given any fixed point z of T, the equation [T ′ (z)]p = 1 can be satisfied by at most one prime number p(z). As T has d + 1 fixed points (counting multiplicities), we conclude that the assumption that T has no point of period p can hold for at most d+1 prime numbers p. That is, T must have periodic points with prime period p for all but at most d + 1 prime numbers p. As the number of periodic points of any fixed period is finite, let z1 and z2 be periodic points of T with prime periods at least 3 and disjoint forward orbits. We end this section with a very useful technical fact. In the following lemma, all distances, derivatives and diameters are considered with respect to the spherical metric ̂ Set ρ on ℂ.

24.1 Fatou set, Julia set, and periodic points

δT :=

� 961

1 1 ̂ distρ (𝒪+ (z1 ), 𝒪+ (z2 )) ∈ (0, diamρ (ℂ)). 8 8

̂→ℂ ̂ be a rational function with deg(T) ≥ 2. Fix δ ∈ (0, δT ). For Lemma 24.1.13. Let T : ℂ every k ∈ ℕ and z ∈ 𝒥 (T), let T∗−k (z) denote the set of all w ∈ T −k (z) for which there exists ̂ of T −k that maps z to w. Thereafter, a (unique) holomorphic branch Tw−k : Bρ (z, 4δ) → ℂ let Fk (δ) be the subset of all points z ∈ 𝒥 (T) such that the set T∗−k (z) is nonempty. Then 󵄨 󵄨 lim inf{󵄨󵄨󵄨(T k )′ (w)󵄨󵄨󵄨ρ : z ∈ Fk (δ), w ∈ T∗−k (z)} = ∞

k→∞

and lim sup{diamρ (Tw−k (Bρ (z, δ))) : z ∈ Fk (δ), w ∈ T∗−k (z)} = 0,

k→∞

with the convention that inf 0 = ∞ while sup 0 = 0. Proof. We first show the second formula. Suppose for a contradiction that this formula fails. Then there exist ε > 0 and a strictly increasing sequence of positive integers (nk )∞ k=1 such that for every k ∈ ℕ there are yk ∈ Fnk (δ) and xk ∈ T∗−nk (yk ) for which k diamρ (Tx−n (Bρ (yk , δ))) > ε. k

(24.6)

Passing to a subsequence if necessary, we may assume that both sequences (yk )∞ k=1 and (xk )∞ converge. Denote their respective limits by y and x. Both of them belong to 𝒥 (T). k=1 Disregarding finitely many terms of the sequence (nk )∞ , we then have that k=1 Bρ (yk , δ) ⊆ Bρ (y, 3δ/2) and

Bρ (y, 3δ) ⊆ Bρ (yk , 4δ),

∀k ∈ ℕ.

Consider the family of inverse branches −n 󵄨 ℱ∗ := {Txk k 󵄨󵄨󵄨B

ρ (y,3δ)

̂ : Bρ (y, 3δ) → ℂ} k∈ℕ .

By the choice of δ, for every k ∈ ℕ the set Txk k (Bρ (y, 3δ)) is disjoint from the forward orbit of at least one periodic point used to define the number δT . Passing to yet another subsequence, we may assume that this is the same periodic orbit for all k ∈ ℕ. But then Montel’s theorem (Theorem 23.2.4) affirms that the family ℱ∗ is normal. So, passing yet again to another subsequence, we may assume that the sequence −n

k 󵄨󵄨 {Tx−n 󵄨 k 󵄨B

ρ (y,2δ)

̂ : Bρ (y, 2δ) → ℂ} k∈ℕ

̂ Therefore, converges uniformly to analytic function f : Bρ (y, 2δ) → ℂ. k f (y) = lim Tx−n (yk ) = lim xk = x. k

k→∞

k→∞

962 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets By Hurwitz’s theorem (Theorem A.3.9), the function f is either constant or univalent. In the former case, k k lim sup diamρ (Tx−n (Bρ (yk , δ))) ≤ lim sup diamρ (Tx−n (Bρ (y, 3δ/2))) k k

k→∞

k→∞

≤ diamρ (f (Bρ (y, 3δ/2))) = 0,

contrary to (24.6). In the latter case, f (Bρ (y, δ)) is an open set containing x and by Theorem A.3.11 there exists Nδ ∈ ℕ such that k Tx−n (Bρ (y, 2δ)) ⊇ f (Bρ (y, δ)), k

∀k ≥ Nδ .

Consequently, T nk (f (Bρ (y, δ))) ⊆ Bρ (y, 2δ),

∀k ≥ Nδ .

Thus, the sequence (T nk |f (Bρ (y,δ)) )∞ k=1 is normal, contrary to the fact that x ∈ 𝒥 (T). The second formula in this lemma is hence proved. The first formula is now an immediate consequence of Koebe’s distortion theorem (Theorem 23.1.9).

24.1.1 Attracting periodic points If w ∈ ℂ is an attracting periodic point of T with prime period ℓ ∈ ℕ, then there exists R > 0 such that 󵄨 ℓ′ 󵄨 󵄨 󵄨 1 + 󵄨󵄨󵄨(T ) (w)󵄨󵄨󵄨 sup 󵄨󵄨󵄨(T ℓ )′ (ξ)󵄨󵄨󵄨 ≤ := χ < 1. 2

ξ∈B(w,R)

By the mean value inequality, 󵄨󵄨 ℓ 󵄨 󵄨 ℓ ′ 󵄨 󵄨󵄨T (z) − w󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨(T ) (ζ (z))󵄨󵄨󵄨 ⋅ |z − w| ≤ χ|z − w| for all z ∈ B(w, R) and some ζ (z) ∈ [w, z] ⊆ B(w, R). For all k ∈ ℕ, we infer by recurrence that 󵄨󵄨 kℓ 󵄨 ℓ 󵄨󵄨T (z) − w󵄨󵄨󵄨 ≤ χ |z − w|,

∀z ∈ B(w, R).

So, lim T kℓ (z) = w for all z ∈ B(w, R). Note further that k→∞

T ℓ (B(w, R)) ⊆ B(w,

󵄨 󵄨 1 + 󵄨󵄨󵄨(T ℓ )′ (w)󵄨󵄨󵄨 R) ⊆ B(w, R). 2

24.1 Fatou set, Julia set, and periodic points

� 963

By Montel’s theorem (Theorem 23.2.4), it follows that B(w, R) ⊆ ℱ (T). Letting 󵄨 ̂ 󵄨󵄨󵄨 lim T kℓ (z) = w}, A(w) := {z ∈ ℂ 󵄨󵄨 k→∞ it ensues from the complete invariance of the Fatou set that B(w, R) ⊆ A(w) ⊆ ℱ (T). The set A(w) is called the basin of attraction of w. The connected component of A(w) that contains w is called the immediate basin of attraction of w and will be denoted by A∗ (w). Obviously, B(w, R) ⊆ A∗ (w) ⊆ A(w) ⊆ ℱ (T). We collect these observations in the following theorem. ̂→ℂ ̂ be a rational function. If w ∈ ℂ ̂ is an attracting periodic Theorem 24.1.14. Let T : ℂ ∗ point of T with prime period ℓ, then A (w) and A(w) are open sets such that w ∈ A∗ (w) ⊆ A(w) ⊆ ℱ (T). Moreover, A∗ (w) is forward T ℓ -invariant whereas A(w) is completely T ℓ -invariant. The open sets ℓ−1

A∗p (w) := ⋃ A∗ (T j (w)) j=0

ℓ−1

and Ap (w) := ⋃ A(T j (w)) j=0

(24.7)

are respectively coined immediate basin of attraction and basin of attraction of the periodic orbit {w, T(w), . . . , T ℓ−1 (w)}. The set A∗p (w) is forward T-invariant while Ap (w) is completely T-invariant. Moreover, {w, T(w), . . . , T ℓ−1 (w)} ⊆ A∗p (w) ⊆ Ap (w) ⊆ ℱ (T). In particular, this asserts that every attracting periodic orbit is in the Fatou set. This observation is the last ingredient needed to establish the following remarkable topological property of the Julia set. ̂ → ℂ ̂ be a rational function with deg(T) ≥ 2. The Julia set Theorem 24.1.15. Let T : ℂ 𝒥 (T) is perfect. Proof. Suppose by way of contradiction that w is an isolated point of 𝒥 (T). Then there exists some open neighborhood U of w such that U ∩ 𝒥 (T) = {w}. It follows from Corollary 24.1.10 that 𝒥 (T) = {T n (w) : 0 ≤ n ≤ n(U)} for some n(U) < ∞. So 𝒥 (T) is finite. Hence, w is a preperiodic or periodic point of T, i. e., there exist a least k ≥ 0 and a least ℓ ∈ ℕ such that T ℓ (T k (w)) = T k (w). But since the map T : 𝒥 (T) → 𝒥 (T) is surjective, it ensues that k = 0, i. e., w is a periodic point, and thus 𝒥 (T) consists of a single periodic orbit. It follows from the complete T-invariance of 𝒥 (T) (cf. Theorem 24.1.2) that T −1 ({w}) = {T ℓ−1 (w)} is a singleton. On the other hand, no point in the periodic orbit

964 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets (T j (w))ℓ−1 j=0 can be a critical point of T since that periodic orbit would then be superattracting, and hence, by Theorem 24.1.14, would be in ℱ (T). This contradicts the fact that deg(T) ≥ 2. Returning to the Fatou set, the local dynamics around attracting but not superattracting fixed points is entirely understood thanks to G. Koenigs’ result that dates back to 1884 (see [64–69]). ̂ →ℂ ̂ be a rational function with deg(T) ≥ 2. If w is an atTheorem 24.1.16. Let T : ℂ tracting but not superattracting fixed point of T, then there exist a neighborhood U of w, a neighborhood V of 0, and a unique (up to multiplication by a nonzero constant) conformal homeomorphism H : U → V such that H ∘ T ∘ H −1 (z) = T ′ (w)z,

∀z ∈ V .

̂ and n ∈ ℕ, Proof. We may assume without loss of generality that w = 0. For every z ∈ ℂ define Hn (z) := (T ′ (0))−n ⋅ T n (z). Then Hn satisfies Hn ∘ T = T ′ (0)Hn+1 ,

∀n ∈ ℕ.

Assume momentarily that (Hn )∞ n=1 converges pointwise to a function H on some neigh′ borhood of 0. Then H ∘ T = T (0)H on that neighborhood and H(0) = 0. It remains to show that H is a univalent holomorphic function on some neighborhood of 0. To obtain holomorphicity of H, we will show that the convergence is uniform on a compact neighborhood of 0. We know from the Maclaurin expansion of T that for some δ > 0 there is a constant C ≥ 0 such that 󵄨󵄨 󵄨 ′ 2 󵄨󵄨T(z) − T (0)z󵄨󵄨󵄨 ≤ C|z| ,

∀z ∈ B(0, δ).

Thus, |T(z)| ≤ |T ′ (0)| |z| + C|z|2 ≤ (|T ′ (0)| + Cδ)|z|. By reducing δ if necessary, we may assume that |T ′ (0)| + Cδ < 1 and we then obtain by induction that n 󵄨󵄨 n 󵄨󵄨 ′ 󵄨󵄨T (z)󵄨󵄨 ≤ (|T (0)| + Cδ) |z|,

∀z ∈ B(0, δ).

In particular, this guarantees that T maps B(0, δ) into itself, and so does all of its iterates. More precisely, T n (z) → 0 uniformly and exponentially fast on B(0, δ). By reducing δ even further if necessary, we may assume that ρ := (|T ′ (0)| + Cδ)2 /|T ′ (0)| < 1. Then n 2 n ′ n 󵄨 󵄨 Cρn |z|2 󵄨󵄨 󵄨 󵄨󵄨 T(T (z)) − T (0)T (z) 󵄨󵄨󵄨 C|T (z)| ≤ ′ , 󵄨󵄨Hn+1 (z) − Hn (z)󵄨󵄨󵄨 = 󵄨󵄨󵄨 󵄨󵄨 ≤ ′ ′ n+1 n+1 󵄨󵄨 |T (0)| 󵄨󵄨 |T (0)| (T (0))

∀z ∈ B(0, δ).

Hence, (Hn )∞ n=1 is uniformly Cauchy, and thus uniformly converges to some function H on B(0, δ). We can set U = B(0, δ) and V = H(U). Let us now show that H is univalent. Shrinking δ if needed ensures that T is univalent on B(0, δ). As T maps B(0, δ) into itself, every iterate T n is univalent on B(0, δ). It

24.1 Fatou set, Julia set, and periodic points

� 965

follows from Hurwitz’s theorem (Theorem A.3.9) that H is univalent on U = B(0, δ). (H cannot be constant on U since H ∘ T = T ′ (0)H on that neighborhood of 0.) It remains to demonstrate the uniqueness of H up to a nonzero multiplicative coñ are two such conjugacies. Then H ∘ T ∘ H −1 = T ′ (0)Id = stant. Suppose that H and H −1 ̃ ̃ ̃ H ∘ T ∘ H . Write h := H ∘ H −1 and observe that ̃ −1 = h ∘ (T ′ (0)Id). (T ′ (0)Id) ∘ h = H ∘ T ∘ H n Let h(z) = ∑∞ n=0 an z be the Maclaurin series of h. Substituting this series into the above conjugation equation and equating coefficients of same power, we deduce that T ′ (0)an = an (T ′ (0))n for all n ≥ 0. As T ′ (0) is neither zero nor a root of unity, we must have an = 0 ̃ ̃ for all n ≠ 1. Therefore, h(z) = a1 z with a1 ≠ 0. Hence, H(z) = h(H(z)) = a1 ⋅ H(z).

The local dynamics around superattracting fixed points is also entirely understood, this time due to L. Böttcher’s result published in 1904 (see [14–18]). ̂ → ℂ ̂ be a rational function with deg(T) ≥ 2. If w is a suTheorem 24.1.17. Let T : ℂ perattracting fixed point of T, then there exist a neighborhood U of w, a neighborhood V of 0, and a unique (up to multiplication by a (degw (T) − 1) root of unity) conformal homeomorphism H : U → V such that H(w) = 0 and H ∘ T ∘ H −1 (z) = zdegw (T) ,

∀z ∈ V .

Proof. Without losing generality, we may assume that w = 0 and let d = deg0 (T). Let 󵄨 ℂ+ := {z ∈ ℂ 󵄨󵄨󵄨 Re(z) > 0}. Let log0 : ℂ+ → ℂ be the principal branch of the logarithm function, i. e., the holomorphic branch of logarithm uniquely determined by the requirement that log0 (1) = 0. We know that T(z) = zd (1 + zf (z)), where f is a holomorphic function defined on B(0, 2R) for some R ∈ (0, 1/4). Set 󵄩 󵄩 B := 󵄩󵄩󵄩f |B(0,R) 󵄩󵄩󵄩∞ < ∞. Decreasing R > 0 if needed, we may assume that 󵄨󵄨 z 󵄨󵄨 4 󵄨 󵄨 4 (24.8) 󵄨󵄨e − 1󵄨󵄨 ≤ |z|, 󵄨󵄨󵄨log0 (1 + z)󵄨󵄨󵄨 ≤ |z|, ∀z ∈ B(0, R), 3 3 󵄨󵄨 1 + z2 1 + z2 󵄨󵄨󵄨󵄨 󵄨 4B = log0 (1 + z2 ) − log0 (1 + z1 ) & 󵄨󵄨󵄨log0 }) log0 󵄨 ≤ 1, ∀z1 , z2 ∈ B(0, R max{1, 3d−4 󵄨󵄨 1 + z1 1 + z1 󵄨󵄨󵄨 (24.9)

966 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets and 󵄨 1 󵄨󵄨 󵄨󵄨T(z)󵄨󵄨󵄨 ≤ |z|, 2

∀z ∈ B(0, R).

(24.10)

Fix r ∈ (0, R) so small that rB < 1/2

and

4 4rB ( + 1) < R. 3d 3d − 4

(24.11)

Define inductively a sequence of holomorphic functions Gn : B(0, r) → ℂ, n ≥ 0, satisfying 4B 󵄨󵄨 󵄨 , 󵄨󵄨Gn (z)󵄨󵄨󵄨 ≤ 3d − 4

∀z ∈ B(0, r)

(24.12)

as follows. Let G0 ≡ 0. For the inductive step, suppose that Gn has been defined satisfying (24.12). Define Gn+1 (z) for z ∈ B(0, r) \ {0} by the formula 1 1 Gn+1 (z) := z−1 [exp( log0 (1 + T(z)Gn (T(z))) + log0 (1 + zf (z))) − 1], d d

(24.13)

which is well-defined thanks to the first inequality in (24.11), (24.10) and (24.12). By the second inequality in (24.8), (24.10), (24.12) and the second inequality in (24.11), we get 󵄨󵄨 4|z| 󵄨 1 󵄨󵄨󵄨 4B 4r 󵄨 󵄨 󵄨 (󵄨󵄨G (T(z))󵄨󵄨󵄨+󵄨󵄨󵄨f (z)󵄨󵄨󵄨) ≤ ( +B) < R. 󵄨󵄨log0 (1+T(z)Gn (T(z)))+log0 (1+zf (z))󵄨󵄨󵄨 ≤ 󵄨 d󵄨 3d 󵄨 n 3d 3d − 4 Using this and the first inequality in (24.8), we deduce that 󵄨 󵄨󵄨 1 󵄨󵄨 󵄨 󵄨 4 −1 󵄨󵄨 1 󵄨󵄨Gn+1 (z)󵄨󵄨󵄨 ≤ |z| 󵄨󵄨󵄨 log0 (1 + T(z)Gn (T(z))) + log0 (1 + zf (z))󵄨󵄨󵄨 󵄨󵄨 d 󵄨󵄨 3 d 4 4B 4 󵄨 4B 󵄨 󵄨 󵄨 ( + 1) = . ≤ (󵄨󵄨󵄨Gn (T(z))󵄨󵄨󵄨 + 󵄨󵄨󵄨f (z)󵄨󵄨󵄨) ≤ 3d 3d 3d − 4 3d − 4 Hence, (24.12) holds for all z ∈ B(0, r) with n replaced by n + 1. Thus, the limit lim Gn+1 (z)

z→0

exists, and denoting its value by Gn+1 (0) completes the inductive step of the proof establishing a sequence (Gn )∞ n=1 of holomorphic functions satisfying (24.12). For every n ≥ 0, define the holomorphic function Hn : B(0, r) → ℂ by Hn (z) := z(1 + zGn (z)).

(24.14)

� 967

24.1 Fatou set, Julia set, and periodic points

Using (24.13), we obtain that d

d

(Hn+1 (z)) = zd (1 + zGn+1 (z))

1 1 = zd [exp( log0 (1 + T(z)Gn (T(z))) + log0 (1 + zf (z)))] d d

d

= zd exp(log0 (1 + T(z)Gn (T(z))) + log0 (1 + zf (z))) = zd exp(log0 (1 + T(z)Gn (T(z)))) exp(log0 (1 + zf (z)))

= zd (1 + T(z)Gn (T(z)))(1 + zf (z))

= T(z)(1 + T(z)Gn (T(z))) = Hn ∘ T(z). Consequently,

d

Hn ∘ T(z) = (Hn+1 (z)) ,

∀z ∈ B(0, r).

(24.15)

By induction, dk

Hn ∘ T k (z) = (Hn+k (z)) ,

∀z ∈ B(0, r), ∀n, k ≥ 0.

(24.16)

For every n ≥ 0, define the function Qn : B(0, r) → ℂ by Qn (z) := 1 + zGn (z). Claim. The sequence of functions (log0 Qn )∞ n=1 converges uniformly on B(0, r). Proof. Given n, k ≥ 0, it follows from (24.16) that dk

H (z) ( n+k+1 ) Hn+k (z)

=

Hn+1 (T k (z)) . Hn (T k (z))

According to (24.14), this means that dk

1 + zGn+k+1 (z) ( ) 1 + zGn+k (z)

=

1 + T k (z)Gn+1 (T k (z)) . 1 + T k (z)Gn (T k (z))

By taking the logarithm of both sides, we get d k log0

1 + zGn+k+1 (z) 1 + T k (z)Gn+1 (T k (z)) = log0 . 1 + zGn+k (z) 1 + T k (z)Gn (T k (z))

It follows from (24.12), the second inequality in (24.11) and the first inequality in (24.9) that

968 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets

log0 (1 + zGn+k+1 (z)) − log0 (1 + zGn+k (z)) = d −k log0

1 + T k (z)Gn+1 (T k (z)) . 1 + T k (z)Gn (T k (z))

Hence, using the second inequality in (24.9) we obtain that 󵄨 󵄨󵄨 󵄨󵄨log (1 + zGn+k+1 (z)) − log (1 + zGn+k (z))󵄨󵄨󵄨 ≤ d −k . 0 󵄨󵄨 󵄨󵄨 0 So, for all m, n ≥ 0 we deduce that 󵄨󵄨 󵄨 n+m−1󵄨 󵄨 󵄨󵄨log (1 + zGn+m (z)) − log (1 + zGn (z))󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨log (1 + zGk+1 (z)) − log (1 + zGk (z))󵄨󵄨󵄨 0 0 󵄨󵄨 0 󵄨󵄨 󵄨󵄨 0 󵄨󵄨 k=n

n+m−1

∞

k=n

k=n

≤ ∑ d −k ≤ ∑ d −k =

d −n . 1 − d −1

Consequently, the sequence (log0 Qn )∞ n=1 is uniformly Cauchy, and thus converges uniformly. The proof of the claim is finished. ◼ Denote by L the uniform limit of the sequence (log0 Qn )∞ n=1 . Then L : B(0, r) → ℂ is holomorphic and L(0) = lim log0 Qn (0) = lim log0 1 = 0. n→∞

n→∞

Let Q := eL : B(0, r) → ℂ. Then Q is holomorphic and Q(0) = eL(0) = e0 = 1.

(24.17)

Let the function H : B(0, r) → ℂ be given by H(z) = zQ(z). Then H is a holomorphic function and, by (24.17), H(z) = z + O(z2 ),

∀z ∈ B(0, r).

(24.18)

By continuity of the exponential function exp : ℂ → ℂ and the dth power, along with the above claim, we obtain

24.1 Fatou set, Julia set, and periodic points

� 969

H(T(z)) = T(z)Q(T(z)) = T(z) exp(L(T(z))) = T(z) exp( lim log0 Qn (T(z))) n→∞

= lim [T(z) exp(log0 Qn (T(z)))] = lim [T(z)Qn (T(z))] = lim Hn (T(z)) n→∞

n→∞ d

d

d

n→∞

= lim (Hn+1 (z)) = [ lim Hn+1 (z)] = [ lim (zQn+1 (z))] n→∞

n→∞

n→∞

d

d

= [z lim exp(log0 Qn+1 (z))] = [z exp( lim log0 Qn+1 (z))] n→∞

= (ze

L(z) d

d

n→∞

d

) = (zQ(z)) = (H(z)) .

Thus, H ∘T = H d on B(0, r) and along with (24.18), the proof of the existence of the sought conjugation is complete. It remains to demonstrate the uniqueness of H up to multiplication by a (degw (T)−1) root of unity. In a similar manner to the uniqueness part of the proof of Theorem 24.1.16, it suffices to show that any conjugation of zd to itself is a multiple of zd and that multiple is a (d − 1)th root of unity. Suppose that h is such a conjugation, i. e., (h(z))d = h(zd ). n Let h(z) = ∑∞ n=0 an z be the Maclaurin series of h. Substituting this series into the above conjugation equation and equating coefficients of same power, we deduce that add−1 = 1 while an = 0 for all n ≠ d. As an immediate consequence of Theorems 24.1.16 and 24.1.17, we get the following two respective results. ̂ → ℂ ̂ be a rational function with deg(T) ≥ 2. If w is an Theorem 24.1.18. Let T : ℂ attracting but not superattracting periodic point of T of period p ∈ ℕ, then there exist a neighborhood U of w, a neighborhood V of 0, and a unique (up to multiplication by a nonzero constant) conformal homeomorphism H : U → V such that H ∘ T p ∘ H −1 (z) = (T p ) (w)z, ′

∀z ∈ V .

̂→ℂ ̂ be a rational function with deg(T) ≥ 2. If w is a superTheorem 24.1.19. Let T : ℂ attracting periodic point of T of period p ∈ ℕ, then there exist a neighborhood U of w, a neighborhood V of 0 and a unique (up to multiplication by a (degw (T p ) − 1) root of unity) conformal homeomorphism H : U → V such that p

H ∘ T p ∘ H −1 (z) = zdegw (T ) ,

∀z ∈ V .

We end our examination of attracting periodic orbits with a demonstration that their immediate basin of attraction must comprise at least one critical point. ̂→ℂ ̂ be a rational function with deg(T) ≥ 2. If w ∈ ℂ ̂ is an Theorem 24.1.20. Let T : ℂ ∗ attracting periodic point of T, then the immediate basin of attraction Ap (w) of the orbit of w contains at least one critical point of T.

970 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets Proof. Assume first that w is an attracting fixed point of T. Suppose for a contradiction that A∗ (w) contains no critical point of T. We shall prove by induction that there exists ∗ an ascending sequence (Un )∞ n=0 of open simply connected subsets of A (w) such that (a) w ∈ U0 . (b) For every n ∈ ℕ, there exists Tn−1 : Un−1 → Un , a unique surjective holomorphic inverse branch of T such that Tn−1 (w) = w. −1 −1 󵄨󵄨 (c) Tn+1 󵄨󵄨U = Tn , ∀n ∈ ℕ. n−1

Indeed, let U0 be an arbitrary open simply connected subset of A∗ (w) containing w. For the inductive step, suppose that open simply connected subsets U0 ⊆ U1 ⊆ ⋅ ⋅ ⋅ ⊆ Un ⊆ A∗ (w) satisfying conditions (a) and (b) have been defined for some n ≥ 0. Since A∗ (w) ∩ T −1 (Un ) contains no critical point of T, the inverse function theorem (Theorem A.3.13) and the monodromy theorem (Theorem A.3.4) yield the existence of a holo−1 morphic inverse branch Tn+1 : Un → A∗ (w) uniquely determined by the requirement −1 −1 that Tn+1 (w) = w. Set Un+1 := Tn+1 (Un ) ⊆ A∗ (w). Readily from this definition, we see that −1 ̂ and that Tn+1 Un+1 is a open simply connected subset of ℂ : Un → Un+1 is the unique sur−1 jective holomorphic inverse branch of T such that Tn+1 (w) = w. As Tn−1 : Un−1 → Un is a holomorphic inverse branch of the same map T such that Tn−1 (w) = w and as Un−1 ⊆ Un −1 −1 by the inductive hypothesis, we infer that Tn+1 |Un−1 = Tn−1 . Hence, Un+1 = Tn+1 (Un ) ⊇ −1 ∞ Tn (Un−1 ) = Un . The inductive construction of the sequence (Un )n=0 is thus complete. It follows from the defining properties of the sequence (Un , Tn+1 )∞ n=0 that for all n ∈ ℕ, the composite function −1 Tw−n := Tn−1 ∘ Tn−1 ∘ ⋅ ⋅ ⋅ ∘ T2−1 ∘ T1−1 : U0 → Un ⊆ A∗ (w)

is a well-defined, surjective and holomorphic inverse branch of T n such that Tw−n (w) = w. Since A∗ (w) ⊆ ℱ (T) and since the Julia set 𝒥 (T) contains at least three points (because it is perfect by Theorem 24.1.15), it follows from Montel’s criterion (Theorem 23.2.4) that the family of maps {Tw−n : U0 → A∗ (w)}∞ n=1 is normal. This, however, produces a contradiction since lim (Tw−n ) (w) = lim ((T n )′ (w))

n→∞

′

n→∞

−1

= lim (T ′ (w)) n→∞

−n

= ∞.

We are thus done when w is a fixed point of T. In general, if w is a periodic point of T with prime period ℓ, then w is a fixed point of T ℓ . By the case just treated, we know that A∗T ℓ (w) contains a critical point cℓ of T ℓ . This critical point of T ℓ has a forward iterate T k (cℓ ) which is a critical point of T (in fact, 0 ≤ k < ℓ). But observe that A∗p (w) ⊇ A∗T ℓ (w) ∋ cℓ . Given that A∗p (w) is forward T-invariant, we conclude that A∗p (w) ∋ T k (cℓ ).

As the number of critical points of T (counted by multiplicities) is 2 deg(T) − 2, the previous theorem yields an upper estimate on the number of attracting cycles.

24.1 Fatou set, Julia set, and periodic points

� 971

̂ → ℂ ̂ with deg(T) ≥ 2 has at most Corollary 24.1.21. Any rational function T : ℂ 2 deg(T) − 2 attracting periodic cycles.

24.1.2 Nonattracting periodic points Recall that there are three different types of nonattracting periodic points: repelling, rationally indifferent, and irrationally indifferent points. 24.1.2.1 Repelling and rationally indifferent periodic points For repelling periodic points, the situation is opposite to that of attracting periodic points (cf. Theorem 24.1.14). ̂ → ℂ ̂ is a rational function with deg(T) ≥ 2, then every reTheorem 24.1.22. If T : ℂ pelling periodic point of T belongs to the Julia set 𝒥 (T). Proof. Suppose for a contradiction that a repelling periodic point w of T with prime period ℓ belongs to the Fatou set ℱ (T). Conjugating by a Möbius map if necessary, we may assume without loss of generality that w ∈ ℂ. Then there exists an open connected neighborhood U ⊆ ℂ of w such that the family {T n }∞ n=1 of iterates of T is normal on U. Hence, the sequence (T ℓn |U )∞ contains a subsequence (T ℓnj |U )∞ n=1 j=1 converging uniformly ̂ Then S(w) = w. Shrinkon compact subsets of U to some meromorphic map S : U → ℂ. ing U if necessary, we may assume that S(U) ⊆ ℂ, i. e., S is holomorphic. On one hand, S ′ (w) ∈ ℂ, i. e., |S ′ (w)| ≠ ∞. On the other hand, 󵄨󵄨 ′ 󵄨󵄨 󵄨 ℓn ′ 󵄨 󵄨 ℓ ′ 󵄨nj 󵄨󵄨S (w)󵄨󵄨 = lim 󵄨󵄨󵄨(T j ) (w)󵄨󵄨󵄨 = lim 󵄨󵄨󵄨(T ) (w)󵄨󵄨󵄨 = ∞. j→∞ j→∞ This is a contradiction. Hence, w ∈ 𝒥 (T). The proof of the corresponding result for rationally indifferent periodic points is somewhat more involved. ̂→ℂ ̂ is a rational function with deg(T) ≥ 2, then every ratioTheorem 24.1.23. If T : ℂ nally indifferent periodic point of T belongs to the Julia set 𝒥 (T). Proof. As in the proof of the previous theorem, through an appropriate Möbius conjugation we may assume without loss of generality that the rationally indifferent periodic point is 0. In view of Theorem 24.1.5, passing to a sufficiently high iterate of T, we may further assume that T(0) = 0 and T ′ (0) = 1. Then the Maclaurin series of T is of the form T(z) = z + azp+1 + O(zp+2 ), where p ∈ ℕ and a ∈ ℂ \ {0}. We prove by induction that

(24.19)

972 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets T n (z) = z + nazp+1 + O(zp+2 ).

(24.20)

Indeed, (24.20) reduces to (24.19) when n = 1 and so holds in this case. For the inductive step, assume that (24.20) holds for some n ∈ ℕ. Then p+1

T n+1 (z) = T n (T(z)) = T(z) + na(T(z))

p+2

+ O((T(z))

) p+1

= z + azp+1 + O(zp+2 ) + na(z + azp+1 + O(zp+2 )) p+1

= z + az

p+1

+ naz

p+2

+ O(z

p+1

) = z + (n + 1)az

p+2

+ O((z + azp+1 + O(zp+2 )) p+2

+ O(z

)

).

Thus, (24.20) is valid. Then the (p + 1)th derivative of T n+1 at 0 is (T n )(p+1) (0) = (p + 1)!na, and hence lim (T n )(p+1) (0) = ∞.

n→∞

Suppose for a contradiction that 0 ∈ ℱ (T). Then there exist an open connected neighborhood U ⊆ ℂ of 0, a strictly increasing sequence (nk )∞ k=1 of positive integers, and ̂ such that a meromorphic function S : U → ℂ, lim T nk = S

k→∞

uniformly on U. Since S(0) = 0, shrinking U if necessary, we may assume that S(U) ⊆ ℂ, i. e., that S is holomorphic. But then ∞ ≠ S (p+1) (0) = lim (T nk )(p+1) (0) = ∞. k→∞

This contradiction shows that 0 ∈ 𝒥 (T). 24.1.2.2 Irrationally indifferent periodic points Peculiarly, irrationally indifferent periodic points are sometimes found in the Fatou set (they are then called Siegel points), sometimes in the Julia set (they are then called Cremer points). Definition 24.1.24. Let w ∈ ℂ be an irrationally indifferent periodic point of a ratiô → ℂ ̂ with prime period ℓ. Then w is called a Siegel point if there nal function T : ℂ exists a holomorphic function H mapping a neighborhood of 0 to ℂ with the following properties: (a) H(0) = w (b) H ′ (0) = 1 (c) H −1 ∘ T ℓ ∘ H(z) = (T ℓ )′ (w)z.

24.1 Fatou set, Julia set, and periodic points

� 973

Property (c) is commonly referred to as the Schröder equation. Compare this definition with Theorem 24.1.18. As H ′ (0) ≠ 0, we know that H is injective on a neighborhood of 0. In fact, H has the following striking property. Proposition 24.1.25. If w ∈ ℂ is a Siegel point of T, then H is injective on any ball that is centered at 0 and contained in the domain of H. Proof. Suppose that B(0, r) is contained in the domain of H. Set λ := (T ℓ )′ (w). Assume that H(z1 ) = H(z2 ) for some points z1 , z2 ∈ B(0, r). For every n ∈ ℕ, property (c) of a Siegel point yields that H(λn z2 ) = T nℓ (H(z2 )) = T nℓ (H(z1 )) = H(λn z1 ). 1 Since {λn }∞ n=0 is a dense subset of the unit circle 𝕊 = {z ∈ ℂ : |z| = 1}, it follows that 1 H(zz1 ) = H(zz2 ) for all z ∈ 𝕊 . Therefore, the holomorphic functions B(0, r/|z1 |) ∋ z 󳨃→ H(zz1 ) ∈ ℂ and B(0, r/|z2 |) ∋ z 󳨃→ H(zz2 ) ∈ ℂ agree on 𝕊1 . (Note: If zi = 0, define r/|zi | = ∞ and B(0, r/|zi |) = ℂ.) Consequently, these functions must agree on B(0, 1) ⊆ B(0, r/ max{|z1 |, |z2 |}), i. e.,

H(zz1 ) = H(zz2 ),

∀z ∈ B(0, 1).

Taking the derivative of both sides at 0 and using property (b) of a Siegel point results in z1 = z1 H ′ (0) = z2 H ′ (0) = z2 . We have thus demonstrated that H is injective on B(0, r). This proposition means that in the vicinity of a Siegel point w of T with prime period ℓ ∈ ℕ, the map T ℓ is topologically conjugate to the rotation by the multiplier of w, i. e., T ℓ is conjugate to z 󳨃→ (T ℓ )′ (w)z. There is a simple but fundamental characterization of Siegel points. Theorem 24.1.26. An irrationally indifferent periodic point of T is a Siegel point if and only if it belongs to the Fatou set ℱ (T). Proof. Set λ = (T ℓ )′ (w). If w is a Siegel point, it follows from properties (a) and (c) and Proposition 24.1.25 that T ℓn (z) = H(λn H −1 (z)) on some neighborhood of w and for all n ∈ ℕ. Therefore, the iterates of T ℓ form a bounded sequence on some sufficiently small neighborhood of w. Montel’s criterion (Theorem 23.2.4) and Theorem 24.1.5 assert that w ∈ ℱ (T ℓ ) = ℱ (T). Conversely, if w ∈ ℱ (T) then as T ℓ (w) = w there exists a neighborhood U of w on which the iterates of T ℓ are uniformly bounded, i. e., there is M > 0 such that 󵄨󵄨 ℓn 󵄨󵄨 󵄨󵄨T (z)󵄨󵄨 ≤ M,

∀z ∈ U, ∀n ≥ 0.

974 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets ̂n : U → ℂ defined by As usual, we may assume that w = 0. Consider the functions H n−1 ̂n (z) := 1 ∑ λ−j T ℓj (z). H n j=0

̂n (z)| ≤ M for all z ∈ U and all n ∈ ℕ. Therefore, the sequence (H ̂n )∞ is normal Then |H n=1 on U. Let n −1

k ̂ := lim 1 ∑ λ−j T ℓj H k→∞ nk j=0

̂ : U → ℂ is analytic, be one of its limit functions. Then H n −1

n −1

1 k −j ℓj 1 k −j ̂ H(0) = lim ∑ λ T (0) = lim ∑ λ ⋅ 0 = 0, k→∞ nk k→∞ nk j=0 j=0 n −1

n −1

k k ̂ ′ (0) = lim 1 ∑ λ−j (T ℓj )′ (0) = lim 1 ∑ λ−j λj = 1, H k→∞ nk k→∞ nk j=0 j=0

(24.21) (24.22)

and n −1

n −1

k k ̂ ̂ ∘ T ℓ (z) = lim 1 ∑ λ−j T ℓj (T ℓ (z)) = λ lim 1 ∑ λ−(j+1) T ℓ(j+1) (z) = λH(z). (24.23) H k→∞ nk k→∞ nk j=0 j=0

̂ −1 satisfies Relations (24.21)–(24.23) directly imply that the holomorphic function H := H conditions (a)–(c) defining 0 as a Siegel point. We now turn our attention to the other class of irrationally indifferent periodic points. Irrationally indifferent periodic points that are not Siegel points are called Cremer points. Obviously, they are characterized by their belonging to the Julia set. Definition 24.1.27. An irrationally indifferent periodic point of a rational function T : ̂→ℂ ̂ is called a Cremer point of T if it lies in the Julia set 𝒥 (T). ℂ Per Theorem 24.1.26, if ξ is a Cremer point of T with prime period ℓ ∈ ℕ, then in the vicinity of ξ the map T ℓ is not topologically conjugate to the rotation by the multiplier of ξ. From all of the above, we have the following. ̂→ Theorem 24.1.28. All attracting and Siegel periodic points of a rational function T : ℂ ̂ ℂ are in the Fatou set of T, while all repelling, rationally indifferent, and Cremer periodic points are in the Julia set of T. Carl Ludwig Siegel [121] proved the existence of Siegel points, which is also of profound importance for the KAM theory; the proof can be found in [21]. The study of Siegel

24.1 Fatou set, Julia set, and periodic points

� 975

points, particularly the nature of the rotational numbers for which they exist, has been the topic of extensive research. About the existence of Cremer points, we shall prove the following fact. Theorem 24.1.29. There exists θ ∈ (0, 1) such that 0 is a Cremer fixed point for every ̂ ∋ z 󳨃󳨀→ e2πiθ z(1 + Q(z)), where Q is a nonconstant polynomial. polynomial of the form ℂ Proof. The statement is equivalent to asserting that there exists λ = e2πiθ so that the Schröder equation has no solution for any polynomial P of the form P(z) = ad zd +⋅ ⋅ ⋅+λz. Suppose for a contradiction that there is a conjugation H defined on B(0, δ) for some δ > 0. Consider the d n fixed points of Pn , i. e., the roots of n

Pn (z) − z = zd + ⋅ ⋅ ⋅ + (λn − 1)z = 0. One root is 0. Label the others z1 , . . . , zdn −1 . As Pn (z) = H(λn H −1 (z)) has only one zero in B(0, δ), we know that zj ∉ B(0, δ) for all 1 ≤ j ≤ d n − 1. Thus, d n −1

n 󵄨 󵄨 δd ≤ ∏ |zj | = 󵄨󵄨󵄨1 − λn 󵄨󵄨󵄨.

j=1

We now construct a λ for which this is impossible, hence contradicting the existence ∞ −qk of H. Take any strictly increasing sequence (qk )∞ k=1 of positive integers. Set θ = ∑k=1 2 2πiθ and λ = e . Then there is a constant Δ ≥ 1 such that 󵄨󵄨 2qk 󵄨 q −q 󵄨󵄨1 − λ 󵄨󵄨󵄨 ≤ Δ ⋅ 2 k k+1 ,

∀k ∈ ℕ.

After using the two relations above and taking logarithms, it can be proved by induction that qk

qk+1 ≤ C(δ)d 2 ,

∀k ∈ ℕ,

q1

d }. If the qk ’s are defined inductively to grow very where C(δ) = max{q2 /d 2 , log(Δ/δ) log 2 d−1 qk rapidly, e. g. if log qk+1 ≥ k2 , this inequality is violated for any d and δ > 0.

An example as above was first given by G. A. Pfeiffer [104] in 1917. His work was continued by Hubert Cremer [30], who proved in 1938 that if λ is not a root of unity but |λ| = 1 and 󵄨 󵄨1/n lim inf󵄨󵄨󵄨λn − 1󵄨󵄨󵄨 = 0, n→∞

then there exists an entire function of the form z 󳨃󳨀→ λz(1 + Q(z)), where Q is a nonconstant entire function. In a seminal paper in 1942, Siegel [121] gave the first example of a θ for which Schröder’s equation is solvable.

976 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets

24.2 Rationally indifferent periodic points In the previous section, we observed that all rationally indifferent periodic points of a rational function lie in the Julia set of that function. We now look at the dynamics around such points not only for rational functions but more generally for holomorphic functions.

24.2.1 Local and asymptotic behavior of holomorphic functions around rationally indifferent periodic points: part I In this section, we present basic results about the qualitative and especially the quantitative asymptotic behaviors of holomorphic functions around their parabolic (i. e., rationally indifferent) periodic points. Definition 24.2.1. A holomorphic map φ defined on a neighborhood of a point ω ∈ ℂ is said to be a locally holomorphic parabolic map at ω (or rationally indifferent at ω) if: (i) ω is a periodic point of φ, i. e., φℓ (ω) = ω for some ℓ ∈ ℕ, (ii) its multiplier (φℓ )′ (ω) is a root of unity, and (iii) no iterate of φ is the identity map. The periodic point ω itself is then called parabolic (or rationally indifferent). Definition 24.2.2. A locally holomorphic parabolic map φ (or the point ω) is called simple parabolic (resp., simple parabolic fixed point) if (a) φ(ω) = ω, (b) φ′ (ω) = 1, and (c) φ is not the identity map. Note that some sufficiently high iterate of any locally holomorphic parabolic map is simple. Therefore, the behavior of locally holomorphic parabolic maps can essentially be analyzed by studying simple parabolic ones. Accordingly, throughout this section the map φ (and its point ω) is always assumed to be locally holomorphic simple parabolic. Within the realm of ideas related to Fatou’s flower theorem (see [1, 8, 21, 94, 123] for historical information), we aim to analyze the behavior of φ in a sufficiently small neighborhood of the simple parabolic fixed point ω. On a sufficiently small neighborhood of ω, the map φ has a Taylor series expansion around ω of the form φ(z) = z + a(z − ω)p+1 + O((z − ω)p+2 ) for some p = p(ω) ∈ ℕ and a = a(ω) ∈ ℂ \ {0}.

(24.24)

24.2 Rationally indifferent periodic points

� 977

The rays coming out of ω and forming the set {z ∈ ℂ : a(z − ω)p < 0} are called attracting directions while the rays forming the set {z ∈ ℂ : a(z − ω)p > 0} are called repelling directions. See Figures 24.1–24.2. Indeed, note that φ′ (z) = 1 + (p + 1)a(z − ω)p + O((z − ω)p+1 ), and hence |φ′ (z)| < 1 for all z close enough to ω that belong to an attracting direction while |φ′ (z)| > 1 for all z close enough to ω that lie along a repelling direction. Fix an attracting direction, say p

A := ω + √−a−1 (0, ∞),

(24.25)

p where √⋅ is a holomorphic branch of the pth radical function defined on ℂ \ a−1 (0, ∞). In order to simplify our analysis, let us change the coordinate system via a conjugation through the affine map

ρA (z) = ω + √−(ap)−1 ⋅ z. p

That is, we define the function φA,0 := ρ−1 A ∘ φ ∘ ρA whose Taylor series expansion around 0 is of the form φA,0 (z) = z −

zp+1 ∞ + ∑ an zp+n p n=2

(24.26)

for some an ∈ ℂ, n ≥ 2. So φA,0 (0) = 0

and

(φA,0 )′ (0) = 1,

i. e., 0 is a simple parabolic fixed point of φA,0 . In addition, ρ−1 A (A) = (0, ∞) is an attracting direction for φA,0 , as in Figures 24.1–24.2. We want to analyze the behavior of φA,0 on sufficiently small neighborhoods of 0. In order to do this, similar to the previous section, we conjugate φA,0 on ℂ \ (−∞, 0] to a p map defined “near” infinity. More precisely, we consider √⋅, the holomorphic branch of

978 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets

π/3

Figure 24.1: Case p = 3 (with a = −1). There are 3 attracting directions (indicated by rays and vectors pointing toward the parabolic point) and 3 repelling directions (indicated by dashed rays and vectors pointing away from the parabolic point).

π/5

Figure 24.2: Case p = 5 (with a = −1). There are 5 attracting directions (indicated by rays and vectors pointing toward the parabolic point) and 5 repelling directions (indicated by dashed rays and vectors pointing away from the parabolic point).

� 979

24.2 Rationally indifferent periodic points

the pth-root function defined on ℂ \ (−∞, 0] which leaves the point 1 fixed. We will also frequently denote this branch by z1/p . Then we define the map H(z) =

1 = z−1/p . √p z

̂→ℂ ̂ defined by the formula This map has a meromorphic inverse H −1 : ℂ H −1 (z) = z−p . Consider the conjugate map ̃ A = H −1 ∘ φA,0 ∘ H = (ρA ∘ H)−1 ∘ φ ∘ (ρA ∘ H), φ

(24.27)

̃ A ) := (ℂ \ (−∞, 0]) ∩ H −1 (Vφ ), where Vφ ⊆ ℂ is a sufficiently small neighdefined on D(φ borhood of 0 on which φA,0 is defined. ̃A 24.2.1.1 Properties of φ ̃ A , its iterates, their derivatives and the limit of those. We We now study the function φ ̃ A and its derivative. shall first prove the following technical but useful result about φ Lemma 24.2.3. If φ is a locally holomorphic simple parabolic map and A is an attracting direction of φ, then (shrinking the neighborhood Vφ if necessary) there exists a holomorphic function B : Vφ → ℂ such that ̃ A (z) = z + 1 + B(H(z)) φ

with

−1 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨B(H(z))󵄨󵄨󵄨 ≤ M 󵄨󵄨󵄨H(z)󵄨󵄨󵄨 = M|z| p ,

̃ A ), ∀z ∈ D(φ

̃ A ) := (ℂ \ (−∞, 0]) ∩ H −1 (Vφ ). for some constant M > 0, where D(φ Reducing Vφ if needed, this yields that ̃ ′A (z) = 1 + (B ∘ H)′ (z) φ

p+1 󵄨󵄨 󵄨 ′ −1 − 󵄨󵄨(B ∘ H) (z)󵄨󵄨󵄨 ≤ Mp |z| p < 1,

with

̃ A ). ∀z ∈ D(φ

̃ A is univalent and In particular, this implies that φ ̃ A (z) ≈ z + 1 and φ ̃ ′A (z) ≈ 1 φ

near ∞,

where f ≈ g near z0 means that lim |f (z) − g(z)| = 0. z→z0

̃ A ), Proof. For all z ∈ D(φ ̃ A (z) = H −1 (φA,0 (H(z))) = H −1 (H(z) − φ − p1

= H (z −1

−

z

−

p+1 p

p

∞

+ ∑ an z n=2

−

p+n p

)

p+1

(H(z)) p

∞

p+n

+ ∑ an (H(z)) n=2

)

980 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets − p1

= H −1 (z = z[1 − Set ζ = H(z) = z

− p1

[1 −

p+n−1 z−1 ∞ − + ∑ an z p ]) p n=2

p+n−1 z−1 ∞ − + ∑ an z p ] . p n=2

−p

(24.28)

and G(ζ ) := 1 −

ζp ∞ + ∑ a ζ p+n−1 , p n=2 n

∀ζ ∈ Vφ .

(24.29)

Clearly, d k G 󵄨󵄨󵄨󵄨 󵄨 = 0, dζ k 󵄨󵄨󵄨0

G(0) = 1,

∀0 ≤ k < p,

d p G 󵄨󵄨󵄨󵄨 󵄨 = −(p − 1)! dζ p 󵄨󵄨󵄨0

and

Therefore, the power function G−p satisfies G−p (0) = 1,

d k (G−p ) 󵄨󵄨󵄨󵄨 󵄨 = 0, dζ k 󵄨󵄨󵄨0

∀0 ≤ k < p,

and

d p (G−p ) 󵄨󵄨󵄨󵄨 󵄨 = p! dζ p 󵄨󵄨󵄨0

Thus, the Taylor series expansion of G−p about 0 is ∞

G−p (ζ ) = 1 + ζ p + ∑ bn ζ p+n

(24.30)

n=1

for some appropriate coefficients bn , n ∈ ℕ, where the tail of the series converges absolutely uniformly on some sufficiently small neighborhood of 0; shrinking Vφ if needed, we may identify this neighborhood with Vφ . Then the associated series ∞

B(ζ ) := ∑ bn ζ n n=1

converges absolutely uniformly on Vφ , and hence constitutes an holomorphic function. Shrinking Vφ even more if necessary, there is a constant M ∈ (0, ∞) such that 󵄨󵄨 󵄨 󵄨󵄨B(ζ )󵄨󵄨󵄨 ≤ M|ζ | and

󵄨󵄨 ′ 󵄨󵄨 󵄨󵄨B (ζ )󵄨󵄨 ≤ M,

∀ζ ∈ Vφ .

(24.31)

̃ A ) := (ℂ \ (−∞, 0]) ∩ H −1 (Vφ ), we get Reverting back to the variable z = ζ −p ∈ D(φ from (24.28)–(24.30) that p

∞

̃ A (z) = zG−p (H(z)) = z[1 + (H(z)) + ∑ bn (H(z)) φ n=1

∞

n

p+n

]

= z[1 + z−1 + z−1 ∑ bn (H(z)) ] = z + 1 + B(H(z)). n=1

24.2 Rationally indifferent periodic points

� 981

It also follows from (24.31) that −1 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨B(H(z))󵄨󵄨󵄨 ≤ M 󵄨󵄨󵄨H(z)󵄨󵄨󵄨 = M|z| p

and

p+1 󵄨 󵄨 ′ 󵄨 󵄨 󵄨 ′ 󵄨󵄨 −1 − ′ 󵄨󵄨(B ∘ H) (z)󵄨󵄨󵄨 = 󵄨󵄨󵄨B (H(z))󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨H (z)󵄨󵄨󵄨 ≤ Mp |z| p .

̃ ′A simply ensues from differentiation of φ ̃A. The expression of φ Given a point x ∈ (0, ∞) and an angle α ∈ (0, π), let 󵄨 S(x, α) := {z ∈ ℂ \ {x} 󵄨󵄨󵄨 − α < arg(z − x) < α} be the open sector with vertex x, central angle α and symmetric about the real axis. By Lemma 24.2.3, for all α ∈ (0, π) and all κ ∈ (0, 1) there is x(α, κ) ∈ (0, ∞) such that ̃ A (S(x, α)) ⊆ S(x + 1 − κ, α) ⊆ S(x, α), φ

∀x ≥ x(α, κ),

(24.32)

and ̃ A (z)) ≥ Re(z) + 1 − κ, Re(φ

∀z ∈ S(x(α, κ), α).

(24.33)

By induction, we obtain the following two statements. They affirm that every sector ̃ A -invariant and describe how the iterates of φ ̃ A map each such sector S(x, α) is forward φ into itself. Lemma 24.2.4. Let φ be a locally holomorphic simple parabolic map and let A be an attracting direction for φ. Fix α ∈ (0, π) and κ ∈ (0, 1). Then there is x(α, κ) ∈ (0, ∞) such ̃ nA : S(x(α, κ), α) → ℂ are well-defined and satisfy that all the iterates φ ̃ nA (S(x, α)) ⊆ S(x + n(1 − κ), α) ⊆ S(x, α), φ

∀x ≥ x(α, κ),

∀n ∈ ℕ.

̃ nA → ∞ uniformly on S(x(α, κ), α). In particular, φ Lemma 24.2.5. If φ is a locally holomorphic simple parabolic map and A is an attracting direction for φ, then for all α ∈ (0, π) and κ ∈ (0, 1) we have 󵄨󵄨 ̃ n 󵄨󵄨 ̃ nA (z)) ≥ Re(z) + n(1 − κ), 󵄨󵄨φA (z)󵄨󵄨 ≥ Re(φ

∀z ∈ S(x(α, κ), α),

∀n ∈ ℕ,

where x(α, κ) arises from (24.32)–(24.33). A consequence of the previous results and yet another description of the behavior ̃ A on the sector S(x(α, κ), α), is the following. of the iterates of φ Lemma 24.2.6. If φ is a locally holomorphic simple parabolic map and A is an attracting direction for φ, then for all α ∈ (0, π), κ ∈ (0, 1), z ∈ S(x(α, κ), α) and n ∈ ℕ, we have 1

̃ nA (z) = z + n + ORe(z) (max{n1− p , log n}), φ

982 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets where Ot is a big O symbol that depends decreasingly on t ∈ ℝ and that converges to 0 when t → ∞. Proof. Fix z ∈ S(x(α, κ), α). By Lemmas 24.2.3–24.2.5, for every n ∈ ℕ we have 1

−p ̃ n+1 ̃n φ A (z) = φA (z) + 1 + ORe(z) (n ).

By induction, n−1

1

1

̃ nA (z) = z + n + ORe(z) ( ∑ k − p ) = z + n + ORe(z) (max{n1− p , log n}). φ k=1

As an immediate ramification of this lemma, we get the following bounds on the distance between the iterates of a point in the sector S(x(α, κ), α) and the point itself. Lemma 24.2.7. If φ is a locally holomorphic simple parabolic map and A is an attracting direction for φ, then for all α ∈ (0, π), κ ∈ (0, 1) and t ∈ ℝ, there exists a constant C = C(α, κ, t) such that for all n ∈ ℕ and all z ∈ S(x(α, κ), α) with Re(z) ≥ t, we have 󵄨̃n 󵄨󵄨 C −1 n ≤ 󵄨󵄨󵄨φ A (z) − z󵄨󵄨 ≤ Cn. For every x ∈ (0, ∞), α ∈ (0, π) and R > 0, let S(x, α, R) := S(x, α) ∩ B(0, R). ̃A. We now establish properties of the derivatives of the iterates of φ Lemma 24.2.8. Let φ be a locally holomorphic simple parabolic map and let A be an attracting direction for φ. Fix α ∈ (0, π), κ ∈ (0, 1) and R > 0. Then 󵄨 ̃ n ′ 󵄨󵄨 0 < inf{󵄨󵄨󵄨(φ A ) (z)󵄨󵄨 : z ∈ S(x(α, κ), α, R), n ∈ ℕ} 󵄨󵄨 ̃ n ′ 󵄨󵄨 ≤ sup{󵄨󵄨(φA ) (z)󵄨󵄨 : z ∈ S(x(α, κ), α, R), n ∈ ℕ} < ∞. Furthermore, for every γ > 1, 󵄨 ̃n ′ 󵄨󵄨 lim sup{󵄨󵄨󵄨(φ A ) (z) − 1󵄨󵄨 : z ∈ S(x, α, γx), n ∈ ℕ} = 0

x→∞

and 󵄨 ̃n ′ 󵄨󵄨 lim sup{󵄨󵄨󵄨(φ A ) (z) − 1󵄨󵄨 : z ∈ S(x, π/2), n ∈ ℕ} = 0.

x→∞

Proof. For every z ∈ S(x(α, κ), α), let ̃ ′A (z) − 1. g(z) = φ

24.2 Rationally indifferent periodic points

� 983

By the chain rule, for every n ∈ ℕ we have n−1

n−1

j

j

̃ nA ) (z) = ∏ φ ̃ ′A (φ ̃ A (z)) = ∏[1 + g(φ ̃ A (z))]. (φ ′

j=0

j=0

(24.34)

󵄨󵄨 󵄨󵄨 ̃ n 󵄨󵄨 󵄨 ̃n According to Exercise 24.8.6 (with an = 1 − 󵄨󵄨󵄨g(φ A (z))󵄨󵄨 and bn = 1 + 󵄨󵄨g(φA (z))󵄨󵄨) and 󵄨󵄨 󵄨󵄨 ̃ j Lemma 24.2.3, it suffices to show that the series ∑∞ j=0 󵄨󵄨g(φA (z))󵄨󵄨 has a finite sum which is independent of z ∈ S(x(α, κ), α, R). By Lemmas 24.2.3–24.2.4, we know that p+1 − 󵄨󵄨 ̃ j 󵄨 −1 󵄨 j ̃ A (z)󵄨󵄨󵄨󵄨 p . 󵄨󵄨g(φA (z))󵄨󵄨󵄨 ≤ Mp 󵄨󵄨󵄨φ

For every x > 0, let r = r(x, α) be the radius of the largest ball centered at 0 which is disjoint from S(x, α). One can show that r(x, α) = ux for some constant u ∈ (0, 1] which depends solely on α (in fact, u = 1 when 0 < α ≤ π/2 and u = sin α when π/2 ≤ α < π). Let kR ≥ 0 be the least integer such that − R + (kR + 1)(1 − κ) ≥ R.

(24.35)

Then −R + kR (1 − κ) < R and, therefore, kR
kR and all z ∈ S(x(α, κ), α, R) we get 󵄨󵄨 ̃ j 󵄨󵄨 󵄨󵄨φA (z)󵄨󵄨 ≥ R + (j − (kR + 1))(1 − κ). Therefore, ∞

kR

j=0

j=0

∞

󵄨 ̃j 󵄨󵄨 󵄨󵄨 ̃ j 󵄨󵄨 󵄨󵄨 ̃ j 󵄨󵄨 ∑󵄨󵄨󵄨g(φ A (z))󵄨󵄨 = ∑󵄨󵄨g(φA (z))󵄨󵄨 + ∑ 󵄨󵄨g(φA (z))󵄨󵄨 j=kR +1

kR

∞

j=0

j=kR +1

󵄨 ̃ j 󵄨󵄨− p+1 󵄨 ̃ j 󵄨󵄨− p+1 p p + ∑ 󵄨󵄨󵄨φ ] ≤ Mp−1 [∑󵄨󵄨󵄨φ A (z)󵄨󵄨 A (z)󵄨󵄨

984 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets

≤ Mp−1 [(ux)

−

p+1 p

(

∞ p+1 2R − + 1) + ∑ [R + (j − (kR + 1))(1 − κ)] p ] 1−κ j=k +1

(

p+1 2R − + 1) + ∑[R + j(1 − κ)] p ] 1−κ j=0

R

= Mp−1 [(ux)

−

p+1 p

∞

∞

p+1 p+1 2R − − ≤ Mp [( + 1)(ux) p + ∫ [R + t(1 − κ)] p dt] 1−κ

−1

0

∞

p+1 p+1 2R 1 − − = Mp [( + 1)(ux) p + ∫ s p ds] 1−κ 1−κ

−1

= Mp−1 [(

2R + 1)(ux) 1−κ

R

p+1 − p

+

p − p1 R ]. 1−κ

This upper estimate proves the first assertion in the lemma. For the second assertion, setting R := γx in the above estimate, we get −1

p+1 pγ p − p1 2γ − p 󵄨 ̃j 󵄨󵄨 −1 −1 + ]x → 0 ∑󵄨󵄨󵄨g(φ A (z))󵄨󵄨 ≤ Mp [( 1 − κ + x )u 1−κ j=0

∞

as

x → ∞.

Taking well-defined branches around 1 of the logarithms of both sides of (24.34) and using the fact that | log(1 + z)| = O(|z|), the second assertion ensues from the above convergence to 0. For the third assertion, if z ∈ S(x, π/2) then Re(z) ≥ x. By Lemma 24.2.5, we obtain (in a similar way as in the first part) 󵄨 ̃j 󵄨󵄨 󵄨󵄨 ̃ j 󵄨󵄨− p+1 −1 p ∑󵄨󵄨󵄨g(φ A (z))󵄨󵄨 ≤ Mp ∑󵄨󵄨φA (z)󵄨󵄨 ∞

∞

j=0

j=0 ∞

≤ Mp−1 ∑(x + j(1 − κ))

−

p+1 p

j=0

≤ Mp−1 (x −1 +

p )x −1/p → 0 1−κ

as

x → ∞.

The third assertion is thus proved. ̃A. Finally, we examine the limit of the derivatives of the iterates of φ Lemma 24.2.9. Let φ be a locally holomorphic simple parabolic map and let A be an attracting direction for φ. Fix α ∈ (0, π) and κ ∈ (0, 1). The following statements hold: (a) For any z ∈ S(x(α, κ), α), the limit ̃ ′A,∞ (z) := lim (φ ̃ nA ) (z) φ n→∞

exists and belongs to ℂ \ {0}.

′

24.2 Rationally indifferent periodic points

� 985

(b) The convergence is uniform on every set S(x(α, κ), α) ∩ {ζ ∈ ℂ : Re(ζ ) ≥ t}, t ∈ ℝ, and in particular on every set S(x(α, κ), α, R), R > 0. ̃ ′A,∞ (z) ∈ ℂ \ {0} is holomorphic. (c) The function S(x(α, κ), α) ∋ z 󳨃󳨀→ φ (d) lim

z∈S(x(α,κ),α) Re(z)→∞

̃ ′A,∞ (z) = 1. φ

(e) In particular, if α ∈ (0, π/2) then lim

S(x(α,κ),α)∋z→∞

(f)

̃ ′A,∞ (z) = 1. φ

̃ ′A,∞ (z) = φ ̃ ′A,∞ (φ ̃ A (z))φ ̃ ′A (z), φ

∀z ∈ S(x(α, κ), α).

(24.36)

̃ ′A,∞ and its modulus are not constant. (g) The holomorphic function φ Proof. Let ε > 0 and t ∈ ℝ. By Lemma 24.2.8, there exists x > 0 such that 󵄨󵄨 ̃ n ′ 󵄨 󵄨󵄨(φA ) (ξ) − 1󵄨󵄨󵄨 < ε,

∀ξ ∈ S(x, π/2), ∀n ∈ ℕ.

(24.37)

̃ nA (z)) ≥ x, i. e., By Lemma 24.2.5, there exists N ∈ ℕ so large that Re(φ ̃ nA (z) ∈ S(x, π/2), φ

∀z ∈ S(x(α, κ), α) ∩ {ζ ∈ ℂ : Re(ζ ) ≥ t}, ∀n ≥ N.

Then for every i ≥ N and every j ≥ 0 we have i+j ′ 󵄨󵄨 (φ 󵄨󵄨 󵄨󵄨 ̃ A ) (z) 󵄨 󵄨 ̃j ′ ̃i 󵄨󵄨 − 1󵄨󵄨󵄨 = 󵄨󵄨󵄨(φ 󵄨󵄨 i ′ A ) (φA (z)) − 1󵄨󵄨 < ε, 󵄨󵄨 ̃ A ) (z) 󵄨󵄨 (φ

∀z ∈ S(x(α, κ), α) ∩ {ζ ∈ ℂ : Re(ζ ) ≥ t}.

̃ nA )′ (z))∞ This means that the sequence ((φ n=0 is uniformly quotient Cauchy on S(x(α, κ), α) ∩ {ζ ∈ ℂ : Re(ζ ) ≥ t} and statements (a) and (b) hold. Assertion (c) follows immediately from (b), while (d) is a direct consequence of (24.37), and (e) is an immediate implication of (d). Statement (f) follows from a straightforward calculation using the definitions of ̃ ′A,∞ (z) and φ ̃ ′A,∞ (φ ̃ A (z)). Finally, (g) clearly ensues from (f) and the openness of holoφ morphic maps. 24.2.1.2 Properties of φ We now return to our original and main goal, namely, the examination of the function φ and its iterates, their derivatives and the limit of those. For every x ∈ (0, ∞), α ∈ (0, π) and R > 0, let S0 (x, α) := H(S(x, α)),

(24.38)

986 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets SφA (x, α) := ρA (S0 (x, α)) = ρA ∘ H(S(x, α))

(24.39)

S0 (x, α, R) := S0 (x, α) \ B(0, R) = H(S(x, α, R ))

(24.40)

−p

SφA (x, α, R) := SφA (x, α) \ B(ω, R) = ρA ∘ H(S(x, α, (|a|p) R )).

(24.41)

−1 −p

The regions S0 (x, α) and SφA (x, α) look like flower petals (see Figures 24.3–24.4) that are p (1) symmetric about the rays (0, ∞) and A = ω + √−a−1 (0, ∞), respectively,

(2) contain initial segments of these rays, and (3) whose tangents at the respective points 0 and ω form a central angle of measure α/p. Indeed, observe that lim arg H(x + te±iα ) = lim arg

ℝ∋t→∞

ℝ∋t→∞

1 p

√x +

te±iα

= lim arg t→∞

1 e

±i α p

√1 + p

x

te±iα

α =∓ . p

The following result is the counterpart of Lemma 24.2.4 for the function φ. It asserts that every flower petal SφA (x, α) is forward φ-invariant and gives a description of the behavior of the iterates of φ on such a petal.

Figure 24.3: Three petals, case p = 3 (with a = −1). Some dynamics are depicted by arrows. There are 3 attracting directions (indicated by rays and vectors pointing toward the parabolic point) with respect to which the petals are symmetric and 3 repelling directions (indicated by dashed rays and vectors pointing away from the parabolic point).

24.2 Rationally indifferent periodic points

� 987

Figure 24.4: Four petals, case p = 4 (with a = −1). There are 4 attracting directions (indicated by rays and vectors pointing toward the parabolic point) with respect to which the petals are symmetric and 4 repelling directions (indicated by dashed rays and vectors pointing away from the parabolic point). Some dynamics in and near one of the petals are depicted by the arrows. Reproduce them in and near the other three petals.

Lemma 24.2.10. Let φ be a locally holomorphic simple parabolic map at ω ∈ ℂ and let A be an attracting direction for φ. Fix α ∈ (0, π) and κ ∈ (0, 1). Then all the iterates φn : SφA (x(α, κ), α) → ℂ are well-defined and satisfy φn (SφA (x, α)) ⊆ SφA (x + n(1 − κ), α) ⊆ SφA (x, α),

∀x ≥ x(α, κ),

∀n ∈ ℕ.

A In particular, the sequence (φn )∞ n=1 converges to ω uniformly on Sφ (x(α, κ), α).

An elementary calculation based on (24.24) yields the next result. Lemma 24.2.11. If φ is a locally holomorphic simple parabolic map at ω ∈ ℂ and A is an attracting direction for φ, then for every 0 ≤ α < π/p and κ ∈ (0, 1) there exists Rα,κ > 0 such that 󵄨󵄨 󵄨 󵄨󵄨φ(z) − ω󵄨󵄨󵄨 < |z − ω|

󵄨 󵄨 and 󵄨󵄨󵄨φ′ (z)󵄨󵄨󵄨 < 1,

∀z ∈ SφA (x(α, κ), α) ∩ B(ω, Rα,κ ) \ {ω}.

988 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets The next statement specifies the asymptotic decay of the distance that separates successive iterates of points in the petal SφA (x(α, κ), α), as well as the asymptotic decay of the distance between these iterates and the point ω. Proposition 24.2.12. Let φ be a locally holomorphic simple parabolic map at ω ∈ ℂ and let A be an attracting direction for φ. Fix α ∈ (0, π) and κ ∈ (0, 1). For all z ∈ SφA (x(α, κ), α), we have lim n

n→∞

p+1 p

− 1 −1 󵄨󵄨 n+1 󵄨 n 󵄨󵄨φ (z) − φ (z)󵄨󵄨󵄨 = (|a|p) p p

(24.42)

1 −1 󵄨 󵄨 lim n p 󵄨󵄨󵄨φn (z) − ω󵄨󵄨󵄨 = (|a|p) p . n→∞

(24.43)

In addition, in both limits, the convergence is uniform on SφA (x(α, κ), α, R) for every R > 0. 1

p Proof. Let z ∈ SφA (x(α, κ), α, R). Then ξ = ρ−1 A (z) ∈ S0 (x(α, κ), α, (|a|p) R) and hence

H −1 (ξ) ∈ S(x(α, κ), α, (|a|p)−1 R−p ). By Lemma 24.2.6, we get that

−1 󵄨󵄨 n 󵄨 󵄨 ̃ nA ∘ H −1 (ξ)󵄨󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨φ ̃ nA (H −1 (ξ))󵄨󵄨󵄨󵄨 p 󵄨󵄨φA,0 (ξ)󵄨󵄨󵄨 = 󵄨󵄨󵄨H ∘ φ 1− 1 󵄨 󵄨− 1 = 󵄨󵄨󵄨H −1 (ξ) + n + ORe(H −1 (ξ)) (max{n p , log n})󵄨󵄨󵄨 p .

It ensues that 1 −1 󵄨 󵄨 󵄨 󵄨− 1 lim n p 󵄨󵄨󵄨φnA,0 (ξ)󵄨󵄨󵄨 = lim 󵄨󵄨󵄨n−1 H −1 (ξ) + 1 + O−(|a|p)−1 R−p (max{n p , n−1 log n})󵄨󵄨󵄨 p = 1, n→∞ n→∞ 1

and the convergence is uniform on ξ ∈ S0 (x(α, κ), α, (|a|p) p R). As n −1 √ φn (z) − ω = ρA (φnA,0 (ρ−1 A (z))) − ω = −(ap) ⋅ φA,0 (ξ), p

relation (24.43) follows along with the corresponding uniform convergence on the set SφA (x(α, κ), α, R). Turning our attention to relation (24.42), we have by (24.24) and (24.43) that lim n

n→∞

p+1 p

󵄨󵄨 n+1 󵄨 n 󵄨󵄨φ (z) − φ (z)󵄨󵄨󵄨

= lim n n→∞

p+1 p

p+1 󵄨󵄨 󵄨 n 󵄨p+2 󵄨 n 󵄨󵄨a(φ (z) − ω) + O(󵄨󵄨󵄨φ (z) − ω󵄨󵄨󵄨 )󵄨󵄨󵄨

p+1 p

p+1 󵄨 󵄨 󵄨󵄨 󵄨 n 󵄨󵄨 n 󵄨󵄨a(φ (z) − ω) 󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨1 + O(󵄨󵄨󵄨φ (z) − ω󵄨󵄨󵄨)󵄨󵄨󵄨 1 1 −1 󵄨 󵄨 p+1 󵄨󵄨 󵄨 󵄨 󵄨󵄨 = |a| lim (n p 󵄨󵄨󵄨φn (z) − ω󵄨󵄨󵄨) ⋅ 󵄨󵄨󵄨1 + lim n p ⋅ lim O(n p 󵄨󵄨󵄨φn (z) − ω󵄨󵄨󵄨)󵄨󵄨󵄨 n→∞ n→∞ 󵄨 n→∞ 󵄨 − p1 p+1 󵄨 − p1 󵄨 = |a|((|a|p) ) ⋅ 󵄨󵄨󵄨1 + 0 ⋅ O((|a|p) )󵄨󵄨󵄨

= lim n n→∞

− p1 −1

= (|a|p)

p .

24.2 Rationally indifferent periodic points

� 989

As for relation (24.43), the convergence is uniform on z ∈ SφA (x(α, κ), α, R). For every z ∈ SφA (x(α, κ), α), let ̃ ′A,∞ ((ρA ∘ H)−1 (z)). φ′A,∞ (z) := φ

(24.44)

The last result about the iterates of φ concerns their derivatives. Proposition 24.2.13. Let φ be a locally holomorphic simple parabolic map at ω ∈ ℂ and let A be an attracting direction for φ at ω. Fix α ∈ (0, π) and κ ∈ (0, 1). Then for every z ∈ SφA (x(α, κ), α) we have lim n

n→∞

p+1 p

p+1 − 󵄨󵄨 n ′ 󵄨󵄨 󵄨 −(p+1) 󵄨󵄨 ′ 󵄨󵄨(φ ) (z)󵄨󵄨 = (|a|p) p |z − ω| 󵄨󵄨φA,∞ (z)󵄨󵄨󵄨,

(24.45)

and the convergence is uniform on SφA (x(α, κ), α, R) for every R > 0. If α ∈ (0, π/2), then lim

SφA (x(α,κ),α)∋z→ω

φ′A,∞ (z) = 1.

(24.46) 1

p Proof. Let z ∈ SφA (x(α, κ), α, R). Then ξ = ρ−1 A (z) ∈ S0 (x(α, κ), α, (|a|p) R) and hence

H −1 (ξ) ∈ S(x(α, κ), α, (|a|p)−1 R−p ). By Lemma 24.2.6, we get that |ξ|p+1 n

p+1 p

p+1 ′ 󵄨󵄨 n ′ 󵄨󵄨 󵄨 ′ ̃ n −1 󵄨󵄨 −1 −1 ′ p+1 ̃n 󵄨󵄨(φA,0 ) (ξ)󵄨󵄨 = |ξ| n p 󵄨󵄨󵄨H (φ A (H (ξ))) ⋅ (φA ) (H (ξ)) ⋅ (H ) (ξ)󵄨󵄨 p+1 p+1 1 󵄨 ̃ n −1 󵄨󵄨− p 󵄨󵄨 ̃ n ′ −1 󵄨 ⋅ 󵄨󵄨(φA ) (H (ξ))󵄨󵄨󵄨 ⋅ p|ξ|−(p+1) = |ξ|p+1 n p 󵄨󵄨󵄨φ A (H (ξ))󵄨󵄨 p

󵄨󵄨 ̃ n −1 󵄨− p+1 󵄨 ̃ n ′ −1 󵄨󵄨 󵄨󵄨φA (H (ξ))󵄨󵄨󵄨 p 󵄨󵄨󵄨(φ A ) (H (ξ))󵄨󵄨 p+1 p+1 󵄨󵄨− p 󵄨 1− p1 󵄨 ̃ n ′ −1 󵄨󵄨 󵄨󵄨󵄨󵄨󵄨󵄨H −1 (ξ) + n + O , log n}) ) (H (ξ)) = n p 󵄨󵄨󵄨(φ (max{n −1 Re(H (ξ)) A 󵄨󵄨 󵄨󵄨󵄨 p+1 󵄨󵄨− p 󵄨 −1 − p1 󵄨󵄨 󵄨󵄨 ̃ n ′ −1 󵄨󵄨󵄨󵄨󵄨 H (ξ) −1 + 1 + O−(|a|p)−1 R−p (max{n , n log n})󵄨󵄨 . = 󵄨󵄨(φA ) (H (ξ))󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 n

=n

p+1 p

Therefore, n

p+1 p

p+1 p+1 󵄨󵄨 n ′ 󵄨󵄨 󵄨 n ′ −1 󵄨 󵄨 n ′ 󵄨 󵄨󵄨(φ ) (z)󵄨󵄨 = n p 󵄨󵄨󵄨(φA,0 ) (ρA (z))󵄨󵄨󵄨 = n p 󵄨󵄨󵄨(φA,0 ) (ξ)󵄨󵄨󵄨 󵄨 ̃ n ′ −1 󵄨󵄨 = |ξ|−(p+1) 󵄨󵄨󵄨(φ A ) (H (ξ))󵄨󵄨⋅ p+1 󵄨󵄨− p 󵄨󵄨 H −1 (ξ) − p1 󵄨󵄨 󵄨󵄨 −1 + 1 + O−(|a|p)−1 R−p (max{n , n log n})󵄨󵄨 ⋅ 󵄨󵄨 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 −1 󵄨󵄨−(p+1) 󵄨󵄨 ̃ n ′ −1 −1 󵄨󵄨 = 󵄨󵄨ρA (z)󵄨󵄨 󵄨󵄨(φA ) (H (ρA (z)))󵄨󵄨⋅ p+1 󵄨󵄨 H −1 (ρ−1 (z)) 󵄨󵄨− p − p1 󵄨󵄨 󵄨󵄨 −1 A ⋅ 󵄨󵄨 + 1 + O−(|a|p)−1 R−p (max{n , n log n})󵄨󵄨 n 󵄨󵄨 󵄨󵄨

990 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets

−

= (|a|p)

p+1 p

󵄨 ̃n ′ 󵄨󵄨 −1 |z − ω|−(p+1) 󵄨󵄨󵄨(φ A ) ((ρA ∘ H) (z))󵄨󵄨⋅

󵄨󵄨 (ρ ∘ H)−1 (z) 󵄨󵄨− −1 󵄨 󵄨 + 1 + O−(|a|p)−1 R−p (max{n p , n−1 log n})󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨 A 󵄨󵄨 󵄨󵄨 n

p+1 p

.

Letting n → ∞ and applying Lemma 24.2.9 leads to the result. 24.2.1.3 Fatou coordinates In the next theorem, we establish the existence of functions, called Fatou coordinates, around simple parabolic fixed points. Though this theorem is of local character, it will eventually and primarily be used to describe the basin of attraction of parabolic points. For every α ∈ (0, π) and t ∈ ℝ, set Qα := S(x(α, 1/2), α)

󵄨 and Qα,t := Qα ∩ {z ∈ ℂ 󵄨󵄨󵄨 Re(z) > t}.

By Lemmas 24.2.4–24.2.5, notice that ̃ A (Qα ) ⊆ Qα φ

and

̃ A (Qα,t ) ⊆ Qα,t , φ

∀t ∈ ℝ.

(24.47)

̃ nA )∞ In particular, all iterates (φ n=1 are well-defined on Qα and Qα,t . Theorem 24.2.14. Let φ be a locally holomorphic simple parabolic map at ω ∈ ℂ and let A be an attracting direction for φ at ω. Then for every α ∈ (0, π) there exists a unique (up to an additive constant) holomorphic function F̃α : Qα → ℂ such that ̃ A = F̃α + 1 F̃α ∘ φ

(24.48)

and F̃α (z) = 1, Qα,t ∋z→∞ z lim

(24.49)

∀t ∈ ℝ.

Furthermore, replacing x(α, 1/2) by a sufficiently large number and keeping the same symbol x(α, 1/2) for that number, renders the function F̃α univalent. ̃A. The functions {F̃α }α∈(0,π) are called the Fatou coordinates of the parabolic map φ Proof. To establish the uniqueness of Fatou coordinates, suppose that a holomorphic function F̃α : Qα → ℂ satisfies (24.48)–(24.49). Fix ε > 0. By definition of Qα and (24.49), there exist β ∈ (0, α) small enough and s > max{2, x(α, 1/2)} large enough that B(z, |z|/2) ⊆ Qα

󵄨󵄨 F̃ (z) 󵄨󵄨 ε 󵄨 󵄨 and 󵄨󵄨󵄨 α − 1󵄨󵄨󵄨 < , 󵄨󵄨 z 󵄨󵄨 3

Cauchy’s integral formula then yields

∀z ∈ Qβ,s .

24.2 Rationally indifferent periodic points

� 991

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 1 F̃α (ξ) F̃α (ξ) − ξ 󵄨󵄨󵄨󵄨 󵄨󵄨 ̃ ′ 󵄨 󵄨󵄨 1 󵄨󵄨 = 󵄨󵄨 dξ − 1 dξ ∫ ∫ 󵄨󵄨 󵄨󵄨Fα (z) − 1󵄨󵄨󵄨 = 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 2πi 󵄨󵄨 2πi 󵄨󵄨 (ξ − z)2 (ξ − z)2 󵄨󵄨 󵄨󵄨 󵄨 󵄨 𝜕B(z,|z|/2) 𝜕B(z,|z|/2) ≤ =

1 2π

∫ 𝜕B(z,|z|/2)

1 π(|z|/2)2

2 ≤ π|z|2

|F̃α (ξ) − ξ| |dξ| |ξ − z|2

󵄨󵄨 ̃ 󵄨 󵄨󵄨Fα (ξ) − ξ 󵄨󵄨󵄨 |dξ|

∫ 𝜕B(z,|z|/2)

∫ 𝜕B(z,|z|/2)

ε |ξ| |dξ| 3

2 |z| ε 3|z| ≤ ⋅ ⋅ = ε, ⋅ 2π 2 3 2 π|z|2

∀z ∈ Qβ,s .

(24.50)

̃ nA (z), φ ̃ nA (ζ ) ∈ Now fix z, ζ ∈ Qα . By Lemma 24.2.6 and (24.47), there is N ∈ ℕ such that φ Qβ,s for all n ≥ N. Using (24.48), we obtain that ̃ nA (z)) − F̃α (φ ̃ nA (ζ )) F̃α (z) − F̃α (ζ ) = F̃α (φ ′ ′ = ∫ F̃α (ξ) dξ = ∫((F̃α (ξ) − 1) + 1) dξ In

In

′ ̃ nA (z) − φ ̃ nA (ζ ) + ∫(F̃α =φ (ξ) − 1) dξ, In

̃ nA (ζ ) to φ ̃ nA (z). So, where In is the oriented segment from φ 󵄨󵄨 ̃ 󵄨 ′ ̃ nA (z) − φ ̃ nA (ζ ))󵄨󵄨󵄨󵄨 ≤ ∫󵄨󵄨󵄨󵄨F̃α (ξ) − 1󵄨󵄨󵄨 |dξ|. 󵄨󵄨(Fα (z) − F̃α (ζ )) − (φ In

̃ nA (z), φ ̃ nA (ζ ) ∈ Qβ,s . The convexity of Qβ,s guarantees that For all n ≥ N, we know that φ In ⊆ Qβ,s for all n ≥ N. Applying (24.50), it follows that 󵄨󵄨 ̃ ̃ nA (z) − φ ̃ nA (ζ ))󵄨󵄨󵄨󵄨 ≤ ε|In | = ε󵄨󵄨󵄨󵄨φ ̃ nA (z) − φ ̃ nA (ζ )󵄨󵄨󵄨󵄨, 󵄨󵄨(Fα (z) − F̃α (ζ )) − (φ

∀n ≥ N.

Thus, 󵄨󵄨 󵄨󵄨 ̃ 󵄨󵄨 󵄨󵄨 Fα (z) − F̃α (ζ ) 󵄨 󵄨󵄨 󵄨󵄨 φ n (z) − φ n (ζ ) − 1󵄨󵄨󵄨 ≤ ε, ̃ ̃ 󵄨󵄨 󵄨󵄨 A A

∀n ≥ N.

̃ nA (z) − φ ̃ nA (ζ ))n=1 exists and Hence, the limit of the sequence (φ ∞

̃ nA (z) − φ ̃ nA (ζ )). F̃α (z) − F̃α (ζ ) = lim (φ n→∞

The uniqueness part is thus established. (Note that ε ↘ 0 ⇒ s ↗ ∞ ⇒ N ↗ ∞.)

992 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets This argument also gives us a hint as to how to prove the existence of Fatou coordinates. Let β ∈ (0, α), t ∈ ℝ and ζ ∈ Qβ,t . Consider the functions ̃ nA (z) − φ ̃ nA (ζ ), Qα ∋ z 󳨃󳨀→ ψn (z) := φ

n ∈ ℕ.

(24.51)

Using Lemma 24.2.3, the mean value inequality, (24.47) and Lemma 24.2.5, we obtain that 󵄨 󵄨 󵄨󵄨 ̃ ̃ n 󵄨󵄨 ̃ nA (z) + 1)] − [φ ̃ A (φ ̃ nA (ζ )) − (φ ̃ nA (ζ ) + 1)]󵄨󵄨󵄨󵄨 (φA (z)) − (φ 󵄨󵄨ψn+1 (z) − ψn (z)󵄨󵄨󵄨 = 󵄨󵄨󵄨[φ A 󵄨 󵄨 󵄨󵄨 󵄨 ̃ nA (z)) − B ∘ H(φ ̃ nA (ζ ))󵄨󵄨󵄨󵄨 = 󵄨󵄨󵄨B ∘ H(φ 󵄨 󵄨 p+1 − p 󵄨󵄨 ̃ n −1 ̃ nA (ζ )󵄨󵄨󵄨󵄨 ≤ Mp sup{|ξ| : ξ ∈ In }󵄨󵄨φA (z) − φ p+1 − 󵄨̃n ̃ n 󵄨󵄨 ≤ Mp−1 (t + n/2) p 󵄨󵄨󵄨φ A (z) − φA (ζ )󵄨󵄨 p+1 − 󵄨 󵄨 ≤ Ct n p 󵄨󵄨󵄨ψn (z)󵄨󵄨󵄨 for some constant Ct > 0 independent of z and n but depending on t in general. Equivalently, 󵄨󵄨 󵄨󵄨 ψ (z) p+1 − 󵄨 󵄨󵄨 n+1 − 1󵄨󵄨󵄨 ≤ Ct n p . 󵄨󵄨 󵄨󵄨 󵄨󵄨 ψn (z)

(24.52)

p+1

p converges, estimate (24.52) ensures that the sequence (ψ )∞ As the series ∑∞ n n=1 n=1 n converges uniformly on Qα,t as it is uniformly quotient Cauchy. So, its limit

−

F̃α := lim ψn

(24.53)

n→∞

is a holomorphic function on Qα,t . Letting t → −∞, we conclude that (24.53) defines a holomorphic function F̃α : Qα → ℂ. Moreover, due to (24.53) and Lemmas 24.2.3 and 24.2.5, for all z ∈ Qα we obtain the following relation: ̃ A (z)) = lim ψn (φ ̃ A (z)) = lim [φ ̃ n+1 ̃n F̃α (φ A (z) − φA (ζ )] n→∞

=

̃ n+1 lim [(φ A (z) n→∞

n→∞

−

̃ n+1 φ A (ζ ))

̃ n+1 ̃n + (φ A (ζ ) − φA (ζ ))]

= lim ψn+1 (z) + 1 n→∞

= F̃α (z) + 1. This proves (24.48). In order to establish (24.49), we must work a little harder. Keep again β ∈ (0, α), t ∈ ℝ fixed and take z ∈ Qα,t . From Lemma 24.2.3, we deduce by induction that 󵄨󵄨 ̃ n 󵄨 −1/p 󵄨󵄨φA (z) − (z + n)󵄨󵄨󵄨 ≤ Lt,n |z| ,

(24.54)

24.2 Rationally indifferent periodic points

� 993

where Lt,n > 0 is some constant independent of z ∈ Qα,t that may depend on t and n. Let ε > 0. By (24.52)–(24.53), there exists k ∈ ℕ so large that 󵄨󵄨 F̃ (z)/z 󵄨󵄨 ε 󵄨󵄨 α 󵄨 − 1󵄨󵄨󵄨 < , 󵄨󵄨 󵄨󵄨 2 󵄨󵄨 ψk (z)/z

∀z ∈ Qα,t .

(24.55)

By (24.51) and (24.54), we infer that k 󵄨󵄨 󵄨󵄨 φ 󵄨󵄨 󵄨󵄨 ψ (z) ̃ kA (ζ ) 󵄨 󵄨 ̃ (z) − φ 󵄨 󵄨󵄨 k − 1󵄨󵄨󵄨 = lim 󵄨󵄨󵄨 A − 1󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 Qα,t ∋z→∞󵄨󵄨 󵄨󵄨 Qα,t ∋z→∞󵄨󵄨 z z k ̃ A (z) − z| |φ = lim Qα,t ∋z→∞ |z|

lim

k + Lt,k |z|−1/p Qα,t ∋z→∞ |z| = 0. ≤

So, limQα,t ∋z→∞

ψk (z) z

lim

= 1, and hence (24.55) yields 󵄨󵄨 F̃ (z) 󵄨󵄨 󵄨󵄨 α 󵄨 − 1󵄨󵄨󵄨 < ε 󵄨󵄨 󵄨󵄨 z 󵄨󵄨

for all z ∈ Qα,t with sufficiently large moduli. Since ε > 0 is arbitrary, the validity of (24.49) is confirmed. It only remains to show that F̃α : Qα → ℂ is univalent. As the main ingredient in the proof of this statement, we establish the following. ̃ A 󵄨󵄨󵄨󵄨Q : Qα → ℂ is univalent. Claim. The map φ α −1

Proof. Assume that x(α, 1/2) ≥ 1 is so large that |z| > 1 and M|z| p < 1/2 for every z ∈ Qα , where M > 0 is the constant in Lemma 24.2.3. In particular, recall that Qα is convex. If ζ ∈ Qα , then it follows from Lemma 24.2.3 that p+1 1 −1 − 󵄨󵄨 ̃ ′ 󵄨 󵄨󵄨φA (ζ ) − 1󵄨󵄨󵄨 ≤ M|ζ | p ≤ M|ζ | p < . 2

̃ ′A (ζ )) > 1/2. In particular, Re(φ ̃ A (ζ ) = φ ̃ A (ξ) for some ζ , ξ ∈ Qα , then it ensues from Lemma 24.2.3 that Now, if φ − p1

|ζ − ξ| ≤ 2M(min{|ζ |, |ξ|})

< 1.

(24.56)

̃ A on the convex set B(ξ, 1) ∩ Qα for an arbitrary So, it suffices to prove the injectivity of φ ξ ∈ Qα . Let ζ1 , ζ2 ∈ B(ξ, 1) ∩ Qα and denote by I the line segment from ζ1 to ζ2 . Then 󵄨󵄨 󵄨󵄨 󵄨󵄨 ̃ ̃ A (ζ1 )󵄨󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨∫ φ ̃ ′A (ζ ) dζ 󵄨󵄨󵄨󵄨 󵄨󵄨φA (ζ2 ) − φ 󵄨󵄨 󵄨󵄨 I

994 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets 󵄨󵄨 󵄨󵄨 1 󵄨󵄨 󵄨󵄨 ′ 󵄨 ̃ = |ζ2 − ζ1 | ⋅ 󵄨󵄨∫ φA (ζ1 + t(ζ2 − ζ1 ))dt 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨0 1

̃ ′A (ζ1 + t(ζ2 − ζ1 ))dt ≥ |ζ2 − ζ1 | ⋅ Re ∫ φ 1

0

̃ ′A (ζ1 + t(ζ2 − ζ1 ))]dt = |ζ2 − ζ1 | ⋅ ∫ Re[φ 0

1 ≥ |ζ2 − ζ1 | ⋅ . 2 ̃ A is injective on B(ξ, 1) ∩ Qα . This completes the proof of the Therefore, the function φ claim. ◼ In view of this claim and of the uniform convergence of the sequence (ψn )∞ n=1 on Qα,t , an application of Hurwitz’s theorem (Theorem A.3.8) yields the univalence of the limit function F̃α |Qα,t for any t ∈ ℝ. Hence, F̃α : Qα → ℂ is univalent. We now establish an important property of the Fatou coordinates, namely that whenever α ≥ π/2 the F̃α -image of every Qα,t contains a vertical half-plane. Lemma 24.2.15. Let φ be a locally holomorphic simple parabolic map at ω ∈ ℂ and let A be an attracting direction for φ at ω. For every α ∈ [π/2, π) and every t ∈ ℝ, there exists s = s(t) > 0 such that F̃α (Qα,t ) ⊇ {ζ ∈ ℂ : Re(ζ ) > s}. Proof. By (24.49), there exists s > max{0, 4[x(α, 1/2) + t]} so large that 󵄨󵄨 F̃ (z) 󵄨󵄨 1 󵄨󵄨 α 󵄨 − 1󵄨󵄨󵄨 ≤ 󵄨󵄨 󵄨󵄨 z 󵄨󵄨 8 whenever Re(z) > s/4. Take any ζ ∈ ℂ with Re(ζ ) > s. Observe that B(ζ , 2|ζ |/3) ⊆ Qα,t .

(24.57)

If z ∈ 𝜕B(ζ , |ζ |/2), then 󵄨󵄨 F̃ (z) 󵄨󵄨 󵄨󵄨 ̃ 󵄨 󵄨 󵄨 󵄨 󵄨 − 1󵄨󵄨󵄨 󵄨󵄨(Fα (z) − ζ ) − (z − ζ )󵄨󵄨󵄨 = 󵄨󵄨󵄨F̃α (z) − z󵄨󵄨󵄨 = |z|󵄨󵄨󵄨 α 󵄨󵄨 z 󵄨󵄨 |z| |ζ | + |ζ |/2 3 |ζ | ≤ = |ζ | < = |z − ζ |. ≤ 8 8 16 2 Thus, Rouché’s theorem (Theorem A.3.7) asserts that the function z 󳨃→ F̃α (z) − ζ has the same number of zeros in B(ζ , |ζ |/2) ⊆ Qα,t as the function z 󳨃→ z − ζ . But this latter

24.2 Rationally indifferent periodic points

� 995

function has exactly one zero, namely z = ζ . Hence, there exists z ∈ B(ζ , |ζ |/2) ⊆ Qα,t such that F̃α (z) = ζ . Fix α ∈ (0, π), κ = 1/2, and denote by Sa1 (ω, α), . . . , Sap (ω, α)

(24.58)

the attracting sectors/flower petals SφA (x(α, 1/2), α) defined in (24.39) and corresponding to the attracting directions A generated by the p holomorphic branches of the pth root p defined in (24.25). See Figures 24.3–24.4 as well as 24.5. function √⋅

Figure 24.5: Four petals, case p = 4 (with a = −1). There are 4 attracting directions (indicated by rays and vectors pointing towards the parabolic fixed point) with respect to which the petals are symmetric and 4 repelling directions (indicated by dashed rays and vectors pointing away from the point). Some dynamics within one of the petals are depicted by the arrows. Reproduce them in the other three petals.

Denote also by Sr1 (ω, α), . . . , Srp (ω, α)

(24.59)

996 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets the analogous attracting sectors for φ−1 . We usually call them the repelling sectors for φ. See Figure 24.6.

Figure 24.6: Four petals, case p = 4 (with a = −1). There are 4 attracting directions (indicated by rays and vectors pointing towards the parabolic fixed point) with respect to which the petals are symmetric and 4 repelling directions (indicated by dashed rays and vectors pointing away from the point). Some dynamics within one of the petals are depicted by the arrows. Reproduce them in the other three petals.

A straightforward consequence of these definitions is the following. Lemma 24.2.16. If φ is a locally holomorphic simple parabolic map at ω ∈ ℂ, then for every α ∈ (0, π) both collections {Sa1 (ω, α), . . . , Sap (ω, α)}

and {Sr1 (ω, α), . . . , Srp (ω, α)}

consist of mutually disjoint sets. ̃A Given the Fatou coordinates F̃α defined through Theorem 24.2.14 for the map φ associated with an attracting direction A, set

24.2 Rationally indifferent periodic points

Fα := F̃α ∘ (ρA ∘ H)−1 .

� 997

(24.60)

The functions {Fα }α∈(0,π) are called the Fatou coordinates of the parabolic map φ. As an immediate consequence of Lemma 24.2.4, Theorem 24.2.14 and Lemma 24.2.15, we obtain the following result. Theorem 24.2.17. If φ is a locally holomorphic simple parabolic map at ω ∈ ℂ, then for every α ∈ (0, π) we have φ(Saj (ω, α)) ⊆ Saj (ω, α),

∀j = 1, . . . , p,

(24.61)

and Fα ∘ φ = Fα + 1 on every Saj (ω, α).

(24.62)

Moreover: (a) With appropriate normalizations (i. e., making the values of the Fα ’s coincide at one point), if 0 < β, γ < π then 󵄨 󵄨 Fγ 󵄨󵄨󵄨Sj (ω,β)∩Sj (ω,γ) = Fβ 󵄨󵄨󵄨Sj (ω,β)∩Sj (ω,γ) . a

a

a

a

󵄨 (b) The map Fα 󵄨󵄨󵄨Sj (ω,α) is univalent. a (c) If α ∈ [π/2, π), then there exists s > 0 such that Fα (Saj (ω, α)) ⊇ {ζ ∈ ℂ : Re(ζ ) > s}.

24.2.2 Leau–Fatou flower petals In this section, we continue our study of parabolic fixed and periodic points undertaken in the previous section. However, up to this point our considerations have had an entirely local character in the sense that the map T (denoted so far by φ) was defined only on a small neighborhood of its simple rationally indifferent (or parabolic) fixed point ω. Instead, let T be defined globally by letting ̂→ℂ ̂ T :ℂ be a rational function with deg(T) ≥ 2 that has a simple rationally indifferent fixed point ω. Let Ω(T) denote the set of rationally indifferent periodic points of T and let Ω0 (T) ⊆ Ω(T) be the set of simple rationally indifferent fixed points of T, i. e., Ω0 (T) = {ω ∈ Ω(T) : T(ω) = ω and T ′ (ω) = 1}.

(24.63)

998 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets Suppose that Ω0 (T) ≠ 0 and ω ∈ Ω0 (T). By Montel’s criterion (Theorem 23.2.4) j and (24.61), the attracting sector/flower petal Sa (ω, α) is a subset of the Fatou set ℱ (T) for every j ∈ {1, . . . , p(ω)} and every α ∈ (0, π). Let δT > 0 be so small that the map T|B(ω,2δT ) is injective and that there exists Tω−1 : B(ω, δT ) → ℂ, a unique holomorphic inverse branch of T that sends ω to ω. j Recall that the sets Sr (ω, α), j ∈ {1, . . . , p(ω)}, are the attracting sectors for the map −1 Tω : B(ω, δT ) → ℂ. For every j ∈ {1, . . . , p(ω)}, let A∗j (ω) be the connected component of the Fatou set j

ℱ (T) that contains Sa (ω, α) for some (equivalently, for every) α ∈ (0, π). The domain

A∗j (ω)

is called the immediate basin of attraction of ω in the direction determined by j. The sets A∗j (ω), j = 1, . . . , p(ω), are also called Leau–Fatou petals or Leau domains. Further define ∞

Aj (ω) := ⋃ T −n (A∗j (ω)). n=0

The open set Aj (ω) is called the basin of attraction of ω in the direction determined by j. This basin is a subset of the Fatou set since A∗j (ω) ⊆ ℱ (T) and the Fatou set is completely T-invariant according to Theorem 24.1.2. Finally, the open set p(ω)

A∗ (ω) := ⋃ A∗j (ω) j=1

is called the immediate basin of attraction of ω, whereas the open set p(ω)

A(ω) := ⋃ Aj (ω) j=1

is said to be the basin of attraction of ω. These two sets are obviously contained in the Fatou set. Notice that p(ω)

p(ω)

j=1

j=1

ω ∈ ⋂ 𝜕A∗j (ω) ∩ ⋂ 𝜕Aj (ω) ∩ 𝜕A∗ (ω) ∩ 𝜕A(ω). The above terminologies make sense because of the following result, which is a somewhat direct consequence of the complete invariance of the Fatou set and Lemma 24.2.10. ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and with a simple Lemma 24.2.18. Let T : ℂ rationally indifferent fixed point ω, i. e., ω ∈ Ω0 (T). Then

24.2 Rationally indifferent periodic points

T(A∗j (ω)) = A∗j (ω),

� 999

∀j ∈ {1, . . . , p(ω)} and thus T(A∗ (ω)) = A∗ (ω).

Moreover, ∀j ∈ {1, . . . , p(ω)} and hence T −1 (A(ω)) = A(ω).

T −1 (Aj (ω)) = Aj (ω), Furthermore,

lim T n (z) = ω,

∀z ∈ A(ω).

n→∞

The immediate basin of attraction of ω is a subset of the backward orbit of the attracting sectors/flower petals of any given angle α. The basin of attraction of ω coincides with the backward orbit of the immediate basin of attraction of ω as well as with the backward orbit of the attracting sectors/flower petals of any given angle α. ̂→ℂ ̂ is a rational function with deg(T) ≥ 2 and ω ∈ Ω0 (T), then Lemma 24.2.19. If T : ℂ for every α ∈ (0, π) we have that p(ω)

∞

p(ω)

j=1

n=0

j=1

A∗ (ω) := ⋃ A∗j (ω) ⊆ ⋃ T −n ( ⋃ Saj (ω, α)) and ∞

∞

p(ω)

n=0

n=0

j=1

A(ω) = ⋃ T −n (A∗ (ω)) = ⋃ T −n ( ⋃ Saj (ω, α)). j

Proof. As A∗j (ω) is the Fatou component containing Sa (ω, α), it is obvious that p(ω)

p(ω)

j=1

j=1

⋃ Saj (ω, α) ⊆ ⋃ A∗j (ω) =: A∗ (ω) ⊆ A(ω).

Since A(ω) is completely T-invariant (see Lemma 24.2.18), one series of inclusions follows immediately. For the opposite series, it suffices to show that ∞

p(ω)

n=0

j=1

A∗ (ω) ⊆ ⋃ T −n ( ⋃ Saj (ω, α)).

(24.64)

Indeed, if ζ ∈ A(ω) then there is 1 ≤ j ≤ p(ω) for which ζ ∈ Aj (ω). This means that there −m ∗ is k ≥ 0 such that ζ ∈ T −k (A∗j (ω)). So, A(ω) ⊆ ⋃∞ (A (ω)). If the above inclusion m=0 T holds, then ∞

∞

p(ω)

∞

p(ω)

m=0

m,n=0

j=1

n=0

j=1

A(ω) ⊆ ⋃ T −m (A∗ (ω)) ⊆ ⋃ T −(m+n) ( ⋃ Saj (ω, α)) = ⋃ T −n ( ⋃ Saj (ω, α)), which is ultimately the series of inclusions sought.

1000 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets Suppose, by way of contradiction, that (24.64) does not hold. This means that there exist some i ∈ {1, . . . , p(ω)} and some ∞

p(ω)

n=0

j=1

z ∈ A∗i (ω)\ ⋃ T −n ( ⋃ Saj (ω, α)).

(24.65)

Since T(A∗i (ω)) = A∗i (ω) and since limk→∞ T k (z) = ω by Lemma 24.2.18, passing to a sufficiently high iterate of z, we may assume without loss of generality that T k (z) ∈ B(ω, δT ),

∀k ≥ 0.

(24.66)

As limk→∞ T k (z) = ω, we deduce from (24.65) that p(ω)

T n (z) ∈ ⋃ Srj (ω, π − α)

(24.67)

j=1

for all n ≥ 0 large enough. But, in light of Lemma 24.2.10, we know that p(ω)

Tω−n ( ⋃ Srj (ω, π − α)) ⊆ B(ω, |z − ω|/2) j=1

(24.68)

for all n ≥ 0 large enough. However, z = Tω−k (T k (z)) for all k ≥ 0 by virtue of (24.66). We thus infer from (24.67) and (24.68) that z ∈ B(ω, |z − ω|/2). This is impossible. ̂ →ℂ ̂ is a rational function with deg(T) ≥ 2 and ω ∈ Ω0 (T), Theorem 24.2.20. If T : ℂ ∗ then the Leau domains Aj (ω), j = 1, . . . , p(ω), are mutually disjoint. Moreover, for every j ∈ {1, . . . , p(ω)} and every α ∈ (0, π), we have that ∞

A∗j (ω) ⊆ ⋃ T −n (Saj (ω, α))

and

n=0

∞

Aj (ω) = ⋃ T −n (Saj (ω, α)). n=0

j

−n Proof. As ⋃∞ n=0 T (Sa (ω, α)), j = 1, . . . , p(ω), is a finite collection of mutually disjoint open sets and as the sets A∗i (ω), i = 1, . . . , p(ω), are each connected, Lemma 24.2.19 im−n j poses that each set A∗i (ω) be contained in exactly one set of the form ⋃∞ n=0 T (Sa (ω, α)). ∗ i Since Ai (ω) ⊇ Sa (ω, α), we deduce that ∞

A∗i (ω) ⊆ ⋃ T −n (Sai (ω, α)). n=0

This not only implies that the Leau domains A∗i (ω), 1 ≤ i ≤ p(ω), are mutually disjoint but also that ∞

∞

∞

m=0

m,n=0

n=0

Ai (ω) := ⋃ T −m (A∗i (ω)) ⊆ ⋃ T −(m+n) (Sai (ω, α)) = ⋃ T −n (Sai (ω, α)).

24.2 Rationally indifferent periodic points

� 1001

On the other hand, Sai (ω, α) ⊆ A∗i (ω) ⊆ Ai (ω). Since Ai (ω) is completely T-invariant (see Lemma 24.2.18), we deduce that ∞

⋃ T −n (Sai (ω, α)) ⊆ Ai (ω).

n=0

Revisiting Theorem 24.2.17 in the context of a globally defined φ, i. e., for a rational function T, it turns out that all the maps Fα extend holomorphically to the entire Leau– Fatou petals A∗j (ω). ̂ →ℂ ̂ is a rational function with deg(T) ≥ 2 and ω ∈ Ω0 (T), Theorem 24.2.21. If T : ℂ then for every j ∈ {1, . . . , p(ω)} there exists a holomorphic function F : A∗j (ω) → ℂ such that F ∘ T = F + 1,

(24.69)

󵄨 and for all α ∈ (0, π) the function F 󵄨󵄨󵄨Sj (ω,α) is univalent. In addition, if α ∈ [π/2, π) then a there exists s > 0 such that F(Saj (ω, α)) ⊇ {ζ ∈ ℂ : Re(ζ ) > s}.

(24.70) j

Proof. Fix z ∈ A∗j (ω). By Theorem 24.2.20, there exists k ≥ 0 such that T k (z) ∈ Sa (ω, α). Define F(z) := Fα (T k (z)) − k.

(24.71)

To see that this definition is independent of k, notice that Theorem 24.2.17 implies that F(z) = Fα (T k+j (z)) − (k + j),

∀j ≥ 0.

So F(z) = lim [Fα (T n (z)) − n] n→∞

(24.72)

is independent of k. It is also easy to show that (24.69) holds. Relation (24.70) is an immediate consequence of Theorem 24.2.17(c). The univalence of F ensues from Theorem 24.2.17(b). Finally, (24.71) directly implies that F is holomorphic. ̂ be a rationally indifferent periodic point of T, i. e., ω ∈ More generally, let ω ∈ ℂ Ω(T). Then there is ℓ ∈ ℕ (ℓ is an appropriate multiple of the prime period of ω) such that ω is a simple rationally indifferent fixed point of T ℓ , i. e., ω ∈ Ω0 (T ℓ ). Define A∗ (ω) to be the immediate basin of attraction of ω under T ℓ . Define also A∗j (ω), j = 1, . . . , p(ω), to

1002 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets be the Leau–Fatou petals of ω considered as a simple parabolic fixed point of T ℓ . Further define ∞

A(ω) := ⋃ T −j (A∗ (ω)), j=0

and ℓ−1

A∗p (ω) := ⋃ A∗ (T j (ω)), j=0

ℓ−1

Ap (ω) := ⋃ A(T j (ω)). j=0

̂→ℂ ̂ is a rational function with deg(T) ≥ 2 and ω ∈ Ω(T), then Theorem 24.2.22. If T : ℂ each Leau–Fatou petal of ω contains at least one critical point of T. Proof. The first part of the proof parallels the corresponding part of the proof of Theorem 24.1.20. However, the argument engendering a contradiction is more complicated. First, assume that ω is a simple rationally indifferent fixed point of T. Fix j ∈ {1, . . . , p(ω)}. Suppose for a contradiction that A∗j (ω) contains no critical point of T. We shall prove by induction that there exists an ascending sequence (Un )∞ n=0 of open simply connected subsets of A∗j (ω) such that (a) ω ∈ U0 . Equivalently, ω ∈ 𝜕Un−1 for all n ∈ ℕ. (b) For every n ∈ ℕ, there exists Tn−1 : Un−1 → Un , a unique surjective holomorphic inverse branch of T such that Tn−1 (ω) = ω. −1 󵄨󵄨 −1 (c) Tn+1 󵄨󵄨U = Tn , ∀n ∈ ℕ. n−1

Indeed, set U0 := Saj (ω, π/2) ⊆ A∗j (ω). It is clear that ω ∈ 𝜕U0 . By relation (24.61) in Theorem 24.2.17, we know that U0 ⊆ Tω−1 (U0 ) ⊆ A∗j (ω), where Tω−1 is the unique local holomorphic inverse branch of T that sends ω to ω. Set 󵄨 T1−1 := Tω−1 󵄨󵄨󵄨U . As U0 is open and simply connected, so is U1 := T1−1 (U0 ). Moreover, U1 ⊇ U0 0 by the above relation. The base step of the induction is complete. For the inductive step, suppose that for some k ∈ ℕ an ascending sequence of open simply connected sets U0 ⊆ U1 ⊆ ⋅ ⋅ ⋅ ⊆ Uk has been constructed such that properties (a) and (b) are satisfied for all 1 ≤ n ≤ k while (c) is satisfied for all 1 ≤ n < k. Since A∗j (ω)∩T −1 (Uk ) contains no critical point of T, since T : A∗j (ω) → A∗j (ω) is a holomorphic

−1 surjective map (per Lemma 24.2.18), and since Uk ⊇ Uk−1 , there exists Tk+1 : Uk → A∗j (ω), a holomorphic inverse branch of T such that −1 󵄨󵄨 −1 Tk+1 󵄨󵄨Uk−1 = Tk .

(24.73)

24.2 Rationally indifferent periodic points

� 1003

So, (c) holds when n = k. Setting −1 Uk+1 := Tk+1 (Uk ),

(24.74)

property (b) holds when n = k + 1. Moreover, Uk+1 is a simply connected open set since −1 so is Uk and Tk+1 is holomorphic. We now check that Uk+1 ⊇ Uk . Successively invoking (24.74), the assumption that Uk ⊇ Uk−1 , (24.73) and property (b) with n = k, we infer that −1 Uk+1 ⊇ Tk+1 (Uk−1 ) = Tk−1 (Uk−1 ) = Uk .

Finally, property (a) holds when n = k + 1 as it holds when n = k and Uk+1 ⊇ Uk but Uk+1 ⊆ ℱ (T) while ω ∈ 𝒥 (T). The inductive step is thus finished. ∗ Since ⋃∞ n=0 Un = Aj (ω) by Theorem 24.2.20, conditions (a)–(c) along with ascendance −1 ∗ ∗ of (Un )∞ n=0 guarantee the existence of a holomorphic bijection T∗ : Aj (ω) → Aj (ω) such

that T∗−1 |Un−1 = Tn−1 for every n ∈ ℕ. As T ∘T∗−1 = IdA∗ (ω) , the restriction T : A∗j (ω) → A∗j (ω) j

is one-to-one. Since we already know that it is onto, we conclude that T : A∗j (ω) → A∗j (ω) is bijective. Iterating formula (24.69) of Theorem 24.2.21 yields F ∘ T n = F + n,

∀n ≥ 0.

(24.75)

Choosing distinct points z, ζ ∈ A∗j (ω), there is n = n(z, ζ ) ≥ 0 so large that T n (z), T n (ζ ) ∈ j

Sa (ω, π/2). It follows from the bijectivity of T n : A∗j (ω) → A∗j (ω), formula (24.75), and the univalence of F|Sj (ω,π/2) in Theorem 24.2.21, that F(z) ≠ F(ζ ). In consequence, the a holomorphic function F : A∗j (ω) → ℂ is univalent. Formula (24.75) further decrees that F ∘ T∗−n = F − n,

∀n ≥ 0.

Therefore, it follows from inclusion (24.70) in Theorem 24.2.21 that F(A∗j (ω)) ⊇ F(Un ) = F(T∗−n (U0 )) = F(U0 ) − n ⊇ {z ∈ ℂ : Re(z) > s − n}. Taking the union over all n ∈ ℕ, it ensues that F(A∗j (ω)) = ℂ. As the map F : A∗j (ω) → ℂ is univalent, its inverse F −1 : ℂ → A∗j (ω) is holomorphic. Because A∗j (ω) ∩ 𝒥 (T) = 0 and 𝒥 (T) is a nonempty perfect set per Theorems 24.1.4 and 24.1.15, the function F −1 : ℂ → A∗j (ω) omits three distinct points and we conclude

from Picard’s theorem that F −1 is constant. This contradiction finishes the proof in the case where ω is a simple parabolic fixed point of T. The general case, in which ω ∈ Ω(T), follows by applying the case of a simple parabolic fixed point to an higher iterate T ℓ such that ω ∈ Ω0 (T ℓ ).

1004 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets Since #Crit(T) ≤ 2 deg(T) − 2, the theorem just proved and Theorem 24.1.20 impose the following stronger constraint than Corollary 24.1.21. ̂ → ℂ ̂ is a rational function with deg(T) ≥ 2, then the toCorollary 24.2.23. If T : ℂ tal number of attracting and rationally indifferent periodic cycles of T does not exceed 2 deg(T) − 2.

24.2.3 Local and asymptotic behavior of rational functions around rationally indifferent periodic points: part II – Fatou’s flower theorem and fundamental domains ̂ → ℂ ̂ is once again a rational function with deg(T) ≥ 2 and ω ∈ In this section, T : ℂ Ω0 (T). The results we present are of fairly classical nature and are scattered widely in the literature; for example, they can be found in [1, 8, 21, 37, 40, 94, 123]. As we have seen in Section 24.2.2, for a sufficiently small radius δT > 0 there exists a unique holomorphic inverse branch Tω−1 : B(ω, δT ) → ℂ which fixes ω. Thus, all the (local) results of Section 24.2.1 apply with both φ = T and φ = Tω−1 . Fix α ∈ (0, π), κ = 1/2, and recall that Sa1 (ω, α), . . . , Sap(ω) (ω, α) are the corresponding attracting sectors for T defined in (24.58) while Sr1 (ω, α), . . . , Srp(ω) (ω, α) are the corresponding repelling sectors for T defined in (24.59). Equivalently, the repelling sectors for T are the attractive sectors for Tω−1 . Since the family of iterates of T is not normal on any neighborhood of any point in the Julia set, we get the following celebrated classical result as a fairly immediate consequence of Lemma 24.2.10. ̂ → ℂ ̂ be a rational function with Theorem 24.2.24 (Fatou’s flower theorem). Let T : ℂ deg(T) ≥ 2. If ω ∈ Ω(T), i. e., if ω is a rationally indifferent periodic point of T, then for every α ∈ (0, π) there exists θα (ω) ∈ (0, δT ) such that p(ω)

j

𝒥 (T) ∩ B(ω, θα (ω)) \ {ω} ⊆ ⋃ Sr (ω, α). j=1

Proof. Passing to a sufficiently high iterate of T, we may assume without loss of generality that ω is a simple parabolic fixed point of T, i. e., ω ∈ Ω0 (T). Seeking a contradiction, p(ω) j suppose that there exists a sequence (zn )∞ n=1 of points in 𝒥 (T) \ ⋃j=1 Sr (ω, α) such that lim |zn − ω| = 0

n→∞

and zn ≠ ω, ∀n ∈ ℕ.

24.2 Rationally indifferent periodic points

� 1005

Set β ∈ (0, α). There would then exist i ∈ {1, . . . , p(ω)} such that zn ∈ Sai (ω, π − β) for every n ∈ ℕ large enough. Fix one such n and i. By Lemma 24.2.10, the attracting petal Sai (ω, π − β) is a subset of ℱ (T). Thus, zn ∉ 𝒥 (T). This is a contradiction. Since the Julia set 𝒥 (T) is completely invariant, we infer from Fatou’s flower theorem (Theorem 24.2.24), Lemma 24.2.10, and (24.26) (actually the derivative of φA,0 ) that for every ω ∈ Ω0 (T) there exists θ(ω) ∈ (0, θπ/4 (ω)) such that for every 0 < R ≤ θ(ω), we have Tω−1 (𝒥 (T) ∩ B(ω, R)) ⊆ 𝒥 (T) ∩ B(ω, R).

(24.76)

Thus, all the iterates Tω−n : 𝒥 (T) ∩ B(ω, R) → 𝒥 (T) ∩ B(ω, R),

n ∈ ℕ,

(24.77)

are well-defined. For all α ∈ (0, π) and all j ∈ {1, 2, . . . , p(ω)}, we know by Lemma 24.2.10 that Tω−1 (Srj (ω, α)) ⊆ Srj (ω, α).

(24.78)

Therefore, all the iterates Tω−n : Srj (ω, α) → Srj (ω, α),

n ∈ ℕ,

(24.79)

are well-defined. It ensues from the complete T-invariance of 𝒥 (T) and (24.79) that Tω−n (𝒥 (T) ∩ Srj (ω, α)) ⊆ 𝒥 (T) ∩ Srj (ω, α),

∀n ∈ ℕ,

∀1 ≤ j ≤ p(ω).

(24.80)

Furthermore, it follows from Lemma 24.2.10 that lim T −n (z) n→∞ ω

=ω

uniformly on z ∈ Srj (ω, α),

(24.81)

while Lemma 24.2.11 states that 󵄨󵄨 −1 󵄨 󵄨󵄨Tω (z) − ω󵄨󵄨󵄨 < |z − ω| and

󵄨󵄨 −1 ′ 󵄨󵄨 󵄨󵄨(Tω ) (z)󵄨󵄨 < 1

(24.82)

j

for every z ∈ Sr (ω, α) ∩ B(ω, Rα,1/2 (ω)) \ {ω} when α < π/p(ω). Set θ = θ(T) := min min{θ(ω), Rπ/4,1/2 (ω)}. ω∈Ω(T)

(24.83)

As an immediate consequence of Fatou’s flower theorem (Theorem 24.2.24) and (24.77), we get the following.

1006 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets ̂ → ℂ ̂ be a rational function with deg(T) ≥ 2. If τ > 0 is Lemma 24.2.25. Let T : ℂ sufficiently small (with θ > 0 decreased if necessary), then for every ω ∈ Ω0 (T) and every z ∈ 𝒥 (T) ∩ B(ω, θ) there exists j ∈ {1, . . . , p(ω)} such that B(z, 2τ|z − ω|) ⊆ B(ω, θ) ∩ Srj (ω, π/2). In addition, all the local holomorphic inverse branches Tω−n : B(z, 2τ|z − ω|) → ℂ are well-defined for all n ∈ ℕ. As a direct repercussion of Proposition 24.2.12 and Fatou’s flower theorem (Theorem 24.2.24), we obtain the next result. ̂ → ℂ ̂ is a rational function with deg(T) ≥ 2, then for every Lemma 24.2.26. If T : ℂ j ω ∈ Ω0 (T), every j ∈ {1, . . . , p(ω) =: p}, and every point z ∈ Sr (ω, α) (including every z ∈ 𝒥 (T) ∩ B(ω, θ) \ {ω}), we have lim n

n→∞

p+1 p

−1 󵄨󵄨 −(n+1) 󵄨 (z) − Tω−n (z)󵄨󵄨󵄨 = (|a|p) p p−1 󵄨󵄨Tω 1 −1 󵄨 󵄨 lim n p 󵄨󵄨󵄨Tω−n (z) − ω󵄨󵄨󵄨 = (|a|p) p .

n→∞

j

In addition, in both limits, the convergence is uniform on the set Sr (ω, α) \ B(ω, R) (and in particular on the set 𝒥 (T) ∩ B(ω, θ) \ B(ω, R)) for every R > 0. As a straightforward consequence of Proposition 24.2.13 and Fatou’s flower theorem (Theorem 24.2.24), we observe the following fact. ̂→ℂ ̂ is a rational function with deg(T) ≥ 2, then for every Proposition 24.2.27. If T : ℂ ω ∈ Ω0 (T) and every α ∈ (0, π) there exists a holomorphic function p(ω)

(Tω−∞ ) : ⋃ Srj (ω, α) =: Sr (ω, α) 󳨀→ ℂ \ {0} ′

j=1

j

with the following properties. Fix j ∈ {1, . . . , p(ω) =: p}. For every z ∈ Sr (ω, α) (including every z ∈ 𝒥 (T) ∩ B(ω, θ) \ {ω}), we have lim n

n→∞

p+1 p

p+1 − 󵄨󵄨 −n ′ 󵄨󵄨 󵄨 −(p+1) 󵄨󵄨 −∞ ′ 󵄨󵄨(Tω ) (z)󵄨󵄨 = (|a|p) p |z − ω| 󵄨󵄨(Tω ) (z)󵄨󵄨󵄨

(24.84)

j

and the convergence is uniform on Sr (ω, α) \ B(ω, R) for every R > 0. If α ∈ (0, π/2), then lim

(Tω−∞ ) (z) = 1.

j Sr (ω,α)∋z→ω

′

(24.85)

24.2 Rationally indifferent periodic points

� 1007

As the last result about the local behavior of a rational function around its parabolic points, we show the following. ̂→ℂ ̂ Proposition 24.2.28 (Local expansiveness at rationally indifferent points). Let T : ℂ be a rational function with deg(T) ≥ 2. If ω ∈ Ω0 (T), z ∈ 𝒥 (T) and T n (z) ∈ B(ω, θ) for all n ≥ 0, then z = ω. Proof. Since z ∈ 𝒥 (T), we know that T n (z) ∈ 𝒥 (T) for every n ≥ 0. It follows from Fatou’s flower theorem (Theorem 24.2.24) and the uniform convergence in (24.81) that ω = lim Tω−n (T n (z)) = lim z = z. n→∞

n→∞

The uniform convergence is crucial because the point T n (z) varies with n in the sector involved in (24.81). We end this section with the following two lemmas, which are interesting on their own and constitute useful tools for applications. Let p(ω)

Sr (ω, α) := ⋃ Srj (ω, α). j=1

̂→ℂ ̂ be a rational function with deg(T) ≥ 2. If ω ∈ Ω0 (T) and Lemma 24.2.29. Let T : ℂ α ∈ (0, π), then the set 𝒥 (T) ∩ (Sr (ω, α) \ Tω−1 (Sr (ω, α))) is a fundamental domain for the map Tω−1 acting on 𝒥 (T) near ω. More precisely: (a) If (i, k), (j, l) ∈ {1, . . . , p(ω)} × {0, 1, 2, . . .} and (i, k) ≠ (j, l), then Tω−k (Sri (ω, α) \ Tω−1 (Sr (ω, α))) ⋂ Tω−l (Srj (ω, α) \ Tω−1 (Sr (ω, α))) = 0. (b)

∞

ω ∈ Int𝒥 (T) (𝒥 (T) ⋂ [{ω} ∪ ⋃ Tω−n (Sr (ω, α) \ Tω−1 (Sr (ω, α)))]). n=0

Proof. (a) Assume first that i = j but k < l. Defining T̃ω−1 = (ρA ∘ H)−1 ∘ Tω−1 (where ρA is an attracting direction for Tω−1 ) and using (24.39) and (24.27), we get Tω−k (Sri (ω, α) \ Tω−1 (Sr (ω, α))) ⋂ Tω−l (Sri (ω, α) \ Tω−1 (Sr (ω, α))) = Tω−k (ρA ∘ H(S(x(α, κ), α)) \ Tω−1 (ρA ∘ H(S(x(α, κ), α)))) ⋂ Tω−l (ρA ∘ H(S(x(α, κ), α)) \ Tω−1 (ρA ∘ H(S(x(α, κ), α)))) = Tω−k (ρA ∘ H(S(x(α, κ), α)) \ ρA ∘ H(T̃ω−1 (S(x(α, κ), α)))) ⋂ Tω−l (ρA ∘ H(S(x(α, κ), α)) \ ρA ∘ H(T̃ω−1 (S(x(α, κ), α))))

1008 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets = Tω−k ∘ ρA ∘ H(S(x(α, κ), α) \ T̃ω−1 (S(x(α, κ), α))) ⋂ Tω−l ∘ ρA ∘ H(S(x(α, κ), α) \ T̃ω−1 (S(x(α, κ), α))) = ρA ∘ H ∘ T̃ω−k (S(x(α, κ), α) \ T̃ω−1 (S(x(α, κ), α))) ⋂ ρA ∘ H ∘ T̃ω−l (S(x(α, κ), α) \ T̃ω−1 (S(x(α, κ), α))) = ρA ∘ H(T̃ω−k (S(x(α, κ), α) \ T̃ω−1 (S(x(α, κ), α))) ⋂ T̃ω−l (S(x(α, κ), α) \ T̃ω−1 (S(x(α, κ), α)))) = ρA ∘ H ∘ T̃ω−k (

(S(x(α, κ), α) \ T̃ω−1 (S(x(α, κ), α))) ⋂ T̃ω−(l−k) (S(x(α, κ), α) \ T̃ω−1 (S(x(α, κ), α)))

)

⊆ ρA ∘ H ∘ T̃ω−k ((S(x(α, κ), α) \ T̃ω−1 (S(x(α, κ), α))) ⋂ T̃ω−(l−k) (S(x(α, κ), α))). As l − k ≥ 1, Lemma 24.2.4 allows us to continue as follows: Tω−k (Sri (ω, α) \ Tω−1 (Sr (ω, α))) ⋂ Tω−l (Sri (ω, α) \ Tω−1 (Sr (ω, α))) ⊆ ρA ∘ H ∘ T̃ω−k ((S(x(α, κ), α) \ T̃ω−1 (S(x(α, κ), α))) ⋂ T̃ω−1 (S(x(α, κ), α))) = ρA ∘ H ∘ T̃ω−k (0) = 0. We have thus ascertained statement (a) in the case i = j but k ≠ l. If i ≠ j, then we are immediately done thanks to Lemma 24.2.16 and relation (24.61) in Theorem 24.2.17. Part (b) is an immediate consequence of (24.78) and Fatou’s flower theorem (Theorem 24.2.24). ̂ →ℂ ̂ be a rational function with deg(T) ≥ 2. Let ω ∈ Ω0 (T) Lemma 24.2.30. Let T : ℂ and α ∈ (0, π). For every u > 0 and every j ∈ {1, 2, . . . , p(ω)}, let Δj (ω, α, u) := [Srj (ω, α) \ Tω−1 (Srj (ω, α))] ∩ [ℂ \ B(ω, u)] and p(ω)

Δ(ω, α, u) := ⋃ Δj (ω, α, u). j=1

Then there exists u > 0 such that the set Δ(ω, α, u) is a fundamental domain for the map Tω−1 acting on 𝒥 (T) near ω. More precisely:

24.3 Nonattracting periodic points revisited

� 1009

(a) If (i, k), (j, l) ∈ {1, 2, . . . , p(ω)} × {0, 1, 2, . . .} and (i, k) ≠ (j, l), then Tω−k (Δj (ω, α, u)) ∩ Tω−l (Δj (ω, α, u)) = 0. −n (b) ω ∈ Int𝒥 (T) (𝒥 (T) ∩ [{ω} ∪ ⋃∞ n=0 Tω (Δ(ω, α, u))]).

Proof. Part (a) is an immediate consequence of Lemma 24.2.29. We now prove (b). Since Tω−1 is an open map (because it is holomorphic), there is u ∈ (0, θα (ω)) such that B(ω, u) ⊆ Tω−1 (B(ω, θα (ω))). It then follows from Fatou’s flower theorem (Theorem 24.2.24) that p(ω)

j

𝒥 (T) ∩ B(ω, u) ⊆ ⋃ Tω (Sr (ω, α)) = Tω (Sr (ω, α)). −1

−1

j=1

(24.86)

In light of Lemma 24.2.29(b), it suffices to show that ∞

∞

𝒥 (T) ∩ [ ⋃ Tω (Sr (ω, α) \ Tω (Sr (ω, α)))] ⊆ ⋃ Tω (Δ(ω, α, u)). −n

−1

n=0

−n

n=0

Let z belong to the left-hand side of this claimed inclusion. Then there is n ≥ 0 such that z ∈ 𝒥 (T) ∩ Tω−n (Sr (ω, α) \ Tω−1 (Sr (ω, α))). By (24.86), we get T n (z) ∈ 𝒥 (T) ∩ [Sr (ω, α) \ Tω−1 (Sr (ω, α))] = 𝒥 (T) ∩ [Sr (ω, α) \ Tω−1 (Sr (ω, α))] ∩ [ℂ \ B(ω, u)]

⊆ Δ(ω, α, u). Hence, z ∈ Tω−n (Δ(ω, α, u)).

24.3 Nonattracting periodic points revisited: total number and denseness of repelling periodic points We now show that the total number of nonrepelling cycles of a rational function does not exceed 6 deg(T) − 6. In fact, we first show the following somewhat stronger result. ̂ → ℂ ̂ with deg(T) ≥ 2 admits at most Theorem 24.3.1. Every rational function T : ℂ 4 deg(T) − 4 indifferent cycles with multipliers different from 1.

1010 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets Proof. Let ω1 , . . . , ωk be indifferent periodic points of T with respective multipliers λ1 , . . . , λk , all different from 1, and all belonging to distinct periodic cycles. By conjugation through a suitably chosen Möbius transformation, we may assume without loss of generality that all the periodic cycles of T containing the points ω1 , . . . , ωk do not visit the point ∞. Let d = deg(T). Write T(z) =

P(z) , Q(z)

where P and Q are relatively prime polynomials. Consider the one-parameter family of rational functions given by the formula Tξ (z) =

(1 − ξ)P(z) + ξzd , (1 − ξ)Q(z) + ξ

z, ξ ∈ ℂ.

(24.87)

Then T0 = T

and T1 (z) = zd .

For every j ∈ {1, . . . , k}, let pj ∈ ℕ be the prime period of ωj under T. Consider the function p

(z, ξ) 󳨃󳨀→ Fj (z, ξ) := Tξ j (z) − z. Then Fj (ωj , 0) = T pj (ωj ) − ωj = ωj − ωj = 0 and 𝜕Fj 𝜕z

(ωj , 0) = (T pj ) (ωj ) − 1 = λj − 1 ≠ 0. ′

So, by the implicit function theorem (Theorem A.3.12), there exist Rj > 0 and a unique holomorphic function Hj : B(0, Rj ) → ℂ such that Hj (0) = ωj

and

Fj (Hj (ξ), ξ) = 0,

∀ξ ∈ B(0, Rj ).

This means that p

Tξ j (Hj (ξ)) = Hj (ξ),

∀ξ ∈ B(0, Rj ).

(24.88)

We may further assume that the Rj ’s are such that the sets Hj (B(0, Rj )), j = 1, . . . , k, are mutually disjoint. Let R = min{R1 , . . . , Rk } > 0. We now identify a finite “bad” set of parameters ξ whose relevance will soon become apparent. Let us declare that ξ ∈ Bad if ̂ which means that the numerator and formula (24.87) is not well-defined for some z ∈ ℂ, the denominator of Tξ have some common zero. For z to be a zero of the denominator of Tξ means that

24.3 Nonattracting periodic points revisited

Q(z) =

−ξ 1−ξ

� 1011

(24.89)

whereas being a zero of the numerator means that P(z) −ξ = . 1−ξ zd

(24.90)

So, if ξ ∈ Bad then P(z) − zd Q(z) = 0 for some z ∈ ℂ (check that z ≠ ∞). This polynomial equation has only finitely many (at most 2d) solutions and, therefore, the corresponding Bad set of parameters ξ determined by (24.89) or (24.90) is finite. p Assume for a contradiction that some multiplier function λj (ξ) := (Tξ j )′ (Hj (ξ)) is

constant on B(0, R). Since T0 = T and T1 (z) = zd , we know that 0, 1 ∉ Bad. Let γ : [0, 1] → ℂ be a smooth curve from γ(0) = 0 to γ(1) = 1 that avoids the set Bad and never returns to B(0, R) if it ever leaves it. We claim that Hj can be extended holomorphically from 0 to 1 along the curve γ. The function Hj is already defined for all t ∈ [0, 1] so small that |γ(t)| < R. Let s be the supremum of all those t ∈ [0, 1] for which an holomorphic pj extension of Hj exists. By continuous dependence on ξ, we see that Tγ(s) (ω) = ω, where ω ∈ ℂ is any limit point of a sequence (Hj (tk ))∞ such that lim t = s. Conjugating k→∞ k k=1 ̂ → ℂ ̂ by a common Möbius transformation if necessary, we may all the maps Tξ : ℂ assume that the orbit of ω under Tγ(s) lies in ℂ. As the function λj (ξ) is constant, using p

j once again the continuous dependence on ξ we infer that (Tγ(s) )′ (ω) = λj (0) = λj ≠ 1. We may therefore apply the implicit function theorem exactly as above and deduce that the function Hj extends uniquely and holomorphically to some open neighborhood of γ(s) so that (24.88) holds. In particular, Hj extends along the curve γ beyond the point γ(s). This implies that s = 1, as otherwise this would contradict the very definition of s. But Tγ(1) (z) = T1 (z) = zd and the only periodic points of T1 are 0 (whose multiplier is zero) and points lying on the unit circle 𝕊1 := {z ∈ ℂ : |z| = 1}. The multipliers of these latter points all have moduli at least equal to d > 1. On the other hand, Tγ(0) = T0 = T and Hj (0) = ωj has a multiplier of modulus 1. This contradicts the constantness of the multiplier function λj (ξ) along γ. In conclusion, no multiplier function λj : B(0, R) → ℂ is constant. Thus, for every 1 ≤ j ≤ k there exists nj ∈ ℕ and a complex number aj ≠ 0 such that the Taylor series expansion of λj /λj (0) about 0 takes the form

λj (ξ)

λj (0)

= 1 + aj ξ nj + O(ξ nj +1 ).

Since the Taylor series expansion of √1 + x about 0 is given by √1 + x = 1 + 1 x + O(x 2 ), 2 we obtain that

1012 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets 󵄨 󵄨 2 2 󵄨 󵄨󵄨 λj (ξ) 󵄨󵄨󵄨 √ 󵄨󵄨 n +1 󵄨 = [1 + Re(aj ξ nj )] + [Im(aj ξ nj )] + O(ξ j ) 󵄨󵄨λj (ξ)󵄨󵄨󵄨 = 󵄨󵄨󵄨 󵄨󵄨 λj (0) 󵄨󵄨󵄨 2

2

= √1 + 2Re(aj ξ nj ) + (Re(aj ξ nj )) + (Im(aj ξ nj )) + O(ξ nj +1 ) 1 1 2 2 = 1 + Re(aj ξ nj ) + (Re(aj ξ nj )) + (Im(aj ξ nj )) + O(ξ nj +1 ) 2 2 = 1 + Re(aj ξ nj ) + O(ξ nj +1 ).

Writing ξ = reiθ , let sj (θ) := sgn(Re(aj ξ nj )). If sj (θ) = 1, then it follows that |λj (reiθ )| > 1 for all r > 0 small enough (depending on θ). While if sj (θ) = −1, then |λj (reiθ )| < 1 for all r > 0 small enough (depending on θ). Obviously, each sj is a step function defined on the circle ℝ/(2πℤ), which takes on the values ±1 everywhere except at 2nj uniformly interspersed jump discontinuities where it is undefined. Hence, 2π

1 ∫ sj (θ) dθ = 0. 2π

(24.91)

0

Therefore, 2π

k 1 ∫ ∑ sj (θ) dθ = 0 2π j=1

(24.92)

0

and, except at its finitely many discontinuities where it is undefined, the sum ∑kj=1 sj (θ) is an integer. For every θ ∈ ℝ/(2πℤ), set 󵄨 I+ (θ) := {1 ≤ j ≤ k 󵄨󵄨󵄨 sj (θ) = +1}

and

󵄨 I− (θ) := {1 ≤ j ≤ k 󵄨󵄨󵄨 sj (θ) = −1}.

If #I+ (θ) > #I− (θ) for some θ ∈ ℝ/(2πℤ), then ∑kj=1 sj (θ) ≥ 1. Hence, if #I+ (θ) > #I− (θ) for all θ ∈ ℝ/(2πℤ) (excepting discontinuities), then (24.92) would yield 0 ≥ 1. This contradiction produces some θ∗ ∈ ℝ/(2πℤ) (which is not a discontinuity) such that #I+ (θ∗ ) ≤ #I− (θ∗ ). Thus, #I− (θ∗ ) ≥ k/2. So, if r > 0 is small enough, then Treiθ∗ has at ∗ least #I− (θ∗ ) ≥ k/2 attracting periodic points Hj (reiθ ). In view of Corollary 24.1.21, this yields k ≤ 2 deg(Treiθ∗ ) − 2 ≤ 2d − 2. 2 Consequently, k ≤ 4d − 4.

24.3 Nonattracting periodic points revisited

� 1013

Combining the above result with Corollary 24.2.23 immediately gives the following result. ̂ → ℂ ̂ with deg(T) ≥ 2 has at most Theorem 24.3.2. Every rational function T : ℂ 6 deg(T) − 6 nonrepelling periodic cycles (i. e., cycles with multipliers of moduli at most 1). In particular, there are finitely many nonrepelling periodic points. Remark 24.3.3. Using quasiconformal surgery, Shishikura proved in [119] that, in fact, the number of nonrepelling periodic cycles does not exceed 2 deg(T) − 2 and this bound is sharp. We are now ready to prove the following dynamical characterization of Julia sets. ̂→ℂ ̂ with deg(T) ≥ 2 is Theorem 24.3.4. The Julia set 𝒥 (T) of a rational function T : ℂ the closure of the set of its repelling periodic points. Proof. Let Per(T) denote the set of all periodic points of T and let Perr (T) denote its subset of all repelling periodic points. As 𝒥 (T) is closed, Theorem 24.1.22 yields that Perr (T) ⊆ 𝒥 (T). Since the Julia set 𝒥 (T) is perfect (by Theorem 24.1.15) and since the set of nonrepelling periodic points is finite (per Theorem 24.3.2), it suffices to show that 𝒥 (T) ⊆ Per(T).

To prove this inclusion, suppose by contradiction that there exists w ∈ 𝒥 (T) \ Per(T). As 𝒥 (T) is perfect, we may select w ≠ ∞ not to be a critical value of T, i. e., not the image under T of a critical point of T. Then T −1 (w) contains at least two distinct points, say a and b. Since neither a nor b are critical points of T, there are two holomorphic inverse branches Ta−1 : D → ℂ and Tb−1 : D → ℂ, where D is some sufficiently small open disk centered at w, uniquely determined by the requirement that Ta−1 (w) = a and Tb−1 (w) = b. Decreasing the disk D if necessary, we may assume that D ∩ Per(T) = 0

(24.93)

Ta−1 (D) ∩ Tb−1 (D) = 0.

(24.94)

and that

Recall that given four complex numbers z1 , z2 , z3 , z4 , their cross ratio is defined by [z1 , z2 , z3 , z4 ] := Consider the sequence of functions

(z1 − z3 )(z2 − z4 ) . (z1 − z4 )(z2 − z3 )

1014 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets ̂ φn := [T n , Ta−1 , Tb−1 , Id] : D → ℂ. That is, φn (z) =

(T n (z) − Tb−1 (z))(Ta−1 (z) − z)

(T n (z) − z)(Ta−1 (z) − Tb−1 (z))

.

(24.95)

Because of (24.93)–(24.94), none of the four terms above ever vanishes (if T n (z) = ∞, then (T n (z) − Tb−1 (z))/(T n (z) − z) = 1). Thus, the sequence (φn )∞ n=1 omits the number 0. If it turned out that φn (z) = 1, then simple algebraic manipulations show that we would have T n (z) = Ta−1 (z) or Tb−1 (z) = z. This would imply that T n+1 (z) = z or T(z) = z, contrary to (24.93). So, (φn )∞ n=1 also omits the number 1. As neither of the terms in the denominator of (24.95) ever vanishes thanks to (24.93)– (24.94), if it happened that φn (z) = ∞ then this would mean that T n (z) − Tb−1 (z) = ∞

or

Ta−1 (z) − z = ∞.

The latter option does not hold since Ta−1 (D) ⊆ ℂ, which only leaves the possibility that T n (z) = ∞ since Tb−1 (D) ⊆ ℂ. But then we would have that φn (z) =

Ta−1 (z) − z

Ta−1 (z) − Tb−1 (z)

≠ ∞.

This contradiction shows that the sequence (φn )∞ n=1 omits the point ∞, too. By Montel’s theorem, we deduce that the sequence (φn )∞ n=1 is normal. Hence, the ∞ sequence (Φn (z))n=1 defined by Φn (z) :=

T n (z) − Tb−1 (z) T −1 (z) − Tb−1 (z) = φn (z) ⋅ a −1 n T (z) − z Ta (z) − z

is also normal. Finally, given that T n (z) =

zΦn (z) − Tb−1 (z) , Φn (z) − 1

the sequence (T n |D )∞ n=1 is normal. So w ∈ ℱ (T). This contradiction finishes the proof. We can now establish that the restriction of T to 𝒥 (T) is topologically exact. ̂→ℂ ̂ be a rational function with deg(T) ≥ 2. Then T : 𝒥 (T) → Corollary 24.3.5. Let T : ℂ 𝒥 (T) is topologically exact. ̂ be an open set such that Proof. By Theorem 24.1.4, we know that 𝒥 (T) ≠ 0. Let U ⊆ ℂ U ∩ 𝒥 (T) ≠ 0. We need to show that there is N = N(U) ∈ ℕ such that T N (U) ⊇ 𝒥 (T). By Theorem 24.3.4, the map T has at least one repelling periodic point in U ∩ 𝒥 (T); call it z.

24.3 Nonattracting periodic points revisited

� 1015

Let q ∈ ℕ be a period of z. Since z is repelling, we know that |(T q )′ (z)| > 1 and the mean value inequality applied to the local inverse branch of T q that maps z to itself entails the existence of R > 0 such that T q (B(z, r)) ⊇ B(z, r),

∀r ∈ (0, R].

(24.96)

Fix r ∈ (0, R] so small that U ⊇ B(z, r).

(24.97)

Applying Corollary 24.1.10 to T q and B(z, r), there is n = n(r) ∈ ℕ such that n

⋃ T kq (B(z, r)) ⊇ 𝒥 (T q ).

k=0

Along with (24.96)–(24.97) and the fact that 𝒥 (T q ) = 𝒥 (T) per Theorem 24.1.5, this implies that n

T nq (U) ⊇ T nq (B(z, r)) = ⋃ T kq (B(z, r)) ⊇ 𝒥 (T q ) = 𝒥 (T). k=0

As an immediate consequence of this corollary we get the following two ramifications. The first one ensues from the fact that any topologically exact system is topologically transitive and that topological transitivity can be characterized by the absence of any proper subsystem with nonempty interior (see Theorem 1.5.11). ̂→ℂ ̂ is a rational function with deg(T) ≥ 2, then either 𝒥 (T) is Corollary 24.3.6. If T : ℂ ̂ ̂ a nowhere dense subset of ℂ or 𝒥 (T) = ℂ. The second repercussion follows from the fact that any topologically exact system is strongly transitive and from the very definition of strong transitivity (see Definition 1.5.14). ̂ → ℂ ̂ is a rational function with deg(T) ≥ 2, then for each Corollary 24.3.7. If T : ℂ z ∈ 𝒥 (T) we have ∞

⋃ T −n (z) = 𝒥 (T)

n=0

and 𝒥 (T) is the only nonempty closed backward T-invariant subset of 𝒥 (T), i. e., the only nonempty closed subset F of 𝒥 (T) satisfying T −1 (F) ⊆ F. We end this section with a useful technical fact.

1016 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets ̂→ℂ ̂ is a rational function with deg(T) ≥ 2, then z ∈ ℱ (T) if Proposition 24.3.8. If T : ℂ ̂ of z and a strictly increasing sequence and only if there exist an open neighborhood Uz ⊆ ℂ nk (nk )∞ of positive integers such that the family {T |Uz }∞ k=1 k=1 is normal. Proof. The direct implication is trivial. For the converse, let ζ ∈ 𝒥 (T). By Lemma 24.1.6, we may assume without loss of generality that ∞ ∉ 𝒥 (T). It suffices to show that for any open neighborhood U of ζ and any strictly increasing sequence (nk )∞ k=1 of positive integers, the sequence (T nk |U )∞ is not normal. Suppose for a contradiction that such k=1 a sequence is normal. Passing to a subsequence if necessary, there exist R > 0 and an ̂ such that B(ζ , R) ⊆ U and the sequence (T nk |B(ζ ,R) )∞ analytic function f : B(ζ , R) → ℂ k=1 converges uniformly to f . Since f is continuous, there exists r ∈ (0, R) such that 1 f (B(ζ , r)) ⊆ B(ζ , diam(𝒥 (T))). 4

(24.98)

But T nk (B(ζ , r)) ⊇ 𝒥 (T) for all k ∈ ℕ large enough, per Corollary 24.3.5. Consequently, f (B(ζ , r)) ⊇ 𝒥 (T). This contradicts (24.98).

24.4 The structure of the Fatou set In this section, we provide a complete description of the Fatou set of any rational funĉ → ℂ ̂ with deg(T) ≥ 2. We first classify the connected components of ℱ (T) tion T : ℂ that are fixed (i. e., forward invariant) under T. From this, we easily derive the classification of the periodic components of ℱ (T). Next, by using this and the Riemannn–Hurwitz formula we show that every periodic component of ℱ (T) is either a Herman ring or is simply connected or infinitely connected. Then we will cite Sullivan’s nonwandering theorem, which asserts that every connected component of the Fatou set is eventually periodic. Several deep and beautiful theorems from complex analysis and from the theory of Riemann surfaces will be crucial in the proofs that follow, including Koebe’s uniformization theorem (indirectly) and Koebe’s distortion theorem discussed in the first two chapters of this volume.

24.4.1 Forward invariant components of the Fatou set The primary goal in front of us is to classify the forward invariant connected components of the Fatou set. This theorem is essentially due to Cremer [29] and Fatou [49]. Fatou [49, Section 56, p. 249] proved that given a rational function T with deg(T) ≥ 2, if {T n |V } has only constant limit functions, then V is an immediate attractive basin or a Leau–Fatou domain. In the case of transcendental functions, his proof shows that the only further possibility is that of a Baker domain. Cremer [29, p. 317] proved that if {T n |V }

24.4 The structure of the Fatou set

� 1017

has nonconstant limit functions, then V is a Siegel disk or a Herman ring. Neither Fatou nor Cremer stated the full classification theorem but Töpfer’s remarks in [135, p. 211] came fairly close to it. The reader may want to reminisce about Subsection 24.1.2.2 prior to reading onward. ̂ → ℂ ̂ be a rational function with deg(T) ≥ 2 and let V be a Theorem 24.4.1. Let T : ℂ forward T-invariant connected component of the Fatou set ℱ (T), i. e., a component such that T(V ) ⊆ V . There are four possibilities: (a) V is the immediate basin of attraction to some attracting fixed point of T. (b) V is the immediate basin of attraction to some simple rationally indifferent fixed point of T lying in 𝜕V . (c) V is a topological disk (i. e., is homeomorphic to the open unit disk 𝔻), and the map T|V : V → V is analytically conjugate to an irrational rotation R2πα : 𝔻 → 𝔻. The domain V is then referred to as a Siegel disk. (d) There exists an annulus Ar (0) := {z ∈ ℂ : r < |z| < 1} with r ∈ (0, 1) such that the map T|V : V → V is analytically conjugate to an irrational rotation R2πα : Ar (0) → Ar (0). The domain V is then called a Herman ring. In either case, T(V ) = V . The proof of this theorem is long and will be broken down into several steps. Let us begin with the following lemma, which is in part a direct proof of Theorem 24.1.8(a). Lemma 24.4.2. If V is a connected component of the Fatou set ℱ (T), then so is T(V ). In addition, V is a connected component of T −1 (T(V )). ̂ and hence Proof. Recall that the completely T-invariant Fatou set ℱ (T) is open in ℂ, ̂ each of its connected components, such as V , is open not only in ℱ (T) but also in ℂ. Recall also that T is an open map, and thus T(V ) is open. Since T(V ) is a connected subset of ℱ (T), there exists a unique connected component W of ℱ (T) such that T(V ) ⊆ W . We shall show that in fact T(V ) = W .

(24.99)

By way of contradiction, suppose that this equality does not hold. As W is connected, we know that 𝜕W (T(V )) ≠ 0, where 𝜕W (T(V )) is the boundary of the set T(V ) with respect to the domain W . Fix some y ∈ 𝜕W (T(V )). Then y ∈ W \ T(V ) and there exists a sequence (yn )∞ n=1 with yn ∈ T(V ) for all n ∈ ℕ such that limn→∞ yn = y. So, for every n ∈ ℕ there exists xn ∈ V such that T(xn ) = yn . Passing to a subsequence if necessary, we may assume ̂ that the sequence (xn )∞ n=1 converges to some point x ∈ ℂ. Of course, x ∈ V . If it turned out that x ∈ V , then we would have that y = lim yn = lim T(xn ) = T(x) ∈ T(V ), n→∞

n→∞

1018 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets and this contradiction shows that y ∈ V \ V = 𝜕V ⊆ 𝒥 (T). But then y ∈ 𝒥 (T) ∩ W ⊆ 𝒥 (T) ∩ ℱ (T) = 0. This contradiction establishes (24.99). Since V ⊆ T −1 (T(V )) ⊆ ℱ (T) and V is a connected component of ℱ (T), we conclude that V is also a connected component of T −1 (T(V )). Corollary 24.4.3. If V is a forward T-invariant connected component of ℱ (T), then T(V ) = V . We now describe the conformal automorphisms of an annulus Ar (0). Proposition 24.4.4. Let r ∈ (0, 1). Every conformal automorphism of the annulus Ar (0) is either a rotation R2πα (z) = e2πiα z or a composition S ∘ R2πα , where α ∈ [0, 1] and S : Ar (0) → Ar (0) is defined by S(z) = r/z. Proof. For every t > 0, let 𝕊t = {z ∈ ℂ : |z| = t}, and for every 0 < i < o let A(0; i, o) denote the open annulus centered at the origin with inner radius i and outer radius o. Let R : Ar (0) → Ar (0) be a conformal automorphism, where we recall that Ar (0) = A(0; r, 1). As R is a homeomorphism, we have that: (a) Either: For every u ∈ 𝕊r and every v ∈ 𝕊1 , if limn→∞ un = u and limn→∞ vn = v, where un , vn ∈ Ar (0) for all n ∈ ℕ, then lim R(un ) = r

n→∞

and

lim R(vn ) = 1,

n→∞

(b) Or: For every u ∈ 𝕊r and every v ∈ 𝕊1 , if limn→∞ un = u and limn→∞ vn = v, where un , vn ∈ Ar (0) for all n ∈ ℕ, then lim R(un ) = 1

n→∞

and

lim R(vn ) = r.

n→∞

If (a) holds, set R∗ := R, whereas if (b) holds, set R∗ := S ∘ R, where we recall that S(z) := r/z. Then (a) holds in both cases if R is replaced by R∗ . With this in mind, by applying the Schwarz reflection principle (Theorem A.3.6) and replacing the complex conjugacy z 󳨃→ z by the inversion z 󳨃→ r 2 /z in the circle 𝕊r , we can extend R∗ to a holomorphic map (denoted by the same symbol R∗ ) from Ar2 (0) to Ar2 (0) such that (a) holds with: (1) R replaced by the extended R∗ ; (2) r replaced by r 2 ; and (3) the annulus Ar (0) replaced by the annulus Ar2 (0). So, proceeding by induction, we construct a holomorphic map R∗∞ : A0 (0) → A0 (0) such that: (c) R∗∞ |Ar (0) = R∗ ; n+1

n

n+1

n

(d) R∗∞ (A(0; r 2 , r 2 )) = A(0; r 2 , r 2 ) for all n ≥ 0; and (e) R∗∞ (𝕊r2n ) = 𝕊r2n .

It follows from (d) that limz→0 R∗∞ (z) = 0 and therefore R∗∞ extends to a holomorphic map from 𝔻 to 𝔻 such that R∗∞ (0) = 0. Because of this and (e), Schwarz lemma (Lemma A.3.5) implies that R∗∞ is a rotation. Thus, either R = R∗ = R2πα |Ar (0) or R = S −1 ∘ R∗ = S ∘ R∗∞ |Ar (0) = S ∘ R2πα |Ar (0) .

24.4 The structure of the Fatou set

�

1019

As an almost immediate consequence of the above proposition, we obtain the following. Corollary 24.4.5. Let r ∈ (0, 1). Every conformal automorphism R : Ar (0) → Ar (0) of infinite order (i. e., such that Rn ≠ Id for all n ∈ ℕ) is an irrational rotation of the annulus Ar (0). Proof. Given the notation from Proposition 24.4.4, observe that S ∘ R2πα = R−1 2πα ∘ S. If 2 2 R = S ∘ R2πα , then R = S = Id. In particular, R is of finite order. It therefore follows from Proposition 24.4.4 that R = R2πα . Since all rational rotations are of finite order, α must be irrational. Let V be a forward T-invariant connected component of ℱ (T). In order to determine the dynamics of the restriction T|V : V → V and ultimately prove Theorem 24.4.1, we first consider two cases characterized by the existence/nonexistence of nonconstant limit functions of the sequence (T n |V )∞ n=0 . Lemma 24.4.6. Let V be a forward T-invariant connected component of ℱ (T). If the sequence (T n |V )∞ n=0 has a nonconstant limit function, then either part (c) or part (d) of Theorem 24.4.1 holds. ̂ Proof. By hypothesis, there exist a nonconstant meromorphic function R : V → ℂ ∞ nk ∞ and an increasing sequence (nk )k=1 such that the sequence (T |V )k=1 converges to R uniformly on compact subsets of V and such that limk→∞ (nk+1 − nk ) = ∞. Given that T n (V ) ⊆ V for all n ∈ ℕ, we deduce that R(V ) ⊆ V . In fact, R(V ) ⊆ V as R(V ) is open. Let mk := nk+1 − nk . Passing to a subsequence if necessary, we may assume that the sequence (T mk |V )∞ k=1 converges uniformly on compact subsets of V to some meromorphic ̂ Since R(V ) ⊆ V , since images of compact sets under R are compact, function S : V → ℂ. nk+1 and since T = T mk ∘ T nk , we deduce that R = S ∘ R. So, S|R(V ) is the identity function. As R(V ) ⊆ V is open, it follows that S = Id|V . Passing to a subsequence, we may assume without loss of generality that (T mk −1 |V )∞ k=1 converges ̂ But then to some meromorphic function M : V → ℂ. Id|V = lim T mk |V = lim T mk −1 |V ∘ T|V = M ∘ T|V . k→∞

k→∞

(24.100)

Therefore, M(T(V )) = V . Since V is forward T-invariant, Corollary 24.4.3 asserts that T(V ) = V . Thus, M(V ) ⊆ V , and hence Id|V = S = lim T ∘ T mk −1 |V = T ∘ M = T|V ∘ M. k→∞

This and (24.100) show that T|V : V → V is a conformal automorphism and (T|V )−1 = M. As Id|V = limk→∞ T mk |V , we thereby see that the topological group of conformal auto-

1020 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets morphisms of the Riemann surface V does not have the discrete topology. Invoking also the fact that the complement of V contains at least three points, we infer (see [136]) that V is conformally equivalent either to the open unit disk 𝔻 or to an annulus Ar (0) for some r ∈ [0, 1). Let us first consider the case where V is conformally equivalent to 𝔻. Recalling that 𝔻 is conformally equivalent to the upper-half plane ℍ := {z ∈ ℂ | Im(z) > 0}, let φ : V → ℍ be a conformal isomorphism. Then φ ∘ T ∘ φ−1 : ℍ → ℍ is a conformal automorphism of ℍ, and hence takes the form of a Möbius transformation ℍ ∋ z 󳨃→ az+b ̂ to ℂ ̂ ∈ ℍ. By the same formula, the map φ ∘ T ∘ φ−1 extends to a Möbius map from ℂ cz+d −1 −1 and φ ∘ T ∘ φ (ℍ) = ℍ. So, φ ∘ T ∘ φ : ℍ → ℍ has a fixed point according to Brouwer’s ̂ and is therefore homeomorphic fixed-point theorem (ℍ is understood as a subset of ℂ to the closed unit disk 𝔻). Suppose for a contradiction that φ∘T ∘φ−1 has no fixed point in ℍ. Then it has either exactly one or exactly two fixed points in 𝜕ℍ satisfying the equation z = (az+b)/(cz+d). In the former case, conjugating φ∘T ∘φ−1 by a Möbius transformation which preserves ℍ, we may assume without loss of generality that the sole fixed point is ∞. But then φ ∘ T ∘ φ−1 (z) = z + b for some b ∈ ℝ \ {0}. Thus, limn→∞ φ ∘ T n ∘ φ−1 (z) = limn→∞ (z + nb) = ∞ for all z ∈ ℍ. However, this implies that the sequence (T n (ζ ))∞ n=1 converges to 𝜕V for every ζ ∈ V , which contradicts the assumption that the sequence (T n |V )∞ n=1 has a nonconstant −1 limit. So, φ ∘ T ∘ φ cannot have only one fixed point in 𝜕ℍ. Assume then that φ ∘ T ∘ φ−1 has exactly two fixed points in 𝜕ℍ. In that case, conjugating φ ∘ T ∘ φ−1 by a Möbius transformation that preserves ℍ, we may assume without loss of generality that these two fixed points are 0 and ∞. But then φ ∘ T ∘ φ−1 (z) = az for some a > 0 because φ ∘ T ∘ φ−1 preserves ℍ. Moreover, a ≠ 1 because φ ∘ T ∘ φ−1 has no fixed point in ℍ by assumption. If a > 1, then we once again get limn→∞ φ ∘ T n ∘ φ−1 (z) = limn→∞ an z = ∞ for all z ∈ ℍ, which leads to a contradiction in precisely the same way as in the previous case. A similar contradiction is reached if 0 < a < 1. In conclusion, the map φ ∘ T ∘ φ−1 : ℍ → ℍ must have a fixed point and, therefore, T : V → V has a fixed point, say w. Let ψ : 𝔻 → V be a conformal isomorphism such that ψ(0) = w. Then ψ−1 ∘ T ∘ ψ(0) = 0. It follows from the classification of conformal automorphisms of the unit disk 𝔻 that ψ−1 ∘ T ∘ ψ : 𝔻 → 𝔻 is a rotation through some angle α, i. e., ψ−1 ∘ T ∘ ψ(z) = e2πiα z. Observe that α must be irrational, as otherwise w ∈ V ⊆ ℱ (T) would be a rationally indifferent periodic point of T and would thereby belong to 𝒥 (T), an impossibility. So V is a Siegel disk, i. e., part (c) of Theorem 24.4.1 holds in this case. Finally, we consider the remaining case of V being conformally isomorphic to an annulus Ar (0) for some 0 ≤ r < 1. Let us first rule out the possibility that r = 0. Indeed, a conformal isomorphism φ : Ar (0) → V would then extend continuously to a map φ : 𝔻 → V ∪ {φ(0)} ⊆ V and φ(0) ∈ 𝜕V would be an isolated point of 𝒥 (T), contrary to the perfectness of Julia sets. So, r > 0. Note that T|V is not of finite order, i. e., there is no k ∈ ℕ such that T k |V = Id. Otherwise, we would have that

24.4 The structure of the Fatou set

� 1021

k T k = Idℂ ̂ , whence 1 = deg(Idℂ ̂ ) = (deg(T)) > 1. Thus, T|V is of infinite order. By Corollary 24.4.5, we conclude that V is a Herman ring, i. e., part (d) of Theorem 24.4.1 holds in this case.

Lemma 24.4.6 affirms that in the case where the sequence (T n |V )∞ n=0 has a nonconstant limit function, the forward T-invariant component V is either a Siegel disk or a Herman ring. What can be said when the sequence (T n |V )∞ n=0 has only constant limit functions? We need to split this case into several subcases, starting with the following. Lemma 24.4.7. Let V be a forward T-invariant connected component of ℱ (T). If the sequence (T n |V )∞ n=0 has a constant limit function whose value belongs to V , then part (a) of Theorem 24.4.1 holds. Proof. The hypothesis means that there exists an increasing sequence (nk )∞ k=1 of positive nk ∞ integers such that the sequence (T |V )k=1 converges to some point w ∈ V uniformly on compact subsets of V . But then, for every z ∈ V we know that T(z) ∈ V , and hence w = lim T nk (T(z)) = T( lim T nk (z)) = T(z) k→∞

k→∞

(24.101)

and limk→∞ (T nk )′ (z) = 0 uniformly on compact subsets of V . In particular, w = T(w) and limk→∞ (T ′ (w))nk = 0, which implies that |T ′ (w)| < 1. In other words, w is an attracting fixed point of T and w ∈ V ∩A∗ (w), where A∗ (w) is the immediate basin of attraction of w. Hence, V and A∗ (w) are connected components of the Fatou set ℱ (T) with nonempty intersection. Thus, V = A∗ (w). The remaining case to consider is when the sequence (T n |V )∞ n=0 has only constant limit functions whose values belong to the boundary 𝜕V = V \ V of the domain V . To deal with this case, we shall first prove another helpful lemma. Lemma 24.4.8. Let V be a forward T-invariant connected component of ℱ (T). If all the limit functions of the sequence (T n |V )∞ n=0 are constant and their values belong to 𝜕V , then T has a fixed point w ∈ 𝜕V such that |T ′ (w)| = 1 and the sequence (T n |V )∞ n=0 converges to w uniformly on compact subsets of V . Proof. Suppose that (nk )∞ k=1 is an increasing sequence of positive integers such that the nk ∞ sequence (T |V )k=1 converges to some point w ∈ 𝜕V uniformly on compact subsets of V . The same calculation as in (24.101) shows that T(z) = w for all z ∈ V and T(w) = w thanks to the continuity of T. Fix R > 0 so small that B(w, 2R) contains no fixed point of T other than w. Fix also an arbitrary point ξ ∈ V . The set A := {n ≥ 0 | T n (ξ) ∈ B(w, R)}

(24.102)

∞ is infinite as it contains a tail of the sequence (nk )∞ k=1 . If (mk )k=1 is any strictly increasing mk ∞ sequence of elements in A such that the sequence (T |V )k=1 converges uniformly on compact subsets of V , then the limit of (T mk |V )∞ k=1 is some constant function ζ , and as

1022 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets above, T(ζ ) = ζ . It follows from (24.102) that ρ(ζ , w) ≤ R < 2R, where ρ is the spherical ̂ and, therefore, ζ = w. If ℕ \ A is infinite, then there is a strictly increasing metric on ℂ sequence (sk )∞ k=1 in ℕ \ A such that sk − 1 ∈ A for all k ∈ ℕ. Passing to a subsequence if necessary, we may assume without loss of generality that the sequence (T sk |V )∞ k=1 converges uniformly on compact subsets of V . By hypothesis, its limit is a constant function c ∈ 𝜕V . By (24.102), we know that ρ(c, w) ≥ R and, in particular, c ≠ w. On the other hand, as sk − 1 ∈ A, with any z ∈ V we obtain that c = lim T sk (z) = T( lim T sk −1 (z)) = T(w) = w. k→∞

k→∞

This contradiction demonstrates that ℕ \ A must be finite and, therefore, the sequence (T n |V )∞ n=0 converges to w uniformly on compact subsets of V . This proves the second assertion of the lemma. To prove the first assertion, note that if |T ′ (w)| < 1, then V ∪ A∗ (w) would be a connected component of the Fatou set ℱ (T) properly containing V . But this would contradict the fact that V is a connected component of ℱ (T). Hence, |T ′ (w)| ≥ 1. If |T ′ (w)| > 1, then for every point z ∈ V we would deduce from the convergence of (T n (z))∞ n=1 to w that T n (z) = w for some n ∈ ℕ, with n depending on z. But then T −n (w) would be uncountable for some n ∈ ℕ, yielding T n ≡ w. This contradiction implies that |T ′ (w)| = 1. The final piece of the puzzle is contained in the following lemma. Lemma 24.4.9. Let V be a forward T-invariant connected component of ℱ (T). If the sequence (T n |V )∞ n=0 has a constant limit function whose value w ∈ 𝜕V is a fixed point for T, ′ then T (w) = 1. Proof. We may assume without loss of generality that w = 0. Let Δ ⊆ Δ ⊆ V be an ̂ and Δ ∪ T(Δ) ⊆ V , there open connected set. Since V is an open connected subset of ℂ exists a smooth arc γ ⊆ V joining Δ and T(Δ). Taking W ⊆ W ⊆ V an open connected neighborhood of γ, the set D := Δ ∪ W ∪ T(Δ) satisfies D ⊆ V and has the following iterative properties for every n ≥ 0: (a) The set T n (D) ⊆ V is open and connected. (b) T n+1 (D) ∩ T n (D) ≠ 0. (c) 0 ∉ T n (D). For every n ≥ 0, let ∞

Dn := ⋃ T k (D). k=n

In light of (a), (b), (c) and Lemma 24.4.8, the sets (Dn )∞ n=0 have the following properties:

24.4 The structure of the Fatou set

(d) (e) (f) (g)

�

1023

The set Dn ⊆ V is open and connected. limn→∞ max{ρ(z, 0) : z ∈ Dn } = 0. T(Dn ) = Dn+1 ⊆ Dn . 0 ∉ Dn .

Since |T ′ (0)| = 1 ≠ 0 by Lemma 24.4.8, there exists s > 0 such that T|B(0,s) is an injection. By (e), we can fix k ≥ 0 such that Dk ⊆ B(0, s) ⊆ ℂ. From this and (f), we see that all the iterates T n |D , n ≥ 0, are injective. Choose arbitrarily ξ̂ ∈ Dk and per (e)+(g) fix q > k k ̂ by such that ξ̂ ∉ Dq . For every n ∈ ℕ, define the meromorphic function ĝn : Dq → ℂ ĝn (z) :=

T n (z) . ̂ T n (ξ)

n The sequence of functions (ĝn )∞ n=1 does not take the value ∞, as T (z) ∈ Dq ⊆ B(0, s) ⊆ ℂ ̂ ∈ D ⊆ V ⊆ ℂ \ {0}. It also omits 0, as T n (z) ∈ D ⊆ V ⊆ ℂ \ {0} and T n (ξ) ̂ ∈ ℂ. and T n (ξ) k

q

And it avoids 1, as T n |Dq is one-to-one. Thus, by Montel’s theorem, the sequence of func′ tions (ĝn )∞ n=1 is normal. As T(0) = 0 and λ := T (0) ≠ 0, the Taylor series expansion of T around 0 can be written as T(z) = λz + azp+1 + O(zp+2 ), with p ∈ ℕ and a ∈ ℂ \ {0}. Then T(z) = λ + azp + O(zp+1 ). z

Therefore, lim

z→0

T(z) =λ z

uniformly on B(0, s)

(24.103)

provided that s > 0 is sufficiently small. Consequently, T j (z) = λj z→0 z lim

uniformly on Dq

(24.104)

̂ ∈ D . For every n ∈ ℕ, define the meromorphic function for every j ∈ ℕ. Set ξ := T q−k (ξ) q ̂ by gn : Dq → ℂ gn (z) :=

T n (z) . T n (ξ)

Then gn (z) =

̂ ̂ T n (z) T n (ξ) T n (ξ) ⋅ n = ĝn (z) ⋅ . ̂ T (ξ) ̂ T n (ξ) T q−k (T n (ξ))

1024 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets It follows from (24.104) that lim

n→∞

gn (z) = λk−q ĝn (z)

uniformly on Dq .

∞ ̂ ∞ So, the sequence (gn )∞ n=1 is, like (gn )n=1 , normal. Fix a strictly increasing sequence (nj )j=1 ∞ ̂ uniformly on compact such that (gnj )j=1 converges to some analytic function g : Dq → ℂ subsets of Dq . Then

gnj +1 (z) = = =

n T nj +1 (z) T(T j (z)) = T nj +1 (ξ) T(T nj (ξ))

λT nj (z) + a(T nj (z))p+1 + O((T nj (z))p+2 )

λT nj (ξ) + a(T nj (ξ))p+1 + O((T nj (ξ))p+2 )

λT nj (z)[1 + aλ−1 (T nj (z))p + O((T nj (z))p+1 )]

λT nj (ξ)[1 + aλ−1 (T nj (ξ))p + O((T nj (ξ))p+1 )]

= gnj (z) ⋅

1 + aλ−1 (T nj (z))p + O((T nj (z))p+1 )

1 + aλ−1 (T nj (ξ))p + O((T nj (ξ))p+1 ) → g(z) as j → ∞ while, by (24.103), gnj +1 (z) =

T nj (T(z)) nj

T (T(ξ))

=

→ g(T(z)) ⋅ λ−1

T nj (T(z)) T nj (ξ) ⋅ nj T (ξ) T(T nj (ξ)) as j → ∞.

Reconciling these two limit functions yields g ∘ T = λg

on Dq .

(24.105)

∞ As the functions (T n |Dq )∞ n=1 are univalent, so are the functions (gn )n=1 . It follows from ̂ is univalent or is constant. Hurwitz’s theorem (Theorem A.3.8) that either g : Dq → ℂ

Assume first that g is univalent. Iterating (24.105) gives g ∘ T n = λn g,

∀n ∈ ℕ.

(24.106)

In particular, g ∘ T n (ξ) = λn g(ξ) = λn ,

∀n ∈ ℕ.

If λ were a root of unity, say λp = 1 for some p ∈ ℕ, then g(T pn (ξ)) = 1 for all n ∈ ℕ, contrary to the facts that the set {T pn (ξ) : n ∈ ℕ} is infinite and the function g is oneto-one. So, λ is not a root of unity. Consequently, the set {λn : n ∈ ℕ} is dense in the unit

24.4 The structure of the Fatou set

�

1025

circle 𝕊1 . Since T n |V tends to 0 uniformly on compact subsets of V and 0 ∉ Dq , there exists j ∈ ℕ so large that T n (T q (D)) ∩ T q (D) = 0,

∀n ≥ j.

(24.107)

Let N ≥ j be such that λN is close enough to 1 that λN g(T q (D)) ∩ g(T q (D)) ≠ 0. It follows from this and (24.106) that g(T N (T q (D))) ∩ g(T q (D)) ≠ 0. ̂ is not one-to-one. This contraIn conjunction with (24.107), this implies that g : Dq → ℂ ̂ diction rules out the case of g : Dq → ℂ being univalent. As the only other outcome of ̂ must be constant. Since g(ξ) = 1, we must Hurwitz’s theorem, the function g : Dq → ℂ have g ≡ 1. But then (24.105) implies that λ = 1. We are finally in a position to establish Theorem 24.4.1. Proof of Theorem 24.4.1. Three of the possibilities, namely (a), (c) and (d), have already been ascertained through Lemmas 24.4.6–24.4.7. The remaining case to consider is when the sequence (T n |V )∞ n=0 has only constant limit functions whose values belong to 𝜕V . In light of Lemmas 24.4.8–24.4.9, the map T has a simple rationally indifferent fixed point w ∈ 𝜕V and V ∩ A∗j (w) ≠ 0 for some j ∈ {1, . . . , p(w)}. Since both V and A∗j (w) are connected components of ℱ (T), it follows that V = A∗j (w) and part (b) describes the fourth possibility.

24.4.2 Periodic components of the Fatou set As a direct consequence of Theorem 24.4.1, we obtain an analogous statement for the periodic components of the Fatou set. ̂→ Theorem 24.4.10 (Classification of the periodic components of the Fatou set). Let T : ℂ ̂ be a rational function with deg(T) ≥ 2 and let V be a periodic component of the Fatou ℂ set ℱ (T), i. e., such that T q (V ) ⊆ V for some q ∈ ℕ. Then T q (V ) = V and one of the following four alternatives holds: (a) V is the immediate basin of attraction to some attracting periodic point of period q for T. (b) V is the immediate basin of attraction to some rationally indifferent periodic point of period q lying in 𝜕V .

1026 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets (c) V is a topological disk (i. e., is homeomorphic to the open unit disk 𝔻), and the map T q |V : V → V is analytically conjugate to an irrational rotation R2πα : 𝔻 → 𝔻. The domain V is then referred to as a Siegel disk. (d) There exists an annulus Ar (0) with r ∈ (0, 1) such that the map T q |V : V → V is analytically conjugate to an irrational rotation R2πα : Ar (0) → Ar (0). The domain V is then called a Herman ring. Each such type of periodic component exists for some rational functions. The existence of rational functions having type (a) or type (b) components is directly evidenced through the examination of the fixed point 0 for the one-parameter family of quadratic polynomials z 󳨃→ z2 +λz. The existence of Siegel disks and Herman rings is subtler. As we have already mentioned in Subsection 24.1.2.2, Carl Ludwig Siegel [121] demonstrated the existence of such disks (which is also of profound importance in the KAM theory); the proof can also be found in [21]. The study of Siegel disks, particularly the nature of the rotational numbers α for which they exist, has been the topic of extensive research led by many mathematicians. With Herman rings, the situation is even more delicate. They cannot exist for polynomials (see Theorem 24.7.3 below). Their existence was shown by Michel Herman [58] in 1984 only. His argument is based on Arnold’s theorem. A proof due to Shishikura [119] can also be found in [21]. We now delimit the connectivity of the periodic components of the Fatou set. ̂→ℂ ̂ is a rational function with deg(T) ≥ 2, then each periodic Theorem 24.4.11. If T : ℂ connected component of ℱ (T) is either simply connected, doubly connected (in which case it is a Herman ring), or infinitely connected. Proof. Let V be a periodic component of the Fatou set ℱ (T), i. e., a component such that T q (V ) ⊆ V for some q ∈ ℕ. Replacing T q by T, we may assume that T(V ) ⊆ V . Suppose that V is finitely connected and denote its connectivity by k. It then follows from formula (22.23) that the Euler characteristic χ(V ) is finite; more precisely χ(V ) = 2 − k. In view of Lemma 24.1.7, we may apply the general Riemann–Hurwitz formula (Theorem 22.5.1) to the map T|V : V → V to get χ(V ) + δT (V ) = deg(T|V ) ⋅ χ(V ). Equivalently, (2 − k) + δT (V ) = deg(T|V ) ⋅ (2 − k).

(24.108)

If it turned out that 3 ≤ k < ∞, then (24.108) and the fact that δT (V ) ≥ 0 would yield deg(T|V ) ⋅ (2 − k) ≥ (2 − k). So deg(T|V ) = 1. But then δT (V ) = 0 per (24.108). This would mean that T has no critical point in V . But according to Theorems 24.1.20 and 24.2.22, this eliminates possibilities (a) and (b) in Theorem 24.4.10. And the remaining possibilities (c)

24.4 The structure of the Fatou set

� 1027

and (d) imply that k = 1 and k = 2. This contradiction rules out the possibility that 3 ≤ k < ∞. If k = 2, then V obviously cannot be a Siegel disk. Moreover, (24.108) reduces to δT (V ) = 0, i. e., T has no critical point in V . Therefore, by virtue of Theorems 24.1.20 and 24.2.22, V cannot be the immediate basin of attraction of an attracting periodic point or of a rationally indifferent periodic point. It thus follows from Theorem 24.4.10 that V must be a Herman ring. The only other possibilities are k = 1 and k = ∞. We now state without proof Sullivan’s nonwandering theorem ([131]) which, along with the classification theorem 24.4.10, gives a complete description of the structure of Fatou components. Sullivan’s nonwandering theorem constituted a breakthrough in the development of the theory of iteration of rational functions. In fact, it brought the theory to its modern state. Proofs of Sullivan’s nonwandering theorem and its various generalizations, particularly to the dynamics of transcendental meromorphic functions, can be found in many sources, one of the clearest appearing in Steinmetz [123]. ̂ →ℂ ̂ be a rational funcTheorem 24.4.12 (Sullivan’s nonwandering theorem). Let T : ℂ tion with deg(T) ≥ 2. Each connected component V of the Fatou set ℱ (T) is eventually periodic, i. e., there are integers 0 ≤ p < q (depending on V ) such that T q−p (T p (V )) = T p (V ). sult.

As an immediate consequence of the last two theorems we get the following re-

̂→ℂ ̂ be a rational function with deg(T) ≥ 2. If K is a compact Corollary 24.4.13. Let T : ℂ subset of the Fatou set ℱ (T) which is disjoint from the basins of attraction of all rationally indifferent periodic points, then ∞

⋃ T n (K) ⊆ ℱ (T)

n=0

and ∞

𝒥 (T) ∩ ⋃ T n (Crit(T) ∩ ℱ (T)) ⊆ Ω(T), n=0

where Ω(T) is the (finite) set of all rationally indifferent periodic points of T.

24.4.3 The postcritical set Recall that the finite set ̂ 󵄨󵄨󵄨 T ′ (z) = 0} Crit(T) := {z ∈ ℂ 󵄨

1028 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets is naturally called the critical set of T while its image T(Crit(T)) is the set of critical values of T. These two sets play important roles for several reasons. One of those reasons is the subject of the following remark. Remark 24.4.14. Every point z outside of T(Crit(T)) admits a neighborhood Nz on which T −1 consists of finitely many (more precisely, deg(T) many) holomorphic inverse branches of T. For instance, Nz = B(z, δz ) where 0 < δz ≤ dist(z, T(Crit(T))) (see Corollary 22.5.6). Dynamically, we are spontaneously led to examine the union of all iterates of the critical set. ̂→ℂ ̂ be a rational function. The countable set Definition 24.4.15. Let T : ℂ ∞

∞

n=1

n=0

PC(T) := ⋃ T n (Crit(T)) = ⋃ T n (T(Crit(T))) is instinctively named the postcritical set of T. Given any N ∈ ℕ, we readily notice that ∞

∞

n=1

k=1

PC(T N ) = ⋃ T Nn (Crit(T N )) = ⋃ T k (Crit(T)) = PC(T).

(24.109)

For a slew of reasons, the postcritical set plays an important role in complex dynamics. The following remark constitutes one of them. Remark 24.4.16. (a) Every point z outside of PC(T) is such that (T n )−1 = T −n consists of finitely many (more precisely, deg(T n ) = (deg(T))n many) holomorphic inverse branches of T n on some neighborhood of z, and this for every n ∈ ℕ. That neighborhood may −k n n depend on n. Given that Crit(T n ) = ⋃n−1 k=0 T (Crit(T)), and thus T (Crit(T )) = n j ⋃j=1 T (Crit(T)) ⊆ PC(T), we can take Nz,n = B(z, δz,n ) where 0 < δz,n ≤ dist(z, T n (Crit(T n ))). (b) Every point z outside of PC(T) is such that every (T n )−1 = T −n consists of finitely many (more precisely, deg(T n ) = (deg(T))n many) holomorphic inverse branches of T n on some neighborhood of z. This time, though, a neighborhood independent of n exists. Set Nz = B(z, δz ) where 0 < δz ≤ dist(z, PC(T)) (see Corollary 22.5.6). It is easy to see that PC(T) = [PC(T) ∩ ℱ (T)] ∪ [PC(T) ∩ 𝒥 (T)] and ∞

PC(T) ∩ ℱ (T) = ⋃ T n (Crit(T) ∩ ℱ (T)) and n=1

∞

PC(T) ∩ 𝒥 (T) = ⋃ T n (Crit(T) ∩ 𝒥 (T)).

Obviously, PC(T) = PC(T) ∩ ℱ (T) ∪ PC(T) ∩ 𝒥 (T). Note also that

n=1

24.5 Cremer points, boundary of Siegel disks and Herman rings

� 1029

PC(T) = PC(T) ∪ ω(Crit(T)). Moreover, Int(PC(T)) = 0 since PC(T) is countable. The second part of Corollary 24.4.13 thereby reduces to the following. ̂→ℂ ̂ is a rational function with deg(T) ≥ 2, then Corollary 24.4.17. If T : ℂ 𝒥 (T) ∩ PC(T) ∩ ℱ (T) = 𝒥 (T) ∩ 𝜕(PC(T) ∩ ℱ (T)) ⊆ Ω(T).

Finally, the sets PC(T), PC(T) ∩ ℱ (T) and PC(T) ∩ 𝒥 (T) are forward T-invariant. So are their closures, due to the continuity of T.

24.5 Cremer points, boundary of Siegel disks and Herman rings In this section, we prove that the boundary of each Siegel disk and Herman ring is contained in the closure of the forward orbit of critical points. We will thereafter prove the same for Cremer points. ̂ →ℂ ̂ be a rational function with deg(T) ≥ 2. If G is a Siegel Theorem 24.5.1. Let T : ℂ disk or a Herman ring for T, then 𝜕G ⊆ ω(Crit(T) ∩ 𝒥 (T)). Proof. Seeking a contradiction, suppose that 𝜕G ⊈ ω(Crit(T) ∩ 𝒥 (T)). Pick a point y ∈ 𝜕G \ ω(Crit(T) ∩ 𝒥 (T)). Then there is δ > 0 such that B(y, 2δ) ∩ PC(T) ∩ 𝒥 (T) ⊆ {y}.

(24.110)

Since 𝜕G consists either of one (if G is a Siegel disk) or two (if G is a Herman ring) connected components, none of which is a singleton, the boundary 𝜕G is perfect. Therefore, there exists a point ξ ∈ (𝜕G ∩ B(y, δ)) \ ({y} ∪ Ω(T)), where Ω(T) is the (finite) set of all rationally indifferent periodic points of T. It then ensues from (24.110), Corollary 24.4.17, and the fact that 𝜕G ⊆ 𝒥 (T), that there exists κ > 0 such that B(ξ, 3κ) ∩ PC(T) = 0. Pick a point w ∈ B(ξ, κ) ∩ G.

(24.111)

1030 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets Then B(w, 2κ) ∩ PC(T) = 0.

(24.112)

Let Δ := 𝔻 = B(0, 1) if G is a Siegel disk and Δ := A(0; r, 1) = Ar (0) with r ∈ (0, 1) coming from Theorem 24.4.10(d) if G is a Herman ring. Let H:Δ→G be the analytic conjugacy arising from Theorem 24.4.10(c)/(d) in the Siegel/Herman case, respectively. Let F := H({z ∈ Δ : |z| = |H −1 (w)|}).

(24.113)

Note that F is a compact set (homeomorphic to a circle) and F ⊆ G. Since w ∈ G, for every n ∈ ℕ the intersection G ∩ T −n (w) must be a singleton (otherwise, this would contradict Theorem 24.4.10(c)/(d)). Denoting its only element by wn , we can readily check that wn ∈ F.

(24.114)

By (24.112) and Remark 24.4.16, for every n ∈ ℕ there is a unique holomorphic branch ̂ Tw−nn : B(w, 2κ) → ℂ

(24.115)

of T −n sending w to wn . As ξ ∈ 𝜕G ⊆ 𝒥 (T), we know that Tw−nn (ξ) ∈ 𝒥 (T) for every n ≥ 0. Thus, Tw−nn (ξ) ∉ G for all n ≥ 0. Using (24.111) and (24.114), we conclude that ̂ \ G) > 0, diam(Tw−nn (B(ξ, κ))) ≥ dist(wn , Tw−nn (ξ)) ≥ dist(F, ℂ

∀n ≥ 0.

Consequently, ̂ \ G) > 0, lim inf diam(Tw−nn (B(ξ, κ))) ≥ dist(F, ℂ n→∞

contrary to Lemma 24.1.13. ̂→ℂ ̂ be a rational function with deg(T) ≥ 2. If ξ is a Cremer Theorem 24.5.2. Let T : ℂ periodic point of T, then ξ ∈ PC(T) ∩ 𝒥 (T).

24.6 Continuity of Julia sets

� 1031

Proof. Let p ∈ ℕ be the prime period of ξ. Seeking a contradiction, suppose that ξ ∉ PC(T) ∩ 𝒥 (T). As ξ ∈ 𝒥 (T)\Ω(T), combining the above assumption with Corollary 24.4.17 allows us to infer that ξ ∉ PC(T). By Remark 24.4.16, there exists r > 0 (e. g., choose r = 21 dist(ξ, PC(T))) such that for every n ∈ ℕ there exists a unique holomorphic branch Tξ

−pn

̂ : B(ξ, 2r) → ℂ

(24.116)

of T −pn that fixes ξ. But because |(T pn )′ (ξ)| = 1 for all n ∈ ℕ and ξ ∈ 𝒥 (T), this contradicts Lemma 24.1.13.

24.6 Continuity of Julia sets If X is a set and (Y , ρ) is a bounded metric space, we consider the metric space (Y X , ρ∞ ) of all functions from X to Y with the metric of uniform convergence ρ∞ given by ρ∞ (f , g) := sup{ρ(f (x), g(x)) : x ∈ X}.

(24.117)

If ρ and d are two bounded metrics on Y , then the topologies induced by ρ∞ and d∞ on Y X are the same whenever ρ and d are uniformly equivalent, i. e., when the identity map IdY : (Y , d) → (Y , ρ) is uniformly bicontinuous. In particular, if Y is a compact metrizable space, then Y X is (canonically) topologized by any metric on Y compatible ̂ ̂ℂ with the topology of Y . In this section, ℂ is understood as a compact metrizable space ̂ topologized in that way. Moreover, we consider the set ℛ of all rational functions on ℂ ̂ ℂ ̂ . We also consider the metric space 𝒦(ℂ) ̂ of all compact as a (metrizable) subspace of ℂ ̂ endowed with the Hausdorff metric ρH induced by the spherical metric subsets of ℂ ̂ We further note that any metric on ℂ ̂ compatible with its standard topology ρ on ℂ. ̂ induces the same topology as ρH on 𝒦(ℂ). ̂ is lower semicontinuous. More Theorem 24.6.1. The function ℛ ∋ T 󳨃󳨀→ 𝒥 (T) ∈ 𝒦(ℂ) precisely, for every S ∈ ℛ and every ε > 0 there exists an open neighborhood U of S in ℛ such that T ∈U

󳨐⇒

𝒥 (S) ⊆ Bρ (𝒥 (T), ε).

In addition, if the Fatou set ℱ (S) consists only of basins of attraction of attracting periodic ̂ points, then S is a continuity point of the function ℛ ∋ T 󳨃→ 𝒥 (T) ∈ 𝒦(ℂ). Proof. Concerning the lower semicontinuity, it follows from Theorem 24.3.4 that there is a finite subset E of the set of repelling periodic points of S in ℂ such that ⋃ Bρ (w, ε/4) ⊇ 𝒥 (S).

w∈E

Let p ∈ ℕ be the product of the prime periods of all points in E. Then

(24.118)

1032 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets S p (w) = w

and

󵄨󵄨 p ′ 󵄨󵄨 󵄨󵄨(S ) (w)󵄨󵄨 > λ

(24.119)

for all w ∈ E and some λ ∈ (1, ∞). Since the function ℛ ∋ T 󳨃→ |(T p )′ (w)| ∈ ℝ is continuous for each w ∈ E, an application of Hurwitz’s theorem (Theorem A.3.8) confirms the existence of δ > 0 and θ ∈ (0, 3ε/4] such that T ∈ Bρ∞ (S, δ)

󳨐⇒

󵄨󵄨 p ′ 󵄨󵄨 √ 󵄨󵄨(T ) (z)󵄨󵄨 > λ,

∀z ∈ ⋃ B(w, 4θ) w∈E

(24.120)

and such that for every w ∈ E the map T p |B(w,4θ) is one-to-one and satisfies 󵄨󵄨 p 󵄨 󵄨󵄨T (w) − w󵄨󵄨󵄨 < θ(√λ − 1).

(24.121)

For each w ∈ E, let Tw−p := (T p |B(w,2θ) )

−1

: T p (B(w, 2θ)) → B(w, 2θ).

For each w ∈ E, we infer from (24.120) that 󵄨󵄨 −p ′ 󵄨󵄨 √ −1 󵄨󵄨(Tw ) (z)󵄨󵄨 < ( λ) < 1,

∀z ∈ B(w, θ).

It ensues from the mean value inequality that Tw−p (B(w, θ)) ⊆ B(w, θ),

∀w ∈ E.

The Banach fixed-point theorem then asserts that for every w ∈ E there is ξw ∈ B(w, θ) −p such that Tw (ξw ) = ξw . Hence, T p (ξw ) = ξw

and

󵄨󵄨 p ′ 󵄨 󵄨󵄨(T ) (ξw )󵄨󵄨󵄨 > √λ > 1.

Therefore, ξw ∈ 𝒥 (T). If z ∈ 𝒥 (S), then in view of (24.118) there is w ∈ E such that ρ(z, w) < ε/4. Consequently, ρ(z, 𝒥 (T)) ≤ ρ(z, ξw ) ≤ ρ(z, w) + ρ(w, ξw ) < ε/4 + θ ≤ ε, and the lower semicontinuity holds. Regarding the continuity points, assume that ε > 0 is so small that Bρ (𝒥 (S), 4ε) ̂ ρ (𝒥 (S), ε) is a compact set contains no attracting periodic points of S. Given that ℂ\B ̂ ρ (𝒥 (S), ε) is contained in the union of the basins of attraction of and, by hypothesis, ℂ\B all (finitely many) attracting periodic points of S, there exists q ∈ ℕ such that ̂ \ Bρ (𝒥 (S), ε)) ⊆ ℂ ̂ \ Bρ (𝒥 (S), 2ε). S q (ℂ Hence, there exists η > 0 such that

24.6 Continuity of Julia sets

T ∈ Bρ∞ (S, η)

󳨐⇒

� 1033

̂ \ Bρ (𝒥 (S), ε)) ⊆ ℂ ̂ \ Bρ (𝒥 (S), ε). T q (ℂ

̂ \ Bρ (𝒥 (S), ε)) ⊆ ℂ ̂ \ Bρ (𝒥 (S), ε) for every n ≥ 0. Thus, the family Therefore, T qn (ℂ ̂ of maps {T qn }∞ n=0 , restricted to the open set ℂ \ Bρ (𝒥 (S), ε), is normal. It follows from Proposition 24.3.8 that

̂ \ Bρ (𝒥 (S), ε) ⊆ ℱ (T). ℂ Equivalently, 𝒥 (T) ⊆ Bρ (𝒥 (S), ε).

If T ∈ Bρ∞ (S, min{η, δ}), where δ > 0 arises from the lower semicontinuity part of the proof, then 𝒥 (S) ⊆ Bρ (𝒥 (T), ε)

and

𝒥 (T) ⊆ Bρ (𝒥 (S), ε).

Consequently, ρH (𝒥 (S), 𝒥 (T)) ≤ ε. This proves that S is a continuity point. In order to make this theorem more concrete and effective, we now show a result which, together with Theorem 24.6.1, entails (semi)continuity of Julia sets in terms of their dependence on the coefficients of the underlying rational functions. Fix an integer d ≥ 2 and consider the space 2d+2

ℛ𝒫 d := {a = (a0 , a1 , . . . , ad , ad+1 , . . . , a2d , a2d+1 ) ∈ ℂ

2d+1

: ∑ |ak | ≠ 0}, k=d+1

with the standard metric inherited from ℂ2d+2 . Consider the map ℛ𝒫 d ∋ a 󳨃󳨀→ Ra ∈ ℛ defined by Ra (z) :=

∑dj=0 aj zj

j−(d+1) ∑2d+1 j=d+1 aj z

̂ ∈ ℂ.

Lemma 24.6.2. For every integer d ≥ 2, the function R : ℛ𝒫 d → ℛ is continuous. ̂ there exist δz > 0 and rz > 0 such that Proof. Fix b ∈ ℛ𝒫 d . Fix ε > 0. For every z ∈ ℂ ρ(Ra (w), Rb (w)) < ε ̂ is compact and for all a ∈ Bℛ𝒫 d (b, δz ) and all w ∈ Bρ (z, rz ). Since the space ℂ ̂ there is a finite set D ⊆ ℂ ̂ such that {Bρ (z, rz )}z∈ℂ ̂ is an open cover of ℂ,

1034 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets ̂ ⋃ Bρ (z, rz ) = ℂ.

z∈D

Let δ := min{δz : z ∈ D}. We then have ρ(Ra (ξ), Rb (ξ)) < ε,

̂ ∀ξ ∈ ℂ.

So ρ∞ (Ra , Rb ) ≤ ε. More about continuity of Julia sets can be found in [92], for instance.

24.7 Polynomials In this section, we prove selected facts about the Fatou and Julia sets of polynomials. Most proofs provided here are inspired by Beardon’s approach in [8]. The first easy but fundamental fact about polynomials P concerns their common fixed point: ∞. Its basin ̂ : limn→∞ Pn (z) = ∞}. Its immediate basin of attraction of attraction is A(∞) = {z ∈ ℂ A∗ (∞) is the connected component of A(∞) that comprises ∞. In fact, these two basins coincide. Moreover, the boundary of that basin is the Julia set. ̂→ℂ ̂ is a polynomial with deg(P) ≥ 2, then A(∞) = A∗ (∞), i. e., Theorem 24.7.1. If P : ℂ −1 ∗ ∗ P (A (∞)) = A (∞). Moreover, 𝒥 (P) = 𝜕A∗ (∞). Proof. Since P−1 ({∞}) = {∞}, the equalities A(∞) = A∗ (∞) and P−1 (A∗ (∞)) = A∗ (∞)

follow from Theorem 24.1.8. So, A∗ (∞) = P−1 (A∗ (∞)). On one hand, the continuity of P and the definition of closure imply that A∗ (∞) = P−1 (A∗ (∞)) ⊆ P−1 (A∗ (∞)).

(24.122)

P−1 (A∗ (∞)) ⊆ P−1 (A∗ (∞)) = A∗ (∞).

(24.123)

On the other hand,

̂\ Let z ∈ 𝜕A∗ (∞). Suppose for a contradiction that P−1 (z) \ A∗ (∞) ≠ 0. Then z ∈ P(ℂ ∗ ∗ ∗ ∗ ̂ \ A (∞)) ∩ A (∞) ≠ 0. So (ℂ ̂\ A (∞)) ∩ 𝜕A (∞). The openness of P implies that P(ℂ A∗ (∞)) ∩ P−1 (A∗ (∞)) ≠ 0. This contradicts (24.123), and thus we deduce that P−1 (z) ⊆ A∗ (∞) for any z ∈ 𝜕A∗ (∞). This means that P−1 (𝜕A∗ (∞)) ⊆ A∗ (∞).

(24.124)

24.7 Polynomials

� 1035

Altogether (24.122)–(24.124) yield P−1 (A∗ (∞)) = P−1 (A∗ (∞)) = A∗ (∞). But as A∗ (∞) = A∗ (∞) ∪ 𝜕A∗ (∞) and P−1 (𝜕A∗ (∞)) ∩ A∗ (∞) = 0, we conclude that P−1 (𝜕A∗ (∞)) ⊆ 𝜕A∗ (∞). Hence, we deduce from Corollary 24.3.7 that 𝒥 (P) = 𝜕A∗ (∞). We now describe the connectivity of the (immediate) basin of attraction of ∞. ̂→ℂ ̂ is a polynomial with deg(P) ≥ 2, then: Theorem 24.7.2. If P : ℂ ∗ (a) A(∞) = A (∞) is either simply connected or infinitely connected. (b) A(∞) = A∗ (∞) is simply connected if and only if the Julia set 𝒥 (P) is connected. ̂ \ 𝒥 (P) different from A∗ (∞) (c) All (necessarily bounded) connected components of ℂ are simply connected. Proof. As ∞ is a superattracting fixed point of P, the basin A(∞) cannot be doubly connected and item (a) is an immediate consequence of Theorem 24.4.11. Since an open con̂ is simply connected if and only if its boundary is connected, item (b) nected subset of ℂ ̂ \ 𝒥 (P) different follows directly from Theorem 24.7.1. All connected components of ℂ ̂ \ A∗ (∞). But since the closure from A∗ (∞) are exactly the connected components of ℂ A∗ (∞) is connected (as A∗ (∞) is), it is a general fact from topology of the plane (see [98], e. g.) that all connected components of its complement are simply connected, meaning that item (c) holds. A straightforward repercussion of this theorem is the following fact. ̂→ℂ ̂ admits a Herman ring. Theorem 24.7.3. No polynomial P : ℂ We now fully characterize the cases where the basin A(∞) = A∗ (∞) is simply connected or infinitely connected. ̂→ℂ ̂ with deg(P) ≥ 2, the following statements Theorem 24.7.4. For a polynomial P : ℂ are equivalent: (a) The basin A(∞) = A∗ (∞) is simply connected. (b) The Julia set 𝒥 (P) is connected. (c) A(∞) = A∗ (∞) contains no finite (i. e., different from ∞) critical point of P. Proof. The equivalence of items (a) and (b) is just Theorem 24.7.2(b). Denote d = deg(P). We now show that (a) entails (c). Suppose that A∗ (∞) is simply connected. Then the Euler characteristic of that basin is χ(A∗ (∞)) = 1 and it follows from Lemma 24.1.7 that the general Riemann–Hurwitz formula (Theorem 22.5.1) applies to the restriction P : A∗ (∞) → A∗ (∞) and yields

1036 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets 1 + δP (A∗ (∞)) = d. Equivalently, δP (A∗ (∞)) = d − 1. But given that deg∞ (P) = d, this implies that P cannot have any other critical point in A∗ (∞), whence item (c) holds. We now show the converse, i. e., that (c) entails (a). Assume that ∞ is the only critical point in A(∞). Since P−1 ({∞}) = {∞} and P−1 (A(∞)) = A(∞), it ensues that for every n ∈ ℕ the polynomial Pn has no finite critical point in A(∞). Since ∞ is a (super)attracting fixed point of P, there is an open disk D centered at ∞ such that ∞ ∈ P(D) ⊆ D ⊆ A(∞). Then {∞} = P−1 ({∞}) ⊆ D ⊆ P−1 (P(D)) ⊆ P−1 (D) ⊆ P−1 (A(∞)) = A(∞) and by induction ∞ ∈ P−(n−1) (D) ⊆ P−n (D) ⊆ A(∞),

∀n ∈ ℕ.

(24.125)

Hence, D ⊆ P−n (D) for every n ∈ ℕ. Let Un be the connected component of P−n (D) containing D. Then the map Pn : Un → D is proper by Lemma 24.1.7 and deg(Pn |Un ) ≤ deg(Pn ) = d n . On the other hand, since deg∞ (P) = d and Un ⊇ D ∋ ∞, it follows that deg(Pn |Un ) ≥ d n . Thus, deg(Pn |Un ) = d n . Consequently, Un is the only connected component of P−n (D), i. e., P−n (D) is connected for every n ∈ ℕ. It ensues from Lemma 24.1.7 that the general Riemann–Hurwitz formula (Theorem 22.5.1) applies to each map Pn : P−n (D) → D to give χ(P−n (D)) + (d n − 1) = χ(P−n (D)) + δPn (P−n (D)) = d n χ(D). As D is simply connected, χ(D) = 1 and we deduce that χ(P−n (D)) = 1. So, every P−n (D) ∞ is simply connected. According to (24.125), the sequence (P−n (D))n=0 is ascending. By the definition of D and as ∞ is an attracting fixed point of P, we infer that ∞

⋃ P−n (D) = A(∞).

n=0

So, A(∞) is the union of an ascending sequence of simply connected sets and thereby is likewise simply connected, meaning that (a) holds. Theorem 24.7.4 implies that if A(∞) contains a finite critical point, then the Julia set

𝒥 (P) is not connected, i. e., it has at least two connected components. Assuming that all

the critical points of P are in A(∞), we can prove much more.

24.7 Polynomials

� 1037

̂→ℂ ̂ is a polynomial with deg(P) ≥ 2 such that Crit(P) ⊆ A(∞), Theorem 24.7.5. If P : ℂ then the Julia set 𝒥 (P) is homeomorphic to the middle-third Cantor set, i. e., it is perfect, metrizable and totally disconnected. Proof. Let D be a compact disk in A(∞) centered at ∞. Since Crit(P) ⊆ A(∞), the set ∞

A := PC(P) \ D = ⋃ Pn (Crit(P)) \ D n=1

is finite. Join each point a ∈ A with 𝜕D via a simple (i. e., without self-intersections) closed polygonal arc γa in A(∞) such that the collection {γa }a∈A consists of mutually disjoint arcs. Then for some ε > 0 small enough, the set D ∪ ⋃ B(γa , ε) a∈A

is a closed topological disk (i. e., a set homeomorphic to 𝔻) contained in A(∞). Denote its boundary by Γ and its interior by W . Then D = W ∪ Γ and, by the Jordan–Schoenflies ̂ \ (W ∪ Γ) is an open topological disk. Observe that theorem, the complement V := ℂ (V ∪ Γ) ∩ PC(P) = 0.

(24.126)

Since W ∪ Γ is a compact set contained in A(∞), there exists N ∈ ℕ such that PN (W ∪ Γ) ⊆ W

(24.127)

P−N (V ∪ Γ) ⊆ V .

(24.128)

and

We deduce from (24.126) and (24.109) that (V ∪ Γ) ∩ PC(PN ) = 0.

(24.129)

Given that 𝒥 (PN ) = 𝒥 (P), replacing PN by P we get from (24.127)–(24.129) that P(W ∪ Γ) ⊆ W ,

P−1 (V ∪ Γ) ⊆ V ,

(V ∪ Γ) ∩ PC(P) = 0,

𝒥 (P) ⊆ V .

Because 𝒥 (P) is compact, V is open and 𝒥 (P) ⊆ V , there exists δ > 0 such that 𝒥 (P) ⊆ V \ B(𝜕V , δ)

and P−1 (V ∪ Γ) ⊆ V \ B(𝜕V , δ).

(24.130)

1038 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets If U is a connected component of P−1 (V ), then it follows from Lemma 24.1.7, Corol̂ that the map P|U : U → V lary 22.5.4, and the fact that V is an open topological disk in ℂ, is a conformal homeomorphism. So, there are d := deg(P) connected components of P−1 (V ). Label them by U1 , U2 , . . . , Ud , and for each j = 1, 2, . . . , d denote φj := P|−1 Uj : V → Uj ⊆ V \ B(𝜕V , δ). The maps φj , 1 ≤ j ≤ d, form a conformal iterated function system (CIFS) Φ in the sense of Chapter 19 once we show that 󵄩󵄩 ′󵄩 󵄩󵄩(φω ) 󵄩󵄩󵄩∞ < 1 for some k ∈ ℕ and all ω ∈ {1, 2, . . . , d}k . This directly follows from Lemma 24.1.13. By construction, the CIFS Φ satisfies the Strong Separation Condition (SSC; also called the Boundary Separation Condition (BSC); see Definitions 19.7.5–19.7.6). It follows from Theorem 19.7.8 that the limit set JΦ of the system Φ is homeomorphic to the middle-third Cantor set. It thus suffices to demonstrate that 𝒥 (P) = JΦ . Since the Julia set 𝒥 (P) is completely P-invariant, it ensues from (24.130) that 𝒥 (P) ⊆

⋂ ω∈{1,2,...,d}k

φω (V \ B(𝜕V , δ))

for every k ∈ ℕ. Therefore, ∞

𝒥 (P) ⊆ ⋂

⋂

k=1 ω∈{1,2,...,d}k

φω (V \ B(𝜕V , δ)) = JΦ .

(24.131)

On the other hand, for every k ∈ ℕ and every ω ∈ {1, 2, . . . , d}k denote by xω the only fixed point of the map φω : V \ B(𝜕V , δ) → V \ B(𝜕V , δ). As ∞

JΦ = ⋃

⋃

k=1 ω∈{1,2,...,d}k

{xω }

and since the points xω are all repelling periodic points of P, we deduce from Theorem 24.1.22 that JΦ ⊆ 𝒥 (P). Along with (24.131), this means that JΦ = 𝒥 (P). So, 𝒥 (P) = JΦ is homeomorphic to the middle-third Cantor set. The following is an immediate consequence of this theorem, Theorem 24.7.4 and the fact that each quadratic polynomial has exactly one finite critical point. ̂→ℂ ̂ is a quadratic polynomial, then the following statements Corollary 24.7.6. If P : ℂ are equivalent: (a) The Julia set 𝒥 (P) is not connected.

24.8 Exercises

� 1039

(b) The Julia set 𝒥 (P) is totally disconnected, i. e., it is homeomorphic to the middle-third Cantor set. (c) The finite critical point of P belongs to A(∞). (d) If c is the finite critical point of P, then lim Pn (c) = ∞. n→∞

24.8 Exercises Exercise 24.8.1. Prove that a map from the Riemann sphere into iself is an analytic endomorphism if and only if it is a rational function. Exercise 24.8.2. Find a rational function T (even a polynomial) of degree one such that 𝒥 (T) = 0. Provide a complete description of the Julia sets of rational functions of degree one. Exercise 24.8.3. Show that any rational function T of degree at least two such that T −1 ({∞}) = {∞} is a polynomial. Exercise 24.8.4. Let a ≠ 0 and d ≥ 2. Show that the Julia set of the polynomial P(z) = azd is the circle C(0, |a|−1/(d−1) ) := 𝜕B(0, |a|−1/(d−1) ). Show also that E(P) = {0, ∞}. Exercise 24.8.5. Suppose that an holomorphic function f has a Maclaurin series expansion of the form f (z) = az + b1 zp+1 + O(z2 )

where

ab1 ≠ 0 and p ∈ ℕ.

(a) Prove that f n (z) = an z + bn zp+1 + O(zp+2 ) with (b) (c) (d) (e)

n−1

bn = an−1 b1 ∑ ak , k=0

∀n ∈ ℕ.

Infer that bn = nb1 when a = 1. Deduce that bn = 0 if and only if ap ≠ 1 and anp = 1. Show that if f n has a single fixed point at 0, then so does f . Prove that f n has at least as many fixed points at 0 as f . Moreover, if f n has more then a ≠ 1 but an = 1.

Exercise 24.8.6. This exercise concerns infinite products and series. (a) Show that log x ≤ x − 1 for all x > 0 and that the equality prevails only when x = 1. (b) Let a > 1. Prove that a(x − 1) ≤ log x for all x ∈ [1/a, 1] and that the equality prevails precisely when x = 1. (c) Let a > 1 and an ∈ [1/a, 1] for all n ∈ ℕ. Deduce from (a) and (b) that ∞

∞

∞

n=1

n=1

n=1

exp(−a ∑ (1 − an )) ≤ ∏ an ≤ exp(− ∑ (1 − an )).

1040 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets ∞ (d) Suppose that 0 < an < 1 for all n ∈ ℕ. Show that ∏∞ n=1 an > 0 ⇐⇒ ∑n=1 (1 − an ) < ∞. ∞ ∞ (e) Let bn > 1 for all n ∈ ℕ. Deduce from (d) that ∏n=1 bn < ∞ ⇐⇒ ∑n=1 (1 − b1 ) < ∞. n

Exercise 24.8.7. Prove relation (24.76). Exercise 24.8.8 (Cf. Exercise 4.2.2, p. 73 in [8]). Suppose that C is a backward T-invariant circle (i. e., such that T −1 (C) ⊆ C). Show that either 𝒥 (T) = C or 𝒥 (T) is a totally disconnected subset of C. This applies to the case when T is a Blaschke product (a finite product of Möbius transformations, each leaving the unit disc 𝔻 = B(0, 1) invariant). Exercise 24.8.9 (Cf. Exercise 6.1.5, p. 101 in [8]). This exercise shows that given any N ∈ ℕ, there is a polynomial with an attracting cycle of prime period at least N. We may assume that N is prime. For each c ∈ ℂ, define the quadratic polynomial Qc (z) = z2 + c. Fix n ∈ ℕ. As the coefficients of Qcn (z) are polynomials in c, the map Pn

:

ℂ c

→ 󳨃 →

ℂ Qcn (c) 2

is a polynomial in c. Moreover, as Qcn+1 (z) = (Qcn (z)) + c, the map P satisfies 2

Pn+1 (c) = (Pn (c)) + c. Prove that Pn (0) = 0,

Pn′ (0) = 1,

Pn (−2) = 2,

∀n ∈ ℕ.

Deduce that for each n ∈ ℕ there is some number γ = γ(n) ∈ (−2, 0) such that Pn (γ) = 0. N Show that Qcn−1 (0) = Pn (c): thus there exists some γ ∈ (−2, 0) with Qγ (0) = 0. This shows that the origin has period k for Qγ , where k divides N. As N is prime, we infer that k ∈ {1, N}, and k ≠ 1 as 0 is not fixed by Qγ ; thus, 0 belongs to a superattracting cycle of period N. Exercise 24.8.10 (Cf. Exercise 6.1.6, p. 101 in [8]). Prove that the fixed points of a polynomial cannot all be attracting. Exercise 24.8.11. Prove that the Julia set of a rational function is either connected or has uncountably many connected components. ̂ which is Exercise 24.8.12. Show that there exists a compact, connected, perfect set in ℂ not the Julia set of any rational function. Exercise 24.8.13. Show that there exists a topological Cantor set (i. e., a compact, totally ̂ which is not the Julia set of any rational function. disconnected, perfect set) in ℂ

24.8 Exercises

� 1041

Exercise 24.8.14. With the notation and terminology of Section 24.6, show that the sub̂ consisting of all compact sets that are not Julia sets of any rational function set of 𝒦(ℂ) ̂ is dense in 𝒦(ℂ). Exercise 24.8.15. Show that there are two rational functions of degree 2 whose Julia sets are homeomorphic but whose restrictions to their Julia sets are not topologically conjugate. Exercise 24.8.16. Show that there are two polynomials whose Julia sets are homeomorphic but whose restrictions to their Julia sets are not topologically conjugate. Exercise 24.8.17 (Cf. Exercise 6.2.1, p. 103 in [8]). Show that every quadratic polynomial P is conjugate via a Möbius transformation to one and only one polynomial Qc (z) = z2 +c. Show also that z2 − z is conjugate to Q−3/4 . Explain why Qc (z) − z divides P2 (z) − z. Using this, show that if Qc has no periodic point of period 2, then c = −3/4. Exercise 24.8.18 (Cf. Exercise 6.2.3, p. 103 in [8]). For a ≠ 1, let Ta (z) =

z2 − z . 1 + az

Show that: (a) Ta has 0, ∞ and α = 2/(1 − a) for fixed points. (b) Ta2 has three fixed points at the origin (i. e., the origin is a fixed point with multiplicity 3). (c) The five fixed points of Ta2 are 0, 0, 0, ∞ and α. (d) Ta has no two-cycle. Prove that Ta is conjugate through a Möbius transformation to Tb if and only if b = a or b = (3 − a)/(1 + a). [Hint: If g ∘ Ta ∘ g −1 = Tb , then Tb must fix g(0), g(∞) and g(α), so these three points must coincide with 0, ∞, β, where β = 2/(1 − b). Further, the multipliers at corresponding points must coincide.] Exercise 24.8.19 (Cf. Exercise 6.2.5, p. 103 in [8]). Let T be a rational function with deg(T) ≥ 2, and let G = {g : g Möbius, g ∘ T ∘ g −1 = T}. Use the fact that T has periodic points of period p for some prime number p to show that G is a finite group. Select a rational function of your choice and attempt to find G explicitly. Exercise 24.8.20 (Cf. Exercise 6.2.6, p. 104 in [8]). Prove that every quadratic polynomial has points of period 3.

1042 � 24 Dynamics and topology of rational functions: their Fatou and Julia sets Exercise 24.8.21 (Cf. Exercise 6.4.2, p. 110 in [8]). Let ζ be a repelling point of T. Show that if T n (z) → ζ as n → ∞, then T n (z) = ζ for some n ≥ 0, and so z ∈ 𝒥 (T). Show, however, that it is possible to have z ∈ ℱ (T) and T n (z) → ξ for some fixed point ξ ∈ 𝒥 (T). This happens, e. g., when T(z) = z/(1 + z2 ) and ξ = 0. Exercise 24.8.22 (Cf. Exercise 6.5.2, p. 132 in [8]). Let P(z) = −z + zp+1 , where p ∈ ℕ. How many petals does P have at the origin? Exercise 24.8.23 (Cf. Exercise 7.1.1, p. 161 in [8]). Let R(z) = z(z − a)/(1 − az), where 0 < |a| < 1. Show that B(0, 1) is a simply connected attracting component of ℱ (R). What can ̂ 1)? be said of ℂ\B(0, Exercise 24.8.24 (Cf. Exercise 7.1.2, p. 161 in [8]). Let S(z) = z/(2 − z2 ). Show that ℱ (S) has an attracting component of infinite connectivity. Exercise 24.8.25 (Cf. Exercise 7.1.4, p. 161, in [8]). Let P(z) = 6z(1 − z). Draw the graph of −n P(x) for x ∈ [0, 1] and show that ⋃∞ n=0 P ([0, 1]) is a Cantor set C. Prove that 𝒥 (P) = C, and deduce that P has a superattracting component of infinite connectivity. Exercise 24.8.26 (Cf. Exercise 9.5.1, p. 209 in [8]). Let P(z) = z2 − 1. For each w ∈ ℱ (P), let Fw be the component of ℱ (P) containing w. Show that the only finite critical point of P is in an attracting two-cycle of P. Deduce that: (i) ℱ (P) has infinitely many components. (ii) Every component of ℱ (P) is simply connected. (iii) P has no Siegel disk. (iv) {F∞ } and {F0 , F−1 } are the only periodic cycles of components of ℱ (P). (v) Every bounded component B of ℱ (P) is mapped by some Pn into F0 ; thus, the se∞ quence (Pn (B))n=0 is eventually alternating between F0 and F−1 . Find the multiplicity of P as a map acting between the given components in each of the following cases: F∞ → F∞ ,

F0 → F−1 ,

F−1 → F0 .

Exercise 24.8.27. Show that if two rational functions commute, then they have the same Julia set. Exercise 24.8.28 (Cf. Exercise 9.5.8, p. 210 in [8]). Let P(z) = z(1 + z2 ) and Q(z) = −P(z). Show that 𝒥 (P) = 𝒥 (Q) but that P and Q are not conjugate. Exercise 24.8.29. Let T : X → X be a self-map of a metric space (X, d). A point w ∈ X is called Lyapunov T-stable if for every ε > 0 there exists δ > 0 such that d(z, w) < δ

󳨐⇒

sup d(T n (z), T n (w)) < ε. n∈ℕ

24.8 Exercises

� 1043

̂→ℂ ̂ be a rational function. Prove that the Fatou set ℱ (T) is equal to the set of Let T : ℂ all Lyapunov T-stable points. Exercise 24.8.30 (Cf. Exercise 5.4.1, p. 89 in [8]). Find examples of domains U and V , and a rational function R mapping U onto V in such a way that 𝜕V is a proper subset of R(𝜕U). Exercise 24.8.31 (Cf. Exercise 5.4.2, p. 89 in [8]). Suppose that R(z) = zd and that V is a Euclidean disk with 0 ∉ 𝜕V . Show that R−1 (V ) is either simply connected (and R is a d-fold map of it onto V ) or it consists of d simply connected domains (each homeomorphic to V under R). What is R−1 (V ) when V = B(1, 1)? ̂→ℂ ̂ be a rational function with deg(T) ≥ 2, and let k ∈ ℕ. Exercise 24.8.32. Let T : ℂ Prove the following statements: −n (a) Crit(T k ) = ⋃k−1 n=0 T (Crit(T)) ⊇ Crit(T). k (b) PC(T ) = PC(T). (c) Ω(T k ) = Ω(T).

25 Selected technical properties of rational functions 25.1 Passing near critical points: results and applications Let D ⊆ ℂ be an open set and let f : D → ℂ be a nonconstant analytic function. For every w ∈ D, let degw (f ) be the local degree of f at w, sometimes called the multiplicity of f at w; more precisely, degw (f ) is the unique d ∈ ℕ such that the Taylor series expansion of f around w has the form ∞

f (z) = f (w) + ∑ ak (z − w)k ,

(25.1)

k=d

with ad :=

1 (d) f (w) d!

≠ 0. So, degw (f ) ≥ 2 if and only if w ∈ Crit(f ). Equivalently, w ∉ Crit(f )

⇐⇒

degw (f ) = 1.

(25.2)

Let c ∈ Crit(f ). Then there exist ̃r (f , c) > 0 and A = A(f , c) ≥ 1 such that 󵄨 󵄨 A−1 |z − c|degc (f ) ≤ 󵄨󵄨󵄨f (z) − f (c)󵄨󵄨󵄨 ≤ A|z − c|degc (f ) ,

∀z ∈ B(c, ̃r (f , c))

(25.3)

and 󵄨 󵄨 A−1 |z − c|degc (f )−1 ≤ 󵄨󵄨󵄨f ′ (z)󵄨󵄨󵄨 ≤ A|z − c|degc (f )−1 ,

∀z ∈ B(c, ̃r (f , c)).

(25.4)

We now prove the following description, well known in complex analysis, of an analytic function’s local behavior. In the vicinity of a function’s critical point, this description is frequently more convenient (even more powerful) and easier to apply than formulas (25.1), (25.3) and (25.4). Theorem 25.1.1. Let D ⊆ ℂ be an open set and f : D → ℂ a nonconstant analytic function. Then for every point w ∈ D the map f restricted to some sufficiently small neighborhood of w is biholomorphically conjugate to the map ξ 󳨃→ ξ degw (f ) on some neighborhood of 0. More precisely, there is a biholomorphic map φw : Uw → Vw whose domain Uw is some neighborhood of w, whose codomain Vw is some neighborhood of 0 and such that φw (w) = 0 and the following diagram commutes: Uw 󳨀󳨀󳨀󳨀󳨀󳨀→ f

↑ ↑ ↓

φw ↑ ↑ ξ󳨃→ξ degw (f )

f (Uw ) ↑ ↑ ↑ ↑Trf (w) ↓

Vw 󳨀󳨀󳨀󳨀󳨀󳨀󳨀󳨀→ f (Uw ) − f (w) = (Vw )degw (f ) where Trf (w) is the translation map Trf (w) (z) = z−f (w) and (Vw )degw (f ) = {zdegw (f ) : z ∈ Vw }. https://doi.org/10.1515/9783110769876-025

1046 � 25 Selected technical properties of rational functions Proof. Let Trw : ℂ → ℂ be the translation Trw (z) = z − w. Consider the analytic map −1 ̂f := Tr f (w) ∘ f ∘ Trw : Trw (D) → Trf (w) (f (D)).

Since f is nonconstant, so is ̂f . Moreover, −1 ̂f (0) = Tr f (w) ∘ f (Trw (0)) = Trf (w) (f (w)) = 0

and deg0 (̂f ) = degf (w) (Trf (w) ) ⋅ degw (f ) ⋅ deg0 (Tr−1 w ) = 1 ⋅ degw (f ) ⋅ 1 = degw (f ). k d ∞ j Alternatively, observe that ̂f (z) = ∑∞ k=d ak z = z ∑j=0 aj+d z . Denote d := degw (f ). Applying (25.1) to ̂f , we deduce that there exist an open simply connected neighborhood

W1 of 0 and a bounded holomorphic function g : W1 → ℂ such that g(0) ≠ 0 and ̂f (z) = zd g(z),

∀z ∈ W1 .

Hence, there exists a bounded holomorphic function h : W1 → ℂ such that d

g(z) = (h(z)) ,

∀z ∈ W1 .

Consider the map ψ : W1 → ℂ defined by ψ(z) = zh(z). Then ψ′ (z) = h(z) + zh′ (z) and, therefore, ψ′ (0) = h(0) ≠ 0. Thus, there exists an open simply connected set W2 ⊆ W1 containing 0 such that the map ψ|W2 is one-to-one and so is biholomorphic. Let Uw := Tr−1 w (W2 ) and φw := ψ ∘ Trw : Uw → ℂ. Then φw is a biholomorphic map, φw (w) = ψ(Trw (w)) = ψ(0) = 0 and d

d

d

d

(φw (z)) = (ψ ∘ Trw (z)) = (Trw (z)h(Trw (z))) = (Trw (z)) g(Trw (z)) = ̂f (Tr (z)) = ̂f ∘ Tr (z) = Tr ∘ f (z). w

w

f (w)

As a direct consequence of this theorem, we get a characterization of the local degree of an analytic function. Theorem 25.1.2. If D ⊆ ℂ is an open set and f : D → ℂ is a nonconstant analytic function, then for every point w ∈ D the local degree degw (f ) is the unique d ∈ ℕ for which there exists a local base {Un (w)}∞ n=1 of topology at w consisting of open sets contained in D such that the restriction

25.1 Passing near critical points: results and applications

󵄨 f 󵄨󵄨󵄨U

n (w)\{w}

�

1047

: Un (w) \ {w} → ℂ

is d-to-1. We now prove several more specific, but interesting on their own, consequences of relations (25.1), (25.3)–(25.4) and Theorem 25.1.1 for rational functions. The first one is obvious: At any positive distance ε away from the critical set, the function is one-to-one on balls of a certain radius α(ε) and the inverse branches are defined on balls of some radius β(ε). ̂ → ℂ ̂ is a rational function, then for every ε > 0 there exist Lemma 25.1.3. If T : ℂ ̂ \ B(Crit(T), ε) the restriction T|B(z,α(ε)) 0 < α(ε) < ε and β(ε) > 0 such that for every z ∈ ℂ ̂ of is injective and there is a unique holomorphic inverse branch Tz−1 : B(T(z), β(ε)) → ℂ −1 T such that Tz (T(z)) = z. The next result concerns (1) the local preservation of simple connectedness for preimages’ components; (2) the control on/uniform boundedness of the components’ diameters; and, finally, (3) the conformal homeomorphy of those components that are disjoint from the critical set. ̂ → ℂ ̂ is a rational function, then for every ε > 0 there exists Lemma 25.1.4. If T : ℂ ̂ δ > 0 such that if V ⊆ ℂ is an open simply connected set with diam(V ) < δ, then each connected component U of T −1 (V ) is simply connected and diam(U) < ε. In addition, if U ∩ Crit(T) = 0 then T|U : U → V is a conformal homeomorphism. ̂ is compact, it Proof. Write d = deg(T). As the map T is open and continuous and ℂ follows from Lemma 4.2.2 and Remark 4.2.3 (in Volume 1) that there is δ > 0 such that T(B(x,

ε )) ⊇ B(T(x), δ), 2d

̂ ∀x ∈ ℂ.

(25.5)

We shall first prove that T −1 (B(w, δ)) ⊆

⋃

z∈T −1 (w)

B(z,

ε ), 2d

̂ ∀w ∈ ℂ.

(25.6)

̂ and let ξ ∈ T −1 (B(w, δ)). This means that T(ξ) ∈ B(w, δ); equivalently, Indeed, fix w ∈ ℂ w ∈ B(T(ξ), δ). By virtue of (25.5), T(B(ξ,

ε )) ⊇ B(T(ξ), δ) ∋ w. 2d

ε ε So, there exists z ∈ B(ξ, 2d ) with T(z) = w. As ξ ∈ B(z, 2d ), the proof of (25.6) is done. It ensues from (25.6) that for each connected component W of T −1 (B(w, δ)) there are points z1 , z2 , . . . , zq , 1 ≤ q ≤ d, in T −1 (w), such that

1048 � 25 Selected technical properties of rational functions q

W ⊆ ⋃ B(zj , j=1

ε ) 2d

and B(zi ,

ε ε ) ∩ B(zi+1 , ) ≠ 0, 2d 2d

∀1 ≤ i < q.

Hence, diam(W ) ≤ q ⋅

ε ≤ ε. d

(25.7)

̂ is a nonempty set with diam(V ) < δ, then fixing any w ∈ V we know If V ⊆ ℂ that V ⊆ B(w, δ). Therefore, each connected component U of T −1 (V ) is contained in a (unique) connected component W of T −1 (B(w, δ)). It follows from (25.7) that diam(U) ≤ diam(W ) ≤ ε.

(25.8)

It just remains to prove that U is simply connected. Let η = min{|c1 − c2 | : c1 , c2 ∈ Crit(T), c1 ≠ c2 }. By the already proven part of the lemma, there is θ > 0 such that diam(U) < η for every ̂ with diam(V ) < θ and for each connected component open simply connected set V ⊆ ℂ −1 U of T (V ). But then U contains at most one critical point of T, and the simple connectedness of U directly follows from Corollaries 22.5.4–22.5.5. Finally, if U contains no critical point, then T|U is a conformal homeomorphism according to Corollary 22.5.4. As a straightforward consequence of this lemma, we get its extension to a finite number of iterates of a rational function. ̂→ℂ ̂ be a rational function. For every n ∈ ℕ and every ε > 0, Lemma 25.1.5. Let T : ℂ ̂ is an open simply connected set with diam(V ) < δ, there exists δ > 0 such that if V ⊆ ℂ then for every 0 ≤ k ≤ n each connected component U of T −k (V ) is simply connected and diam(U) < ε. In addition, if U ∩ Crit(T k ) = 0 then T k |U : U → V is a conformal homeomorphism. In particular, it follows from Lemma 25.1.4 and (25.3)–(25.4) that there exist R = R(T, c) > 0 and A = A(T, c) ≥ 1 such that 󵄨 󵄨 A−1 |z − c|degc (T) ≤ 󵄨󵄨󵄨T(z) − T(c)󵄨󵄨󵄨 ≤ A|z − c|degc (T)

(25.9)

󵄨 󵄨 A−1 |z − c|degc (T)−1 ≤ 󵄨󵄨󵄨T ′ (z)󵄨󵄨󵄨 ≤ A|z − c|degc (T)−1

(25.10)

and

for every z ∈ Comp(c, T, R), where Comp(c, T, R) denotes the connected component of T −1 (B(T(c), R)) containing c. Furthermore, T(Comp(c, T, R)) = B(T(c), R)

25.1 Passing near critical points: results and applications

1049

�

and there exists 0 < r(T, c) ≤ ̃r (T, c) such that B(c, r(T, c)) ⊆ Comp(c, T, R(T, c)).

(25.11)

Notably, relations (25.9)–(25.10) hold for all z ∈ B(c, r(T, c)). The next result provides, at any positive distance R away from the critical set, a uniform bound on the diameters of the components of the preimages of balls. In addition to depending on R, that bound depends linearly on the radius r of the balls. ̂ →ℂ ̂ is a rational function, then for every R > 0 there exists a Lemma 25.1.6. If T : ℂ ̂ \ B(Crit(T), R) we have for every r > 0 that constant KR ≥ 1 such that for every z ∈ ℂ diam(Uz,r ) < KR r, where Uz,r is the connected component of T −1 (B(T(z), r)) containing z. ̂ \ B(Crit(T), R). Using Lemma 25.1.4 with ε = R/2 and Proof. Let R > 0 and let z ∈ ℂ ̂ ̂ and every connected defining R = δ/2, we know that diam(U) < R/2 for every w ∈ ℂ −1 ̂ component U of T (B(w, R)). In particular, diam(Uz,R̂ ) < R/2. ̂ \ B(Crit(T), R)], we deduce that Since z ∈ Uz,R̂ ∩ [ℂ ̂ \ B(Crit(T), R/2). Uz,R̂ ⊆ ℂ ̂ Then Assume first that r ≥ R. ̂ ≤ diam(Uz,r ) ≤ diam(ℂ)

̂ diam(ℂ) r. ̂ R

̂ Given that Uz,r ⊆ U ̂ ⊆ ℂ ̂ \ B(Crit(T), R/2), it ensues that Now assume that r ≤ R. z,R ̂ \ B(Crit(T), R/2)} > 0 inf{|T ′ (x)| : x ∈ Uz,r } ≥ ‖T ′ ‖R := inf{|T ′ (y)| : y ∈ ℂ and Uz,r ∩ Crit(T) = 0. By Lemma 25.1.4, the restriction T|Uz,r : Uz,r → B(T(z), r) is a biholomorphic map. Applying the mean value inequality to T|−1 Uz,r , we deduce that ′ 󵄨 󵄨󵄨 diam(Uz,r ) ≤ sup{󵄨󵄨󵄨(T|−1 Uz,r ) (w)󵄨󵄨 : w ∈ B(T(z), r)} ⋅ 2r

= inf{|T ′ (ζ )|−1 : ζ ∈ Uz,r } ⋅ 2r ≤ 2‖T ′ ‖−1 R r.

Set

1050 � 25 Selected technical properties of rational functions

KR := max{

̂ diam(ℂ) , 2‖T ′ ‖−1 R }. ̂ R

The following lemma in some way generalizes the previous one. It offers a uniform bound on the diameters of the components of the preimages of balls. That bound evidently depends on the radius r of the balls though not linearly this time. ̂ → ℂ ̂ is a rational function, then for every r ∈ (0, 1/2] small Lemma 25.1.7. If T : ℂ ̂ we have that enough and all w ∈ ℂ 1

diam(U) ≤ r 2dT for every connected component U of T −1 (B(w, r)), where dT := max{degc (T) : c ∈ Crit(T)}. ̂ Recalling the definition of r(T, c) from (25.11), let Proof. Pick w ∈ ℂ. ρ :=

1 min{r(T, c) : c ∈ Crit(T)} > 0. 2

(25.12)

Using Lemma 25.1.4, assume that r ∈ (0, 1/2] is so small that diam(V ) < ρ/2

(25.13)

̂ and every connected component V of T −1 (B(ζ , r)). for every ζ ∈ ℂ Fix a point z ∈ U ∩ T −1 (w). If dist(z, Crit(T)) ≥ ρ/2, then Lemma 25.1.6 asserts that 1

diam(U) ≤ Kρ/2 r ≤ r 2dT ,

(25.14)

provided that r ≤ 1/2 is small enough. So, assume that dist(z, Crit(T)) < ρ/2. There exists c ∈ Crit(T) such that |z − c| < ρ/2. Using (25.13) and (25.11), we get that U ⊆ B(c,

ρ + diam(U)) ⊆ B(c, ρ) ⊆ Comp(c, T, R(T, c)). 2

(25.15)

Suppose first that 1 r ≤ |w − T(c)|. 2

(25.16)

Denote dc := degc (T). With A = max{A(T, x) : x ∈ Crit(T)}, it follows from (25.9)–(25.10) and (25.15) that for every point ξ ∈ U, −(1+ d1 )

|T ′ (ξ)| ≥ A−1 |ξ − c|dc −1 ≥ A

c

1− d1

|T(ξ) − T(c)|

c

25.1 Passing near critical points: results and applications

−(1+ d1 )

≥A

c

1− d1

(|w − T(c)| − |T(ξ) − w|)

−(1+ d1 )

≥A

c

c

1− d1

�

1051

−(1+ d1 ) 1− d1

(2r − r)

c

=A

1

1

c

r

c

.

By the mean value inequality, we deduce that 1+ d1

diam(U) ≤ A

c

1

r dc

−1

⋅ (2r) = 2A

1+ d1

c

1

r dc ≤ r 2dc ≤ r 2dT ,

(25.17)

where the second inequality sign was written assuming that r ≤ 1/2 is small enough. Suppose now that r > 21 |w − T(c)|. Then B(w, r) ⊆ B(T(c), 3r).

(25.18)

There exists a point y ∈ U such that |y − c| ≥ 31 diam(U) (otherwise, U ⊆ B(c, 31 diam(U)), which implies that diam(U) ≤ 2 ⋅ 31 diam(U), an impossibility). Using (25.9) and (25.17)– (25.18), we infer that d

3r ≥ |T(y) − T(c)| ≥ A−1 |y − c|dc ≥ 3−dc A−1 (diam(U)) c . Thus, 1+ d1

diam(U) ≤ 3

c

1

1

1

A dc r dc ≤ r 2dT ,

where the second inequality sign was written assuming that r ≤ 1/2 is small enough. This inequality along with (25.14) and (25.17) complete the proof. The upcoming two results apply to any analytic function. The first one pertains to their injectivity on balls disjoint from the critical set. Unlike in previous results, the balls do not stay uniformly away from the critical points. Lemma 25.1.8. Let D ⊆ ℂ be an open set and f : D → ℂ a nonconstant analytic function. For every c ∈ Crit(f ), there exist two numbers r1 (f , c) ∈ (0, 21 ̃r (f , c)) and s(f , c) ∈ (0, 21 ], this latter depending only on degc (f ), such that for any w ∈ B(c, r1 (f , c)) the restriction 󵄨 f 󵄨󵄨󵄨B(w,s(f ,c)|w−c|) : B(w, s(f , c)|w − c|) → ℂ is one-to-one. (Note: ̃r (f , c) arises from (25.3)–(25.4).) Proof. We define s∗ = s∗ (f , c) to be the largest number in (0, 21 ] such that ∠(1 + s∗ i, 0, 1 − s∗ i) ≤ Denote dc := degc (f ).

π . degc (f )

1052 � 25 Selected technical properties of rational functions For every ξ ∈ ℂ, the map ℂ ∋ z 󳨃→ Edc (z) := zdc ∈ ℂ restricted to the ball B(ξ, s∗ |ξ|) is oneto-one. Decreasing the neighborhood Uc of c produced in Theorem 25.1.1 if necessary, we may assume that 0 < mc := inf{|φ′c (ξ)| : ξ ∈ Uc } ≤ sup{|φ′c (ξ)| : ξ ∈ Uc } =: Mc < ∞. Let s = mc Mc−1 s∗ and let r1 (f , c) ∈ (0, 21 ̃r (f , c)) be so small that B(c, 2r1 (f , c)) ⊆ Uc . Let w ∈ B(c, r1 (f , c)). By the mean value inequality, we get for every z ∈ B(w, s|w − c|) that |φc (z) − φc (w)| ≤ Mc |z − w| ≤ Mc s|w − c| and 󵄨 󵄨󵄨 −1 −1 −1 |w − c| = 󵄨󵄨󵄨φ−1 c (φc (w)) − φc (φc (c))󵄨󵄨 ≤ mc |φc (w) − φc (c)| = mc |φc (w)|. Hence, |φc (z) − φc (w)| ≤ Mc smc−1 |φc (w)| = s∗ |φc (w)|. Therefore, φc (B(w, s|w − c|)) ⊆ B(φc (w), s∗ |φc (w)|). 󵄨 So, the map Edc 󵄨󵄨󵄨φ (B(w,s|w−c|)) is one-to-one. Consequently, by Theorem 25.1.1 the map c

󵄨 󵄨󵄨 f 󵄨󵄨󵄨B(w,s|w−c|) = Tr−1 f (c) ∘ Edc ∘ φc 󵄨󵄨B(w,s|w−c|) is one-to-one, too. Set s(f , c) = s. As a fairly easy consequence of this lemma, we get an upper bound on the ratio of the derivatives of the function on balls disjoint from the critical points. Lemma 25.1.9. Let D ⊆ ℂ be an open set and f : D → ℂ a nonconstant analytic function. For any η > 0 and c ∈ Crit(f ), there is rη (f , c) ∈ (0, 21 r1 (f , c)) such that if w ∈ B(c, rη (f , c)) then |f ′ (z)| ≤ (1 + η)(1 + s)degc (f )−1 |f ′ (w)|

25.1 Passing near critical points: results and applications

�

1053

for every s ∈ (0, s(f , c)] and every z ∈ B(w, s|w − c|). (Note: r1 (f , c) and s(f , c) come from Lemma 25.1.8.) Proof. A variant of this lemma, which would frequently be sufficient in this book, would be an immediate consequence of Lemma 25.1.8 and Koebe’s distortion theorem. We rather provide an elementary proof of Lemma 25.1.9, i. e., a proof that neither relies on Lemma 25.1.8 nor on Koebe’s distortion theorem but merely on the Taylor series expansion formula (25.1). As before, denote dc := degc (f ). By (25.1), there exist ̂r = ̂r (f , c) ∈ (0, 21 ̃r (f , c)] and a bounded holomorphic function ̂f : B(c, ̂r ) → ℂ with bounded derivative such that f (z) = f (c) + adc (z − c)dc + (z − c)dc +1 ̂f (z),

∀z ∈ B(c, ̂r ).

Differentiating this formula, we get f ′ (z) = adc dc (z − c)dc −1 + (z − c)dc [(dc + 1)̂f (z) + (z − c)̂f ′ (z)],

∀z ∈ B(c, ̂r ).

(25.19)

Define the function ̃f : B(c, ̂r ) → ℂ by ̃f (z) := (a d )−1 [(d + 1)̂f (z) + (z − c)̂f ′ (z)]. dc c c

(25.20)

r Let rη ∈ (0, 1+s ] be so small that ̂

1 + 2‖̃f ‖∞ rη < 1 + η. 1 − ‖̃f ‖ r ∞ η

(25.21)

Let w ∈ B(c, rη ) and z ∈ B(w, s|w − c|). Then |z − c| < (1 + s)|w − c| < (1 + s)rη ≤ ̂r . Using (25.19)–(25.21), we get that 󵄨󵄨 󵄨 󵄨󵄨adc dc (z − c)dc −1 + (z − c)dc ⋅ adc dc ̃f (z)󵄨󵄨󵄨 |f ′ (z)| = |f ′ (w)| 󵄨󵄨󵄨ad dc (w − c)dc −1 + (w − c)dc ⋅ ad dc ̃f (w)󵄨󵄨󵄨 󵄨 c 󵄨 c ̃ 󵄨󵄨󵄨 z − c 󵄨󵄨󵄨dc −1 |1 + ̃f (z)(z − c)| dc −1 1 + 2‖f ‖∞ rη 󵄨󵄨 ≤ (1 + s) = 󵄨󵄨󵄨 󵄨 󵄨󵄨 w − c 󵄨󵄨 |1 + ̃f (w)(w − c)| 1 − ‖̃f ‖∞ rη ≤ (1 + s)dc −1 (1 + η).

We now prove a global version of the previous lemma for rational functions. This time, we do rely on Koebe’s distortion theorem.

1054 � 25 Selected technical properties of rational functions ̂→ℂ ̂ be a rational function. For every ε > 0 there is sε ∈ (0, 1 ] Lemma 25.1.10. Let T : ℂ 2 ̂ and z ∈ B(w, sε dist(w, Crit(T))) we have such that for any w ∈ ℂ |T ′ (z)| ≤ 1 + ε. |T ′ (w)| Proof. We stick to the notation adopted in Lemmas 25.1.8–25.1.9 and in their proofs. Fix ε > 0. Let 󵄨 dT := max{degc (T) 󵄨󵄨󵄨 c ∈ Crit(T)} and

󵄨 s(T) := min{s(T, c) 󵄨󵄨󵄨 c ∈ Crit(T)}.

Given 0 < η < ε, let ̂s ∈ (0, s(T)) be so small that (1 + η)(1 + ̂s)dT −1 ≤ 1 + ε.

(25.22)

r (T,c) For every c ∈ Crit(T), let rη (T, c) ∈ (0, 1+s(T,c) ] be so small that ̂

̃ ∞ rη (T, c) 1 + 2‖T‖ < 1 + η. ̃ ∞ rη (T, c) 1 − ‖T‖ Thereafter, let 󵄨 rη (T) := min{rη (T, c) 󵄨󵄨󵄨 c ∈ Crit(T)}. Let ̂ \ B(Crit(T), rη (T)). X := ℂ Then T ′ (ξ) ≠ 0 for all ξ ∈ X. Thus, for every ξ ∈ X there exists ρξ > 0 such that the map T|B(ξ,ρξ ) is one-to-one. Since X is compact, the open cover {B(ξ, ρξ ) : ξ ∈ X} of X has a Lebesgue number ρ > 0. If x ∈ X, then there exists x̂ ∈ X such that B(x, ρ/2) ⊆ B(x̂, ρx̂ ). Hence, the map T|B(x,ρ/2) is one-to-one. So, by virtue of Koebe’s distortion theorem (Theorem 23.1.7), there exists ̃s ∈ (0, ̂s) such that |T ′ (x)| ≤1+ε |T ′ (ξ)|

(25.23)

whenever ξ ∈ X and x ∈ B(ξ, ̃sρ/2). Set ̂ ̃sρ/2}. sε := min{̃s, diam−1 (ℂ) ̂ and z ∈ B(w, sε dist(w, Crit(T))). If w ∈ B(Crit(T), rη (T)), then there exists Take w ∈ ℂ c ∈ Crit(T) such that w ∈ B(c, rη (T, c)). Moreover, z ∈ B(w, ̂s|w − c|). It follows from Lemma 25.1.9 and (25.22) that

25.1 Passing near critical points: results and applications

�

1055

|T ′ (z)| ≤ 1 + ε. |T ′ (w)| On the other hand, if w ∈ X then ̂ ≤ ̃sρ/2, |z − w| < sε dist(w, Crit(T)) ≤ sε diam(ℂ) whence |T ′ (z)| ≤1+ε |T ′ (w)| by virtue of (25.23). We now show that, whenever the iterates of a point stay away from the critical set for some time and the modulus of the map’s derivative exceeds one at all these iterates, there exists a unique inverse branch on a small enough ball. ̂→ℂ ̂ be a rational function with deg(T) ≥ 2. For every ε > 0 Lemma 25.1.11. Let T : ℂ and every λ > 1, there exists θ = θ(ε, λ) > 0 such that if for some n ∈ ℕ and some point ̂ we have z∈ℂ ̂ \ B(Crit(T), ε) and 󵄨󵄨󵄨T ′ (T j (z))󵄨󵄨󵄨 ≥ λ, {z, T(z), . . . , T n−1 (z)} ⊆ ℂ 󵄨 󵄨

∀0 ≤ j < n,

̂ of T −n sending then there is a unique holomorphic branch Tz−n : B(T n (z), 2θ(ε, λ)) → ℂ n T (z) to z. Proof. Without losing generality, we may assume that 0 < ε ≤ π/4. Let δ > 0 be ascribed to ε = π/4 per Lemma 25.1.4. Let also β(ε) > 0 arise from Lemma 25.1.3. Set β′ (ε) := min{δ, β(ε)}. Take t ∈ (0, 1) so small that λ−1 k2R,s (t) < 1, where k2R,s (t) is the function produced in Theorem 23.1.9 (Koebe’s distortion theorem, spherical version) with R = β′ (ε) and s = π/4. It follows from Lemma 25.1.3 that for every j = 1, 2, . . . , n there exists ̂ satisfying T −1 (T j (z)) = a unique holomorphic inverse branch Tj−1 : B(T j (z), β′ (ε)) → ℂ j

T j−1 (z). Hence, in view of Theorem 23.1.9, we get

Tj−1 (B(T j (z), tβ′ (ε))) ⊆ B(T j−1 (z), λ−1 k2R,s (t)tβ′ (ε)) ⊆ B(T j−1 (z), tβ′ (ε)). Therefore, the composition T1−1 ∘ T2−1 ∘ ⋅ ⋅ ⋅ ∘ Tn−1 is well-defined on B(T n (z), tβ′ (ε)) and T1−1 ∘ T2−1 ∘ ⋅ ⋅ ⋅ ∘ Tn−1 (T n (z)) = z. Set θ = tβ′ (ε)/2 and Tz−n = T1−1 ∘ T2−1 ∘ ⋅ ⋅ ⋅ ∘ Tn−1 . In a similar vein, we establish the following fact about inverse branches. ̂ → ℂ ̂ is a rational function with deg(T) ≥ 2, then there exists Lemma 25.1.12. If T : ℂ τ ∈ (0, 1/2) such that for every c ∈ Crit(T) and every z ∈ B(c,

1 min{|c1 − c2 | : c1 , c2 ∈ Crit(T), c1 ≠ c2 }) 2

1056 � 25 Selected technical properties of rational functions ̂ such there exists a unique holomorphic inverse branch Tz−1 : B(T(z), 2τ|T(z) − T(c)|) → ℂ −1 that Tz (T(z)) = z. Proof. Recall that Comp(ζ , T, r) denotes the connected component of T −1 (B(T(ζ ), r)) containing ζ . By virtue of Lemma 25.1.4, there exists τ ∈ (0, 1/2) so small that diam( Comp(w, T, 2τ|T(w) − T(c)|) )
η :=

1 min{α(R/4), K −1 ‖T ′ ‖−1 ∞ β(R/4)}. 4

Employing (25.26) and (25.30), we obtain an upper bound on the ratio diam(T(F))

dc (T)−1

diam(F) min{(Dist(c, F))

: c ∈ Crit(T)}

≤

̂ diam(ℂ) . d η(R/4) T −1

̂ \ B(Crit(T), R/4)] we know that On the other hand, for any ξ ∈ F ∩ [ℂ η/2 < diam(F ∩ B(ξ, η/2)) ≤ η. Using this and (25.29) with F replaced by F ∩ B(ξ, η/2), we get a lower bound:

(25.30)

1058 � 25 Selected technical properties of rational functions diam(T(F)) diam(F) min{(Dist(c, F)) ≥ ≥ ≥

dc (T)−1

.

: c ∈ Crit(T)}

diam(T(F ∩ B(ξ, η/2)))

̂ ⋅ (diam(ℂ)) ̂ dT −1 diam(ℂ)

K −1 |T ′ (ξ)|diam(F ∩ B(ξ, η/2)) dT

̂ (diam(ℂ))

̂ \ B(Crit(T), R/4)} ⋅ η/2 K −1 inf{|T ′ (z)| : z ∈ ℂ dT

̂ (diam(ℂ))

.

Thus, the result holds in Subcase 1b, too, which means that it holds in Case 1 altogether. Case 2: Assume now that ̂ \ B(Crit(T), R/4)] = 0. F ∩ [ℂ

(25.31)

This means that F ⊆ B(Crit(T), R/4). As F is connected, the definition of R imposes the existence of a unique cF ∈ Crit(T) such that F ⊆ B(cF , R/4). Then D := Dist(cF , F) ≤ R/4. Given that F ⊆ B(cF , D), it ensues from the right inequality in (25.4) that diam(T(F)) ≤ diam(F) sup{|T ′ (z)| : z ∈ B(cF , D)} ≤ ADdcF (T)−1 diam(F). If c ≠ cF , then Dist(c, F) ≥ max{D, R/4} and thus DdcF (T)−1 Dist(c, F)dcF (T)−1 ≤ = Dist(c, F)dcF (T)−dc (T) ≤ max{π dT , (R/4)−dT }. Dist(c, F)dc (T)−1 Dist(c, F)dc (T)−1 So diam(T(F)) diam(T(F)) DdcF (T)−1 = ≤ A max{π dT , (R/4)−dT }. d (T)−1 d (T)−1 diam(F) Dist(c, F) c Dist(c, F)dc (T)−1 diam(F) D cF This results in an upper bound for the entire case: diam(T(F))

dc (T)−1

diam(F) min{(Dist(c, F))

: c ∈ Crit(T)}

≤ A max{π dT , (R/4)−dT }.

(25.32)

25.1 Passing near critical points: results and applications

1059

�

For a lower bound, we will soon split the case into subcases. Since F is connected, diam(F) = diam(F) and diam(T(F)) = diam(T(F)) = diam(T(F)), we may assume without loss of generality that F is a closed set, whence compact. Let ξ ∈ F be such that |ξ − cF | = D.

(25.33)

Let τ ∈ (0, 1/2) come from Lemma 25.1.12. Subcase 2a: Suppose first that 1 diam(T(F)) ≤ τ|T(ξ) − T(cF )|. 2 Then F=

⋃

z∈T −1 (T(ξ))∩B(cF ,D)

Tz−1 (T(F)).

As the set F is connected and all the sets Tz−1 (T(F)), z ∈ T −1 (T(ξ)), are closed and mutually disjoint, there exists a unique z ∈ T −1 (T(ξ)) such that F ⊆ Tz−1 (T(F)). In fact, z = ξ since ξ ∈ F ∩ T −1 (T(ξ)) ∩ B(cF , D). It then follows from Koebe’s distortion theorem and the left inequality in (25.4) that diam(F) ≤ diam(Tz−1 (T(F))) ≤ diam(Tz−1 (B(T(z), diam(T(F))))) ≤ K|T ′ (z)|−1 ⋅ 2diam(T(F)) ≤ K[A−1 DdcF (T)−1 ]

−1

= 2KAD

−(dcF (T)−1)

diam(T(F)),

⋅ 2diam(T(F))

hence leading to a lower bound: diam(T(F))

dc (T)−1

diam(F) min{(Dist(c, F))

: c ∈ Crit(T)}

=

diam(T(F))

diam(F) DdcF (T)−1

We are done in Subcase 2a. Subcase 2b: Suppose now that 1 diam(T(F)) ≥ τ|T(ξ) − T(cF )|. 2 By definition of ξ and the left inequality in (25.3), we have that 1 1 diam(T(F)) ≥ τA−1 |ξ − cF |dcF (T) = τA−1 DdcF (T) . 2 2 As F ⊆ B(cF , D), we know that diam(F) ≤ 2D and, therefore,

≥ (2KA)−1 .

1060 � 25 Selected technical properties of rational functions 1 τA−1 DdcF (T) 1 −1 d (T)−1 diam(T(F)) ≥ 2 = τA D cF , diam(F) 2D 4

and we are done in Subcase 2b, too. So, Case 2 holds overall and the proof of Lemma 25.1.13 is complete. Finally, we establish the integrability of some logarithmic functions on the Julia set based on the integrability of the logarithm of the modulus of the derivative of the map on that set. ̂ → ℂ ̂ be a rational function with deg(T) ≥ 2. If μ is a finite Lemma 25.1.14. Let T : ℂ Borel measure on 𝒥 (T) and if the function log |T ′ | is μ-integrable, then the function 𝒥 (T) ∋ z 󳨃󳨀→ log |z − c|

is μ-integrable for every c ∈ Crit(T)∩ 𝒥 (T). If in addition μ is T-invariant, then the function 𝒥 (T) ∋ z 󳨃󳨀→ log |z − T(c)|

is μ-integrable. Proof. The first assertion follows immediately from (25.4) and since outside of any neighborhood of Crit(T), the function |T ′ | is uniformly bounded away from zero and infinity. In order to prove the second part of the lemma, consider a closed curve R emanating from T(c) such that μ(R) = 0 and a disk B(T(c), r) with r ∈ (0, R(T, c)) so small that a holomorphic inverse branch of T, Tc−1 : B(T(c), r) \ R → ℂ with the property that

lim

B(T(c),r)∋z→T(c)

Tc−1 (z) = c, is well-defined. Let

D := B(T(c), r) \ R. By virtue of (25.9), we have |z − T(c)| ≍ |Tc−1 (z) − c|degc (T) ,

∀z ∈ Tc−1 (B(T(c), r)),

where ≍ is the usual symbol of multiplicative comparability: A ≍ B means that C −1 A ≤ B ≤ CA for some constant C ≥ 1. It suffices to show that ∫ log |z − T(c)| dμ(z) D

(25.34)

25.2 Two rules for critical points

� 1061

is finite. Using the T-invariance of μ, we get ∫ log |z − T(c)| dμ(z) = ∫ 1D (z) log |z − T(c)| dμ(z) D

𝒥 (T)

≍+ ∫ 1D (z) log |Tc−1 (z) − c|degc (T) dμ(z) 𝒥 (T)

= ∫ (1D ∘ T)(z) log |z − c|degc (T) dμ(z) 𝒥 (T)

= degc (T) ∫ 1T −1 (D) (z) log |z − c| dμ(z). 𝒥 (T)

Notice that the functions z 󳨃→ 1D (z) log |z − T(c)| and

z 󳨃→ 1D (z) log |Tc−1 (z) − c|degc (T)

are well-defined on 𝒥 (T) and that the sign ≍+ stands for additive comparability. The finiteness of the last integral follows from the first part of the lemma.

25.2 Two rules for critical points In this section, we first prove a lower bound on the “first-return time” of any ball centered on a critical point that lies in the Julia set. This “first-return time” obviously depends on the radius of the ball: it tends to infinity as the radius decreases to zero. We call it Rule I. Note that this result generally does not hold for critical points in the Fatou set; for instance, c = 0 is a superattractive fixed point of z 󳨃→ z2 . Rule I was proved in [107]. Later we prove Rule II. It is kind of a strengthening of Rule I and was proved in [33]. We also derive consequences of these two rules. ̂ →ℂ ̂ there is a constant C1 > 0 Lemma 25.2.1 (Rule I). For any rational function T : ℂ n such that if T (B(c, ε)) ∩ B(c, ε) ≠ 0 for some c ∈ Crit(T) ∩ 𝒥 (T), ε > 0 and n ∈ ℕ, then n ≥ −C1 log ε. Proof. As #Crit(T) < ∞, by Taylor’s theorem (see (25.3)) there are δ ∈ (0, 1/2) and Γ > 0 such that diam(T(B(c, 2ε))) ≤ Γε2 ,

∀ε ∈ (0, δ], ∀c ∈ Crit(T).

Using the mean value inequality, we then get diam(T k (B(c, 2ε))) ≤ Γε2 ‖T ′ ‖k−1 ∞ ,

∀k ∈ ℕ, ∀ε ∈ (0, δ], ∀c ∈ Crit(T).

1062 � 25 Selected technical properties of rational functions Increasing Γ and decreasing δ if necessary, we may assume that Γ ≥ ‖T ′ ‖∞

and

C :=

log Γ − log ‖T ′ ‖∞ 1 [1 + ] > 0. log ‖T ′ ‖∞ log δ

(25.35)

Suppose that T n (B(c, ε)) ∩ B(c, ε) ≠ 0 for some c ∈ Crit(T) ∩ 𝒥 (T), some ε > 0 and some n ∈ ℕ. If ε ∈ (0, δ] and it turned out that Γε2 ‖T ′ ‖n−1 ∞ < ε, then we would deduce from the above that T n (B(c, 2ε)) ⊆ B(c, 2ε). By induction, we would have T nk (B(c, 2ε)) ⊆ B(c, 2ε),

∀k ∈ ℕ.

n Hence, the family of iterates {T nk |B(c,2ε) }∞ k=1 would be normal, whence c ∈ ℱ (T ) = ℱ (T), 2 ′ n−1 which would contradict the hypothesis that c ∈ 𝒥 (T). Therefore, Γε ‖T ‖∞ ≥ ε. Then

n≥

log Γ − log ‖T ′ ‖∞ − log ε − log Γ − log ε +1= [1 + ] ≥ −C log ε. ′ ′ log ‖T ‖∞ log ‖T ‖∞ log ε

On the other hand, if ε > δ then n≥1=

− log δ 1 ≥ (− log ε). − log δ − log δ

Set C1 := min{C,

1 }. − log δ

We will not really need the next lemma, but it is interesting in itself and is a fairly immediate consequence of Lemma 25.2.1. For every n ∈ ℕ, denote n

Critv(T n ) := T n (Crit(T n )) = ⋃ T k (Crit(T)), k=1

i. e., Critv(T n ) is the set of all critical values of T n . Then #Critv(T n ) ≤ #Crit(T) ⋅ n ≤ [2 deg(T) − 2]n. ̂→ℂ ̂ is a rational function, then there exists Lemma 25.2.2 (Finiteness lemma). If T : ℂ M ∈ ℕ such that #[B(Crit(T) ∩ 𝒥 (T), e−n ) ∩ Critv(T n )] ≤ M,

∀n ∈ ℕ.

Proof. Suppose that T k (̃c), T l (̃c) ∈ B(c, e−n ) for some c ∈ Crit(T) ∩ 𝒥 (T), n ∈ ℕ, l > k ≥ 1 and some ̃c ∈ Crit(T). It then follows from Lemma 25.2.1 that k − l ≥ −C1 log(e−n ) = C1 n. So, the iterate T n has at most (C1−1 + 1)#Crit(T) critical values in B(c, e−n ). Take M := (C1−1 + 1)(#Crit(T))2 .

25.2 Two rules for critical points

� 1063

̂→ℂ ̂ be a rational function. Fix any a ∈ (0, 1) such that Let T : ℂ B(Crit(T) ∩ 𝒥 (T), a) ∩ Crit(T) = Crit(T) ∩ 𝒥 (T).

(25.36)

̂ → ℕ0 ∪ {∞} = {0, 1, 2, . . .} ∪ {∞} For each c ∈ Crit(T) ∩ 𝒥 (T), define the function kc : ℂ by the formula kc (x) := inf{k ∈ ℕ0 : x ∉ B(c, ae−(k+1) )},

(inf(0) = ∞).

For every x ∈ B(c, a) \ {c}, this is equivalent to kc (x) being the unique k ∈ ℕ0 such that x ∈ B(c, ae−k ) \ B(c, ae−(k+1) ). Obviously, kc (c) = ∞ and kc (x) = 0 for all x ∉ B(c, a). For any x, set also k(x) := max{kc (x) : c ∈ Crit(T) ∩ 𝒥 (T)}. As a rather straightforward repercussion of Lemma 25.2.1, we prove the following. ̂→ℂ ̂ is a rational function, then for every c ∈ Crit(T) ∩ Lemma 25.2.3 (Rule I’). If T : ℂ ̂ and every n ∈ ℕ, we have that 𝒥 (T), every x ∈ ℂ min{kc (x), kc (T n (x))} ≤ C2 n, where C2 = (1 + C1 )/C1 and C1 > 0 is the constant produced in Lemma 25.2.1. Proof. By way of contradiction, suppose that min{kc (x), kc (T n (x))} > C2 n for some c ∈ ̂ and n ∈ ℕ. Then x, T n (x) ∈ B(c, ae−(C2 n+1) ), where a comes Crit(T) ∩ 𝒥 (T), x ∈ ℂ from (25.36). So, T n (x) ∈ T n (B(c, ae−(C2 n+1) )) ∩ B(c, ae−(C2 n+1) ). In particular, T n (B(c, ae−(C2 n+1) )) ∩ B(c, ae−(C2 n+1) ) ≠ 0. It follows from Lemma 25.2.1 that n ≥ −C1 log(ae−(C2 n+1) ) = −C1 log a + C1 (C2 n + 1) ≥ C1 C2 n + C1 (1 − log a) 1 + C1 n + C1 (1 − log a) = (1 + C1 )n + C1 (1 − log a). > C1 C1 Hence, C1 n < −C1 (1 − log a). So n < −(1 − log a) < 0, a contradiction. As a direct consequence of Taylor’s theorem (see (25.4)), we record the following. ̂ → ℂ ̂ is a rational function, then there exists θ > 0 (deObservation 25.2.4. If T : ℂ pending on a) such that exp(−θ − (degc (T) − 1)kc (x)) ≤ |T ′ (x)| ≤ exp(θ − (degc (T) − 1)kc (x))

(25.37)

1064 � 25 Selected technical properties of rational functions ̂ for every c ∈ Crit(T) and every point x ∈ ℂ. ̂ → ℂ ̂ there exists a constant Lemma 25.2.5 (Rule II). For any rational function T : ℂ ̂ satisfy H > 0 such that if c ∈ Crit(T) ∩ 𝒥 (T), n ∈ ℕ and x ∈ ℂ kc (T j (x)) ≤ kc (T n (x)),

∀1 ≤ j < n,

(25.38)

then n−1

min{kc (x), kc (T n (x))} + ∑ kc (T j (x)) ≤ Hn.

(25.39)

j=1

Proof. The proof is by induction with respect to n ∈ ℕ. For n = 1, the statement is Lemma 25.2.3. The procedure for the inductive step will be as follows: Given x and n ∈ ℕ satisfying (25.38) we shall decompose the string (trajectory) x, T(x), . . . , T n (x) into two blocks: (a) x, T(x), . . . , T m (x), m ≤ n, for which we shall prove (25.39); (b) T m (x), . . . , T n (x) for which we can apply the inductive hypothesis. Summing up these two estimates will establish (25.39) for the string x, T(x), . . . , T n (x). We will demonstrate that it suffices to take H := α−1 (m(θ + α) + log 2 + 1 − log(e − 1)),

(25.40)

where α := degc (T) − 1 and θ arises from Observation 25.2.4. Denote k ′ := min{kc (x), kc (T n (x))} and

′

B := B(c, ae−(k −1) ),

where a comes from (25.36). If k ′ ≥ 1, denote by m the least positive integer such that kc (T m (x)) > min{kc (T m (z)) : z ∈ B} + 1

(25.41)

kc (T m (x)) ≥ k ′ .

(25.42)

or

In that case, observe that 1 ≤ m ≤ n and {x, T n (x)} ⊆ B(c, ae−k ) ⊆ B. If k ′ = 0, set ′

m = 1.

(25.43)

In all these cases, the sequence y = T m (x), T(y), . . . , T n−m (y) satisfies (25.38) and thereby kc (y) = min{kc (y), kc (T n−m (y))}.

25.2 Two rules for critical points

� 1065

Hence, by the inductive hypothesis, n−1

∑ kc (T j (x)) ≤ H(n − m).

(25.44)

j=m

Assume first that k ′ ≥ 1. By definition of m, neither (25.41) nor (25.42) holds for any 1 ≤ j < m. Therefore, for every 1 ≤ j < m and for every z ∈ B we have that kc (T j (x)) ≤ kc (T j (z)) + 1. Hence, by Observation 25.2.4, m−2

m−2

󵄨󵄨 m−1 ′ 󵄨 󵄨 ′ k+1 󵄨 k+1 󵄨󵄨(T ) (T(z))󵄨󵄨󵄨 = ∏ 󵄨󵄨󵄨T (T (z))󵄨󵄨󵄨 ≤ ∏ exp(θ − (degc (T) − 1)kc (T (z))) k=0

k=0

m−1

≤ exp((m − 1)θ − α ∑ [kc (T j (x)) − 1]),

∀z ∈ B.

j=1

By definition of B, we also know that kc (z) ≥ k ′ − 1 for every z ∈ B. By the same Observation 25.2.4, we then get |T ′ (z)| ≤ exp(θ − (degc (T) − 1)kc (z)) ≤ exp(θ − α(k ′ − 1)),

∀z ∈ B.

Thus, m−1 diam(T m (B)) ≤ sup |(T m )′ (z)| ≤ exp(mθ + mα − α[k ′ + ∑ kc (T j (x))]). diam(B) z∈B j=1

(25.45)

Moreover, for every z ∈ B we have T m (z) ∈ B(c, ae−kc (T

m

T m (x) ∈ B(c, ae−kc (T

m

(z))

) \ B(c, ae−(kc (T

m

(x))

) \ B(c, ae−(kc (T

m

(z))+1)

).

In particular, (x))+1)

).

If (25.41) holds but (25.42) does not, then there is z ∈ B for which kc (T m (x)) > kc (T m (z)) + 1 and we deduce that diam(T m (B)) ≥ |T m (z) − T m (x)| ≥ ae−(kc (T

m

≥ a(e−(kc (T = ae

m

(z))+1)

m

− ae−kc (T

(x))−1)

−kc (T (x))

− e−kc (T

(e − 1)

m m

(x))

(x))

)

1066 � 25 Selected technical properties of rational functions ′

≥ ae−k (e − 1). Combining this with (25.45) gives m−1 e−1 ≤ exp(m(θ + α) − α[k ′ + ∑ kc (T j (x))]). 2e j=1

Consequently, m−1

k ′ + ∑ kc (T j (x)) ≤ α−1 (m(θ + α) + log 2 + 1 − log(e − 1)) = H. j=1

(25.46)

If (25.42) holds, we obtain diam(T m (B)) ≥ ae−k (e − 1). ′

(25.47)

Otherwise, given any w ∈ T m (B) and having already asserted that x ∈ B, we would have |w − T m (x)| ≤ diam(T m (B)) < ae−k (e − 1). ′

Furthermore, as kc (T m (x)) ≥ k ′ by (25.42), the definition of kc would yield that T m (x) ∈ B(c, ae−kc (T

m

(x))

′

) ⊆ B(c, ae−k ),

i. e., |T m (x) − c| < ae−k . Then we would deduce that ′

|w − c| ≤ |w − T m (x)| + |T m (x) − c| < ae−k (e − 1) + ae−k = ae−(k −1) , ′

′

′

i. e., w ∈ B(c, ae−(k −1) ) =: B. So, we would infer that T m (B) ⊆ B. Since diam(B) ≤ 2a < ̂ the set ℂ ̂ \ B contains at least 3 points. Therefore, the family of iterates (T mj |B : diam(ℂ), ∞ ̂ B → ℂ) would be normal according to Montel’s theorem, which would contradict the ′

j=1

hypothesis that c ∈ 𝒥 (T). Consequently, (25.47) holds, and hence (25.46) is valid in this case, too. Finally, if k ′ = 0 then (25.43) holds and (25.46) is trivially satisfied. The combination of (25.44) and (25.46) means that (25.39) holds with H defined by (25.40). We now prove the following consequence of this lemma, i. e., of Rule II. ̂ → ℂ, ̂ there exist constants κ ∈ (0, 1] Lemma 25.2.6. For every rational function T : ℂ and β > 0 such that if x ∈ 𝒥 (T) and n > 2#(Crit(T) ∩ 𝒥 (T)), then there are integers j and l such that 0 ≤ j < j + l ≤ n, l ≥ nκ, and 󵄨󵄨 l ′ j 󵄨 −βn 󵄨󵄨(T ) (T (x))󵄨󵄨󵄨 ≥ e .

25.2 Two rules for critical points

� 1067

Proof. Temporarily fix c ∈ Crit(T) ∩ 𝒥 (T). Set i0 = 0. Let q(c) = i1 denote an integer in {0, 1, . . . , n} for which kc (T i (x)) attains its maximum; recall that even kc (T i (x)) = ∞ is possible if c = T i (x), but there exists at most one such i. Define inductively il to be the integer in {il−1 + 1, il−1 + 2, . . . , n} where kc (T i (x)) attains its maximum. This procedure terminates after finitely many steps, say u, and of course iu = n. Decompose the trajectory x, T(x), . . . , T n (x) into the following pieces (with overlapping ends): (x, . . . , T i1 (x)), (T i1 (x), . . . , T i2 (x)), . . . , (T iu−1 (x), . . . , T iu (x)). Observe that each of these pieces satisfies the hypotheses of Lemma 25.2.5 and kc (T i1 (x)) ≥ kc (T i2 (x)) ≥ ⋅ ⋅ ⋅ ≥ kc (T iu−1 (x)) ≥ kc (T iu (x)) = kc (T n (x)). Applying Lemma 25.2.5 to all these pieces and remembering that i1 = q(c), we obtain ik+1 −1

min{kc (T ik (x)), kc (T ik+1 (x))} + ∑ kc (T j (x)) ≤ H(ik+1 − ik ) j=ik +1

for all 0 ≤ k < u and thereby ik+1

i1 −1

u−1

j=0

k=1 j=ik +1

u−1

∑ kc (T j (x)) + ∑ ∑ kc (T j (x)) ≤ ∑ H(ik+1 − ik ) k=0

iu

−kc (T i1 (x)) + ∑ kc (T j (x)) ≤ H(iu − i0 ) j=i0 n

−kc (T q(c) (x)) + ∑ kc (T j (x)) ≤ Hn. j=0

(25.48)

Denote N := #(Crit(T) ∩ 𝒥 (T))

and Zn := {0, 1, . . . , n} \ {q(c) : c ∈ Crit(T) ∩ 𝒥 (T)}.

By hypothesis, n > 2N. Recalling that k(x) = max{kc (x) : c ∈ Crit(T) ∩ 𝒥 (T)}, we get from (25.48) that ∑ k(T j (x)) ≤ NHn.

j∈Zn

(25.49)

As the set Zn is a union of at most N + 1 blocks, there is at least one such block whose length is at least

1068 � 25 Selected technical properties of rational functions n−N +1 . N +1 Denote this block’s first element by j and the last one by j + l. Invoking (25.49), we have 0 ≤ j < j + l ≤ n,

l≥

n−N +1 n/2 1 ≥ = n N +1 N + 1 2(N + 1)

and

j+l−1

∑ k(T j (x)) ≤ NHn. i=j

From the chain rule and Observation 25.2.4, we know that j+l−1

󵄨󵄨 l ′ j 󵄨−1 j 󵄨󵄨(T ) (T (x))󵄨󵄨󵄨 ≤ exp(θl + (deg(T) − 1) ∑ k(T (x))). i=j

We therefore obtain that 󵄨󵄨 l ′ j 󵄨−1 󵄨󵄨(T ) (T (x))󵄨󵄨󵄨 ≤ exp(θl + (deg(T) − 1)NHn) ≤ exp([θ + (deg(T) − 1)NH]n). Hence, 󵄨󵄨 l ′ j 󵄨 󵄨󵄨(T ) (T (x))󵄨󵄨󵄨 ≥ exp(−[θ + (deg(T) − 1)NH]n). Take κ :=

1 2(N + 1)

and

β := θ + (deg(T) − 1)NH.

25.3 Expanding subsets of Julia sets ̂ →ℂ ̂ be a rational function. A forward T-invariant closed Definition 25.3.1. Let T : ℂ (so compact) subset X of the Julia set 𝒥 (T) is called expanding (or hyperbolic) if there exists q ∈ ℕ such that |(T q )′ (z)| > 1 for all z ∈ X. The proof of the following characterizations of expanding sets is left to the reader. ̂ → ℂ ̂ be a rational function. If X is a forward T-invariant Theorem 25.3.2. Let T : ℂ closed subset of the Julia set 𝒥 (T), then the following statements are equivalent: (a) The set X is expanding. (b) There exist λ > 1 and n ∈ ℕ such that |(T n )′ (z)| ≥ λ for all z ∈ X. (c) For each λ > 1, there exists n ∈ ℕ such that |(T n )′ (z)| ≥ λ for all z ∈ X. (d) There exist λ > 1 and c > 0 such that |(T n )′ (z)| ≥ cλn for all z ∈ X and all n ∈ ℕ. (e) For each λ > 1, there exists c > 0 such that |(T n )′ (z)| ≥ cλn for all z ∈ X and all n ∈ ℕ. ̂ → ℂ ̂ is called expanding (or hyperbolic) Definition 25.3.3. A rational function T : ℂ if the Julia set 𝒥 (T) is an expanding set for T, i. e., if there exists q ∈ ℕ such that |(T q )′ (z)| > 1 for all z ∈ 𝒥 (T).

25.3 Expanding subsets of Julia sets

� 1069

In this section, we will prove selected results about expanding subsets of Julia sets and identify significant subsets which are not expanding. We first provide basic examples of expanding sets. ̂ → ℂ ̂ be a rational function. Every finite collection of reProposition 25.3.4. Let T : ℂ pelling periodic orbits is an expanding set. Moreover, if X ⊆ 𝒥 (T) is an expanding set, then any forward T-invariant closed set Y ⊆ X is expanding as well. In particular, if T is expanding then every forward T-invariant closed subset of 𝒥 (T) is expanding. We now record two obvious observations, the latter of which is actually a reformulation of Proposition 25.3.4. ̂→ℂ ̂ be a rational function. If X ⊆ 𝒥 (T) is an expanding Observation 25.3.5. Let T : ℂ set for T, then X is an expanding set for every higher iterate T k , k ∈ ℕ. ̂ → ℂ ̂ be a rational function. If a forward T-invariant Observation 25.3.6. Let T : ℂ closed set X ⊆ 𝒥 (T) is not expanding, then no forward T-invariant closed set X ⊆ Y ⊆ 𝒥 (T) is expanding. Next, we prove the following result. ̂ →ℂ ̂ is a rational function with deg(T) ≥ 2, then none of the Theorem 25.3.7. If T : ℂ following forward T-invariant closed subsets of the Julia set 𝒥 (T) is expanding: (a) Rationally indifferent periodic orbits. (b) Boundaries of cycles of Siegel disks. (c) Boundaries of cycles of Herman rings. Proof. Item (a) is obvious. We shall prove items (b) and (c) simultaneously. The proof is similar to that of Theorem 24.5.1 but this time the holomorphic inverse branches originate from Lemma 25.1.11. Let G be either a Siegel disk or a Herman ring for T. Denote p−1 its period by p. Seeking a contradiction, suppose that 𝜕(⋃i=0 T i (G)) is an expanding set for T. By Observation 25.3.5, it is also an expanding set for T p . Since T p (𝜕G) ⊆ 𝜕G, Proposition 25.3.4 asserts that 𝜕G is an expanding set for T p . Replacing T by T ip with a sufficiently large integer i ∈ ℕ, we may assume that T(𝜕G) ⊆ 𝜕G

and

|T ′ (z)| ≥ 2,

∀z ∈ 𝜕G.

(25.50)

Then Crit(T) ∩ 𝜕G = 0, and let ε = dist(Crit(T), 𝜕G). Let Δ = B(0, 1) if G is a Siegel disk and

(25.51)

1070 � 25 Selected technical properties of rational functions Δ = A(0; r, 1) if G is a Herman ring, with r ∈ (0, 1) coming from Theorem 24.4.10(d). Let H:Δ→G be the analytic conjugacy resulting from Theorem 24.4.10(c)/(d) in the Siegel/Herman case, respectively. Fix w ∈ G such that 1 dist(w, 𝜕G) < θ(ε, 2), 2

(25.52)

where θ(ε, 2) > 0 comes from Lemma 25.1.11. Fix a point ξ ∈ 𝜕G such that |w − ξ| = dist(w, 𝜕G).

(25.53)

Let γ : [0, 1] → ℂ be the function γ(t) = w + t(ξ − w). Then γ(0) = w, γ(1) = ξ, γ([0, 1]) is the closed line segment from w to ξ, and γ([0, 1)) ⊆ G. Let F := H({z ∈ Δ : |z| = |H −1 (w)|}).

(25.54)

Note that F is a compact set (homeomorphic to a circle) and F ⊆ G. Therefore, dist(F, 𝜕G) > 0.

(25.55)

Since the map T|G : G → G is a holomorphic homeomorphism, its inverse T|−1 G : G → G is also a holomorphic homeomorphism. Fix n ∈ ℕ. By way of contradiction, suppose n −n that T|−n G (γ([0, 1))) ⊆ G. As T (T|G (γ([0, 1)))) is a compact set, it is closed and we would then have n ξ ∈ γ([0, 1]) = T n (T|−n G (γ([0, 1)))) ⊆ T (G) = G.

This contradiction shows that T|−n G (γ([0, 1))) ⊈ G. Hence, there exists a point −n ξn ∈ [T|−n G (γ([0, 1))) \ T|G (γ([0, 1)))] ∩ 𝜕G.

Then

25.3 Expanding subsets of Julia sets

� 1071

T n (ξn ) ∈ T n (T|−n G (γ([0, 1)))) \ G = γ([0, 1]) \ G = {ξ}. So, T n (ξn ) = ξ. By (25.50)–(25.51) and Lemma 25.1.11, there is a unique holomorphic ̂ of T −n sending ξ to ξn . By definition of ξn , there exists a branch Tξ−n : B(ξ, 2θ(ε, 2)) → ℂ n ∞ sequence (tk )k=1 in [0, 1) such that lim tk = 1

k→∞

and

lim T|−n G (γ(tk )) = ξn .

k→∞

Consequently, for every k ∈ ℕ large enough we have 1 −n T|−n G (γ(tk )) ∈ Tξn (B(ξ, θ(ε, 2))). 2 Hence, Tξ−n (γ(tk )) = T|−n G (γ(tk )) n

(25.56)

for all k large enough. It follows from (25.52)–(25.53) that γ([0, 1)) ⊆ B(w, |w − ξ|) ⊆ B(ξ, θ(ε, 2)) ∩ G. −n −n Thus, both holomorphic branches T|−n are defined on B(w, |w − ξ|). As G and Tξn of T both of them, restricted to B(w, |w − ξ|), are uniquely determined by their value at any point in B(w, |w − ξ|), we infer from (25.56) that

󵄨 󵄨󵄨 (Tξ−n )󵄨󵄨 = (T|−n G )󵄨󵄨B(w,|w−ξ|) . n 󵄨B(w,|w−ξ|) In particular, Tξ−n (w) = T|−n G (w). n

(25.57)

Using also the fact that T|−n G (w) ∈ F, we deduce that (w)| = |ξn − T|−n diam(Tξ−n (B(ξ, θ(ε, 2)))) ≥ |ξn − Tξ−n G (w)| ≥ dist(𝜕G, F). n n By (25.55), we conclude that lim inf diam(Tξ−n (B(ξ, θ(ε, 2)))) ≥ dist(𝜕G, F) > 0, n n→∞

contrary to Lemma 24.1.13 (recall that ξ ∈ 𝜕G ⊆ 𝒥 (T)). We now prove the following result in a “positive” direction, i. e., about expanding sets. The proof stems from [110].

1072 � 25 Selected technical properties of rational functions ̂ → ℂ ̂ be a rational function with deg(T) ≥ 2. If X ⊆ 𝒥 (T) Theorem 25.3.8. Let T : ℂ is a forward T-invariant closed set, homeomorphic to the Cantor set, such that the map T|X : X → X is expanding, then for every η > 0 there exists a forward T-invariant closed set X ⊆ Xη ⊆ B(X, η), homeomorphic to the Cantor set, such that the map T|Xη : Xη → Xη is open and expanding. In addition, if T|X is topologically transitive, then the map T|Xη is likewise topologically transitive. Proof. Since X ⊆ 𝒥 (T), we may assume that X ⊆ ℂ by performing a Möbius change of coordinates. As the set X is expanding, there exists q ∈ ℕ such that 󵄨󵄨 q ′ 󵄨󵄨 󵄨󵄨(T ) (x)󵄨󵄨 ≥ 3,

∀x ∈ X.

Denote by (⟨⋅, ⋅⟩x )x∈ℂ the standard Euclidean Riemannian metric on ℂ and define a new Riemannian metric (⟨⋅, ⋅⟩∗x )x∈ℂ by q−1

q−1

j=0

j=0

′ ′ ′ 󵄨 󵄨2 ⟨v, w⟩∗x := ∑ ⟨(T j ) (x)v, (T j ) (x)w⟩T j (x) = ∑ 󵄨󵄨󵄨(T j ) (x)󵄨󵄨󵄨 ⟨v, w⟩x .

Claim 1. There exist R > 0 and λ > 1 such that |T ′ (z)|∗ ≥ λ,

∀z ∈ B∗ (X, R),

where |T ′ (z)|∗ is the modulus of the derivative (scaling factor) given by |T ′ (z)|∗ =

‖T ′ (z)v‖T(z) . ‖v‖z

Proof of Claim 1. As X is compact, a continuity argument yields the existence of some R > 0 such that 󵄨󵄨 q ′ 󵄨󵄨 󵄨󵄨(T ) (z)󵄨󵄨 ≥ 2,

∀z ∈ B∗ (X, 2R).

(25.58)

For every point z ∈ ℂ and every tangent vector v ∈ Tz ℂ, we have q−1

󵄨󵄨 j ′ 󵄨󵄨2 2 ‖v‖∗2 z = ‖v‖z ∑ 󵄨󵄨(T ) (z)󵄨󵄨 .

(25.59)

j=0

Therefore, for every z ∈ B∗ (X, R), q−1

q−1

󵄨󵄨 j ′ 󵄨󵄨2 󵄨󵄨 j ′ 󵄨󵄨2 ′ 2 2 ′ 2 ‖T ′ (z)v‖∗2 T(z) = ‖T (z)v‖T(z) ∑ 󵄨󵄨(T ) (T(z))󵄨󵄨 = |T (z)| ‖v‖z ∑ 󵄨󵄨(T ) (T(z))󵄨󵄨 j=0

j=0

q−1

q−1

j=0

j=0

′ ′ ′ 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 = ‖v‖2z ∑ 󵄨󵄨󵄨(T j+1 ) (z)󵄨󵄨󵄨 = ‖v‖2z ( ∑ 󵄨󵄨󵄨(T j ) (z)󵄨󵄨󵄨 − 1 + 󵄨󵄨󵄨(T q ) (z)󵄨󵄨󵄨 )

25.3 Expanding subsets of Julia sets

󵄨󵄨 q ′ 󵄨󵄨2 ∗2 2 2 = ‖v‖∗2 z + ‖v‖z ⋅ (󵄨󵄨(T ) (z)󵄨󵄨 − 1) ≥ ‖v‖z + 3‖v‖z .

� 1073

(25.60)

It ensues from (25.59) that q−1

2 ′ 2j ‖v‖∗2 z ≤ ‖v‖z ∑ ‖T ‖∞ = j=0

‖T ′ ‖2q ∞−1 ⋅ ‖v‖2z , ‖T ′ ‖2∞ − 1

where ‖T ′ ‖∞ := sup{|T ′ (w)| : w ∈ B∗ (X, R)} < ∞. Inserting this into (25.60), we get ‖T ′ (z)v‖∗2 T(z) ≥ (1 + 3

‖T ′ ‖2∞ − 1 2q

‖T ′ ‖∞ − 1

)‖v‖∗2 z .

Taking λ = (1 + 3

‖T ′ ‖2∞ − 1 2q

‖T ′ ‖∞ − 1

1/2

)

completes the proof of Claim 1.

>1

◼

Denote by |w − z|∗ the distance between any two points w, z ∈ ℂ generated by the Riemannian metric (⟨⋅, ⋅⟩∗x )x∈ℂ . Fix any r ∈ (0, R] and define a relation ∼r on B∗ (X, r) by declaring that z ∼r w if and only if there are finitely many points x1 , x2 , . . . , xj in X such that z ∈ B∗ (x1 , r), w ∈ B∗ (xj , r), and B∗ (xi , r) ∩ B∗ (xi+1 , r) ≠ 0 for all i = 1, 2, . . . , j − 1. It is immediate that ∼r is an equivalence relation on X. Claim 2. Each equivalence class of the relation ∼r is open in ℂ and intersects X. Moreover, it is a union of open ∗ balls of radius r centered at points in X. Proof of Claim 2. The latter assertion is obvious. For the former one, suppose that z ∈ B∗ (X, r) and [z]r is the equivalence class of z. Let w ∈ [z]r . This means that z ∼r w. Let x1 , . . . , xk ∈ X be points witnessing this relation between z and w. Then w ∈ B∗ (xk , r) and each element of B∗ (xk , r) is in relation with z through the same points x1 , . . . , xk . So, xk ∈ B∗ (xk , r) ⊆ [z]r . The proof of Claim 2 is complete. ◼ It follows from this claim and the compactness of X that the number N(r) of equivalence classes of ∼r is finite. It is obvious that if 0 < r ≤ s ≤ R and z ∼r w then z ∼s w, whence each equivalence class of the relation ∼r is contained in a unique equivalence class of the relation ∼s . Thus, N(s) ≤ N(r). This means that the function (0, R] ∋ r 󳨃→ N(r)

1074 � 25 Selected technical properties of rational functions is monotone decreasing. Since it takes only positive integer values, the set Y of its discontinuity points, or equivalently its points of nonconstantness, is a countable set whose sole accumulation point may be 0. Claim 3. If r ∈ (0, R]\Y , then the closures of any two different equivalence classes of the relation ∼r are disjoint. Proof of Claim 3. Let δ > 0 be so small that B(r, 2δ) ∩ Y = 0. Then N(r + δ) = N(r).

(25.61)

Seeking a contradiction, suppose that A and B are two distinct classes of the relation ∼r such that A ∩ B ≠ 0. Let z ∈ A ∩ B. Then A ∩ B∗ (z, δ) ≠ 0 and B ∩ B∗ (z, δ) ≠ 0. Let x ∈ A ∩ B∗ (z, δ)

and

y ∈ B ∩ B∗ (z, δ).

By Claim 2, there are two points x ′ , y′ ∈ X such that x ∈ B∗ (x ′ , r) and

y ∈ B∗ (y′ , r).

Hence, x ′ ∈ A, y′ ∈ B and z ∈ B∗ (x ′ , r + δ) ∩ B∗ (y′ , r + δ) ≠ 0. Thus, x ′ ∼r+δ y′ . Therefore, A and B are contained in the same class of the relation ∼r+δ . So, N(r + δ) ≤ N(r) − 1, contrary to (25.61). The proof of Claim 3 is complete. ◼ By set Y ’s structure, there exists a strictly decreasing sequence (rk )∞ k=1 converging to 0 such that rk ∈ (0, R]\Y for all k ∈ ℕ. Denote by Ek the collection of all equivalence classes of the relation ∼rk . Claim 4. lim max{diam∗ (A) : A ∈ Ek } = 0. k→∞

Proof of Claim 4. Fix ε > 0. Since X is a topological Cantor set, there exists a finite partition α of X consisting of closed sets such that diam∗ (A) < ε,

∀A ∈ α.

(25.62)

As lim rk = 0, there is N ∈ ℕ such that k→∞

rk
ε for some D ∈ Ek . Then there are two points z, w ∈ D such that |z − w|∗ > ε. In view of (25.62), there are then two different atoms B, C ∈ α such that z ∈ B and w ∈ C. As B ∩ D ≠ 0 and C ∩ D ≠ 0, we also have ̂ contrary to (25.64). Thus, (25.65) holds for all ̂ ∩ D ≠ 0 and Ĉ ∩ D ≠ 0. Hence, B ̂ = D = C, B k ≥ N. This finishes the proof of Claim 4. ◼ Now, it follows from (25.58) that T ′ (x) ≠ 0 for all x ∈ X. So it ensues from the inverse function theorem (Theorem A.3.13) along with a standard compactness argument that there exists R1 ∈ (0, R] such that for every x ∈ X there is a unique holomorphic branch Tx−1 : B∗ (T(x), R1 ) → ℂ of T −1 sending T(x) to x. Decreasing R1 if necessary, we have that Tx−1 (B∗ (T(x), R1 )) ⊆ B∗ (x, R1 ).

(25.66)

By Claim 4, there exists q ∈ ℕ such that max{rq , diam∗ (A)} < min{η∗ , R1 /4},

∀A ∈ Eq ,

where η∗ ∈ (0, η) is so small that B∗ (X, η∗ ) ⊆ B(X, η). Given A ∈ Eq , fix an element yA ∈ A ∩ X. Pick also any x ∈ X ∩ T −1 (yA ). If z, w ∈ A, then z ∼rq w and Claim 1 affirms

that Tx−1 (z) ∼λ−1 rq Tx−1 (w). Hence, Tx−1 (z) ∼rq Tx−1 (w). Thus, there is a unique element Ax ∈ Eq such that Tx−1 (A) ⊆ Ax .

(25.67)

Take V := Eq for (finite) vertex set. For each vertex v = A ∈ Eq = V , define Xv := A and Wv := B∗ (yA , R1 ). Take E := {(A, x) : A ∈ Eq , x ∈ X ∩ T −1 (yA )} for (finite) alphabet (i. e., set of edges). For each letter/edge e = (A, x) ∈ E, respectively define the initial and terminal vertices of e by i(e) = Ax and t(e) = A, where Ax arises from (25.67). Then associate the map (generator) φe = φA,x := Tx−1 : B∗ (yA , R1 ) → ℂ. In light of Claim 1 and (25.66)–(25.67), the family Φ := {φe }e∈E forms a CGDS (i. e., a CGDMS whose matrix solely reflects the system’s underlying graph; for more information on CGDMSs, see Chapter 19). By Claim 3, this CGDMS satisfies the Strong Separation Condition (SSC; also called the Boundary Separation Condition (BSC); see Definitions 19.7.5–19.7.6). Consequently, the limit set 𝒥 of the system Φ is compact and the corresponding coding map π : EG∞ → 𝒥 is a homeomorphism, where G is the incidence matrix of the system Φ. We will prove several claims about 𝒥 . The first is immediate. Claim 5. 𝒥 ⊆ B∗ (X, η∗ ) ⊆ B(X, η).

1076 � 25 Selected technical properties of rational functions Claim 6. X ⊆ 𝒥 . Proof of Claim 6. Let x ∈ X. For every n ≥ 0, consider the holomorphic inverse branch TT−1n (x) : B∗ (T n+1 (x), R1 ) → ℂ and let xn := TT−1n (x) (yA n+1 ). By Claim 1, we conclude that T

(x)

π(((AT n+1 (x) , xn ))n=0 ) = x. ∞

So x ∈ 𝒥 . Thus, X ⊆ 𝒥 and Claim 6 is proved.

◼

Claim 7. T(𝒥 ) ⊆ 𝒥 . Proof of Claim 7. Let x ∈ 𝒥 . Then there exists a (unique) sequence ω ∈ EG∞ such that x = π(ω) = φω1 (π(σ(ω))). Therefore, T(x) = T ∘ φω1 (π(σ(ω))) = π(σ(ω)) ∈ 𝒥 . Claim 7 is established.

◼

As an immediate consequence of Claims 1 and 7, we get the following. Claim 8. The map T : 𝒥 → 𝒥 is expanding. Claim 9. The maps T : 𝒥 → 𝒥 and σ : EG∞ → EG∞ are topologically conjugate. More precisely, the map π : EG∞ → 𝒥 is a topological conjugacy, i. e., a homeomorphism such that the following diagram commutes: EG∞

σ ? ∞ EG π

π

?

𝒥

T

? ? 𝒥

i. e., π ∘ σ = T ∘ π. Proof of Claim 9. We have already observed that π is a homeomorphism. Let ω ∈ EG∞ . Then T ∘ π(ω) = T ∘ φω1 (π(σ(ω))) = π(σ(ω)). So T ∘ π = π ∘ σ, and Claim 9 is proved. ◼ As the map σ : EG∞ → EG∞ is open, Claim 9 entails the following. Claim 10. The map T : 𝒥 → 𝒥 is open. If the map T : X → X is transitive, then the incidence matrix G is irreducible, whence the shift map σ : EG∞ → EG∞ is transitive. Thus, invoking Claim 9 again, we obtain the following. Claim 11. If the map T : X → X is transitive, then the map T : 𝒥 → 𝒥 is also transitive. Setting Xη := 𝒥 completes the proof of Theorem 25.3.8.

25.3 Expanding subsets of Julia sets

� 1077

In Definition 16.2.1, we adopted a definition a little too narrow. In fact, all results derived about conformal repellers in Volume 2 are valid when part (d) of their definition is replaced by the following weaker condition: (d ′ ) There exists λ > 1 and q ∈ ℕ such that |(T q )′ (x)| ≥ λ for all x ∈ X. As a complementary result to Theorem 25.3.8, we shall prove the following. ̂ →ℂ ̂ is a rational function with deg(T) ≥ 2 and X ⊆ 𝒥 (T) is Theorem 25.3.9. If T : ℂ an expanding set such that the restriction T|X : X → X is open and transitive, then there ̂ for which the triple (X, U, T) is a conformal expanding exists an open set X ⊆ U ⊆ ℂ repeller. Proof. Since X is expanding, there are λ > 1 and q ∈ ℕ such that |(T q )′ (x)| ≥ λ,

∀x ∈ X.

(25.68)

By standard compactness arguments, there exists δ > 0 such that T q |B(x,4δ) is 1-to-1,

∀x ∈ X

(25.69)

and |(T q )′ (z)| ≥ √λ,

∀z ∈ B(X, 4δ) := ⋃ B(x, 4δ). x∈X

(25.70)

Consequently, for every x ∈ X there is a unique holomorphic inverse branch Tx ̂ such that Tx−q (T q (x)) = x. By (25.70), we have that B(T q (x), 4δ) → ℂ

−q

T q (B(x, r)) ⊇ B(T q (x), √λ r),

|(Tx−q )′ (z)| ≤ λ−1/2 ,

and ρ(Tx−q (w), Tx−q (z)) ≤ λ−1/2 ρ(w, z),

:

∀r ∈ [0, 4δ], ∀z ∈ B(x, 4δ), ∀x ∈ X (25.71)

∀w, z ∈ B(x, 4δ), ∀x ∈ X.

(25.72)

Applying Corollary 4.3.6 to the open distance expanding map T q : X → X, there exists α ∈ (0, δ) such that every (1 + Lip(T q ))α-pseudo-orbit for T q |X is δ-shadowed by a point in X, where Lip(T q ) is a Lipschitz constant for T q . Let U = B(X, α). −n All we need to do for the proof of this theorem is to show that ⋂∞ n=0 T (U) ⊆ X. Of course, it is sufficient to establish that ∞

⋂ T −qn (U) ⊆ X.

n=0

1078 � 25 Selected technical properties of rational functions So, let ∞

z ∈ ⋂ T −qn (U). n=0

Then, for every n ≥ 0 there exists xn ∈ X such that T qn (z) ∈ B(xn , α). Therefore ρ(xn+1 , T q (xn )) ≤ ρ(xn+1 , T q(n+1) (z)) + ρ(T q(n+1) (z), T q (xn )) < α + ρ(T q (T qn (z)), T q (xn )) ≤ α + Lip(T q ) ρ(T qn (z), xn ) < [1 + Lip(T q )]α.

q q This means that (xn )∞ n=0 is a (1 + Lip(T ))α-pseudo-orbit for T |X . Thus, it is δ-shadowed by a point x ∈ X. Hence, for all n ≥ 0,

ρ(T qn (z), T qn (x)) ≤ ρ(T qn (z), xn ) + ρ(xn , T qn (x)) < α + δ < 2δ.

(25.73)

It follows from (25.69) and (25.72) that T qn (z) = TT qn (x) (T q(n+1) (z)). −q

Hence, by applying (25.72) again, we get ρ(T qn (z), T qn (x)) ≤ λ−1/2 ρ(T q(n+1) (z), T q(n+1) (x)). So, by induction and (25.73), we obtain that ρ(z, x) ≤ 2δλ−n/2 ,

∀n ≥ 0.

Thus ρ(z, x) = 0, whence z = x ∈ X. We will also need the following general result about topological pressure. Theorem 25.3.10. Let T : X → X be a topological dynamical system, i. e. T is a continuous self-map of a compact metrizable space X. For any (continuous) potential φ : X → ℝ, P(T, φ) = sup{P(T|F , φ|F ) : F ⊆ X closed, forward T-invariant and T|F transitive}. Proof. Obviously, P(T, φ) ≥ sup{P(T|F , φ|F )}. We now prove the opposite inequality. If μ is an ergodic T-invariant Borel probability measure on X, then supp(μ) is a closed, forward T-invariant subset of X. Moreover, according to Theorem 8.2.27, the subsystem T|supp(μ) is topologically transitive. By the variational principle (Theorem 12.1.1), we further know that

25.4 Lyubich’s geometric lemma

P(T|supp(μ) , φ|supp(μ) ) ≥ hμ (T|supp(μ) ) +

∫

� 1079

φ dμ = hμ (T) + ∫ φ dμ. X

supp(μ)

It therefore follows from this same variational principle and Remark 12.1.2 that P(T, φ) = sup{hμ (T) + ∫ φ dμ} ≤ sup{P(T|supp(μ) , φ|supp(μ) )} ≤ sup{P(T|F , φ|F )}, μ

X

μ

F

where the leftmost two suprema are taken over all ergodic T-invariant Borel probability measures μ on X while the rightmost supremum is taken over all closed, forward T-invariant subsets F of X such that the subsystem T|F is transitive. We now introduce some quantities that are offshoots of the topological pressure function. ̂ → ℂ ̂ be a rational function with deg(T) ≥ 2. For every Definition 25.3.11. Let T : ℂ t ∈ ℝ, we denote 󵄨 Pexp (t) := sup{P(T|X , −t log |T ′ |) 󵄨󵄨󵄨 X ⊆ 𝒥 (T) expanding forward T-invariant closed set} 󵄨󵄨 X ⊆ 𝒥 (T) expanding forward T-invariant closed set 󵄨 P∗exp (t) := sup{P(T|X , −t log |T ′ |) 󵄨󵄨󵄨 } 󵄨󵄨 and T|X : X → X transitive 󵄨 ′ 󵄨󵄨󵄨 X ⊆ 𝒥 (T) expanding forward T-invariant closed set P∗∗ (t) := sup{P(T| , −t log |T |) } 󵄨󵄨 X exp 󵄨󵄨 and T|X : X → X open and transitive As a direct consequence of Theorems 25.3.8–25.3.10, we get the following assertion on the pressure function. ̂→ℂ ̂ is a rational function with deg(T) ≥ 2, then for every Proposition 25.3.12. If T : ℂ t ∈ ℝ we have Pexp (t) = P∗exp (t) = P∗∗ exp (t).

25.4 Lyubich’s geometric lemma Let (X, ρ) be a metric space and let Z ⊆ X be a finite set. Fix R > ε > 0 and β ∈ (0, 1). Let #[Z; R, ε, β] be the minimum of the cardinalities of all sets F ⊆ X such that B(Z, R) \ B(Z, ε) ⊆ ⋃ B(x, βρ(x, Z)). x∈F

The next lemma’s proof and formulation are adapted from [79].

1080 � 25 Selected technical properties of rational functions Lemma 25.4.1. Let Z ⊂ ℂ be a nonempty finite set. If R > ε > 0 and β ∈ (0, 1), then #[Z; R, ε, β] ≤ A(β)(#Z)2 log(R/ε), where A(β) is a constant which depends only on β (and thus is independent of Z, R and ε). Proof. Fix r > 0. For every w ∈ Z, we have 𝜕B(w, r) \ B(Z, r) = 𝜕B(w, r) \ B(Z\{w}, r) =

⋃

L∈𝒜w (r)

L,

where 𝒜w (r) is a finite collection of mutually disjoint closed topological arcs in 𝜕B(w, r). Observe that #𝒜w (r) ≤ #Z − 1. Consider an arbitrary arc L ∈ 𝒜w (r). Denote its endpoints by aL and bL . Let u1 , u2 , . . . , usL , where sL ∈ ℕ, be a unique sequence of points in L such 󵄨 󵄨 󵄨 󵄨 that u1 = aL , 󵄨󵄨󵄨∠(uj , w, uj+1 )󵄨󵄨󵄨 = β for all 1 ≤ j < sL and 0 < 󵄨󵄨󵄨∠(usL , w, bL )󵄨󵄨󵄨 ≤ β. So (sL − 1)β + 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨∠(usL , w, bL )󵄨󵄨 = 󵄨󵄨∠(aL , w, bL )󵄨󵄨. Then 󵄨󵄨 󵄨 󵄨∠(a , w, bL )󵄨󵄨󵄨 + 1. sL < 󵄨 L β Thus, ∑

L∈𝒜w (r)

sL ≤

2π 2π#Z 2π + #𝒜w (r) ≤ + #Z − 1 ≤ . β β β

Hence, ∑

∑

w∈Z L∈𝒜w (r)

sL ≤

2π (#Z)2 . β

(25.74)

As 𝜕B(Z, r) ⊆ ⋃w∈Z [𝜕B(w, r) \ B(Z, r)], it follows from our construction that 𝜕B(Z, r) ⊆ ⋃

sL

⋃

⋃ B(uj , βr).

w∈Z L∈𝒜w (r) j=1

But since the set on the left is open while the set on the right is compact, there exists η = η(β) > 0 such that sL

⋃

⋃

⋃ B(uj , βr) ⊇ B(𝜕B(Z, r), ηr) ⊇ B(Z, (1 + η)r) \ B(Z, r).

w∈Z L∈𝒜w (r) j=1

(25.75)

Obviously, 1 + η is independent of R and ε. We can clearly have it independent of Z and, by a similarity argument, independent of r.

25.5 Two auxiliary partitions

� 1081

Let N be the largest integer such that (1 + η)N ε < R. Then N
0 there exist a number L > 0 and a Lebesgue measurable set A ⊆ I such that Leb(I \ A) < ε and the function k : I → ℝ is Lipschitz continuous at each point of A with Lipschitz constant not exceeding L. Proof. See Lemma 16.5.4. Corollary 25.5.2. For every Borel probability measure ν on a compact metric space (X, ρ) and for every r > 0 there exists a finite Borel partition 𝒫 = {Pj }M j=1 of X with the following three properties:

1082 � 25 Selected technical properties of rational functions (a)

ν(P) > 0,

(b)

(25.77)

∀P ∈ 𝒫 .

diam(𝒫 ) < r.

(25.78)

(c) There exists a constant C > 0 such that for every η > 0 it holds that ν(𝜕η 𝒫 ) ≤ Cη,

where

M

𝜕η 𝒫 := ⋂ ⋃ B(Pk , η). j=1 k =j̸

(25.79)

Proof. See Corollary 16.5.5. 25.5.2 Exponentially large partition Corollary 25.5.3. Let ν be a Borel probability measure on a compact metric space (X, ρ) and let T : X → X be a Borel endomorphism of X preserving the measure ν. Then for every r > 0 there exists a finite partition 𝒫 of X into Borel sets of positive ν-measure with diam(𝒫 ) < r such that for every δ > 0 and ν–a. e. x ∈ X there exists an integer N(x) ≥ 0 for which B(T n (x), e−nδ ) ⊆ 𝒫 (T n (x)),

∀n ≥ N(x),

(25.80)

where 𝒫 (y) denotes the atom P ∈ 𝒫 such that y ∈ P. Proof. See Corollary 16.5.7. 25.5.3 Mañé’s partition In this subsection, basically following Mañé’s book [83], we construct the so-called Mañé partition, which will play an important role in the proof of part of the volume lemma given in the next section. We begin with the following elementary fact. Lemma 25.5.4. If xn ∈ (0, 1) for every n ∈ ℕ and ∑∞ n=1 nxn < ∞, then ∞

∑ −xn log xn < ∞.

n=1

Proof. Let S := {n ∈ ℕ | − log xn ≥ n}. Then ∞

∞

∑ −xn log xn = ∑ −xn log xn + ∑ −xn log xn ≤ ∑ nxn + ∑ −xn log xn .

n=1

n∉S

n∈S

n=1

n∈S

25.5 Two auxiliary partitions

� 1083

Since n ∈ S means that xn ≤ e−n and since log t ≤ 2√t for all t ≥ 1, we have ∑ xn log

n∈S

∞ ∞ 1 1 ≤ 2 ∑ xn √ ≤ 2 ∑ e−n/2 < ∞. xn xn n=1 n=1

The next lemma is the main, and simultaneously the last, result of this subsection. Lemma 25.5.5 (Mañé’s partition). If μ is a Borel probability measure supported on a bounded subset M of a (finite-dimensional) Euclidean space and ρ : M → (0, 1] is a Borel measurable function such that log ρ is integrable with respect to μ (i. e., ∫M log ρ dμ > −∞), then there exists a countable Borel partition 𝒫 of M such that Hμ (𝒫 ) < ∞

and diam(𝒫 (x)) ≤ ρ(x) for μ-a.e. x ∈ M,

where Hμ (𝒫 ) is the measure-theoretic entropy of 𝒫 with respect to μ (cf. Definition 9.4.1). Proof. Let d be the dimension of the Euclidean space containing M. Since M is bounded, there is a constant C > 0 such that for every 0 < r < 1 there exists a Borel partition 𝒫r of M with diam(𝒫r ) ≤ r and #𝒫r ≤ Cr −d . For every n ≥ 0, set Un := {x ∈ M | e−(n+1) < ρ(x) ≤ e−n }. The family 𝒰 := {Un }∞ n=0 is clearly a Borel partition of M. Since log ρ is a nonpositive integrable function, we have ∞

∞

n=1

n=1 U n

∑ −nμ(Un ) ≥ ∑ ∫ log ρ dμ = ∫ log ρ dμ > −∞,

(25.81)

M

so that ∞

∑ nμ(Un ) < ∞.

(25.82)

n=1

For every n ≥ 0, let rn := e−(n+1) and 𝒬n := 𝒫rn |Un be the Borel partition of Un whose atoms are of the form Q ∩ Un , where Q ∈ 𝒫rn . Thereafter, define the Borel partition ∞ 𝒫 := ⋃n=0 𝒬n . Then ∞

Hμ (𝒫 ) := − ∑ μ(P) log μ(P) = ∑ (− ∑ μ(P) log μ(P)). P∈𝒫

n=0

P∈𝒬n

For every n ≥ 0, let μUn be the conditional measure of μ on Un . Then − ∑ μ(P) log μ(P) = μ(Un ) ∑ − P∈𝒬n

P∈𝒬n

μ(P) μ(P) μ(P) log − μ(Un ) ∑ log μ(Un ) μ(Un ) μ(Un ) μ(U n) P∈𝒬 n

1084 � 25 Selected technical properties of rational functions = μ(Un )HμU (𝒬n ) − μ(Un ) log μ(Un ) n

≤ μ(Un ) log #𝒬n − μ(Un ) log μ(Un )

≤ μ(Un ) log #𝒫rn − μ(Un ) log μ(Un )

≤ μ(Un )(log C − d log rn ) − μ(Un ) log μ(Un )

= μ(Un ) log C + d(n + 1)μ(Un ) − μ(Un ) log μ(Un ). Summing over all n ≥ 0, we obtain ∞

∞

n=0

n=0

Hμ (𝒫 ) ≤ log C + d + d ∑ nμ(Un ) + ∑ −μ(Un ) log μ(Un ). In light of (25.82) and Lemma 25.5.4, we conclude that Hμ (𝒫 ) is finite. Moreover, if x ∈ Un then the atom 𝒫 (x) is contained in some atom of 𝒫rn . Therefore, diam(𝒫 (x)) ≤ rn = e−(n+1) < ρ(x). As ⋃∞ n=0 Un = M, the relation diam(𝒫 (x)) < ρ(x) holds for all x ∈ M.

25.6 Miscellaneous facts 25.6.1 Miscellaneous general facts Lemma 25.6.1. Let d ∈ ℕ and a ∈ ℝd . If μ is a finite Borel measure on ℝd such that the function ℝd ∋ z 󳨃󳨀→ log ‖z − a‖ is μ-integrable, then for every C > 0 and every 0 < t < 1 we have ∞

∑ μ(B(a, Ct n )) < ∞.

n=1

Proof. As μ is finite and as for any 0 < t < s < 1 there exists N ∈ ℕ such that Ct n ≤ sn for all n ≥ N, we may assume that C = 1 without losing any generality. Given b ∈ ℝd and two numbers 0 ≤ r < R, consider the “half-closed, half-open” annulus A(b; r, R) := {z ∈ ℝd : r ≤ ‖z − b‖ < R}. Since − log(t n ) ≤ − log ‖z − a‖ for every z ∈ B(a, t n ), we get ∞

∞

n=1

n=1

∑ μ(B(a, t n )) = ∑ μ(A(a; t n+1 , t n )) ≤ ≤

−1 ∞ ∑ − log(t n )μ(A(a; t n+1 , t n )) log t n=1

−1 ∫ − log ‖z − a‖ dμ(z) < ∞. log t B(a,t)

25.6 Miscellaneous facts

� 1085

The following lemma is obvious and we state it without proof. Lemma 25.6.2. Let (X, ρ) be a metric space and let ν be a Borel probability measure on X. Fix x ∈ X and α > 0. Suppose that there exist C1 , C2 ≥ 1 and r0 > 0 with the following property: For every 0 < r < r0 there are 0 < r1 ≤ C1 r and r2 ≥ C2−1 r for which ν(B(x, r1 )) ≥ C1−1 r α

and

ν(B(x, r2 )) ≤ C2 r α .

Then there exists C3 ≥ 1 such that C3−1 ≤

ν(B(x, r)) ≤ C3 , rα

∀0 < r ≤ 1,

i. e., ν is a geometric measure with exponent α (cf. Definition 15.6.12).

25.6.2 Miscellaneous facts about rational functions The following fact is well known for all topologically exact, open, distance expanding systems, including T : 𝒥 (T) → 𝒥 (T) (cf. Proposition 13.8.6). We repeat its short proof in this complex-analytic setting. ̂→ℂ ̂ is a rational function with deg(T) ≥ 2, then Proposition 25.6.3. If T : ℂ lim sup{dist(z, T −n (ζ )) : z, ζ ∈ 𝒥 (T)} = 0.

n→∞

Proof. Fix ε > 0. As 𝒥 (T) is compact, there is a finite set J ⊆ 𝒥 (T) such that ⋃ B(w, ε/2) ⊇ 𝒥 (T).

w∈J

Since the map T : 𝒥 (T) → 𝒥 (T) is topologically exact (per Corollary 24.3.5), for every w ∈ J there exists Nw ≥ 0 such that T Nw (B(w, ε/2)) ⊇ 𝒥 (T). Setting Nε = max{Nw : w ∈ J} yields T n (B(w, ε/2)) ⊇ 𝒥 (T),

∀w ∈ J, ∀n ≥ Nε .

Take z, ζ ∈ 𝒥 (T). Then z ∈ B(w, ε/2) for some w ∈ J. For every n ≥ Nε , there exists ζn ∈ B(w, ε/2) such that T n (ζn ) = ζ . Thus, dist(z, T −n (ζ )) ≤ |z − ζn | ≤ |z − w| + |w − ζn | < ε/2 + ε/2 = ε. For every rational function, we now establish the existence of finite Borel partitions of the Riemann sphere of arbitrarily small diameters on the atoms of which the function is injective.

1086 � 25 Selected technical properties of rational functions ̂→ℂ ̂ be a rational function with deg(T) ≥ 2. For every δ > 0, Lemma 25.6.4. Let T : ℂ ̂ such that diam(α(δ)) < δ and the restriction there exists a finite Borel partition α(δ) of ℂ ̂ is injective on every A ∈ α(δ). T|A : A → ℂ Proof. Fix δ > 0. Let c ∈ Crit(T). It follows from Theorem 25.1.1 that there exist ηc ∈ (0, δ) deg (T) and degc (T) mutually disjoint Borel sets {Bk (c)}k=1c such that degc (T)

⋃ Bk (c) = B(c, ηc ) and T|Bk (c) is injective for every 1 ≤ k ≤ degc (T).

k=1

̂ On the other hand, if z ∈ ℂ\Crit(T) then there exists ηz ∈ (0, δ) such that the map T|B(z,ηz ) is injective. ̂ is compact, there is a finite subcover of the open cover {B(w, ηw )} ̂ . ThereSince ℂ w∈ℂ fore, there is a finite subcover ℬ = {B1 , . . . , Bn } of the Borel cover ̂ {B(z, ηz ) : z ∈ ℂ\Crit(T)} ∪ {Bk (c) : c ∈ Crit(T), 1 ≤ k ≤ degc (T)}. ̂ from the cover ℬ in the following We form the sought-after Borel partition α(δ) of ℂ standard way: A1 := B1

and Aj := Bj \ ⋃ Bi , i 1 for all z ∈ 𝒥 (T). As a direct consequence of Theorem 25.3.2, we get the following. ̂ →ℂ ̂ is a rational function, then the following statements are Theorem 26.1.1. If T : ℂ equivalent: ̂→ℂ ̂ is expanding. (a) The map T : ℂ (b) There exist λ > 1 and n ∈ ℕ such that |(T n )′ (z)| ≥ λ for all z ∈ 𝒥 (T). (c) For each λ > 1, there exists n ∈ ℕ such that |(T n )′ (z)| ≥ λ for all z ∈ 𝒥 (T). (d) There exist λ > 1 and c > 0 such that |(T n )′ (z)| ≥ cλn for all z ∈ 𝒥 (T) and all n ∈ ℕ. (e) For each λ > 1, there exists c > 0 such that |(T n )′ (z)| ≥ cλn for all z ∈ 𝒥 (T) and all n ∈ ℕ. As a straightforward repercussion, we have the following. ̂ → ℂ ̂ is a rational function, then the following statements Proposition 26.1.2. If T : ℂ are equivalent: ̂→ℂ ̂ is expanding. (a) The map T : ℂ ̂→ℂ ̂ is expanding. (b) There exists n ∈ ℕ such that the iterate T n : ℂ n ̂ ̂ is expanding. (c) For every n ∈ ℕ, the iterate T : ℂ → ℂ Apart from Theorem 26.1.1 itself, this proposition adds the decisive argument to coin the concerned functions as being expanding. Recall that the finite set ̂ | T ′ (z) = 0} Crit(T) := {z ∈ ℂ is the critical set of T while the countable set https://doi.org/10.1515/9783110769876-026

1088 � 26 Expanding (or hyperbolic), subexpanding, and parabolic rational functions ∞

PC(T) := ⋃ T n (Crit(T)) n=1

is the postcritical set of T. Note that PC(T) = PC(T) ∪ ω(Crit(T)). We now characterize expandingness as the disjointness of the Julia set from the closure of the postcritical set. This provides further insight into the role played by the critical points and their orbits. ̂→ℂ ̂ with deg(T) ≥ 2 is expanding if and only Theorem 26.1.3. A rational function T : ℂ if 𝒥 (T) ∩ PC(T) = 0.

̂ →ℂ ̂ is expanding. The very definition of this Proof. Assume first that the map T : ℂ property implies that 𝒥 (T) ∩ Crit(T) = 0. Since 𝒥 (T) is completely T-invariant, it follows that 𝒥 (T) ∩ PC(T) = 0. Suppose for a contradiction that 𝒥 (T) ∩ PC(T) ≠ 0. As PC(T) = PC(T)∪ω(Crit(T)), this means that 𝒥 (T)∩ω(Crit(T)) ≠ 0. The finiteness of Crit(T) implies that there exists a point c ∈ Crit(T) ⊆ ℱ (T) such that 𝒥 (T) ∩ ω(c) ≠ 0.

(26.1)

By virtue of Sullivan’s nonwandering theorem (Theorem 24.4.12), there then exist an integer n ≥ 0 and a periodic connected component V of ℱ (T) such that T n (c) ∈ V . The classification theorem 24.4.10 then affirms that the only possibility for (26.1) to hold is that V be the immediate basin of attraction to some simple rationally indifferent periodic point ζ ∈ 𝜕V ⊆ 𝒥 (T). But then statement (e) in Theorem 26.1.1 fails for z = ζ . This contradiction shows that 𝒥 (T) ∩ PC(T) = 0. This proves the direct implication. For the converse, assume that 𝒥 (T) ∩ PC(T) = 0. The compactness of the sets 𝒥 (T) and PC(T) implies that there is δ > 0 for which B(𝒥 (T), 2δ) ∩ PC(T) = 0.

(26.2)

̂→ℂ ̂ is not expanding. This means By way of contradiction, suppose that the map T : ℂ ∞ that there exist a strictly increasing sequence (nk )k=1 of positive integers and a sequence (ξk )∞ k=1 of points in 𝒥 (T) such that 󵄨󵄨 nk ′ 󵄨 󵄨󵄨(T ) (ξk )󵄨󵄨󵄨 < 2,

∀k ∈ ℕ.

(26.3)

Passing to a subsequence if necessary, we may assume that both of the sequences (ξk )∞ k=1 and (T nk (ξk ))∞ in 𝒥 (T) respectively converge to, say, ξ ∈ 𝒥 (T) and w ∈ 𝒥 (T). Disrek=1 garding finitely many terms, we may further assume that

26.1 Expanding rational functions

|ξk − ξ| < δ/16,

∀k ∈ ℕ

� 1089

(26.4)

and that 󵄨󵄨 nk 󵄨 󵄨󵄨T (ξk ) − w󵄨󵄨󵄨 < δ,

∀k ∈ ℕ.

(26.5)

By (26.2), we also know that B(w, 2δ) ∩ PC(T) = 0.

(26.6)

Per Remark 24.4.16 and (26.5)–(26.6), there exists for every k ∈ ℕ a unique analytic −n ̂ of T −nk that sends T nk (ξk ) to ξk . By Koebe’s 1 –theorem branch Tξ k : B(w, 2δ) → ℂ 4 k (Theorem 23.1.3) and (26.3)–(26.5), we get −nk

Tξ

k

−nk

(B(w, 2δ)) ⊇ Tξ

k

(B(T nk (ξk ), δ)) −nk

⊇ B(Tξ

k

1 󵄨 −n ′ 󵄨 (T nk (ξk )), 󵄨󵄨󵄨(Tξ k ) (T nk (ξk ))󵄨󵄨󵄨δ) k 4

1󵄨 ′ 󵄨−1 = B(ξk , 󵄨󵄨󵄨(T nk ) (ξk )󵄨󵄨󵄨 δ) 4 ⊇ B(ξk , δ/8) ⊇ B(ξ, δ/16),

∀k ∈ ℕ.

Consequently, T nk (B(ξ, δ/16)) ⊆ B(w, 2δ) for all k ∈ ℕ. Therefore, the sequence (T nk |B(ξ,δ/16) )∞ k=1 is normal. By virtue of Proposition 24.3.8, this tells us that ξ ∈ ℱ (T). On the other hand, ξ = limk→∞ ξk ∈ 𝒥 (T). This contradiction confirms the converse implication. This theorem, when taken in conjunction with the classification theorem 24.4.10 and Sullivan’s nonwandering theorem (Theorem 24.4.12), leads to the following result. ̂→ℂ ̂ with deg(T) ≥ 2 is expanding if and only Theorem 26.1.4. A rational function T : ℂ if its critical set Crit(T) is contained in a (finite) union of connected components of basins of attraction to attracting periodic points of T. As belonging to a connected component of the basin of attraction to an attracting periodic point is an open condition and as, by Corollary 24.1.21, there are at most 2d −2 (a finite number depending only on d) attracting periodic cycles for any rational function of degree d, an immediate ramification of Theorem 26.1.4 is the following. Corollary 26.1.5. Expanding rational functions of a given degree d ≥ 2 form an open subset of the space of all rational functions of degree d. ̂→ℂ ̂ with deg(T) ≥ 2 admits no Theorem 26.1.6. An expanding rational function T : ℂ Cremer periodic point, rationally indifferent periodic point, Siegel disk or Herman ring. In

1090 � 26 Expanding (or hyperbolic), subexpanding, and parabolic rational functions particular, the Fatou set ℱ (T) consists only of basins of attraction to attracting periodic points of T. Proof. The presence of rationally indifferent periodic points is ruled out by Definition 25.3.3. The rest of the first assertion is an immediate consequence of Theorems 26.1.3, 24.5.2 and 24.5.1. The last assertion immediately follows from the first one and Theorem 24.4.10. With the terminology and notation of Section 24.6 and as an immediate consequence of the previous theorem and Theorem 24.6.1, we get the following (cf. [92]). ̂ → ℂ ̂ with deg(T) ≥ 2 is a Theorem 26.1.7. Every expanding rational function T : ℂ ̂ continuity point of the function ℛ ∋ S 󳨃󳨀→ 𝒥 (S) ⊆ 𝒦(ℂ). We can actually say more: expanding rational functions are structurally stable on their Julia sets. To see this, we start with the following repercussion of Theorem 26.1.4 and of the very definition of expandingness. ̂ →ℂ ̂ is an expanding rational function with deg(T) ≥ 2, Observation 26.1.8. If T : ℂ ̂ containing the Julia set 𝒥 (T) and whose closure U conthen for every open set U ⊆ ℂ tains no attracting periodic point of T, the triple (𝒥 (T), U, T) is a topologically exact conformal expanding repeller in the sense of Definition 16.2.1. Consequently, it is a distance expanding map. (For more information, see Corollary 24.3.5 and Chapters 16 and 4.) We can now prove the following remarkable result. ̂→ℂ ̂ be an expanding rational function with deg(T) ≥ 2 and Theorem 26.1.9. Let T : ℂ ̂ → ℂ) ̂ λ∈Λ is a collection of rational functions of the let Λ be an open subset of ℂ. If (Tλ : ℂ ̂ the map same degree such that Tλ0 = T for some λ0 ∈ Λ and if for every z ∈ ℂ ̂ Λ ∋ λ 󳨃󳨀→ Tλ (z) ∈ ℂ is analytic (in fact, it suffices to know this for every z in some neighborhood of 𝒥 (T)), then ̂ ⊆ Λ of λ0 and an open neighborhood U ̂ of 𝒥 (T) such there exist an open neighborhood Λ ̂ that for every λ ∈ Λ the following hold: ̂ is a conformal ̂ → ℂ ̂ is expanding and (𝒥 (Tλ ), Tλ , U) (a) The rational function Tλ : ℂ expanding repeller. (b) The maps Tλ : 𝒥 (Tλ ) → 𝒥 (Tλ ) and T : 𝒥 (T) → 𝒥 (T) are topologically conjugate. More precisely, there exists a homeomorphism τλ : 𝒥 (T) → 𝒥 (Tλ ) with the following properties: (i) For every z ∈ 𝒥 (T), the map ̂ ∋ λ 󳨃󳨀→ τλ (z) ∈ ℂ ̂ Λ is holomorphic.

26.1 Expanding rational functions

� 1091

(ii) The map τλ0 is the identity map and the map 𝒥 (T) ∋ z 󳨃󳨀→ τλ (z) ∈ 𝒥 (Tλ )

is quasiconformal and Hölder continuous with exponent converging to 1 uniformly as λ → λ0 . (iii) The homeomorphism τλ conjugates the maps T and Tλ , i. e., Tλ ∘ τλ = τλ ∘ T|𝒥 (T) . ̂ × 𝒥 (T) ∋ (λ, z) 󳨃󳨀→ τλ (z) ∈ U ̂ is continuous. (iv) The map Λ ̂ → ℂ) ̂ λ∈Λ inProof. Having Observation 26.1.8 and noticing that the family (Tλ : ℂ duces an analytic perturbation for every set U appearing in this observation (cf. Definition 16.4.1), our theorem almost entirely follows from Theorem 16.4.7. All we need to show is that τλ (𝒥 (T)) = 𝒥 (Tλ ),

̂ ∀λ ∈ Λ.

(26.7)

In view of Theorem 16.4.7, each set τλ (𝒥 (T)) is a conformal expanding repeller for the ̂ → ℂ. ̂ Therefore, τλ (𝒥 (T)) is the closure of a set of repelling periodic points map Tλ : ℂ of Tλ (cf. Definition 16.2.1 and Corollary 4.3.10). Thus, τλ (𝒥 (T)) ⊆ 𝒥 (Tλ ). Since the maps T|𝒥 (T) and Tλ |τλ (𝒥 (T)) are conjugate (by virtue of Theorem 16.4.7), we have that deg(Tλ |τλ (𝒥 (T)) ) = deg(T|𝒥 (T) ) = deg(T). But by hypothesis, deg(T) = deg(Tλ ), whence we get deg(Tλ |τλ (𝒥 (T)) ) = deg(Tλ ). This implies that Tλ−1 (τλ (𝒥 (T))) ⊆ τλ (𝒥 (T)). Thus, (26.7) directly follows from Corollary 24.3.7 and the proof is over. Note: In fact, by item (iii) we also know that Tλ (τλ (𝒥 (T))) = τλ (𝒥 (T)) ⊆ Tλ−1 (τλ (𝒥 (T))). So, Tλ (τλ (𝒥 (T))) = τλ (𝒥 (T)) = Tλ−1 (τλ (𝒥 (T))).

1092 � 26 Expanding (or hyperbolic), subexpanding, and parabolic rational functions 26.1.1 The Mandelbrot set ̂→ℂ ̂ be the quadratic polynomial For every c ∈ ℂ, let Qc : ℂ Qc (z) := z2 + c. The Mandelbrot set (see Figure 26.1) is possibly the most famous fractal. It is commonly denoted by ℳ and is defined as the set of all c ∈ ℂ such the set {Qcn (0) : n ≥ 0} is bounded or, equivalently, 󵄨 󵄨 lim inf󵄨󵄨󵄨Qcn (0)󵄨󵄨󵄨 < ∞. n→∞

The Mandelbrot set was defined and studied numerically by Benoit Mandelbrot [81]. Then came Douady and Hubbard’s pioneering work [43, 44] on the theoretical understanding of the topological and conformal structures of this set. This was followed by a huge quantity of research aiming to prove various properties of the Mandelbrot set, including typicality of expandingness and local connectedness of the boundary of ℳ. We only mention here the breakthrough concept of Yoccoz puzzles and Shishikura’s work [120] on the Hausdorff dimension of the boundary of the Mandelbrot set, which both had enormous influence on the direction of development of the research on that set. Denote by ℳE the set of all parameters c ∈ ℂ for which the map Qc is expanding. Obviously, 0 ∈ ℳE . In view of Corollary 26.1.5, the set ℳE is open, and hence c ∈ ℳE for every c ∈ ℂ sufficiently small in modulus. The main cardioid ℳ0 of the Mandelbrot set ℳ is the connected component of ℳE containing 0. So, the main cardioid ℳ0 of the Mandelbrot set ℳ contains a neighborhood of 0. As a nearly immediate consequence of Theorem 26.1.9, we can prove the following. Theorem 26.1.10. If c belongs to the main cardioid ℳ0 of the Mandelbrot set ℳ, then the Julia set Jc of the quadratic polynomial Qc is a Jordan (i. e., a simple closed) curve. Furthermore, Jc is a quasicircle, i. e., it is a quasiconformal image of the unit circle. Proof. By Theorem 26.1.9, for every s ∈ ℳ0 there exists rs > 0 such that for every c ∈ B(s, rs ) the maps Qc |Jc and Qs |Js are topologically conjugate via a quasiconformal homeomorphism. Since ℳ0 is a connected set, it follows that all maps Qc |Jc are topologically conjugate to the map Q0 |J0 via a quasiconformal homeomorphism. As J0 is the unit circle 𝕊1 , the proof of Theorem 26.1.10 is complete. Remark 26.1.11. An analogous theorem holds for the one-parameter family of polynomials z 󳨃→ zd + c for any integer d ≥ 2.

26.2 Expansive and parabolic rational functions

� 1093

Figure 26.1: The Mandelbrot set.

26.2 Expansive and parabolic rational functions ̂→ℂ ̂ is called expansive if the map T|𝒥 (T) : Definition 26.2.1. A rational function T : ℂ 𝒥 (T) → 𝒥 (T) is expansive. Clearly, all expanding rational functions are expansive but this latter class is bigger and their difference will be the primary consideration in this section and, most notably, in Chapter 31. The current section can be partly viewed as a preparation for that chapter. We now characterize expansive rational functions in terms that are purely topological and less dynamical than general expansiveness. The statement is as simple as those of Theorems 26.1.3–26.1.4. The proof, however, is somewhat more complex. It was first provided in [37]. We begin with an intermediate result. Recall that Ω(T) denotes the set of all rationally indifferent periodic points of T, which is finite according to Theorem 24.3.2. ̂ →ℂ ̂ is a rational function with deg(T) ≥ 2 and such that Proposition 26.2.2. If T : ℂ 𝒥 (T) ∩ Crit(T) = 0, then for every η > 0 there exists θη ∈ (0, η) for which B(𝒥 (T)\B(Ω(T), η), 2θη ) ∩ PC(T) = 0. In particular, given any z ∈ 𝒥 (T)\B(Ω(T), η), the analytic inverse branches of T n are welldefined on B(z, 2θη ) for every n ∈ ℕ.

1094 � 26 Expanding (or hyperbolic), subexpanding, and parabolic rational functions Proof. Since 𝒥 (T) ∩ Crit(T) = 0, we know that 𝒥 (T) ∩ PC(T) = 0. It also follows from the classification theorem 24.4.10 and Sullivan’s nonwandering theorem (Theorem 24.4.12) that ω(Crit(T)) ∩ 𝒥 (T) ⊆ Ω(T). These two facts taken together imply that [𝒥 (T)\B(Ω(T), η)] ∩ PC(T) = 0. Since the sets 𝒥 (T)\B(Ω(T), η) and PC(T) are compact, the existence of the required θη > 0 ensues. We can now derive the sought characterization of expansiveness. ̂ → ℂ ̂ is a rational function with deg(T) ≥ 2, then the map Theorem 26.2.3. If T : ℂ T|𝒥 (T) : 𝒥 (T) → 𝒥 (T) is expansive if and only if 𝒥 (T) ∩ Crit(T) = 0. Proof. Suppose that 𝒥 (T)∩Crit(T) ≠ 0 and let c ∈ 𝒥 (T)∩Crit(T). By Theorem 24.1.15, the Julia set 𝒥 (T) is perfect. Therefore, there exists a sequence (ξn )∞ n=1 of points in 𝒥 (T)\{c} converging to c. Since c ∈ Crit(T), for all n ∈ ℕ large enough (removing finitely many ̂ such that ξn′ ≠ ξn , T(ξn′ ) = T(ξn ), ξn ’s, we get that for all n ∈ ℕ) there are points ξn′ ∈ ℂ ′ ′ and limn→∞ ξn = c. But then ξn ∈ 𝒥 (T), and we hence constructed a sequence of pairs of arbitrarily close but distinct points ξn and ξn′ in 𝒥 (T) that are mapped to the same points. Thus, T|𝒥 (T) fails to be expansive. The direct implication has been established. To prove the converse, note first that 𝒥 (T) ∩ Crit(T) = 0 entails the existence of a constant χ > 0 such that for every z ∈ 𝒥 (T) there is a unique holomorphic branch Tz−1 : B(T(z), χ) → ℂ of T −1 mapping T(z) to z. A standard argument (cf. Lemma 4.2.2) shows that δ1 := sup{r > 0 : B(z, r) ⊆ Tz−1 (B(T(z), χ)), ∀z ∈ 𝒥 (T)} > 0. Let δ2 be the θ from (24.83) and Proposition 24.2.28. Subsequently, let θδ2 /2 arise from Proposition 26.2.2. Finally, let δ = 21 min{χ, δ1 , δ2 , θδ2 /2 }. We shall prove that δ is an expansive constant for T : 𝒥 (T) → 𝒥 (T). Assume that x, y ∈ 𝒥 (T) and |T n (x) − T n (y)| < δ for all n ≥ 0. As δ < δ2 /2, it follows from Proposition 24.2.28 and the finiteness of Ω(T) that the set 󵄨 Z := {n ≥ 0 󵄨󵄨󵄨 {T n (x), T n (y)} ⊈ B(Ω(T), δ2 /2)} is infinite. This means that at least one of the sets Zx := {n ≥ 0 | T n (x) ∉ B(Ω(T), δ2 /2)} or Zy := {n ≥ 0 | T n (y) ∉ B(Ω(T), δ2 /2)} is infinite. Assume without loss of generality that Zx is infinite. By virtue of Proposition 26.2.2, for every n ∈ Zx there exists a unique analytic ̂ of T −n sending T n (x) to x. branch Tx−n : B(T n (x), θδ2 /2 ) → ℂ Claim. If n ∈ Zx , then Tx−n (T n (y)) = y. Proof of Claim. Note that T n (y) ∈ B(T n (x), δ) ⊆ B(T n (x), 21 θδ2 /2 ), and thus Tx−n (T n (y)) is well-defined. Suppose for a contradiction that n ∈ Zx and Tx−n (T n (y)) ≠ y. Let k ∈ {0, 1, . . . , n} be the largest integer such that T k (y) ≠ T k (Tx−n (T n (y))). Then k < n and T k+1 (y) = T k+1 (Tx−n (T n (y))).

(26.8)

26.2 Expansive and parabolic rational functions

� 1095

In addition, from the definitions of δ and δ1 , we know that T k (y) ∈ B(T k (x), δ) ⊆ B(T k (x), δ1 ) ⊆ TT−1k (x) (B(T k+1 (x), χ)).

(26.9)

Finally, from (26.8) notice that T k (Tx−n (T n (y))) ∈ T −1 (T k+1 (y)). As |T j (x) − T j (y)| < δ for all j ≥ 0 (and in particular for j = k + 1), there exists z ∈ T −1 (T k+1 (x)) for which T k (Tx−n (T n (y))) = Tz−1 (T k+1 (y)). But then T k (x) = Tz−1 (T k+1 (x)) = z. Therefore, T k (Tx−n (T n (y))) = TT−1k (x) (T k+1 (y)) ⊆ TT−1k (x) (B(T k+1 (x), δ))

⊆ TT−1k (x) (B(T k+1 (x), χ)).

(26.10)

In sum, (26.9), (26.10) and (26.8) assert that T k (y) and T k (Tx−n (T n (y))) both lie in TT−1k (x) (B(T k+1 (x), χ)) and have the same image under T. The injectivity of T on

TT−1k (x) (B(T k+1 (x), χ)) imposes that T k (y) = T k (Tx−n (T n (y))). This contradicts the definition of k and concludes the proof of the claim. ◼ Write Zx as a strictly increasing sequence (nk )∞ k=1 . By the −nk Tx (B(T nk (x), 21 θδ2 /2 )) for all k ∈ ℕ. However, since x ∈ 𝒥 (T),

ma 24.1.13 that

above claim, x, y ∈ it follows from Lem-

1 lim diam(Tx−nk (B(T nk (x), θδ2 /2 ))) = 0. 2

k→∞

So, |y − x| = 0. This means that x = y. We now describe interesting ramifications of expansiveness. ̂→ℂ ̂ admits no Cremer periodic Theorem 26.2.4. An expansive rational function T : ℂ point, Siegel disk or Herman ring. In particular, the Fatou set ℱ (T) consists only of basins of attraction to attracting periodic points of T and basins of attraction to rationally indifferent periodic points. Proof. The first assertion is an immediate consequence of Theorems 24.5.1, 24.5.2 and 26.2.3. The last assertion immediately follows from the first one and Theorem 24.4.10. Let ∞

Ω−∞ (T) := ⋃ T −n (Ω(T)) n=0

be the completely T-invariant backward orbit of Ω(T). ̂ → ℂ ̂ be an expansive rational function with deg(T) ≥ 2. If Lemma 26.2.5. Let T : ℂ z ∈ 𝒥 (T), then

1096 � 26 Expanding (or hyperbolic), subexpanding, and parabolic rational functions ′ 󵄨 󵄨 lim sup󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 > 0. n→∞

More precisely: (a) If z ∈ Ω−∞ (T), then ′ ′ 󵄨 󵄨 󵄨 󵄨 0 < lim inf󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 ≤ lim sup󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 < ∞. n→∞

n→∞

(b) If z ∉ Ω−∞ (T), then ′ 󵄨 󵄨 lim sup󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 = ∞. n→∞

In fact, if δT > 0 is an expansive constant for T : 𝒥 (T) → 𝒥 (T) so small that [T(B(ω, δT ))] ∩ [B(Ω(T)\{T(ω)}, δT )] = 0 for all ω ∈ Ω(T), then ′ 󵄨 󵄨 lim 󵄨󵄨󵄨(T nk (z) ) (z)󵄨󵄨󵄨 = ∞,

k→∞

n where (nk (z))∞ k=1 is the sequence of all n ∈ ℕ such that T (z) ∉ B(Ω(T), δT ).

Proof. Assertion (a) is obvious since 𝒥 (T)∩Crit(T) = 0 by Theorem 26.2.3, and T q (ω) = ω and (T q )′ (ω) = 1 for some q ∈ ℕ and all ω ∈ Ω(T). For (b), let δT and (nk (z))∞ k=1 be as in ∞ the statement (observe that (nk (z))k=1 does exist). By Proposition 26.2.2, for every k ∈ ℕ −n (z) ̂ of T −nk (z) sending there exists a unique analytic branch Tz k : B(T nk (z) (z), 2θδT ) → ℂ T nk (z) (z) to z. Lemma 24.1.13 then entails that limk→∞ |(T nk (z) )′ (z)| = ∞. ̂ → ℂ ̂ is called parabolic if it is expansive Definition 26.2.6. A rational function T : ℂ but not expanding. Together, Theorems 26.2.3, 26.1.3, 24.4.12 and 24.4.10 provide a characterization of parabolic rational functions. ̂→ℂ ̂ with deg(T) ≥ 2 is parabolic if and only Theorem 26.2.7. A rational function T : ℂ if 𝒥 (T) ∩ Crit(T) = 0

and

Ω(T) ≠ 0.

It is worth noticing that an iterate T k of a rational function T is parabolic if and only if the map T is parabolic too (see Exercise 24.8.32). Parabolic rational functions abound. Perhaps the most famous one is the quadratic polynomial Q1/4 (z) = z2 + 1/4 whose Julia set is commonly called the cauliflower. In fact, every quadratic polynomial of the form z 󳨃→ λz(1 + z), where λ is a root of unity, is a parabolic rational function.

26.3 Subexpanding rational functions

� 1097

26.3 Subexpanding rational functions ̂→ℂ ̂ be a rational function. Define the compact set Let R : ℂ ω(R) := 𝒥 (R) ∩ ω(Crit(R)), where ω(Crit(R)) = ⋃ζ ∈Crit(R) ω(ζ ) is the set of limit points of the forward orbits of the critical points of R. The set ω(R) is forward R-invariant. ̂→ℂ ̂ is a rational function, then ω(R) ⊆ R−1 (ω(R)). EquivaProposition 26.3.1. If R : ℂ lently, R(ω(R)) ⊆ ω(R). This forward invariance ensures that the following concept is well-defined. ̂→ℂ ̂ is said to be subexpanding if the map Definition 26.3.2. A rational function R : ℂ R|ω(R) : ω(R) → ω(R) is expanding, i. e., if there exists s ∈ ℕ such that 󵄨󵄨 s ′ 󵄨󵄨 󵄨󵄨(R ) (z)󵄨󵄨 > 1,

∀z ∈ ω(R).

(26.11)

Given that the (immediate) basin of attraction of any rationally indifferent periodic point would contain a critical point, a subexpanding rational function cannot admit such a periodic point. The following statement is then a simple consequence of Theorem 24.2.22 and Corollary 24.4.17. ̂ →ℂ ̂ is a subexpanding rational function with deg(R) ≥ 2, Proposition 26.3.3. If R : ℂ then Ω(R) = 0 and 𝒥 (R) ∩ ω(Crit(R) ∩ ℱ (R)) = 0,

i. e.,

ω(R) = ω(Crit(R) ∩ 𝒥 (R)).

Observe also that any subexpanding rational function R satisfies Crit(R) ∩ ω(R) = 0.

(26.12)

In fact, relation (26.12) is equivalent to the subexpanding property. This easily follows from Mañé’s theorem [85]. Of course, all expanding rational functions are subexpanding (cf. Definition 25.3.3) but the latter class is bigger. Among others, let us draw the attention of the reader to the family of critically finite rational functions. A rational function R is called critically finite if R has no rationally indifferent periodic point and the forward orbit of each critical point in the Julia set of R is finite. In this case, Mañé’s theorem is not needed; a direct argument establishes that each critically finite rational function is subexpanding. Indeed, ω(R) then consists of finitely many repelling periodic orbits, whence it is an expanding set. Any critically finite rational function whose Julia set contains a critical point is thus

1098 � 26 Expanding (or hyperbolic), subexpanding, and parabolic rational functions subexpanding but not expanding. Examples of such maps include the quadratic polynomials Qi (z) = z2 + i, Q−2 (z) = z2 − 2, Lattés examples from [74] and especially from John Milnor’s article [95]. In [97], Michal Misiurewicz proved the existence of uncountably (in fact, continuum) many real numbers c such that the quadratic polynomial Qc (z) = z2 + c is subexpanding. For such c’s, he showed that c ∉ ω(c) and the Julia set of Qc is a compact subset of ℝ (see [28] and [93] for more on this). The difference between the classes of expanding and subexpanding rational functions will be the main concern in this section and, most notably, in Chapter 32. The present section can be partially considered as a preparation for that chapter. Both this section and Chapter 32 are substantial alterations and developments of the paper [39]. We begin with a topological property of ω(R) as a subset of 𝒥 (R). ̂ →ℂ ̂ is a subexpanding rational function with deg(R) ≥ 2, Proposition 26.3.4. If R : ℂ then the compact set ω(R) is nowhere dense in 𝒥 (R). Proof. If Crit(R) ∩ 𝒥 (R) = 0 then ω(R) = 0 by Proposition 26.3.3, and the result is trivially true. Otherwise, by subexpandingness (see (26.11)), the compact set ω(R) is a proper subset of 𝒥 (R). Hence, the result is an immediate consequence of the forward invariance of ω(R) (see Proposition 26.3.1) and the topological exactness of the map R : 𝒥 (R) → 𝒥 (R) (see Corollary 24.3.5). As a repercussion of Theorems 24.5.1 and 24.4.10, we deduce the next property, which subexpanding rational functions share with expanding ones (cf. Theorem 26.1.6). ̂→ℂ ̂ admits no Cremer periodic Theorem 26.3.5. A subexpanding rational function R : ℂ point, rationally indifferent periodic point, Siegel disk or Herman ring. In particular, the Fatou set ℱ (R) consists only of basins of attraction to attracting periodic points of R. With the terminology and notation from Section 24.6 and as an immediate consequence of Theorems 26.3.5 and 24.6.1, we identify another property that extends to the class of subexpanding rational functions (cf. Theorem 26.1.7). ̂→ℂ ̂ is a continuity point Theorem 26.3.6. Every subexpanding rational function R : ℂ ̂ of the function ℛ ∋ S 󳨃󳨀→ 𝒥 (S) ⊆ 𝒦(ℂ). We now turn our attention toward the existence of, and the domain and range of, holomorphic inverse branches of subexpanding rational functions in the vicinity of various points. ̂→ℂ ̂ be a subexpanding rational function. Set Let R : ℂ T := Rs , where s comes from (26.11). Since ω(T) ⊆ ω(R), we have 󵄨󵄨 ′ 󵄨󵄨 󵄨󵄨T (z)󵄨󵄨 > 1,

∀z ∈ ω(T).

(26.13)

26.3 Subexpanding rational functions

� 1099

So T is also subexpanding, i. e., Crit(T) ∩ ω(T) = 0. As the map T ′ is continuous and the set ω(T) is compact, it follows from Proposition 26.3.3 that there exist η ∈ (0, 2−6 dist(𝒥 (T), [Crit(T) ∪ PC(T)] ∩ ℱ (T)))

and

λ>1

(26.14)

such that 󵄨󵄨 ′ 󵄨󵄨 󵄨󵄨T (z)󵄨󵄨 ≥ λ,

∀z ∈ B(ω(T), 2η).

(26.15)

By definition of ω(T), there exists p ≥ 2 such that ̂ \ B(𝒥 (T), 26 η)], T j (Crit(T) ∪ PC(T)) ⊆ B(ω(T), η/2) ∪ [ℂ

∀j ≥ p − 1.

(26.16)

̂ and setting Therefore, denoting ‖T ′ ‖∞ := sup{|T ′ (z)| : z ∈ ℂ} γ :=

η

p−1

4‖T ′ ‖∞

(26.17)

,

we infer the following about holomorphic inverse branches near some points of the Julia set. ̂ →ℂ ̂ be a subexpanding rational function with deg(R) ≥ 2. If Lemma 26.3.7. Let R : ℂ n z ∈ 𝒥 (T), n ≥ p − 1 and T (z) ∉ B(ω(T), η) then T n−p+1 (z) ∉ B(PC(T), 2γ), where T, p, η and γ are defined in (26.13)–(26.17). Thus, there is a unique holomorphic −(n−p+1) ̂ of T −(n−p+1) such that Tz−(n−p+1) (T n−p+1 (z)) = z. branch Tz : B(T n−p+1 (z), 2γ) → ℂ Proof. Otherwise, there are z ∈ 𝒥 (T), n ≥ p − 1 and T n (z) ∉ B(ω(T), η) for which T n−p+1 (z) ∈ B(PC(T), 2γ). Then T n (z) = T p−1 (T n−p+1 (z)) ∈ B(ω(T), η)

by (26.16)–(26.17).

This is a contradiction. By virtue of (26.15) and (25.3)–(25.4), there exists ε ∈ (0, η) so small that conditions (26.18)–(26.21) are satisfied: B(Crit(T), ε) ∩ B(ω(T), 2η) = 0;

(26.18)

there exists A ≥ 1 such that for every c ∈ Crit(T p ) and every z ∈ B(c, ε), we have p p 󵄨 󵄨 A−1 |z − c|degc (T ) ≤ 󵄨󵄨󵄨T p (z) − T p (c)󵄨󵄨󵄨 ≤ A|z − c|degc (T )

(26.19)

1100 � 26 Expanding (or hyperbolic), subexpanding, and parabolic rational functions A−1 |z − c|degc (T

p

)−1

p ′ 󵄨 󵄨 ≤ 󵄨󵄨󵄨(T p ) (z)󵄨󵄨󵄨 ≤ A|z − c|degc (T )−1 ;

(26.20)

and finally B(c1 , 2ε) ∩ B(c2 , 2ε) = 0,

∀c1 , c2 ∈ Crit(T p ), c1 ≠ c2 .

(26.21)

We continue with a few more results about analytic inverse branches. As a direct consequence of Lemmas 25.1.12 and 25.1.5, we obtain the following fact. ̂→ℂ ̂ is a subexpanding rational function with deg(R) ≥ 2, then Lemma 26.3.8. If R : ℂ there is τ ∈ (0, 1/2) such that for each c ∈ Crit(T p ) and each z ∈ B(c, ε) there exists a −p unique holomorphic inverse branch Tz : B(T p (z), 2τ|T p (z) − T p (c)|) → B(z, γ) such that −p Tz (T p (z)) = z, where T, p and γ are defined in (26.13)–(26.17). Proof. By virtue of Lemmas 25.1.12 and 25.1.5, there exists τ ∈ (0, 1/2) so small that diam( Comp(w, T p , 2τ|T p (w) − T p (c)|) ) < min{γ, ε} ̂ where the reader will recall that Comp(w, T p , 2τ|T p (w) − T p (c)|) is for all w ∈ ℂ, the connected component of T −p (B(T p (w), 2τ|T p (w) − T p (c)|)) containing w. It follows from (26.21) that Crit(T p ) ∩ Comp(z, T p , 2τ|T p (z) − T p (c)|) = 0. Thus, in view of Corollary 22.5.4, the restriction of T p to Comp(z, T p , 2τ|T p (z) − T p (c)|) is a conformal homeomorphism onto B(T p (z), 2τ|T p (z) − T p (c)|). Its inverse is the required −p map Tz . ̂→ℂ ̂ is a subexpanding rational function with deg(R) ≥ 2, then Lemma 26.3.9. If R : ℂ there exists ΓR > 0 such that lim sup sup diam(V ) = 0, n→∞

(z,V )

where the supremum is taken over all z ∈ 𝒥 (R) and all connected components V of R−n (B(z, 4ΓR )). Proof. Because of the uniform continuity of R (and all its iterates) it suffices to prove this lemma with R replaced by T. Suppose first that z ∈ 𝒥 (T) \ B(ω(T), η). It follows from Lemma 26.3.7 that for any w ∈ T −(p−1) (z), any n ≥ 0 and any x ∈ T −n (w), ̂ of T −n sending w to x. It there exists a unique holomorphic branch Tx−n : B(w, 2γ) → ℂ then ensues from Lemma 24.1.13 that

26.3 Subexpanding rational functions

� 1101

lim sup{diam(Tx−n (B(w, γ))) : z ∈ 𝒥 (T)\B(ω(T), η), w ∈ T −(p−1) (z), x ∈ T −n (w)} = 0.

n→∞

On the other hand, Lemma 25.1.4 implies that there exists Δ1 > 0 such that for every z ∈ 𝒥 (T) \ B(ω(T), η) and every w ∈ T −(p−1) (z), the connected component of T −(p−1) (B(z, Δ1 )) containing w is contained in B(w, γ). Therefore, the lemma is proved for the supremum taken over all points z ∈ 𝒥 (T) \ B(ω(T), η). Let’s now look at 𝒥 (T) ∩ B(ω(T), η). Fixing δ > 0, there exists Nδ ≥ p such that diam(U) < δ

(26.22)

for every y ∈ 𝒥 (T) \ B(ω(T), η), every n ≥ Nδ and every connected component U of T −n (B(z, Δ1 )). Now take Δ2 ∈ (0, min{Δ1 , γ, θ(ε, λ)})

(26.23)

so small (see Lemma 25.1.4) that diam(D) < Δ1

(26.24)

for every y ∈ 𝒥 (T), every 0 ≤ q ≤ N and every connected component D of T −q (B(y, Δ2 )), and take N ∈ ℕ so large that θ(ε, λ)λ−N < Δ1 .

(26.25)

Also, note that because of Lemma 25.1.4 there exists δ̃ ∈ (0, δ] such that diam(W ) < δ

(26.26)

̃ for every y ∈ 𝒥 (T), every 0 ≤ j ≤ Nδ and every connected component W of T −j (B(y, δ)). Finally, take M ∈ ℕ so large that ̃ θ(ε, λ)λ−M < δ.

(26.27)

n ≥ N + Nδ + M

(26.28)

Fix any

and suppose that z ∈ 𝒥 (T) ∩ B(ω(T), η). Set w ∈ T −n (z). If T j (w) ∈ B(ω(T), η) for all j = 0, 1, . . . , n, then it follows from (26.18) and Lemma 25.1.11 that the diameter of the connected component G of T −n (B(z, θ(ε, λ))) containing w satisfies

1102 � 26 Expanding (or hyperbolic), subexpanding, and parabolic rational functions diam(G) ≤ θ(ε, λ)λ−n < δ,

(26.29)

where the last inequality holds thanks to (26.28), (26.27) and the fact that δ̃ ≤ δ. Otherwise, let k be the largest integer in {0, 1, . . . , n − 1} such that T k (w) ∉ B(ω(T), η). Let V be the connected component of T −n (B(z, Δ2 )) containing w. Consider several cases. Case 1: k ≥ n − N. Then n − k ≤ N and it ensues from (26.24) that diam(T k (V )) < Δ1 . Since k ≥ n − N ≥ Nδ by (26.28), it follows from (26.22) that diam(V ) < δ.

(26.30)

Case 2: k < n − N. Consider two subcases: Subcase 2a: k ≥ Nδ . Since n − k > N, we infer from (26.25), (26.23) and (26.29) that diam(T k (V )) < Δ1 . As k ≥ Nδ , it ensues from (26.22) that diam(V ) < δ.

(26.31)

Subcase 2b: k < Nδ . Then n − (k + 1) ≥ n − Nδ ≥ M, whence by (26.29) and (26.27), ̃ diam(T k+1 (V )) ≤ θ(ε, λ)λ−[n−(k+1)] ≤ θ(ε, λ)λ−M < δ. As k + 1 ≤ Nδ , it follows from (26.26) that diam(V ) < δ. This, along with (26.29)–(26.31), shows that the lemma holds if the supremum is taken over all z ∈ 𝒥 (T) ∩ B(ω(T), η). As an immediate reverberation of this lemma, of Lemma 25.1.5, and of the fact that c ∉ ω(c) for every c ∈ Crit(R) ∩ 𝒥 (R) per (26.12), we deduce the following. ̂ → ℂ ̂ is a subexpanding rational function with deg(R) ≥ 2, Corollary 26.3.10. If R : ℂ then for every z ∈ 𝒥 (R), every n ≥ 0, and every connected component V of R−n (B(z, 4ΓR )), the number of integers 0 ≤ k ≤ n such that Rk (V ) ∩ Crit(R) ≠ 0 is bounded above by #Crit(R) and each such nonempty intersection is a singleton. ̂→ℂ ̂ is a subexpanding rational function with deg(R) ≥ 2, then Lemma 26.3.11. If R : ℂ there exist κ ∈ (0, 1] and ϑ ∈ (0, 1] such that the following holds: Let z ∈ 𝒥 (R) and let n ≥ 0 be an integer. Suppose that Q(1) ⊆ Q(2) ⊆ B(Rn (z), κΓR ) are connected sets. Let Qn(1) be a connected component of R−n (Q(1) ) contained in the connected component Vn (z, κΓR ) of R−n (B(Rn (z), κΓR )) containing z and let Qn(2) be the connected component of R−n (Q(2) ) containing Qn(1) . Then

26.3 Subexpanding rational functions

diam(Qn(1) )

diam(Qn(2) )

≥ϑ

diam(Q(1) ) . diam(Q(2) )

� 1103

(26.32)

Proof. Since 𝒥 (R) contains no superattracting periodic point of R, there exists q ∈ ℕ such that Crit(R) ∩ Rq−1 (PC(R) ∩ 𝒥 (R)) = 0.

(26.33)

Otherwise, there would exist a strictly increasing sequence (nk )∞ k=1 such that Crit(R) ∩ Rnk (Crit(R) ∩ 𝒥 (R)) ≠ 0. Due to the finiteness of Crit(R), there would then be c1 ∈ Crit(R), c2 ∈ Crit(R) ∩ 𝒥 (R) and k < l ∈ ℕ for which c1 = Rnk (c2 ) = Rnl (c2 ). Then we would have Rnl −nk (c1 ) = Rnl −nk (Rnk (c2 )) = Rnl (c2 ) = c1 . Moreover, the complete invariance of the Julia set would impose that c1 ∈ 𝒥 (R), just like c2 . This would contradict the fact that 𝒥 (R) contains no superattracting periodic point of R. As Crit(R) ∩ ω(Crit(R) ∩ 𝒥 (R)) = Crit(R) ∩ ω(R) = 0, there is 0 < Δ < π/2 such that B(Crit(R), Δ) ∩ Rq−1 (PC(R) ∩ 𝒥 (R)) = 0

(26.34)

and B(c, Δ) ∩ (Crit(R) ∪ PC(R)) = {c},

∀c ∈ Crit(R).

(26.35)

Fix n ≥ 0 and z ∈ 𝒥 (R). For every r ∈ (0, 4ΓR ), let Vn (z, r) := Comp(z, Rn , r) be the connected component of R−n (B(Rn (z), r)) containing z. In view of Lemma 25.1.5 and Lemma 26.3.9, there exists τ ∈ (0, 1] such that diam(Rk (Vn (z, 4τΓR ))) < Δ,

∀0 ≤ k ≤ n.

(26.36)

By virtue of Theorem 25.1.1, there exist Δ1 ∈ (0, Δ] and A ∈ [1, ∞) such that 1

1

B(c, A−1 r dc (Rk ) ) ⊆ Comp(c, Rk , r) ⊆ B(c, Ar dc (Rk ) )

(26.37)

for every 1 ≤ k ≤ q + 1, every c ∈ Crit(Rk ) and every r ∈ (0, Δ1 ], where dc (Rk ) = degc (Rk ). Finally, let κ := ρτ, where ρ ∈ (0, 1] is so small that 1

1

2A(2Kρ) dw (Rq+1 ) ≤ A−1 (K −1 − Kρ) dw (Rq+1 ) ,

∀w ∈ Crit(Rq+1 )

and K := k2τΓR ,π/2 (1/2) is the Koebe constant from Theorem 23.1.9. We will consider several cases. Case 1: For every 0 ≤ k < n, Rk (Vn (z, 4τΓR )) ∩ Crit(R) = 0. It then follows from Koebe’s distortion theorem that

(26.38)

1104 � 26 Expanding (or hyperbolic), subexpanding, and parabolic rational functions diam(Qn(1) )

diam(Qn(2) )

≥ K −2

diam(Q(1) ) , diam(Q(2) )

and we are done with Case 1. Case 2: There exists 0 ≤ u < n such that Ru (Vn (z, 4τΓR )) ∩ Crit(R) ≠ 0

(26.39)

and assume that u is the largest number with that property. It then ensues from Theorem 23.1.9 that diam(Ru+1 (Qn(1) ))

diam(Ru+1 (Qn(2) ))

≥ K −2

diam(Q(1) ) . diam(Q(2) )

(26.40)

̂ such that In light of Corollary 26.3.10, there exists c ∈ ℂ Ru (Vn (z, 4τΓR )) ∩ Crit(R) = {c}.

(26.41)

Subcase 2a: c ∉ Ru (Vn (z, 2ρτΓR )). In light of (26.41), this implies that Ru (Vn (z, 2ρτΓR )) ∩ Crit(R) = 0.

(26.42)

Suppose for a contradiction that Ru (Vn (z, 2ρτΓR )) ∩ PC(R) ≠ 0.

(26.43)

As diam(Ru (Vn (z, 4τΓR ))) < Δ by (26.36) and as c ∈ Ru (Vn (z, 4τΓR )) by (26.41), we infer that B(c, Δ) ⊇ Ru (Vn (z, 4τΓR )) ⊇ Ru (Vn (z, 2ρτΓR )).

(26.44)

It follows from (26.43)–(26.44) that B(c, Δ) ∩ PC(R) ≠ 0. We deduce from (26.35) that B(c, Δ) ∩ PC(R) = {c}. It ensues from this and (26.43)–(26.44) that Ru (Vn (z, 2ρτΓR )) ∩ PC(R) = {c}.

(26.45)

26.3 Subexpanding rational functions

� 1105

In particular, c ∈ Ru (Vn (z, 2ρτΓR )). But this is not the case in Subcase 2a. So, (26.43) is impossible in that subcase, and thus Ru (Vn (z, 2ρτΓR )) ∩ PC(R) = 0.

(26.46)

Now, if there were some 0 ≤ j < u such that Rj (Vn (z, 2ρτΓR )) ∩ Crit(R) ≠ 0, then (26.46) would be contradicted. Hence, Rj (Vn (z, 2ρτΓR )) ∩ Crit(R) = 0,

∀0 ≤ j < u.

(26.47)

It follows from (26.47), (26.42) and Koebe’s distortion theorem (Theorem 23.1.9) that diam(Qn(1) )

diam(Qn(2) )

≥ K −2

diam(Ru+1 (Qn(1) ))

diam(Ru+1 (Qn(2) ))

(26.48)

.

We deduce from (26.40) and (26.48) that diam(Qn(1) )

diam(Qn(2) )

≥ K −4

diam(Q(1) ) , diam(Q(2) )

and we are done in Subcase 2a. Subcase 2b: c ∈ Ru (Vn (z, 2ρτΓR )). Let k := min{q + 1, u + 1}. Since both sets Ru+1−k (Qn(1) ) and Ru+1−k (Qn(2) ) are connected, we get for j = 1, 2 from Lemma 25.1.13 that E −1 ≤

diam(Ru+1 (Qn )) (j)

(j)

(j)

dc (Rk )−1

diam(Ru+1−k (Qn )) ⋅ [Dist(c, Ru+1−k (Qn ))]

≤ E.

where E := max{ERk′ : 1 ≤ k ′ ≤ q + 1} with ERk′ ∈ [1, ∞), 1 ≤ k ′ ≤ q + 1, being the constants produced in Lemma 25.1.13. Hence, diam(Ru+1−k (Qn(1) ))

diam(Ru+1−k (Qn(2) ))

≥E

−2

≥ E −2

diam(Ru+1 (Qn(1) ))

dc (Rk )−1

[Dist(c, Ru+1−k (Qn(1) ))] diam(Ru+1 (Qn(1) ))

diam(Ru+1 (Qn(2) ))

Invoking (26.40), we thereby get that

.

⋅

[Dist(c, Ru+1−k (Qn(2) ))]

dc (Rk )−1

diam(Ru+1 (Qn(2) ))

1106 � 26 Expanding (or hyperbolic), subexpanding, and parabolic rational functions diam(Ru+1−k (Qn(1) ))

diam(Ru+1−k (Qn(2) ))

≥ (EK)−2

diam(Q(1) ) . diam(Q(2) )

(26.49)

Sub-subcase 2b1: k = u + 1. Then (26.49) yields diam(Qn(1) )

diam(Qn(2) )

≥ (EK)−2 ,

and we are done in this sub-subcase. Sub-subcase 2b2: k = q + 1. Recall that Comp(ζ , f , r) denotes the connected component of f −1 (B(f (ζ ), r)) containing ζ . Remember also that Vn (z, 2ρτΓR ) := Comp(z, Rn , 2ρτΓR ). In Subcase 2b, by hypothesis c ∈ Ru (Vn (z, 2ρτΓR )). By definition, n Ru+1 (Vn (z, 2ρτΓR )) ⊆ Ru+1 ∘ R−n z (B(R (z), 2ρτΓR )).

Applying Koebe’s distortion theorem (Theorem 23.1.9) to the right-hand side yields Ru+1 (Vn (z, 2ρτΓR )) ⊆ B(Ru+1 (z), 2ρτΓR K|(Rn−(u+1) )′ (Ru+1 (z))|−1 ). Thus, diam( Ru+1 (Vn (z, 2ρτΓR )) ) ≤ 4ρτΓR K|(Rn−(u+1) )′ (Ru+1 (z))|−1 . Since R(c) ∈ Ru+1 (Vn (z, 2ρτΓR )), we have Ru+1 (Vn (z, 2ρτΓR )) ⊆ B(R(c), diam(Ru+1 (Vn (z, 2ρτΓR ))))

⊆ B(R(c), 4ρτΓR K|(Rn−(u+1) )′ (Ru+1 (z))|−1 ).

Fix a point c∗ ∈ R(u+1)−(q+1) (Vn (z, 2ρτΓR )) ∩ R−q (c).

(26.50)

Then (Rq+1 )′ (c∗ ) = 0, i. e., c∗ ∈ Crit(Rq+1 ). Moreover, Ru+1 (Vn (z, 2ρτΓR )) ⊆ B(Rq+1 (c∗ ), 4ρτΓR K|(Rn−(u+1) )′ (Ru+1 (z))|−1 ). It follows from this, (26.50) and (26.37) that R(u+1)−(q+1) (Vn (z, 2ρτΓR )) = Comp(c∗ , Rq+1 , Ru+1 (Vn (z, 2ρτΓR ))) 1

⊆ B(c∗ , A[4KρτΓR |(Rn−(u+1) )′ (Ru+1 (z))|−1 ] dc∗ (R

q+1 )

).

(26.51)

On the other hand, by the definition of u and Koebe’s distortion theorem we have Ru+1 (Vn (z, 2τΓR )) ⊇ B(Ru+1 (z), 2τΓR K −1 |(Rn−(u+1) )′ (Ru+1 (z))|−1 )

26.3 Subexpanding rational functions

� 1107

⊇ B(R(c), 2τΓR K −1 |(Rn−(u+1) )′ (Ru+1 (z))|−1 − |R(c) − Ru+1 (z)|)

⊇ B(R(c), 2τΓR K −1 |(Rn−(u+1) )′ (Ru+1 (z))|−1 − diam(Vn (z, 2τΓR ))) ⊇ B(R(c), 2τΓR (K −1 − Kρ)|(Rn−(u+1) )′ (Ru+1 (z))|−1 ).

(26.52)

Relations (26.37) and (26.52) imply that 1

R(u+1)−(q+1) (Vn (z, 2τΓR )) ⊇ B(c∗ , A−1 [2τΓR (K −1 − Kρ)|(Rn−(u+1) )′ (Ru+1 (z))|−1 ] dc∗ (R It follows from (26.38) and (26.53) that 1

R(u+1)−(q+1) (Vn (z, 2τΓR )) ⊇ B(c∗ , 2A[4KρτΓR |(Rn−(u+1) )′ (Ru+1 (z))|−1 ] dc∗ (R

q+1 )

).

q+1 )

). (26.53)

(26.54)

Seeking a contradiction, suppose that R(u+1)−(q+1) (Vn (z, 2τΓR )) ∩ PC(R) ≠ 0. Applying Rq , we obtain that Ru (Vn (z, 2τΓR )) ∩ Rq (PC(R)) ≠ 0. But c ∈ Ru (Vn (z, 2τΓR )), as we are in Subcase 2b. Using (26.36), we get B(c, Δ) ∩ Rq (PC(R)) ≠ 0. However, this contradicts (26.34) and establishes that R(u+1)−(q+1) (Vn (z, 2τΓR )) ∩ PC(R) = 0.

(26.55)

But if there were some 0 ≤ j < u − q such that Rj (Vn (z, 2τΓR )) ∩ Crit(R) ≠ 0, then (26.55) would be contradicted. Hence, Rj (Vn (z, 2τΓR )) ∩ Crit(R) = 0,

∀0 ≤ j < u − q.

(26.56)

Having this, (26.51) and (26.54), it follows from Koebe’s distortion theorem (Theorem 23.1.9) that diam(Qn(1) )

diam(Qn(2) )

≥ K −2

diam(R(u+1)−(q+1) (Qn(1) ))

diam(R(u+1)−(q+1) (Qn(2) ))

= K −2

diam(R(u+1)−k (Qn(1) ))

diam(R(u+1)−k (Qn(2) ))

We deduce from (26.57) and (26.49) that diam(Qn(1) )

diam(Qn(2) )

≥ (EK 2 )

and the proof of Lemma 26.3.11 is complete.

(1) −2 diam(Q ) diam(Q(2) )

.

(26.57)

1108 � 26 Expanding (or hyperbolic), subexpanding, and parabolic rational functions

26.4 Exercises Exercise 26.4.1. Let P(z) = (3z − z3 )/2. Show that all the critical points of P are superattracting fixed points, and hence that P is expanding on its Julia set. Show also that all components of its Fatou set are simply connected. Exercise 26.4.2. Let Qc (z) = z2 + c. Show that if |z| ≥ |c|, then |Qc (z)| ≥ (|c| − 1)|z|. Deduce that if |c| > 2, then 𝒥 (Qc ) is a Cantor set and Qc is expanding and c ∉ ℳ, where ℳ denotes the Mandelbrot set. Conclude that ℳ ⊆ B(0, 2). Exercise 26.4.3. Show that (1/4, ∞) ⊆ ℂ \ ℳ. Exercise 26.4.4. Let c1 , c2 ∈ ℂ \ ℳ. Show that the polynomials Qc1 and Qc2 are topô by Möbius logically conjugate on their Julia sets. Which of them are conjugate on ℂ transformations? Exercise 26.4.5. Prove that the complement of the immediate basin of attraction of a parabolic point is either connected or else has uncountably many connected components. Exercise 26.4.6. Show that every quadratic polynomial of the form z 󳨃→ λz(1 + z), where λ is a root of unity, is a parabolic rational function. Exercise 26.4.7. Note that every quadratic polynomial has at most one rationally indifferent periodic orbit. Show that for every p ∈ ℕ there exists a parabolic quadratic polynomial with a rationally indifferent periodic point of prime period p. Exercise 26.4.8. Find an expansive constant for the parabolic polynomial Q1/4 . Exercise 26.4.9 (Cf. [94], Exercise 10e). Show that the only fixed point of the rational function T(z) = z − 1/z is the point at infinity, with local degree 3. Show that the two parabolic basins are the upper and lower half-planes and that 𝒥 (T) = ℝ ∪ {∞}. Is this rational function parabolic? Exercise 26.4.10 (Cf. [94], Exercise 10e). Show that ∞ is a rationally indifferent periodic point of the rational function T(z) = z − 1/z + 1, with multiplicity 2, and that 𝒥 (T) is a Cantor set within ℝ ∪ {∞}. Is this rational function parabolic? Exercise 26.4.11 (Cf. [94], Exercise 10e). Show that the rational function T(z) = z − 1/z − 2 has a rationally indifferent periodic point at ∞ with multiplicity 2. Show further that 𝒥 (T) = [0, ∞]. Is this rational function parabolic? Exercise 26.4.12 (Cf. [94], Exercise 10e). Show that T(z) = z + 1/(1 + z2 ) has a rationally indifferent periodic point at ∞ with multiplicity 4. Show that one of the three immediate basins of attraction of ∞ contains ℝ, and hence nearly disconnects the Riemann sphere. Is this rational function parabolic? Exercise 26.4.13. Consider the family of polynomials z 󳨃→ z(1 − zp ), p ∈ ℕ. Which of them are parabolic?

27 Equilibrium states for rational functions and Hölder continuous potentials with pressure gap ̂ → ℂ ̂ is an arbitrary rational function with deg(T) ≥ 2 while In this chapter, T : ℂ φ : 𝒥 (T) → ℝ is a Hölder continuous function (a potential) for which P(T, φ) > sup φ. Our ultimate goal is to prove the existence and uniqueness of an equilibrium state for each such potential and to explore some of that state’s ergodic and dynamical properties. The full statement is given in Theorem 27.8.3, first proved by Denker and Urbański [36]. Another proof was later provided by Przytycki [106]. Our proof basically stems from [36], although we incorporate some reasoning from [106] and [146]. We discuss the proof’s structure and main ingredients in greater detail in a number of remarks in the last section. We also discuss in these remarks many matters related to Theorem 27.8.3.

27.1 Bad and good inverse branches ̂→ℂ ̂ be a rational function with d := deg(T) ≥ 2. For any z ∈ ℂ, ̂ observe that Let T : ℂ degz (T) ≤ deg(T) =: d

(27.1)

and n−1

degz (T n ) = ∏ degT j (z) (T), j=0

∀n ∈ ℕ.

(27.2)

Let z ∈ 𝒥 (T). As T has no superattracting periodic point in 𝒥 (T), it holds that 󵄨 #{n ≥ 0 󵄨󵄨󵄨 T n (z) ∈ Crit(T)} ≤ #Crit(T) ≤ 2d − 2.

(27.3)

It follows from (25.2) and (27.1)–(27.3) that degz (T n ) ≤ d 2d−2 ,

∀n ∈ ℕ, ∀z ∈ 𝒥 (T).

(27.4)

In fact, a much better upper estimate than (27.4) is degz (T n ) ≤ Δ(T) :=

∏

c∈Crit(T)∩𝒥 (T)

degc (T),

∀n ∈ ℕ, ∀z ∈ 𝒥 (T).

(27.5)

̂ normalized so that Let ρ be the spherical metric on ℂ ̂ = 1, Aρ (ℂ) ̂ induced by ρ. where Aρ is the two-dimensional volume (area) on ℂ https://doi.org/10.1515/9783110769876-027

(27.6)

1110 � 27 Equilibrium states for rational functions Let R > 0 be so small that Bρ (𝒥 (T), 2R) ∩ (Crit(T) \ 𝒥 (T)) = 0.

(27.7)

The starting point of all our considerations in this section and on which virtually all hinges in this chapter, is the following technical fact. ̂ → ℂ ̂ be a rational function with d := deg(T) ≥ 2. Let also Lemma 27.1.1. Let T : ℂ γ ∈ (0, 1) and s ∈ ℕ. For every z ∈ 𝒥 (T), there exist rz = rz (γ, s) ∈ (0, R] and Cz = Cz (γ, s) ≥ 1 such that for every n ≥ 0 there is a family Wn (γ, s; z) of connected components of T −n (B(z, rz )) with the following properties: (an ) If 0 ≤ n ≤ s, then the family Wn (γ, s; z) consists of all the connected components of T −n (B(z, rz )). (bn ) max{diam(V ) | V ∈ Wn (γ, s; z)} ≤ Cz γn/2 . (b∗n ) If n > s, −1 ′ 󵄨 󵄨 min{inf 󵄨󵄨󵄨(T n−s ) (ξ)󵄨󵄨󵄨ρ : V ∈ Wn (γ, s; z)} ≥ (Cz γn/2 ) . ξ∈V

(cn )

Wn (γ, s; z) ⊆ Zn (γ, s; z)

(27.8)

#(Zn (γ, s; z) \ Wn (γ, s; z)) ≤ dγ−n + d(2d − 2),

(27.9)

and

where Zn (γ, s; z) is the family of all connected components of all sets of the form T −1 (V ) with V ∈ Wn−1 (γ, s; z). (dn ) If 0 ≤ n ≤ s, then Zn (γ, s; z) = Wn (γ, s; z). ̂ (en ) Each V ∈ Wn (γ, s; z) is a simply connected open subset of ℂ. (fn ) If n > s, then V ∩ Crit(T) = 0,

∀V ∈ Wn (γ, s; z).

(gn ) If n > s, then the map T|V : V → T(V ) is a conformal homeomorphism for every V ∈ Wn (γ, s; z). (hn ) If V ∈ Wn (γ, s; z), then deg(T n |V : V → T n (V )) ≤ Δ(T) := ∏c∈Crit(T)∩𝒥 (T) degc (T). Proof. Let δ > 0 come from the application of Lemma 25.1.5 with n = s + 1 and ε = γ. Since #Crit(T s ) < ∞ and #(T −s (z)) < ∞, that lemma implies the existence of rz ∈ (0, δ/4) such that Crit(T s ) ∩ W ⊆ T −s (z)

and

#(W ∩ T −s (z)) = 1

for all connected components W of T −s (B(z, 2rz )). Then

27.1 Bad and good inverse branches

Crit(T k ) ∩ W ⊆ T −k (z)

and

#(W ∩ T −k (z)) = 1

� 1111

(27.10)

for every connected component W of T −k (B(z, 2rz )) and every 0 ≤ k ≤ s. In particular, for such k’s, Crit(T k ) ∩ (V + \ V ) = 0

(27.11)

for every connected component V of T −k (B(z, rz )), where V + is the unique connected component of T −k (B(z, 2rz )) containing V . We first construct a family Wn′ (γ, s; z) of connected components of T −n (B(z, 2rz )) with the following properties: (a′n ) If 0 ≤ n ≤ s, then the family Wn′ (γ, s; z) consists of all the connected components of T −n (B(z, 2rz )). 󵄨 ′ (bn ) max{Aρ (V ′ ) 󵄨󵄨󵄨 V ′ ∈ Wn′ (γ, s; z)} ≤ γn . (c′n ) Wn′ (γ, s; z) ⊆ Zn′ (γ, s; z)

(27.12)

#(Zn′ (γ, s; z) \ Wn′ (γ, s; z)) ≤ γ−n + (2d − 2),

(27.13)

and

where Zn′ (γ, s; z) is the family of all connected components of all sets of the form ′ T −1 (V ′ ) with V ′ ∈ Wn−1 (γ, s; z). (d′n ) If 0 ≤ n ≤ s, then Zn′ (γ, s; z) = Wn′ (γ, s; z). ̂ (e′n ) Each V ′ ∈ Wn′ (γ, s; z) is a simply connected open subset of ℂ. (f′n ) If n > s, then V ′ ∩ Crit(T) = 0,

∀V ′ ∈ Wn′ (γ, s; z).

(g′n ) If n > s, then the map T|V ′ : V ′ → T(V ′ ) is a conformal homeomorphism for every V ′ ∈ Wn′ (γ, s; z). ′ (hn ) If V ′ ∈ Wn′ (γ, s; z), then deg(T n |V ′ : V ′ → T n (V ′ )) ≤ Δ(T). Starting the construction, for every 0 ≤ n ≤ s let Wn′ (γ, s; z) consist of all connected components of T −n (B(z, 2rz )) and set Zn′ (γ, s; z) = Wn′ (γ, s; z),

∀0 ≤ n ≤ s.

(27.14)

Thus, for all 0 ≤ n ≤ s, properties (a′n ), (c′n ) and (d′n ) are trivially satisfied; property (b′n ) can be fulfilled by reducing rz if necessary so that by Lemma 25.1.5 all sets in Wn′ (γ, s; z) have areas not exceeding γs ; property (e′n ) follows from Lemma 25.1.5 as 2rz < δ; and (h′n ) is a consequence of (27.10) and (27.5); finally, properties (f′n ) and (g′n ) are vacuously fulfilled. In summary, properties (a′n )–(h′n ) hold for every 0 ≤ n ≤ s.

1112 � 27 Equilibrium states for rational functions Proceeding by induction, suppose that the family Wn′ (γ, s; z) has been constructed ′ for some n ≥ s. Let Zn+1 (γ, s; z) be the family of all connected components of all sets of −1 ′ the form T (V ) with V ′ ∈ Wn′ (γ, s; z). Define ′ ′ (γ, s; z) := {V ′ ∈ Zn+1 (γ, s; z) : Aρ (V ′ ) ≤ γn+1 and V ′ ∩ Crit(T) = 0}. Wn+1

(27.15)

We want to show that properties (a′n+1 )–(h′n+1 ) hold. Properties (a′n+1 ) and (d′n+1 ) are vacuously satisfied. Properties (b′n+1 ) and (f′n+1 ) follow directly from the definition of ′ Wn+1 (γ, s; z). Concerning property (c′n+1 ), observe that 󵄨 ′ ′ ′ Zn+1 (γ, s; z) \ Wn+1 (γ, s; z) = {V ′ ∈ Zn+1 (γ, s; z) 󵄨󵄨󵄨 Aρ (V ′ ) > γn+1 or V ′ ∩ Crit(T) ≠ 0}. (27.16) ′ Since all sets in Zn+1 (γ, s; z) are mutually disjoint, relation (27.6) imposes that

󵄨 ′ #{V ′ ∈ Zn+1 (γ, s; z) 󵄨󵄨󵄨 Aρ (V ′ ) > γn+1 } ≤ γ−(n+1) .

(27.17)

󵄨 ′ #{V ′ ∈ Zn+1 (γ, s; z) 󵄨󵄨󵄨 V ′ ∩ Crit(T) ≠ 0} ≤ #Crit(T) ≤ 2d − 2.

(27.18)

′ ′ #(Zn+1 (γ, s; z) \ Wn+1 (γ, s; z)) ≤ γ−(n+1) + (2d − 2),

(27.19)

Moreover,

Therefore,

′ which means that (27.13) in (c′n+1 ) is fulfilled. Of course, by the definitions of Zn+1 (γ, s; z) ′ ′ and Wn+1 (γ, s; z), all of the other requirements of (cn+1 ) also hold. It remains to establish (e′n+1 ), (g′n+1 ) and (h′n+1 ). Both (e′n+1 ) and (g′n+1 ) follow from ′ Corollary 22.5.4 by invoking (e′n ) and the definition of Wn+1 , while (h′n+1 ) follows from (h′n ) and (g′n+1 ). The inductive construction of the family Wn′ (γ, s; z) is complete. Now, for every n ≥ 0 the family Wn (γ, s; z) is defined to be comprised of all connected components of the sets of the form V ′ ∩ T −n (B(z, rz )), where V ′ ∈ Wn′ (γ, s; z). Properties (an ), (fn ) and (gn ) are then automatically satisfied. For each 0 ≤ n ≤ s, property (en ) follows in exactly the same way as (e′n ) did, i. e., from Lemma 25.1.5 and the choice of V ′ , while for n > s it follows from (en−1 ) and (gn ). Property (hn ) ensues directly from (h′n ). Left to show are (bn )–(b∗n ), (cn ) and (dn ). We shall first demonstrate that (27.8) in (cn ) holds. Let V ∈ Wn (γ, s; z). Then there exists a unique V ′ ∈ Wn′ (γ, s; z) such that V is a connected component of V ′ ∩ T −n (B(z, rz )). Therefore,

T(V ) ⊆ T(V ′ ) ∩ T −(n−1) (B(z, rz )).

(27.20)

′ By virtue of (27.12) and the definition of Zn′ (γ, s; z), we know that T(V ′ ) ∈ Wn−1 (γ, s; z). Thus, there exists a unique element W ∈ Wn−1 (γ, s; z) such that

27.1 Bad and good inverse branches

� 1113

T(V ) ⊆ W ⊆ T(V ′ ) ∩ T −(n−1) (B(z, rz )). ′ (γ, s; z) such that W is a As W ∈ Wn−1 (γ, s; z), there exists a unique element W ′ ∈ Wn−1 ′ −n connected component of W ∩ T (B(z, rz )). In particular,

W ⊆ W ′. Since V ⊆ T −1 (W ) and V is connected, there exists a unique connected component V + of T −1 (W ) containing V . In particular, V + ∈ Zn (γ, s; z)

(27.21)

and V + ⊆ T −1 (W ′ ). Because V + is connected and V + ⊆ T −1 (W ′ ), there exists a unique connected component V ++ of T −1 (W ′ ) containing V + . In particular, V ++ ∈ Zn′ (γ, s; z). Given that V ++ ⊇ V + ⊇ V and V ∈ Wn (γ, s; z), we deduce that V ++ ∈ Wn′ (γ, s; z). As, moreover, V + is connected and V + ⊆ V ++ ∩ T −1 (W ) ⊆ V ++ ∩ T −n (B(z, rz )), ̂ of Wn (γ, s; z) containing V + . Hence, there is a unique element W ̂. V ⊆ V+ ⊆ W ̂ ∈ Wn (γ, s; z), it follows that V = W ̂ and thereby V = V + . By virtue Given that V , W of (27.21), we conclude that V ∈ Zn (γ, s; z), i. e., (27.8) is established. We will return to the other parts of (cn ) shortly. For now, we shall prove that (dn ) holds. In view of (27.8), it suffices to show that Zn (γ, s; z) ⊆ Wn (γ, s; z),

∀0 ≤ n ≤ s.

(27.22)

Fix 0 ≤ n ≤ s and V ∈ Zn (γ, s; z). Then T(V ) ∈ Wn−1 (γ, s; z).

(27.23)

1114 � 27 Equilibrium states for rational functions ′ So, there exists a unique H ∈ Wn−1 (γ, s; z) such that T(V ) is a connected component of −(n−1) H ∩T (B(z, rz )). In particular,

V ⊆ T −1 (H) ∩ T −n (B(z, rz )). As V is connected, there is a unique connected component G of T −1 (H) containing V . Hence, G ∈ Zn′ (γ, s; z) and, as V ∈ Zn (γ, s; z), we have that G ∈ Wn′ (γ, s; z). Also, since V is connected and V ⊆ G ∩ T −n (B(z, rz )), there exists a unique connected component V + of G ∩ T −n (B(z, rz )) containing V . Thus, V + ∈ Wn (γ, s; z).

(27.24)

Since T(V + ) ⊆ T −(n−1) (B(z, rz )) ∩ T(G) = T −(n−1) (B(z, rz )) ∩ H, ′ since T(V + ) is connected and since H ∈ Wn−1 (γ, s; z), there is a unique element of + Wn−1 (γ, s; z) containing T(V ). But since also T(V + ) ∩ T(V ) ≠ 0 (in fact, T(V ) ⊆ T(V + )), it follows from (27.23) that T(V + ) ⊆ T(V ). Thus,

T(V ) = T(V + ). Therefore, V ⊆ V + ⊆ T −1 (T(V + )) = T −1 (T(V )).

(27.25)

But as V ∈ Zn (γ, s; z), it ensues from (27.23) and the definition of Zn (γ, s; z) that V is a connected component of T −1 (T(V )). Since V + is connected, it follows from (27.25) that V = V + . This and (27.24) yield that V ∈ Wn (γ, s; z). This means that (27.22) is proved, and so is (dn ) simultaneously. Now, aiming to prove (27.9) in (cn ), we shall first show by induction that #Wn (γ, s; z) = #Wn′ (γ, s; z),

∀n ≥ 0.

(27.26)

For n = 0, this is true since W0 (γ, s; z) = {B(z, rz )} and W0′ (γ, s; z) = {B(z, 2rz )}. So, sup′ pose that (27.26) holds for some n ≥ 0. Since each element of Wn+1 (γ, s; z) contains at least one element of Wn+1 (γ, s; z) and since each element of Wn+1 (γ, s; z) is contained in ex′ ′ actly one element of Wn+1 (γ, s; z), it suffices to show that each element V ′ of Wn+1 (γ, s; z) contains exactly one element of Wn+1 (γ, s; z). For this purpose, let

27.1 Bad and good inverse branches

� 1115

󵄨 Wn+1 (V ′ ) := {U ∈ Wn+1 (γ, s; z) 󵄨󵄨󵄨 U ⊆ V ′ }. Then T(U) ∈ Wn (γ, s; z) and T(U) ⊆ T(V ′ ) ∈ Wn′ (γ, s; z). By the inductive hypothesis, all the images T(U), U ∈ Wn+1 (V ′ ), are then equal. Denote them by V . Then the degrees deg(T|V ′ : V ′ → T(V ′ )) and deg(T|U : U → V ), U ∈ Wn+1 (V ′ ), are well-defined by virtue of Theorem 22.1.3. Thus, ∑

U∈Wn+1 (V ′ )

deg(T|U ) = deg(T|V ′ ).

(27.27)

If n ≥ s, then we deduce from (27.27) and (f′n+1 ) (or (g′n+1 )) that #Wn+1 (V ′ ) = 1. So, let’s consider the case 0 ≤ n < s. By (e′n+1 ), (e′n ), (en+1 ) and (en ), respectively, the sets V ′ , T(V ′ ), U and V are simply connected for every U ∈ Wn+1 (V ′ ). It follows from Corollary 22.5.3 that ∑

[degc (T) − 1] = deg(T|V ′ ) − 1.

c∈V ′ ∩Crit(T)

(27.28)

Similarly, it ensues from Corollary 22.5.3 and (27.27) that ∑

U∈Wn+1

[degc (T) − 1] =

∑

(V ′ )

c∈U∩Crit(T)

[deg(T|U ) − 1] = deg(T|V ′ ) − #Wn+1 (V ′ ).

∑

U∈Wn+1

(V ′ )

(27.29) But, because of (27.11), we have the disjoint union ⋃U∈Wn+1 (V ′ ) U ∩ Crit(T) = V ′ ∩ Crit(T) and thus ∑

∑

U∈Wn+1 (V ′ ) c∈U∩Crit(T)

[degc (T) − 1] =

∑

[degc (T) − 1].

c∈V ′ ∩Crit(T)

(27.30)

We infer from (27.28)–(27.30) that #Wn+1 (V ′ ) = 1. The proof of (27.26) is complete. By virtue of (27.26), we get the following. Claim 1. For every n ≥ 0, each element of Wn (γ, s; z) is contained in exactly one element of Wn′ (γ, s; z) and each element of Wn′ (γ, s; z) contains exactly one element of Wn (γ, s; z). We are now ready to prove (27.9). As each element of Zn′ (γ, s; z) contains at most d elements of Zn (γ, s; z), we conclude by means of Claim 1 and (27.13) that #(Zn (γ, s; z) \ Wn (γ, s; z)) ≤ d #(Zn′ (γ, s; z) \ Wn′ (γ, s; z)) ≤ dγ−n + d(2d − 2), meaning that (27.9) holds. Since (27.8) was already established, we know that (cn ) is satisfied.

1116 � 27 Equilibrium states for rational functions We are thus left to show that (bn )–(b∗n ) are valid. For every n ≥ 0 and V ∈ Wn (γ, s; z), denote by V + the unique element of Wn′ (γ, s; z) such that V is a connected component of V + ∩ T −n (B(z, rz )). So V + is a connected component of T −n (B(z, 2rz )). Since rz < 2rz , we have V ⊆ V +.

(27.31)

Recall that #Ws (γ, s; z) < ∞. Using (27.31) and (e′s ), we gather from Koebe’s distortion theorem (Theorem 23.1.9) that there exists a constant K ≥ 1 such that K −1 ≤

|H ′ (x)| ≤K |H ′ (y)|

(27.32)

̂ For any for all W ∈ Ws (γ, s; z), all points x, y ∈ W and all univalent maps H : W + → ℂ. ̂ connected open set G ⊆ ℂ and any points a, b ∈ G, denote by ℓG (a, b) the infimum of the lengths of all rectifiable curves joining a and b in G. Denote also 󵄨 ℓ(G) := sup{ℓG (a, b) 󵄨󵄨󵄨 a, b ∈ G} ∈ [0, ∞]. As the boundary of a set W ∈ Ws (γ, s; z) is a real-analytic Jordan curve, we know that ℓ(W ) < ∞. Then 󵄨 D := max{ℓ(W ) 󵄨󵄨󵄨 W ∈ Ws (γ, s; z)} < ∞.

(27.33)

Continuing the proof of (bn ), we next note that we can make this property hold for all 0 ≤ n ≤ s by taking Cz ≥ 1 large enough. So, suppose that n > s and V ∈ Wn (γ, s; z). As the analytic map T n−s : V + → T n−s (V + ) ∈ Ws′ (γ, s; z) is injective by (gk ), k = n, n−1, . . . , s+1, we get from (b′n ) that ′ 󵄩−2 ′ 󵄩−2 󵄩 󵄩 γn ≥ Aρ (V + ) ≥ Aρ (V ) ≥ 󵄩󵄩󵄩(T n−s |V ) 󵄩󵄩󵄩∞ Aρ (T n−s (V )) ≥ A󵄩󵄩󵄩(T n−s |V ) 󵄩󵄩󵄩∞ ,

where 󵄨 A := min{Aρ (W ) 󵄨󵄨󵄨 W ∈ Ws (γ, s; z)} > 0. Equivalently, 󵄩󵄩 n−s ′ 󵄩󵄩−1 −1/2 n/2 󵄩󵄩(T |V ) 󵄩󵄩∞ ≤ A γ .

(27.34)

Using (27.33), and (27.32) applied to the univalent map T n−s : V + → T(V + ) ∈ Ws′ (γ, s; z), we obtain that ′ 󵄩−1 ′ 󵄩−1 󵄩 󵄩 diam(V ) ≤ ℓ(V ) ≤ K 󵄩󵄩󵄩(T n−s |V ) 󵄩󵄩󵄩∞ ℓ(T n−s (V )) ≤ KD󵄩󵄩󵄩(T n−s |V ) 󵄩󵄩󵄩∞ ≤ KDA−1/2 γn/2 .

27.1 Bad and good inverse branches

� 1117

This proves item (bn ), while combining (27.34) with (27.32) proves (b∗n ). The proof of Lemma 27.1.1 is complete. Since all V ∈ Zn (γ, s; z), n ≥ 0, have real-analytic Jordan curves for boundary and since the union ⋃sn=0 Zn (γ, s; z) is a finite set, there exists a constant L(z) = L(γ, s; z) ≥ 1 such that ℓV (x, y) ≤ L(z)|x − y|

(27.35)

for all 0 ≤ n ≤ s, all V ∈ Zn (γ, s; z) and all x, y ∈ V , where ℓV (x, y) is the infimum of the lengths of all rectifiable curves joining x to y within V . Recall that (Hα (𝒥 (T)), ‖ ⋅ ‖α ) denotes the Banach space of all Hölder continuous functions with exponent α defined on 𝒥 (T) and equipped with the norm ‖φ‖α := vα (φ)+ ‖φ‖∞ , where vα (φ) stands for the least constant for which φ is Hölder with exponent α while (C(𝒥 (T)), ‖ ⋅ ‖∞ ) is the Banach space of all continuous functions on 𝒥 (T) endowed with the usual supremum norm (see Section 13.1 and Exercise 13.11.2). Lemma 27.1.1 will now start bearing fruits. The first one is the following technical fact about the ergodic sums of Hölder continuous potentials, which will play a crucial role in subsequent considerations. Lemma 27.1.2. In addition to Lemma 27.1.1’s hypotheses, let φ : 𝒥 (T) → ℝ be a Hölder continuous potential. For every z ∈ 𝒥 (T) there is a constant Γz (φ) ≥ 1 such that 󵄨󵄨 󵄨 󵄨󵄨Sn φ(y) − Sn φ(x)󵄨󵄨󵄨 ≤ Γz (φ)

(27.36)

for every n ∈ ℕ, every V ∈ Wn (γ, s; z) and all x, y ∈ V . Furthermore, for every ε > 0 there exists δz (φ) > 0 such that 󵄨󵄨 󵄨 󵄨󵄨Sn φ(y) − Sn φ(x)󵄨󵄨󵄨 < ε,

(27.37)

for every n ∈ ℕ, every V ∈ Wn (γ, s; z) and all x, y ∈ V for which |T n (y) − T n (x)| ≤ δz (φ). Proof. Fix n ∈ ℕ, V ∈ Wn (γ, s; z) and x, y ∈ V . Recall that V ∈ Wn (γ, s; z) implies that T j (V ) is contained in an element of Wn−j (γ, s; z), for every 0 ≤ j ≤ n. We first prove (27.36). Use of Lemma 27.1.1(bn ) and of the Hölder continuity of φ (say, with exponent α and Hölder constant vα (φ)) yields that 󵄨󵄨 n−1 󵄨󵄨n−1 󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨 j j j j 󵄨󵄨Sn φ(y) − Sn φ(x)󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∑ (φ ∘ T (y) − φ ∘ T (x))󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨φ(T (y)) − φ(T (x))󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 j=0 󵄨 j=0 n−1

n−1

󵄨 󵄨α ≤ vα (φ) ∑ 󵄨󵄨󵄨T j (y) − T j (x)󵄨󵄨󵄨 ≤ vα (φ) ∑ diamα (T j (V )) j=0

j=0

n−1

≤ vα (φ)Czα ∑ γ(n−j)α/2 ≤ Γz (φ), j=0

1118 � 27 Equilibrium states for rational functions where Γz (φ) = vα (φ)Czα (1 − γα/2 ) . −1

(27.38)

This establishes (27.36). The proof of (27.37) is somewhat similar but subtler. Set Σα T := vα (φ)

n−1

󵄨α 󵄨󵄨 j j 󵄨󵄨T (y) − T (x)󵄨󵄨󵄨 .

∑

j=max{0,n−s}

Using Lemma 27.1.1(b∗n ), the Hölder continuity of φ and (27.35), we get (with the convention that if n − s − 1 < 0, then any summation from 0 to n − s − 1 is null): n−s−1

󵄨󵄨 󵄨 󵄨 j 󵄨α j α 󵄨󵄨Sn φ(y) − Sn φ(x)󵄨󵄨󵄨 ≤ vα (φ) ∑ 󵄨󵄨󵄨T (y) − T (x)󵄨󵄨󵄨 + ΣT j=0

n−s−1

≤ vα (φ) ∑ ℓTαj (V ) (T j (y), T j (x)) + Σα T j=0

n−s−1

′ 󵄨 󵄨 −α ≤ vα (φ) ∑ [ inf 󵄨󵄨󵄨(T n−s−j ) (ξ)󵄨󵄨󵄨] ℓTαn−s (V ) (T n−s (y), T n−s (x)) + Σα T ξ∈T j (V ) j=0

n−s−1

≤ vα (φ)ℓTαn−s (V ) (T n−s (y), T n−s (x)) ∑ Czα/2 γ(n−j)α/2 + Σα T j=0

≤ vα (φ)Czα/2 (1 − ≤ vα (φ)Czα/2 (1 −

−1 γα/2 ) ℓTαn−s (V ) (T n−s (y), T n−s (x)) + Σα T α 󵄨 n−s α 󵄨 α/2 −1 n−s γ ) (L(z)) 󵄨󵄨󵄨T (y) − T (x)󵄨󵄨󵄨 + Σα T.

But Lemma 25.1.7 affirms that lim max{diam(V ) : V connected component of T −k (B(z, r)) for some 0 ≤ k ≤ s} = 0.

r→0

So, recalling that V ∈ Wn (γ, s; z) implies that T j (V ) is an element of Wn−j (γ, s; z), for any 0 ≤ j ≤ n, there exists δz (φ) > 0 such that −1 α󵄨 󵄨α vα (φ)Czα/2 (1 − γα/2 ) (L(z)) 󵄨󵄨󵄨T n−s (y) − T n−s (x)󵄨󵄨󵄨 < ε/2

and Σα T := vα (φ)

n−1

∑

󵄨󵄨 j 󵄨α j 󵄨󵄨T (y) − T (x)󵄨󵄨󵄨 < ε/2

j=max{0,n−s}

whenever |T n (y) − T n (x)| ≤ δz (φ). Then we conclude that 󵄨󵄨 󵄨 󵄨󵄨Sn φ(y) − Sn φ(x)󵄨󵄨󵄨 < ε/2 + ε/2 = ε,

27.1 Bad and good inverse branches

� 1119

and (27.37) is proved. As an immediate consequence of this lemma, we get the following. Corollary 27.1.3. Under Lemma 27.1.2’s hypotheses and notation, we have that e−Γz (φ) ≤

exp(Sn φ(y))

exp(Sn φ(x))

≤ eΓz (φ)

(27.39)

for every n ∈ ℕ, every V ∈ Wn (γ, s; z) and all x, y ∈ V . In Section 27.6, we will need a generalization of the previous two results. Let φ :

𝒥 (T) → ℝ be a Hölder continuous potential. Given two numbers η, H ≥ 0, write

Dη,H (φ) := {ψ ∈ Hα (𝒥 (T)) : ‖ψ − φ‖∞ ≤ η and vα (ψ) ≤ H}.

(27.40)

Observe that Dη,H (φ) is a compact subset of the Banach space (C(𝒥 (T)), ‖ ⋅ ‖∞ ). Unless otherwise stated, we consider Dη,H (φ) to be endowed with the topology inherited from (C(𝒥 (T)), ‖ ⋅ ‖∞ ). Lemma 27.1.4. In addition to Lemma 27.1.1’s hypotheses, let φ : 𝒥 (T) → ℝ be a Hölder continuous potential and η, H ≥ 0. Then for every z ∈ 𝒥 (T) there exists a constant Γz (φ, η, H) ≥ 1 such that 󵄨󵄨 󵄨 󵄨󵄨Sn ψ(y) − Sn ψ(x)󵄨󵄨󵄨 ≤ Γz (φ, η, H)

(27.41)

for all ψ ∈ Dη,H (φ), every n ∈ ℕ, every V ∈ Wn (γ, s; z) and all x, y ∈ V . Furthermore, for every ε > 0 there exists δz (φ, η, H) > 0 such that 󵄨󵄨 󵄨 󵄨󵄨Sn ψ(y) − Sn ψ(x)󵄨󵄨󵄨 < ε

(27.42)

for all ψ ∈ Dη,H (φ), every n ∈ ℕ, every V ∈ Wn (γ, s; z) and all x, y ∈ V for which |T n (y) − T n (x)| ≤ δz (φ, η, H). Proof. The key is to observe in (27.38) that the sole dependence on φ of the constant Γz (φ) is the term vα (φ). But like the norms ‖φ‖α and ‖φ‖∞ , the term vα (φ) depends continuously on φ. Given the compactness of Dη,H (φ), this implies that Γz (φ, η, H) := maxψ∈Dη,H (φ) Γz (ψ) < ∞. A similar argument holds to secure δz (φ, η, H) > 0. Corollary 27.1.5. Under Lemma 27.1.4’s hypotheses and notation, we have that e−Γz (φ,η,H) ≤

exp(Sn φ(y))

exp(Sn φ(x))

≤ eΓz (φ,η,H)

for all ψ ∈ Dη,H (φ), every n ∈ ℕ, every V ∈ Wn (γ, s; z) and all x, y ∈ V .

(27.43)

1120 � 27 Equilibrium states for rational functions

27.2 The transfer operator ℒφ : C(𝒥 (T)) → C(𝒥 (T)): its lower and upper bounds ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ be a continuous Let T : ℂ potential. As in Chapter 13, we define a transfer operator ℒφ : C(𝒥 (T)) → C(𝒥 (T)) associated to the potential φ by the formula ℒφ (g)(z) :=

∑

w∈T −1 (z)

g(w)eφ(w) degw (T),

∀g ∈ C(𝒥 (T)), ∀z ∈ 𝒥 (T).

(27.44)

It is immediate that ℒφ is a positive bounded linear operator on C(𝒥 (T)). Iterating (27.44) yields for every n ∈ ℕ that n

ℒφ (g)(z) =

=

n−1

∑

g(w) exp(Sn φ(w)) ∏ degT j (w) (T)

∑

g(w) exp(Sn φ(w)) degw (T n ).

w∈T −n (z) w∈T −n (z)

j=0

(27.45)

Furthermore, observe that 󵄩󵄩 n 󵄩󵄩 󵄩 n 󵄩 󵄩󵄩ℒφ 󵄩󵄩∞ = 󵄩󵄩󵄩ℒφ (1)󵄩󵄩󵄩∞ < ∞,

∀n ∈ ℕ.

(27.46)

In this section, we ultimately aim at finding lower and upper bounds on the iterates

ℒnφ (1), n ∈ ℕ. These bounds will depend on n exponentially. This will be accomplished ̂φ of ℒφ and by bounding the iterates ℒ ̂n (1) by two by introducing the normalization ℒ φ

positive constants. But this requires several preliminary results. 27.2.1 First lower bounds on ℒφ

̂ → ℂ ̂ is a rational function with d := deg(T) ≥ 2, then for all Lemma 27.2.1. If T : ℂ δ > 0, all ε > 0, and all n = n(δ, ε) ∈ ℕ large enough, there exists a finite open cover 𝒱n (δ, ε) of 𝒥 (T) consisting of open balls such that: (a) #𝒱n (δ, ε) ≤ eδn , and (b) diam(T i (U)) < ε for every V ∈ 𝒱n (δ, ε), every connected component U of T −n (V ), and all 0 ≤ i ≤ n. Proof. Recall that Critv(T n ) := T n (Crit(T n )) = ⋃nk=1 T k (Crit(T)) is the set of all critical values of T n and satisfies #Critv(T n ) ≤ #Crit(T) ⋅ n ≤ (2d − 2)n. Set δ ∈ (0, 1) and χn := exp(−enδ/4 ),

∀n ∈ ℕ.

(27.47)

27.2 The transfer operator ℒφ : C(𝒥 (T )) → C(𝒥 (T )): its lower and upper bounds

�

1121

Let C ≥ 3 be a constant whose value will be determined later in the course of the proof. In particular, it will depend on ε. Claim 1. There exists a finite set Q ⊆ 𝒥 (T) such that for every n ∈ ℕ large enough there ̂ and radii sz > 0, z ∈ Fn , such that are a finite set Fn ⊆ ℂ #Fn ≤ enδ/2 ,

n

⋃ B(z, sz ) ⊇ 𝒥 (T) \ B(Critv(T ) ∪ Q, χn ),

z∈Fn

Critv(T n ) ∩ ⋃ B(z, Csz ) = 0,

𝒥 (T) ∩ B(z, sz ) ≠ 0,

(27.49) (27.50)

z∈Fn

and

(27.48)

∀z ∈ Fn .

(27.51)

̂ and Q ⊆ 𝒥 (T) be a finite set such that Proof of Claim 1. Let R′ = 41 diam(ℂ) B(Q, R′ ) ⊇ 𝒥 (T) ̂ < δT so small that (thanks to Lemma 24.1.13) and 0 < δ := sup{dist(w, Q) : w ∈ ℂ} sup sup{diamρ (Tw−k (Bρ (z, δ))) : z ∈ Fk (δ), w ∈ T∗−k (z)} ≤ π/2. k∈ℕ

Fix n ∈ ℕ. Apply Lemma 25.4.1 with ε = χn , β = C −1 , and Z = Zn := Critv(T n ) ∪ Q.

(27.52)

This produces a finite set Fn′ ⊆ B(Zn , R′ ) such that ⋃ B(z, C −1 dist(z, Zn )) ⊇ B(Zn , R′ ) \ B(Zn , χn ) ⊇ B(Q, R′ ) \ B(Zn , χn ) ⊇ 𝒥 (T) \ B(Zn , χn )

z∈Fn′

(27.53)

and #Fn′ ≤ A(C −1 ) ⋅ (#Zn )2 log(R′ /χn ) = A(C −1 ) ⋅ (#Zn )2 (log R′ + enδ/4 ) ≤ enδ/2 , where, thanks to (27.47), the last inequality holds for all n ∈ ℕ large enough. Let 󵄨 Fn := {z ∈ Fn′ 󵄨󵄨󵄨 𝒥 (T) ∩ B(z, C −1 dist(z, Zn )) ≠ 0}. By virtue of (27.52)–(27.53), we have that ⋃ B(z, C −1 dist(z, Zn )) ⊇ 𝒥 (T) \ B(Zn , χn ).

z∈Fn

Setting sz := C −1 dist(z, Zn ) for every z ∈ Fn , Claim 1 ensues.

◼

1122 � 27 Equilibrium states for rational functions Now define 󵄨

󵄨

n

𝒱n (δ, ε) := {B(w, χn ) 󵄨󵄨󵄨 w ∈ Critv(T ) ∪ Q} ∪ {B(z, sz ) 󵄨󵄨󵄨 z ∈ Fn }.

(27.54)

By (27.49), observe that 𝒱n (δ, ε) is a finite open cover of 𝒥 (T). By (27.51) and the definition of Q, we also know that all atoms of this cover intersect the Julia set. Let us show that item (a) holds. It follows from (27.47)–(27.48) that #𝒱n (δ, ε) ≤ (2d − 2)n + #Q + enδ/2 ≤ eδn for all n ∈ ℕ large enough, which confirms (a)’s validity. Moving on to (b), let n ∈ ℕ be so large as required thus far in this proof. First, we will prove (b) for those V ∈ 𝒱n (δ, ε) of the form V = B(w, χn ), where w ∈ Critv(T n ) ∪ Q. Take any connected component U of T −n (B(w, χn )). Pick 0 < κ < δ/[16 log(2dT )], where dT originates from Lemma 25.1.7. We show by induction that diam(T n−i (U)) ≤ (2χn )(2dT )

−κi


0, there exists for each n ∈ ℕ large enough a point xn ∈ 𝒥 (T) for which n

ℒφ (1)(xn ) ≥ exp(n(P(φ) − δ)).

Proof. By Theorem 11.2.1 (Bowen’s pressure) and the uniform continuity of φ, there exists ε > 0 such that for every n ∈ ℕ large enough there is a maximal (n, 2ε)-separated set En (2ε) ⊆ 𝒥 (T) for which ∑

y∈En (2ε)

exp(Sn φ(y)) ≥ exp(n(P(φ) − δ/3))

(27.59)

and w, z ∈ 𝒥 (T), |w − z| ≤ ε

|φ(w) − φ(z)| < δ/3.

󳨐⇒

(27.60)

For every n ∈ ℕ so large as required in Lemma 27.2.1, let 𝒱n (δ/3, ε) be the finite open cover of 𝒥 (T) constructed in that lemma. By (27.59) and Lemma 27.2.1(a), there exists V ∈ 𝒱n (δ/3, ε) such that ∑

y∈En (2ε)∩T −n (V )

exp(Sn φ(y)) ≥

1 ∑ exp(Sn φ(y)) #𝒱n (δ/3, ε) y∈E (2ε) n

≥ exp(n(P(φ) − 2δ/3)).

(27.61)

Let xn be an arbitrary point in V ∩ 𝒥 (T). For every y ∈ En (2ε) ∩ T −n (V ), denote by ̂y an arbitrary point of the set T −n (xn ) belonging to the same connected component of T −n (V ) as y. By Lemma 27.2.1(b), we have |T i (y) − T i (̂y)| ≤ ε,

∀i = 0, . . . , n.

Hence, the function En (2ε) ∩ T −n (V ) ∋ y 󳨃󳨀→ ̂y ∈ T −n (xn ) is 1-to-1. With the use of (27.60), we also know that Sn φ(̂y) − Sn φ(y) > −nδ/3.

(27.62)

27.2 The transfer operator ℒφ : C(𝒥 (T )) → C(𝒥 (T )): its lower and upper bounds

�

1125

Using this, (27.62) and (27.61), we get n

ℒφ (1)(xn ) ≥

∑

exp(Sn φ(̂y))

∑

exp(Sn φ(y)) exp(−nδ/3)

y∈En (2ε)∩T −n (V )

≥

y∈En (2ε)∩T −n (V )

≥ exp(n(P(φ) − δ)). 27.2.2 Auxiliary operators Lβ Throughout this section, assume that β > sup φ.

(27.63)

γ := γβ ∈ (esup φ−β , 1).

(27.64)

Take any

Let d and Δ(T) be as defined in (27.1) and (27.5). Pick s ∈ ℕ so large that γ−s ≥ d(2d − 2)Δ(T)

(27.65)

2d(2d − 2)Δ(T)(1 − e−η ) e−η(s+1) < 1/2,

(27.66)

η := log γ − sup φ + β > 0.

(27.67)

and −1

where

For every n ∈ ℕ and every z ∈ 𝒥 (T), define the function Bn (β, z, ⋅) : B(z, rz ) → ℝ by Bn (β, z, ξ) := e−βn

∑

∑

V ∈Zn (γ,s;z)\Wn (γ,s;z) w∈T −n (ξ)∩V

exp(Sn φ(w)),

(27.68)

where Wn (γ, s; z), Zn (γ, s; z) and rz originate from Lemma 27.1.1. The symbol B in this definition alludes to the fact that the above sum is taken over “bad” inverse branches/components of iterates of T in the sense that there is no distortion control over them. Using (27.67) and items (cn ) and (hn ) of Lemma 27.1.1, we get for every ξ ∈ B(z, rz ) that Bn (β, z, ξ) ≤ e−βn ⋅ exp(n sup φ) ⋅ #(Zn (γ, s; z) \ Wn (γ, s; z)) ⋅ Δ(T) ≤ exp(n(sup φ − β)) ⋅ 2d(2d − 2)γ−n ⋅ Δ(T)

= exp(n(sup φ − β − log γ)) ⋅ 2d(2d − 2)Δ(T) = 2d(2d − 2)Δ(T)e−ηn .

(27.69)

1126 � 27 Equilibrium states for rational functions Similarly, define the function Gn (β, z, ⋅) : B(z, rz ) → ℝ by Gn (β, z, ξ) := e−βn

∑

∑

V ∈Wn (γ,s;z) w∈T −n (ξ)∩V

exp(Sn φ(w)).

(27.70)

The symbol G alludes to the fact that the above sum is taken over “good” inverse branches/components of iterates of T in the sense that there is good distortion control over them. We now define a multiple of the transfer operator ℒφ by Lβ := e−β ℒφ : C(𝒥 (T)) → C(𝒥 (T)). For every n > s and ξ ∈ B(z, rz ), observe that n

Lnβ (1)(ξ) = ∑

∑

k=s+1 V ∈Zk (γ,s;z)\Wk (γ,s;z)

e−βk

∑ w∈T −k (ξ)∩V

exp(Sk φ(w))Ln−k β (1)(w)

+ Gn (β, z, ξ).

(27.71)

Indeed, recall that Lnβ (1)(ξ) consists of a sum over all the preimages of ξ under T n . These preimages can be split according to the first time k at which they belong to a “bad” branch (from time 0 at ξ and going backwards in time). The sum Gn (β, z, ξ) accounts for all the preimages that belong only to “good” branches (i. e., that belong to a component in Wn (γ, s; ξ) by Lemma 27.1.1(gn )). The rest is the sum over all preimages that belong to “bad” branches. Each such preimage w belongs to a “good” branch until some maximal time s ≤ k − 1 < n and belongs to a “bad” branch for the first time at time k (so belongs to Zk (γ, s; z)\Wk (γ, s; z) by Lemma 27.1.1(ck )); from that time on, i. e., for the remaining −βk time n − k, one takes Ln−k exp(Sk φ(w)) to reckon for β (1)(w) and one multiplies it by e the trajectory along “good” branches from time 0 until time k − 1. Since 𝒥 (T) is compact, there exists a finite set Yγ,s ⊆ 𝒥 (T) such that ⋃ B(z, rz ) ⊇ 𝒥 (T).

z∈Yγ,s

(27.72)

Under a uniform boundedness condition on the family of functions {Gn (β, z, ⋅)}∞ n=1 , we now derive a uniform upper bound on the supremum norms of the {Lnβ (1)}∞ . n=1 ̂ → ℂ ̂ be a rational function with d := deg(T) ≥ 2 and let Lemma 27.2.3. Let T : ℂ φ : 𝒥 (T) → ℝ be a Hölder continuous potential. If 󵄨󵄨 Kβ := sup{ sup Gn (β, z, ξ) 󵄨󵄨󵄨 n ∈ ℕ, z ∈ Yγ,s } < ∞ 󵄨 ξ∈B(z,r ) z

and

(27.73)

27.2 The transfer operator ℒφ : C(𝒥 (T )) → C(𝒥 (T )): its lower and upper bounds

�

󵄩 j 󵄩 Mβ (φ) := max{Kβ , max󵄩󵄩󵄩Lβ (1)󵄩󵄩󵄩∞ }, 0≤j≤s

1127 (27.74)

then −1 −1 󵄩 󵄩 sup󵄩󵄩󵄩Lnβ (1)󵄩󵄩󵄩∞ ≤ Nβ (φ) := Mβ (φ)[1 − 2d(2d − 2)Δ(T)(1 − e−η ) e−η(s+1) ] . n≥0

(27.75)

Proof. The first crucial observation is that for every n > s, every z ∈ Yγ,s and every ξ ∈ B(z, rz ), we obtain by means of (27.68)–(27.69) and (27.71) that n

Lnβ (1)(ξ) = ∑

∑

k=s+1 V ∈Zk (γ,s;z)\Wk (γ,s;z)

e−βk

∑ w∈T −k (ξ)∩V

exp(Sk φ(w))Ln−k β (1)(w)

+ Gn (β, z, ξ) n

≤ ∑

∑

k=s+1 V ∈Zk (γ,s;z)\Wk (γ,s;z)

e−βk

∑ w∈T −k (ξ)∩V

󵄩 󵄩󵄩 exp(Sk φ(w))󵄩󵄩󵄩Ln−k β (1)󵄩 󵄩∞

+ Gn (β, z, ξ) n

󵄩 󵄩󵄩 ≤ ∑ Bk (β, z, ξ)󵄩󵄩󵄩Ln−k β (1)󵄩 󵄩∞ + Gn (β, z, ξ) k=s+1

n

󵄩 󵄩󵄩 ≤ 2d(2d − 2)Δ(T) ∑ e−ηk 󵄩󵄩󵄩Ln−k β (1)󵄩 󵄩∞ + Gn (β, z, ξ). k=s+1

(27.76)

Proceeding by induction, note that (27.75) holds for all 0 ≤ n ≤ s by the very definition of Nβ (φ). So, suppose that n > s and that (27.75) holds for all integers from 0 to n − 1. We can then continue (27.76) as follows: n

Lnβ (1)(ξ) ≤ 2d(2d − 2)Δ(T)Nβ (φ) ∑ e−ηk + Mβ (φ) k=s+1

≤ 2d(2d − 2)Δ(T)Nβ (φ)(1 − e−η ) e−η(s+1) + Mβ (φ) −1

= [Nβ (φ) − Mβ (φ)] + Mβ (φ) = Nβ (φ). 󵄩 󵄩 Invoking (27.72), we conclude that 󵄩󵄩󵄩Lnβ (1)󵄩󵄩󵄩∞ ≤ Nβ (φ).

We now demonstrate that the uniform boundedness condition (27.73) on the family of functions {Gn (β, z, ⋅)}∞ n=1 is not fulfilled when sup φ < β < P(φ). This happens whenever the potential φ exhibits a pressure gap (cf. Definition 13.7.18 and Remarks 27.9.1–27.9.3). According to Theorem 13.7.30, every Hölder continuous potential generates a pressure gap on a transitive, open, distance expanding system (on an infinite space). So, this property holds for all expanding rational functions. ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Lemma 27.2.4. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. If sup φ < β < P(φ), then

1128 � 27 Equilibrium states for rational functions 󵄨󵄨 Kβ := sup{ sup Gn (β, z, ξ) 󵄨󵄨󵄨 n ∈ ℕ, z ∈ Yγ,s } = ∞. 󵄨 ξ∈B(z,r )

(27.77)

z

Proof. Seeking a contradiction, suppose that Kβ < ∞. It follows from Lemma 27.2.3 that ‖Lnβ (1)‖∞ ≤ Nβ (φ) for every n ≥ 0. Equivalently, n

βn

ℒφ (1)(ξ) ≤ Nβ (φ)e ,

∀n ≥ 0, ∀ξ ∈ 𝒥 (T).

(27.78)

Take δ > 0 so small that P(φ) − δ > β. According to Lemma 27.2.2, there exists a constant C > 0 such that for every n ∈ ℕ there is xn ∈ 𝒥 (T) for which n

ℒφ (1)(xn ) ≥ C exp(n(P(φ) − δ)).

Combining this with (27.78), we deduce that C exp(n(P(φ) − δ)) ≤ Nβ (φ)eβn ,

∀n ∈ ℕ.

Equivalently, exp(n(P(φ) − δ − β)) ≤ C −1 Nβ (φ),

∀n ∈ ℕ.

However, this is impossible since P(φ) − δ − β > 0.

̂φ ; the eigenmeasure mφ 27.2.3 Final bounds on ℒφ and its normalization ℒ Per the notation used throughout this book, let M(𝒥 (T)) denote the topological space of all Borel probability measures on 𝒥 (T) endowed with the weak∗ topology inherited from the topological vector space C(𝒥 (T), ℝ)∗ of all real-valued bounded linear functionals on 𝒥 (T). We know that C(𝒥 (T), ℝ)∗ is locally convex with respect to the weak∗ topology and M(𝒥 (T)) is a convex compact subset of C(𝒥 (T), ℝ)∗ (see Theorem A.1.24). Let ℒ∗φ : C(𝒥 (T), ℝ)∗ → C(𝒥 (T), ℝ)∗ be the dual operator of ℒφ ; it is defined by ℒφ (m)(g) := m(ℒφ (g)), ∗

∀m ∈ C(𝒥 (T), ℝ)∗ , ∀g ∈ C(𝒥 (T), ℝ).

Since ℒφ (1) is a positive continuous function on 𝒥 (T), the operator M(𝒥 (T)) ∋ m 󳨃󳨀→

ℒ∗φ (m) ℒ∗φ (m)(1)

∈ M(𝒥 (T))

is positive, linear and continuous. This was already checked in another context (cf. Theorem 13.6.2). Therefore, the Schauder–Tychonov fixed-point theorem asserts that there is mφ ∈ M(𝒥 (T)) such that

27.2 The transfer operator ℒφ : C(𝒥 (T )) → C(𝒥 (T )): its lower and upper bounds

ℒ∗φ (mφ ) ℒ∗φ (mφ )(1)

�

1129

= mφ .

Writing λφ := ℒ∗φ (mφ )(1) > 0, this means that ℒφ (mφ ) = λφ mφ .

(27.79)

∗

The corresponding normalized operator ̂φ := λ ℒφ : C(𝒥 (T)) → C(𝒥 (T)) ℒ φ −1

(27.80)

̂∗ which fixes mφ : has a dual ℒ φ ̂ (mφ ) = mφ . ℒ φ

(27.81)

∗

Equivalently, ̂φ (g)) = mφ (g), mφ (ℒ

∀g ∈ C(𝒥 (T)).

(27.82)

That is, ̂φ (g) dmφ = ∫ g dmφ , ∫ ℒ 𝒥 (T)

∀g ∈ C(𝒥 (T)).

𝒥 (T)

̂φ = Lβ when β = log λφ . Observe that ℒ Iterating (27.81)–(27.82) yields ̂ (mφ ) = mφ ℒ φ ∗n

and

̂n (g)) = mφ (g), mφ (ℒ φ

∀n ∈ ℕ, ∀g ∈ C(𝒥 (T)).

(27.83)

̂∗n = (ℒ ̂n )∗ . So, ℒ φ φ In Subsection 13.6.2, we have observed that the eigenmeasures of the normalized transfer operator’s dual are conformal measures. This is the case here too. For the sake of completeness, let us repeat this classical demonstration (cf. last part of the proof of ̂∗ Lemma 13.6.13.) Let n ∈ ℕ and B be a Borel set such that T n |B is injective. Because ℒ φ

fixes mφ , we obtain that (using (27.83) and (27.45))

̂n∗ (mφ ) = λ−n ∫ ℒn (1B ⋅g) dmφ = λ−n ∫ ∫ g dmφ = ∫ g d ℒ φ φ φ φ B

B

Letting g = λnφ e−Sn φ results in

𝒥 (T)

∑ 1B (y)g(y)eSn φ(y) dmφ (x).

−n 𝒥 (T) y∈T (x)

1130 � 27 Equilibrium states for rational functions ∫ λnφ e−Sn φ dmφ = ∫ B

n

∑

y∈T −n (x)

𝒥 (T)

1B (y) dmφ (x) = mφ (T (B)).

(27.84)

That is, mφ is (λφ e−φ )-conformal in the sense of Chapter 13. On the other hand, for any set A ⊆ T n (B) we have −n (Sn φ)∘T mφ (T n |−1 B (A)) = λφ ∫ e

n −1 |B

dmφ .

(27.85)

A

In Chapters 13, 16, 19 and 21, we witnessed the crucial role played by conformal measures. We will see their power at work later in this chapter (in the proof of Lemma 27.5.2) and much more substantially in Chapters 29–32. Our next goal is to show that λφ ≥ eP(φ) . We proceed as follows. Fix z ∈ 𝒥 (T) and ξ1 , ξ2 ∈ 𝒥 (T) ∩ B(z, rz ). For every n ≥ 0 and every V ∈ Wn (γ, s; z), pick zn (V ) ∈ V . By virtue of (27.36) in Lemma 27.1.2 and Lemma 27.1.1(hn ), we get for every n ≥ 0 that Gn (0, z, ξ2 ) ≤ eΓz (φ)

∑

∑

V ∈Wn (γ,s;z) w∈T −n (ξ2 )∩V

≤ Δ(T)eΓz (φ)

∑

V ∈Wn (γ,s;z)

≤ Δ(T)e2Γz (φ)

exp(Sn φ(zn (V )))

exp(Sn φ(zn (V )))

∑

∑

V ∈Wn (γ,s;z) w∈T −n (ξ1 )∩V

exp(Sn φ(ξ1 ))

≤ Δ(T)e2Γz (φ) Gn (0, z, ξ1 ).

Therefore, for every β ∈ ℝ, every n ≥ 0, every z ∈ 𝒥 (T), and all ξ1 , ξ2 ∈ 𝒥 (T) ∩ B(z, rz ), we obtain that (Δ(T)e2Γz (φ) )

−1

≤

Gn (β, z, ξ2 ) ≤ Δ(T)e2Γz (φ) . Gn (β, z, ξ1 )

−1

≤

Gn (β, z, ξ2 ) ≤ Cγ,s (φ) Gn (β, z, ξ1 )

In particular, (Cγ,s (φ))

(27.86)

for every β ∈ ℝ, every n ≥ 0, every z ∈ Yγ,s , and all ξ1 , ξ2 ∈ 𝒥 (T) ∩ B(z, rz ), where 󵄨 Cγ,s (φ) := Δ(T) max{e2Γw (φ) 󵄨󵄨󵄨 w ∈ Yγ,s } ∈ [1, ∞).

(27.87)

󵄨 Eφ := min{mφ (B(w, rw )) 󵄨󵄨󵄨 w ∈ Yγ,s } > 0.

(27.88)

Denote

The positivity of this quantity is a consequence of the fact that supp(mφ ) = 𝒥 (T). (Note: While T|𝒥 (T) is not necessarily expanding, it is strongly transitive (in fact, topologically

27.2 The transfer operator ℒφ : C(𝒥 (T )) → C(𝒥 (T )): its lower and upper bounds

�

1131

exact), open, and admits finitely many inverses branches of any given order defined on sufficiently small balls that form a cover of 𝒥 (T). These are the fundamental properties that ensure that mφ has full support. For more information, see Proposition 13.6.14, Corollary 13.6.5 and Proposition 13.5.2.) ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Lemma 27.2.5. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. Then 󵄨󵄨 Klog λφ := sup{ sup Gn (log λφ , z, ξ) 󵄨󵄨󵄨 n ∈ ℕ, z ∈ Yγ,s } ≤ Cγ,s (φ)Eφ−1 . 󵄨 ξ∈B(z,r ) z

Proof. Applying (27.83) with g = 1 and using (27.71) and (27.86), we get for every n ≥ 0 and every z ∈ Yγ,s that ̂n (1)) = ∫ ℒ ̂n (1) dmφ ≥ ∫ ℒ ̂n (1) dmφ 1 = mφ (1) = mφ (ℒ φ φ φ 𝒥 (T)

B(z,rz )

≥ ∫ Gn (log λφ , z, ξ) dmφ (ξ) B(z,rz )

≥ [ inf

ξ∈B(z,rz )

Gn (log λφ , z, ξ)] ⋅ mφ (B(z, rz ))

≥ (Cγ,s (φ)) [ sup Gn (log λφ , z, ξ)] ⋅ Eφ . −1

ξ∈B(z,rz )

Hence, sup Gn (log λφ , z, ξ) ≤ Cγ,s (φ)Eφ−1 .

ξ∈B(z,rz )

We draw two repercussions of the uniform boundedness of the functions {Gn (log λφ , z, ⋅)}∞ n=1 . First, the eigenvalue λφ is larger than or equal to the exponential of the pressure of the system under the potential φ. Second, the supremum norms of the iterates of the ̂n (1)}∞ are uniformly bounded. normalized operator {ℒ φ n=1 ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Lemma 27.2.6. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. Then log λφ ≥ P(φ).

(27.89)

Proof. By way of contradiction, suppose that log λφ < P(φ). It follows from Lemma 27.2.4 that 󵄨󵄨 sup{ sup Gn (log λφ , z, ξ) 󵄨󵄨󵄨 n ∈ ℕ, z ∈ Yγ,s } = ∞. 󵄨 ξ∈B(z,r ) z

This contradicts Lemma 27.2.5.

1132 � 27 Equilibrium states for rational functions ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Lemma 27.2.7. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. Then 󵄩 ̂n 󵄩󵄩 uφ := sup󵄩󵄩󵄩ℒ φ (1)󵄩 󵄩∞ < ∞.

(27.90)

n≥0

Proof. Lemma 27.2.5 ensures that Lemma 27.2.3 applies with β = log λφ and yields 󵄩 ̂n 󵄩󵄩 sup󵄩󵄩󵄩ℒ φ (1)󵄩 󵄩∞ ≤ Nlog λφ (φ) < ∞. n≥0

An immediate consequence of this lemma is the following. ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Corollary 27.2.8. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. Then 󵄩 ̂n 󵄩󵄩 sup󵄩󵄩󵄩ℒ φ (g)󵄩 󵄩∞ ≤ uφ ‖g‖∞ < ∞, n≥0

∀g ∈ C(𝒥 (T)).

We now prove the lower bound counterpart of Lemma 27.2.7. ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Lemma 27.2.9. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. Then ̂n (1)(ξ) > 0. inf ℒ φ

ℓφ := inf

(27.91)

n≥0 ξ∈𝒥 (T)

Proof. Set β = log λφ and γ according to (27.64). Regarding s, we require that it satisfy (27.65) as well as 2d(2d − 2)Δ(T)(1 − e−η )−1 e−η(s+1) ⋅ uφ < 1/2 rather than (27.66). As before, η is defined per (27.67). From (27.76) and Lemma 27.2.7, we get for every n > s, every z ∈ Yγ,s and every ξ ∈ B(z, rz ), that n

n

̂ (1)(ξ) ≤ 2d(2d − 2)Δ(T) ∑ e ℒ φ k=s+1 n

−ηk 󵄩 ̂n−k (1)󵄩󵄩󵄩 󵄩󵄩󵄩ℒ φ 󵄩∞

+ Gn (log λφ , z, ξ)

≤ 2d(2d − 2)Δ(T)uφ ∑ e−ηk + Gn (log λφ , z, ξ) k=s+1

≤ 2d(2d − 2)Δ(T)uφ (1 − e−η ) e−η(s+1) + Gn (log λφ , z, ξ) −1

≤

1 + Gn (log λφ , z, ξ). 2

(27.92)

̂n (1)) = 1, there exists ξn ∈ 𝒥 (T) such that Since mφ (ℒ φ n

̂ (1)(ξn ) ≥ 1. ℒ φ

(27.93)

By (27.72), there is zn ∈ Yγ,s such that ξn ∈ B(zn , rzn ). Using this, (27.92) and (27.93), we deduce that

̂φ ; “Gibbs states” mφ and μφ 27.3 Equicontinuity of iterates of ℒ

∀n > s.

Gn (log λφ , zn , ξn ) ≥ 1/2,

� 1133

(27.94)

As the map T : 𝒥 (T) → 𝒥 (T) is topologically exact (per Corollary 24.3.5), for every z ∈ Yγ,s there exists qz > s such that T qz (B(z, rz )) ⊇ 𝒥 (T). Letting q = max{qz : z ∈ Yγ,s } > s, we infer that T q (B(z, rz )) ⊇ 𝒥 (T),

∀z ∈ Yγ,s .

Consequently, for every w ∈ 𝒥 (T) and every n ≥ 0 there exists wn ∈ B(zn , rzn ) ∩ T −q (w).

(27.95)

Using this, (27.71) with β = log λφ , (27.86) and (27.94), we obtain that n+q

n

̂ (1)(w) ≥ λ exp(Sq φ(wn ))ℒ ̂ (1)(wn ) ≥ λ exp(q inf φ)Gn (log λφ , zn , wn ) ℒ φ φ φ φ −q

−q

≥ λ−q φ exp(q inf φ)(Cγ,s (φ)) Gn (log λφ , zn , ξn ) −1

1 −1 ≥ (Cγ,s (φ)) λ−q φ exp(q inf φ). 2

Hence, 1 −1 ℓφ ≥ min{ (Cγ,s (φ)) λ−q φ exp(q inf φ), min 0≤k≤q 2

̂k (1)(ξ)} > 0. inf ℒ φ

ξ∈𝒥 (T)

We end this section with a straightforward but important ramification of Lemmas 27.2.7 and 27.2.9. ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Lemma 27.2.10. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. Then lim

n→∞

1 log ℒnφ (1)(z) = log λφ , n

∀z ∈ 𝒥 (T).

(27.96)

̂φ ; “Gibbs states” mφ and μφ 27.3 Equicontinuity of iterates of ℒ We now look at the equicontinuity of families of functions generated by the iterates of the normalized operator, like we did for distance expanding maps (cf. Theorem 13.7.4). ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ a Lemma 27.3.1. Let T : ℂ Hölder continuous potential such that P(φ) > sup φ. Then for every function g ∈ C(𝒥 (T)), ̂n (g)}∞ is equicontinuous and pointwise bounded in C(𝒥 (T)). That is, it is the family {ℒ φ n=0

1134 � 27 Equilibrium states for rational functions ̂φ : C(𝒥 (T)) → C(𝒥 (T)) is relatively compact in C(𝒥 (T)). In other terms, the operator ℒ almost periodic. Proof. The pointwise boundedness was the object of Corollary 27.2.8. Once equicontinuity is established, the rest will follow from the Arzelà–Ascoli theorem (Theorem A.2.3) and the definition of almost periodicity of an operator. As regards equicontinuity, fix ε > 0. Set β = log λφ and γ according to (27.64). Concerning s, we require that it satisfy (27.65) as well as 2d(2d − 2)Δ(T)(1 − e−η ) e−η(s+1) ⋅ uφ ‖g‖∞ < ε/4, −1

(27.97)

where uφ comes from (27.90) in Lemma 27.2.7 while η is still defined per (27.67). Generalizing naturally (27.70), define for every n ∈ ℕ, every z ∈ 𝒥 (T) and every ξ ∈ B(z, rz ) the function Gn (log λφ , z, ξ; g) := λ−n φ

∑

∑

V ∈Wn (γ,s;z) w∈T −n (ξ)∩V

exp(Sn φ(w))g(w).

(27.98)

For every n > s, every z ∈ Yγ,s and every ξ ∈ B(z, rz ), relation (27.71) generalizes to n

n

̂ (g)(ξ) = ∑ ℒ φ

∑

k=s+1 V ∈Zk (γ,s;z)\Wk (γ,s;z)

λ−k φ

+ Gn (log λφ , z, ξ; g).

∑ w∈T −k (ξ)∩V

̂n−k (g)(w) exp(Sk φ(w))ℒ φ (27.99)

Denote the first summand by Hn (log λφ , z, ξ; g). Using Corollary 27.2.8, (27.69) and (27.97), we infer that Hn (log λφ , z, ξ; g) ≤ uφ ‖g‖∞ Hn (log λφ , z, ξ; 1) n

= uφ ‖g‖∞ ∑ Bk (log λφ , z, ξ) k=s+1

n

≤ uφ ‖g‖∞ ⋅ 2d(2d − 2)Δ(T) ∑ e−ηk k=s+1

≤ uφ ‖g‖∞ ⋅ 2d(2d − 2)Δ(T)(1 − e−η ) e−η(s+1) −1

< ε/4.

(27.100)

We now deal with Gn (log λφ , z, ξ; g), the second summand in (27.99). Keep n > s and z ∈ Yγ,s . Fix two points x, y ∈ B(z, rz ). Consider V ∈ Wn (γ, s; z). As the degree of the map T n |V : V → T n (V ) is well-defined, there exists a bijection bV ,x,y : V ∩ T −n (x) → V ∩ T −n (y),

̂φ ; “Gibbs states” mφ and μφ 27.3 Equicontinuity of iterates of ℒ

� 1135

where each element in V ∩ T −n (x) and V ∩ T −n (y) is counted according to its degree with respect to the map T n . Starting with (27.98) and using the mean value theorem and Corollary 27.1.3, we can write: 󵄨 󵄨󵄨 󵄨󵄨Gn (log λφ , z, y; g) − Gn (log λφ , z, x; g)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨λ−n ∑ ∑ [eSn φ(w) g(w) − eSn φ(bV ,x,y (w)) g(bV ,x,y (w))]󵄨󵄨󵄨 φ 󵄨󵄨 󵄨󵄨 −n V ∈Wn (γ,s;z) w∈T

󵄨󵄨 󵄨󵄨 = 󵄨󵄨󵄨󵄨λ−n ∑ 󵄨󵄨 φ V ∈W (γ,s;z) n 󵄨 ≤ λ−n φ

≤ λ−n φ ≤ λ−n φ

∑

V ∈Wn (γ,s;z)

∑

V ∈Wn (γ,s;z)

∑

V ∈Wn (γ,s;z)

(x)∩V

󵄨󵄨 eSn φ(w) ⋅ [g(w) − g(bV ,x,y (w))] 󵄨󵄨󵄨 󵄨 [ ∑ Sn φ(w) Sn φ(bV ,x,y (w)) ]󵄨󵄨 + g(bV ,x,y (w)) ⋅ [e −e ] 󵄨󵄨 w∈T −n (x)∩V 󵄨 󵄨 󵄨 eSn φ(w) 󵄨󵄨󵄨g(w) − g(bV ,x,y (w))󵄨󵄨󵄨 [ ∑ 󵄨 󵄨 ] + ‖g‖∞ 󵄨󵄨󵄨eSn φ(w) − eSn φ(bV ,x,y (w)) 󵄨󵄨󵄨 w∈T −n (x)∩V 󵄨 󵄨 eSn φ(w) 󵄨󵄨󵄨g(w) − g(bV ,x,y (w))󵄨󵄨󵄨 [ ] ∑ ‖g‖∞ max{eSn φ(w) , eSn φ(bV ,x,y (w)) } [ ] + [ ] −n 󵄨 󵄨 w∈T (x)∩V 󵄨Sn φ(w) − Sn φ(bV ,x,y (w))󵄨󵄨 ⋅ 󵄨 󵄨 󵄨 [ ] 󵄨󵄨 󵄨󵄨 g(w) − g(b (w)) 󵄨 V ,x,y 󵄨󵄨 eSn φ(w) [ 󵄨 ∑ 󵄨󵄨 󵄨 ], Γ(φ) + e ‖g‖ S φ(w) − Sn φ(bV ,x,y (w))󵄨󵄨󵄨 󵄨 −n ∞ n 󵄨 w∈T (x)∩V (27.101)

where Γ(φ) = max{Γz (φ) : z ∈ Yγ,s } and Γz (φ), z ∈ Yγ,s , come from Lemma 27.1.2. Because the function g : 𝒥 (T) → ℝ is uniformly continuous, because of (27.37) in Lemma 27.1.2 and because #Yγ,s < ∞, there exists δ > 0 such that 󵄨 󵄨 |a − b| < δ 󳨐⇒ uφ 󵄨󵄨󵄨g(a) − g(b)󵄨󵄨󵄨 < ε/4 and 󵄨 󵄨 eΓ(φ) ‖g‖∞ uφ 󵄨󵄨󵄨Sn φ(a) − Sn φ(b)󵄨󵄨󵄨 < ε/4, if, in addition, a, b ∈ V for the same z ∈ Yγ,s and the same V ∈ Wn (γ, s; z). Using this and (27.90) in Lemma 27.2.7, we can continue (27.101) as follows: 󵄨󵄨 󵄨 󵄨󵄨Gn (log λφ , z, y; g) − Gn (log λφ , z, x; g)󵄨󵄨󵄨 ε ε ε −1 ε −1 ≤ Gn (log λφ , z, x) [ uφ−1 + eΓ(φ) ‖g‖∞ e−Γ(φ) ‖g‖−1 ∞ uφ ] ≤ uφ [ uφ ] = . 4 4 2 2 Combining this estimate with (27.100) and (27.99) yields 󵄨󵄨 ̂n ̂n (g)(x)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨Hn (log λφ , z, y; g)󵄨󵄨󵄨 + 󵄨󵄨󵄨Hn (log λφ , z, x; g)󵄨󵄨󵄨 󵄨󵄨ℒφ (g)(y) − ℒ φ 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 + 󵄨󵄨󵄨Gn (log λφ , z, y; g) − Gn (log λφ , z, x; g)󵄨󵄨󵄨 < ε/4 + ε/4 + ε/2 = ε

1136 � 27 Equilibrium states for rational functions ̂n (g)}∞ is thereby uniformly for all n > s, z ∈ Yγ,s , and x, y ∈ B(z, rz ). The family {ℒ φ n=0 equicontinuous. We now identify a fixed point for the normalized operator. This standard argument, already given in the context of distance expanding maps (cf. Lemma 13.7.5), is included here for the sake of completeness and independence of this section. ̂→ℂ ̂ be a rational function of deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Proposition 27.3.2. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. Then there is ρφ ∈ C(𝒥 (T)) such that ̂φ (ρφ ) = ρφ , ℒ

0 < ℓφ ≤ ρφ ≤ uφ < ∞,

and

mφ (ρφ ) := ∫ ρφ dmφ = 1. 𝒥 (T)

Proof. It ensues from Lemmas 27.2.7 and 27.2.9 that ℓφ ≤

1 n−1 ̂j ∑ ℒ (1)(x) ≤ uφ , n j=0 φ

(27.102)

∀n ∈ ℕ, ∀x ∈ 𝒥 (T).

∞ ̂j Hence, the family { n1 ∑n−1 j=0 ℒφ (1)}n=1 is uniformly bounded and thereby pointwise ̂jφ (1)}∞ is bounded. Moreover, it follows from Lemma 27.3.1 that the family { 1 ∑n−1 ℒ n

j=0

n=1

equicontinuous. Thus, all the hypotheses of the Arzelà–Ascoli theorem (Theorem A.2.3) are satisfied. According to that theorem, there exists a strictly increasing sequence ∞ nk −1 ̂j 1 (nk )∞ k=1 such that the sequence { nk ∑j=0 ℒφ (1)}k=1 converges uniformly. Denote its limit by ρφ : 𝒥 (T) → [0, ∞). As a uniform limit of continuous functions on a compact metric space, the function ρφ is also continuous, i. e., ρφ ∈ C(𝒥 (T)). It immediately follows from (27.102) that ℓφ ≤ ρφ ≤ uφ . Furthermore, using (27.83) with g = 1 and Lebesgue’s dominated convergence theorem, we have that mφ (ρφ ) = lim mφ ( k→∞

n −1

n −1

n −1

k 1 k ̂j 1 k ̂j (1)) = lim 1 ∑ mφ (1) = 1. ∑ ℒφ (1)) = lim ∑ mφ (ℒ φ k→∞ nk k→∞ nk nk j=0 j=0 j=0

̂φ , we obtain that Finally, using Lemma 27.2.7 and the continuity and linearity of ℒ n −1

n −1

1 k ̂j 1 k ̂j+1 ∑ ℒφ (1)) = lim ∑ ℒφ (1) k→∞ nk k→∞ nk j=0 j=0

̂φ (ρφ ) = ℒ ̂φ ( lim ℒ

n −1

n −1

k k 1 ̂nk ̂j (1) − 1] = lim 1 ∑ ℒ ̂j (1) = ρφ . [ℒφ (1) + ∑ ℒ φ φ k→∞ nk k→∞ nk j=0 j=0

= lim

̂φ and their dynamical consequences 27.4 Spectral properties of the transfer operator ℒ

�

1137

Thanks to this proposition, the usual reasoning, already outlined for distance expanding maps (cf. Theorem 13.7.7), yields an invariant measure. ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Theorem 27.3.3. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. Then μφ := ρφ mφ is a T-invariant Borel probability measure on 𝒥 (T) which is equivalent to mφ . Proof. It follows from the second and third relations in Proposition 27.3.2 that μφ is equivalent to mφ and that μφ ∈ M(𝒥 (T)). Using (27.82) and the first relation in Proposition 27.3.2, we deduce for every g ∈ C(𝒥 (T)) that ̂φ (ρφ ⋅ g ∘ T)) (μφ ∘ T −1 )(g) = μφ (g ∘ T) = mφ (ρφ ⋅ g ∘ T) = mφ (ℒ ̂φ (ρφ )) = mφ (ρφ g) = mφ (g ⋅ ℒ = μφ (g).

Therefore, μφ ∘ T −1 = μφ . We sometimes refer to the measures mφ and μφ as “Gibbs states” for the potential φ : 𝒥 (T) → ℝ because they resemble, to some extent, the Gibbs states exposed for distance expanding maps in Chapter 13. They behave exactly as ordinary Gibbs states with respect to good inverse branches Gn and are genuine Gibbs states whenever the rational function T is expanding. Indeed, in Section 27.5 we will prove that μφ is an equilibrium state for φ, and if T is expanding then μφ is a Gibbs state according to Chapter 13.

̂φ and their 27.4 Spectral properties of the transfer operator ℒ dynamical consequences We can now quite easily prove the following about the eigenvalue 1 and the eigenvector ̂φ (cf. Theorem 13.8.18). ρφ of ℒ ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Lemma 27.4.1. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. Then the geometric multiplicity of ̂φ : C(𝒥 (T)) → C(𝒥 (T)) is equal to 1. More precisely, the eigenvalue 1 of the operator ℒ ̂φ − Id) = ℂρφ . Ker(ℒ Proof. We first establish the following fact. ̂φ (h) = h and h(w) = 0 Claim 1∗ . If h : 𝒥 (T) → [0, ∞) is a continuous function such that ℒ for some w ∈ 𝒥 (T), then h ≡ 0. Proof of Claim 1∗ . Using (27.45), we have for every n ∈ ℕ that ̂n (h)(w) = λ−n 0 = h(w) = ℒ φ φ

∑

z∈T −n (w)

h(z) exp(Sn φ(z)) degz (T n ).

1138 � 27 Equilibrium states for rational functions −n Therefore, h(z) = 0 for every z ∈ ⋃∞ n=0 T (w). Since h is continuous, it ensues that ∞ −n h(z) = 0 for every z ∈ ⋃n=0 T (w). As T|𝒥 (T) is topologically exact, it is strongly transitive, i. e., the backward orbit of any Julia point is dense in the Julia set (see Definition 1.5.14). So, the claim follows. ◼

Moving on to the proof of the lemma itself, assume that g ∈ C(𝒥 (T)) is an eigenfunĉφ : C(𝒥 (T)) → C(𝒥 (T)). Given tion associated with the eigenvalue 1 of the operator ℒ ̂ that ℒφ preserves the vector subspace C(𝒥 (T), ℝ) consisting of all real-valued functions, both the real and imaginary parts of g are also eigenfunctions associated with the eigenvalue 1. So, we may assume without loss of generality that g is real-valued. Because the function g/ρφ : 𝒥 (T) → ℝ is continuous and 𝒥 (T) is compact, this function attains its minimum value; denote this latter by s. Then (g − sρφ )(z) ≥ 0 for all z ∈ 𝒥 (T) and there is ξ ∈ 𝒥 (T) such that (g − sρφ )(ξ) = 0. Since g − sρφ is an eigenfunction associated with the eigenvalue 1, too, it follows from Claim 1∗ that g − sρφ ≡ 0; equivalently, g = sρφ . ̂φ , namely We now introduce a fully normalized version of the operator ℒ Λφ (g) :=

1 ̂ ℒ (ρ g). ρφ φ φ

(27.103)

̂φ̂ in Section 13.8 but we lighten notation here. Like ℒ ̂φ , This operator was denoted by ℒ this positive bounded linear operator acts on the Banach space C(𝒥 (T)). It is conjugate ̂φ via the Banach isomorphism g 󳨃→ ρφ g. By induction on n ∈ ℕ, with ℒ Λnφ (g) =

1 ̂n ℒ (ρ g). ρφ φ φ

(27.104)

Using this and (27.83) (with g replaced by ρφ g), notice that ̂n (ρφ g) dmφ = ∫ ρφ g dmφ = μφ (g). μφ (Λnφ (g)) = ∫ ρφ Λnφ (g) dmφ = ∫ ℒ φ 𝒥 (T)

𝒥 (T)

(27.105)

𝒥 (T)

̂φ . So, μφ takes the role for Λφ that mφ played for ℒ Substituting the function 1 in (27.103) and using Proposition 27.3.2 results in Λφ (1) =

ρφ 1 ̂ = 1, ℒφ (ρφ ) = ρφ ρφ

(27.106)

i. e., 1 is an eigenfunction corresponding to the eigenvalue 1 of Λφ . From this, we obtain for every g ∈ C(𝒥 (T)) and n ∈ ℕ that 󵄩󵄩 n 󵄩󵄩 󵄩 n 󵄩 󵄩 n 󵄩 󵄩󵄩Λφ (g)󵄩󵄩∞ ≤ 󵄩󵄩󵄩Λφ (‖g‖∞ 1)󵄩󵄩󵄩∞ = 󵄩󵄩󵄩Λφ (1)󵄩󵄩󵄩∞ ‖g‖∞ = ‖g‖∞ .

(27.107)

In Theorem 13.6.3, we established that the linear operator ℒφ , and hence Λφ , can be extended to L1 (μφ ). From (27.105), we have for any g ∈ L1 (μφ ) that

̂φ and their dynamical consequences 27.4 Spectral properties of the transfer operator ℒ

󵄨 n 󵄨 󵄩󵄩 n 󵄩󵄩 n 󵄩󵄩Λφ (g)󵄩󵄩L1 (μφ ) := μφ (󵄨󵄨󵄨Λφ (g)󵄨󵄨󵄨) ≤ μφ (Λφ (|g|)) = μφ (|g|) =: ‖g‖L1 (μφ ) .

� 1139

(27.108)

The reader may want to show that Lemmas 27.2.7–27.3.1, as well as 27.4.1, have analogs for the fully normalized operator Λφ . Among those, the following is an immediate consequence of Lemma 27.3.1, relation (27.104) and the continuity of ρφ from Proposition 27.3.2. ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ a Lemma 27.4.2. Let T : ℂ Hölder continuous potential such that P(φ) > sup φ. Then for every function g ∈ C(𝒥 (T)), the family of iterates {Λnφ (g)}∞ n=0 is equicontinuous and pointwise bounded in C(𝒥 (T)). That is, it is relatively compact in C(𝒥 (T)). In other terms, the operator Λφ : C(𝒥 (T)) → C(𝒥 (T)) is almost periodic. It turns out that the sequence (Λnφ (g))∞ n=0 converges uniformly and we now identify its limit (cf. Lemma 13.8.7). ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ a Lemma 27.4.3. Let T : ℂ Hölder continuous potential such that P(φ) > sup φ. Then for every function g ∈ C(𝒥 (T)), the sequence (Λnφ (g))∞ n=0 converges uniformly in the Banach space C(𝒥 (T)) and lim Λnφ (g) = μφ (g)1.

n→∞

Proof. Splitting g into its real and imaginary parts, we may assume without loss of generality that g ∈ C(𝒥 (T), ℝ). Fix a strictly increasing sequence (nk )∞ k=1 of natural numbers. By Lemma 27.4.2, there exists a strictly increasing subsequence (nkj )∞ j=1 such nk

that the sequence (Λφ j (g))∞ j=1 converges uniformly in C(𝒥 (T), ℝ). Denote its limit by g ∗ ∈ C(𝒥 (T), ℝ). For every f ∈ C(𝒥 (T), ℝ) and every z ∈ 𝒥 (T), we have that Λφ (f )(z) ≤ Λφ (sup f ⋅ 1)(z) = sup f ⋅ Λφ (1)(z) = sup f ⋅ 1(z) = sup f . Thus, sup Λφ (f ) ≤ sup f . Consequently, the sequence (sup Λnφ (f ))∞ n=0 is monotone decreasing. In particular, the sen ∗ ∞ quence (sup Λφ (g ))n=0 is monotone decreasing and sup Λnφ (g ∗ ) ≤ sup g ∗ ,

∀n ≥ 0.

(27.109) nk

Furthermore, by the uniform convergence, sup g ∗ = limj→∞ sup Λφ j (g). It follows that lim sup Λnφ (g) = inf sup Λnφ (g) = sup g ∗ .

n→∞

n≥0

1140 � 27 Equilibrium states for rational functions Hence, sup Λnφ (g ∗ ) ≥ sup g ∗ ,

∀n ≥ 0.

(27.110)

Inequalities (27.109)–(27.110) yield sup Λnφ (g ∗ ) = sup g ∗ ,

∀n ≥ 0.

Since 𝒥 (T) is compact, for every n ≥ 0 there exists a point ξn ∈ 𝒥 (T) such that sup Λnφ (g ∗ ) = Λnφ (g ∗ )(ξn ). Therefore, sup g ∗ = Λnφ (g ∗ )(ξn ) =

λ−n φ

∑

z∈T −n (ξ

n

ρ (ξ ) ) φ n

ρφ (z) exp(Sn φ(z)) degz (T n )g ∗ (z).

By (27.106), the right-hand side of the above equation is a convex combination of numbers smaller than or equal to sup g ∗ . Consequently, g ∗ (z) = sup g ∗ ,

∀z ∈ T −n (ξn ), ∀n ≥ 0.

As the function g ∗ : 𝒥 (T) → ℝ is continuous, it follows from Proposition 25.6.3 that g ∗ is constant. From (27.105) and Lebesgue’s dominated convergence theorem, we then get that nk

nk

g ∗ = μφ (g ∗ ) = μφ ( lim Λφ j (g)) = lim μφ (Λφ j (g)) = lim μφ (g) = μφ (g). j→∞

j→∞

j→∞

∞ So, we have proved that every subsequence (nk )∞ k=1 contains a subsubsequence (nkj )j=1 nk

such that (Λφ j (g))∞ j=1 converges to μφ (g)1 in C(𝒥 (T)). This means that the sequence (Λnφ (g))∞ converges uniformly. n=0 Let 󵄨 ℂ⊥ := {g ∈ C(𝒥 (T)) 󵄨󵄨󵄨 μφ (g) = 0}. As a straightforward repercussion of Lemma 27.4.3 and relations (27.105)–(27.106), the space C(𝒥 (T)) is the direct sum of two simply expressed, Λφ -invariant subspaces and the Λφ -orbits of its points converge as follows (cf. Theorem 13.8.8). ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Theorem 27.4.4. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. Then: (a) ℂ1 and ℂ⊥ are both closed vector subspaces (so are Banach subspaces) of C(𝒥 (T)). (b) Λφ (ℂ1) = ℂ1 while Λφ (ℂ⊥ ) ⊆ ℂ⊥ .

̂φ and their dynamical consequences 27.4 Spectral properties of the transfer operator ℒ

�

1141

(c) C(𝒥 (T)) = ℂ1 ⨁ ℂ⊥ . (d) If g ∈ C(𝒥 (T)), then the sequence (Λnφ (g))∞ n=0 converges uniformly, i. e., in the Banach space C(𝒥 (T)), and lim Λnφ (g) = μφ (g)1.

n→∞

(e) In particular, if g ∈ ℂ⊥ then lim Λnφ (g) = 0.

n→∞

Let 󵄨 (ℂρφ )⊥ := {g ∈ C(𝒥 (T)) 󵄨󵄨󵄨 mφ (g) = 0}. As a direct ramification of Theorem 27.4.4 and (27.104), the space C(𝒥 (T)) is the dîφ -invariant subspaces and the ℒ ̂φ -orbits of its points rect sum of two simply expressed, ℒ converge as follows (cf. Theorem 13.8.9). ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Theorem 27.4.5. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. Then: (a) ℂρφ and (ℂρφ )⊥ are closed vector subspaces (so are Banach subspaces) of C(𝒥 (T)). ̂φ (ℂρφ ) = ℂρφ while ℒ ̂φ ((ℂρφ )⊥ ) ⊆ (ℂρφ )⊥ . (b) ℒ (c) C(𝒥 (T)) = (ℂρφ ) ⨁(ℂρφ )⊥ . ̂n (g))∞ converges uniformly and (d) If g ∈ C(𝒥 (T)) then the sequence (ℒ φ n=0 ̂n (g) = mφ (g)ρφ . lim ℒ φ

n→∞

(e) In particular, if g ∈ (ℂρφ )⊥ then ̂n (g) = 0. lim ℒ φ

n→∞

In Theorem 27.4.4, we described the uniform convergence of the iterates of a function g ∈ C(𝒥 (T)) under the operator Λφ . We now establish the L1 (μφ )-convergence of the iterates of a function g ∈ L1 (μφ ) (cf. Theorem 13.8.12). ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Theorem 27.4.6. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. For every g ∈ L1 (μφ ), 󵄩 lim 󵄩󵄩Λn (g) n→∞󵄩 φ

󵄩 − μφ (g)1󵄩󵄩󵄩L1 (μ ) = 0. φ

Proof. Fix g ∈ L1 (μφ ) and ε > 0. Since C(𝒥 (T)) is a dense subset of the Banach space L1 (μφ ), there exists f ∈ C(𝒥 (T)) such that ‖g − f ‖L1 (μφ ) < ε/2.

1142 � 27 Equilibrium states for rational functions Then 󵄨 󵄨󵄨 󵄨󵄨μφ (g) − μφ (f )󵄨󵄨󵄨 < ε/2. Applying Lemma 27.4.3 and relation (27.108), we obtain that 󵄩 󵄩 lim sup󵄩󵄩󵄩Λnφ (g) − μφ (g)1󵄩󵄩󵄩L1 (μ

φ)

n→∞

󵄩 󵄩 = lim sup󵄩󵄩󵄩Λnφ (g − f ) + (Λnφ (f ) − μφ (f )1) + (μφ (f ) − μφ (g))1󵄩󵄩󵄩L1 (μ

φ)

n→∞

󵄩 󵄩 󵄩 󵄩 ≤ lim sup󵄩󵄩󵄩Λnφ (g − f )󵄩󵄩󵄩L1 (μ ) + lim sup󵄩󵄩󵄩Λnφ (f ) − μφ (f )1󵄩󵄩󵄩L1 (μ ) φ φ n→∞ n→∞ 󵄩󵄩 󵄩󵄩 + 󵄩󵄩(μφ (f ) − μφ (g))1󵄩󵄩L1 (μ ) φ 󵄩 󵄩 󵄨 󵄨 ≤ ‖g − f ‖L1 (μφ ) + lim sup󵄩󵄩󵄩Λnφ (f ) − μφ (f )1󵄩󵄩󵄩∞ + 󵄨󵄨󵄨μφ (f ) − μφ (g)󵄨󵄨󵄨 n→∞ < ε/2 + 0 + ε/2 = ε. The arbitrariness of ε > 0 leads to the conclusion. As a fairly immediate consequence of this theorem, we obtain the following remarkable result (cf. Theorem 13.8.13). ̂→ℂ ̂ be a rational function of deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Theorem 27.4.7. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. Then the measure-preserving dynamical system (T, μφ ) is totally ergodic. That is, all iterates T q : 𝒥 (T) → 𝒥 (T), q ∈ ℕ, are ergodic with respect to the measure μφ , including T itself. Similarly, the measure mφ is totally ergodic. Proof. Fix q ∈ ℕ. Suppose that A ⊆ 𝒥 (T) is a completely T q -invariant Borel set, i. e., T −q (A) = A. Then T −qn (A) = A for all n ≥ 0. Equivalently, 1A ∘ T qn = 1A for all n ≥ 0. Therefore, qn qn qn Λqn φ (1A ) = Λφ (1A ∘ T ) = 1A ⋅ Λφ (1) = 1A .

It follows from Theorem 27.4.6 that 1A = μφ (A)1 μφ -almost everywhere. Thus, μφ (A) ∈ {0, 1}, yielding the ergodicity of T q . The total ergodicity of mφ ensues from that of μφ and Theorem 27.3.3. In fact, we can easily prove more (cf. Theorem 13.8.14). ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Theorem 27.4.8. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. Then the measure-preserving dynamical system (𝒥 (T), T, μφ ) is metrically exact, and hence its Rokhlin natural extension is a K-system.

27.5 Equilibrium states for Hölder potentials φ : 𝒥 (T ) → ℝ

� 1143

Proof. Let ℬ = ℬ(𝒥 (T)) be the Borel σ-algebra of 𝒥 (T). By Definition 8.4.4, the measurepreserving dynamical system (𝒥 (T), T, μφ ) is metrically exact if limn→∞ μφ (T n (B)) = 1 for all B ∈ ℬ. Per Proposition 8.4.5, this holds if and only if the tail σ-algebra ∞

TailT (ℬ) := ⋂ T −n (ℬ) n=0

consists only of sets of μφ -measure 0 or 1. So, let A ∈ TailT (ℬ). For every n ≥ 0, there exists a set An ∈ ℬ such that A = T −n (An ). Then Λnφ (1A ) = Λnφ (1An ∘ T n ) = 1An ⋅ Λnφ (1) = 1An . It ensues from Theorem 27.4.6 that 1An converges in L1 (μφ ) to the constant function μφ (A)1. But since 1An has only two values, namely 0 and 1, we conclude that μφ (A) ∈ {0, 1}. Thus, the metric exactness of the measure-preserving system (𝒥 (T), T, μφ ) is ascertained. That its Rokhlin natural extension is K-mixing is the object of Theorem 8.4.6.

27.5 Equilibrium states for Hölder potentials φ : 𝒥 (T) → ℝ 27.5.1 The “Gibbs state” μφ is an equilibrium state for φ As the only invariant Borel probability measure absolutely continuous with respect to an ergodic invariant Borel probability measure, is this ergodic measure itself, we establish from Theorems 27.4.7 and 27.3.3 the following fact. ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Theorem 27.5.1. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. Then μφ is the only T-invariant Borel probability measure on 𝒥 (T) which is absolutely continuous with respect to mφ . Our next objective, the main one in this subsection and one of the chief results in this chapter, is to show that μφ is an equilibrium state for the potential φ. The following lemma provides the desired lower bound on the free energy of T with respect to μφ . ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Lemma 27.5.2. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. Then hμφ (T) + ∫ φ dμφ ≥ log λφ . 𝒥 (T)

(27.111)

1144 � 27 Equilibrium states for rational functions Proof. Fix ε > 0. As the functions φ : 𝒥 (T) → ℝ, ρφ : 𝒥 (T) → [ℓφ , uφ ] ⊆ (0, ∞) and 1/ρφ : 𝒥 (T) → [1/uφ , 1/ℓφ ] are uniformly continuous on the compact set 𝒥 (T), there exists δ > 0 such that 󵄨 󵄨󵄨 φ(y) − eφ(x) 󵄨󵄨󵄨 < ε inf eφ 󵄨󵄨e

󵄨󵄨 󵄨 󵄨󵄨ρφ (y) − ρφ (x)󵄨󵄨󵄨 < ε inf ρφ

and

(27.112)

whenever x, y ∈ 𝒥 (T) and |x − y| < δ. Let α = α(δ) be the Borel partition of 𝒥 (T) ̂ is injective for every constructed per Lemma 25.6.4. So, diam(α) < δ and T|A : A → ℂ A ∈ α. Fix z ∈ 𝒥 (T). Recall that α(z) is the atom in α where z lies. Using Chapter 9’s notation, for every n ∈ ℕ we then have n−1

n−2

j=1

j=0

αn1 (z) := ⋂ T −j (α(T j (z))) = T −1 ( ⋂ T −j (α(T j+1 (z)))) and αn (z) := αn0 (z) = α(z) ∩ αn1 (z). Using (27.85) and the left relation in (27.112), we obtain that n−2

mφ (αn (z)) = mφ (α(z) ∩ T −1 ( ⋂ T −j (α(T j+1 (z))))) j=0

n−2

= mφ ((T|α(z) ) (T(α(z)) ∩ ⋂ T −j (α(T j+1 (z))))) −1

j=0

=

λ−1 φ exp(φ

∫

∘ (T|α(z) )−1 ) dmφ

−j j+1 (z))) T(α(z))∩⋂n−2 j=0 T (α(T

n−2

φ(z) ≤ λ−1 mφ (T(α(z)) ∩ ⋂ T −j (α(T j+1 (z)))) φ (1 + ε)e j=0

n−2

φ(z) ≤ (1 + ε)λ−1 mφ ( ⋂ T −j (α(T j+1 (z)))). φ e j=0

Using the right relation in (27.112) and the T-invariance of μφ , we get that μφ (αn (z)) = ∫ ρφ dmφ ≤ (1 + ε)ρφ (z)mφ (αn (z)) αn (z)

n−2

φ(z) ≤ (1 + ε)2 λ−1 ρφ (z)mφ ( ⋂ T −j (α(T j+1 (z)))) φ e j=0

= (1 +

φ(z) ε)2 λ−1 φ e

∫ ⋂n−2 j=0

T −j (α(T j+1 (z)))

ρφ (z) dmφ (w)

27.5 Equilibrium states for Hölder potentials φ : 𝒥 (T ) → ℝ

φ(z) ≤ (1 + ε)2 λ−1 φ e

(1 + ε)ρφ (w) dmφ (w)

∫ ⋂n−2 j=0

T −j (α(T j+1 (z)))

⋂n−2 j=0

T −j (α(T j+1 (z)))

φ(z) ≤ (1 + ε)3 λ−1 φ e

� 1145

dμφ (w)

∫ n−2

φ(z) = (1 + ε)3 λ−1 μφ ( ⋂ T −j (α(T j+1 (z)))) φ e j=0

n−2

φ(z) = (1 + ε)3 λ−1 μφ ( ⋂ T −(j+1) (α(T j+1 (z)))) φ e j=0

= (1 +

φ(z) ε)3 λ−1 μφ (αn1 (z)). φ e

Thus, (μφ )αn1 (z) (αn (z)) :=

μφ (αn (z) ∩ αn1 (z)) μφ (αn1 (z))

=

μφ (αn (z))

μφ (αn1 (z))

φ(z) ≤ (1 + ε)3 λ−1 . φ e

Hence, 󵄨 Iμφ (αn 󵄨󵄨󵄨αn1 )(z) := − log((μφ )αn1 (z) (αn (z))) ≥ −3 log(1 + ε) + log λφ − φ(z). So, 󵄨 󵄨 Hμφ (αn 󵄨󵄨󵄨αn1 ) := ∫ Iμφ (αn 󵄨󵄨󵄨αn1 ) dμφ ≥ −3 log(1 + ε) + log λφ − ∫ φ dμφ . 𝒥 (T)

𝒥 (T)

Therefore (cf. Definition 9.4.10), 󵄨 hμφ (T) ≥ hμφ (T, α) := lim Hμφ (αn 󵄨󵄨󵄨αn1 ) ≥ −3 log(1 + ε) + log λφ − ∫ φ dμφ . n→∞

𝒥 (T)

Letting ε ↘ 0 leads to the result. As an immediate consequence of this lemma, Lemma 27.2.6 and the variational principle (Theorem 12.1.1), we reach our objective (cf. Proposition 13.2.9). ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Theorem 27.5.3. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. Then P(φ) = log λφ and μφ is an equilibrium state for the potential φ. Finally, we confirm the uniqueness of mφ , μφ and ρφ (cf. Proposition 13.8.10). ̂→ℂ ̂ be a rational function of deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Proposition 27.5.4. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. Then:

1146 � 27 Equilibrium states for rational functions (a) There exists a unique pair (λ, m), with λ ∈ [0, ∞) and m ∈ M(𝒥 (T)) such that ℒ∗φ (m) = λm; namely, λ = eP(φ) and m = mφ . In particular, λ > 0. ̂φ (ρ) = ρ, ρ ≥ 0, and ∫ (b) There exists a unique ρ ∈ C(𝒥 (T)) such that ℒ ρ dmφ = 1; 𝒥 (T) namely ρ = ρφ . Proof. From Subsection 27.2.3 on, all that was assumed about mφ and λφ was that mφ is an eigenmeasure of the dual operator ℒ∗φ : C(𝒥 (T))∗ → C(𝒥 (T))∗ with eigenvalue λφ > 0, that is, ℒ∗φ (mφ ) = λφ mφ (cf. (27.79)). Thus, all the results from that point on apply to any pair (λ, m), with λ ∈ (0, ∞) and m ∈ M(𝒥 (T)) such that ℒ∗φ (m) = λm.

The fact that λ = eP(φ) , and hence is unique, follows from the observation that, when proving that λφ = eP(φ) , all that we assumed about λφ was that ℒ∗φ (mφ ) = λφ mφ for some mφ ∈ M(𝒥 (T)). Likewise, assuming only the equality ℒ∗φ (mφ ) = λφ mφ for some mφ ∈ M(𝒥 (T)), we derived Theorem 27.4.5. From part (d) of that result, we deduce that ̂n (1), ρφ = lim ℒ φ n→∞

which is completely independent of mφ (given that λφ has already been shown to be unique). So ρφ is unique, i.e. ρ = ρφ , and the above is a formula for it. This yields part (b). From this and Theorem 27.4.5(d), we infer that mφ (g) = lim

n→∞

̂n (g)(z) ℒ φ ̂n (1)(z) ℒ φ

,

∀z ∈ 𝒥 (T).

This establishes the uniqueness of mφ .

27.6 Continuous dependence on φ of the “Gibbs states” mφ , μφ and of the density ρφ Our ultimate goal now is to show that μφ is the unique equilibrium state for the potential φ. We will achieve this by proving in the next section that the pressure function is differentiable. It is a fairly general fact that such differentiability gives rise to uniqueness of equilibria. As a preparation, we establish in this section some interesting results leading to the continuous dependence on the potential φ of the “Gibbs states” mφ , μφ , and of the Radon–Nikodym derivative ρφ . ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ a Hölder Let T : ℂ continuous potential such that P(φ) > sup φ. As the function C(𝒥 (T)) ∋ ψ 󳨃→ P(ψ) ∈ ℝ is continuous (even Lipschitz continuous with Lipschitz constant 1; see Exercise 11.5.7 or Corollary 12.1.11), there exists ηφ > 0 such that P(ψ) > sup ψ

for every

ψ ∈ B∞ (φ, ηφ ) = {ψ ∈ C(𝒥 (T)) : ‖ψ − φ‖∞ ≤ ηφ }.

27.6 Continuous dependence on φ of the “Gibbs states” mφ , μφ and of the density ρφ

�

1147

Prior to stating the next result, recall the definition of Dη,H (φ) in (27.40). When η = ηφ , we shorten the notation Dηφ ,H (φ) to DH (φ). ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Lemma 27.6.1. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. For any H ≥ 0, the function DH (φ) ∋ ψ 󳨃󳨀→ mψ ∈ M(𝒥 (T)) is continuous when M(𝒥 (T)) is equipped with the weak∗ topology. Proof. Let (ψn )∞ n=1 be a sequence of functions in DH (φ) converging uniformly to some function ψ : 𝒥 (T) → ℝ. Let (mψn )∞ k=1 be any convergent subsequence of the sequence k ∞ (mψn )n=1 . Denote its limit by m. The function C(𝒥 (T)) ∋ χ 󳨃󳨀→ ℒχ ∈ L(C(𝒥 (T))) is obviously continuous. Recalling that C(𝒥 (T)) ∋ χ 󳨃→ P(χ) ∈ ℝ is continuous too, we deduce that m = lim mψn = lim e−P(ψnk ) ℒ∗ψn (mψn ) = e−P(ψ) ℒ∗ψ (m). k→∞

k

k→∞

k

k

So, ℒ∗ψ (m) = eP(ψ) m, whence m = mψ by Proposition 27.5.4(a). In Lemma 27.2.7, we established that the supremum norms of the iterates of the normalized operator are uniformly bounded from above and called that bound uφ . We now show that there is a bound that works for all ψ ∈ DH (φ) and call it uφ,H . ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Lemma 27.6.2. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. For any H ≥ 0, 󵄩 ̂n 󵄩󵄩 uφ,H := sup{󵄩󵄩󵄩ℒ ψ (1)󵄩 󵄩∞ : ψ ∈ DH (φ), n ≥ 0} < ∞. Proof. It follows from Lemma 27.6.1 and the compactness of M(𝒥 (T)) that 󵄨 Eφ,H := inf{Eψ 󵄨󵄨󵄨 ψ ∈ DH (φ)} > 0, where the quantities Eψ are defined by (27.88). Moreover, by definition of DH (φ), the function DH (φ) ∋ ψ 󳨃→ P(ψ) − sup ψ is strictly positive at each point. The continuity of that function and the compactness of DH (φ) imply that this function attains a strictly positive minimum. Call this minimum β′ . Take ′ γ ∈ (e−β , 1) in lieu of (27.64). Pick s according to (27.65)–(27.66). Observe that the results brought about in Subsection 27.2.2 apply with β = log λψ = P(ψ) (and corresponding η per (27.67)) for every ψ ∈ DH (φ). Additionally, thanks to Lemma 27.1.4 we can replace Cγ,s (φ) in (27.87) by the constant

1148 � 27 Equilibrium states for rational functions 󵄨 C(φ, H) = Cγ,s (φ, H) := Δ(T) max{e2Γw (φ,ηφ ,H) 󵄨󵄨󵄨 w ∈ Yγ,s } ∈ [1, ∞). It ensues from Lemma 27.2.5 that 󵄨 −1 Gφ,H := sup{Gn (log λψ , z, ξ) 󵄨󵄨󵄨 n ∈ ℕ, ψ ∈ DH (φ), z ∈ Yγ,s , ξ ∈ B(z, rz )} ≤ Eφ,H C(φ, H). We can further replace Mβ (φ) and Nβ (φ) from (27.74)–(27.75) by 󵄨 Mφ,H := sup{Mlog λψ (ψ) 󵄨󵄨󵄨 ψ ∈ DH (φ)} ≤ max{Gφ,H ,

󵄩 ̂j 󵄩 sup max󵄩󵄩󵄩ℒ (1)󵄩󵄩󵄩∞ } < ∞ ψ

ψ∈DH (φ) 0≤j≤s

and Nφ,H := Mφ,H [1 − 2d(2d − 2)Δ(T)(1 − e−η ) e−η(s+1) ] −1

−1

< ∞.

Lemma 27.2.3 implies that uφ,H ≤ Nφ,H < ∞. Aiming at generalizing Lemma 27.3.1, we now turn our attention to the relative com̂n (g). pactness of sets of functions ℒ ψ ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Lemma 27.6.3. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. Then for every H ≥ 0 and every relatively compact set K ⊆ C(𝒥 (T)), the set ̂n (g) : ψ ∈ DH (φ), g ∈ K, n ≥ 0} ⊆ C(𝒥 (T)) {ℒ ψ is relatively compact. Proof. By Lemma 27.6.2, the said set is uniformly bounded and thereby pointwise bounded. Suppose first that K is a singleton, i. e., K = {g} for some g ∈ C(𝒥 (T)). With slight appropriate modifications which we indicate below, the proof of Lemma 27.3.1 establishes that the set ̂n (g) : ψ ∈ DH (φ), n ≥ 0} ⊆ C(𝒥 (T)) {ℒ ψ is equicontinuous, and hence relatively compact according to the Arzelà–Ascoli theorem ′ (Theorem A.2.3). Indeed, as in the proof of Lemma 27.6.2, take γ ∈ (e−β , 1). However, pick s so that it satisfy (27.65) as well as (27.97) with uφ replaced by uφ,H from Lemma 27.6.2. Observe once again that the results brought about in Subsection 27.2.2 apply with β = log λψ = P(ψ) for every ψ ∈ DH (φ). Accordingly, replace φ by ψ ∈ DH (φ) in the rest of the proof of Lemma 27.3.1. Next, replace all occurrences of uφ by uφ,H . Replace also Γ(φ) by Γ(φ, H) := max{Γz (φ, ηφ , H) : z ∈ Yγ,s } and correspondingly use Lemma 27.1.4 rather than Lemma 27.1.2.

27.6 Continuous dependence on φ of the “Gibbs states” mφ , μφ and of the density ρφ

� 1149

Now consider the general case. Choose an arbitrary sequence n

̂ j (gj )) , (ℒ j=1 ψ ∞

j

∞ where (nj )∞ j=1 is a sequence of nonnegative integers, (ψj )j=1 a sequence in DH (φ) and ∞ (gj )j=1 a sequence in K. As the set K is relatively compact and DH (φ) is compact, by passing to a subsequence, we may assume without loss of generality that (gj )∞ j=1 converges uniformly to some g ∈ C(𝒥 (T)) and (ψj )∞ converges uniformly to some ψ ∈ DH (φ). j=1 Using Lemma 27.6.2, we then have that

󵄩󵄩 ̂nj 󵄩󵄩 ̂nj (g)󵄩󵄩󵄩 = 󵄩󵄩󵄩ℒ ̂nj 󵄩󵄩ℒψ (gj ) − ℒ 󵄩∞ 󵄩 ψj (gj − g)󵄩󵄩∞ ≤ uφ,H ‖gj − g‖∞ . ψj j Since limj→∞ uφ,H ‖gj −g‖∞ = 0 and since, in view of the already proved singleton part of ̂nj (g))∞ is relatively compact, we conclude that (ℒ ̂nj (gj ))∞ our lemma, the sequence (ℒ ψj

j=1

ψj

contains a convergent subsequence.

j=1

Lemma 27.6.3 (applied with K = {1}) combined with Proposition 27.4.5(e) (applied with g := 1) allow us to attest the relative compactness of the set of Radon–Nikodym derivatives ρψ = dμψ /dmψ . ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Lemma 27.6.4. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. Then for every H ≥ 0 the set {ρψ : ψ ∈ DH (φ)} ⊆ C(𝒥 (T)) is relatively compact. We can now assert the continuity of the Radon–Nikodym derivatives ρψ as well as the continuity of the “Gibbs states” mψ and μψ (cf. Theorem 13.8.21). ̂→ℂ ̂ be a rational function of deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Proposition 27.6.5. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. Then for every H ≥ 0 the function DH (φ) ∋ ψ 󳨃󳨀→ ρψ ∈ C(𝒥 (T)) is continuous with respect to the topology of uniform convergence on C(𝒥 (T)) while the functions DH (φ) ∋ ψ 󳨃󳨀→ mψ ∈ M(𝒥 (T))

and DH (φ) ∋ ψ 󳨃󳨀→ μψ ∈ M(𝒥 (T))

are continuous with respect to the weak∗ topology on the set M(𝒥 (T)) of Borel probability measures on 𝒥 (T). Proof. Let (ψn )∞ n=1 be a sequence of functions in DH (φ) converging uniformly to some function ψ : 𝒥 (T) → ℝ, which belongs to DH (φ) as this set is compact.

1150 � 27 Equilibrium states for rational functions The assertion about χ 󳨃→ mχ has already been proved in Lemma 27.6.1. The statement on χ 󳨃→ μχ will follow once we establish the continuity of χ 󳨃→ ρχ . To this end, suppose for a contradiction that this latter fails at a function ψ ∈ DH (φ). This means that there exist δ > 0 and a sequence (ψk )∞ k=1 of functions in DH (φ) that converges uniformly to the function ψ and such that ‖ρψk − ρψ ‖∞ > δ,

(27.113)

∀k ∈ ℕ.

By virtue of Lemma 27.6.4 and by passing to a subsequence, we may assume that the sequence (ρψk )∞ k=1 converges uniformly to some function ρ ∈ C(𝒥 (T)). By (27.113), ρ = ̸ ρψ .

(27.114)

On the other hand, the continuity of the functions χ 󳨃→ ℒχ and χ 󳨃→ P(χ) entails that ℒψ (ρ) = e

P(ψ)

ρ.

Moreover, using the continuity of χ 󳨃→ mχ , we conclude that ∫ ρ dmψ = lim ∫ ρψk dmψk = lim 1 = 1. k→∞

𝒥 (T)

k→∞

𝒥 (T)

So, ρ = ρψ according to Proposition 27.5.4(b), contrary to (27.114). The continuity of χ 󳨃→ μχ is a direct consequence of the continuity of the eigenmeasures χ 󳨃→ mχ and of the eigenfunctions χ 󳨃→ ρχ . Indeed, if (ψn )∞ n=1 is a sequence in DH (φ) converging uniformly to ψ ∈ DH (φ), then for every function g ∈ C(𝒥 (T)) we get lim ∫ g dμψn = lim ∫ gρψn dmψn = ∫ gρψ dmψ = ∫ g dμψ .

n→∞

n→∞

𝒥 (T)

𝒥 (T)

𝒥 (T)

𝒥 (T)

∗ So, the sequence (μψn )∞ n=1 converges to μψ in the weak topology on M(C(𝒥 (T))).

27.7 Differentiability of topological pressure In this section, we demonstrate that the pressure function is differentiable. This will be the key ingredient to prove the uniqueness of equilibrium states. ̂→ℂ ̂ be a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Lemma 27.7.1. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. Fix H ≥ 0. If ψ ∈ DH (φ) and g ∈ C(𝒥 (T)), then 󵄩󵄩 1 ℒn (S g) 󵄩󵄩 󵄩󵄩 n ψ n 󵄩󵄩 󵄩󵄩 = 0. lim 󵄩󵄩󵄩 − ( g dμ ) 1 ∫ ψ 󵄩󵄩 n n→∞󵄩 󵄩󵄩 ℒψ (1) 󵄩󵄩∞ 𝒥 (T)

27.7 Differentiability of topological pressure

� 1151

Proof. Observe that 1 n ℒ (S g) n ψ n ℒnψ (1)

=

1 n

j ̂n ∑n−1 j=0 ℒψ (g ∘ T )

̂n (1) ℒ ψ

=

1 n

n−j

j

̂ ̂ ∑n−1 j=0 ℒψ (g ℒψ (1)) ̂n (1) ℒ ψ

.

(27.115)

It follows from Lemma 27.6.3 that ̂j (1) : j ≥ 0} ⊆ C(𝒥 (T)) is relatively compact. the set {g ℒ ψ Applying Lemma 27.6.3 again, we deduce that ̂n−j (g ℒ ̂j (1)) : n ∈ ℕ, 0 ≤ j < n} is relatively compact. the set {ℒ ψ ψ Therefore, the set {

1 n−1 ̂n−j ̂j ∑ ℒ (g ℒψ (1)) : n ∈ ℕ} is relatively compact. n j=0 ψ

Using Lemma 27.6.2, we obtain that 󵄩󵄩 󵄩󵄩 󵄩󵄩 ̂n−j ̂j 󵄩 󵄩󵄩ℒψ (g ℒψ (1)) − (∫ g dμψ )ρψ 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩∞ 󵄩󵄩 󵄩󵄩 󵄩 ̂n−j ̂j ̂n−j (gρψ ) − (∫ g dμψ )ρψ 󵄩󵄩󵄩 = 󵄩󵄩󵄩ℒ (g ℒψ (1) − gρψ ) + ℒ 󵄩󵄩 ψ ψ 󵄩󵄩 󵄩∞ 󵄩󵄩 󵄩󵄩 󵄩 ̂n−j 󵄩󵄩 󵄩󵄩 ̂j 󵄩 󵄩 ̂n−j 󵄩󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩ℒ g( ℒ ( 1 ) − ρ ) + ℒ (gρ ) − (∫ g dμ )ρ 󵄩 󵄩 󵄩 󵄩 ψ 󵄩∞ 󵄩 ψ ψ ψ󵄩 󵄩󵄩 ψ 󵄩∞ 󵄩 ψ 󵄩󵄩 ψ 󵄩∞ 󵄩󵄩 󵄩󵄩 j n−j 󵄩̂ 󵄩 󵄩̂ 󵄩 ≤ uφ,H ‖g‖∞ 󵄩󵄩󵄩ℒ (1) − ρψ 󵄩󵄩󵄩∞ + 󵄩󵄩󵄩ℒ (gρψ ) − (∫ g dμψ )ρψ 󵄩󵄩󵄩 . ψ 󵄩󵄩 ψ 󵄩󵄩∞ By virtue of Theorem 27.4.5(d), 󵄩 ̂n 󵄩󵄩 lim 󵄩󵄩󵄩ℒ ψ (1) − ρψ 󵄩 󵄩∞ = 0

n→∞

and

󵄩󵄩 󵄩󵄩 󵄩 ̂n−j 󵄩 lim 󵄩󵄩󵄩ℒ (gρψ ) − (∫ g dμψ )ρψ 󵄩󵄩󵄩 = 0. ψ 󵄩󵄩∞ n−j→∞󵄩 󵄩

Therefore, 󵄩󵄩 󵄩󵄩 󵄩 ̂n−j ̂j 󵄩󵄩 lim 󵄩󵄩󵄩ℒ (g ℒ ( 1 )) − (∫ g dμ )ρ 󵄩󵄩 = 0. ψ ψ ψ j→∞ 󵄩 ψ 󵄩󵄩∞ 󵄩 n−j→∞ Consequently, 󵄩󵄩 󵄩󵄩 n−1 󵄩 󵄩󵄩 1 ̂n−j (g ℒ ̂j (1)) − (∫ g dμψ )ρψ 󵄩󵄩󵄩 = 0. lim 󵄩󵄩󵄩 ∑ ℒ 󵄩󵄩 ψ ψ n→∞󵄩 󵄩󵄩∞ 󵄩󵄩 n j=0

(27.116)

1152 � 27 Equilibrium states for rational functions It follows from this and Theorem 27.4.5(d) (applied with g = 1) that n−j 󵄩󵄩 1 ∑n−1 ℒ 󵄩󵄩 ̂j (1)) 󵄩󵄩 n j=0 ̂ψ (g ℒ 󵄩󵄩 ψ 󵄩󵄩 = 0. lim 󵄩󵄩󵄩 − (∫ g dμ ) 1 ψ 󵄩󵄩 n→∞󵄩 ̂n (1) ℒ 󵄩󵄩 󵄩󵄩∞ ψ

We are done by invoking (27.115). Finally, we assert the differentiability of the pressure function (cf. Theorem 13.10.6). ̂→ℂ ̂ be a rational function of deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Proposition 27.7.2. Let T : ℂ a Hölder continuous potential such that P(φ) > sup φ. If g : 𝒥 (T) → ℝ is a Hölder continuous function, then the function ℝ ∋ t 󳨃󳨀→ P(φ + tg) ∈ ℝ is differentiable on some sufficiently small open neighborhood of 0, and d P(φ + tg) = ∫ g dμφ+tg . dt 𝒥 (T)

Proof. Set φt := φ + tg. Obviously, there are H > 0 and δ > 0 such that φt ∈ DH (φ) for all t ∈ [−δ, δ]. It then ensues from Lemma 27.2.10 and Theorem 27.5.3 that for every z ∈ 𝒥 (T), 1 log ℒnφt (1)(z) n 1 = lim log ∑ exp(Sn φ(w) + tSn g(w)) degw (T n ). n→∞ n w∈T −n (z)

P(φt ) = lim

n→∞

(27.117)

Fix an arbitrary z ∈ 𝒥 (T) and for every n ∈ ℕ set Fn (t) =

1 log ℒnφt (1)(z), n

so that P(φt ) = lim Fn (t). n→∞

The following claim follows from Hölder’s inequality. Claim 1. For every n ∈ ℕ, the function [−δ, δ] ∋ t 󳨃→ Fn (t) ∈ ℝ is convex. The reader can further convince themselves of the following fact.

(27.118)

27.8 Uniqueness of equilibrium states for Hölder continuous potentials

� 1153

Claim 2. For every n ∈ ℕ, Fn′ (t) =

1 n ℒ (S g)(z) n φt n , ℒnφt (1)(z)

∀t ∈ (−δ, δ).

The functions ( (−δ, δ) ∋ t 󳨃󳨀→ Fn′ (t) ∈ ℝ )n=1 ∞

are each monotone by Claim 1. By Claim 2 and Lemma 27.7.1, they converge pointwise to the function (−δ, δ) ∋ t 󳨃󳨀→ ∫ g dμφt , 𝒥 (T)

which is continuous in view of Proposition 27.6.5. Hence, by virtue of Dini’s theorem (Theorem A.1.10), this convergence is uniform on any compact subset of (−δ, δ). Combining this with (27.118), we conclude that the function (−δ, δ) ∋ t 󳨃󳨀→ P(φ + tg) ∈ ℝ d is differentiable and its derivative dt P(φ+tg) is equal to ∫ g dμφ+tg , the limit of the deriva′ tives Fn (t).

27.8 Uniqueness of equilibrium states for Hölder continuous potentials In order to make use of Proposition 27.7.2 to conclude the proof of the uniqueness of equilibrium states, we need one general fact. We formulate and prove it now. Suppose that (V , ‖ ⋅ ‖) is a real normed space. Fix a point x ∈ V and let U be an open neighborhood of x in V . Let P:U →ℝ be a function. We call a continuous linear functional F:V →ℝ tangent to P at a point x ∈ V if there exists a number η = η(F) > 0 such that x + BV (0, η) ⊆ U

and

F(y) ≤ P(x + y) − P(x),

∀y ∈ BV (0, η).

We denote the set of all such functionals by V ∗ (P; x). Sometimes the term supporting functional instead of tangent is used in the literature.

1154 � 27 Equilibrium states for rational functions The following fact is well known. For instance, its short proof is provided as the proof of the second part (implication) of Theorem 3.6.5 in [110]. We include it here for the sake of completeness and convenience of the reader. Lemma 27.8.1. Let (V , ‖ ⋅ ‖) be a real normed space, let x ∈ V and let U be an open neighborhood of x in V . Let P : U → ℝ be a function. If the function P is differentiable at x in every direction in V , then the set V ∗ (P; x) contains at most one element. Proof. Suppose for a contradiction that V ∗ (P; x) contains at least two distinct bounded ̂ Then F(y) − F(y) ̂ linear functionals, say F and F. > 0 for some y ∈ V . Since F, F̂ are linear, ̂ By hypothesis, P is differentiable we can find such a point y with ‖y‖ < min{η(F), η(F)}. at x in the direction given by y. Hence, lim t→0

P(x − ty) − P(x) P(x + ty) − P(x) = lim . t→0 t −t

Consequently, lim t→0

P(x + ty) + P(x − ty) − 2P(x) = 0. t

On the other hand, as F and F̂ are supporting functionals of P at x, we have for every 0 < t ≤ 1 that ̂ ̂ P(x + ty) − P(x) ≥ F(ty) = tF(y) and P(x − ty) − P(x) ≥ F(−ty) = −t F(y). Thus, lim sup t→0

P(x + ty) + P(x − ty) − 2P(x) ̂ ≥ F(y) − F(y) > 0. t

This contradiction finishes the proof. Finally, we can confirm the uniqueness of equilibrium states. It is worth recalling that the uniqueness of equilibrium states was established in Chapter 13 for transitive, open, distance expanding systems. But not all rational functions are expanding. The proof in the current setting is quite different. It relies on the differentiability of the pressure function. In Chapter 13, we opted for a more straightforward approach. We could have proved uniqueness of equilibrium states through the pressure’s differentiability as well but that property (in fact, the pressure’s real-analyticity) was derived at a later stage. ̂→ℂ ̂ is a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Theorem 27.8.2. If T : ℂ is a Hölder continuous potential such that P(φ) > sup φ, then μφ is the unique equilibrium state for φ.

27.8 Uniqueness of equilibrium states for Hölder continuous potentials

� 1155

Proof. Consider the topological pressure function P : Hα (𝒥 (T)) → ℝ. Let ν be an equilibrium state for the potential φ. Restricting ν to the Banach space Hα (𝒥 (T)), we know that ν is a bounded linear functional. Since ν(φ + ψ) = ν(φ) + ν(ψ), by subtracting the equality hν (T) + ν(φ) = P(φ) from both sides of the inequality hν (T) + ν(φ + ψ) ≤ P(φ + ψ), we get ν(ψ) ≤ P(φ + ψ) − P(φ). So, since, by Proposition 27.7.2, the pressure function P : Hα (𝒥 (T)) → ℝ is differentiable at φ in every direction in Hα (𝒥 (T)), we deduce from Lemma 27.8.1 that there exists at most one equilibrium state for the potential φ. As μφ is an equilibrium state, the proof of Theorem 27.8.2 is complete. Invoking Theorem 12.2.2 according to which hμ (T) > 0 for every equilibrium state μ of φ whenever P(φ) > sup φ, we can collect the main results of this chapter in the following statements. ̂→ℂ ̂ is a rational function with deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ Theorem 27.8.3. If T : ℂ is a Hölder continuous potential such that P(φ) > sup φ, then: (a) There is a unique equilibrium state for the potential φ. It is μφ . (b) There is a unique pair (λ, m), with λ ∈ [0, ∞), m ∈ M(𝒥 (T)) such that ℒ∗φ (m) = λm. (c) (d)

(e) (f)

Namely, λ = eP(φ) and m = mφ . There is a unique ρ ∈ C(𝒥 (T)) such that ℒφ (ρ) = eP(φ) ρ, ρ ≥ 0, and ∫𝒥 (T) ρ dmφ = 1. Namely, ρ = ρφ = dμφ /dmφ > 0. The measure-preserving dynamical system (𝒥 (T), T, μφ ) is metrically exact and its Rokhlin natural extension is a K-system. In particular, this system is totally ergodic, so ergodic. This implies that mφ is metrically exact. Note: Metric exactness was introduced in Definition 8.4.4 for invariant measures, but this notion makes sense for non-invariant measures equally. eP(φ) is a simple eigenvalue of the operator ℒφ : C(𝒥 (T)) → C(𝒥 (T)) and it is the only eigenvalue of ℒφ with modulus equal to eP(φ) . If g ∈ C(𝒥 (T)), then the sequence ((e−P(φ) ℒφ )n (g))∞ n=0 converges uniformly and lim e−nP(φ) ℒnφ (g) = mφ (g)ρφ .

n→∞

In particular, lim e−nP(φ) ℒnφ (1) = ρφ .

n→∞

(g) hμφ (T) > 0.

1156 � 27 Equilibrium states for rational functions We now derive a consequence of Theorem 27.8.3. That consequence is very important on its own but will also be used a few times in the sequel. ̂→ℂ ̂ is a rational function with deg(T) ≥ 2, then Theorem 27.8.4. If T : ℂ htop (T) = log(deg(T)). Proof. The inequality htop (T) ≥ log(deg(T)) directly follows from the Misiurewicz– Przytycki theorem (Theorem 7.5.1) while the inequality htop (T) ≤ log(deg(T)) directly follows from Lemma 27.2.2 and (27.45) applied to the potential φ ≡ 0 since then n

ℒ0 (1)(z) = (deg(T))

n

for every n ≥ 0 and every z ∈ 𝒥 (T). According to Theorem 27.8.4, P(0) = htop (T) = log(deg(T)) ≥ log 2 > 0 = sup 0.

(27.119)

Therefore, Theorem 27.8.3 applies to φ ≡ 0. This special case yields the following statement. ̂→ℂ ̂ is a rational function with deg(T) ≥ 2, then: Theorem 27.8.5. If T : ℂ (a) There is a unique measure of maximal entropy for T. It is μ0 . (b) There is a unique pair (λ, m), with λ ∈ [0, ∞), m ∈ M(𝒥 (T)) such that ℒ∗0 (m) = λm. Namely, λ = deg(T) and m = μ0 . (c) There is a unique ρ ∈ C(𝒥 (T)) such that ℒ0 (ρ) = deg(T)ρ, ρ ≥ 0, and ∫𝒥 (T) ρ dμ0 = 1. Namely, ρ = 1. Explicitly, ℒ0 (1) = deg(T)1.

(d) The measure-preserving dynamical system (𝒥 (T), T, μ0 ) is metrically exact and its Rokhlin natural extension is a K–system. In particular, this system is totally ergodic, so ergodic. (e) deg(T) is a simple eigenvalue of the operator ℒ0 : C(𝒥 (T)) → C(𝒥 (T)) and it is the only eigenvalue of ℒ0 with modulus equal to deg(T). ∞ 1 (f) If g ∈ C(𝒥 (T)), then the sequence (( deg(T) ℒ0 )n (g))n=0 converges uniformly and lim

n→∞

1 (deg(T))

n n ℒ0 (g)

= μ0 (g)1.

(g) hμ0 (T) = htop (T) = P(0) = log(deg(T)) ≥ log 2 > 0. Proof. This theorem is an immediate consequence of Theorem 27.8.3, Theorem 27.8.4, formula (27.119) and the obvious equality ℒ0 (1) = deg(T)1.

27.9 Assorted remarks

� 1157

27.9 Assorted remarks Remark 27.9.1. The hypothesis P(φ) > sup φ can be replaced by a weaker one, namely P(Sk φ) > sup(Sk φ)

(27.120)

for some k ∈ ℕ. Only a few proofs in this chapter would require minor modifications. Remark 27.9.2. The hypothesis P(φ) > sup φ is not so restrictive. For instance, it is entailed by the hypothesis sup φ − inf φ < log(deg(T)). Indeed, in light of Theorem 27.8.4, this inequality means that sup φ − inf φ < htop (T).

(27.121)

As φ ≥ ψ ⇒ P(φ) ≥ P(ψ), as P(φ + c) = P(φ) + c for any c ∈ ℝ and as P(0) = htop (T), we deduce that P(φ) ≥ htop (T) + inf φ > sup φ. Remark 27.9.3. The hypothesis P(φ) > sup φ is necessary in general. Indeed, consider the quadratic polynomial Q1/4 (z) = z2 + 1/4 already encountered in the context of the definition of the Mandelbrot set (see Subsection 26.1.1) and which we will come across again, for instance in Example 31.4.6. Let h = HD(𝒥 (Q1/4 )) and introduce the Hölder potential φh : 𝒥 (Q1/4 ) → ℝ defined by ′ φh (z) = −h log |Q1/4 (z)| = −h log(2|z|).

By the forthcoming Bowen’s formula (Theorem 31.2.1(i)), P(φh ) = 0. Thus, sup φh ≥ φh (1/2) = 0 = P(φh ). So, the inequality P(φh ) > sup φh fails. The potential φh admits more than one equilibrium state. One of them is the Dirac measure δ1/2 concentrated at 1/2 since it is Q1/4 –invariant (1/2 being a fixed point of Q1/4 ) and

1158 � 27 Equilibrium states for rational functions hδ1/2 (Q1/4 ) +

∫

φh dδ1/2 = 0 + 0 = 0 = P(φh ).

𝒥 (Q1/4 )

Another one is the atomless, ergodic, Q1/4 –invariant Borel probability measure μQ1/4 arising from Theorem 31.4.1; see also Corollary 31.4.4 and Example 31.4.6. Indeed, it follows from Theorems 28.4.1–28.4.2 that hμQ (Q1/4 ) + 1/4

∫ 𝒥 (Q1/4 )

φh dμQ1/4 = [HD(μQ1/4 ) − h]

∫

′ log |Q1/4 | dμQ1/4 = 0 = P(φh ).

𝒥 (Q1/4 )

Remark 27.9.4. According to Theorem 13.7.20, condition (27.120) holds for all expanding (hyperbolic) rational functions and all Hölder continuous potentials. For rational functions in general, Inoquio-Renteria and Rivera-Letelier [60] proved that this condition holds if and only if the Lyapunov exponent of each equilibrium state of φ is strictly positive. As observed in [60], it then immediately follows that if the Lyapunov exponent of each T-invariant Borel probability measure on 𝒥 (T) is strictly positive, then inequality (27.120) holds for all Hölder continuous potentials. As noted in [60], there is one large celebrated class of rational functions, called topological Collet–Eckmann maps, for which this condition holds. This class has many characterizations; one of them is the existence of a positive number χ such that the Lyapunov exponent of each T-invariant Borel probability measure on 𝒥 (T) is larger than χ. Another one, also extremely useful, is commonly referred to as the exponential shrinking property. Topological Collet– Eckmann rational functions allow critical points in their Julia set. They comprise all semi-hyperbolic (in particular, all subhyperbolic) rational functions. A systematic treatment of topological Collet–Eckmann maps and their statistical properties can be found in [108]. Remark 27.9.5. The existence and uniqueness of the measure of maximal entropy observed in Theorem 27.8.5(g) was explicitly stated and proved by Misha Lyubich in [79]. Actually, the whole Theorem 27.8.5 is therein. The existence, uniqueness and T-invariance of a Borel probability measure m satisfying ℒ∗0 (m) = deg(T)m was also proved in [52]. Remark 27.9.6. Theorem 27.8.3 in particular asserts that each Hölder continuous potential such that P(φ) > sup φ admits an equilibrium state. In fact, Misha Lyubich proved in [79] that every continuous potential admits an equilibrium state. He actually proved more, namely that the entropy function μ 󳨃→ hμ (T) is upper semicontinuous. Even more ̂→ℂ ̂ with deg(T) ≥ 2 is asympspecifically, he proved that each rational function T : ℂ totically h-expansive. This property of a topological dynamical system introduced and explored by Michal Misiurewicz in [96] was shown therein to entail upper semicontinuity of the entropy function, thus yielding the existence of equilibrium states for all continuous potentials.

27.9 Assorted remarks

� 1159

Remark 27.9.7. In this remark, we discuss the proof of Theorem 27.8.3, its steps and ingredients. The key ingredient is Lemma 27.1.1. This lemma stems from an earlier form in [36], which can in turn be traced back to Ricardo Mañé’s seminal paper [82]. There are however substantial differences and improvements partly emanating from, but not confined to, progress made in [146] and in [133, 134]. The main difference is that in this volume we constructed “good” inverse branches locally (on small balls) rather than globally (on a large topological disk). The other major difference is that we dealt with the iterates of the map T itself rather than some of its iterated power T q . The second key ingredient is Lemma 27.2.6. The proof in [36] was somewhat obscure, tedious and complicated; likewise in [106]. On the other hand, the papers [133, 134, 146] relied on the stronger hypothesis (27.121), which trivializes Lemma 27.2.6. In this volume, that lemma is a derivative product of a series of preparatory lemmas, starting with Lemma 27.2.1. It is conceptually simplified and clarified through those lemmas. In particular, we formulated and proved Lemma 25.4.1, which Lemma 27.2.1 calls upon. Lemma 25.4.1 was reworked from [79]. Perhaps more crucially, we adapted Lemma 27.2.2 and its proof from [106]. We simplified it, clarified it and provided a detailed proof. The second important component of Lemma 27.2.6’s proof is Lemma 27.2.3, which is based on the split of each iterate of the transfer operator Lβ into “bad” parts Bn and “good” parts Gn . This way, we avoided the heaviness of the approaches adopted in [106] and [36]. The idea of splitting the iterates of the transfer operator Lβ comes from [36]. In fact, the resulting recursive formula (27.71) is one of the most important, perhaps the most important, relation exploited in the proof of Lemma 27.2.6 and thereby in the proof of Theorem 27.8.3. In this book, we adapted this splitting to the proof of Lemma 27.2.3, which is central to establishing Lemmas 27.2.4 and 27.2.6. The third key ingredient in the proof of Theorem 27.8.3 is Lemma 27.5.2 (and thereafter Theorem 27.5.3). This lemma was proved in [36] by means of heavy duty methods from the theory of Jacobians. Indeed, it used the highly nontrivial existence of countable generators with finite entropy for T-invariant measures μ on 𝒥 (T) proved in [82]. This is an indispensable fact to establish the formula hμ (T) = ∫ log Jμ (T) dμ,

(27.122)

𝒥 (T)

where Jμ (T) is the Jacobian of T with respect to μ. The complex proof of this formula exploits the full power of the entropy theory for uncountable measurable partitions, which we did not cover in this book. Formula (27.122) was also extensively used in [36] to provide a fairly simple proof of the uniqueness of equilibrium states. In this volume, we provided instead a longer but self-contained proof of that uniqueness based on the differentiability of the pressure function. Its fundamental constituents comprise Section 27.7 as well as Lemma 27.8.1. Those were revamped from an unpublished part of [146].

1160 � 27 Equilibrium states for rational functions Finally, the proof of ergodicity and K-mixing provided in this volume is based on a simpler, much more elegant and transparent method than that used in [36]. It utilizes the ̂φ , derived from much simpler spectral/convergent properties of the transfer operator ℒ means, mainly Lemma 27.4.1 whose proof was adapted from [99]. Remark 27.9.8. There has been substantial progress in the field since Theorem 27.8.3 was proved and the paper [36] was published. Some information can be found in [141]. First, as a relatively easy consequence of results proved in [106], the eigenfunction ρφ of the operator ℒφ was shown in [33] to be Hölder continuous. This is by no means obvious since, in the presence of critical points in the Julia set, the operator ℒφ does not generally preserve any space of Hölder continuous functions with a fixed exponent. Furthermore, it was shown in [33] that if g : 𝒥 (T) → ℝ is Hölder continuous with nonzero variance, then the stationary sequence of random variables ((g ∘ T n , μφ ))∞ n=1 satisfies the central limit theorem. Along this line, substantial progress was realized in [133] and [146] where, based on a new method of fine inducing and on a refined method of Young’s towers from [148, 149], the law of iterated logarithm and the exponential decay of correlations were demonstrated. Also, real-analyticity of the pressure function t 󳨃→ P(tφ) was shown in [133, 134], and the conditions under which a measure μφ can be of maximal Hausdorff dimension were identified. Moreover, some research on transfer operators and stochastic properties of equilibrium measures was done in [56, 57]. We further bring the reader’s attention to the fact that [133, 146] additionally dealt with the higher-dimensional case of holomorphic endomorphisms of complex projective spaces. ̂ were Nearly the same results as in the one-dimensional case of the Riemann sphere ℂ achieved. Finally, we mention the recent preprint [9] in which holomorphic endomorphisms of complex projective spaces are also examined. Using different methods, the preprint’s authors defined a Banach space B, containing Hölder continuous functions, which is preserved by the transfer operator ℒφ and such that the operator ℒφ : B → B has eP(φ) as a simple eigenvalue and the rest of its spectrum is contained in a disk with radius smaller than eP(φ) . This opens the door to developing a full thermodynamic formalism for the dynamical system T : 𝒥 (T) → 𝒥 (T) and Hölder continuous potentials with pressure gap. We would have loved to provide a full account of the main results of [133, 146] with proofs but the needed methods would go too far beyond the scope of this book and would make this volume too long.

27.10 Exercises Exercise 27.10.1. Prove (27.45). Exercise 27.10.2. Prove (27.71). Exercise 27.10.3. Prove Claims 1–2 in the proof of Proposition 27.7.2. Exercise 27.10.4. Prove Remark 27.9.1.

28 Invariant measures: fractal and dynamical properties ̂→ℂ ̂ is a rational function with deg(T) ≥ 2. Let M(T) be Throughout this chapter, T : ℂ the set of all T-invariant Borel probability measures supported on 𝒥 (T). Let Me (T) be its subset of ergodic measures. Let also M + (T) be the subset of all measures with positive entropy (see Chapter 9) and Me+ (T) = Me (T) ∩ M + (T).

28.1 Lyapunov exponents are nonnegative In this section, the derivatives are understood to be with respect to the spherical metric ̂ For every μ ∈ M(T), the Lyapunov exponent χμ (T) of the measure μ is defined to on ℂ. be (cf. Definition 16.5.1) χμ (T) := ∫ log |T ′ | dμ. 𝒥 (T)

This integral is well-defined since the function log |T ′ | is bounded above by log ‖T ′ ‖∞ . However, it generally would not be well-defined if we worked with the Euclidean metric. A priori, χμ (T) might be equal to −∞. But we will establish that this is never the case, i. e., χμ (T) > −∞. In fact, we will show that χμ (T) ≥ 0. This is Przytycki’s theorem, first proved in [107]. Our proof follows Przytycki’s general strategy but is substantially simplified by the use of Rules I–II from Section 25.2; more precisely, we will use one of those rules’ consequences, namely Lemma 25.2.6. As a matter of fact, we will prove a somewhat stronger result. We start with a first intermediate result. ̂→ℂ ̂ is a rational function with deg(T) ≥ 2, then χμ (T) > −∞ for Lemma 28.1.1. If T : ℂ every μ ∈ M(T). Proof. By Birkhoff’s ergodic theorem (Theorem 8.2.11), there is a Borel set X1 ⊆ 𝒥 (T) such that μ(X1 ) = 1, T −1 (X1 ) = X1 , and the limit χ(z) := lim

n→∞

1 log |(T n )′ (z)| n

exists for every point z ∈ X1 (including the possibility that χ(z) = −∞ for some z). Then χ(T i (z)) = χ(z),

∀z ∈ X1 , ∀i ∈ ℕ.

(28.1)

Set M := https://doi.org/10.1515/9783110769876-028

4 κ (β + log ‖T ′ ‖∞ ), κ 2

(28.2)

1162 � 28 Invariant measures: fractal and dynamical properties where κ ∈ (0, 1] and β > 0 come from Lemma 25.2.6. Let ε > 0. By Egorov’s theorem (Theorem A.1.12), there are a Borel set X2,ε ⊆ X1 and N2,ε ∈ ℕ such that μ(X2,ε ) ≥ 1 − ε and 1 log |(T n )′ (z)| < max{χ(z) + 1, −M}, n

∀z ∈ X2,ε , ∀n ≥ N2,ε .

(28.3)

By Birkhoff’s ergodic theorem once again, there is a Borel set X3,ε ⊆ X2,ε such that μ(X3,ε ) = μ(X2,ε ) and 1 n−1 ∑ 1X2,ε (T i (z)) > 0, n→∞ n i=0 lim

∀z ∈ X3,ε .

i Let z ∈ X3,ε . If (nk (z))∞ k=1 is the sequence of all i ∈ ℕ such that T (z) ∈ X2,ε , then

lim

k→∞

nk+1 (z) − nk (z) = 0. nk (z)

(28.4)

Take an integer n ≥ max{2#(Crit(T) ∩ 𝒥 (T)), N2,ε }. Let jn = j and ln = l be the numbers produced in Lemma 25.2.6 for the point z. By virtue of (28.4), there exists N3,ε ≥ max{2#(Crit(T) ∩ 𝒥 (T)), N2,ε } such that for any n ≥ N3,ε there is 0 ≤ in < nκ/2 for which T jn +in (z) ∈ X2,ε .

(28.5)

ln − in > nκ − (nκ/2) = nκ/2 > 0,

(28.6)

Noting that

it follows from the chain rule that 󵄨󵄨 ln ′ jn 󵄨 󵄩 ′ 󵄩in 󵄨 l −i ′ j +i 󵄨 󵄩 ′ 󵄩nκ/2 󵄨 l −i ′ j +i 󵄨 󵄨󵄨(T ) (T (z))󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩T 󵄩󵄩󵄩∞ ⋅ 󵄨󵄨󵄨(T n n ) (T n n (z))󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩T 󵄩󵄩󵄩∞ ⋅ 󵄨󵄨󵄨(T n n ) (T n n (z))󵄨󵄨󵄨. Invoking Lemma 25.2.6, we get that 󵄨󵄨 ln −in ′ jn +in 󵄨 󵄩 󵄩−nκ/2 ) (T (z))󵄨󵄨󵄨 ≥ 󵄩󵄩󵄩T ′ 󵄩󵄩󵄩∞ e−βn . 󵄨󵄨(T It follows from (28.1), (28.3) and (28.5) that 󵄩 󵄩−nκ/2 exp(max{χ(z) + 1, −M}(ln − in )) ≥ 󵄩󵄩󵄩T ′ 󵄩󵄩󵄩∞ e−βn . Taking the logarithm of both sides and then dividing by n, we have max{χ(z) + 1, −M} Assume now that

ln − in κ ≥ −(β + log ‖T ′ ‖∞ ). n 2

28.1 Lyapunov exponents are nonnegative

� 1163

χ(z) + 1 < 0.

(28.7)

Using (28.6), we obtain κ κ max{χ(z) + 1, −M} ≥ −(β + log ‖T ′ ‖∞ ). 2 2 By (28.2), we deduce that max{χ(z) + 1, −M} ≥ − So, χ(z) + 1 ≥ − M2 . Equivalently, χ(z) ≥ −1 − even if (28.7) fails, we conclude that χ(z) ≥ −1 − Let X3 = ⋃ε>0 X3,ε . Then χ(z) ≥ −1 −

M 2

M . 2

M . Observing now that this inequality holds 2

M , 2

∀z ∈ X3,ε .

for all z ∈ X3 . As μ(X3 ) = 1, we finally get

χμ (T) = ∫ χ(z) dμ(z) ≥ −(1 + X3

M ) > −∞. 2

A second intermediate result is also indispensable. ̂→ℂ ̂ be a rational function with deg(T) ≥ 2. If z ∈ 𝒥 (T) is such Lemma 28.1.2. Let T : ℂ that dist(T n (z), Crit(T) ∩ 𝒥 (T)) ≥ e−nδ for all n ∈ ℕ large enough, then χ(z) := lim sup n→∞

1 log |(T n )′ (z)| ≥ −δ. n

Proof. Seeking a contradiction, suppose that χ(z) < −δ. Then there exists γ > 0 with χ(z) + γ < −δ.

(28.8)

For every n ∈ ℕ, let Bn := B(T n (z), |(T n )′ (z)|eγn ). By our hypothesis and (28.8), diam(Bn ) 1 1 1 1 ≤ log 2 + log |(T n )′ (z)| + γ + δ ≤ (χ(z) + γ + δ) < 0 log n n n n 2 dist(T (z), Crit(T) ∩ 𝒥 (T))

1164 � 28 Invariant measures: fractal and dynamical properties for all n ∈ ℕ large enough, say n ≥ N1 . Equivalently, diam(Bn ) n ≤ exp( (χ(z) + γ + δ)). 2 dist(T n (z), Crit(T) ∩ 𝒥 (T)) It follows from Lemma 25.1.10 that for all n ≥ N1 sufficiently large, say n ≥ N2 ≥ N1 , |T ′ (w)| ≤ eγ , |T ′ (T n (z))|

∀w ∈ Bn .

Therefore, T(Bn ) ⊆ B(T n+1 (z), |(T n )′ (z)|eγn eγ |T ′ (T n (z))|) = B(T n+1 (z), |(T n+1 )′ (z)|eγ(n+1) ) = Bn+1

(28.9)

for all n ≥ N2 . But invoking (28.8), we know that lim sup n→∞

1 1 log(2|(T n )′ (z)|eγn ) = lim sup log |(T n )′ (z)| + γ = χ(z) + γ < −δ < 0. n n→∞ n

We obtain from this and (28.9) that lim sup diam(T n (BN2 )) ≤ lim sup diam(Bn+1 ) ≤ lim sup(2|(T n )′ (z)|eγn ) = 0. n→∞

n→∞

n→∞

̂∞ Hence, the family of maps (T n |BN : BN2 → ℂ) n=1 is equicontinuous, which implies that 2

T N2 (z) ∈ ℱ (T). So, z ∈ ℱ (T), which is a contradiction.

With these two intermediate results at our fingertips, we can now establish the main result of this section, i. e., the nonnegativity of Lyapunov exponents. ̂→ℂ ̂ is a rational function with deg(T) ≥ 2 Theorem 28.1.3 (Przytycki’s theorem). If T : ℂ and μ ∈ M(T), then χ(z) := lim

n→∞

1 log |(T n )′ (z)| ≥ 0 n

for μ-a.e. z ∈ 𝒥 (T).

Consequently, χμ (T) ≥ 0. Note: The existence of the limit for μ-almost every Julia point directly follows from Birkhoff’s ergodic theorem. Proof. Let δ > 0. Temporarily fix c ∈ Crit(T) ∩ 𝒥 (T). For every integer n ≥ 0, denote Bn (c) := B(c, e−nδ ).

28.1 Lyapunov exponents are nonnegative

� 1165

Pick N ∈ ℕ so large that e−Nδ ≤ r(T, c), where r(T, c) arises from (25.9)–(25.11). Using Lemma 28.1.1, (25.9) and Taylor’s theorem, we get ∞

−∞ < ∫ log |T ′ | dμ = ∑

n=N

BN (c) ∞

≤ log A(T, c) + ∑

n=N

log |T ′ | dμ

∫ Bn (c)\Bn+1 (c)

log(|z − c|degc (T)−1 ) dμ(z)

∫ Bn (c)\Bn+1 (c)

∞

≤ log A(T, c) + (degc (T) − 1) ∑ (−nδ)μ(Bn (c)\Bn+1 (c)) n=N

∞

= log A(T, c) − (degc (T) − 1)δ[Nμ(BN (c)) + ∑ μ(Bn (c))]. n=N+1

(28.10)

∞ Thus, the series ∑∞ n=N+1 μ(Bn (c)) converges. So, the series ∑n=0 μ(Bn (c)) converges too. Therefore, by the T-invariance of μ, ∞

∑ μ(T −n (Bn (c))) < ∞.

n=0

Summing over all points c ∈ Crit(T) ∩ 𝒥 (T), we obtain that ∞

∑ μ(T −n (B(Crit(T) ∩ 𝒥 (T), e−nδ ))) =

n=0

∑

∞

∑ μ(T −n (Bn (c))) < ∞.

c∈Crit(T)∩𝒥 (T) n=0

Therefore, by the Borel–Cantelli lemma (Lemma A.1.1), T n (z) ∉ B(Crit(T) ∩ 𝒥 (T), e−nδ ) for μ-almost every z ∈ 𝒥 (T), say z ∈ Jδ with μ(Jδ ) = 1, and all n ∈ ℕ sufficiently large. Hence, χ(z) ≥ −δ according to Lemma 28.1.2. So, ∞

μ( ⋂ J1/k ) = 1 k=1

and χ(z) ≥ 0 for all z in this intersection. As a fairly easy consequence of this theorem, we get the following. ̂→ℂ ̂ is a rational function with deg(T) ≥ 2 and μ ∈ M(T), then Corollary 28.1.4. If T : ℂ lim sup |(T n )′ (z)| ≥ 1 n→∞

for μ-a.e. z ∈ 𝒥 (T).

Proof. Seeking a contradiction, suppose that there is a Borel set J1 ⊆ 𝒥 (T) such that μ(J1 ) > 0 and

1166 � 28 Invariant measures: fractal and dynamical properties lim sup |(T n )′ (z)| < 1,

∀z ∈ J1 .

n→∞

Thus, there exist 0 < γ < 1, a Borel set J2 ⊆ J1 and N2 ∈ ℕ such that μ(J2 ) > 0 and |(T n )′ (z)| ≤ γ,

∀z ∈ J2 , ∀n ≥ N2 .

(28.11)

By Birkhoff’s ergodic theorem, there then are a Borel set J3 ⊆ J2 , α ∈ (0, 1], and an integer N3 ≥ N2 such that μ(J3 ) > 0 and #{0 ≤ j < n : T j (z) ∈ J2 } ≥ αn,

∀z ∈ J3 , ∀n ≥ N3 .

(28.12)

Given any z ∈ J3 , represent the set {j ≥ 0 : T j (z) ∈ J2 } as a strictly increasing sequence ∞ (nk )∞ k=0 = (nk (z))k=0 . Then n0 = 0 and using the chain rule and (28.11), we obtain for every k ∈ ℕ that k−1

′ n 󵄨󵄨 nkN3 ′ 󵄨󵄨 󵄨 n 󵄨 −n k 󵄨󵄨(T ) (z)󵄨󵄨 = ∏ 󵄨󵄨󵄨(T (j+1)N3 jN3 ) (T jN3 (z))󵄨󵄨󵄨 ≤ γ . j=0

(28.13)

But if k ≥ N3 , then nk ≥ N3 and it ensues from (28.12) that k ≥ αnk . In particular, kN3 ≥ αnkN3 for every k ∈ ℕ. Thus, lim inf n→∞

α log γ 1 k 1 ′ ′ 󵄨 󵄨 󵄨 󵄨 log󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 ≤ lim inf log󵄨󵄨󵄨(T nkN3 ) (z)󵄨󵄨󵄨 ≤ log γ lim inf ≤ < 0. k→∞ nkN k→∞ nkN n N3 3 3

As μ(J3 ) > 0, this contradicts Przytycki’s theorem (Theorem 28.1.3).

28.2 Ruelle’s inequality The next famous result connects the Lyapunov exponent with the metric entropy. This inequality was originally proved by David Ruelle [113] in the context of smooth diffeomorphisms of multidimensional smooth manifolds. In this section, all the metrics and ̂ Our approach derivatives are understood to be with respect to the spherical metric on ℂ. stems from [110]. ̂→ℂ ̂ is a rational function with deg(T) ≥ 2 Theorem 28.2.1 (Ruelle’s inequality). If T : ℂ and μ ∈ Me (T), then hμ (T) ≤ 2 χμ (T). Proof. Conjugating T by a Möbius transformation if necessary, we may assume without loss of generality that μ({∞}) = 0. Observe (trivially) that there exist a sequence of ∞ positive numbers (ak )∞ k=1 decreasing to 0 and a sequence (𝒫k )k=1 of increasingly finer

28.2 Ruelle’s inequality

� 1167

̂ consisting of atoms satisfyBorel partitions of the complex plane ℂ (or the sphere ℂ) ing diam(𝒫1 ) ≤

1 , 100

diam(𝒫k ) ≤ ak

and A(𝒫k ) ≥

1 2 a , 4 k

∀k ∈ ℕ,

(28.14)

̂ For every z ∈ 𝒥 (T) and every k ∈ ℕ, where A refers to the spherical area on ℂ. let N(T, z, k) := #{P ∈ 𝒫k : T(𝒫k (z)) ∩ P ≠ 0}. Let k(T) ∈ ℕ be so large that for all k ≥ k(T) the Lipschitz constant of T|𝒫k (z) is smaller than |T ′ (z)| + 1. Then the set T(𝒫k (z)) is contained in the ball B(T(z), (|T ′ (z)| + 1)ak ). Thus, T(𝒫k (z)) ∩ P ≠ 0 implies that P ⊆ B(T(z), (|T ′ (z)| + 2)ak ). As also 2

A( B(T(z), (|T ′ (z)| + 2)ak ) ) = 2π[1 − cos((|T ′ (z)| + 2)ak )] ≤ 2π[(|T ′ (z)| + 2)ak ] , we get with the help of (28.14) that N(T, z, k) ≤

2

2π[(|T ′ (z)| + 2)ak ] ak2 /4

2

= 8π(|T ′ (z)| + 2) ,

∀k ≥ k(T).

(28.15)

Let N(T, z) := sup N(T, z, k). k≥k(T)

In view of (28.15), 2

N(T, z) ≤ 8π(|T ′ (z)| + 2) .

(28.16)

Observe that (cf. (9.2)) Hμ𝒫

k (z)

󵄨 (T −1 (𝒫k )󵄨󵄨󵄨𝒫k (z)) ≤ log #{P ∈ 𝒫k : T −1 (P) ∩ 𝒫k (z) ≠ 0} = log N(T, z, k).

Using Theorems 9.4.17 and 9.4.11(b), we then get 󵄨 hμ (T) = lim hμ (T, 𝒫k ) ≤ lim inf Hμ (T −1 (𝒫k )󵄨󵄨󵄨𝒫k ) k→∞

k→∞

= lim inf ∫ Hμ𝒫 k→∞

k (z)

𝒥 (T)

󵄨 (T −1 (𝒫k )󵄨󵄨󵄨𝒫k (z)) dμ(z) ≤ lim inf ∫ log N(T, z, k) dμ(z) k→∞

𝒥 (T)

1168 � 28 Invariant measures: fractal and dynamical properties ≤ ∫ log N(T, z) dμ(z). 𝒥 (T)

Applying this inequality to T n and employing Theorem 9.4.13 and (28.16), we obtain that hμ (T) =

1 1 1 2 hμ (T n ) ≤ log[8π(|(T n )′ (z)| + 2) ] dμ(z). ∫ log N(T n , z) dμ(z) ≤ ∫ n n n 𝒥 (T)

𝒥 (T)

As 𝒥 (T) contains at least one (in fact, a dense set of) repelling periodic point, we know that ‖T ′ ‖∞ > 1, and hence there exists M ∈ ℕ (in fact, M = 2 works) such that 0≤

1 2 log(‖(T n )′ ‖∞ + 2) ≤ 2(log ‖T ′ ‖∞ + 1), n

∀n ≥ M.

Moreover, the ergodic case of Birkhoff’s ergodic theorem (Corollary 8.2.14) affirms that lim

n→∞

1 󵄨 󵄨 log󵄨󵄨󵄨(T n )′ (z)󵄨󵄨󵄨 = ∫ log |T ′ | dμ = χμ (T) n

for μ-a.e z ∈ 𝒥 (T).

𝒥 (T)

In light of Przytycki’s theorem (Theorem 28.1.3), we deduce that lim

n→∞

1 log(|(T n )′ (z)| + 2) = χμ (T) n

for μ-a.e z ∈ 𝒥 (T).

Applying Lebesgue’s dominated convergence theorem, we conclude that hμ (T) ≤ lim ∫ n→∞

𝒥 (T)

1 2 log(|(T n )′ (z)| + 2) dμ(z) = 2χμ (T). n

Ruelle’s inequality is thus proved.

28.3 Pesin’s theory in a conformal setting In this section, we present a special version of a theory whose foundations were laid in Yasha Pesin’s fundamental works [102, 103]. Since then, Pesin’s theory has been extended, generalized, refined and cited in numerous articles, research books and textbooks. Our exposition follows Section 11.2 of the book [110]. Theorem 28.3.1. Let (Z, ℱ , ν) be a finite measure space with an ergodic measure-prê→ℂ ̂ be a rational function with deg(T) ≥ 2. serving automorphism S : Z → Z. Let T : ℂ Suppose that μ is an ergodic T-invariant Borel probability measure on 𝒥 (T) with positive Lyapunov exponent χμ (T). Suppose also that h : Z → 𝒥 (T) is a measurable map such that ν ∘ h−1 = μ

and h ∘ S = T ∘ h,

ν-a.e.

28.3 Pesin’s theory in a conformal setting

� 1169

Then there exists a measurable function r : Z → (0, ∞) such that for ν-a. e. z ∈ Z the following is satisfied: ̂ For every n ∈ ℕ, there exists an holomorphic inverse branch Tx−n : B(x0 , r(z)) → ℂ n n −n of T that sends x0 := h(z) to xn := h(S (z)). In addition, given any χ ∈ (−χμ (T), 0), there is a measurable function K : Z → (0, ∞) such that −1 nχ

K(z) e

󵄨󵄨 −n ′ 󵄨󵄨 󵄨󵄨(Txn ) (w)󵄨󵄨 󵄨󵄨 −n ′ 󵄨󵄨 nχ < 󵄨󵄨(Txn ) (y)󵄨󵄨 < K(z)e and 󵄨 −n ≤ K, 󵄨󵄨(T )′ (y)󵄨󵄨󵄨 󵄨 xn 󵄨

∀y, w ∈ B(x0 , r(z)), ∀n ∈ ℕ, (28.17)

where K is the Koebe constant corresponding to the scale 1/2. n Proof. Suppose that μ(PC(T)) > 0, where PC(T) = ⋃∞ n=1 T (Crit(T)) is the postcritical set of T. As μ is ergodic, this implies that μ must be concentrated on the orbit of a periodic point w ∈ PC(T). This means that w = T q (c) = T q+k (c) for some q, k ∈ ℕ and c ∈ Crit(T), and

μ({T q (c), T q+1 (c), . . . , T q+k−1 (c)}) = 1. Since χμ (T) := ∫ log |T ′ | dμ > 0, we infer that |(T k )′ (T q (c))| > 1. Thus, the theorem is obviously true for the set h−1 ({T q (c), T q+1 (c), . . . , T q+k−1 (c)}) of full ν-measure. Now suppose that μ(PC(T)) = 0. Fix λ ∈ (eχ/4 , 1). Consider z ∈ Z such that: (1) x0 := h(z) ∉ PC(T), (2) xn := h(S −n (z)) ∈ B(T(Crit(T)), λn ) for finitely many n’s only, and 󵄨 󵄨 (3) lim n1 log󵄨󵄨󵄨(T n )′ (xn )󵄨󵄨󵄨 = χμ (T). n→∞

The set of points z satisfying these three conditions is of full ν-measure. Indeed, the first condition is satisfied on a set of full ν-measure because ν(h−1 (PC(T))) = μ(PC(T)) = 0 by our hypothesis. In order to prove that the set of points respecting the second condition has full ν-measure, recall that S is a ν-measure-preserving automorphism and hence ∞

∞

n=1

n=1 ∞

∑ ν(S n (h−1 (B(T(Crit(T)), λn )))) = ∑ ν(h−1 (B(T(Crit(T)), λn ))) = ∑ μ(B(T(Crit(T)), λn )) n=1

< ∞,

1170 � 28 Invariant measures: fractal and dynamical properties the finiteness being due to Lemmas 25.1.14 and 25.6.1. An application of the Borel-Cantelli lemma (Lemma A.1.1) finishes the demonstration of the second condition. Finally, the third condition is fulfilled on a set of full ν-measure according to Birkhoff’s ergodic theorem (Theorem 8.2.11). Let n1 = n1 (z) be the least n ∈ ℕ such that xn := h(S −n (z)) ∉ B(T(Crit(T)), λn ),

∀n ≥ n1 .

Note that the function z 󳨃→ n1 (z) is Borel measurable. Because of our choices of χ and λ as well as condition (3), there is a least n2 = n2 (z) ≥ n1 such that 󵄨󵄨 n ′ 󵄨−1/4 < λn , 󵄨󵄨(T ) (xn )󵄨󵄨󵄨

∀n ≥ n2 .

(28.18)

Again, observe that the function z 󳨃→ n2 (z) is Borel measurable. Set ∞

󵄨 󵄨−1/4 Δ := ∑ 󵄨󵄨󵄨(T n )′ (xn )󵄨󵄨󵄨 ,

(28.19)

n=1

bn :=

1 󵄨󵄨 n+1 ′ 󵄨−1/4 󵄨(T ) (xn+1 )󵄨󵄨󵄨 , 2Δ 󵄨

(28.20)

∀n ∈ ℕ,

and ∞

Π := ∏(1 − bn )−1 .

(28.21)

n=1

This infinite product converges since the series ∑∞ n=1 bn does (see Exercise 24.8.6). Let r(z) := min{(16ΠKΔ3 ) , −1

n

∞ 2 1 1 󵄨󵄨 n2 ′ 󵄨 n dist(h(z), ⋃ T j (Crit(T))), 󵄨󵄨(T ) (xn2 )󵄨󵄨󵄨λ 2 ∏ (1 − bk )}. 4Π 2K j=1 k=n 2

̂ are well-defined for all 1 ≤ n ≤ n2 , Then all the inverse branches Tx−n : B(x0 , Πr(z)) → ℂ n and by Koebe’s distortion theorem ∞

2 diam(Tx−n (B(x0 , r(z) ∏ (1 − bk )−1 ))) ≤ λn2 . n 2

k=n2

We show by induction that for every n ≥ n2 there exists an analytic inverse branch −1 ̂ Tx−n : B(x0 , r(z)Π∞ k=n (1 − bk ) ) → ℂ sending x0 to xn and such that n ∞

diam(Tx−n (B(x0 , r(z) ∏(1 − bk )−1 ))) ≤ λn . n k=n

(28.22)

Indeed, for n = n2 this immediately follows from our requirements imposed on r(z). So, suppose that the claim is true for some n ≥ n2 . Since

28.3 Pesin’s theory in a conformal setting

� 1171

xn = Tx−n (x0 ) ∉ B(T(Crit(T)), λn ), n ̂ mapping xn to xn+1 . By (28.22), which there exists an inverse branch Tx−1n+1 : B(xn , λn ) → ℂ is part of the induction hypothesis, the composition ∞

̂ Tx−1n+1 ∘ Tx−n : B(x0 , r(z) ∏(1 − bk )−1 ) → ℂ n k=n

is then well-defined and forms the inverse branch of T n+1 that sends x0 to xn+1 . By Koebe’s distortion theorem (Theorem 23.1.6) and using (28.18)–(28.21), we now estimate that ∞

diam(Tx−(n+1) (B(x0 , r(z) ∏ (1 − bk )−1 ))) n+1 k=n+1

∞

−1 󵄨󵄨

󵄨−1 ≤ 2r(z) ∏ (1 − bk ) 󵄨󵄨(T n+1 )′ (xn+1 )󵄨󵄨󵄨 Kb−3 n k=n+1

󵄨 󵄨−1 󵄨 󵄨3/4 ≤ 16r(z)ΠKΔ3 󵄨󵄨󵄨(T n+1 )′ (xn+1 )󵄨󵄨󵄨 󵄨󵄨󵄨(T n+1 )′ (xn+1 )󵄨󵄨󵄨 󵄨 󵄨−1/4 = 16r(z)ΠKΔ3 󵄨󵄨󵄨(T n+1 )′ (xn+1 )󵄨󵄨󵄨 < 16r(z)ΠKΔ3 λn+1 ≤ λn+1 , where the last inequality ensues from our choice of r(z). Replacing r(z) by r(z)/2, the assertion about the existence of K(z) then follows from a combined application of condition (3) and Koebe’s distortion theorem (Theorem 23.1.8). As an immediate consequence of Theorem 28.3.1, we deduce the following. Corollary 28.3.2. Assume the same notation and hypotheses as Theorem 28.3.1. Fix ε > 0. Then there exist a measurable set Z(ε) ⊆ Z and numbers r(ε) ∈ (0, 1) and K(ε) ≥ 1 such that: (1) μ(Z(ε)) > 1 − ε, (2) r(z) ≥ r(ε) for all z ∈ Z(ε), (3) ′ 󵄨 󵄨 K(ε)−1 exp(−n[χμ (T) + ε]) ≤ 󵄨󵄨󵄨(Tx−n ) (y)󵄨󵄨󵄨 ≤ K(ε) exp(−n[χμ (T) − ε]) n

and 󵄨󵄨 −n ′ 󵄨󵄨 󵄨󵄨(Tx ) (y)󵄨󵄨 K −1 ≤ 󵄨 n ′ 󵄨 ≤ K 󵄨󵄨(T −n ) (w)󵄨󵄨 󵄨 xn 󵄨 for all n ∈ ℕ, all z ∈ Z(ε) and all y, w ∈ B(x0 , r(ε)), where K is the Koebe constant corresponding to the scale 1/2.

1172 � 28 Invariant measures: fractal and dynamical properties In future applications, particularly in the next section, the system (Z, S, ν) will usu? ̃ μ ̃ ) of the ally be given by (see Theorem 8.4.3) the Rokhlin’s natural extension (𝒥 (T), T, ? holomorphic system (𝒥 (T), T, μ). In that context and with h = π0 : 𝒥 (T) → 𝒥 (T), Corollary 28.3.2 yields the following. ̂→ℂ ̂ be a rational function with deg(T) ≥ 2. If μ is an ergodic Theorem 28.3.3. Let T : ℂ T-invariant Borel probability measure on 𝒥 (T) with positive Lyapunov exponent χμ (T), ? ̃ -a. e. ̃z ∈ 𝒥 then for μ (T) there exists r(̃z) > 0 such that the function ̃z 󳨃→ r(̃z) is measurable

and the following is satisfied: −n ̂ ̃ Writing ̃z = (zk )∞ k=0 , for every n ≥ 0 there exists Tzn : B(z0 , r(z)) → ℂ, an holomorn phic inverse branch of T mapping z0 to zn . In addition, given any χ ∈ (−χμ (T), 0), there exists a constant K(̃z) ∈ (0, ∞), independent of n, measurable as a function of ̃z, and such that −1 nχ

K(̃z) e

′ 󵄨 󵄨 < 󵄨󵄨󵄨(Tz−n ) (y)󵄨󵄨󵄨 < K(̃z)enχ n

and

󵄨󵄨 −n ′ 󵄨󵄨 󵄨󵄨(Tzn ) (y)󵄨󵄨 󵄨󵄨 −n ′ 󵄨󵄨 ≤ K, 󵄨󵄨(Tzn ) (w)󵄨󵄨

∀y, w ∈ B(z0 , r(̃z)), (28.23)

where K is the Koebe constant for the map T corresponding to the scale 1/2. ? ? Consequently, for every ε > 0 there exist a closed set 𝒥 (T)(ε) ⊆ 𝒥 (T) and numbers r(ε) ∈ (0, 1) and K(ε) ≥ K such that: ? ̃ (𝒥 (1) μ (T)(ε)) > 1 − ε, ? (2) r(̃z) ≥ r(ε) for all ̃z ∈ 𝒥 (T)(ε), and (3)

′ 󵄨 󵄨 K(ε)−1 exp(−n[χμ (T) + ε]) ≤ 󵄨󵄨󵄨(Tz−n ) (y)󵄨󵄨󵄨 ≤ K(ε) exp(−n[χμ (T) − ε]) n

and K

−1

󵄨󵄨 −n ′ 󵄨󵄨 󵄨󵄨(Tz ) (y)󵄨󵄨 ≤ 󵄨 n ′ 󵄨 ≤K ) (w)󵄨󵄨󵄨 󵄨󵄨󵄨(Tz−n n

? for all n ∈ ℕ, all ̃z ∈ 𝒥 (T)(ε) and all y, w ∈ B(z0 , r(ε)).

28.4 Volume lemmas; Hausdorff and packing dimensions of invariant measures This section is devoted to providing a closed formula for the Hausdorff and packing dimensions of Borel probability measures that are ergodic and invariant under a rational function. For historical reasons, partly justified, this formula is frequently referred to as a volume lemma. Its first forms can be traced back to the works of Eggleston [46] and

28.4 Volume lemmas; Hausdorff and packing dimensions of invariant measures

� 1173

Billingsley [11]. From a dynamical viewpoint, a breakthrough was achieved by Lai–Sang Young [148]. Since then, a multitude of papers appeared. We mention only some of the early ones: [84, 86, 105]. Also, the early papers [19, 111] shed some light on the nature of dimensions of measures. The main result in this section is Theorem 28.4.1. It has exactly the same form as Theorem 16.5.2 for conformal expanding repellers (CERs) and Theorem 19.8.30 for conformal graph directed Markov systems (CGDMSs). However, there is a substantial difference: while both CERs and CGDMSs are uniformly expanding/contracting, rational functions are generally not, and while the proofs of Theorems 16.5.2 and 19.8.30 relied respectively on Brin–Katok’s local entropy formula for expansive systems (Theorem 9.6.4) and on the ergodic case of Shannon–McMillan–Breiman’s theorem (Corollary 9.5.5), for rational functions we call on Rokhlin’s natural extension through Theorem 28.3.3, in addition of Shannon–McMillan–Breiman’s theorem. In all cases, Birkhoff’s ergodic theorem is used. ̂→ℂ ̂ be a rational function with deg(T) ≥ 2. Theorem 28.4.1 (Volume lemma). Let T : ℂ If μ is an ergodic T-invariant Borel probability measure on 𝒥 (T) with χμ (T) > 0, then lim

r→0

log μ(B(x, r)) hμ (T) = log r χμ (T)

for μ-a. e. x ∈ 𝒥 (T).

In particular, the measure μ is dimensionally exact and, by Corollary 15.6.11, HD(μ) = PD(μ) =

hμ (T) χμ (T)

.

Proof. In view of Corollary 15.6.11, the second formula follows from the first one and we therefore only need to prove that one. Let us first show that lim inf r→0

log μ(B(x, r)) hμ (T) ≥ log r χμ (T)

for μ-a. e. x ∈ 𝒥 (T).

(28.24)

Let ε > 0. By Corollary 25.5.3, there exists a finite partition 𝒫 of 𝒥 (T) into Borel sets of positive μ-measure such that hμ (T, 𝒫 ) ≥ hμ (T) − ε and for every x in a set X0 ⊆ 𝒥 (T) of full μ-measure there exists n(x) ≥ 0 such that B(T n (x), e−nε ) ⊆ 𝒫 (T n (x)),

∀n ≥ n(x).

(28.25)

? ? ̃ μ ̃ ). Let 𝒥 Let us work from now on with the Rokhlin’s natural extension (𝒥 (T), T, (T)(ε) and r(ε) be given by Theorem 28.3.3. In view of the ergodic case of Birkhoff’s ergodic the? ̃ ⊆𝒥 ̃ ̃ (F(ε)) orem (Corollary 8.2.14), there exists a measurable set F(ε) (T)(ε) such that μ = ? ̃ (𝒥 μ (T)(ε)) and 1 n−1 ? ̃ (𝒥 ∘ T̃ n (x̃) = μ (T)(ε)), ∑ 1𝒥 ? (T)(ε) n→∞ n j=0 lim

̃ ∀x̃ ∈ F(ε).

1174 � 28 Invariant measures: fractal and dynamical properties ̃ Let F(ε) = π0 (F(ε)). Then ? ̃ ̃ (π0−1 (F(ε))) ≥ μ ̃ (F(ε)) ̃ (𝒥 μ(F(ε)) = μ =μ (T)(ε)) > 1 − ε. Consider an arbitrary point x ∈ F(ε) ∩ X0 ̃ such that x = π0 (x̃). By the above, there exists a strictly increasing and take x̃ ∈ F(ε) sequence (nk = nk (x̃))∞ k=1 such that ? T̃ nk (x̃) ∈ 𝒥 (T)(ε)

and

nk+1 − nk ≤ ε, nk

(28.26)

∀k ∈ ℕ.

Moreover, we can assume that n1 ≥ n(x). Pick an integer n ≥ n1 and the ball B(x, Cr(ε) exp(−n[χμ (T) + ε(log ‖T ′ ‖∞ + 2)])), where 0 < C < K(ε)−1 is a constant (possibly depending on x), so small that T q (B(x, Cr(ε) exp(−n[χμ (T) + ε(log ‖T ′ ‖∞ + 2)]))) ⊆ 𝒫 (T q (x)),

∀q ≤ n1 ,

(28.27)

with K(ε) ≥ 1 as the constant appearing in Theorem 28.3.3. Take any n1 ≤ q ≤ n and ? associate k such that nk ≤ q < nk+1 . As T̃ nk (x̃) ∈ 𝒥 (T)(ε) and as π0 (T̃ nk (x̃)) = T nk (x), Theorem 28.3.3 produces an holomorphic inverse branch ̂ Tx−nk : B(T nk (x), r(ε)) → ℂ of T nk such that Tx k (T nk (x)) = x. This theorem also yields −n

Tx−nk (B(T nk (x), r(ε))) ⊇ B(x, K(ε)−1 r(ε) exp(−nk [χμ (T) + ε])). Since n ≥ nk , we have B(x, Cr(ε) exp(−n[χμ (T) + ε(log ‖T ′ ‖∞ + 2)])) ⊆ B(x, K(ε)−1 r(ε) exp(−nk [χμ (T) + ε])) and it follows from Theorem 28.3.3 that T nk (B(x, Cr(ε) exp(−n[χμ (T) + ε(log ‖T ′ ‖∞ + 2)]))) ⊆ B(T nk (x), CK(ε)r(ε)e−χμ (T)(n+nk ) exp(ε[nk − n(log ‖T ′ ‖∞ + 2)])). Applying T q−nk and using the fact that nk ≤ q ≤ n and q−nk ≤ εnk (by (28.26)), we deduce that

28.4 Volume lemmas; Hausdorff and packing dimensions of invariant measures

� 1175

T q (B(x, Cr(ε) exp(−n[χμ (T) + ε(log ‖T ′ ‖∞ + 2)]))) k ⊆ B(T q (x), CK(ε)r(ε)e−χμ (T)(n+nk ) exp(ε[nk − n(log ‖T ′ ‖∞ + 2)]) ⋅ ‖T ′ ‖q−n ∞ )

⊆ B(T q (x), CK(ε)r(ε) exp(ε[nk − n log ‖T ′ ‖∞ − 2n + nk log ‖T ′ ‖∞ ]))

⊆ B(T q (x), CK(ε)r(ε)e−εn )

⊆ B(T q (x), e−qε ).

Combining this with (28.25) yields T q (B(x, Cr(ε) exp(−n[χμ (T) + ε(log ‖T ′ ‖∞ + 2)]))) ⊆ 𝒫 (T q (x)),

∀n1 ≤ q ≤ n.

Using this and (28.27), we infer that n

B(x, Cr(ε) exp(−n[χμ (T) + ε(log ‖T ′ ‖∞ + 2)])) ⊆ ⋁ T −j (𝒫 )(x) =: 𝒫 n+1 (x). j=0

Applying Corollary 9.5.5 (the ergodic case of the Shannon–McMillan–Breiman theorem), we get 1 lim inf − log μ(B(x, Cr(ε) exp(−n[χμ (T) + ε(log ‖T ′ ‖∞ + 2)]))) ≥ hμ (T, 𝒫 ) ≥ hμ (T) − ε. n→∞ n Denoting Cr(ε) exp(−n[χμ (T) + ε(log ‖T ′ ‖∞ + 2)]) by rn , we thereby have lim inf n→∞

hμ (T) − ε log μ(B(x, rn )) ≥ . log rn χμ (T) + ε(log ‖T ′ ‖∞ + 2)

Since (rn )∞ n=1 is a geometric sequence and since ε > 0 can be taken arbitrarily small, we conclude that lim inf r→0

log μ(B(x, r)) hμ (T) ≥ log r χμ (T)

for μ-a. e. x ∈ 𝒥 (T).

This completes the proof of (28.24). Let us now prove that lim sup r→0

log μ(B(x, r)) hμ (T) ≤ log r χμ (T)

for μ-a. e. x ∈ 𝒥 (T).

(28.28)

? ̃ μ ̃ ) and apply Pesin’s theory. We again work with Rokhlin’s natural extension (𝒥 (T), T, ? ? ̃ In particular, the sets 𝒥 (T)(ε), F(ε) ⊆ 𝒥 (T)(ε) and the radius r(ε) produced via Theorem 28.3.3 have the same meaning as in the proof of (28.24).

1176 � 28 Invariant measures: fractal and dynamical properties To begin, notice that because of Lemma 25.1.8 there exist two numbers R > 0 and 0 < Q < min{1, r(ε)/2} such that the following two conditions are satisfied: (1) If z ∉ B(Crit(T), R), then T|B(z,Q) is injective. (2) If z ∈ B(Crit(T), R), then T|B(z,Qdist(z,Crit(T))) is injective. Observe also that if z is sufficiently close to a critical point c, then T ′ (z) is of order degc (T) − 1. In particular, the quotient of T ′ (z) and (z − c)degc (T)−1 remains bounded away from 0 and ∞ and, therefore, there exists B > 1 such that |T ′ (z)| ≤ Bdist(z, Crit(T)). In view of Lemma 25.1.14, the logarithm of the function ρ(z) := Q min{1, dist(z, Crit(T))} is then integrable and, consequently, Lemma 25.5.5 applies. Let 𝒫 be the Mañé’s partition produced by that lemma. Then B(x, ρ(x)) ⊇ 𝒫 (x) for μ-a. e. x ∈ 𝒥 (T), say for all x in a set Xρ ⊆ 𝒥 (T) of full μ–measure. Therefore, n−1

n−1

j=0

j=0

Bn (x, ρ(x)) := ⋂ T −j (B(T j (x), ρ(T j (x)))) ⊇ ⋁ T −j (𝒫 )(x) =: 𝒫 n (x)

(28.29)

for every n ∈ ℕ and every x ∈ Xρ . By our choice of Q and the definition of ρ, the function T is injective on all balls B(T j (x), ρ(T j (x))), j ≥ 0, and hence T i is injective on the set Bn (x, ρ(x)) for every 0 ≤ i ≤ n − 1. Let x ∈ F(ε) ∩ Xρ . ̃ such that x = π0 (x̃), and pick a sequence (nk = nk (x̃))∞ satisfying (28.26). Take x̃ ∈ F(ε) k=1 −q ′ Let k be the greatest subscript such that q := nk ′ ≤ n − 1. Denote by Tx the unique q holomorphic inverse branch of T produced by Theorem 28.3.3 which sends T q (x) to x. By (28.29), Bn (x, ρ(x)) ⊆ T −q (B(T q (x), ρ(T q (x)))). As T q is injective on Bn (x, ρ(x)), we even have that Bn (x, ρ(x)) ⊆ Tx−q (B(T q (x), ρ(T q (x)))). By Theorem 28.3.3, diam(Tx−q (B(T q (x), ρ(T q (x))))) ≤ K(ε) exp(−q[χμ (T) − ε]). By (28.26), n ≤ nk ′ +1 ≤ q(1 + ε). So, we finally deduce from the above that

28.4 Volume lemmas; Hausdorff and packing dimensions of invariant measures

� 1177

Bn (x, ρ(x)) ⊆ B(x, K(ε) exp(−n(1 + ε)−1 [χμ (T) − ε])). In light of (28.29), B(x, K(ε) exp(−n(1 + ε)−1 [χμ (T) − ε])) ⊇ 𝒫 n (x). Therefore, setting sn = K(ε) exp(−n(1 + ε)−1 [χμ (T) − ε]), it follows from Corollary 9.5.5 (the ergodic case of the Shannon–McMillan–Breiman theorem) that for μ-a. e. x ∈ 𝒥 (T), 1 lim sup − log μ(B(x, sn )) ≤ hμ (T, 𝒫 ) ≤ hμ (T). n n→∞ So, lim sup n→∞

hμ (T) log μ(B(x, sn )) ≤ (1 + ε). log sn χμ (T) − ε

As (sn )∞ n=1 is a geometric sequence and as ε > 0 can be taken arbitrarily small, we conclude that lim sup r→0

log μ(B(x, r)) hμ (T) ≤ log r χμ (T)

for μ-a. e. x ∈ 𝒥 (T).

This completes the proof of (28.28). Combining together (28.24) and (28.28) finishes the proof of Theorem 28.4.1. In order to apply Theorem 28.4.1, we need to know when the Lyapunov exponent of a measure is positive. Recall that Ω(T) denotes the set of all rationally indifferent periodic points of T, which is finite according to Theorem 24.3.2. ̂→ℂ ̂ be a rational function with deg(T) ≥ 2. If μ is an ergodic Theorem 28.4.2. Let T : ℂ T-invariant Borel probability measure on 𝒥 (T), then χμ (T) > 0

󳨐⇒

μ(Ω(T)) = 0.

⇐⇒

μ(Ω(T)) = 0.

Moreover, if T is parabolic then χμ (T) > 0

Proof. Assume first that μ(Ω(T)) > 0. As μ is ergodic, there is ω ∈ Ω(T) such that μ(𝒪+ (ω)) = 1. Let p be the prime period of ω. The T-invariance of μ imposes that μ({T k (ω)}) = 1/p for every 0 ≤ k < p. It immediately follows that 1 ′ 󵄨 󵄨 󵄨 󵄨 χμ (T) := ∫ log󵄨󵄨󵄨T ′ 󵄨󵄨󵄨 dμ = log󵄨󵄨󵄨(T p ) (ω)󵄨󵄨󵄨 = 0. p 𝒥 (T)

1178 � 28 Invariant measures: fractal and dynamical properties Assume now that T is parabolic and that μ(Ω(T)) = 0. Pick any ξ ∈ supp(μ)\Ω(T) ≠ 0. As T is parabolic, Proposition 26.2.2 asserts that there is r > 0 with B(ξ, 4r) ∩ PC(T) = 0. Let κ > 0 be such that κK ≤ r, where K is the usual constant of bounded distortion. Per Lemma 24.1.13, there exists ℓ ∈ ℕ such that diam(V ) < κ for every connected component V of T −q (B(ξ, 2r)) and for every q ≥ ℓ. As μ(B(ξ, r)) > 0, the ergodic case of Birkhoff’s ergodic theorem (Corollary 8.2.14) asserts that there exists z ∈ B(ξ, r) such that lim

n→∞

1 #{0 ≤ k < n : T k (z) ∈ B(ξ, r)} = μ(B(ξ, r)) n

and

lim

n→∞

1 ′ 󵄨 󵄨 log󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 = χμ (T). n

n Let (nj )∞ j=1 be the sequence of all n ≥ ℓ such that T (z) ∈ B(ξ, r) and such that nj+1 −nj ≥ ℓ. ̂ of T −q (B(ξ, r)) is contained Take any integer q ≥ ℓ. As any connected component V

in a unique connected component V of T −q (B(ξ, 2r)), we have by means of Koebe’s distortion theorem that ̂)) ≤ diam(V ̂)󵄩󵄩󵄩(T q )′ 󵄩󵄩󵄩 ̂ ≤ diam(V )K 󵄨󵄨󵄨(T q )′ (y)󵄨󵄨󵄨 ≤ κK 󵄨󵄨󵄨(T q )′ (y)󵄨󵄨󵄨 2r = diam(T q (V 󵄨 󵄨 󵄨 󵄨 󵄩 󵄩∞,V ̂. Consequently, inf ̂ |(T q )′ (y)| ≥ 2r(κK)−1 ≥ 2. for any y ∈ V y∈V

We claim that |(T nj )′ (z)| ≥ 2j for all j large enough. If this is the case, then the result will follow because Birkhoff’s ergodic theorem will then yield χμ (T) = lim

j→∞

j 1 ′ 󵄨 󵄨 log󵄨󵄨󵄨(T nj ) (z)󵄨󵄨󵄨 ≥ log 2 lim = log 2 ⋅ μ(B(ξ, r)) > 0. j→∞ nj nj

Clearly, |(T n1 )′ (z)| ≥ 2. For the inductive step, assume that |(T nj )′ (z)| ≥ 2j . Then ′ 󵄨󵄨 nj+1 ′ 󵄨󵄨 󵄨󵄨 nj+1 −nj ′ nj 󵄨 󵄨 󵄨 ) (T (z))󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨(T nj ) (z)󵄨󵄨󵄨 ≥ 2 ⋅ 2j = 2j+1 . 󵄨󵄨(T ) (z)󵄨󵄨 = 󵄨󵄨(T

28.5 HD(𝒥 (T)) > 0 and radial Julia sets 𝒥r (T), 𝒥er (T), 𝒥uer (T) It immediately follows from Theorem 27.8.5(d,g), Theorem 28.2.1 (Ruelle’s inequality) ̂ → ℂ ̂ is a rational function with and Theorem 28.4.1 (volume lemma) that if T : ℂ deg(T) ≥ 2, then HD(𝒥 (T)) ≥ HD(μ0 ) =

hμ0 (T) χμ0 (T)

=

htop (T) χμ0 (T)

=

log(deg(T)) > 0. χμ0 (T)

(28.30)

But we can prove that HD(𝒥 (T)) > 0 in a more elementary way (in a sense), while simultaneously obtaining a result which will be needed in the next chapter. We first need some concepts which are of importance on their own and which will also play an important role in the next chapter.

28.5 HD(𝒥 (T )) > 0 and radial Julia sets 𝒥r (T ), 𝒥er (T ), 𝒥uer (T )

�

1179

̂→ℂ ̂ be a rational function. A point z ∈ 𝒥 (T) is said to be Definition 28.5.1. Let T : ℂ radial (or conical) if there exists η > 0 such that for infinitely many n ∈ ℕ the restriction of T n to Comp(z, T n , η) is one-to-one, where the reader will recall that Comp(z, T n , η) stands for the connected component of T −n (B(T n (z), η)) containing z. Denote the set of all such n’s by Nz (T). A radial point z ∈ 𝒥 (T) is called expanding if there exists λ > 1 such that 󵄨󵄨 n ′ 󵄨󵄨 n 󵄨󵄨(T ) (z)󵄨󵄨 ≥ λ

(28.31)

for infinitely many n’s in Nz (T). An expanding radial point z is uniformly expanding if the set ℕ \ Nz (T) is finite and inequality (28.31) holds for all n ∈ Nz (T) large enough. We respectively denote by 𝒥r (T), 𝒥er (T), and 𝒥uer (T) the sets of radial, expanding radial, and uniformly expanding radial, points (in 𝒥 (T)). Given η > 0, define 𝒥r (T)(η) to be the set of all points in 𝒥r (T) that are radial by satisfying that definition with the number 2η. Thereafter, define 𝒥er (T)(η) := 𝒥r (T)(η) ∩ 𝒥er (T)

and

𝒥uer (T)(η) := 𝒥r (T)(η) ∩ 𝒥uer (T).

The radial Julia set was informally introduced in [80] and independently, but formally, in [139] by analogy to radial/conical sets in the theory of Kleinian groups (see also [32, 92]). We record the following straightforward fact. ̂→ℂ ̂ be a rational function with deg(T) ≥ 2. If z ∈ ℂ ̂ and Observation 28.5.2. Let T : ℂ there exist η > 0, λ > 1, and a strictly increasing sequence (nj )∞ of positive integers with j=1 bounded gaps (i. e., sup{nj+1 − nj : j ∈ ℕ} < ∞) such that for each j ∈ ℕ the restriction of T nj to Comp(z, T nj , η) is 1-to-1 and 󵄨󵄨 nj ′ 󵄨󵄨 j 󵄨󵄨(T ) (z)󵄨󵄨 ≥ λ ,

∀j ∈ ℕ,

then z ∈ 𝒥uer (T). We now prove that the uniformly expanding radial Julia set has positive Hausdorff dimension. As a by-product, we will know that the expanding radial Julia set, the radial Julia set and the Julia set itself all share that property. The strategy consists in building an IFS whose limit set stands within the uniformly expanding radial Julia set. Because the generators of an IFS are contractions, it is natural to look at the inverse branches of a repelling periodic point. To make sure that inverse branches of all orders exist, it is necessary to take a point that lies outside of the postcritical set. That point will be denoted by ξ. To apply the theory of CGDMSs (which covers IFSs) derived in Chapter 19, the first-level sets of the IFS must not overlap beyond their boundaries. This means that a second repelling periodic point x near ξ is needed.

1180 � 28 Invariant measures: fractal and dynamical properties ̂→ℂ ̂ is a rational function with deg(T) ≥ 2, then Theorem 28.5.3. If T : ℂ HD(𝒥uer (T)) > 0. In consequence, HD(𝒥 (T)) > 0. Proof. Fix a repelling periodic point ξ ∈ 𝒥 (T) \ PC(T). Let p ∈ ℕ be a period of ξ. Fix β ∈ (1, |(T p )′ (ξ)|). According to Remark 24.4.16, there is R > 0 such that there exists a −p ̂ of T −p fixing ξ, and unique holomorphic branch Tξ : B(ξ, R) → ℂ 󵄨󵄨 −p ′ 󵄨󵄨 −1 󵄨󵄨(Tξ ) (z)󵄨󵄨 ≤ β < 1,

∀z ∈ B(ξ, R).

(28.32)

By the mean value inequality, Tξ (B(ξ, r)) ⊆ B(ξ, β−1 r), −p

∀r ∈ (0, R].

(28.33)

−n As the set ⋃∞ n=0 T (ξ) is dense in 𝒥 (T) per Corollary 24.3.7, there exist k ∈ ℕ and a point

x ∈ (T −k (ξ) ∩ B(ξ, R)) \ {ξ}.

(28.34)

Since ξ ∉ PC(T), we know that |(T k )′ (x)| ≠ 0. So, there exists r ∈ (0, R/2] for which there ̂ of T −k sending ξ to x and is a unique holomorphic branch Tx−k : B(ξ, 2r) → ℂ Tx−k (B(ξ, 2r)) ⊆ B(ξ, R) \ B(ξ, η)

(28.35)

for some η > 0. By (28.32)–(28.33), there exists ℓ ∈ ℕ so large that 󵄨󵄨 −ℓ ′ 󵄨󵄨 󵄨 −2 󵄨 k ′ 󵄨󵄨(Tξ ) (x)󵄨󵄨 ≤ K 󵄨󵄨󵄨(T ) (x)󵄨󵄨󵄨, Tξ−ℓ

∘

Tx−k (B(ξ, 2r))

(28.36)

⊆ B(ξ, r/2) \ B(ξ, η ) ′

(28.37)

and Tξ−ℓ ∘ Tx−k (B(ξ, r/2)) ⊆ B(ξ, r/4)

(28.38)

for some η′ ∈ (0, r/2), where K is the Koebe constant corresponding to the scale 1/2. It then follows from Theorem 23.1.9 (Koebe’s distortion theorem, spherical version) and (28.36) that 󵄨󵄨 −ℓ 󵄨 󵄨 −ℓ 󵄨 󵄨 −ℓ ′ 󵄨 󵄨 −k ′ 󵄨 −k ′ −k ′ −1 󵄨󵄨(Tξ ∘ Tx ) (z)󵄨󵄨󵄨 ≤ K 󵄨󵄨󵄨(Tξ ∘ Tx ) (ξ)󵄨󵄨󵄨 = K 󵄨󵄨󵄨(Tξ ) (x)󵄨󵄨󵄨 󵄨󵄨󵄨(Tx ) (ξ)󵄨󵄨󵄨 ≤ K < 1

(28.39)

for every z ∈ B(ξ, r). By virtue of (28.33), there exists q ∈ ℕ such that Tξ

−qp

(B(ξ, r)) ⊆ B(ξ, η′ /2) ⊆ B(ξ, r/4).

(28.40)

28.5 HD(𝒥 (T )) > 0 and radial Julia sets 𝒥r (T ), 𝒥er (T ), 𝒥uer (T )

� 1181

Set φξ := Tξ

−qp

and

φx := Tξ−ℓ ∘ Tx−k .

Let V = {v} be a singleton vertex set and the pair E = {ξ, x} serve as an alphabet. Let A : E × E → {0, 1} be the constant function 1, so that EA∞ = E ∞ = {ξ, x}ℕ . Finally, let X = Xv = B(ξ, r/2) and W = Wv = B(ξ, r). Together, relations (28.32)–(28.33) and (28.38)–(28.40) show that the collection Φ := {V , E, A; X, W ; φξ , φx } forms a conformal IFS (CIFS). Denote its limit set by J. By Bowen’s formula (Theorem 19.6.4), HD(J) > 0.

(28.41)

By (28.37) and (28.40), the system Φ satisfies the strong separation condition (SSC; see Definition 19.7.6). According to Theorem 19.7.8, the limit set J is a topological Cantor set. By (28.38) and (28.40), it is further clear that J ⊆ B(ξ, r/4). For every z ∈ J = π(E ∞ ), there is a unique ω ∈ {ξ, x}ℕ such that z = π(ω) and π(σ j (ω)) = T j∗ (z),

∀j ≥ 0,

where j∗ := (qp)#{1 ≤ i ≤ j : ωi = ξ} + (ℓ + k)#{1 ≤ i ≤ j : ωi = x}. ̂ is a holomorphic branch Moreover, B(T j∗ (z), r/2) ⊆ B(ξ, r) and φω|j : B(T j∗ (z), r) → ℂ

of T −j∗ sending T j∗ (z) to z. Since the sequence (j∗ )∞ j=1 has bounded gaps and, by (28.32)– (28.33) and (28.38)–(28.39), ′ 󵄨󵄨 j∗ ′ 󵄨󵄨 󵄨󵄨 󵄨−1 j qp j 󵄨󵄨(T ) (z)󵄨󵄨 = 󵄨󵄨(φω|j ) (π(σ (ω)))󵄨󵄨󵄨 ≥ (min{K, β }) ,

it follows from Observation 28.5.2 that z ∈ 𝒥uer (T). Thus, J ⊆ 𝒥uer (T). Combined with (28.41), this completes the proof of Theorem 28.5.3. In the case of polynomials, something more can be said. Anna Zdunik proved in [151] that if a polynomial has a connected Julia set then either it induces a parabolic orbifold (equivalently meaning that it is either a Möbius conjugate of a map z 󳨃→ z±d or of a Tchebyschev polynomial) and then the Julia set is a closed arc of a geometric circle (including the circle itself), or else its Hausdorff dimension is strictly larger than 1. It is worth noting that 𝒥uer (T) contains all expanding subsets of 𝒥 (T). As mentioned during the examination of expanding sets in Section 25.3 (just before Theorem 25.3.9), Definition 16.2.1 is a little too narrow for expanding sets. In fact, all results derived about

1182 � 28 Invariant measures: fractal and dynamical properties conformal repellers in Volume 2 are valid when part (d) of their definition is replaced by the following weaker condition: (d ′ ) There exists λ > 1 and q ∈ ℕ such that |(T q )′ (x)| ≥ λ for all x ∈ X. ̂→ℂ ̂ is a rational function with deg(T) ≥ 2 and X ⊆ 𝒥 (T) is Proposition 28.5.4. If T : ℂ an expanding set, then the pressure function ℝ ∋ t 󳨃→ P(T|X , t) = P(T|X , −t log |T ′ |) ∈ ℝ possesses the following properties: (a) It is Lipschitz continuous, with Lipschitz constant L = L(X) := log ‖T ′ ‖X,∞ and where ‖T ′ ‖X,∞ := supx∈X |T ′ (x)|. (b) It is strictly decreasing. (c) lim P(T|X , t) = ∞ whereas lim P(T|X , t) = −∞. t→−∞

t→∞

(d) P(T|X , 0) = htop (T|X ). If T is transitive and X is not a single periodic orbit of T, then P(T|X , 0) = htop (T|X ) > 0. (e) There exists a unique h = h(X) ∈ ℝ such that P(T|X , h(X)) = 0. If T is transitive and X is not a single periodic orbit of T, then h(X) > 0. (f) The pressure function ℝ ∋ t 󳨃→ P(T|X , t) ∈ ℝ is convex. N.B.: This result is analogous to Proposition 16.3.1 for conformal expanding repellers. Parts (d,e) of that proposition were not entirely correctly stated. The underlined condition on X was lacking; this condition is obviously satisfied when X is infinite. (The transitivity of T was part of the definition of a conformal expanding repeller.) Proof. Let q ≥ 1 be an integer attesting of the expandingness of T|X . The proof of Proposition 16.3.1 applies verbatim to T|X except that when proving items (a)–(c) one req places “a unique equilibrium state” by “an equilibrium state”. As P(T|X , −t log |(T q )′ |) = ′ qP(T|X , −t log |T |), the result ensues. ̂→ℂ ̂ is a rational function with deg(T) ≥ 2 and X ⊆ 𝒥 (T) is an Theorem 28.5.5. If T : ℂ expanding set, then HD(X) = h, where h comes from Proposition 28.5.4. Proof. Fix any t > h. Then P(T|X , t) < 0. Since X is expanding, there are λ > 1 and q ∈ ℕ such that |(T q )′ (x)| ≥ λ,

∀x ∈ X.

(28.42)

By standard compactness arguments, there exists δ > 0 such that T q |B(x,4δ) is 1-to-1, and

∀x ∈ X

(28.43)

28.5 HD(𝒥 (T )) > 0 and radial Julia sets 𝒥r (T ), 𝒥er (T ), 𝒥uer (T )

|(T q )′ (z)| ≥ √λ,

∀z ∈ B(X, 4δ) := ⋃ B(x, 4δ).

� 1183

(28.44)

x∈X

Consequently, for every x ∈ X there is a unique holomorphic inverse branch Tx ̂ such that Tx−q (T q (x)) = x. By (28.44), we have that B(T q (x), 4δ) → ℂ

−q

T q (B(x, r)) ⊇ B(T q (x), √λ r),

|(Tx−q )′ (z)| ≤ λ−1/2 ,

and ρ(Tx−q (w), Tx−q (z)) ≤ λ−1/2 ρ(w, z),

:

∀r ∈ [0, 4δ], ∀z ∈ B(x, 4δ), ∀x ∈ X (28.45)

∀w, z ∈ B(x, 4δ), ∀x ∈ X.

(28.46)

It follows from (28.46) that Tx−q (B(T q (x), r)) ⊆ B(x, λ−1/2 r) ⊆ B(x, r) for all r ∈ [0, 4δ] and all x ∈ X. Hence, for every n ∈ ℕ and x ∈ X the composition ̂ Tx−qn := Tx−q ∘ TT(x) ∘ ⋅ ⋅ ⋅ ∘ TT q(n−2) (x) ∘ TT q(n−1) (x) : B(T qn (x), 4δ) → ℂ −q

−q

−q

is well defined, satisfies Tx−qn (B(T qn (x), r)) ⊆ B(x, λ−n/2 r) ⊆ B(x, r),

∀r ∈ [0, 4δ],

and is the unique analytic inverse branch of T qn defined on B(T qn (x), 4δ) that sends T qn (x) to x. −qn Let D ⊆ X be a maximal δ-separated set. Then T|X (D) is a (n, δ)-separated set for q T|X for all n ∈ ℕ and {Tw−qn (B(z, δ)) : z ∈ D, w ∈ T|X (z)} −qn

is an open cover of X with diam(Tw−qn (B(z, δ))) ≤ δλ−n/2 . Moreover, by bounded distortion (entirely dynamical; no Koebe distortion is needed), diam(Tw−qn (B(z, δ))) ≤ Cδ|(T qn )′ (w)|−1 with distortion constant C ≥ 1 depending only on T. Therefore the t-dimensional Hausdorff measure of X satisfies Ht (X) ≤ lim inf ∑ n→∞

∑

t

z∈D w∈T|−qn (z) X

[diam(Tw−qn (B(z, δ)))]

1184 � 28 Invariant measures: fractal and dynamical properties ≤ Cδ lim inf ∑ n→∞

∑

z∈D w∈T|−qn (z) X

󵄨󵄨 qn ′ 󵄨󵄨−t 󵄨󵄨(T ) (w)󵄨󵄨 .

But by Bowen’s definition of the pressure we have lim inf n→∞

1 log ∑ n z∈D

q 󵄨󵄨 qn ′ 󵄨󵄨−t q ′ ′ 󵄨󵄨(T|X ) (w)󵄨󵄨 ≤ P(T|X , −t log |(T ) |) = qP(T|X , −t log |T |) < 0,

∑ −qn w∈T|X (z)

we deduce that Ht (X) = 0. Thus, HD(X) ≤ t for all t > h. Hence, HD(X) ≤ h. For the converse inequality, it is clear that HD(X) ≥ h if h = 0. So we may assume that h > 0. Fix any t ∈ (0, h). By Proposition 28.5.4, P(T|X , t) > 0. Per the variational principle (Theorem 12.1.1 and Remark 12.1.2), there exists an ergodic T|X -invariant Borel probability measure μ on X such that hμ (T|X ) − tχμ (T|X ) > 0. But χμ (T|X ) > 0 since T|X is expanding. Using the volume lemma (Theorem 28.4.1), we infer from the above inequality that HD(X) ≥ HD(μ) =

hμ (T|X ) χμ (T|X )

> t.

Given that this holds for all t ∈ (0, h), we conclude that HD(X) ≥ h. It is noteworthy to recall that the pressure and the Hausdorff dimension functions were shown to be real analytic under analytic perturbations of a conformal repeller. See Theorems 16.4.9–16.4.11. Other papers treating of the real analyticity of the Hausdorff dimension include [2, 114, 125, 132, 143, 147].

28.6 Conformal Katok’s theory of expanding sets and topological pressure In this section, we provide a conformal account of Katok’s theory, which describes the high complexity (expanding/hyperbolic subsets and periodic points) of many classes of dynamical systems having positive entropy. Its full exposition is given in [62]. Our approach to the conformal case stems from [110].

28.6.1 Pressure-like definition of the functional hμ (T) + ∫ φ dμ In this subsection, we develop general tools that will be used in the next subsection to approximate the topological pressure on expanding sets. These tools are reminiscent of those that emerged in Sections 7.3, 11.2 and 12.1 of the first volume. The reader is

28.6 Conformal Katok’s theory of expanding sets and topological pressure � 1185

encouraged to review those sections (especially the latter) to identify the similarities and differences between the various tools. Unlike in the previous sections of this chapter, no smoothness/conformality is needed in this subsection. Just like in Sections 7.3, 11.2 and 12.1, we work in a metric space setting. Let T : X → X be a topological dynamical system, i. e., a continuous selfmap of a compact metric space (X, d), and let μ be a Borel probability measure on X. Given ε > 0 and 0 ≤ δ ≤ 1, a set E ⊆ X is said to be μ − (n, ε, δ)-spanning if μ(⋃ Bn (x, ε)) ≥ 1 − δ, x∈E

where n−1

Bn (x, ε) := ⋂ T −j (B(T j (x), ε)) j=0

consists of those points whose iterates stay within a distance ε from the corresponding iterates of x until time n − 1 at least. Recall that Bn (x, ε) = {y ∈ X : dn (x, y) < ε}, where dn is the dynamical metric (also called Bowen metric) dn (x, y) = max{d(T j (x), T j (y)) : 0 ≤ j < n}. The ball Bn (x, ε) is accordingly called the dynamical (n, ε)-ball centered at x. Let φ : X → ℝ be a potential, i. e., a continuous real-valued function on X. Define Qμ (T, φ, n, ε, δ) = inf{ ∑ exp(Sn φ(x))}, E

x∈E

where the infimum is taken over all μ − (n, ε, δ)-spanning sets E. The main result of this subsection is a formula for the free energy. Theorem 28.6.1. If T : (X, d) → (X, d) is a topological dynamical system and φ : X → ℝ is a potential on X, then for every 0 < δ < 1 and every ergodic T-invariant Borel probability measure μ on X, we have hμ (T) + ∫ φ dμ = lim lim inf X

ε→0 n→∞

1 1 log Qμ (T, φ, n, ε, δ) = lim lim sup log Qμ (T, φ, n, ε, δ). ε→0 n→∞ n n

Proof. Denote the quantity following the first equality sign by Pμ (T, φ, δ) and the quantity following the second equality sign by Pμ (T, φ, δ). Following essentially the proof of Part I of the variational principle (Theorem 12.1.1), we shall first show that

1186 � 28 Invariant measures: fractal and dynamical properties Pμ (T, φ, δ) ≥ hμ (T) + ∫ φ dμ.

(28.47)

X

Indeed, in a similar way to that proof, consider a finite Borel partition 𝒰 = {A1 , . . . , As } of X and, using Lemma 12.1.4, take compact sets Bi ⊆ Ai , i = 1, 2, . . . , s, such that the partition 𝒱 = {B1 , . . . , Bs , X \ (B1 ∪ ⋅ ⋅ ⋅ ∪ Bs )}

satisfies Hμ (𝒰 |𝒱 ) ≤ 1. Let θ > 0. For every q ∈ ℕ, set 1 Xq := {x ∈ X : − log μ(𝒱 n (x)) ≥ hμ (T, 𝒱 ) − θ n

and

1 S φ(x) ≥ ∫ φ dμ − θ, n n

∀n ≥ q}.

X

Fix 0 < δ < 1. It follows from the ergodic cases of the Shannon–McMillan–Breiman theorem (Corollary 9.5.5) and of Birkhoff’s ergodic theorem (Corollary 8.2.14) that the sets (Xq )∞ q=1 form an ascending sequence covering μ-almost everywhere X. Therefore, μ(Xq ) > δ

(28.48)

for all q ∈ ℕ large enough. Fix such a q. Since the potential φ : X → ℝ is uniformly continuous, there exists 0 0 2

so small that 󵄨 󵄨 d(x, y) < ε 󳨐⇒ 󵄨󵄨󵄨φ(x) − φ(y)󵄨󵄨󵄨 < θ.

(28.49)

Take any integer n ≥ q. For every x ∈ X, the set Bn (x, ε) ∩ Xq can be covered by at most 2n atoms of 𝒱 n because at every iterate this set can intersect at most one Bi and X \ ⋃sj=1 Bj . For every w ∈ Xq , we also know that − n1 log μ(𝒱 n (w)) ≥ hμ (T, 𝒱 ) − θ. Consequently, μ(Bn (x, ε) ∩ Xq ) ≤ 2n max{μ(𝒱 n (w)) : w ∈ Xq } ≤ exp(n[log 2 − hμ (T, 𝒱 ) + θ]).

(28.50)

Let E ⊆ X be a μ − (n, ε, δ)-spanning set, and consider the set E ′ := {x ∈ E : Bn (x, ε) ∩ Xq ≠ 0}. Take a point y ∈ Bn (x, ε) ∩ Xq . By the choice of ε in (28.49), we have |Sn φ(x) − Sn φ(y)| < nθ. As y ∈ Xq , we also know that Sn φ(y) − n ∫ φ dμ ≥ −nθ. Therefore,

28.6 Conformal Katok’s theory of expanding sets and topological pressure

� 1187

∑ exp(Sn φ(x)) exp(−n[hμ (T, 𝒱 ) + ∫ φ dμ − 3θ − log 2])

x∈E

X

≥ ∑ exp(Sn φ(x) − n ∫ φ dμ) exp(−n[hμ (T, 𝒱 ) − 3θ − log 2]) because E ′ ⊆ E x∈E ′

X

= ∑ exp(Sn φ(x) − Sn φ(y)) exp(Sn φ(y) − n ∫ φ dμ) exp(−n[hμ (T, 𝒱 ) − 3θ − log 2]) x∈E ′

X

≥ ∑ exp(−nθ) exp(−nθ) exp(−n[hμ (T, 𝒱 ) − 3θ − log 2]) x∈E ′

= ∑ exp(n[log 2 − hμ (T, 𝒱 ) + θ]) ≥ ∑ μ(Bn (x, ε) ∩ Xq ) by (28.50) x∈E ′

x∈E ′

≥ μ( ⋃ Bn (x, ε) ∩ Xq ) = μ([⋃ Bn (x, ε)] ∩ Xq ) x∈E

x∈E ′

≥ μ(Xq ) − δ >0

because E is μ − (n, ε, δ)-spanning

by (28.48).

This implies that 1 log Qμ (T, φ, n, ε, δ) ≥ hμ (T, 𝒱 ) + ∫ φ dμ − 3θ − log 2, n

∀n ≥ q.

X

Taking the lim inf as n → ∞ followed by the limit as ε → 0, we deduce that Pμ (T, φ, δ) ≥ hμ (T, 𝒱 ) + ∫ φ dμ − 3θ − log 2. X

Since θ > 0 is an arbitrary number and since hμ (T, 𝒰 ) ≤ hμ (T, 𝒱 )+Hμ (𝒰 |𝒱 ) ≤ hμ (T, 𝒱 )+1, we get that Pμ (T, φ, δ) ≥ hμ (T, 𝒰 ) − 1 + ∫ φ dμ − log 2. X

Given the arbitrariness of the partition 𝒰 , upon passing to the supremum over all partitions we infer that Pμ (T, φ, δ) ≥ hμ (T) + ∫ φ dμ − log 2 − 1. X

Using now the standard trick consisting in replacing T by an arbitrary iterate T k and φ by Sk φ, we obtain that kPμ (T, φ, δ) = Pμ (T k , Sk φ, δ) ≥ hμ (T k ) + ∫ Sk φ dμ − log 2 − 1 = khμ (T) + k ∫ φ dμ − log 2 − 1. X

X

1188 � 28 Invariant measures: fractal and dynamical properties Dividing the extremes of this inequality by k and letting k → ∞, we conclude that Pμ (T, φ, δ) ≥ hμ (T) + ∫ φ dμ. X

Let us now demonstrate that Pμ (T, φ, δ) ≤ hμ (T) + ∫ φ dμ.

(28.51)

X

Fix 0 < δ < 1. Let ε > 0 and θ > 0. Let 𝒫 be a finite Borel partition of X with diam(𝒫 ) ≤ ε. By the ergodic cases of the Shannon–McMillan–Breiman theorem (Corollary 9.5.5) and of Birkhoff’s ergodic theorem (Corollary 8.2.14), there exists a Borel set Z ⊆ X such that μ(Z) > 1 − δ and −

1 log μ(𝒫 n (x)) ≤ hμ (T) + θ n

and

1 S φ(x) ≤ ∫ φ dμ + θ n n

(28.52)

X

for every x ∈ Z and all n = n(x) large enough. Take such an x and n. From each atom of 𝒫 n intersecting Z, choose one point, hence generating a finite set {x1 , x2 , . . . , xk }. Then Bn (xj , ε) ⊇ 𝒫 n (xj ) for every j = 1, 2, . . . , k and, therefore, the set {x1 , x2 , . . . , xk } is μ − (n, ε, δ)-spanning. By the first part of (28.52), we have k ≤ exp(n[hμ (T) + θ]). Using this and the second part of (28.52), we get k

∑ exp(Sn φ(xj )) ≤ exp(n[hμ (T) + θ + ∫ φ dμ + θ]). j=1

X

Therefore, Qμ (T, φ, n, ε, δ) ≤ exp(n[hμ (T) + ∫ φ dμ + 2θ]). X

Applying the logarithm to both sides, dividing by n and taking successively the lim sup as n → ∞ and the limit as ε → 0, we deduce that Pμ (T, φ, δ) ≤ hμ (T) + ∫ φ dμ + 2θ. X

Since θ > 0 is arbitrary, (28.51) ensues. Joining (28.51) to (28.47) completes the proof of Theorem 28.6.1. 28.6.2 Conformal Katok’s theory This subsection is dedicated to proving the following result.

28.6 Conformal Katok’s theory of expanding sets and topological pressure � 1189

̂→ℂ ̂ be a rational function with deg(T) ≥ 2. If μ is an ergodic Theorem 28.6.2. Let T : ℂ T-invariant Borel probability measure on 𝒥 (T) with positive Lyapunov exponent χμ (T) and if φ : 𝒥 (T) → ℝ is a (continuous) potential, then there exists a sequence (Xk )∞ k=1 of forward T-invariant closed subsets of 𝒥 (T) that are all topological Cantor sets such that for every k ∈ ℕ the restriction T|Xk : Xk → Xk is a conformal topologically transitive open expanding dynamical system and lim inf P(T|Xk , φ) ≥ hμ (T) + ∫ φ dμ. k→∞

(28.53)

𝒥 (T)

Furthermore, if for every k ∈ ℕ there is an ergodic T-invariant Borel probability ∗ measure μk on Xk , then the sequence (μk )∞ k=1 converges to μ in the weak topology on M(𝒥 (T)). Moreover, k→∞ 󵄨 󵄨 󵄨 󵄨 χμk (T|Xk ) := ∫ log󵄨󵄨󵄨T ′ 󵄨󵄨󵄨 dμk 󳨀→ ∫ log󵄨󵄨󵄨T ′ 󵄨󵄨󵄨 dμ =: χμ (T). Xk

𝒥 (T)

In particular, this is true if the measures (μk )∞ k=1 are each supported on individual periodic orbits of T in Xk . Proof. By Tietze’s extension theorem, we may assume that the continuous function φ is ̂ Whatever the T-invariant closed sets Xk ⊆ 𝒥 (T) are, the variational defined on all of ℂ. principle (Theorem 12.1.1) ensures that for every c ∈ ℝ, P(T|Xk , φ + c) = P(T|Xk , φ) + c. Moreover, hμ (T) + ∫ (φ + c) dμ = hμ (T) + ∫ φ dμ + c. 𝒥 (T)

𝒥 (T)

Therefore, by adding a sufficiently large constant c if necessary, we can assume without loss of generality that φ is positive, i. e., inf φ > 0. ̂ Fix a countable basis {ψj }∞ j=1 of the Banach space C(ℂ, ℝ) of all real-valued continuous ̂ Fix θ > 0 and s ∈ ℕ. In view of Theorem 28.6.1 and the uniform contifunctions on ℂ. nuity of all the functions φ and ψj , j ∈ ℕ, there exists ε > 0 so small that lim inf n→∞

1 log Qμ (T, φ, n, ε, δ) > (hμ (T) + ∫ φ dμ) − θ, n

󵄨 󵄨 |x − y| < ε 󳨐⇒ 󵄨󵄨󵄨φ(x) − φ(y)󵄨󵄨󵄨 < θ

(28.54)

𝒥 (T)

(28.55)

1190 � 28 Invariant measures: fractal and dynamical properties and 󵄨 󵄨 |x − y| < ε 󳨐⇒ 󵄨󵄨󵄨ψi (x) − ψi (y)󵄨󵄨󵄨 < θ/2,

∀i = 1, 2, . . . , s.

(28.56)

? ̃ μ ̃ ). Given δ > 0, We work once more with Rokhlin’s natural extension (𝒥 (T), T, ? ? let 𝒥 (T)(δ) and r(δ) be produced through Theorem 28.3.3. Then the set π0 (𝒥 (T)(δ)) is ? compact as a continuous image of the compact set 𝒥 (T)(δ). That theorem also produces a constant χ ′ > 0 (possibly with a smaller radius r(δ)) such that ′

diam(Tx−n (B(π0 (x̃), r(δ)))) ≤ e−nχ , n

? ∀x̃ ∈ 𝒥 (T)(δ), ∀n ≥ 0.

(28.57)

Set β := r(δ)/2 ? and fix a finite (β/2)-spanning set {x1 , . . . , xt } of π0 (𝒥 (T)(δ)). This means that ? B(x1 , β/2) ∪ ⋅ ⋅ ⋅ ∪ B(xt , β/2) ⊇ π0 (𝒥 (T)(δ)). Let 𝒰 be a finite Borel partition of 𝒥 (T) with diam(𝒰 ) < β/2 and let n1 ∈ ℕ be so large that e−n1 χ < min{β/3, K −1 }, ′

(28.58)

where K is the Koebe constant corresponding to the scale 1/2. Given n ∈ ℕ, define q

? ? ? 𝒥 (T)n,s := {x̃ ∈ 𝒥 (T)(δ) : T̃ (x̃) ∈ 𝒥 (T)(δ), π0 (T̃ q (x̃)) ∈ 𝒰 (π0 (x̃)) for some q ∈ [n + 1, (1 + θ)n], 󵄨󵄨 θ 󵄨󵄨 1 󵄨 󵄨 and 󵄨󵄨󵄨 Sk (ψi )(π0 (x̃)) − ∫ ψi dμ󵄨󵄨󵄨 < , ∀k ≥ n, ∀1 ≤ i ≤ s}. 󵄨󵄨 k 󵄨󵄨 2 𝒥 (T)

By the ergodic case of Birkhoff’s ergodic theorem (Corollary 8.2.14), we know that ? ? ̃ (𝒥 ̃ (𝒥 limn→∞ μ (T)n,s ) = μ (T)(δ)) > 1 − δ. Therefore, there exists n ≥ n1 so large that ? ̃ (𝒥 (T)n,s ) > 1 − δ. Let μ ? 𝒥 (T)n,s := π0 (𝒥 (T)n,s ). Then μ(𝒥 (T)n,s ) > 1 − δ.

28.6 Conformal Katok’s theory of expanding sets and topological pressure

� 1191

Let En ⊆ 𝒥 (T)n,s be a maximal (n, ε)-separated subset of 𝒥 (T)n,s . Then En is a (n, ε)spanning set of 𝒥 (T)n,s . It follows from (28.54) that for all integers n ≥ n1 large enough 1 log ∑ exp(Sn φ(x)) > (hμ (T) + ∫ φ dμ) − θ. n x∈E n

𝒥 (T)

Equivalently, ∑ exp(Sn φ(x)) > exp(n[hμ (T) + ∫ φ dμ − θ]).

x∈En

𝒥 (T)

For every integer q ∈ [n + 1, (1 + θ)n], set Vq := {x ∈ En : T q (x) ∈ 𝒰 (x)}. (1+θ)n Then let m = m(n) be a value of q that maximizes ∑x∈Vq exp(Sn φ(x)). As ⋃q=n+1 Vq = En , we thus obtain that

∑ exp(Sn φ(x)) ≥

x∈Vm

1 1 ∑ ∑ exp(Sn φ(x)) ≥ ∑ exp(Sn φ(x)) nθ q=n+1 x∈V nθ x∈E (1+θ)n

q

n

≥ exp(n[hμ (T) + ∫ φ dμ − 2θ]). 𝒥 (T)

Consider the sets Vm ∩B(xj , β/2), 1 ≤ j ≤ t, and choose a value i = i(m) of j that maximizes the sum ∑

x∈Vm ∩B(xj ,β/2)

exp(Sn φ(x)).

Denoting the intersection Vm ∩ B(xi(m) , β/2) by Dm , we have t

Vm = ⋃ Vm ∩ B(xj , β/2) j=1

and ∑ exp(Sn φ(x)) ≥

x∈Dm

1 1 ∑ exp(Sn φ(x)) ≥ exp(n[hμ (T) + ∫ φ dμ − 2θ]). t x∈V t m

𝒥 (T)

As φ is positive and m ≥ n + 1, this implies that ∑ exp(Sm φ(x)) ≥

x∈Dm

1 exp(n[hμ (T) + ∫ φ dμ − 2θ]). t 𝒥 (T)

(28.59)

1192 � 28 Invariant measures: fractal and dynamical properties For any x ∈ Dm , we know that x ∈ B(xi(m) , β/2), i. e., |x − xi(m) | < β/2. We further know that x ∈ Vm , and hence T m (x) ∈ 𝒰 (x). As diam(𝒰 ) < β/2, it ensues that |T m (x) − x| < β/2. Then |T m (x) − xi(m) | ≤ |T m (x) − x| + |x − xi(m) | < β/2 + β/2 = β and, therefore, T m (x) ∈ B(xi(m) , β) ⊆ B(T m (x), 2β).

(28.60)

? Choose x̃ ∈ π0−1 (x) ∩ 𝒥 (T)n,s . Given that m ≥ n ≥ n1 , it ensues from (28.57)–(28.58) that diam(Tx−m (B(T m (x), 2β))) ≤ e−mχ < β/3. ′

By this and (28.60), we deduce that Tx−m (B(xi(m) , β)) ⊆ B(x, β/3) ⊆ B(xi(m) , β/2 + β/3) = B(xi(m) , 5β/6). In particular, this implies that Tx−m (B(xi(m) , β)) ⊆ B(xi(m) , β).

(28.61)

Observe that Ty−m (B(xi(m) , 2β)) ∩ Ty−m (B(xi(m) , 2β)) = 0, 1 2

∀y1 , y2 ∈ Dm , y1 ≠ y2 .

In particular, Ty−m (B(xi(m) , β)) ∩ Ty−m (B(xi(m) , β)) = 0, 1 2

∀y1 , y2 ∈ Dm , y1 ≠ y2 .

Let ξ := min{β, min{dist( Ty−m (B(xi(m) , β)) , Ty−m (B(xi(m) , β)) ) : y1 , y2 ∈ Dm , y1 ≠ y2 }}. 1 2 Define inductively the sequence of sets (X (j) )∞ j=0 by setting X (0) := B(xi(m) , β) and

X (j+1) := ⋃ Tx−m (X (j) ). x∈Dm

By (28.61), (X (j) )∞ j=0 is a descending sequence of nonempty compact sets. Therefore, the intersection ∞

X ∗ (θ, s) := ⋂ X (j) j=0

is also a nonempty compact set. Moreover, by construction: (a∗ ) X ∗ (θ, s) is a topological Cantor set.

28.6 Conformal Katok’s theory of expanding sets and topological pressure � 1193

(b∗ ) T m (X ∗ (θ, s)) = X ∗ (θ, s), i. e., T m |X ∗ (θ,s) : X ∗ (θ, s) → X ∗ (θ, s) is a surjective dynamical system. ∗ (c ) The system T m |X ∗ (θ,s) is topologically conjugate to the one-sided full shift generated by an alphabet counting #Dm letters. As such, the system T m |X ∗ (θ,s) is topologically exact. (d∗ ) The system/map T m |X ∗ (θ,s) is open. (e∗ ) The system T m |X ∗ (θ,s) is distance expanding. (This follows from Theorem 28.3.3; see (28.57).) (f∗ ) Consequently, X ∗ (θ, s) ⊆ 𝒥 (T). Set m−1

X(θ, s) := ⋃ T l (X ∗ (θ, s)). l=0

We readily see that: (a) The set X(θ, s) is a topological Cantor set. (b) T|X(θ,s) : X(θ, s) → X(θ, s) is a surjective dynamical system. (c) The system T|X(θ,s) is topologically transitive. (d) The system T|X(θ,s) is distance expanding. (e) X(θ, s) ⊆ 𝒥 (T). It follows from Theorem 25.3.8 that for every η > 0 there exists a forward T-invariant closed set ̂ s; η) ⊆ B(X(θ, s), η) X(θ, s) ⊆ X(θ,

(28.62)

such that ̂ s; η) → X(θ, ̂ s; η) is a topologically transitive, open and expanding (f) T|X(θ,s;η) : X(θ, ̂ system. Fix j ∈ ℕ. For any j-tuple (z0 , z1 , . . . , zj−1 ) of points in Dm , choose exactly one point y in the set Tz−m ∘ Tz−m ∘ ⋅ ⋅ ⋅ ∘ Tz−m (X ∗ (θ, s)) and denote the set consisting of all those points y j−1 j−2 0 j−1

by Aj . By (28.55) and (28.57), Sjm φ(y) ≥ ∑l=0 Sm φ(zl ) − jmθ and we get that ∑ exp(Sjm φ(y)) ≥

y∈Aj

j−1

∑ j

(z0 ,z1 ,...,zj−1 )∈Dm

= e−jmθ

∑

exp(∑ Sm φ(zl ) − jmθ) l=0

j−1

∏ exp(Sm φ(zl ))

j (z0 ,z1 ,...,zj−1 )∈Dm l=0

j

= e−jmθ ( ∑ exp(Sm φ(x))) . x∈Dm

1194 � 28 Invariant measures: fractal and dynamical properties Equivalently, 1 log ∑ exp(Sjm φ(y)) ≥ log ∑ exp(Sm φ(x)) − mθ. j y∈A x∈D j

m

In view of the definition of ξ, the set Aj is (j, ξ)-separated for T m and ξ is an expansive constant for T m . Hence, invoking (28.59) and letting j → ∞, we obtain via Theorem 11.2.8 that P(T m |X ∗ (θ,s) , Sm φ) ≥ log ∑ exp(Sm φ(x)) − mθ x∈Dm

≥ n[hμ (T) + ∫ φ dμ − 2θ] − log t − mθ. 𝒥 (T)

Since n + 1 ≤ m ≤ n(1 + θ) and since inf φ > 0 (and consequently hμ (T) + ∫ φ dμ > 0), we deduce that 1 1 P(T m |X(θ,s) , Sm φ) ≥ P(T m |X ∗ (θ,s) , Sm φ) m m log t 1 ≥ [hμ (T) + ∫ φ dμ − 2θ] − − θ. 1+θ m

P(T|X(θ,s;η) , φ) ≥ P(T|X(θ,s) , φ) = ̂

𝒥 (T)

Assuming now that n and, consequently, m, was chosen sufficiently large, we get P(T|X(θ,s;η) , φ) ≥ ̂

1 [h (T) + ∫ φ dμ] − 4θ. 1+θ μ 𝒥 (T)

If η > 0 is sufficiently small, and we shall denote such an η by ηθ,s , then for every ̂ s; ηθ,s ), it follows from (1) the ergodic T-invariant Borel probability measure ν on X(θ, ? definition of the set 𝒥 (T)n,s , (2) the construction of the set X(θ, s), (3) the ergodic case of Birkhoff’s ergodic theorem (Corollary 8.2.14) and (4) relations (28.56)–(28.57), that 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∫ ψi dν − ∫ ψi dμ󵄨󵄨󵄨 < θ, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝒥 (T) 𝒥 (T)

∀i = 1, 2, . . . , s.

A similar estimate for log |T ′ | ensues from the definition of 𝒥 (T)(δ) and Corollary 28.3.2 with ε = θ/2. Therefore, all the assertions of Theorem 28.6.2 are satisfied for the sets ̂ Xk := X(1/k, k; η1/k,k ), k ∈ ℕ. The proof of Theorem 28.6.2 is complete. Remark 28.6.3. The sets Xk in Theorem 28.6.2 can be found independent of φ. Indeed, take Xk constructed under the constant potential φ ≡ 0. Then Theorem 28.6.2 states that lim sup htop (T|Xk ) ≥ hμ (T). k→∞

28.6 Conformal Katok’s theory of expanding sets and topological pressure � 1195

For every k ∈ ℕ, let μk be a measure of maximal entropy for the map T|Xk : Xk → Xk , i. e., hμk (T) = htop (T|Xk ). Theorem 28.6.2 asserts that limk→∞ μk = μ weakly∗ . Consider now an arbitrary potential φ : 𝒥 (T) → ℝ. Then ∫ φ dμk → ∫ φ dμ. Therefore, with the use of the variational principle (Theorem 12.1.1), lim inf P(T|Xk , φ) ≥ lim inf(hμk (T) + ∫ φ dμk ) ≥ hμ (T) + ∫ φ dμ. k→∞

k→∞

𝒥 (T)

𝒥 (T)

One more note: although lim inf hμk (T) ≥ hμ (T) k→∞

∞ for the maximal entropy measures (μk )∞ k=1 , this need not be true for all sequences (μk )k=1 .

̂→ℂ ̂ be a rational function of deg(T) ≥ 2 and φ : 𝒥 (T) → ℝ a Corollary 28.6.4. Let T : ℂ Hölder continuous potential such that P(φ) > sup φ. Then there exists a sequence (Xk )∞ k=1 of forward T-invariant closed subsets of 𝒥 (T) that are all topological Cantor sets such that for every k ∈ ℕ the restriction T|Xk : Xk → Xk is a conformal topologically transitive open expanding map and lim P(T|Xk , φ) = P(T, φ).

k→∞

Proof. By the variational principle (Theorem 12.1.1), P(T, φ) = lim (hνk (T) + ∫ φ dνk ) k→∞

(28.63)

𝒥 (T)

for some sequence of ergodic T-invariant Borel probability measures νk on 𝒥 (T). So, as P(T, φ) > sup φ, there exists N ∈ ℕ such that hνk (T) + ∫ φ dνk > sup φ 𝒥 (T)

for all k ≥ N. For such k’s, we get hνk (T) > sup φ − ∫ φ dνk ≥ 0. 𝒥 (T)

Hence, χνk (T) > 0 for all such k’s according to Ruelle’s inequality (Theorem 28.2.1). For each such k ∈ ℕ, apply Theorem 28.6.2 with μ = νk to get a set Xk satisfying the conclusion of that theorem, namely, for which P(T|Xk , φ) ≥ hνk (T) + ∫ φ dνk − 𝒥 (T)

1 . k

1196 � 28 Invariant measures: fractal and dynamical properties It follows from this and (28.63) that lim inf P(T|Xk , φ) ≥ P(T, φ). k→∞

(28.64)

Obviously, P(T|Xk , φ) ≤ P(T, φ) for every k ∈ ℕ, so the proof of Corollary 28.6.4 is complete.

28.7 Exercises ̂→ℂ ̂ be a rational function with deg(T) ≥ 2. Exercise 28.7.1. Let T : ℂ − −n (a) Show that 𝒥r (T) ∩ Crit∞ (T) = 0, where Crit−∞ (T) := ⋃∞ n=0 T (Crit(T)) is the backward orbit of the critical set of T. (b) Prove that T −1 (𝒥r (T)) \ Crit(T) = 𝒥r (T) and T(𝒥r (T)) ⊆ 𝒥r (T). (c) Identify a sufficient condition for T(𝒥r (T)) = 𝒥r (T). (d) Establish corresponding relations for 𝒥er (T) and 𝒥uer (T). ̂→ℂ ̂ be a parabolic rational function with deg(T) ≥ 2. Show Exercise 28.7.2. Let T : ℂ that 𝒥r (T) = 𝒥 (T) \ Ω∞ (T), −

(28.65)

−n where Ω−∞ (T) := ⋃∞ n=0 T (Ω(T)) is the backward orbit of the set Ω(T) of all rationally indifferent periodic points of T. Then prove that 𝒥r (T) is backward T-invariant, i. e. T −1 (𝒥r (T)) ⊆ 𝒥r (T).

29 Sullivan’s conformal measures for rational functions In Section 29.1, we present the general concept of conformal measure introduced in [34]. We have already encountered such measures in Chapter 13 in the important, nevertheless quite special, context of open distance expanding systems under Hölder continuous potentials. We now deal with such measures for much more general systems and potentials. As we already saw in Chapter 13 and as we will see in the first section, this concept is closely related to the notion of quasi-invariance. A substantial difference now is that the maps need not be open nor do the potentials need be continuous or even bounded, whence the method based on the Schauder–Tychonov fixed-point theorem for proving the existence of such measures is not applicable. Conformal measures were first defined and introduced by Samuel Patterson in his seminal paper [100] (see also [101]) in the context of Fuchsian groups. Dennis Sullivan extended this concept to all Kleinian groups in [126, 128, 130]. In the papers [127, 129], Sullivan defined, and proved the existence of, conformal measures for all rational funĉ Both Patterson and Sullivan came up with conformal tions of the Riemann sphere ℂ. measures in order to get an understanding of geometric measures such as Hausdorff and packing measures. Although Sullivan already noticed that there are conformal measures for Kleinian groups that are not equal, nor even equivalent, to any Hausdorff or packing (generalized) measure, the main purpose to study them is still to understand Hausdorff and packing measures. But their use goes beyond that purpose. Conformal measures for rational functions, also called Sullivan’s conformal measures, constitute the theme of Section 29.2. They have been studied in numerous research works. We cite only few of them: [5, 37, 39, 40, 42, 108, 138, 140, 141]. They have been studied in greater detail in [35] where, in particular, the structure of the set of their exponents was examined. In Section 29.2, we describe a construction whose objective is to establish the existence of such measures. We further provide several characterizations of the minimal exponent for which such measures exist, in particular its equality with the Hausdorff dimension of the Julia set and of the three radial Julia subsets, as well as with four dynamically-defined dimensions. The intrigued reader may peek ahead to Theorem 29.2.12. The concept of conformal measures, still in the sense of Sullivan, was subsequently extended to countable-alphabet iterated function systems in [87] and to conformal graph directed Markov systems in [88]. These systems were treated at length in Chapters 19 and 21 of the second volume of this book. This concept was also extended to transcendental meromorphic dynamics in [71, 89, 144]; see also [6, 7, 90, 145]. Last but not least, this concept found its place in random dynamics; among others, see [91]. This chapter ends with a short Section 29.3 on Pesin’s formula, which affirms that the Hausdorff dimension of an ergodic invariant measure which has positive entropy and is absolutely continuous with respect to a Sullivan conformal measure, is equal to https://doi.org/10.1515/9783110769876-029

1198 � 29 Sullivan’s conformal measures for rational functions the exponent of that measure, and that this exponent is nothing else than the minimal exponent alluded to in the above description of Section 29.2.

29.1 General concept of conformal measures In this section, we deal at length with the concept, properties and construction of general conformal measures in the sense of [34]. Our approach stems from and develops the one of [34]. As noted at the beginning of this chapter, the method for proving the existence of such measures, based on the Schauder–Tychonov fixed-point theorem is not applicable. Instead, we construct them as limit measures of a Patterson–Sullivan type of construction.

29.1.1 Motivation for and definition of general conformal measures Let T : X → X be a Borel measurable self-map of a completely metrizable space X. Assume that T is (at most) countable-to-one, i. e., X = ⋃ Xj , j∈I

where the sets Xj , j ∈ I, are Borel, pairwise disjoint and for each j ∈ I the restriction Tj := T|Xj : Xj → T(Xj ) is one-to-one. Per Theorem 4.5.4 in Srivastava [122], the sets T(Xj ), j ∈ I, are Borel as well. Now assume that m is a quasi-T-invariant Borel probability measure. Recall that 1

1

ℒm : L (m) 󳨀→ L (m)

denotes the transfer operator defined in Section 13.4 (T need not be open distance expanding for ℒm to be well-defined). As a reminder, for any non-negative function g : X → ℝ in L1 (m) we defined the finite Borel measure gm(A) := ∫ g dm,

∀A ∈ ℬ(X).

A

We observed that gm ≺≺ m and thereby gm ∘ T −1 ≺≺ m ∘ T −1 ≺≺ m. Using the Radon-Nikodym Theorem (Theorem A.1.18), the transfer operator ℒm could then be defined by

29.1 General concept of conformal measures � 1199

ℒm (g) :=

d(gm ∘ T −1 ) ∈ L1 (m). dm

For a general g ∈ L1 (m), define ℒm (g) := ℒm (g + )− ℒm (g − ). It is easy to see that (cf. (13.21)) ∫ ℒm (g) dm = ∫ g dm,

∀g ∈ L1 (m).

(29.1)

∀g ∈ L1 (m),

(29.2)

X

X

Defining the measure ℒ∗m (m) by the formula ℒm (m)(g) := m(ℒm (g)), ∗

equality (29.1) means that ℒm (m)(g) = m(g), ∗

∀g ∈ L1 (m).

(29.3)

This equality signifies that ℒm (m) = m.

(29.4)

∗

As the restrictions Tj : Xj → T(Xj ), j ∈ I, of T are bijective measurable maps, we further have that m ∘ Tj−1 ≺≺ m|T(Xj ) ,

∀j ∈ I,

that is, m is quasi-Tj -invariant for all j ∈ I. We can thus define the Jacobian of m with respect to T by J−1 m (T)

:

X

󳨀→

[0, ∞)

x

󳨃󳨀→

J−1 m (T)(x) :=

dm ∘ Tj−1 dm|T(Xj )

(T(x))

whenever x ∈ Xj .

By the change-of-variables formula (13.19), we obtain for any non-negative g ∈ L1 (m) and Borel set A ⊆ X that gm ∘ T −1 (A) = gm(⋃ Tj−1 (A)) j∈I

= ∑ gm(Tj−1 (A)) j∈I

∫ g dm

=∑ j∈I

Tj−1 (A)

−1 = ∑ ∫(g ∘ Tj−1 ) ⋅ (J−1 m (T) ∘ Tj ) dm j∈I A

1200 � 29 Sullivan’s conformal measures for rational functions −1 = ∫ ∑ g(Tj−1 (x)) J−1 m (T)(Tj (x)) dm(x). A j∈I

Since this is true for all Borel sets A ⊆ X, we deduce that d(gm ∘ T −1 ) −1 (x) = ∑ g(Tj−1 (x)) ⋅ J−1 ∑ g(y) J−1 m (T)(Tj (x)) = m (T)(y) dm j∈I y∈T −1 (x) for m-a.e. x ∈ X. Consequently, ℒm (g)(x) =

∑

y∈T −1 (x)

g(y)J−1 m (T)(y)

for m-a. e. x ∈ X.

(29.5)

For an arbitrary g ∈ L1 (m), the equality follows from the above and the fact that ℒm (g) = ℒm (g + ) − ℒm (g − ) by definition. Definition 29.1.1. A Borel set A ⊆ X is called a special set for T if T|A is injective. Remark 29.1.2. (a) Every Borel subset of a special set is special. (b) If A is a special set, then T(A) is a Borel set according to Theorem 4.5.4 in [122]. Assume additionally that m is nonsingular with respect to T, meaning that m(T((J−1 m (T)) (0))) = 0, or equivalently, m(T((Jm (T)) (∞))) = 0. −1

−1

−1 (T) yield If A ⊆ X is a special set, then (29.4), (29.2), (29.5) and the fact that Jm (T) = 1/Jm

∫ Jm (T) dm = ∫ 1A ⋅ Jm (T) dm = ∫ 1A ⋅ Jm (T) d ℒ∗m (m) = ∫ ℒm (1A ⋅ Jm (T)) dm A

X

=∫

X

∑

−1 X y∈T (x)

= ∫

∑

X

−1 1A (y)Jm (T)(y)Jm (T)(y) dm(x)

−1 T(A) y∈T (x)

1A (y) dm(x)

= ∫ 1 dm = m(T(A)).

(29.6)

T(A)

We single out this property by saying that the nonsingular measure m is Jm (T)-conformal. This prompts us to introduce the following notion. Definition 29.1.3. Let T : X → X be a Borel measurable self-map of a completely metrizable topological space X and let ψ : X → [0, ∞) be a Borel measurable function. A Borel probability measure m on X is said to be ψ-conformal for T : X → X if

29.1 General concept of conformal measures � 1201

m(T(A)) = ∫ ψ dm

(29.7)

A

for every special set A ⊆ X. Observation 29.1.4. If ψ > 0, then any ψ-conformal measure m is nonsingular with respect to the map T, and J−1 m (T) = 1/ψ. Given φ ∈ L1 (m), define (cf. Definition 13.5.1) ℒφ (g)(x) :=

∑

y∈T −1 (x)

g(y)eφ(y) ,

∀x ∈ X, ∀g ∈ L1 (m).

Observe that ℒm (g)(x) = ℒlog J−1 (T) (g)(x) m

for m-a. e. x ∈ X, ∀g ∈ L1 (m).

Notice that even if T and φ are continuous, the operator ℒφ need not map Cb (X) into Cb (X) in general. But there are natural hypotheses, such as T open and distance expanding, under which this is the case. Nevertheless, if we just assume that ℒφ (C(X)) ⊆ C(X), then the linear operator ℒφ : C(X) → C(X) is bounded, and so is its dual ℒ∗φ : C(X)∗ → C(X)∗ . The following result states that the conformality of a measure can be characterized by this measure being a fixed point (an eigenmeasure) of the dual of an appropriate transfer operator (cf. Lemma 13.6.13). Proposition 29.1.5. Let T : X → X be a countable-to-one open continuous self-map of a completely metrizable topological space X and let ψ : X → (0, ∞) be a positive continuous function. Then a Borel probability measure m is ψ-conformal if and only if ℒ− log ψ (m) = m. ∗

As we may encounter trouble with the operator ℒ∗ for the maps T that are not open, rather than looking for theorems that would assure the existence of fixed points of ℒ∗ , we shall provide another general method of construction of conformal measures. That method stems from the Patterson–Sullivan method [100] (see also [101]) in the context of Fuchsian groups, [126, 128, 130] for Kleinian groups and [127, 129] for rational functions.

29.1.2 Selected properties of general conformal measures Definition 29.1.6. Let T : X → X be a self-map of a topological space X. A point x ∈ X is said to be singular for T if at least one of the following two conditions is satisfied: (a) There is no open neighborhood U of x such that the restriction T|U is one-to-one.

1202 � 29 Sullivan’s conformal measures for rational functions (b) For every open neighborhood U of x, there exists an open set V ⊆ U whose image T(V ) is not open. The set of all singular points will be denoted by Sing(T) and will be referred to as the set of singular points of T or the singular set of T. The set of all points satisfying condition (a) will be denoted by Crit(T) and will be referred to as the set of critical points of T or the critical set of T. The set of all points satisfying condition (b) will be denoted by X∗ (T). First, let us record the following immediate facts. Observation 29.1.7. Let T : X → X be a self-map of a topological space X. Then: (a) Sing(T) = Crit(T) ∪ X∗ (T). (b) All three sets Crit(T), X∗ (T) and Sing(T) are closed. (c) X \ X∗ (T) is the largest (in the sense of inclusion) open subset of X over which the map T is open. It is easy to give examples of continuous maps T for which Crit(T) ∩ X∗ (T) ≠ 0. We will need the following, somewhat obvious, topological fact whose proof is provided for the sake of completeness and convenience of the reader. Lemma 29.1.8. Let T : X → X be a continuous self-map of a regular topological space X. For every x ∈ X\Sing(T), there exists an open neighborhood U(x) of x with the following properties: (a) U(x) ⊆ X\Sing(T); (b) The map T|U(x) is one-to-one; (c) The set T(U(x)) is open and the map T|U(x) : U(x) → X is open; (d) If G ⊆ U(x) is an open set, then 𝜕(T(G)) = T(𝜕G). Proof. As x ∈ X\Sing(T) and Sing(T) is closed, there exists W ⊆ X\Sing(T), an open neighborhood of x, such that the map T|W : W → X is one-to-one and open. In particular, T(W ) is an open neighborhood of T(x). Since the space X is regular, there exists an open neighborhood V of T(x) such that V ⊆ T(W ).

(29.8)

As T is continuous, there is an open neighborhood U ′ (x) of x such that U ′ (x) ⊆ W and T(U ′ (x)) ⊆ V .

(29.9)

Claim 1∘ . If F ⊆ U ′ (x) is a closed subset of X, then T(F) ⊆ X is also closed. Proof of Claim 1∘ . Since W \F ⊆ X is open and T|W is open, the set T(W \F) is open. As T|W is one-to-one, it is also clear that T(W \F) = T(W )\T(F). So, T(F) is a closed subset of T(W ). That is, there exists a closed subset H of X such that T(F) = H ∩ T(W ). By (29.9),

29.1 General concept of conformal measures

� 1203

we know that T(F) ⊆ T(U ′ (x)) ⊆ V ⊆ V . So, T(F) = H ∩ T(W ) ∩ V . By (29.8), we get T(F) = H ∩ V . Thus, T(F) is closed in X and the claim holds. ◼ By the regularity of X, there is an open neighborhood U(x) of x such that U(x) ⊆ U ′ (x).

(29.10)

As a fairly immediate consequence of Claim 1∘ , we obtain the following fact. Claim 2∘ . If A ⊆ U(x), then T(A) = T(A). Proof of Claim 2∘ . On one hand, the continuity of T guarantees that T(A) ⊆ T(A). On the other hand, the set T(A) is closed in X per Claim 1∘ and (29.10). As T(A) ⊆ T(A), we deduce that T(A) ⊆ T(A). ◼ To finish the proof of the lemma, let G ⊆ U(x) ⊆ U ′ (x) ⊆ W ⊆ X\Sing(T) be an open set. Then T(G) is also open and, using Claim 2∘ and the injectivity of T|W , we conclude that 𝜕(T(G)) = T(G)\T(G) = T(G)\T(G) = T(G\G) = T(𝜕G). Lemma 29.1.9. Let T : X → X be a continuous map of a Polish (i. e., completely metrizable and separable) topological space X, and let m be a Borel probability measure on X. Let Γ ⊆ X be a closed set containing Sing(T). If g : X → [0, ∞) is in L1 (m) and if (29.7) holds for all special sets A such that A ∩ Γ = 0 and m(𝜕A) = m(𝜕T(A)) = 0, then (29.7) continues to hold for all special sets A such that A ∩ Γ = 0. Proof. Fix a complete metric d compatible with the topology of X. Let A be a special set disjoint from Γ. Let ε > 0. Since A ⊆ X \Γ, since the set X \Γ is open and since the measure m is outer regular, there exists an open set A ⊆ V ⊆ X \ Γ such that ∫ g dm < ∫ g dm + ε. V

(29.11)

A

For every x ∈ V there exists r(x) > 0 such that B(x, 2r(x)) ⊆ V ∩ U(x),

(29.12)

where U(x) is the open neighborhood of x produced in Lemma 29.1.8. Since 𝜕B(x, r) ⊆ {y ∈ X : d(y, x) = r} for every r > 0 and since the sets {y ∈ X : d(y, x) = r}, r > 0, are mutually disjoint, the set R1 (x) := {r ∈ (0, r(x)] : m(𝜕B(x, r)) > 0} is countable. In addition, by Lemma 29.1.8(b,d) the sets 𝜕(T(B(x, r))), r ∈ (0, r(x)], are mutually disjoint, whence the set

1204 � 29 Sullivan’s conformal measures for rational functions R2 (x) := {r ∈ (0, r(x)] : m(𝜕(T(B(x, r)))) > 0} is countable. Thus, the set R1 (x) ∪ R2 (x) is countable and, therefore, there exists s(x) ∈ (0, r(x)] such that m(𝜕B(x, s(x))) = 0

and

m(𝜕(T(B(x, s(x))))) = 0.

(29.13)

Since {B(x, s(x)) : x ∈ V } is an open cover of V and since V is Lindelöf (as a separable metrizable space), there exists a countable set {xn }∞ n=1 ⊆ V such that ∞

⋃ B(xn , s(xn )) = V .

(29.14)

n=1

Define inductively a partition {Gn }∞ n=1 of V in the standard way: n

G1 := B(x1 , s(x1 )) and Gn+1 := B(xn+1 , s(xn+1 )) \ ⋃ B(xk , s(xk )), k=1

∀n ∈ ℕ.

As n

𝜕Gn ⊆ ⋃ 𝜕B(xk , s(xk )), k=1

∀n ∈ ℕ,

it follows from (29.13) that m(𝜕Gn ) = 0,

(29.15)

∀n ∈ ℕ.

Likewise, by Lemma 29.1.8(b), n−1

T(Gn ) = T(B(xn , s(xn ))) \ ⋃ T(B(xk , s(xk ))), k=1

∀n ∈ ℕ.

It ensues that n

𝜕(T(Gn )) ⊆ ⋃ 𝜕(T(B(xk , s(xk )))), k=1

∀n ∈ ℕ.

It follows from (29.13) that m(𝜕(T(Gn ))) = 0,

∀n ∈ ℕ.

(29.16)

Moreover, Gn ⊆ B(xn , r(xn )) for all n ∈ ℕ. By (29.12), Gn ∩ Γ = 0,

∀n ∈ ℕ.

(29.17)

29.1 General concept of conformal measures

� 1205

Finally, T|Gn is one-to-one since Gn ⊆ B(xn , r(xn )) ⊆ V ∩ U(xn ) ⊆ X \ Γ ⊆ X \ Sing(T). So, every Gn is a special set satisfying (29.15)–(29.17). From the hypothesis on such special sets, we know that m(T(Gn )) = ∫ g dm,

(29.18)

∀n ∈ ℕ.

Gn

By construction and (29.14), we also know that ⋃∞ n=1 Gn = V ⊇ A. From this, (29.11) and (29.18), we obtain that ∞

∞

∞

n=1

n=1

n=1 G n

m(T(A)) = m( ⋃ T(A ∩ Gn )) ≤ ∑ m(T(Gn )) = ∑ ∫ g dm = ∫ g dm < ∫ g dm + ε. V

A

Since ε > 0 is arbitrary, we infer that m(T(A)) ≤ ∫ g dm

(29.19)

A

for every special set A ⊆ X such that A ∩ Γ = 0. Using (29.19) on each Gn \A and (29.18), we get ∞

∞

n=1

n=1

m(T(A)) = m( ⋃ T(A ∩ Gn )) = ∑ m(T(A ∩ Gn )) ∞

= ∑ [m(T(Gn )) − m(T(Gn \A))] n=1 ∞

≥ ∑ [ ∫ g dm − ∫ g dm] n=1 G n

=

Gn \A

∫ g dm − ∪∞ n=1 Gn

g dm = ∫ g dm − ∫ g dm

∫ ∪∞ n=1 Gn \A

V

V \A

= ∫ g dm = ∫ g dm. V ∩A

A

Along with (29.19), this gives m(T(A)) = ∫ g dm. A

29.1.3 The limit construction and PS limit measures In this subsection, we provide a construction aimed at producing conformal measures. It relies on the following simple fact. For a real sequence (an )∞ n=1 , the number

1206 � 29 Sullivan’s conformal measures for rational functions c := lim sup n→∞

an n

(29.20)

is called the sequence’s transition parameter and is uniquely determined by the property that the series ∞

∑ exp(an − ns)

n=1

converges when s > c but diverges when s < c. (If s = c, the series may converge or not.) By a simple argument, one obtains the following. Lemma 29.1.10. Given a real sequence (an )∞ n=1 and its transition parameter c := lim sup n→∞

there exists a sequence (bn )∞ n=1 of positive reals such that ∞ < ∞ if s > c ∑ bn exp(an − ns) { = ∞ if s ≤ c n=1

and

lim

n→∞

an , n

bn+1 = 1. bn

∞

Moreover, ∑ bn exp(an − ns) depends continuously on s. n=1

∞ Proof. If ∑∞ n=1 exp(an − nc) = ∞, set bn = 1 for every n ∈ ℕ. If ∑n=1 exp(an − nc) < ∞, choose a sequence (nk )∞ k=1 of positive integers such that

εk :=

ank nk

k→∞

− c 󳨀→ 0

and

nk = 0. k→∞ nk+1 lim

Letting bn = exp(n[

n − nk−1 nk − n ε + ε ]) for every nk−1 ≤ n < nk , nk − nk−1 k−1 nk − nk−1 k

it is easy to check that the lemma follows. The demonstration of the continuity on s is left to the reader as an exercise. The current subsection’s actual setting is this: (a) T : X → X is a continuous self-map of a completely metrizable topological space X. (b) (En )∞ n=1 is a sequence of nonempty finite subsets of X such that T −1 (En ) ⊆ En+1 ,

∀n ∈ ℕ.

(29.21)

(c) φ : X → (−∞, ∞] is a bounded below continuous function such that ∞

φ( ⋃ En ) ⊆ ℝ. n=1

(29.22)

29.1 General concept of conformal measures � 1207

We adopt the standard convention that e−∞ = 0

and e∞ = ∞.

Under this convention, we have the following obvious fact. Observation 29.1.11. The function e−φ : X → [0, ∞) is bounded and continuous. Let an := log ∑ exp(Sn φ(x)) x∈En

where

n−1

Sn φ = ∑ φ ∘ T k . k=0

(29.23)

∞ Denote by c the transition parameter of the sequence (an )∞ n=1 . Choose a sequence (bn )n=1 of positive reals per Lemma 29.1.10. For every s > c, define ∞

Ms := ∑ bn exp(an − ns) n=1

(29.24)

and the normalized measure ms =

1 ∞ ∑ ∑ b exp(Sn φ(x) − ns)δx , Ms n=1 x∈E n

(29.25)

n

where δx denotes the Dirac δ-measure supported at the point x ∈ X. Let A ⊆ X be a special set. Using (29.21) and (29.25), it follows that ms (T(A)) =

1 ∞ ∑ Ms n=1

x∈En ∩T(A)

=

1 ∞ ∑ Ms n=1

y∈A∩T −1 (En )

=

1 ∞ ∑ Ms n=1

x∈A∩En+1

−

∑

∑

1 ∞ ∑ Ms n=1

∑

bn exp(Sn φ(x) − ns) bn exp(Sn φ(T(y)) − ns)

bn exp(Sn+1 φ(x) − (n + 1)s) exp(s − φ(x)) ∑

x∈A∩(En+1 \T −1 (En ))

bn exp(Sn φ(T(x)) − ns).

(29.26)

Set 󵄨󵄨 󵄨󵄨 1 ∞ ΔA (s) := 󵄨󵄨󵄨 ∑ 󵄨󵄨 Ms n=1 󵄨 Observe that

󵄨󵄨 󵄨󵄨 bn exp(Sn+1 φ(x) − (n + 1)s) exp(s − φ(x)) − ∫ exp(c − φ) dms 󵄨󵄨󵄨. 󵄨󵄨 x∈A∩En+1 󵄨 A ∑

1208 � 29 Sullivan’s conformal measures for rational functions 󵄨󵄨 󵄨󵄨 exp(Sn+1 φ(x) − (n + 1)s)es−φ(x) [bn − bn+1 ec−s ] − b1 ∑ ec−s 󵄨󵄨󵄨 󵄨󵄨 x∈A∩En+1 x∈A∩E1 󵄨 ∞ c−s 󵄨󵄨 b 󵄨󵄨 b e #E 1 󵄨 󵄨 1 ≤ es−φ(x) . ∑ ∑ 󵄨󵄨 n − ec−s 󵄨󵄨󵄨bn+1 exp(Sn+1 φ(x) − (n + 1)s) + 1 󵄨󵄨 Ms n=1 x∈E 󵄨󵄨󵄨 bn+1 Ms

ΔA (s) =

󵄨 1 󵄨󵄨󵄨󵄨 ∞ 󵄨∑ Ms 󵄨󵄨󵄨󵄨n=1

∑

n+1

By Lemma 29.1.10, we know that lim (bn+1 /bn ) = 1 and lim Ms = ∞. Therefore, n→∞

s↘c

lim ΔA (s) = 0

(29.27)

s↘c

uniformly over all special sets A. Definition 29.1.12. Any weak accumulation point, when s ↘ c, of the measures (ms )s>c , defined by (29.25), is called a PS limit measure (associated to the function φ and the sequence (En )∞ n=1 ). The letters PS are in honor of Patterson and Sullivan.

29.1.4 Conformality properties of PS limit measures In order to find conformal measures among the PS limit measures, it is necessary to examine (29.26) in greater detail. To begin with, for a Borel set B ⊆ X consider the following condition: lim s↘c

1 ∞ ∑ ∑ Ms n=1 x∈B∩(E \T −1 (E n+1

bn exp(Sn φ(T(x)) − ns) = 0.

(29.28)

n ))

As a direct consequence of (29.26)–(29.27), we get the following. Lemma 29.1.13. Let T : X → X be a continuous self-map of a completely metrizable topological space X. Let (En )∞ n=1 be a sequence of finite subsets of X such that (29.21) holds and let φ : X → (−∞, ∞] be a bounded below continuous function satisfying (29.22). If a Borel set B ⊆ X satisfies (29.28), then the sets of all accumulation points of the functions (c, ∞) ∋ s 󳨃󳨀→ ms (T(B)) and (c, ∞) ∋ s 󳨃󳨀→ ∫ exp(s − φ) dms B

as s ↘ c, coincide. More precisely, if (sn )∞ n=1 is a sequence of real numbers in (c, ∞) such that lim sn = c, then n→∞

lim msn (T(B)) exists

n→∞

⇐⇒

lim ∫ exp(sn − φ) dmsn exists

n→∞

B

and if any one of those limits exists, then they coincide.

29.1 General concept of conformal measures

� 1209

Lemma 29.1.14. Let T : X → X be a continuous self-map of a completely metrizable topological space X. Let (En )∞ n=1 be a sequence of finite subsets of X such that (29.21) holds and let φ : X → (−∞, ∞] be a bounded below continuous function satisfying (29.22). Let m be a PS limit measure (per Definition 29.1.12) and let Γ be a closed set containing Sing(T). Assume that every special set B ⊆ X such that B∩Γ = 0 and m(𝜕B) = m(𝜕T(B)) = 0 further satisfies condition (29.28). Then m(T(A)) = ∫ exp(c − φ) dm A

for every special set A such that A ∩ Γ = 0. Proof. It immediately follows from Lemma 29.1.13 and the boundedness and continuity of the function e−φ that m(T(B)) = ∫ exp(c − φ) dm B

for every special set B ⊆ X for which B ∩ Γ = 0 and m(𝜕B) = m(𝜕T(B)) = 0. Applying Lemma 29.1.9 completes the proof. Lemma 29.1.15. Let T : X → X be a continuous self-map of a completely metrizable topological space X. Let (En )∞ n=1 be a sequence of finite subsets of X such that (29.21) holds and let φ : X → (−∞, ∞] be a bounded below continuous function satisfying (29.22). Suppose that condition (29.28) holds for every special set B such that B∩X∗ (T) = 0. Suppose also that w ∈ X \ X∗ (T) is a point for which there exist finitely many mutually disjoint special sets W1 , W2 , . . . , Wq such that W1 ∪ W2 ∪ . . . ∪ Wq is an open neighborhood of w. If m is a PS limit measure, then m(T({w})) ≤ exp(c − φ(w))m({w}). Proof. By the same standard argument as the one yielding countability of the set R1 (x) in the proof of Lemma 29.1.9 and since w ∉ X∗ (T) and X∗ (T) is a closed subset of X, there exists a sequence (rn )∞ n=1 in (0, ∞) converging to 0 such that q

B(w, rn ) ⊆ ⋃ Wi \ X∗ (t) i=1

and

m(𝜕B(w, rn )) = 0,

∀n ∈ ℕ.

(29.29)

For every i ∈ {1, . . . , q} and n ∈ ℕ, let Wi (n) = Wi ∩ B(w, rn ). Then for each n ∈ ℕ the sets W1 (n), W2 (n), . . . , Wq (n) are mutually disjoint special sets whose union is

1210 � 29 Sullivan’s conformal measures for rational functions q

⋃ Wi (n) = B(w, rn ).

(29.30)

i=1

From the definition of m, there is a sequence (sk )∞ k=1 in (c, ∞) such that limk→∞ sk = c and limk→∞ msk = m. Fix n ∈ ℕ. Since the function e−φ : X → [0, ∞) is bounded and continuous, using the second property in (29.29) the Portmanteau theorem (Theorem A.1.23) affirms that lim

k→∞

∫ esk −φ dmsk =

B(w,rn )

∫ ec−φ dm.

(29.31)

B(w,rn )

As the sequence ∞

( ∫ esk −φ dmsk )

k=1

W1 (n)

∞ is bounded, there is a subsequence (sk(1) )∞ k=1 of the sequence (sk )k=1 such that the limit

∫ esk

(1)

lim

k→∞

−φ

dms(1) k

W1 (n)

exists. Proceeding by induction, suppose that for some 1 ≤ j < q a subsequence (sk )∞ k=1 of the sequence (sk )∞ k=1 is such that the limit (j)

lim ∫ esk

(j)

k→∞

−φ

dms(j) k

Wi (n)

exists for every i = 1, . . . , j. Since the sequence ( ∫ esk

(j)

∞

−φ

dms(j) ) k

Wj (n)

is bounded, there is a subsequence (sk

(j+1) ∞ )k=1

lim

k→∞

∫

k=1

of the sequence (sk )∞ k=1 such that the limit

esk

(j)

(j+1)

−φ

dms(j+1) k

Wj+1 (n)

exists. Then the limit lim ∫ esk

(q)

k→∞

Wj (n)

−φ

dms(q) k

29.2 Sullivan’s conformal measures

� 1211

exists for all j = 1, . . . , q. By virtue of the hypotheses and Lemma 29.1.13, for all j = 1, . . . , q the limit lim ms(q) (T(Wj (n)))

k→∞

k

also exists and lim ms(q) (T(Wj (n))) = lim ∫ esk

(q)

k→∞

k→∞

k

−φ

dms(q) .

(29.32)

k

Wj (n)

Recall that B(w, rn ) ⊆ X\X∗ (t) by the first property in (29.29). Thus, T(B(w, rn )) is open by Observation 29.1.7(c). Using successively the Portmanteau theorem, (29.30), (29.32) and (29.31), we deduce that m({T(w)}) ≤ m(T(B(w, rn ))) ≤ lim inf ms(q) (T(B(w, rn ))) k→∞

q

k

q

= lim inf ms(q) (T(⋃ Wj (n))) = lim inf ms(q) (⋃ T(Wj (n))) k→∞

k

k→∞

j=1

k

j=1

q

≤ lim inf ∑ ms(q) (T(Wj (n))) k→∞

k

j=1

q

q

= ∑ lim ms(q) (T(Wj (n))) = ∑ lim j=1

k→∞

k

q

j=1

= lim ∑ ∫ esk k→∞

=

(q)

−φ

dms(q) = lim k

j=1 W (n) j

k→∞

k→∞

∫ esk

(q)

−φ

∫ esk

(q)

B(w,rn )

dms(q) k

Wj (n) −φ

dms(q) k

∫ ec−φ dm. B(w,rn )

Hence, m({T(w)}) ≤ lim

n→∞

∫ ec−φ dm = ec−φ(w) m({w}). B(w,rn )

The proof of Lemma 29.1.15 is complete.

29.2 Sullivan’s conformal measures This section is devoted to a comprehensive study of more special, though very important, conformal measures called Sullivan conformal measures. An extended historical and motivational discussion of these measures as well as more general ones was given at

1212 � 29 Sullivan’s conformal measures for rational functions the outset of this chapter and in the previous Section 29.1. We accordingly dive right into the actual mathematics. ̂ → ℂ ̂ is a rational function with deg(T) ≥ 2. We describe a In this section, T : ℂ construction whose objective is to establish the existence of measures that are called Sullivan t-conformal (or just t-conformal) measures. In fact, we will prove the existence of measures with slightly weaker properties. Then we will prove the existence of genuine Sullivan t-conformal measures and we will provide several characterizations of the minimal exponent for which such measures exist. Fix ξ ∈ 𝒥 (T) \ PC−∞ (T)

where

∞

PC−∞ (T) := ⋃ T −n (PC(T)) n=0

and set En := T −(n−1) (ξ),

∀n ∈ ℕ.

En+1 = T −1 (En ),

∀n ∈ ℕ.

Then

Therefore the sequence (En )∞ n=1 satisfies (29.21) and (29.28) with all Borel sets B ⊆ 𝒥 (T). Fix an arbitrary t ≥ 0 and observe that the function φt := −t log |T ′ | : 𝒥 (T) → (−∞, ∞] is continuous, bounded below, satisfies (29.22), and the function e−φt = |T ′ |t is continuous. Let cξ (t) be the transition parameter associated to the sequence (En )∞ n=1 and the function φt according to (29.20) and (29.23), i. e., aξ,n (t) = log

󵄨󵄨 n−1 ′ 󵄨󵄨−t 󵄨󵄨(T ) (x)󵄨󵄨

∑ x∈T −(n−1) (ξ)

(29.33)

and cξ (t) = lim sup n→∞

aξ,n (t) n

.

(29.34)

Because the map T : 𝒥 (T) → 𝒥 (T) is open, it turns out that 𝒥∗ (T) = 0. Moreover, |T ′ (c)| = 0 precisely when c ∈ Crit(T). Then, as a fairly immediate consequence of Lemmas 29.1.14–29.1.15, we get the following. ̂→ℂ ̂ is a rational function with deg(T) ≥ 2, then for every t ≥ 0 Lemma 29.2.1. If T : ℂ there exists a PS limit measure mt on 𝒥 (T) such that

29.2 Sullivan’s conformal measures

� 1213

󵄨 󵄨t mt (T(A)) = ∫ ecξ (t) 󵄨󵄨󵄨T ′ 󵄨󵄨󵄨 dmt A

for every special set A ⊆ 𝒥 (T). Proof. Fix t ≥ 0 and any PS limit measure mt per Definition 29.1.12, i. e. any weak accumulation point of the measures (ms )s>cξ (t) when s ↘ cξ (t), where ms is defined by (29.25). Recall that 𝒥∗ (T) = 0, as the map T|𝒥 (T) is open. It follows from Lemma 29.1.14 that 󵄨 󵄨t mt (T(A)) = ∫ ecξ (t) 󵄨󵄨󵄨T ′ 󵄨󵄨󵄨 dmt

(29.35)

A

for every special set A ⊆ 𝒥 (T) such that A ∩ Crit(T) = 0. But as an aftermath of Theorem 25.1.1, the hypotheses of Lemma 29.1.15 are satisfied for every point c ∈ Crit(T). For every such c, it ensues from that lemma that 󵄨 󵄨t 0 ≤ mt ({T(c)}) ≤ ecξ (t) 󵄨󵄨󵄨T ′ (c)󵄨󵄨󵄨 mt ({c}) = 0. Hence, mt ({T(c)}) = 0 for all c ∈ Crit(T). We conclude from this and (29.35) that this latter in fact holds for all special sets A ⊆ 𝒥 (T). Write mt for mt . ̂→ℂ ̂ be a rational function. A point z ∈ 𝒥 (T) is said to be Definition 29.2.2. Let T : ℂ safe if z ∉ PC(T)

and

lim inf n→∞

1 log dist(z, T n (Crit(T))) ≥ 0. n

(29.36)

Denote the set of all safe points of T, also called the safe set of T, by Safe(T). This notion first appeared in [110] (Definition 12.5.7). The simple proof of the next lemma stems from that of Proposition 12.5.10 in [110]. This result states that the set of “nonsafe” Julia points has 0 for Hausdorff dimension. ̂→ℂ ̂ is a rational function, then Lemma 29.2.3. If T : ℂ HD(𝒥 (T)\Safe(T)) = 0. Proof. Fix ε > 0 and k ∈ ℕ. Set NS(T; ε, k) := {z ∈ 𝒥 (T) :

1 log dist(z, T n (Crit(T))) < −ε, ∀n ≥ k}. n

In other terms, z ∈ NS(T; ε, k) if and only if dist(z, T n (Crit(T))) < e−εn , So

∀n ≥ k.

1214 � 29 Sullivan’s conformal measures for rational functions ∞

NS(T; ε, k) ⊆ ⋂ B(T n (Crit(T)), e−εn ). n=k

Fix any t > 0. For every n ≥ k, the family {B(T n (c), e−εn ) : c ∈ Crit(T)} is a finite (2e−εn )-cover of NS(T; ε, k). Therefore, t

0 ≤ Ht2e (NS(T; ε, k)) ≤ #Crit(T) ⋅ (2e−εn ) . −εn

Letting n → ∞, we deduce that Ht (NS(T; ε, k)) = 0. As t > 0 is arbitrary, we conclude that HD(NS(T; ε, k)) = 0. Hence, HD(NS(T, ε)) = 0

where

∞

NS(T, ε) := ⋃ NS(T; ε, k). k=1

(29.37)

Since ∞

𝒥 (T)\Safe(T) = PC(T) ∪ ⋃ NS(T, 1/j) j=1

and since the set PC(T) is countable, by invoking (29.37) we conclude that HD(𝒥 (T)\Safe(T)) = 0. We now want to establish properties of the function cξ (t), and especially get a parameter t ≥ 0 such that cξ (t) = 0. We start with the following lemma. Its proof stems from that of Theorem 12.5.11 in [110], which in turn originates from an appropriate proof in [109]. In this latter paper, a tally of variants of the topological pressure for potentials of the form −t log |T ′ | is developed, and all such variants are shown to be equal. This is partly reproduced in the last chapter of the book [110]. We would love to present this beautiful theory in full but it would far exceed the space allotted to this volume. ̂→ℂ ̂ is a rational function with deg(T) ≥ 2, then for every point Lemma 29.2.4. If T : ℂ ξ ∈ Safe(T) ∩ 𝒥uer (T) and every t ≥ 0 we have that cξ (t) ≤ P∗∗ exp (t)

29.2 Sullivan’s conformal measures

� 1215

where P∗∗ exp (t) was introduced in Definition 25.3.11. Proof. By virtue of Proposition 25.3.12, it suffices to show that cξ (t) ≤ P∗exp (t). Further note that the set Safe(T) ∩ 𝒥uer (T) is nonempty because of Theorem 28.5.3 and Lemma 29.2.3. Since ξ ∈ 𝒥uer (T), there exists η > 0 such that ξ ∈ 𝒥uer (T)(η). Fix any α ∈ (0, 1). Let λ > 1 be a constant witnessing the radial uniform expandingness of T at the point ξ (see Definition 28.5.1). Since λ > 1, α > 0 and ξ ∈ Safe(T), it follows from (29.36) that 2n

B(ξ, 2λ−αn ) ∩ ⋃ T i (Crit(T)) = 0 i=1

for all n ∈ ℕ large enough, say n ≥ N1 . According to Corollary 22.5.6, for every 1 ≤ j ≤ 2n −j ̂ and every x ∈ T −j (ξ) there exists a unique holomorphic branch Tx : B(ξ, 2λ−αn ) → ℂ −j of T sending ξ to x. By Theorem 23.1.9 (Koebe’s distortion theorem, spherical version), we have for every r ∈ (0, 1/2] and every 1 ≤ j ≤ 2n that diam(Tx (B(ξ, rλ−αn ))) −j

diam(Tx (B(ξ, λ−αn ))) −j

≤ K2

diam(B(ξ, rλ−αn )) diam(B(ξ, λ−αn ))

= K 2 r.

−j ̂ < ∞, there exists ε ∈ (0, 1/2] such that As diam(Tx (B(ξ, λ−αn ))) ≤ diam(ℂ)

Tx−j (B(ξ, ελ−αn )) ⊆ B(x,

η ), 16K

∀x ∈ T −j (ξ), ∀1 ≤ j ≤ 2n.

(29.38)

Since ξ ∈ 𝒥uer (T)(η), there exists N2 ≥ N1 such that for every n ≥ N2 there exists a unique ̂ of T −2[αn] sending T 2[αn] (ξ) to ξ and holomorphic branch Tξ−2[αn] : B(T 2[αn] (ξ), 2η) → ℂ ξ 󵄨󵄨 2[αn] ′ 󵄨󵄨 ) (ξ)󵄨󵄨 ≥ λ2[αn] . With the help of Lemma 23.1.11 and enlarging N2 if needed, we get 󵄨󵄨(T Tξ−2[αn] (B(T 2[αn] (ξ), 2η)) ⊆ B(ξ, ελ−αn ),

∀n ≥ N2 .

(29.39)

As the map T : 𝒥 (T) → 𝒥 (T) is topologically exact per Corollary 24.3.5, there exists m(η) ≥ 0 such that T m(η) (B(y,

η )) ⊇ 𝒥 (T), 16K

∀y ∈ 𝒥 (T).

Consequently, for every n ∈ ℕ and every z ∈ T −n (ξ) ⊆ 𝒥 (T) there exists a point z′ ∈ T −m(η) (z) ∩ B(T 2[αn] (ξ),

η ). 16K

(29.40)

1216 � 29 Sullivan’s conformal measures for rational functions Assuming now that n ≥ max{N2 , m(η)}, we have 2n ≥ m(η) + n, and hence there exists a −(m(η)+n) ̂ of T −(m(η)+n) sending ξ to z′ . It unique holomorphic branch Tz′ : B(ξ, 2λ−αn ) → ℂ then follows from (29.38) and (29.40) that Tz′

−(m(η)+n)

(B(ξ, ελ−αn )) ⊆ B(z′ ,

η η ) ⊆ B(T 2[αn] (ξ), ). 16K 8K

Looking up at (29.39), we see that the composition φ(n) := Tz′ z′

−(m(η)+n)

∘ Tξ−2[αn] : B(T 2[αn] (ξ), 2η) → B(T 2[αn] (ξ),

η ) 8K

(29.41)

is well-defined and holomorphic. It ensues from this and Koebe’s 41 -theorem that B(T 2[αn] (ξ),

η 1󵄨 ′ 2[αn] 󵄨 ) ⊇ φ(n) (B(T 2[αn] (ξ), η)) ⊇ B(T 2[αn] (ξ), 󵄨󵄨󵄨(φ(n) (ξ))󵄨󵄨󵄨η). ′ ) (T z′ 8K 4 z

Hence, η 1 󵄨󵄨 (n) ′ 2[αn] 󵄨 (ξ))󵄨󵄨󵄨η ≤ . 󵄨󵄨(φz′ ) (T 4 8K Equivalently, 1 󵄨󵄨 (n) ′ 2[αn] 󵄨 (ξ))󵄨󵄨󵄨 ≤ . 󵄨󵄨(φz′ ) (T 2K Theorem 23.1.9 (Koebe’s distortion theorem, spherical version) then implies that 󵄨󵄨 (n) ′ 󵄨󵄨 1 󵄨󵄨(φz′ ) (w)󵄨󵄨 ≤ , 2

∀w ∈ B(T 2[αn] (ξ), η).

(29.42)

Let V = {v} be a vertex set and E = T −n (ξ) serve as an alphabet. Let A : E × E → {0, 1} be the constant function 1, so that EA∞ = E ∞ = T −n (ξ)ℕ . Let X = Xv = B(T 2[αn] (ξ), η/4) and W = Wv = B(T 2[αn] (ξ), η/2). By construction, φ(n) (B(T 2[αn] (ξ), η/2))∩φ(n) (B(T 2[αn] (ξ), η/2)) = 0, z′ w′

∀z′ , w′ ∈ T −n (ξ), z′ ≠ w′ . (29.43)

By (29.41)–(29.43), the collection Φn := {V , E, A; X, W ; {φ(n) } } z′ z∈T −n (ξ) forms a finite conformal IFS (a CIFS) that satisfies the strong separation condition (SSC; see Definition 19.7.6). Per Theorem 19.7.8, its limit set 𝒥n is a topological Cantor set and T m(η)+n+2[αn] (𝒥n ) ⊆ 𝒥n .

(29.44)

29.2 Sullivan’s conformal measures

� 1217

(Note: As stated, Theorem 19.7.8 is not entirely true; more precisely, J might not be perfect. However, the perfectness of J does occur when A is irreducible and J is infinite. If A is primitive and J has at least two points, then J is infinite and (according to the previous sentence) perfect; this is the case for the CIFS Φn . A weaker condition than the infiniteness of J, similar to that of the non single periodic orbit given in Proposition 28.5.4(d,e), exists and we leave it to the reader as an exercise to identify it.) Along with (29.42), this shows the following. Claim 1. The set 𝒥n is invariant and expanding for the map Fn := T m(η)+n+2[αn] . Fix an arbitrary t ≥ 0. As Fn : 𝒥n → 𝒥n is expansive, Theorem 11.1.26 guarantees that P(Fn , −t log |Fn′ |) = P(Fn , −t log |Fn′ |, 𝒰 ) for any finite open partition 𝒰 of 𝒥n with diam(𝒰 ) ≤ δ, where δ is any expansive constant for Fn . Let (n)

𝒰 = {φz′ (B(T

2[αn]

(ξ), η/2)) ∩ 𝒥n : z ∈ T −n (ξ)}.

By (29.43) (the OSC) and since 𝒥n ⊆ ⋃z∈T −n (ξ) φ(n) (B(T 2[αn] (ξ), η/2)), it is obvious that 𝒰 is z′ a finite open partition of 𝒥n . However, it is unclear whether diam(𝒰 ) ≤ δ. Nevertheless, thanks to (29.42) observe that diam(𝒰 k ) ≤ 2−(k−1) diam(𝒰 ) for all k ∈ ℕ. Thus there exists ℓ ∈ ℕ such that diam(𝒰 ℓ ) ≤ δ. Lemma 11.1.15 affirms that P(Fn , −t log |Fn′ |, 𝒰 ℓ ) = P(Fn , −t log |Fn′ |, 𝒰 ). So it suffices to compute P(Fn , −t log |Fn′ |, 𝒰 ). By Definition 11.1.11, the topological pressure of the potential −t log |Fn′ | with respect to the open cover 𝒰 of 𝒥n is 1 log Zk (Fn , −t log |Fn′ |, 𝒰 ), k→∞ k

P(Fn , −t log |Fn′ |, 𝒰 ) = lim

where, per Definition 11.1.2, the k-th partition function of 𝒰 is Zk (Fn , −t log |Fn′ |, 𝒰 ) = inf{ ∑ exp(S k (−t log |Fn′ |)(U ′ )) : 𝒰 ′ is a subcover of 𝒰 k }. U ′ ∈𝒰 ′

As 𝒰 is a partition of 𝒥n , so is 𝒰 k , and thus Zk (Fn , −t log |Fn′ |, 𝒰 ) reduces to Zk (Fn , −t log |Fn′ |, 𝒰 ) = ∑ exp(S k (−t log |Fn′ |)(U)) U∈𝒰 k

= ∑ exp(S k (−t log |Fn′ |)(φω (B(T 2[αn] (ξ), η/2)) ∩ 𝒥n )) ω∈E k

󵄨 󵄨−t = ∑ sup{󵄨󵄨󵄨(Fnk )′ (x)󵄨󵄨󵄨 : x ∈ φω (B(T 2[αn] (ξ), η/2)) ∩ 𝒥n } ω∈E k

󵄩 󵄩t ≥ K −t ∑ 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩𝒥 n ω∈E k

󵄩 󵄩t k ≥ K −(k+1)t [∑ 󵄩󵄩󵄩φ′e 󵄩󵄩󵄩𝒥 ] , e∈E

n

1218 � 29 Sullivan’s conformal measures for rational functions where K is the Koebe constant corresponding to the scale 1/2. (This argument above is inspired from the proof of Theorem 16.3.4. Alternatively, use Lemma 19.4.1 to get the same expression.) Consequently, P(Fn , −t log |Fn′ |) = P(Fn , −t log |Fn′ |, 𝒰 )

1 󵄩 󵄩t k log(K −(k+1)t [∑ 󵄩󵄩󵄩φ′e 󵄩󵄩󵄩𝒥 ] ) n k→∞ k e∈E

≥ lim

󵄩 󵄩t = − t log K + log ∑ 󵄩󵄩󵄩φ′e 󵄩󵄩󵄩𝒥 e∈E

≥ − t log K + log = − t log K + log ≥ − t log K + log

n

′ 󵄨 󵄨t ) (T 2[αn] (ξ))󵄨󵄨󵄨 ∑ 󵄨󵄨󵄨(φ(n) z′

z∈T −n (ξ)

′ ′ ′ 󵄨 󵄨−t 󵄨 󵄨−t 󵄨 󵄨−t ∑ 󵄨󵄨󵄨(T 2[αn] ) (ξ)󵄨󵄨󵄨 󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 󵄨󵄨󵄨(T m(η) ) (z′ )󵄨󵄨󵄨

z∈T −n (ξ)

′ 󵄨 󵄨−t ∑ 󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 ‖T ′ ‖−t(m(η)+2[αn]) ∞

z∈T −n (ξ)

= − t log K − t(m(η) + 2[αn]) log ‖T ′ ‖∞ + log

′ 󵄨−t 󵄨 ∑ 󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 .

z∈T −n (ξ)

(29.45)

Setting ∗

𝒥n :=

m(η)+n+2[αn]−1

⋃ i=0

T i (𝒥n ),

Claim 1 and (29.44) yield the following. Claim 2. The set 𝒥n∗ is invariant and expanding for the map T. Using the continuity of the iterates T i , i = 0, ..., m(η) + n + 2[αn] − 1, it directly follows from the construction of the sets 𝒥n and 𝒥n∗ that the map T : 𝒥n∗ → 𝒥n∗ is transitive. Recall from Proposition 11.1.26 that P(T k , Sk φ) = kP(T, φ). It then follows from (29.45) that 1 ′󵄨 󵄨 P(T m(η)+n+2[αn] |𝒥n∗ , −t log󵄨󵄨󵄨(T m(η)+n+2[αn] ) 󵄨󵄨󵄨) m(η) + n + 2[αn] 1 P(Fn , −t log |Fn′ |) ≥ m(η) + n + 2[αn] 󵄨 󵄨−t −t log K − t(m(η) + 2[αn]) log ‖T ′ ‖∞ + log ∑z∈T −n (ξ) 󵄨󵄨󵄨(T n )′ (z)󵄨󵄨󵄨 ≥ . m(η) + n + 2[αn]

P(T|𝒥n∗ , −t log |T ′ |) =

Thus, for every α ∈ (0, 1) and every n ≥ max{m(η), N2 } we obtain that P∗exp (t) ≥ P(T|𝒥n∗ , −t log |T ′ |)

29.2 Sullivan’s conformal measures

≥

� 1219

󵄨−t 󵄨 1 log ∑z∈T −n (ξ) 󵄨󵄨󵄨(T n )′ (z)󵄨󵄨󵄨 n . 2 [αn] n

K − t( m(η) + 2 [αn] ) log ‖T ′ ‖∞ + − t log n n n m(η) n

+1+

Taking the lim sup as n → ∞, we deduce that P∗exp (t) ≥

−2tα log ‖T ′ ‖∞ + cξ (t) 1 + 2α

.

Finally, letting α ↘ 0, we conclude that P∗exp (t) ≥ cξ (t) and the proof of Lemma 29.2.4 is complete. Throughout the rest of this section, we keep ξ ∈ Safe(T)∩𝒥uer (T), as in Lemma 29.2.4. The standard convexity arguments showing continuity of the topological pressure also lead to the following result. Lemma 29.2.5. The function [0, ∞) ∋ t 󳨃󳨀→ cξ (t) ∈ ℝ is convex and thereby continuous. Set sξ (T) := inf{t ≥ 0 : cξ (t) ≤ 0}.

(29.46)

We now introduce new dimensions for the Julia set. ̂ → ℂ ̂ be a rational function. The dynamical dimension of Definition 29.2.6. Let T : ℂ 𝒥 (T) is the nonnegative number DD(𝒥 (T)) := sup HD(μ), μ

where the supremum is taken over all ergodic T-invariant Borel probability measures μ on 𝒥 (T) such that hμ (T) > 0. ̂ → ℂ ̂ be a rational function. The expanding dimension of Definition 29.2.7. Let T : ℂ 𝒥 (T) is the nonnegative number ExpD(𝒥 (T)) := sup HD(X), X

where the supremum is taken over all forward T-invariant closed subsets X of 𝒥 (T) for which the map T|X : X → X is expanding. If, additionally, the map T|X : X → X is required to be transitive, then the resulting supremum is called the strict expanding dimension of 𝒥 (T) and is denoted by ExpD∗ (𝒥 (T)). If, on top of this, T|X : X → X is required to be open, then the resulting supremum is called the strictest expanding dimension of 𝒥 (T) and is denoted by

1220 � 29 Sullivan’s conformal measures for rational functions ExpD∗∗ (𝒥 (T)). We now describe relations between the different types of dimension of the Julia set. ̂→ℂ ̂ is a rational function with deg(T) ≥ 2, then Lemma 29.2.8. If T : ℂ sξ (T) ≤ ExpD∗∗ (𝒥 (T)) ≤ ExpD∗ (𝒥 (T)) ≤ ExpD(𝒥 (T)) ≤ DD(𝒥 (T)) ≤ HD(𝒥 (T)) ≤ 2. Proof. All but the first and fourth inequalities are obvious. To demonstrate the first one, suppose on the contrary that ExpD∗∗ (𝒥 (T)) < sξ (T). Take ExpD∗∗ (𝒥 (T)) < t < sξ (T). From this choice, Lemma 29.2.4 and Proposition 25.3.12, we have that 0 < cξ (t) ≤ P∗∗ exp (t).

(29.47)

So, using Theorem 25.3.9, we deduce that there exists a forward T-invariant closed set X ⊆ 𝒥 (T) and an open set U ⊇ X such that: (1) T|X : X → X is a topologically transitive, open, expanding dynamical system; (2) the triple (X, U, T) is a conformal expanding repeller; and (3) 1 P(T|X , −t log |T ′ |) ≥ cξ (t) > 0. 2 It follows from Theorem 16.3.2 and Proposition 16.3.1 that t ≤ HD(X) ≤ ExpD∗∗ (𝒥 (T)), which contradicts the assumption that ExpD∗∗ (𝒥 (T)) < t. This finishes the proof of the first inequality. To establish the fourth inequality, first notice that it clearly holds if ExpD(𝒥 (T)) = 0. So, suppose that ExpD(𝒥 (T)) > 0. Pick any 0 ≤ t < ExpD(𝒥 (T)). Then there exists an expanding forward T-invariant closed subset Y of 𝒥 (T) such that HD(Y ) > t. But by Proposition 28.5.4 and Theorem 28.5.5, P(T|Y , −t log |T ′ |) > 0. Therefore, by the variational principle (Theorem 12.1.1), there exists an ergodic T-invariant Borel probability measure μ on Y such that hμ (T) − tχμ (T) > 0.

(29.48)

Using Przytycki’s theorem (Theorem 28.1.3; χμ (T) ≥ 0), we deduce that hμ (T) > 0. Applying Ruelle’s inequality (Theorem 28.2.1; hμ (T) ≤ 2χμ (T)), we infer that χμ (T) > 0. Hence, it follows from the volume lemma (Theorem 28.4.1) and (29.48) that t
0. It follows from Ruelle’s inequality (Theorem 28.2.1) that χμ (T) > 0. So, we will be able to apply Theorem 28.6.2 later. Moreover, HD(μ) > 0 according to the volume lemma (Theorem 28.4.1). For every integer n ≥ 0, define on 𝒥 (T) the continuous function φn := max{−n, log |T ′ |}.

1222 � 29 Sullivan’s conformal measures for rational functions Then φn ≥ log |T ′ | and φn ↘ log |T ′ | pointwise on 𝒥 (T). In addition, φn ≤ log ‖T ′ ‖∞ . The Lebesgue monotone convergence theorem then affirms that lim ∫ φn dμ = ∫ log |T ′ | dμ =: χμ (T) > 0.

n→∞

𝒥 (T)

𝒥 (T)

Fix 0 < ε < 1. Then there is nε ∈ ℕ such that 0 < ∫ φn dμ ≤ 𝒥 (T)

χμ (T) 1−ε

,

∀n ≥ nε .

(29.51)

Along with the volume lemma (Theorem 28.4.1), this implies that hμ (T) = HD(μ) χμ (T) ≥ (1 − ε)HD(μ) ∫ φn dμ.

(29.52)

𝒥 (T)

Let (Xk (ε))∞ k=1 be the sequence of expanding forward T-invariant closed sets produced in Theorem 28.6.2 for the measure μ and the function φ = −HD(μ)φnε . For every k ∈ ℕ, Theorem 12.2.5 guarantees that there is an ergodic equilibrium state με,k for the expansive map T|Xk (ε) : Xk (ε) → Xk (ε) and the potential −HD(μ)φnε restricted to Xk (ε). It follows from the second part of Theorem 28.6.2 that the measures (με,k )∞ k=1 converge to μ in the weak∗ topology on M(𝒥 (T)). Using this and (29.51), we deduce that lim ∫ φnε dμε,k = ∫ φnε dμ > 0.

k→∞

𝒥 (T)

(29.53)

𝒥 (T)

By the variational principle (Theorem 12.1.1), Theorem 28.6.2 and (29.52), we obtain that lim inf[hμε,k (T) − HD(μ) ∫ φnε dμε,k ] = lim inf P(T|Xk (ε) , −HD(μ)φnε ) k→∞

𝒥 (T)

k→∞

≥ hμ (T) − HD(μ) ∫ φnε dμ 𝒥 (T)

≥ −εHD(μ) ∫ φnε dμ. 𝒥 (T)

Using this, (29.51)–(29.53) and the definition of φn , we get for all k ∈ ℕ large enough that hμε,k (T) ≥ HD(μ) ∫ φnε dμε,k − 2εHD(μ) ∫ φnε dμ 𝒥 (T)

𝒥 (T)

≥ HD(μ) ∫ φnε dμε,k − 3εHD(μ) ∫ φnε dμε,k 𝒥 (T)

𝒥 (T)

29.2 Sullivan’s conformal measures

� 1223

= (1 − 3ε)HD(μ) ∫ φnε dμε,k 𝒥 (T)

≥ (1 − 3ε)HD(μ) ∫ log |T ′ | dμε,k = (1 − 3ε)HD(μ)χμε,k (T). 𝒥 (T)

Invoking the volume lemma (Theorem 28.4.1) once again, we infer that ExpD∗∗ (𝒥 (T)) ≥ HD(Xk (ε)) ≥ HD(με,k ) =

hμε,k (T) χμε,k (T)

≥ (1 − 3ε)HD(μ).

Letting ε → 0, we conclude that ExpD∗∗ (𝒥 (T)) ≥ HD(μ). Taking the supremum over all ergodic T-invariant Borel probability measures μ on 𝒥 (T) for which hμ (T) > 0, we get the first inequality of the current lemma. We now prove the last inequality. The main idea is to get to a large scale. Fix t ≥ 0 and assume that a t-conformal measure m exists for T. Fix η > 0 and consider any point x ∈ 𝒥r (T)(η). (See Definition 28.5.1 for a recollection of notation and meaning.) For every ̂ of n ∈ Nx (T), there exists a unique holomorphic inverse branch Tx−n : B(T n (x), 2η) → ℂ n −n n T such that Tx (T (x)) = x. It follows from Lemma 23.1.11 that Tx−n (B(T n (x), η)) ⊆ B(x, K|(T n )′ (x)|−1 η).

(29.54)

rn (x) = K|(T n )′ (x)|−1 η.

(29.55)

Set

By Theorem 28.3.3 and the t-conformality of the measure m, we get m(B(T n (x), η)) = m(T n (Tx−n (B(T n (x), η)))) 󵄨󵄨󵄨(T n )′ 󵄨󵄨󵄨t dm = ∫ 󵄨 󵄨 Tx−n (B(T n (x),η))

≤

∫ B(x,rn (x))

󵄨󵄨 n ′ 󵄨󵄨t 󵄨󵄨(T ) 󵄨󵄨 dm

by (29.54)–(29.55)

󵄨 󵄨t ≤ K t 󵄨󵄨󵄨(T n )′ (x)󵄨󵄨󵄨 m(B(x, rn (x))). Hence, 󵄨 󵄨−t m(B(x, rn (x))) ≥ K −t 󵄨󵄨󵄨(T n )′ (x)󵄨󵄨󵄨 m(B(T n (x), η)) ≥ Mm (η)−1 K −2t η−t (rn (x))t , where Mm (η) := inf{m(B(y, η)) : y ∈ 𝒥 (T)} > 0

1224 � 29 Sullivan’s conformal measures for rational functions since supp(m) = 𝒥 (T). As #Nx (T) = ∞, we deduce that lim sup r→0

m(B(x, rn (x))) m(B(x, r)) −1 ≥ lim sup ≥ (Mm (η)K 2t ηt ) . t rt (r (x)) Nx (T)∋n→∞ n

It follows from Theorem 15.5.3(a) (Frostman converse theorem) that Ht (𝒥r (T)(η)) ≤ 10t Mm (η)K 2t ηt < ∞. Hence, HD(𝒥r (T)(η)) ≤ t. As ∞

𝒥r (T) = ⋃ 𝒥r (T)(1/k), k=1

we conclude from Theorem 15.3.5 (σ–stability of HD) that HD(𝒥r (T)) ≤ t. Since t ≥ 0 is arbitrary as long as a t-conformal measure exists, we thus get that HD(𝒥r (T)) ≤ δ(T). As a direct consequence of this lemma, Lemma 29.2.8 and Theorem 29.2.9, we obtain the main result of this section. ̂→ℂ ̂ is a rational function with deg(T) ≥ 2, then Theorem 29.2.12. If T : ℂ δ(T) = ExpD∗∗ (𝒥 (T)) = ExpD∗ (𝒥 (T)) = ExpD(𝒥 (T)) = DD(𝒥 (T)) = HD(𝒥uer (T)) = HD(𝒥er (T)) = HD(𝒥r (T)) = sξ (T)

≤ HD(𝒥 (T)) ≤ 2

for every ξ ∈ Safe(T) ∩ 𝒥uer (T), and T admits a Sullivan δ(T)-conformal measure. We end this section with two results shedding more light on the nature of the objects we have dealt with in this section. In [20], Xavier Buff and Arnaud Chéritat showed the existence of quadratic polynomials whose Julia sets have positive planar Lebesgue measure (but of course are nowhere dense). The situation is quite different for radial Julia sets 𝒥r (T). We prove the following fact related to Corollary 24.3.6. ̂→ℂ ̂ is a rational function of deg(T) ≥ 2 and 𝒥 (T) ≠ ℂ, ̂ then Theorem 29.2.13. If T : ℂ Leb2 (𝒥r (T)) = 0. Proof. By a Möbius change of coordinates, we may assume without loss of generality ̂ it is a nowhere dense subset of ℂ according to that 𝒥 (T) ⊆ ℂ. Fix l ∈ ℕ. Since 𝒥 (T) ≠ ℂ, Corollary 24.3.6. Then there exists ε > 0 such that for every y ∈ ℂ there is yε ∈ B(y, (2l)−1 ) for which B(yε , ε) ⊆ B(y, (2l)−1 ) \ 𝒥 (T).

(29.56)

29.2 Sullivan’s conformal measures

� 1225

Fix an arbitrary point z ∈ 𝒥r (T)(1/l). For every n ∈ Nz (T), there exists a unique holô of T n such that Tz−n (T n (z)) = z. On morphic inverse branch Tz−n : B(T n (z), (2l)−1 ) → ℂ one hand, by Koebe’s distortion theorem (Theorem 23.1.8) and (29.56) we have Tz−n (B(T n (z)ε , ε)) ⊆ Tz−n (B(T n (z), (2l)−1 ) \ 𝒥 (T)) ⊆ B(z, K|(T n )′ (z)|−1 (2l)−1 ) \ 𝒥 (T). On the other hand, by Koebe’s 41 –theorem and (29.56), we get 1 󵄨 󵄨−1 Tz−n (B(T n (z)ε , ε)) ⊇ B(Tz−n (T n (z)ε ), ε󵄨󵄨󵄨(T n )′ (z)󵄨󵄨󵄨 ). 4 Therefore, Leb2 (B(z, K|(T n )′ (z)|−1 (2l)−1 ) \ 𝒥 (T)) Leb2 (B(z, K|(T n )′ (z)|−1 (2l)−1 ))

≥(

εl 2 ) > 0. 2K

So, z is not a Lebesgue density point for the set 𝒥r (T)(1/l), and thus Leb2 (𝒥r (T)(1/l)) = 0. Hence, ∞

∞

l=1

l=1

Leb2 (𝒥r (T)) = Leb2 (⋃ 𝒥r (T)(1/l)) ≤ ∑ Leb2 (𝒥r (T)(1/l)) = 0. The proof of Theorem 29.2.13 is complete. We already know from Lemma 29.2.11 and Theorem 28.5.3 that δ(T) > 0. We now show that the mere presence of rationally indifferent periodic points entails an explicit positive lower bound on δ(T). Recall that Ω(T) denotes the set of rationally indifferent periodic points of T while Ω0 (T) is the set of simple rationally indifferent fixed points of T (see (24.63)). For every ω ∈ Ω(T), let q(ω) be the smallest q ∈ ℕ such that ω a simple fixed point for T q , i. e., the smallest q ∈ ℕ for which ω ∈ Ω0 (T q ). Define p(ω) to be the p ∈ ℕ that comes from (24.24) with φ = T q(ω) . Let pT := max{p(ω) : ω ∈ Ω(T)} ∈ ℕ.

(29.57)

̂→ℂ ̂ is a rational function with deg(T) ≥ 2, then Theorem 29.2.14. If T : ℂ δ(T) >

pT > 0. pT + 1

Proof. In view of Theorems 29.2.9 and 29.2.12, there exists a δ(T)-conformal measure mδ(T) . Fix ω ∈ Ω(T) so that p(ω) = pT . Passing to a sufficiently high iterate of T, we may assume without loss of generality that ω is a simple parabolic fixed point of T. Aiming to use Proposition 24.2.27, fix z ∈ 𝒥 (T) ∩ B(ω, θ) \ {ω} as required in that proposition and where θ comes from (24.83). Per (24.77), there exists r > 0 such that all the iterates

1226 � 29 Sullivan’s conformal measures for rational functions Tω−n are well-defined on B(z, 2r), where Tω−n is a local holomorphic inverse branch of T n fixing ω. Claim. If r > 0 is sufficiently small, then the sets Tω−n (B(z, 2r)), n ≥ 0, are mutually disjoint. Proof of Claim. As limn→∞ Tω−n (z) = ω according to Lemma 24.2.26, we do know that z ∉ {Tω−n (z) : n ∈ ℕ}. Then Δ := dist(z, {Tω−n (z) : n ∈ ℕ}) > 0.

(29.58)

Take r1 ∈ (0, τ/2], where τ > 0 arises from Lemma 24.2.25. It follows from that lemma, relation (24.84) in Proposition 24.2.27 and Koebe’s distortion theorem that lim sup diam(Tω−n (B(z, 2r1 ))) = 0. n→∞

So, there exists N ∈ ℕ such that diam(Tω−n (B(z, 2r1 ))) < Δ/2,

∀n ≥ N.

Therefore, B(z, Δ/2) ∩ Tω−n (B(z, 2r1 )) = 0,

∀n ≥ N.

It also follows from (29.58) that there exists r2 ∈ (0, r1 ] such that B(z, Δ/2) ∩ Tω−n (B(z, 2r2 )) = 0,

∀1 ≤ n < N.

Thus, for every r ∈ (0, min{Δ/4, r2 }) we have that B(z, 2r) ∩ Tω−n (B(z, 2r)) = 0,

∀n ∈ ℕ.

(29.59)

By way of contradiction, suppose that Tω−n1 (B(z, 2r)) ∩ Tω−n2 (B(z, 2r)) ≠ 0 for some n2 > n1 ≥ 0. Then Tω−n1 (B(z, 2r) ∩ Tω−(n2 −n1 ) (B(z, 2r))) ≠ 0, and hence B(z, 2r) ∩ Tω−(n2 −n1 ) (B(z, 2r)) ≠ 0, contradicting (29.59). This establishes the claim.

◼

29.3 Pesin’s formula

�

1227

Furthermore, the δ(T)-conformality of mδ(T) yields that mδ(T) (B(z, r)) = mδ(T) (T n (Tω−n (B(z, r)))) = 󵄨δ(T)

∫ Tω−n (B(z,r))

󵄨󵄨 n ′ 󵄨󵄨δ(T) dmδ(T) 󵄨󵄨(T ) 󵄨󵄨

󵄨 ≍ 󵄨󵄨󵄨(T ) (Tω−n (z))󵄨󵄨󵄨 mδ(T) (Tω−n (B(z, r))) ′ 󵄨−δ(T) 󵄨 mδ(T) (Tω−n (B(z, r))), = 󵄨󵄨󵄨(Tω−n ) (z)󵄨󵄨󵄨 n ′

where ≍ is the usual symbol of multiplicative comparability (see (25.34)). From this and Proposition 24.2.27, we deduce that p +1 ′ − T δ(T) 󵄨 󵄨δ(T) mδ(T) (Tω−n (B(z, r))) ≍ 󵄨󵄨󵄨(Tω−n ) (z)󵄨󵄨󵄨 mδ(T) (B(z, r)) ≍ (n + 1) pT Mmδ(T) (r),

where 1 ≥ Mmδ(T) (r) := inf{mδ(T) (B(w, r)) : w ∈ 𝒥 (T)} > 0 since supp(mδ(T) ) = 𝒥 (T). Using the claim, we infer that ∞

∞

∞

n=0

n=0

n=1

1 ≥ mδ(T) ( ⋃ Tω−n (B(z, r))) = ∑ mδ(T) (Tω−n (B(z, r))) ≍ ∑ n Consequently,

pT +1 δ(T) pT

−

pT +1 δ(T) pT

.

> 1, or equivalently, δ(T) >

pT . pT + 1

29.3 Pesin’s formula ̂→ℂ ̂ be a rational function with deg(T) ≥ 2. Theorem 29.3.1 (Pesin’s formula). Let T : ℂ If m is a t-conformal measure for T and μ is an ergodic T-invariant Borel probability measure on 𝒥 (T) absolutely continuous with respect to m and such that hμ (T) > 0, then HD(μ) = t = δ(T). Proof. In light of Theorem 29.2.12, we only need to prove that t ≤ HD(μ). To do this, we bring up the arguments from the proof of (28.24). We work with Rokhlin’s natural ex? ̃ μ ̃ ). Since hμ (T) > 0, it follows from Ruelle’s inequality (Theorem 28.2.1) tension (𝒥 (T), T, ? that χμ (T) > 0. So, we can fix 0 < ε < χμ (T)/3 and let 𝒥 (T)(ε) and r(ε) > 0 be given

by Theorem 28.3.3. In view of the ergodic case of Birkhoff’s ergodic theorem (Corol? ̃ ̃ ̃ (F(ε)) lary 8.2.14 or 8.2.15), there exists a measurable set F(ε) ⊆ 𝒥 (T)(ε) such that μ = ? ̃ (𝒥 μ (T)(ε)) ≥ 1 − ε and

1228 � 29 Sullivan’s conformal measures for rational functions 1 n−1 ? ̃ (𝒥 ∘ T̃ j (x̃) = μ (T)(ε)), ∑ 1𝒥 ? (T)(ε) n→∞ n j=0 lim

̃ ∀x̃ ∈ F(ε).

̃ Let F(ε) = π0 (F(ε)). Then ̃ ̃ (π0−1 (F(ε))) ≥ μ ̃ (F(ε)) μ(F(ε)) = μ ≥ 1 − ε. ̃ Consider a Borel set X ⊆ 𝒥 (T) with μ(X) = 1. Take any point x ∈ F(ε) ∩ X and fix x̃ ∈ F(ε) such that x = π0 (x̃). By the above, there then exists a strictly increasing sequence (nk = nk (x))∞ k=1 such that ? T̃ nk (x̃) ∈ 𝒥 (T)(ε)

and

nk+1 − nk ≤ ε, nk

∀k ∈ ℕ.

(29.60)

Moreover, Theorem 28.3.3 proclaims the existence of holomorphic inverse branches −n ̂ of T nk such that Tx−nk (T nk (x)) = x and Tx k : B(T nk (x), r(ε)) → ℂ ′ 󵄨 󵄨 Tx−nk (B(T nk (x), r(ε))) ⊇ B(x, K(ε)−1 󵄨󵄨󵄨(Tx−nk ) (T nk (x))󵄨󵄨󵄨r(ε)) = B(x, rk ),

(29.61)

where rk = rk (x) := K(ε)−1 |(T nk )′ (x)|−1 r(ε). Take any 0 < r ≤ r1 and let k ∈ ℕ be the unique integer such that rk+1 < r ≤ rk . Using the t-conformality of m and Theorem 28.3.3, we get m(B(x, r)) ≤ m(B(x, rk ))

≤ m(Tx−nk (B(T nk (x), r(ε)))) by (29.61) 󵄨󵄨 −nk ′ 󵄨󵄨t = ∫ 󵄨󵄨(Tx ) 󵄨󵄨 dm by the t-conformality of m B(T nk (x),r(ε))

′ 󵄨 󵄨t ≤ K t 󵄨󵄨󵄨(Tx−nk ) (T nk (x))󵄨󵄨󵄨 m(B(T nk (x), r(ε))) by the Koebe distortion theorem ′ 󵄨 󵄨−t ≤ K(ε)t 󵄨󵄨󵄨(T nk ) (x)󵄨󵄨󵄨 as K ≤ K(ε)

≤ K(ε)2t r(ε)−t rkt

from definition of rk .

(29.62)

We seek an upper bound in terms of r rather than rk , and thus we need to bound rk from above in terms of r, which can be accomplished by comparing rk with rk+1 . Using successively Theorem 28.3.3, (29.60) and the choice of ε, we obtain the intermediate estimate ′ 󵄨 󵄨 rk = rk+1 󵄨󵄨󵄨(T nk+1 −nk ) (T nk (x))󵄨󵄨󵄨 󵄨 −(nk+1 −nk ) ′ nk+1 󵄨−1 = rk+1 󵄨󵄨󵄨(TT nk (x) ) (T (x))󵄨󵄨󵄨

≤ rk+1 K(ε) exp((nk+1 − nk )[χμ (T) + ε])

≤ rK(ε) exp(nk ε[χμ (T) + ε])

≤ rK(ε) exp(2nk ε[χμ (T) − ε])

by Theorem 28.3.3 by (29.60) and rk+1 < r

since ε < χμ (T)/3.

(29.63)

29.3 Pesin’s formula

� 1229

It remains to bound nk from above in terms of rk . By Theorem 28.3.3, ′ ′ 󵄨 󵄨−1 󵄨 󵄨 rk := K(ε)−1 󵄨󵄨󵄨(T nk ) (x)󵄨󵄨󵄨 r(ε) = K(ε)−1 󵄨󵄨󵄨(Tx−nk ) (T nk (x))󵄨󵄨󵄨r(ε) ≤ exp(−nk [χμ (T) − ε])r(ε).

So 2ε

exp(2nk ε[χμ (T) − ε]) ≤ (rk−1 r(ε)) . Substituting this into (29.63) yields 2ε

rk ≤ rK(ε)(rk−1 r(ε)) . Equivalently, rk1+2ε ≤ rK(ε)r(ε)2ε . Replacing that into (29.62) gives m(B(x, r)) ≤ K(ε)2t r(ε)−t (rK(ε)r(ε)2ε )

t/(1+2ε)

.

Hence, lim sup r→0

m(B(x, r)) ≤ K(ε)2t+t/(1+2ε) r(ε)2εt/(1+2ε)−t . r t/(1+2ε)

This holds for all x ∈ F(ε) ∩ X. It follows from Theorem 15.5.3(b) (Frostman converse theorem) that Ht/(1+2ε) (F(ε) ∩ X) ≥ 2−t/(1+2ε) K(ε)−2t−t/(1+2ε) r(ε)−2εt/(1+2ε)+t > 0, whence HD(X) ≥ HD(F(ε) ∩ X) ≥

t . 1 + 2ε

Letting ε → 0 results in HD(X) ≥ t. Taking the infimum over all Borel sets X ⊆ 𝒥 (T) for which μ(X) = 1, we conclude that HD(μ) ≥ t. This proves Theorem 29.3.1. In higher dimensions for smooth manifolds with m as the Lebesgue measure (Riemannian volume), the analog of Theorem 29.3.1 is usually called Pesin’s formula (see [103], cf. [82]). The next result is a converse to Theorem 29.3.1. We will not prove it in this book. We refer the reader to [78] and [42]. ̂ → ℂ ̂ be a rational function with deg(T) ≥ 2. If m is a Theorem 29.3.2. Let T : ℂ t-conformal measure for T and μ is an ergodic T-invariant Borel probability measure on

1230 � 29 Sullivan’s conformal measures for rational functions 𝒥 (T) such that χμ (T) > 0 and HD(μ) ≥ t, then μ is absolutely continuous with respect to m. Moreover, the Radon–Nikodym derivative dμ/dm is bounded away from 0. In particular, μ is the unique measure satisfying these conditions.

29.4 Exercises Exercise 29.4.1. Find examples of continuous self-maps T : X → X for which X∗ (T) ∩ Crit(T) ≠ 0. ̂→ℂ ̂ be a rational function with deg(T) ≥ 2. Show that the set Exercise 29.4.2. Let T : ℂ Safe(T) is backward T-invariant, i. e. T −1 (Safe(T)) ⊆ Safe(T). Exercise 29.4.3. In this exercise, we outline the original construction of conformal measures given by Dennis Sullivan in [127] and [129]. ̂ → ℂ ̂ be a rational function with deg(T) ≥ 2. If 𝒥 (T) = ℂ, ̂ then the Let T : ℂ Lebesgue measure (i. e., the Riemannian volume) generated by the spherical metric is a ̂ then proceed as follows: 2-conformal measure. If 𝒥 (T) ≠ ℂ, (a) Prove that there exists a connected component V of ℱ (T) which is neither a Siegel disk nor a Herman ring. (b) Show that V \ Crit(T) ∪ PC(T) ≠ 0. (c) Pick any ξ ∈ V \ Crit(T) ∪ PC(T). Carry out the limit construction of Section 29.2 with this point ξ. Show that all resulting PS limit measures are supported on 𝒥 (T) and that all of them are conformal with the resulting transition parameter. (Note that this construction gives no a priori information about the minimal exponent of conformal measures.) ̂ →ℂ ̂ be a rational function with deg(T) ≥ 2. Show that for Exercise 29.4.4. Let T : ℂ every t ≥ 0 the set CM(T, t) of all t-conformal measures for T is a convex and compact subset of the set M(𝒥 (T)) of all Borel probability measures on 𝒥 (T) endowed with the weak∗ topology. ̂→ℂ ̂ be a rational function with deg(T) ≥ 2. Exercise 29.4.5. Let T : ℂ (a) Show that for every t ≥ 0 the set CMe (T, t) of all ergodic t-conformal measures for T coincides with the set of all extreme points of CM(T, t). (b) By applying the Krein–Milman theorem, deduce that CMe (T, t) ≠ 0 whenever CM(T, t) ≠ 0. (c) By means of Theorem 29.2.12, conclude that there exists an ergodic δ(T)-conformal measure for T. ̂ → ℂ ̂ be a rational function with deg(T) ≥ 2. Prove that if Exercise 29.4.6. Let T : ℂ t ≥ 0 and m is a t-conformal measure for T such that m(𝒥r (T)) = 1, then t = δ(T). ̂→ℂ ̂ be a rational function with deg(T) ≥ 2. Exercise 29.4.7. Let T : ℂ

29.4 Exercises

� 1231

(a) Show that any two δ(T)-conformal measures supported on 𝒥r (T) are equivalent. (b) Using Exercise 29.4.5, prove that there exists at most one δ(T)-conformal measure supported on 𝒥r (T). ̂→ℂ ̂ be a rational function and suppose that m is a (Sullivan) Exercise 29.4.8. Let T : ℂ t-conformal measure for T. (a) Prove that m has full topological support, i. e., supp(m) = 𝒥 (T). (b) Let n ∈ ℕ. Show that m is t-conformal for T n , i. e. ′ 󵄨t 󵄨 m(T n (A)) = ∫󵄨󵄨󵄨(T n ) 󵄨󵄨󵄨 dm A

for every Borel set A ⊆ 𝒥 (T n ) = 𝒥 (T) such that T n |A is injective. (c) Assume that x ∈ 𝒥 (T), r > 0 and n ∈ ℕ are such that there exists an holomorphic ̂ of T n such that Tx−n (T n (x)) = x. Show that inverse branch Tx−n : B(T n (x), r) → ℂ ′ 󵄨t 󵄨 m(Tx−n (A)) = ∫󵄨󵄨󵄨(Tx−n ) 󵄨󵄨󵄨 dm A

for every Borel set A ⊆ B(T n (x), r).

30 Conformal measures, invariant measures and fractal geometry of expanding rational functions 30.1 Fundamental fractal geometry: Bowen’s formula From its very definition (see Definition 25.3.3 and results 26.1.1–26.1.2), every expanding ̂→ℂ ̂ restricted to its Julia set 𝒥 (T) is a topologically exact conforrational function T : ℂ mal expanding repeller and in particular an open distance expanding map. Therefore, the results established in Chapters 13 and 16 hold for all expanding rational functions. According to Remark 27.9.1, the results from Chapter 27 apply as well. Employing those and results from Chapter 29, we easily obtain the following characteristics of the pressure function. Theorem 30.1.1 (Fundamental fractal geometry for expanding rational functions). If T : ̂→ℂ ̂ is an expanding rational function with deg(T) ≥ 2, then the following statements ℂ hold: (a) The pressure function ℝ ∋ t 󳨃󳨀→ P(t) := P(T|𝒥 (T) , −t log |T ′ |) ∈ ℝ is convex, Lipschitz continuous, real-analytic, strictly decreasing and lim P(t) = ∞

t→−∞

while

lim P(t) = −∞.

t→∞

(b) P(0) = htop (T) = log(deg(T)). (c) (Bowen’s formula) The pressure function t 󳨃󳨀→ P(t) has a unique zero hT > 0 and hT = δ(T) = ExpD∗∗ (𝒥 (T)) = ExpD∗ (𝒥 (T)) = ExpD(𝒥 (T)) = DD(𝒥 (T)) = HD(𝒥uer (T)) = HD(𝒥er (T)) = HD(𝒥r (T))

= HD(𝒥 (T)) = PD(𝒥 (T)) = BD(𝒥 (T)) ∈ (

pT , 2). pT + 1

(d) mhT := m−hT log |T ′ | is the only Sullivan conformal measure for T and its exponent is hT . (e) Both mhT and μhT := μ−hT log |T ′ | (the only T-invariant Gibbs state for the potential −hT log |T ′ |) are geometric measures with exponent hT . (f) The measures mhT , Hausdorff HhT |𝒥 (T) and packing PhT |𝒥 (T) , are finite and positive, and are equal up to a multiplicative constant. Proof. (a) The real-analyticity of the pressure function ensues from Corollary 13.10.4. The rest of (a) is a direct consequence of Proposition 16.3.1. (b) The first equality is by definition of the topological entropy and pressure, while the second equality is Theorem 27.8.4. (c) The existence and uniqueness of a zero is a direct consequence of item (a). The long chain of equalities follows from Theorems 29.2.12 and 16.3.2, and (d) below. https://doi.org/10.1515/9783110769876-030

1234 � 30 Conformal measures, invariant measures and fractal geometry The inequality δ(T) > pT /(pT + 1) is the object of Theorem 29.2.14. It only remains to show that hT < 2, which will be accomplished after (f). (d) As P(hT ) = 0, item (d) follows from the definition of the measure mhT (see Proposition 13.6.9, Corollary 13.6.10 and Proposition 13.8.10). (e, f) Both items ensue from Theorem 16.3.2. ̂ Theorem 29.2.13 then states (c) Proof of hT < 2: Exercise 30.3.1 affirms that 𝒥 (T) ≠ ℂ. that Leb2 (𝒥 (T)) = 0. Since the measure H2 is a constant multiple of Leb2 , we conclude from item (f) that hT < 2. A sketch of the graph of the pressure function P(t) is given in Figure 30.1.

Figure 30.1: Graph of the pressure function t 󳨃→ P(t) for an expanding rational function.

Bowen’s formula states that the Hausdorff (as well as the packing, box, dynamical and other) dimension of the Julia set of an expanding rational function is equal to the unique zero of the natural pressure function. The Hausdorff dimension of the Julia set gives us an idea of the global fractal structure of the Julia set. More precisely, it describes the complexity there is in covering the Julia set at infinitesimal scales. Nevertheless, two fractal sets (for instance, Julia sets) with similar global structures may have substantially different local structures. The pointwise or local dimension of a geometric measure contributes to capturing the local structure of a set, and potentially helps in distinguishing that set from another one which may be globally similar (for instance, for distinguishing two Julia sets sharing the same Hausdorff dimension). Multifractal analysis of the Julia set of a rational function consists in: (1) determining the range of local dimensions that the Julia set admits, and (2) for each value in that range, determining the size of the subset of Julia points where the local dimension is equal to that value.

30.1 Fundamental fractal geometry: Bowen’s formula

�

1235

The resulting map is called multifractal spectrum of the Julia set, as it describes the relative sizes of a multitude of Julia subsets which form a splitting of the Julia set. Formally, to avoid any confusion with the temperature function that will naturally ̂→ℂ ̂ be denoted by T, the rational function will be denoted by R temporarily. So let R : ℂ be a rational function and ν be a Borel probability measure on 𝒥 (R). The measure ν is said to have pointwise or local dimension α at a Julia point z ∈ 𝒥 (R) if (cf. Definition 15.6.2) dν (z) := lim

r→0

log ν(B(z, r)) = α. log r

For each number 0 ≤ α ≤ ∞, let 󵄨

𝒥 (R)ν (α) := {z ∈ 𝒥 (R) 󵄨󵄨󵄨 dν (z) = α}

be the subset of all Julia points z ∈ 𝒥 (R) where ν has local dimension α. The domain of the pointwise dimension (function) dν , namely the set 𝒥 (R)ν :=

⋃ 𝒥 (R)ν (α),

0≤α≤∞

is called the regular part of 𝒥 (R). Its complement in the Julia set 󵄨

c

𝒥 (R)ν := 𝒥 (R) \ 𝒥 (R)ν = {z ∈ 𝒥 (R) 󵄨󵄨󵄨 dν (z) ∄},

i. e., the subset of all Julia points z ∈ 𝒥 (R) where the measure ν does not have a local dimension, is called the singular part of 𝒥 (R). The decomposition (i. e., the partition) of the Julia set 𝒥 (R) as 𝒥 (R) =

⋃ 𝒥 (R)ν (α) ∪ 𝒥 (R)cν

0≤α≤∞

is called multifractal decomposition of 𝒥 (R) with respect to the pointwise dimension induced by ν. Let Fν (α) = HD(𝒥 (R)ν (α)) be the Hausdorff dimension of the Julia subset 𝒥 (R)ν (α). The function α 󳨃󳨀→ Fν (α), whose domain is {0 ≤ α ≤ ∞ : 𝒥 (R)ν (α) ≠ 0}, is called (fine Hausdorff ) multifractal spectrum of the measure ν.

1236 � 30 Conformal measures, invariant measures and fractal geometry Returning to the pointwise dimension spectrum, a priori there is no reason for the function Fν to behave nicely. Its domain of definition obviously depends on ν. If ν is a geometric measure (see Definition 15.6.12), then the domain of Fν is a singleton, namely, the exponent t of the geometric measure. Its range is the singleton hR = HD(𝒥 (R)). For instance, by Theorem 30.1.1 we deduce that for the geometric measures mhR := m−hR log |R′ | and μhR := μ−hR log |R′ | , the domains and the ranges of Fmh and Fμh are R R all {hR }. If ν is an ergodic R-invariant Borel probability measure on 𝒥 (R), such as μhR , then the volume lemma (Theorem 28.4.1) states that dν (z) = α0 for ν-a. e. z ∈ 𝒥 (R), where α0 = HD(ν). In this case, we know that the domain of Fν is nonempty. However, for any α ≠ α0 the set 𝒥 (R)ν (α) is not visible to the measure ν since ν(𝒥 (R)ν (α)) = 0. Whereas the Hentschel–Procaccia spectrum HPq (ν) (see Section 16.6) can be determined by statistical properties of ν-typical (a. e.) trajectories, the function Fν (α) seems intractable. Nevertheless, if ν = μφ is the unique R-invariant Gibbs state and unique equilibrium state associated with a Hölder continuous potential φ on a neighborhood U of 𝒥 (R), then miraculously the dimension spectra α 󳨃→ Fμφ (α) and q 󳨃→ HPq (μφ ) happen to be realanalytic functions and −Fμφ (−p) and HPq (μφ ) are mutual Legendre transforms. Indeed, the multifractal analysis of Gibbs states for conformal expanding repellers developed in Section 16.6 applies to expanding rational functions. Accordingly, let φ : U → ℝ be a Hölder continuous potential on some neighbourhood U of 𝒥 (R), and per Proposition 13.7.12, let μφ be its associated, unique R-invariant Gibbs state and unique equilibrium state. For q, t ∈ ℝ, consider the two-parameter family of auxiliary Hölder continuous potentials φq,t : 𝒥 (R) → ℝ defined by φq,t = qφ − t log |R′ |. The unique R-invariant Gibbs state and unique equilibrium state for the potential φq,t , normally denoted by μφq,t , will be abbreviated to μq,t . Lemma 16.6.1 then proclaims the following. ̂ →ℂ ̂ be an expanding rational function with deg(R) ≥ 2 and Lemma 30.1.2. Let R : ℂ φ : U → ℝ be a Hölder continuous potential on some neighborhood U of 𝒥 (R). For every q ∈ ℝ, the pressure function ℝ ∋ t 󳨃→ P(q, t) := P(R, φq,t ) = P(R, qφ − t log |R′ |) ∈ ℝ possesses the following properties: ′ (a) It is Lipschitz with Lipschitz constant L := log ‖R′ ‖∞ , where ‖R′ ‖∞ := supz∈ℂ ̂ |R (z)|. (b) It is strictly decreasing. (c) lim P(q, t) = ∞ whereas lim P(q, t) = −∞. t→−∞

t→∞

(d) P(q, 0) = P(R, qφ). (e) There is a unique T(q) ∈ ℝ such that P(q, T(q)) = 0. The resulting function ℝ ∋ q 󳨃→ T(q) ∈ ℝ is called the temperature function. Moreover: (f) The two-variable pressure function (q, t) 󳨃→ P(q, t) is convex.

30.1 Fundamental fractal geometry: Bowen’s formula

�

1237

The unique R-invariant Gibbs state and unique equilibrium state for the potential φq,T(q) , normally denoted by μφq,T(q) or μq,T(q) , will be further abbreviated to μq . The relation between the measures μq , q ∈ ℝ, and the measure μφ is explained in the following special case of Theorem 16.6.5. ̂→ℂ ̂ be an expanding rational function with deg(R) ≥ 2 and Theorem 30.1.3. Let R : ℂ φ : U → ℝ be a Hölder continuous potential on some neighborhood U of 𝒥 (R) and such that P(R, φ) = 0. (a) For μφ -a. e. z ∈ 𝒥 (R), dμφ (z) =

−μφ (φ) χμφ (R)

=

hμφ (R) χμφ (R)

= HD(μφ ) = PD(μφ ).

(b) The temperature function ℝ ∋ q 󳨃→ T(q) ∈ ℝ is real analytic, and satisfies T(0) = HD(𝒥 (R)),

T(1) = 0,

T ′ (q) =

μq (φ)

χμq (R)

≤

−hμq (R) χμq (R)

< 0,

and

T ′′ (q) ≥ 0,

where μq is the R-invariant Gibbs measure for the potential φq,T(q) . (c) For all q ∈ ℝ, we have μq (𝒥 (R)μφ (−T ′ (q))) = 1 and HD(μq ) = HD(𝒥 (R)μφ (−T ′ (q))).

(d) For every q ∈ ℝ, we have Fμφ (−T ′ (q)) = T(q) − qT ′ (q), i. e., p 󳨃󳨀→ −Fμφ (−p) is the Legendre transform of T(q). In particular, Fμφ is continuous at the boundary points −T ′ (±∞) of its domain, as the Legendre transform is. Furthermore, the sets 𝒥 (R)μφ (α) are empty for all α ∉

[−T ′ (∞), −T ′ (−∞)], so these α’s do not lie in the domain of Fμφ , as they do not belong to the domain of the Legendre transform. (This emptiness property is called the completeness of the Fμφ -spectrum.) Recall that hR = HD(𝒥 (R)). If μφ = μhR := μ−hR log |R′ | (equivalently, if φ is cohomologous to hR in the class of bounded functions on 𝒥 (R)), then T(q) is affine and the domain of Fμφ (α) is one point, namely, hR = −T ′ (q) for all q ∈ ℝ. If μφ ≠ μhR , then T ′′ (q) > 0 and Fμ′′φ (α) < 0, i. e., the functions T(q) and Fμφ (α) are

respectively strictly convex on ℝ and strictly concave on [−T ′ (∞), −T ′ (−∞)], which is a bounded interval in ℝ+ = {α ∈ ℝ : α > 0}. (e) For every q ≠ 1 the HP and Rényi spectra exist (i. e., limits in their definitions exist) = HPq (μφ ) = Rq (μφ ). For q = 1, the information dimension I(μφ ) exists and and T(q) 1−q lim

q→1, q=1̸

T(q) = −T ′ (1) = HD(μφ ) = PD(μφ ) = I(μφ ). 1−q

For more information on multifractal analysis, please see Section 16.6.

1238 � 30 Conformal measures, invariant measures and fractal geometry

30.2 Geometric rigidity ̂ →ℂ ̂ is an expanding rational function whose Julia set Throughout this section, T : ℂ 𝒥 (T) is a Jordan curve. It follows from the Jordan–Schoenflies theorem that the Fatou ̂ \ 𝒥 (T) has exactly two connected components A1 (T) and A2 (T) which are set ℱ (T) = ℂ both open topological disks while their closures are closed topological disks. The proof of the following theorem is straightforward. We provide it for the sake of completeness and convenience to the reader. ̂→ℂ ̂ is an expanding rational function whose Julia set 𝒥 (T) is Theorem 30.2.1. If T : ℂ a Jordan curve, then: (1) either T(A1 (T)) = A1 (T) and T(A2 (T)) = A2 (T) (2) or T(A1 (T)) = A2 (T) and T(A2 (T)) = A1 (T). In either case, T 2 (A1 (T)) = A1 (T)

and

T 2 (A2 (T)) = A2 (T).

In addition, both A1 (T) and A2 (T) are immediate basins of attraction of T to attracting periodic points of minimal period either 1 or 2. Denote those periodic points by a1 (T) and a2 (T), respectively. Proof. We first prove items (1) and (2). Suppose first that T(A1 (T)) = A1 (T). If T(A2 (T)) = ̂ → ℂ. ̂ Thus, A1 (T), then T −1 (A2 (T)) = 0, hence contradicting the surjectivity of T : ℂ T(A2 (T)) = A2 (T) and we are done in this case. So, suppose that T(A1 (T)) = A2 (T). It then follows from the just considered case (with A1 replaced by A2 ) that T(A2 (T)) = A1 (T). Items (1) and (2) are therefore proved. The remaining assertions follow from items (1)–(2) and Theorem 26.1.6. We can now prove the main result of this section. The idea behind the proof goes back to [19] and [127]. Since then, more theorems of this kind and their generalizations have been proved; for instance, see [137] for parabolic rational functions and [72, 142] for class 𝒮 meromorphic functions. ̂→ℂ ̂ is an Theorem 30.2.2 (Geometric rigidity for expanding rational functions). If T : ℂ expanding rational function whose Julia set 𝒥 (T) is a Jordan curve, then either 𝒥 (T) is a geometric circle or HD(𝒥 (T)) > 1. Proof. Suppose that HD(𝒥 (T)) ≤ 1. We are to show that 𝒥 (T) is a geometric circle. Since 𝒥 (T) is connected, it holds that HD(𝒥 (T)) ≥ 1 and consequently HD(𝒥 (T)) = 1. It follows from Theorem 30.1.1(f) that the Hausdorff measure H1 (𝒥 (T)) is finite. Let 𝔻 be the open

unit disk and let 𝕊1 := 𝜕𝔻 be the unit circle. By the Riemann mapping theorem, there ̂ \ 𝔻 → A2 (T) such that are two conformal homeomorphisms R1 : 𝔻 → A1 (T) and R2 : ℂ R1 (0) = a1 (T)

and

R2 (∞) = a2 (T).

30.2 Geometric rigidity

� 1239

Consider the conjugations of T 2 2 g1 := R−1 1 ∘ T ∘ R1 : 𝔻 → 𝔻 and

2 ̂ ̂ g2 := R−1 2 ∘ T ∘ R2 : ℂ \ 𝔻 → ℂ \ 𝔻.

(30.1)

By the choice of R1 and R2 , these conjugate maps fix 0 and ∞, respectively: g1 (0) = 0

and

g2 (∞) = (∞).

(30.2)

As the boundary of A1 (T) and A2 (T) is the Jordan curve 𝒥 (T), Carathéodory’s extension theorem (Theorem A.3.15) asserts that the Riemann conformal homeomorphisms R1 and R2 extend to homeomorphisms, denoted by the same symbols R1 and R2 , of the closed ̂ \ 𝔻, respectively. This implies that g1 and g2 have continuous extensions disks 𝔻 and ℂ ̂ \ 𝔻, respectively. Keeping the same notation g1 and g2 for these extensions, to 𝔻 and ℂ we have in particular that g1 (𝕊1 ) = 𝕊1

and

g2 (𝕊1 ) = 𝕊1 .

It ensues from Schwarz’ reflection principle (also called Schwarz’ symmetry principle; see the complex-analytic version in Theorem A.3.6) that both g1 and g2 uniquely extend ̂ Thus, both of these extensions are to meromorphic functions of the Riemann sphere ℂ. rational functions. We still keep the same notation g1 and g2 for them. It immediately follows from the Schwarz’ reflection that g1 (∞) = ∞

and

g2 (0) = (0).

(30.3)

By (30.2)–(30.3) and Theorem 30.2.1, the points 0 and ∞ are attracting fixed points for ̂ \ 𝔻 are their immediate basins of attraction, respectively. both g1 and g2 , and 𝔻 and ℂ ̂ \ 𝕊1 , so 𝒥 (g1 ) = 𝒥 (g2 ) = 𝕊1 , and it follows from Theorem 26.1.3 Hence, ℱ (g1 ) = ℱ (g2 ) = ℂ that both rational functions g1 and g2 are expanding. Let ℓ be the normalized Lebesgue measure on 𝕊1 . Of course, ℓ is a 1-conformal measure for both g1 and g2 . There then exist two (ergodic) invariant Gibbs states μ− log |g1′ | and μ− log |g ′ | for g1 and g2 , respectively, both equivalent to ℓ. This actually would suffice 2 for us but we can do better; we can precisely say what these measures are. Indeed, for every continuous function u : 𝕊1 → ℝ let ũ : 𝔻 → ℝ be the unique harmonic function (given by the Poisson kernel formula) that has a continuous extension to 𝔻 (and whose restriction to 𝕊1 is thus equal to u). Then for both i = 1, 2, the function ũ ∘ gi : 𝔻 → ℝ is continuous and its restriction to 𝔻 is harmonic. Applying twice the mean value theorem for harmonic functions, we get ∫ u ∘ gi dℓ = ũ ∘ gi (0) = ũ (0) = ∫ u dℓ. 𝕊1

𝕊1

This means that both maps R1 |𝕊1 : 𝕊1 → 𝕊1 and R2 |𝕊1 : 𝕊1 → 𝕊1 preserve the Lebesgue measure on 𝕊1 , i. e.,

1240 � 30 Conformal measures, invariant measures and fractal geometry

ℓ(g1−1 (F)) = ℓ(F)

and

ℓ(g2−1 (F)) = ℓ(F)

for every Borel set F ⊆ 𝕊1 . In consequence, μ− log |g1′ | = ℓ = μ− log |g ′ | . 2

(30.4)

Since H1 (𝒥 (T)) < ∞, Riesz’s theorem (Theorem A.2.4) yields that the homeomorphisms R1 |𝕊1 and R2 |𝕊1 , as well as their inverses, are absolutely continuous with respect to ℓ on 𝕊1 and with respect to H1 on 𝒥 (T). Denoting by μ the T-invariant Gibbs state for the potential − log |(T 2 )′ |, which is equivalent to H1 |𝒥 (T) , and applying Riesz’s theorem again we see that both measures μ ∘ R1 and μ ∘ R2 are absolutely continuous with respect to ℓ on 𝕊1 and −2 μ ∘ Ri ∘ gi−1 = μ ∘ Ri ∘ R−1 ∘ Ri = μ ∘ T −2 ∘ Ri = μ ∘ Ri i ∘T

for both i = 1, 2. This means that each measure μ ∘ Ri is gi -invariant. By (30.4), the gi invariant measure ℓ is ergodic (as a Gibbs state). Hence, all gi -invariant measures absolutely continuous with respect to ℓ must be equal by Theorem 8.2.22. So μ∘R1 = ℓ = μ∘R2 . 1 1 Consider the composite map R−1 2 ∘ R1 : 𝕊 → 𝕊 . Then ℓ ∘ (R−1 2 ∘ R1 )

−1

−1 = ℓ ∘ R−1 1 ∘ R2 = μ ∘ R1 ∘ R1 ∘ R2 = μ ∘ R2 = ℓ,

1 which means that the map R−1 2 ∘R1 preserves the measure ℓ on 𝕊 . Since both R1 and R2 are 1 −1 homeomorphisms of the circle 𝕊 , so is R2 ∘R1 . As it preserves the Lebesgue measure ℓ, it is a rotation of 𝕊1 . Thus, modifying R1 by precomposing it with an appropriate rotation, 1 we can make the composition R−1 2 ∘R1 to be the identity map. Consequently, R1 = R2 on 𝕊 . Thus, the map defined by the formula

R(z) := {

R1 (z) R2 (z)

if z ∈ 𝔻 ̂\𝔻 if z ∈ ℂ

̂ \ 𝔻) = ℂ ̂ \ 𝕊1 and continuous on ℂ. ̂ A straightforward apis meromorphic on 𝔻 ∪ (ℂ ̂→ℂ ̂ is meromorphic. As R is a plication of Morera’s theorem shows that the map R : ℂ homeomorphism, it must be a Möbius map. Since 𝒥 (T) = R1 (𝕊1 ) = R(𝕊1 ), we conclude that 𝒥 (T) is a geometric circle. The proof of Theorem 30.2.2 is complete. Remark 30.2.3. In the proof of Theorem 30.2.2, we eventually showed that g1 = g2 . Dê → ℂ ̂ is a rational function prenote this function by g. We demonstrated that g : ℂ ̂ serving 𝔻 and ℂ \ 𝔻. Thus, g is a finite Blaschke product. Furthermore, (30.1) states that the maps T 2 and g are conjugate via the Möbius map R. Moreover, if we knew that the connected components A1 (T) and A2 (T) were fixed under T, then we would not have to pass to T 2 in the proof and we would conclude that the map T and the finite Blaschke product g are conjugate through the Möbius map R. (For more information on Blaschke products, see [53].)

30.3 Exercises

� 1241

We can now easily prove the following. Theorem 30.2.4. If c ∈ ℂ \ {0} belongs to the main cardioid ℳ0 of the Mandelbrot set ℳ, then HD(Jc ) > 1, where Jc is the Julia set of the quadratic polynomial Qc (z) = z2 + c. Proof. Using the notation from Theorem 30.2.2 and Remark 30.2.3, we have Qc ∘ R = R ∘ g. Consequently, Qcn ∘ R = R ∘ g n ,

(30.5)

∀n ∈ ℕ.

Differentiating this equality yields (Qcn ) (R(z))R′ (z) = R′ (g n (z))(g n ) (z), ′

′

∀z ∈ ℂ.

(30.6)

If ξ is a repelling periodic point of Qc of period n ∈ ℕ, then (30.5)’s conjugation requires that R−1 (ξ) be a periodic point of g with the same period n. Substituting R−1 (ξ) for z in (30.6) leads to (Qcn ) (ξ) = (g n ) (R−1 (ξ)). ′

′

Therefore, R−1 (ξ) is a repelling periodic point of period n for the map g, whence R−1 (ξ) ∈ 𝒥 (g) = 𝕊1 . Since g(𝕊1 ) = 𝕊1 , it follows that (g n )′ (R−1 (ξ)) ∈ ℝ. That is, (Qcn )′ (ξ) ∈ ℝ. Looking at the repelling periodic points of Qc of periods 1 and 2, we conclude from a direct calculation that c ∈ (−∞, −3/4]. Hence, c ∉ ℳ0 , contrary to our hypothesis.

30.3 Exercises ̂ for every expanding rational function T : ℂ ̂ → ℂ. ̂ Exercise 30.3.1. Prove that 𝒥 (T) ≠ ℂ ̂ →ℂ ̂ be an expanding rational function and suppose that μ Exercise 30.3.2. Let T : ℂ is an ergodic T-invariant Borel probability measure such that HD(μ) = HD(𝒥 (T)). Show that μ = μHD(𝒥 (T)) = μhT .

31 Conformal measures, invariant measures and fractal geometry of parabolic rational functions Although there are some similarities, the picture for parabolic rational functions given in this chapter is quite different from that of expanding and subexpanding ones treated in the previous and the next chapters, respectively. The potential −t log |T ′ | is welldefined and even Hölder continuous but, because of the presence of rationally indifferent periodic points, it lacks pressure gap. In particular, the results proved in Chapter 27 do not apply. Nevertheless, Section 31.2 enunciates a variation of Bowen’s formula, which identifies the Hausdorff dimension hT of the Julia set 𝒥 (T) with the smallest zero of the pressure function P(t). Unlike the expanding case, P(t) = 0 for all t > hT . In Section 31.3, we show that hT is equal to the minimal exponent for which a t-conformal measure exists. As in the subhyperbolic case (treated in Chapter 32), there exists a unique Sullivan hT -conformal measure and that measure is atomless; furthermore, it is ergodic and conservative. Also as in the subhyperbolic case, there exist t-conformal measures for all t > hT . All of them are purely atomic and are convex combinations of atomic measures supported on the backward orbits of rationally indifferent periodic points. In Section 31.4, we establish the existence and uniqueness of an invariant measure which is absolutely continuous with respect to the hT -conformal measure, and we provide a simple criterion for this measure’s finiteness/infiniteness. In particular, this measure is finite for the quadratic polynomial Q1/4 (z) = z2 + 1/4, whereas it is infinite for parabolic Blaschke products. Finally, in Section 31.5, we provide a complete description of the hT -dimensional Hausdorff and packing measures on the Julia sets of parabolic rational functions. These measures are sometimes finite, sometimes positive and sometimes infinite, exclusively depending on the value of the Hausdorff dimension hT . Chapter 31 combines, develops, goes beyond and improves the results established in several papers written by Manfred Denker and the first named author of this book; see [1, 37, 38, 40].

31.1 General conformal measures for expansive topological dynamical systems In order to fully meaningfully deal with various kinds of conformal measures for expansive (especially parabolic) rational functions, it is convenient to recall the concept of transfer operator from Chapters 13 (Gibbs states and transfer operators for open, distance expanding systems) and 27 (Equilibrium states for Hölder continuous potentials with pressure gap). Remember that in this book a topological dynamical system is a conhttps://doi.org/10.1515/9783110769876-031

1244 � 31 Conformal measures, invariant measures and fractal geometry tinuous self-map T : X → X of a compact metrizable space X. Let T : X → X be an open, expansive topological dynamical system and let φ ∈ C(X) be a continuous potential. Define the transfer operator ℒφ in the standard way (cf. (13.26)): ℒφ (g)(x) =

∑ x∈T −1 (x)

g(x)eφ(x) ,

∀x ∈ X, ∀g ∈ C(X).

We emphasize that in this section the potential φ is not assumed to be Hölder continuous, but merely continuous. Also, unlike in Chapter 27, we do not require φ to exhibit a pressure gap. So, we cannot really rely on most of the results in Chapters 13 and 27. Nevertheless, the fact that every expansive map is expanding with respect to a metric compatible with the topology (see Theorem 5.3.1 in the first volume) helps us establish results, starting with the following assertion (cf. Lemma 13.6.1). Proposition 31.1.1. If T : X → X is an open, expansive topological dynamical system and φ ∈ C(X) a potential, then: (a) ℒφ is positive, linear and bounded. (b) ℒφ (C(X)) ⊆ C(X) and ‖ℒφ ‖∞ ≤ #T −1 ⋅ esup φ . (c) ℒnφ (g)(x) = ∑ g(x) exp(Sn φ(x)), ∀x ∈ X, ∀g ∈ C(X), ∀n ≥ 0. x∈T −n (x)

From now on, we assume that d is a metric compatible with the topology of X and with respect to which T is expanding. Accordingly, the letters δ, λ and ξ will be the constants from Definition 4.1.1 and relations (4.29)–(4.31); alternatively, see (13.1)–(13.2). Lemma 31.1.2. If T : X → X is an open, expansive topological dynamical system and φ ∈ C(X), then for every ε > 0 there exists ζ > 0 such that d(y, x) < ζ

󳨐⇒

e−εn ≤

ℒnφ (1)(y)

ℒnφ (1)(x)

≤ eεn ,

∀n ≥ 0.

Proof. Take ζ ∈ (0, ξ) so small that |φ(y) − φ(x)| < ε whenever d(y, x) < ζ . Let 0 ≤ k ≤ n. For every x ∈ T −n (x), we then have d(T k (Tx−n (y)), T k (x)) < λ−(n−k) ζ < ζ . Thus, n

ℒφ (1)(y) =

≤

∑ y∈T −n (y)

∑ x∈T −n (x)

g(y) exp(Sn φ(y)) =

∑ x∈T −n (x)

exp(Sn φ(Tx−n (y)))

exp(Sn φ(x) + εn) = eεn ℒnφ (1)(x).

So, the right inequality is proved. Since the left inequality is equivalent to the right one, the proof is complete.

31.1 General conformal measures for expansive topological dynamical systems

� 1245

The reader may want to compare the previous and next results with Lemma 13.7.1, as well as with Lemma 13.1.1 and Corollary 13.1.2. Lemma 31.1.3. If T : X → X is a topologically transitive, open, expansive dynamical system and φ ∈ C(X), then for every ε > 0 there exists a constant cε ≥ 1 such that cε−1 e−εn ≤

ℒnφ (1)(y)

≤ cε eεn ,

ℒnφ (1)(x)

∀x, y ∈ X, ∀n ≥ 0.

(31.1)

Proof. Fix ε > 0 and ascribe ζ > 0 to it according to Lemma 31.1.2. Let F be a finite (ζ /2)-spanning set of X. According to Corollary 5.3.4 (see also Lemma 4.2.10 and Definition 1.5.15), any open, expansive and transitive dynamical system is very strongly transitive. As F is finite, this implies that there exists M ∈ ℕ such that M

⋃ T m (B(w, ζ /2)) = X,

m=0

∀w ∈ F.

(31.2)

Fix any x, y ∈ X. Since F is (ζ /2)-spanning, there exists w ∈ F such that x ∈ B(w, ζ /2). But then B(x, ζ ) ⊇ B(w, ζ /2), and hence (31.2) yields 0 ≤ l ≤ M such that y ∈ T l (B(x, ζ )). In particular, there exists z ∈ B(x, ζ ) such that T l (z) = y. Let n ≥ l. By Lemma 31.1.2, we obtain that n

n−l

ℒφ (1)(y) ≥ exp(Sl φ(z))ℒφ (1)(z) ≥ exp(−l‖φ‖∞ )e

−εn n−l ℒφ (1)(x).

On the other hand, n

n−l

l

n−l

l

l

n−l

ℒφ (1)(x) = ℒφ (ℒφ (1))(x) ≤ ℒφ (‖ℒφ (1)‖∞ ⋅ 1)(x) = ‖ℒφ ‖∞ ℒφ (1)(x).

Therefore, n

ℒφ (1)(y) ≥ exp(−l‖φ‖∞ )e

−εn

n ‖ℒlφ ‖−1 ∞ ℒφ (1)(x).

The left inequality in (31.1) is proved for all n ≥ l. For all n < l, a sheer compactnesscontinuity argument yields that 0 < min inf{ 0≤n 0, Lemma 31.1.3 then yields 1 1 log eSn φ(y) = log ∑ ∑ n n y∈E(ε) y∈T −n (E(ε)) = ≤ =

∑ y∈T −n (y)

eSn φ(y)

1 log ∑ ℒnφ (1)(y) n y∈E(ε)

1 log ∑ cγ eγn ℒnφ (1)(x) n y∈E(ε)

1 1 log(cγ #E(ε)) + γ + log ℒnφ (1)(x). n n

Consequently, P(φ) = lim lim inf ε→0 n→∞

1 1 log exp(Sn φ(y)) ≤ γ + lim inf log ℒnφ (1)(x). ∑ n→∞ n n −n y∈T (E(ε))

As γ > 0 is arbitrary, we deduce that

31.1 General conformal measures for expansive topological dynamical systems

P(φ) ≤ lim inf n→∞

1 log ℒnφ (1)(x). n

� 1247

(31.4)

The result ensues from (31.3)–(31.4). In exactly the same way as in Theorem 13.6.2, i. e. by applying the Schauder– Tychonov fixed-point theorem, we obtain a Borel probability measure mφ on X such that ℒφ (mφ ) = Λφ mφ ∗

(31.5)

with the same Λφ > 0. According to Proposition 13.5.2, such an eigenmeasure is quasi-Tinvariant (and even more). As demonstrated through Proposition 13.6.9–Corollary 13.6.11 (see also Definition 13.6.12), the measure mφ is (Λφ e−φ )-conformal, i. e., −n mφ (Tx−n (A)) = Λ−n φ ∫ exp((Sn φ) ∘ Tx ) dmφ

(31.6)

A

for every x ∈ X, every n ∈ ℕ and every Borel set A ⊆ B(T n (x), ξ). One can easily deduce from this that mφ (T n (B)) = Λnφ ∫ exp(−Sn φ) dmφ

(31.7)

B

for every n ∈ ℕ and every Borel set B ⊆ X such that T n |B is one-to-one. In fact, it was observed in Lemma 13.6.13 that relations (31.5)–(31.7) are all equivalent. If T is transitive, open and expansive, then it is (very) strongly transitive, which −n means that ⋃∞ n=0 T (x) = X for any x ∈ X. From (31.6), we readily infer the following fact (cf. Proposition 13.6.14). Proposition 31.1.5. If T : X → X is a topologically transitive, open, expansive system and φ ∈ C(X), then the measure mφ is positive on nonempty open sets. In other words, supp(mφ ) = X. According to Lemma 13.6.15, measures with full support in a compact space confer to balls of any given radius weights that are uniformly bounded away from 0. Lemma 31.1.6. If T : X → X is a topologically transitive, open and expansive system and φ ∈ C(X), then for every r > 0 we have 󵄨 Mmφ (r) := inf{mφ (B(x, r)) 󵄨󵄨󵄨 x ∈ X} > 0. We now identify the value of Λφ . Proposition 31.1.7. Let T : X → X be a topologically transitive, open, expansive system and φ ∈ C(X). If Λφ satisfies (31.5), then log Λφ = P(φ).

1248 � 31 Conformal measures, invariant measures and fractal geometry Proof. Fix x ∈ X and ε > 0. Using Lemma 31.1.3, we notice that for every n ∈ ℕ, n −1 −εn n εn n Λnφ = Λnφ mφ (1) = ℒ∗n ℒφ (1)(x), cε e ℒφ (1)(x)] φ (mφ )(1) = ∫ ℒφ (1) dmφ ∈ [cε e X

Consequently, −ε + lim sup n→∞

1 1 log ℒnφ (1)(x) ≤ log Λφ ≤ ε + lim inf log ℒnφ (1)(x). n→∞ n n

Theorem 31.1.4 then implies that −ε + P(φ) ≤ log Λφ ≤ ε + P(φ). Letting ε → 0 results in P(φ) = log Λφ .

31.2 Geometric topological pressure and generalized geometric conformal measures for parabolic rational functions ̂ →ℂ ̂ is an expansive rational function. Then T : 𝒥 (T) → 𝒥 (T) is Suppose that T : ℂ a topologically exact, open, expansive topological dynamical system, whence all the results from the previous section apply for continuous potentials on 𝒥 (T). In this section, we will deal with the usual geometric potentials: given any t ∈ ℝ, let φt := −t log |T ′ | : 𝒥 (T) → ℝ. This potential is well defined in light of Theorem 26.2.3. Denote the operator ℒφt , resulting from the previous section, by ℒt and, likewise, mφt by mt . (Note: Though the measure mt in Lemma 29.2.1 and the one just produced are formally different, we will use the same notation to represent them; the context will distinguish which of these measures is being used.) Also, denote P(t) := P(T|𝒥 (T) , φt ) = P(T|𝒥 (T) , −t log |T ′ |), i. e., P(t) is the topological pressure generated by the system T : 𝒥 (T) → 𝒥 (T) under the potential φt (z) = −t log |T ′ (z)|. Formulas (31.5)–(31.7) in conjunction with Proposition 31.1.7 respectively become ℒt (mt ) = e ∗

P(t)

mt ,

󵄨 󵄨t mt (Tz−n (A)) = e−nP(t) ∫󵄨󵄨󵄨(Tz−n )′ 󵄨󵄨󵄨 dmt

(31.8) (31.9)

A

for all z ∈ 𝒥 (T), all n ∈ ℕ and all Borel sets A ⊆ B(T n (z), ξ), and 󵄨 󵄨t mt (T n (B)) = enP(t) ∫󵄨󵄨󵄨(T n )′ 󵄨󵄨󵄨 dmt B

(31.10)

31.2 Geometric topological pressure and generalized geometric conformal measures

� 1249

for all n ∈ ℕ and all special Borel sets B ⊆ 𝒥 (T) for the map T n , i. e., such that the map T n |B is one-to-one. Relations (31.8)–(31.10) are all equivalent and by virtue of (31.9), the measure mt is eP(t)−φt -conformal. Recall that it is also quasi-T-invariant. Throughout the rest of this chapter, we deal with parabolic rational functions. In̂ → ℂ ̂ troduced in Section 26.2, they are per Definition 26.2.6 rational functions T : ℂ whose restriction T : 𝒥 (T) → 𝒥 (T) is expansive but not expanding. Equivalently, by Theorem 26.2.7, they fulfill two conditions: 𝒥 (T) ∩ Crit(T) = 0

and Ω(T) ≠ 0

where Ω(T) is the (finite) set of all rationally indifferent periodic points of T. In Theorem 30.1.1, we described the main properties of the natural pressure function associated with an expanding rational function. We will do the same for parabolic functions, and thereafter identify similarities and differences between them. Denote 󵄨 hT := inf{t ∈ ℝ 󵄨󵄨󵄨 P(t) = 0}.

(31.11)

Recall from the end of Section 29.2 (see (29.57)) that 󵄨 pT := max{p(ω) 󵄨󵄨󵄨 ω ∈ Ω(T)} ∈ ℕ. The next result is the counterpart of Theorem 30.1.1(a–c). ̂→ℂ ̂ is a parabolic rational function with deg(T) ≥ 2, then the Theorem 31.2.1. If T : ℂ pressure function ℝ ∋ t 󳨃→ P(t) ∈ ℝ has the following properties: (a) It is convex and Lipschitz continuous. (b) It is nonincreasing. (c) P(t) ≥ 0 for all t ∈ ℝ. (d) P(t) > 0 for all t < hT . (e) P(t) = 0 for all t ≥ hT . (f) P|(−∞,hT ] is strictly decreasing. (g) lim P(t) = ∞. t→−∞

(h) P(0) = htop (T) = log(deg(T)). (i) (Bowen’s formula) The pressure function t 󳨃󳨀→ P(t) has a least zero hT > 0 and hT = δ(T) = ExpD∗∗ (𝒥 (T)) = ExpD∗ (𝒥 (T)) = ExpD(𝒥 (T)) = DD(𝒥 (T)) = HD(𝒥uer (T)) = HD(𝒥er (T)) = HD(𝒥r (T))

= HD(𝒥 (T)) ∈ (

pT , 2]. pT + 1

1250 � 31 Conformal measures, invariant measures and fractal geometry Proof. (a) Both properties can be proved directly per Exercise 11.5.7 (see also Exercise 11.5.10) or by means of the variational principle (Theorem 12.1.1) per Corollary 12.1.11 and Exercise 12.4.9. (b) Let s ≤ t. For any μ ∈ M(T), Przytycki’s theorem (Theorem 28.1.3; χμ (T) ≥ 0) implies that hμ (T) + ∫ φt dμ = hμ (T) − tχμ (T) ≤ hμ (T) − sχμ (T) = hμ (T) + ∫ φs dμ. 𝒥 (T)

𝒥 (T)

Hence, P(t) ≤ P(s) by the variational principle (Theorem 12.1.1). (c) By Theorem 26.2.7, Ω(T) ≠ 0. Let ω ∈ Ω(T) and consider the atomic measure μω equidistributed on the (finite) orbit of ω. Of course, μω ∈ M(T) and by the variational principle we have P(t) ≥ hμω (T) − tχμω (T) = 0 − t ⋅ 0 = 0. (d) Recall from Definition 29.2.6 and Theorems 29.2.12 and 29.2.14 that 󵄨 DD(𝒥 (T)) := sup{HD(μ) 󵄨󵄨󵄨 μ ∈ Me+ (T)} ∈ (0, 2] is the dynamical dimension of the Julia set 𝒥 (T). If t < DD(𝒥 (T)), there is μ ∈ Me+ (T) such that HD(μ) > t. By Ruelle’s inequality (Theorem 28.2.1; hμ (T) ≤ 2χμ (T)) and the volume lemma (Theorem 28.4.1; HD(μ) =

hμ (T) ), χμ (T)

this means that

hμ (T) χμ (T)

> t. Equiva-

lently, hμ (T) − tχμ (T) > 0. By the variational principle, this implies that P(t) > 0. In particular, this shows that hT ≥ DD(𝒥 (T)). Item (d) will hold after demonstrating that hT ≤ DD(𝒥 (T)) under (e). (e) Let t ≥ DD(𝒥 (T)) and suppose for a contradiction that P(t) > 0. By the variational principle, there then exists μ ∈ Me (T) such that hμ (T) − tχμ (T) > 0. By Przytycki’s theorem, it ensues that hμ (T) > tχμ (T) ≥ 0. So μ ∈ Me+ (T). By Ruelle’s inequality, we deduce that χμ (T) > 0. Hence the volume lemma applies and leads to DD(𝒥 (T)) ≤ t
0, then sχν (T) < tχν (T). Using the variational principle, this implies that

31.3 Sullivan conformal measures for parabolic rational functions

� 1251

P(t) = P(s) ≥ hν (T) − sχν (T) > hν (T) − tχν (T) = P(t), which is impossible. Therefore, χν (T) = 0. But Ruelle’s inequality then implies that hν (T) = 0. Hence, P(s) = P(t) = 0 by (31.12). This contradicts (d) as s < hT . (g) This ensues from (f) and the convexity in (a). (h) The first equality is by definition of the topological entropy and pressure, while the second equality is Theorem 27.8.4. (i) The first two lines in this item follow from Theorem 29.2.12 and the earlier observation that hT = DD(𝒥 (T)) while the last equality sign ensues from the obvious fact that 𝒥r (T) ⊇ 𝒥 (T) \ Ω∞ (T) −

(31.13)

−n and that Ω−∞ (T) := ⋃∞ n=0 T (Ω(T)) is countable. The inequality δ(T) > pT /(pT + 1) is simply Theorem 29.2.14.

As pointed out earlier, any iterate T k of a parabolic rational function T is parabolic, too. Moreover, per Theorem 11.1.22, the pressure function satisfies P(T k ) = k P(T). It follows from Theorem 31.2.1 that hT k = hT . A depiction of the graph of P is given in Figure 31.1. Compare it with Figure 30.1.

Figure 31.1: Graph of t 󳨃→ P(t) for a parabolic rational function.

31.3 Sullivan conformal measures for parabolic rational functions In Chapter 29, we presented a fairly comprehensive account of Sullivan’s conformal measures for general rational functions (in particular, see Theorem 29.2.12). In Chapter 30, more precisely in Theorem 30.1.1(d), we chronicled the existence and uniqueness of Sullivan’s conformal measures for expanding rational functions. We will provide in Theorem 31.3.11 a counterpart to Theorem 30.1.1(d), i. e., a full description of Sullivan’s

1252 � 31 Conformal measures, invariant measures and fractal geometry conformal measures for parabolic rational functions. That description is quite different from the case of expanding rational functions. Nonetheless, we will prove in Subsection 31.3.2 the existence, uniqueness, ergodicity and conservativity of an atomless hT conformal measure mT . 31.3.1 Technical preparations By definition, every measure mt , t ≥ 0, is eP(t)−φt -conformal per (31.10). However, according to Theorem 31.2.1(d), no measure mt , t < hT , is Sullivan t-conformal. Nevertheless, by Theorem 31.2.1(e), each measure mt , t ≥ hT , is Sullivan t-conformal. Employing mt ’s eP(t)−φt -conformality (formula (31.10)) with a singleton set B, we obtain from Lemma 26.2.5 and Theorem 31.2.1 the following pointwise properties of mt . ̂→ℂ ̂ be a parabolic rational function with deg(T) ≥ 2. Lemma 31.3.1. Let T : ℂ − −n (a) If t ≥ 0 and z ∉ Ω∞ (T) := ⋃∞ n=0 T (Ω(T)), then (b) If t < hT , then

mt ({z}) = 0. mt (Ω−∞ (T)) = 0.

In other terms, for any t < hT the eP(t)−φt -conformal measure mt is atomless (and lives on 𝒥 (T) \ Ω−∞ (T)). For t ≥ hT , the t-conformal measure mt may have atoms on Ω−∞ (T) only. Now, recall that whereas Ω(T) denotes the set of rationally indifferent periodic points of T, the set of simple rationally indifferent fixed points of T is denoted by Ω0 (T) (see (24.63)): 󵄨 Ω0 (T) := {ω ∈ Ω(T) 󵄨󵄨󵄨 T(ω) = ω and T ′ (ω) = 1} For every ω ∈ Ω(T), let q(ω) be the smallest q ∈ ℕ such that ω ∈ Ω0 (T q ). Define p(ω) to be the number p ∈ ℕ that comes from (24.24) with φ = T q(ω) and pT := max{p(ω) : ω ∈ Ω(T)} (cf. (29.57)). For every t ≥ 0 and ω ∈ Ω(T), let αt (ω) := t + p(ω)(t − 1) = −p(ω) + t(p(ω) + 1).

(31.14)

We now observe that the measure mt , t ≥ hT , behaves like a geometric measure with exponent αt (ω) on every small enough, pointed neighborhood of any rationally indifferent periodic point ω. When t < hT , the upper bound still prevails, and in fact holds on the entire neighborhood of ω. ̂→ℂ ̂ is a parabolic rational function with deg(T) ≥ 2, then there Lemma 31.3.2. If T : ℂ exists R > 0 such that for every t ≥ hT there is a constant c = c(t) ≥ 1 for which

31.3 Sullivan conformal measures for parabolic rational functions

c−1 ≤

mt (B(ω, r) \ {ω}) ≤ c, r αt (ω)

∀r ∈ (0, R], ∀ω ∈ Ω(T).

� 1253

(31.15)

On the other hand, there exists R > 0 such that for every t < hT there is a constant c = c(t) ≥ 1 for which mt (B(ω, r)) ≤ c, r αt (ω)

∀r ∈ (0, R], ∀ω ∈ Ω(T).

(31.16)

Furthermore, for any pT /(pT + 1) < h < hT , it turns out that ch := sup{c(t) : h ≤ t < hT } < ∞.

(31.17)

̂ per Exercise 31.6.1, we may through a Möbius conjugation assume Proof. As 𝒥 (T) ≠ ℂ that ∞ ∉ 𝒥 (T). Given that mt is t-conformal for any iterate of T and given the definition of p(ω), we may further assume that ω ∈ Ω0 (T) without loss of generality. Let θ be from (24.83). Take 0 < R < θ and let ′ P := {z ∈ ℂ : R(2‖Tω,θ ‖∞ )

−1

≤ |z − ω| ≤ R} ∩ 𝒥 (T),

(31.18)

where ′ ‖Tω,θ ‖∞ := sup{|T ′ (z)| : z ∈ B(ω, θ) ∩ 𝒥 (T)}.

The set [B(ω, R) \ {ω}] ∩ 𝒥 (T) is nonempty since 𝒥 (T) is perfect by Theorem 24.1.15. We will soon show that P ≠ 0. But first fix τ > 0 so small that Lemma 24.2.25 applies. Let also δ = τ inf{|z − ω| : z ∈ P} > 0. As P is compact, there are finitely many points z1 , . . . , zq ∈ P such that q

P ⊆ ⋃ B(zi , δ) i=1

and

B(zi , δ) ∩ P ≠ 0, ∀1 ≤ i ≤ q.

(31.19)

By virtue of Lemma 24.2.26, we may further assume that δ is so small that Tω−k (B(zi , δ)) ∩ B(zi , δ) = 0,

∀k ∈ ℕ, ∀i = 1, . . . , q.

(31.20)

By the local behavior of T around a parabolic point (more precisely, Proposition 24.2.28), for every z ∈ [B(ω, R) \ {ω}] ∩ 𝒥 (T) there exists a least integer ̃l(z) ≥ 0 such that −1

< |T l (z) − ω| < R.

−1

< |z − ω| < R} ∩ 𝒥 (T) ≠ 0.

′ R(2‖Tω,θ ‖∞ )

̃

So, T l (z) ∈ P ≠ 0 and in fact ̃

′ {z ∈ ℂ : R(2‖Tω,θ ‖∞ )

1254 � 31 Conformal measures, invariant measures and fractal geometry Since this latter set is open in 𝒥 (T), we deduce that for some 1 ≤ j ≤ q the set B(zj , δ) ∩ P has nonempty interior in 𝒥 (T). Hence, M(t) := mt (B(zj , δ) ∩ P) > 0. Let p = p(ω) and a be from (24.24). For every n ∈ ℕ, define 1 −1 −1 −1 −1 󵄨 󵄨 Pn := {z ∈ B(ω, θ) ∩ 𝒥 (T) : (|a|p) p n p < 󵄨󵄨󵄨Tω−n (z) − ω󵄨󵄨󵄨 < 2(|a|p) p n p }. 2 By Lemma 24.2.26, there is n0 ∈ ℕ such that Tω−n (z) ∈ Pn for every n ≥ n0 and every z ∈ P. Thus, for all n ≥ n0 we deduce that Tω−n (P) ⊆ Pn and − p1 − p1

B(ω, 2(|a|p)

n

∞

∞

k=n

k=n

q

∞

) \ {ω} ⊇ ⋃ Pk ⊇ ⋃ Tω−k (P) ⊇ ⋃ ⋃ Tω−k (B(zi , δ) ∩ P). k=n i=1

(31.21)

If t ≥ hT , then P(t) = 0. As the sets {Tω−k (B(zi , δ))}∞ k=0 are mutually disjoint for every 1 ≤ i ≤ q per (31.20), it follows from the t-conformality of mt in the form of (31.9), Koebe’s distortion theorem (Theorem 23.1.8) and Proposition 24.2.27, that − p1 − p1

mt (B(ω, 2(|a|p)

n

∞

) \ {ω}) ≥ ∑ mt (Tω−k (B(zj , δ) ∩ P)) k=n

2n p+1 p+1 t 1 − t − ≥ (|a|p) p R−(p+1)t K −t M(t) ∑ (k p ) 2 k=n

̃ t) ⋅ n ⋅ (2n) ≥ C(ω,

−

p+1

p+1 t p

=2

−

p+1 t p

1

̃ t)(n− p )αt (ω) , (31.22) C(ω,

t

̃ t) = 1 (|a|p) p R−(p+1)t K −t M(t). This holds for all n large enough. where C(ω, 2 On the other hand, according to Lemma 24.2.26 and Proposition 24.2.28, every point −

−1 −1

z ∈ [B(ω, R) \ {ω}] ∩ 𝒥 (T) admits an integer l ≥ 0 such that z ∉ B(ω, 21 (|a|p) p l p ) and T l (z) ∈ P. Let l(z) ≥ 0 be the smallest such integer. Take n1 ≥ n0 so large that if 1 − p1 − p n1 )

z ∈ B(ω, 2(|a|p)

\ {ω}, then l(z) ≥ n0 . Consider any − p1 − p1

z ∈ [B(ω, 2(|a|p)

n

) \ {ω}] ∩ 𝒥 (T)

with n ≥ n1 . Since l(z) ≥ n0 and T l(z) (z) ∈ P, we see that z = Tω−l(z) (T l(z) (z)) ∈ Pl(z) . − p1

Consequently, 21 (|a|p) − p1 − p1

B(ω, 2(|a|p)

n

− p1

l(z)

− p1 − p1

≤ 2(|a|p)

n

and thus l(z) ≥ 4−p n. Hence, q

) ∩ 𝒥 (T) ⊆ {ω} ∪ ⋃ Tω−l (P) ⊆ {ω} ∪ ⋃ ⋃ Tω−l (B(zi , δ) ∩ 𝒥 (T)). l≥4−p n

i=1 l≥4−p n

It follows from the eP(t)−φt -conformality of mt , Theorem 23.1.8 (Koebe’s distortion theorem), Proposition 24.2.27 and the series integral test, that

31.3 Sullivan conformal measures for parabolic rational functions

− p1 − p1

mt (B(ω, 2(|a|p)

n

� 1255

q

) \ {ω}) ≤ mt (⋃ ⋃ Tω−l (B(zi , δ))) i=1 l≥4−p n

≤ q ⋅ 2(|a|p)

−

≤ 2qK

t

≤ 2qK

t

∑ l

−(p+1)t

′ [R(2‖Tω,θ ‖∞ )−1 ]

K t ∑ e−lP(t) l

−

p+1 t p

l≥4−p n

p+1 − p t

l≥4−p n

1

αt (ω) −p (4 n p

≤ C(ω, t)(n p+1

p+1 t p

− 1)

− p1 αt (ω)

)

,

αt (ω) p

∀n ≥ max{n1 , 2 ⋅ 4p },

(31.23)

t

′ where K := (|a|p) p [R(2‖Tω,θ ‖∞ )−1 ] K and C(ω, t) := 4pqK /αt (ω). This estimate is valid provided that αt (ω) > 0, i. e., t > p(ω)/(p(ω) + 1). We neglected the pressure factor in the series (as if t ≥ hT ). When t < hT , we can work with the pressure factor and obtain a similar inequality. We omit the details. The proof in the case when t ≥ hT is now concluded from (31.22)–(31.23) by recalling that the set Ω0 (T) is finite. The proof in the case when t < hT follows from (31.23) and Lemma 31.3.1. The proof of the last part of the lemma follows from a study of the dependence of C(ω, t) on t. −

−(p+1)

In the remainder of this chapter, let δT > 0 be an expansive constant for T : 𝒥 (T) → 𝒥 (T) so small that [T(B(ω, δT ))] ∩ [B(Ω(T) \ {T(ω)}, δT )] = 0 for every ω ∈ Ω(T). Given z ∈ 𝒥 (T) \ Ω−∞ (T), let (as in Lemma 26.2.5(b)) (nj (z))j=1 ⊆ {n ≥ 0 : T n (z) ∉ B(Ω(T), δT )}, ∞

(31.24)

where (nj (z))∞ j=1 is understood as a strictly increasing sequence. Passing to a subsequence if necessary, we may further assume that lim T nj (z) (z) = ̂z

j→∞

(31.25)

for some point ̂z ∈ 𝒥 (T) \ B(Ω(T), δT ). Let θT := θδT be ascribed to δT according to Proposition 26.2.2. Further, set rj,u (z) :=

1 󵄨󵄨 nj (z) ′ 󵄨󵄨−1 ) (z)󵄨󵄨 u 󵄨(T 4󵄨

(31.26)

for every j ∈ ℕ and u ∈ (0, θT ], and let rj (z) := rj,θT (z).

(31.27)

1256 � 31 Conformal measures, invariant measures and fractal geometry The next result is a description of some “weak” form of geometric behavior of mt around any Julia point that lies outside the backward orbit of the set of rationally indifferent periodic points. Indeed, note that the constants ct,u ≥ 1 depend on u. ̂ →ℂ ̂ is a parabolic rational function with deg(T) ≥ 2, then for Lemma 31.3.3. If T : ℂ every t ≥ hT and every u ∈ (0, θT ] there exists a constant ct,u ≥ 1 such that −1 ct,u ≤

mt (B(z, rj,u (z))) t (z) rj,u

≤ ct,u ,

∀j ∈ ℕ, ∀z ∈ 𝒥 (T) \ Ω−∞ (T).

(31.28)

Proof. By Koebe’s 41 –theorem (Theorem 23.1.3), we have 1 󵄨 −n (z) ′ −n (z) 󵄨 B(z, rj,u (z)) = B(z, 󵄨󵄨󵄨(Tz j ) (T nj (z) (z))󵄨󵄨󵄨u) ⊆ Tz j (B(T nj (z) (z), u)). 4 Remember that P(t) = 0 by Theorem 31.2.1, as t ≥ hT . It follows from the t-conformality (31.9) and Koebe’s distortion theorem (Theorem 23.1.8) that 󵄨 −n (z) ′ 󵄨t mt (B(z, rj,u (z))) ≤ K t 󵄨󵄨󵄨(Tz j ) (T nj (z) (z))󵄨󵄨󵄨 mt (B(T nj (z) (z), u)) 4K t t ′ 󵄨 󵄨−t ≤ K t 󵄨󵄨󵄨(T nj (z) ) (z)󵄨󵄨󵄨 = ( ) ⋅ rj,u (z). u

(31.29)

On the other hand, by Koebe’s distortion theorem again, we get B(z, rj,u (z)) ⊇ Tz

−nj (z)

(B(T nj (z) (z), u/(4K))).

(31.30)

Together, the t-conformality (31.9), Koebe’s distortion theorem (Theorem 23.1.8) and Lemma 31.1.6 yield mt (B(z, rj,u (z))) ≥ mt (Tz

−nj (z)

(B(T nj (z) (z), u/(4K))))

󵄨 −n (z) ′ 󵄨t ≥ K −t 󵄨󵄨󵄨(Tz j ) (T nj (z) (z))󵄨󵄨󵄨 mt (B(T nj (z) (z), u/(4K))) ′ 󵄨 󵄨−t ≥ K −t 󵄨󵄨󵄨(T nj (z) ) (z)󵄨󵄨󵄨 Mmt (u/(4K)) 4 t t (z). = ( ) Mmt (u/(4K)) ⋅ rj,u Ku

(31.31)

Combining (31.29) with (31.31) results in (31.28). The next result outlines a relation between conformal measures with exponents greater than or equal to hT . Lemma 31.3.4. Let hT ≤ s ≤ t. If mt′ and ms′′ are t- and s-conformal measures respectively (i. e., they satisfy (31.8) for t and s, respectively), then mt′ |𝒥 (T)\Ω−∞ (T) is absolutely continuous with respect to ms′′ |𝒥 (T)\Ω−∞ (T) . In fact, 2

′′ ′′ mt′ (A) ≤ 4s (cs,θ ) cs,θ c′ ⋅ ms′′ (A) T T /4 t,θT

� 1257

31.3 Sullivan conformal measures for parabolic rational functions

for every Borel set A ⊆ 𝒥 (T) \ Ω−∞ (T). If additionally s < t, then mt′ (𝒥 (T) \ Ω−∞ (T)) = 0. Proof. We deduce from Lemma 31.3.3 that ′′ ′ t−s (cs,u ct,u ) rj,u (z) ≤ −1

mt′ (B(z, rj,u (z)))

ms′′ (B(z, rj,u (z)))

′′ ′ t−s ≤ (cs,u ct,u )rj,u (z)

(31.32)

for all j ∈ ℕ, all u ∈ (0, θT ] and all z ∈ 𝒥 (T) \ Ω−∞ (T). Observe that (31.26)–(31.27) yield rj,θT /4 (z) =

1 r (z), 4 j

∀j ∈ ℕ, ∀z ∈ 𝒥 (T) \ Ω−∞ (T).

(31.33)

Let A be any Borel subset of 𝒥 (T) \ Ω−∞ (T) and let ε > 0. Since the measure ms′′ is outer ̂ such that regular, there exists an open set A ⊆ G ⊆ ℂ ms′′ (G) ≤ ms′′ (A) + ε.

(31.34)

For every z ∈ A, take j(z) ∈ ℕ so large that r(z) ≤ ε

and

B(z, r(z)) ⊆ G,

(31.35)

where r(z) := rj(z) (z).

(31.36)

It follows from the 4r-covering theorem (Theorem 15.4.1 and Remark 15.4.2(e)) that there exists a countable set A∗ ⊆ A such that the family {B(z, r(z)/4)}z∈A∗ consists of mutually disjoint sets and the family {B(z, r(z))}z∈A∗ covers A. Using (31.32)–(31.36) and Lemma 31.3.3 (twice), we obtain that mt′ (A) ≤ mt′ ( ⋃ B(z, r(z))) ≤ ∑ mt′ (B(z, r(z))) z∈A∗

≤

′′ cs,θ c′ T t,θT

t−s

∑ (r(z))

z∈A∗

z∈A∗

ms′′ (B(z, r(z))) 2

′′ ′′ ′ ≤ cs,θ c′ εt−s ∑ ms′′ (B(z, r(z))) ≤ (cs,θ ) ct,θ εt−s ∑ (r(z)) T t,θT T T 2

z∈A∗

s

′′ ′ = 4s (cs,θ ) ct,θ εt−s ∑ (r(z)/4) T T z∈A∗

≤4

s

2 ′′ ′′ (cs,θ ) cs,θ c′ εt−s T T /4 t,θT

∑ ms′′ (B(z, r(z)/4))

z∈A∗

z∈A∗

s

1258 � 31 Conformal measures, invariant measures and fractal geometry 2

′′ ′′ = 4s (cs,θ ) cs,θ c′ εt−s ms′′ ( ⋃ B(z, r(z)/4)) T T /4 t,θT

≤ ≤

z∈A∗ 2 ′′ ′′ 4s (cs,θ ) cs,θ c′ εt−s ms′′ (G) T T /4 t,θT 2 ′′ ′′ 4s (cs,θ ) cs,θ c′ εt−s (ms′′ (A) + T T /4 t,θT

ε).

Letting ε → 0 leads to the result. In fact, in Theorem 31.3.11(a) we will need a more general version of Lemmas 31.3.3– 31.3.4. Given t ≥ 0 and λt > 0, a Borel probability measure νt on 𝒥 (T) is called (t, λt )conformal if it satisfies (31.8), or equivalently (31.9), with eP(t) replaced by λt , i. e. if ℒt (νt ) = λt νt , ∗

󵄨󵄨 −n ′ 󵄨󵄨t or equivalently, νt (Tz−n (A)) = λ−n t ∫󵄨󵄨(Tz ) 󵄨󵄨 dνt ,

(31.37)

A

for all z ∈ 𝒥 (T), all n ∈ ℕ, and all Borel sets A ⊆ B(T n (z), ξ). Making evident adjustments to the proof of Lemma 31.3.3, we get the following generalization. ̂→ℂ ̂ be a parabolic rational function with deg(T) ≥ 2. Let also Lemma 31.3.5. Let T : ℂ t ≥ 0 and λt > 0. If νt is a (t, λt )-conformal measure, then for every u ∈ (0, θT ] there exists a constant ct,u ≥ 1 such that −1 ct,u ≤

νt (B(z, rj,u (z))) −n (z) t λt j rj,u (z)

≤ ct,u ,

∀j ∈ ℕ, ∀z ∈ 𝒥 (T) \ Ω−∞ (T).

(31.38)

We then obtain a generalization of Lemma 31.3.4. ′ ′ ′′ ′ ′′ Lemma 31.3.6. Let 0 ≤ s ≤ t and 0 < λ′′ s ≤ λt . If νt and νs are (t, λt )- and (s, λs )conformal measures respectively, then νt′ |𝒥 (T)\Ω−∞ (T) is absolutely continuous with respect to νs′′ |𝒥 (T)\Ω−∞ (T) . In fact, 2

′′ ′′ νt′ (A) ≤ 4s (cs,θ ) cs,θ c′ ⋅ νs′′ (A) T T /4 t,θT ′ for every Borel set A ⊆ 𝒥 (T) \ Ω−∞ (T). If additionally s < t or λ′′ s < λt , then

νt′ (𝒥 (T) \ Ω−∞ (T)) = 0. Proof. We deduce from Lemma 31.3.5 that ′′ ′ ct,u ) ( (cs,u −1

nj (z)

λ′′ s ) λ′t

t−s rj,u (z) ≤

νt′ (B(z, rj,u (z)))

νs′′ (B(z, rj,u (z)))

′′ ′ ≤ (cs,u ct,u )(

nj (z)

λ′′ s ) λ′t

t−s rj,u (z)

(31.39)

31.3 Sullivan conformal measures for parabolic rational functions

� 1259

for all j ∈ ℕ, all u ∈ (0, θT ] and all z ∈ 𝒥 (T) \ Ω−∞ (T). This is a generalization of (31.32). Following the lines in the proof of Lemma 31.3.4 yields 2

′′ ′′ νt′ (A) ≤ 4s (cs,θ ) cs,θ c′ εt−s (νs′′ (A) + ε). T T /4 t,θT

(31.40)

′ If λ′′ s < λt , then pick 0 < δ ≤ 1. We may choose j(z) ∈ ℕ so that not only (31.35) is satisfied but also that nj(z) (z) ∈ ℕ (cf. (31.24)–(31.25)) is so large that nj(z) (z)

(

λ′′ s ) λ′t

≤ δ.

Going through nearly identical lines to the proof of Lemma 31.3.4 results in 2

′′ ′′ νt′ (A) ≤ 4s (cs,θ ) cs,θ c′ εt−s δ(νs′′ (A) + ε). T T /4 t,θT

(31.41)

Letting ε → 0 in (31.40) and δ → 0 in (31.41) completes the proof.

31.3.2 The atomless hT -conformal measure mT : existence, uniqueness, ergodicity and conservativity We define mT to be a weak cluster measure (i. e., a weak accumulation point) of the measures (mt )t 0 and P(hT ) = 0 per Theorem 31.2.1. According to (31.42), the measure mT is a fixed point of the dual of the transfer operator ℒhT , and hence is hT conformal. Employing Lemma 26.2.5 and formula (31.10) with B being a singleton, we deduce that mT ({z}) = 0 for every z ∈ 𝒥 (T) \ Ω−∞ (T). Now let us turn our attention to ω ∈ Ω(T). By passing to a sufficiently high iterate of T, we may assume without loss of

1260 � 31 Conformal measures, invariant measures and fractal geometry generality that ω ∈ Ω0 (T). Let (tn )∞ n=1 be a sequence such that tn ↗ hT , 0 ≤ tn < hT , and such that the sequence (mtn )∞ converges weakly to the measure mT . The Portmanteau n=1 theorem (Theorem A.1.23) and Lemma 31.3.2 yield that mT (B(ω, r)) ≤ lim inf mtn (B(ω, r)) ≤ lim inf c(tn )r αtn (ω) ≤ ch r αhT (ω) n→∞

n→∞

for all r ∈ (0, R], where h = 21 ( ppT+1 + hT ). This establishes the right-hand side of (31.43). T Consequently, mT ({ω}) = 0. So, mT is atomless. On the other hand, the left-hand side of (31.43) follows from the left-hand side of (31.15) in Lemma 31.3.2. Indeed, (31.42) affirms that the measure mT is a fixed point of the conjugate transfer operator and is hence eP(hT )−φhT -conformal. So, (31.15) holds for mT , too. The following facts are repercussions of this theorem. ̂→ℂ ̂ be a parabolic rational function with deg(T) ≥ 2. Lemma 31.3.8. Let T : ℂ (a) If hT ≥ 1, then for every β > 0 there exists C2 ≥ 1 such that mT (B(z, β|z − ω|)) ≤ C2 |z − ω|hT ,

∀ω ∈ Ω(T), ∀z ∈ 𝒥 (T).

(b) If hT ≤ 1, then for every β > 0 there exists C3 ≥ 1 such that mT (B(z, β|z − ω|)) ≥ C3 |z − ω|hT ,

∀ω ∈ Ω(T), ∀z ∈ 𝒥 (T).

Proof. Through a Möbius conjugation if necessary, we may assume that ∞ ∉ Ω(T). Since Ω(T) is finite, it suffices to show the result for some fixed ω ∈ Ω(T). To establish (a), let β > 0 and take z ∈ ℂ such that (1 + β)|z − ω| ≤ R, where R comes from Theorem 31.3.7. In view of that theorem, we have mT (B(z, β|z − ω|)) ≤ mT (B(ω, (1 + β)|z − ω|)) ≤ c[(1 + β)|z − ω|]

αhT (ω)

≤ c(1 + β)hT |z − ω|hT .

If (1+β)|z−ω| > R, then we can simply rely on the obvious estimate mT (B(z, β|z−ω|)) ≤ 1. Part (a) holds. For (b), let 0 < β < 1. For z ∈ ℂ, consider the annulus A(z) := {x ∈ ℂ : (1 − β)|z − ω| ≤ |x − ω| ≤ (1 + β)|z − ω|}. j

Observe that there exists 0 < α < π/(2p(ω)) such that if z ∈ Sr (ω, α), then A(z) ∩ Srj (ω, α) ⊆ B(z, (2 + β)|z − ω|). Now apply the construction in the proof of Theorem 31.3.7. We use the same notation as in Lemma 31.3.2 but assume additionally that P ⊆ B(ω, θα (ω)) and that the radius δ of the balls B(z1 , δ), . . . , B(zq , δ) is so small that each of these balls is contained in exactly

31.3 Sullivan conformal measures for parabolic rational functions

� 1261

j

one sector Sr (ω, α). This is possible in light of Fatou’s flower theorem (Theorem 24.2.24), the definition of P and since α < π/(2p(ω)). Let e := (|a|p(ω))−1/p(ω) . Choose ε > 0 so small that f := (

p(ω)

e+ε ) (1 + β)(e − ε)

1.

By this and Lemma 24.2.26, there is n0 ∈ ℕ so large that for all n ≥ fn0 and all z ∈ P, g−f n 2

[gn] − ([fn] + 1) ≥

and

󵄨 󵄨 (e − ε)n−1/p(ω) ≤ 󵄨󵄨󵄨Tω−n (z) − ω󵄨󵄨󵄨 ≤ (e + ε)n−1/p(ω) ,

where [t] is the integer part of t. For every n ∈ ℕ, let An := {x ∈ ℂ : (1 − β)(e + ε)n−1/p(ω) ≤ |x − ω| ≤ (1 + β)(e − ε)n−1/p(ω) }. By Theorem 24.2.24, for every z ∈ 𝒥 (T) ∩ B(ω, θα (ω)) \ {ω} there exists exactly one j(z) 1 ≤ j(z) ≤ p(ω) such that z ∈ Sr (ω, α). Moreover, j(Tω−n (z)) = j(z) for every z ∈ 𝒥 (T) ∩ B(ω, θα (ω)) \ {ω}. Hence, for every z ∈ P and every n ≥ fn0 we have An ∩ Srj(z) (ω, α) ⊆ B(Tω−n (z), (2 + β)|Tω−n (z) − ω|). Now consider an arbitrary x ∈ ⋃n≥n0 Tω−n (P). Then x ∈ Tω−n (P) for some n ≥ n0 . Let z = T n (x) and choose 1 ≤ i ≤ q such that z ∈ B(zi , δ). As the radius δ is small enough, we conclude from Theorem 24.2.24 and (31.21) that Tω−l (𝒥 (T) ∩ B(zi , δ)) ⊆ Srj(z) (ω, α),

∀l ≥ 0.

If y ∈ P and fn0 ≤ fn ≤ l ≤ gn, then by the choice of ε and n0 we have that 󵄨󵄨 −l 󵄨 −1/p(ω) ≤ (e + ε)f −1/p(ω) n−1/p(ω) = (1 + β)(e − ε)n−1/p(ω) 󵄨󵄨Tω (y) − ω󵄨󵄨󵄨 ≤ (e + ε)l and 󵄨󵄨 −l 󵄨 −1/p(ω) ≥ (e − ε)g −1/p(ω) n−1/p(ω) = (1 − β)(e + ε)n−1/p(ω) . 󵄨󵄨Tω (y) − ω󵄨󵄨󵄨 ≥ (e − ε)l Thus, Tω−l (P) ⊆ An for all fn0 ≤ fn ≤ l ≤ gn, whence ⋃ Tω−l (B(zi , δ) ∩ P) ⊆ An ∩ Srj(z) (ω, α)

fn≤l≤gn

⊆ B(Tω−n (z), (2 + β)|Tω−n (z) − ω|) = B(x, (2 + β)|x − ω|).

(31.44)

By (31.19), for every 1 ≤ j ≤ q there exist ζj ∈ B(zj , δ) ∩ P and δj > 0 such that B(ζj , δj ) ⊆ B(zj , δ) ∩ P. Set

1262 � 31 Conformal measures, invariant measures and fractal geometry δ := min δj , 1≤j≤q

d := min |ζj − ω|, 1≤j≤q

and

󵄨 −∞ ′ 󵄨 󵄨󵄨 −∞ ′ 󵄨󵄨 󵄨󵄨(T ω ) 󵄨󵄨 := min 󵄨󵄨󵄨(Tω ) (ζi )󵄨󵄨󵄨. 1≤j≤q

As the sets Tω−l (B(zi , δ)), l = [fn] + 1, . . . , [gn], are mutually disjoint, it follows from the combination of the hT -conformality of mT with Koebe’s distortion theorem, Proposition 24.2.27 and Lemma 31.1.6 that mT ( ⋃ Tω−l (B(zi , δ) ∩ P)) ≥ fn≤l≤gn

[gn]

∑ mT (Tω−l (B(zi , δ) ∩ P))

l=[fn]+1 [gn]

≥

∑ mT (Tω−l (B(ζi , δi )))

l=[fn]+1 [gn]

≥

h

∑ [K −1 |(Tω−l )′ (ζi )|] T mT (B(ζi , δ))

l=[fn]+1 [gn]

≥

∑ [ l=[fn]+1

p(ω)+1 1 − ′ 󵄨 hT 󵄨 (|a|pl) p(ω) |ζi − ω|−(p(ω)+1) 󵄨󵄨󵄨(Tω−∞ ) (ζi )󵄨󵄨󵄨] MmT (δ) 2K

[gn]

≥ C4 ∑ l

−

p(ω)+1 hT p(ω)

,

l=[fn]+1

p(ω)+1 ′ 󵄨 hT − 󵄨 󵄨 where C4 := [(2K)−1 (|a|p) p(ω) d −(p(ω)+1) 󵄨󵄨󵄨(T −∞ ω ) 󵄨󵄨] MmT (δ) > 0. Since hT ≤ 1, we deduce from the above and (31.44) that p(ω)+1 g−f − h n(gn) p(ω) T 2 g − f − p(ω)+1 −p(ω)+(p(ω)+1)hT h = C4 g p(ω) T (n−1/p(ω) ) 2 p(ω)+1 g−f − h 󵄨󵄨 −n 󵄨αh (ω) ≥ C4 g p(ω) T 󵄨󵄨Tω (z) − ω󵄨󵄨󵄨 T 2(e + ε)αhT (ω) p(ω)+1 g−f − h |x − ω|hT −p(ω)(1−hT ) = C4 g p(ω) T 2(e + ε)αhT (ω) p(ω)+1 g−f − h ≥ C4 g p(ω) T |x − ω|hT . 2(e + ε)αhT (ω)

mT (B(x, (2 + β)|x − ω|)) ≥ C4

This proves (b) for all points in the set G := ⋃n≥n0 Tω−n (P). As G ⊇ [B(ω, R) \ {ω}] ∩ 𝒥 (T) and as for ω itself there is nothing to prove, statement (b) holds on the neighborhood {ω} ∪ G of ω. Since mT is positive on open sets, we deduce that m(B(x, (2 + β)|x − ω|)) ≥ C for some constant C > 0 and all x ∉ {ω} ∪ G, proving (b). The second main result of this section states that mT is ergodic, conservative, and unique. ̂ → ℂ ̂ is a parabolic rational function with deg(T) ≥ 2, then Theorem 31.3.9. If T : ℂ the dynamical system T : 𝒥 (T) → 𝒥 (T) is conservative and ergodic with respect to the

31.3 Sullivan conformal measures for parabolic rational functions

� 1263

hT -conformal measure mT . In addition, supp(mT ) = 𝒥 (T) and mT is the unique measure satisfying (31.8) with P(hT ) = 0, i. e., the unique hT -conformal measure for T. In other terms, mT = mhT . Proof. The topological exactness of T|𝒥 (T) (per Corollary 24.3.5) and the hT -conformality of mT (in the form (31.10)) guarantee that supp(mT ) = 𝒥 (T). Claim 1. Any hT -conformal measure m′ for T is equivalent to mT . In particular, m′ (Ω−∞ (T)) = 0. Proof of Claim 1. If it turned out that m′ (Ω−∞ (T)) > 0, then the measure m′′ defined by m′′ (A) :=

m′ (A ∩ Ω−∞ (T)) m′ (Ω−∞ (T))

,

∀A ∈ ℬ(𝒥 (T)),

would be well-defined, hT -conformal, and m′′ (𝒥 (T) \ Ω−∞ (T)) = 0. It would then follow from Lemma 31.3.4 that m′ (𝒥 (T) \ Ω−∞ (T)) = 0. In conjunction with Theorem 31.3.7, this would result in mT (𝒥 (T)) = 0. This contradiction shows that m′ (Ω−∞ (T)) = 0. Therefore, the measures m′ and mT are equivalent with bounded Radon–Nikodym derivatives by virtue of Theorem 31.3.7 and Lemma 31.3.4. Claim 1 is proved. ◼ Claim 2. mT is ergodic. Proof of Claim 2. Suppose on the contrary that mT is not ergodic. This means that there exists a Borel set F ⊆ 𝒥 (T) such that 0 < mT (F) < 1

and T −1 (F) = F.

Then the conditional measure of mT on F defined by (mT )F (A) :=

mT (A ∩ F) , mT (F)

∀A ∈ ℬ(𝒥 (T)),

satisfies (31.8) with P(hT ) = 0 but is not equivalent to mT , contrary to Claim 1. Claim 2 is thus established. ◼ Claim 3. If m′ is an hT -conformal measure for T and ρ := dm′ /dmT , then ρ is mT -a. e. T-invariant, i. e., ρ∘T =ρ

mT -a. e.

Proof of Claim 3. By virtue of Claim 1, the function ρ is well-defined mT -almost everywhere. For every z ∈ 𝒥 (T) \ Ω−∞ (T), the set 󵄨 Q(z) := {n ≥ 0 󵄨󵄨󵄨 T n (z) ∉ B(Ω(T), δT )} is infinite. For every n ∈ Q(z), let Vn (z) := Tz−n ( B(T n (z), θT /(6K 2 )) ).

1264 � 31 Conformal measures, invariant measures and fractal geometry Since infn≥0 diam(Vn (z)) = 0, the family V = {Vn (z) : z ∈ 𝒥 (T) \ Ω−∞ (T), n ∈ Q(z)} is (in the terminology of Section 2.8 in Federer [50]) fine at every z ∈ 𝒥 (T) \ Ω−∞ (T). Set ̂n (z) := { ⋃ W : W ∩ Vn (z) ≠ 0 and diam(W ) ≤ 2 diam(Vn (z))}. V W ∈V

(31.45)

From relations established in the proof of Lemma 31.3.3, we deduce the following analogous ones: Tz−n (B(T n (z), θT )) ⊇ B(z, K −1 θT |(T n )′ (z)|−1 ), ′ 󵄨 󵄨−h mT (Tz−n (B(T n (z), θT ))) ≤ K hT 󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 T , θ ′ 󵄨 󵄨−h mT (Vn (z)) ≥ MmT ( T2 )K −hT 󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 T . 6K

(31.46) (31.47) (31.48)

From Lemma 23.1.10, we further have that diam(Vn (z)) ≤

θT 󵄨󵄨 n ′ 󵄨󵄨−1 󵄨(T ) (z)󵄨󵄨 . 3K 󵄨

(31.49)

Using (31.45), (31.49) and (31.46) successively, we get ̂n (z) ⊆ B(z, 3diam(Vn (z))) ⊆ B(z, K −1 θT |(T n )′ (z)|−1 ) ⊆ T −n (B(T n (z), θT )). V z Therefore, by (31.47), ̂n (z)) ≤ K hT 󵄨󵄨󵄨(T n )′ (z)󵄨󵄨󵄨−hT . mT (V 󵄨 󵄨 Along with (31.48), this yields ̂n (z)) mT (V

mT (Vn (z))

≤ K 2hT [MmT (

θT )] . 6K 2 −1

Thus, lim sup (diam(Vn (z)) +

Q(z)∋n→∞

̂n (z)) mT (V mT (Vn (z))

) ≤ K 2hT [MmT (

θT )] 6K 2

−1

< ∞.

So, the hypotheses of Theorem 2.8.17 in Federer [50] are satisfied. That theorem along with Theorem 2.9.8 in [50] imply that ρ(z) :=

m′ (Vn (z)) dm′ (z) = lim Q(z)∋n→∞ mT (Vn (z)) dmT

for mT -a. e. z ∈ 𝒥 (T) \ Ω−∞ (T).

Since Vn (z) = Tz−1 (Vn−1 (T(z))) for all z ∈ 𝒥 (T) \ Ω−∞ (T) and all n ∈ Q(z)\{0}, it ensues that 󵄨󵄨 −1 ′ 󵄨󵄨hT ′ 󵄨󵄨(Tz ) 󵄨󵄨 dm ρ(z) = lim = lim . ′ 󵄨h 󵄨 Q(z)∋n→∞ mT (T −1 (Vn−1 (T(z)))) Q(z)∋n→∞ ∫V (T(z)) 󵄨󵄨󵄨(Tz−1 ) 󵄨󵄨󵄨 T dmT z n−1 m′ (Tz−1 (Vn−1 (T(z))))

∫V

n−1 (T(z))

(31.50)

31.3 Sullivan conformal measures for parabolic rational functions

� 1265

As ′ ′ 󵄨 󵄨 󵄨 󵄨 inf{󵄨󵄨󵄨(Tz−1 ) (w)󵄨󵄨󵄨 : w ∈ Vn−1 (T(z))} sup{󵄨󵄨󵄨(Tz−1 ) (w)󵄨󵄨󵄨 : w ∈ Vn−1 (T(z))} = 1 = lim , 󵄨 󵄨 󵄨󵄨 −1 ′ 󵄨󵄨 −1 ′ Q(z)∋n→∞ Q(z)∋n→∞ 󵄨󵄨(Tz ) (T(z))󵄨󵄨󵄨 󵄨󵄨(Tz ) (T(z))󵄨󵄨󵄨

lim

we can continue (31.50) as follows: ρ(z) =

lim

Q(z)∋n→∞

m′ (Vn−1 (T(z)))

mT (Vn−1 (T(z)))

=

lim

Q(T(z))∋k→∞

m′ (Vk (T(z)))

mT (Vk (T(z)))

= ρ(T(z))

for mT -a. e. z ∈ 𝒥 (T) \ Ω−∞ (T). As mT (𝒥 (T) \ Ω−∞ (T)) = 0 according to Claim 1, we conclude that ρ ∘ T = ρ mT -almost everywhere. Claim 3 is thus proved. ◼ Claim 4. mT is the unique hT -conformal measure for T. Proof of Claim 4. Let m′ be an hT -conformal measure for T. As mT is quasi-T-invariant, it follows from Claims 2–3 that the Radon–Nikodym derivative ρ = dm′ /dmT is constant mT -almost everywhere. Since both measures m′ and mT are probabilistic, the derivative ρ must be equal to 1 mT -a. e. and can be taken to be identically 1. So, m′ = mT and Claim 4 is proved. ◼ Our next goal is to show that the measure mT is conservative. This will easily ensue from the following claim. Claim 5. If E ⊆ 𝒥 (T) is a forward or backward T-invariant Borel set, then mT (E) ∈ {0, 1}. Proof of Claim 5. By passing to the complement, it suffices to prove the statement for a forward T-invariant set E. Since mT (Ω−∞ (T)) = 0 and the set Ω−∞ (T) is completely T-invariant, we may assume that E ⊆ 𝒥 (T) \ Ω−∞ (T). Suppose for a contradiction that 0 < mT (E) < 1. By Lebesgue’s density theorem, there exists at least one point z ∈ E (in fact, the set of such points has mT -measure equal to mT (E)) such that mT (E ∩ B(z, r)) = 1. r→0 mT (B(z, r)) lim

(31.51)

Seeking a contradiction, suppose that mT (B(̂z, θT /(8K)) \ E) = 0, where the reader will recall that ̂z is defined by (31.25). Given that T|𝒥 (T) is topologically exact according to Corollary 24.3.5, let N ∈ ℕ be such that T N (B(̂z, θT /(8K))) ⊇ 𝒥 (T). By partitioning the ball B(̂z, θT /(8K)) into finitely many special Borel sets for T N (cf. Lemma 29.1.15), the hT -conformality of mT entails that mT (T N (B(̂z, θT /(8K)) \ E)) = 0. From the forward T-invariance of E, we deduce that 0 = mT (T N (B(̂z, θT /(8K)) \ E)) ≥ mT (T N (B(̂z, θT /(8K))) \ T N (E))

≥ mT (T N (B(̂z, θT /(8K))) \ E) ≥ mT (T N (B(̂z, θT /(8K)))) − mT (E)

= 1 − mT (E).

1266 � 31 Conformal measures, invariant measures and fractal geometry Hence, mT (E) ≥ 1. This contradiction shows that γ := mT (B(̂z, θT /(8K)) \ E) > 0. As |T nj (z) (z) − ̂z| < θT /(8K) for j ∈ ℕ large enough, say j ≥ N(z), we infer that B(̂z, θT /(8K)) ⊆ B(T nj (z) (z), θT /(4K)),

∀j ≥ N(z).

Using (31.30) and the forward T-invariance of E, we deduce that Tz

−nj (z)

(B(̂z, θT /(8K)) \ E) ⊆ Tz

−nj (z)

(B(T nj (z) (z), θT /(4K)) \ E) ⊆ B(z, rj (z)) \ E,

∀j ≥ N(z).

By virtue of the hT -conformality of mT and of the Koebe distortion theorem (Theorem 23.1.9), we thereby get that mT (B(z, rj (z)) \ E) ≥ mT (Tz j (B(̂z, θT /(8K)) \ E)) 󵄨 −n (z) ′ 󵄨h ≥ K −hT 󵄨󵄨󵄨(Tz j ) (T nj (z) (z))󵄨󵄨󵄨 T mT (B(̂z, θT /(8K)) \ E) ′ 󵄨 󵄨−h = γK −hT 󵄨󵄨󵄨(T nj (z) ) (z)󵄨󵄨󵄨 T , ∀j ≥ N(z). −n (z)

Combining this with (31.29), we obtain that mT (B(z, rj (z)) \ E) mT (B(z, rj (z)))

′ 󵄨 󵄨−h γK −hT 󵄨󵄨󵄨(T nj (z) ) (z)󵄨󵄨󵄨 T ≥ = γK −2hT > 0, ′ 󵄨 󵄨−h K hT 󵄨󵄨󵄨(T nj (z) ) (z)󵄨󵄨󵄨 T

∀j ≥ N(z).

Hence, lim sup j→∞

mT (E ∩ B(z, rj (z))) mT (B(z, rj (z)))

≤ 1 − γK −2hT < 1,

contrary to (31.51). So, mT (E) ∈ {0, 1}. This finishes the proof of Claim 5.

◼

Claim 6. mT is conservative. Proof of Claim 6. According to Theorem 10.1.11 from Volume 1, we need to show that the quasi-T-invariant measure mT is such that mT (B) > 0

󳨐⇒

mT (𝒥 (T) \ B∞ ) = 0,

where ∞ ∞

B∞ := ⋂ ⋃ T −k (B). n=0 k=n

So, let B be a Borel subset of 𝒥 (T) such that mT (B) > 0. Since mT is atomless and conformal per Theorem 31.3.7, it turns out that mT (T −k (B)) > 0 for every k ≥ 0. Indeed,

31.3 Sullivan conformal measures for parabolic rational functions

� 1267

as #T k (Crit(T k )) < ∞ and mT is atomless, we know that mT (T k (Crit(T k ))) = 0. Thus, mT (B \ T k (Crit(T k ))) = mT (B) > 0. In light of Remark 24.4.16(a), we can partition the set B \ T k (Crit(T k )) into finitely many Borel subsets over each of which T k has an inverse branch. At least one of these subsets has positive mT -measure. Call one such subset SB . It ensues from the conformality of mT (in the form of (31.9)) that mT (T −k (SB )) > 0. Hence mT (T −k (B)) > 0,

∀k ≥ 0.

This immediately implies that ∞

mT ( ⋃ T −k (B)) > 0, k=n

∀n ≥ 0.

−k As each set ⋃∞ k=n T (B) is backward T-invariant, we deduce from Claim 5 that ∞

mT ( ⋃ T −k (B)) = 1, k=n

∀n ≥ 0.

Consequently, mT (B∞ ) = 1. So mT (𝒥 (T) \ B∞ ) = 0, and mT is conservative.

◼

This finishes the proof of Theorem 31.3.9. This theorem entails the set of transitive points to be of full mT -measure. ̂→ℂ ̂ is a parabolic rational function with deg(T) ≥ 2, then the Corollary 31.3.10. If T : ℂ mT -measure of the set of transitive points of the dynamical system T : 𝒥 (T) → 𝒥 (T) is equal to 1. Proof. Let {On }∞ n=1 be a countable base for the relative topology of 𝒥 (T). Fix temporarily n ∈ ℕ. Since supp(mT ) = 𝒥 (T), we know that mT (On ) > 0. Then mT (𝒥 (T) \ (On )∞ ) = 0 according to Theorems 31.3.9 and 10.1.11. This means that mT ((On )∞ ) = 1. As the intersection of a countable family of sets of measure 1 has measure 1, we conclude that the set ∞ ∞ ∞

⋂ ⋂ ⋃ T −ℓ (On )

n=1 k=0 ℓ=k

has measure 1. This set is precisely the set of transitive points of T|𝒥 (T) . 31.3.3 The complete structure of Sullivan’s conformal measures for parabolic rational functions: the atomless measure mT and purely atomic measures We can now collect together the results of the previous sections and provide a short concluding proof of the main theorem of the current section. It gives a full description of the structure of the Sullivan conformal measures of a parabolic rational function.

1268 � 31 Conformal measures, invariant measures and fractal geometry ̂→ℂ ̂ be a parabolic rational function with deg(T) ≥ 2. Recall Theorem 31.3.11. Let T : ℂ that Bowen’s formula (Theorem 31.2.1(i)) states that hT = δ(T) = ExpD∗∗ (𝒥 (T)) = ExpD∗ (𝒥 (T)) = ExpD(𝒥 (T)) = DD(𝒥 (T)) = HD(𝒥uer (T)) = HD(𝒥er (T)) = HD(𝒥r (T))

= HD(𝒥 (T)) ∈ (

pT , 2]. pT + 1

The following situations prevail: (a) If t < hT , then T does not admit any Sullivan t-conformal measure. (b) T has exactly one hT -conformal measure, namely mT = mhT . This measure is atomless, ergodic and conservative. (c) For every t > hT and every ζ ∈ 𝒥 (T), it holds that Σt (ζ ) :=

′ 󵄨 󵄨−t ∑ 󵄨󵄨󵄨(T k(z) ) (z)󵄨󵄨󵄨 < ∞,

z∈𝒪− (ζ )

(31.52)

−n k(z) where 𝒪− (ζ ) := ⋃∞ (z) = ζ . n=0 T (ζ ) and k(z) ≥ 0 is the least integer such that T Accordingly, let

νt,ω :=

1 ′ 󵄨 󵄨−t ∑ 󵄨󵄨(T k(z) ) (z)󵄨󵄨󵄨 δz , Σt (ω) z∈𝒪 (ω)󵄨

(31.53)

−

where δz denotes the Dirac measure concentrated at the point z. Then the set of all t-conformal measures for T coincides with the set of all convex combinations of measures {νt,ω }ω∈Ω(T) . In particular, these conformal measures are atomic. Proof. (a) Suppose for a contradiction that t ∈ [0, hT ) and m is a Sullivan t-conformal measure. As 1 < eP(t) (per Theorem 31.2.1), applying Lemma 31.3.6 to the (t, 1)-conformal measure m and the (t, eP(t) )-conformal measure mt = mφt produced at the beginning of Section 31.2 (see (31.8)), we get mt (𝒥 (T) \ Ω−∞ (T)) = 0. Equivalently, mt (Ω−∞ (T)) = 1. The conformality of mt implies that mt (Ω(T)) > 0. So, there exists ω ∈ Ω(T) for which mt ({ω}) > 0. Take n ∈ ℕ such that T n (ω) = ω. Then |(T n )′ (ω)| = 1 and we deduce from (31.10) that ′ 󵄨 󵄨t mt ({ω}) = mt ({T n (ω)}) = enP(t) 󵄨󵄨󵄨(T n ) (ω)󵄨󵄨󵄨 mt ({ω}) > mt ({ω}).

This contradiction completes the proof of (a). (b) This follows from the combination of Theorems 31.3.7 and 31.3.9.

31.3 Sullivan conformal measures for parabolic rational functions

� 1269

(c) For every t ≥ 0 and every x ∈ 𝒥 (T), let Σt (x) :=

′ 󵄨−t 󵄨 ∑ 󵄨󵄨󵄨(T k(z) ) (z)󵄨󵄨󵄨 ,

z∈𝒪− (x)

where k(z) ≥ 0 is the least integer k such that T k (z) = x. If x is not periodic, then ∞

′ 󵄨−t 󵄨 ∑ 󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 .

Σt (x) = ∑

n=0 z∈T −n (x)

The following three results are obvious. Claim 1. For every t ≥ 0 and every x ∈ 𝒥 (T), Σt (x) < ∞

⇐⇒

Σt (z) < ∞, ∀z ∈ T −1 (x) \ 𝒪+ (x).

̂ the set T −1 (x) \ 𝒪+ (x) contains no periodic point of T. Claim 2. For every x ∈ ℂ, Claim 3. If Σt (x) < ∞ for some t ≥ 0 and x ∈ 𝒥 (T), then Σt (z) < ∞ for all z ∈ 𝒪− (x). The next fact deserves a proof. Claim 4. If Σt (x) < ∞ for some t ≥ 0 and x ∈ 𝒥 (T), then Σt (y) < ∞ for every y ∈ 𝒥 (T) \ PC(T). Proof of Claim 4. In light of Claims 1–2, we may assume without loss of generality that neither x nor y is periodic. Recalling Remark 24.4.16(b), there exists r = r(y) > 0 such that B(y, 2r) ∩ PC(T) = 0 and for every n ≥ 0 and every z ∈ T −n (y) there is a unique holomorphic inverse branch Tz−n : B(y, 2r) → ℂ of T n that sends y to z. Given the topological exactness of T|𝒥 (T) , there exists k ≥ 0 such that T k (B(y, r)) ⊇ 𝒥 (T). Therefore, there is y0 ∈ B(y, r) such that T k (y0 ) = x. By Claim 3, Σt (y0 ) < ∞. As y0 (like x) is not periodic and as ⋃z∈T −n (y) Tz−n (y0 ) = T −n (y0 ), we get from Koebe’s distortion theorem that ∞

Σt (y) = ∑

n=0

∞

′ 󵄨 󵄨t ∑ 󵄨󵄨󵄨(Tz−n ) (y)󵄨󵄨󵄨 ≍ ∑

z∈T −n (y)

′ 󵄨 󵄨t ∑ 󵄨󵄨󵄨(Tz−n ) (y0 )󵄨󵄨󵄨 = Σt (y0 ) < ∞,

n=0 z∈T −n (y)

where ≍ is the usual symbol of multiplicative comparability (see (25.34)). Thus, Claim 4 holds. ◼ Claim 5. If Σt (x) < ∞ for some t ≥ 0 and x ∈ 𝒥 (T), then Σt (y) < ∞ for every y ∈ 𝒥 (T) \ Ω(T). Proof of Claim 5. As T is parabolic, it is expansive, i. e., Crit(T) ∩ 𝒥 (T) = 0 per Theorem 26.2.3. Therefore PC(T) ∩ 𝒥 (T) = 0. It also follows from the classification theorem 24.4.10 and Sullivan’s nonwandering theorem (Theorem 24.4.12) that ω(Crit(T)) ∩ 𝒥 (T) ⊆ Ω(T). Together, these two facts imply that PC(T) ∩ 𝒥 (T) ⊆ Ω(T).

1270 � 31 Conformal measures, invariant measures and fractal geometry (Proposition 26.2.2 offers a stronger assertion of this fact.) The claim follows from Claim 4. ◼ As an immediate consequence of Claims 5 and 1, we get the following. Claim 6. Given any t ≥ 0 and x ∈ 𝒥 (T), Σt (x) < ∞

⇐⇒

Σt (y) < ∞, ∀y ∈ 𝒥 (T).

We finally prove a claim for the specific range of t’s we are interested in. Claim 7. If t > hT , then there is ω ∈ Ω(T) such that Σt (ω) < ∞. Proof of Claim 7. Recall that mt = mφt is the measure produced at the beginning of Section 31.2 (see (31.8)) with P(t) = 0 per Theorem 31.2.1. Thus, mt is t-conformal. Taking any s ∈ [hT , t), it follows from Lemma 31.3.4 that mt (𝒥 (T) \ Ω−∞ (T)) = 0. So there exists ω ∈ Ω(T) for which mt (𝒪− (ω)) > 0 and thereby mt ({ω}) > 0. But mt (𝒪− (ω)) = mt ({ω})Σt (ω), whence Σt (ω) = (mt ({ω})) mt (𝒪− (ω)) ≤ (mt ({ω})) −1

−1

< ∞.

◼

Together, Claims 6–7 conclude the proof of (31.52). By definition (31.53), all measures {νt,ω }ω∈Ω(T) are t-conformal. Obviously, so are all their convex combinations. On the other hand, let m be a t-conformal measure. It ensues from Lemma 31.3.4 (applied with s = hT ) that m(Ω−∞ (T)) = 1. That is, ∑ω∈Ω(T) m(𝒪− (ω)) = 1. Using the t-conformality of m, we readily deduce that m is a convex combination of measures νt,ω , ω ∈ Ω(T). Thus, (c) holds.

31.4 Invariant measures equivalent to mT : existence, uniqueness, ergodicity and the finite–infinite dichotomy In this section, we establish the existence and uniqueness of a T-invariant measure which is absolutely continuous with respect to the hT -conformal measure mT = mhT , and we provide a simple criterion for that invariant measure to be either finite or infinite. This is the counterpart of Theorem 30.1.1(e). ̂ → ℂ ̂ is a parabolic rational function with deg(T) ≥ 2, then Theorem 31.4.1. If T : ℂ there exists a T-invariant σ-finite Borel measure μT on 𝒥 (T) which is absolutely continuous with respect to the hT -conformal measure mT . In fact, the measure μT is unique up to a multiplicative constant, equivalent to mT , ergodic and conservative. Furthermore, μT (𝒥 (T) \ B(Ω(T), R)) < ∞ for every R > 0. Proof. Fix any R ∈ (0, diam(𝒥 (T))). By Proposition 26.2.2, for every z ∈ 𝒥 (T) \ Ω(T) there exists η(z) > 0 such that

31.4 Invariant measures equivalent to mT

� 1271

B(z, 2η(z)) ∩ PC(T) = 0. As there are countably many r > 0 for which mT (𝜕B(z, r)) = 0, we may further assume that each η(z) is such that mT (𝜕B(z, η(z))) = 0. Since 𝒥 (T) \ B(Ω(T), R) is compact, there is a finite set ER ⊆ 𝒥 (T) \ B(Ω(T), R) such that ⋃ B(z, η(z)) ⊇ 𝒥 (T) \ B(Ω(T), R).

(31.54)

z∈ER

Out of the cover {B(z, η(z))}z∈ER of 𝒥 (T) \ B(Ω(T), R), form in the standard way (see, for instance, the last step in the proof of Lemma 25.6.4) a finite partition {Pz }z∈ER of 𝒥 (T) \ B(Ω(T), R) such that Pz ⊆ B(z, η(z)),

∀z ∈ ER .

As (𝒥 (T) \ Ω(T)) ∩ B(Ω(T), R) is a Lindelöf space, there is a countable set FR ⊆ B(Ω(T), R) such that ⋃ B(z, η(z)) ⊇ (𝒥 (T) \ Ω(T)) ∩ B(Ω(T), R).

z∈FR

In the same standard way, construct out of the cover {B(z, η(z))}z∈FR a countable partition {Pz }z∈FR of (𝒥 (T) \ Ω(T)) ∩ B(Ω(T), R) such that Pz ⊆ B(z, η(z)),

∀z ∈ FR .

In fact, in the standard construction of the partitions {Pz }z∈ER and {Pz }z∈FR we may subtract from each element of the cover the closure of the preceding elements (we obviously throw away all empty sets generated by this construction). This way, the elements of the partition {Pz }z∈ER ∪FR will be nonempty, open and the complement of their union will be a set of mT -measure zero. This will be useful in part (c) below. The countable partition {Pz }z∈ER ∪FR of 𝒥 (T) \ Ω(T) is a Marco–Martens cover for the Marco–Martens map T : 𝒥 (T) → 𝒥 (T) per Definition 10.4.3: (a) All the atoms of {Pz }z∈ER ∪FR are Borel sets. (b) As mT is atomless according to Theorem 31.3.11 and Ω(T) is finite, it is trivial that mT (Ω(T)) = 0. (c) Since each Pz has nonempty interior by construction and since T|𝒥 (T) is topologically exact while supp(mT ) = 𝒥 (T), it ensues that mT (Pz ∩ T −k (Pw )) > 0 for every z, w ∈ ER ∪ FR and some k = k(z, w) ≥ 0. (d) For each z ∈ ER ∪ FR , Koebe’s distortion theorem (Theorem 23.1.8) and the hT conformality of mT ensure that mT (T −n (A))mT (B) ≤ KmT (A)mT (T −n (B)), for all Borel sets A, B ⊆ Pz .

∀n ≥ 0

1272 � 31 Conformal measures, invariant measures and fractal geometry (e) Given that mT is quasi-T-invariant, conservative, and mT (Pz ) > 0 for all z ∈ ER ∪ FR , −n it follows from Corollary 10.1.10 that ∑∞ n=0 mT (T (Pz )) = ∞ for every z ∈ ER ∪ FR . (f) This property holds since T is Borel measurable. (g) This property holds per Remark 10.4.4(5) since T is finite-to-one. Let μT be the T-invariant Borel σ-finite measure resulting from Theorem 10.4.5 with reference measure mT . The uniqueness up to a multiplicative constant, the ergodicity and the conservativity of μT ensue from Theorems 31.3.9 and 10.4.5. Moreover, the σ-finiteness of μT imposes that μT (Pz ) ≤ μT (B(z, η(z))) < ∞ for all z ∈ ER ∪ FR . Since the set ER is finite and since {Pz }z∈ER is a partition of 𝒥 (T) \ B(Ω(T), R), it follows that μT (𝒥 (T) \ B(Ω(T), R)) < ∞. We now provide a characterization of the finiteness of μT . ̂→ℂ ̂ is a parabolic rational function with deg(T) ≥ 2, then the Theorem 31.4.2. If T : ℂ measure μT from Theorem 31.4.1 is finite if and only if hT >

2pT , pT + 1

where pT = max{p(ω) : ω ∈ Ω(T)}. Proof. As this measure is invariant for all the iterates of T, replacing T by a sufficiently high iterate allows us to assume that Ω(T) = Ω0 (T), i. e. that Ω(T) consists of simple fixed points of T only. As the map T : 𝒥 (T) → 𝒥 (T) is expansive, by virtue of Corollary 5.3.2 there exists a finite Markov partition ℛ for T : 𝒥 (T) → 𝒥 (T) with diam(ℛ) < δT /2, where we recall that δT is an expansive constant for T|𝒥 (T) . For every ω ∈ Ω(T), let ℛ(ω) = {R ∈ ℛ : ω ∈ R}

and

V (ω) :=

⋃ R.

R∈ℛ(ω)

For every ω ∈ Ω(T), let Tω−1 be the local holomorphic inverse branch of T fixing ω. Because all elements of Ω(T) are simple fixed points, all the iterates Tω−n , ω ∈ Ω(T), n ≥ 0, are well-defined on V (ω) and Tω−n (R) ⊆ R,

∀R ∈ ℛ(ω), ∀ω ∈ Ω(T), ∀n ≥ 0.

Then Tω−n (V (ω)) ⊆ V (ω),

∀ω ∈ Ω(T), ∀n ≥ 0.

We further set Vn (ω) := Tω−n (V (ω)) \ Tω−(n+1) (V (ω)) = Tω−n (V0 (ω)), and

∀n ≥ 0,

(31.55)

31.4 Invariant measures equivalent to mT

�

1273

V c := 𝒥 (T) \ V .

V := ⋃ V (ω) and ω∈Ω(T)

By Theorem 31.4.1, 0 < μT (V c ) < ∞

μT (V c \ (V c )∞ ) = 0,

and

(31.56)

and there exists Q ≥ 1 such that Q−1 ≤

dμT 󵄨󵄨󵄨󵄨 󵄨 ≤ Q. dmT 󵄨󵄨󵄨V c

(31.57)

Let τV c : V c → ℕ and

TV c : V c → V c

be respectively the first return time and the first return map to V c (see (10.1)–(10.2)). All the iterates of TV c are well-defined on V c \ Ω−∞ (T). The conditional measure (μT )V c of μT on V c is TV c -invariant according to Theorem 10.2.1. By Corollary 10.1.14 and Kac’s lemma (Lemma 10.2.6), the measure μT is finite if and only if ∫ τV c dμT < ∞. Vc

Thus, our task boils down to characterizing the finiteness of the integral of the first return time to V c . For every ω ∈ Ω(T) and every n ≥ 0, let Vn∗ (ω) :=

⋃ z∈T −1 (ω)\{ω}

Tz−1 (Vn (ω)) ⊆ V c .

Then τV c |Vn∗ (ω) = n + 2

and

τV c |V c \T −1 (V ) = 1.

(31.58)

Using (31.58), we obtain that ∫ τV c dμT = Vc

∫ V c \T −1 (V )

=

∫ V c \T −1 (V )

∞

τV c dμT + ∑

∑

∫ τV c dμT

ω∈Ω(T) n=0 V ∗ (ω) n

1 dμT + ∑

∞

∑

∫ (n + 2)dμT

ω∈Ω(T) n=0 V ∗ (ω) n

= μT (V c \T −1 (V )) + ∑

∞

∑ (n + 2)μT (Vn∗ (ω)).

ω∈Ω(T) n=0

(31.59)

1274 � 31 Conformal measures, invariant measures and fractal geometry We need to estimate μT (Vn∗ (ω)). Let H := min{mT (V0 (ω)) : ω ∈ Ω(T)} > 0. By (24.84) in Proposition 24.2.27, there exists Γ ≥ 1 such that Γ−1 (n + 2)

−

p(ω)+1 p(ω)

p(ω)+1 ′ − 󵄨 󵄨 ≤ 󵄨󵄨󵄨(Tω−n ) (z)󵄨󵄨󵄨 ≤ Γ(n + 2) p(ω)

for all n ≥ 0, all ω ∈ Ω(T) and all z ∈ V0 (ω). Combining this with (31.55) and the hT conformality of mT , it follows that HΓ−1 ≤

mT (Vn (ω)) (n + 2)

−

≤ Γ,

p(ω)+1 hT p(ω)

∀ω ∈ Ω(T), ∀n ≥ 0.

(31.60)

󵄨 󵄨 Since the derivatives 󵄨󵄨󵄨(Tx−1 )′ (z)󵄨󵄨󵄨, x ∈ T −1 (ω)\{ω}, z ∈ V , are uniformly bounded away from zero and infinity, it ensues from the hT -conformality of mT and (31.60) that there is C ≥ 1 such that C −1 ≤

mT (Vn∗ (ω)) (n + 2)

−

p(ω)+1 hT p(ω)

≤ C,

∀ω ∈ Ω(T), ∀n ≥ 0.

Along with (31.57), this yields (QC)−1 ≤

μT (Vn∗ (ω)) (n + 2)

−

p(ω)+1 hT p(ω)

≤ QC,

∀ω ∈ Ω(T), ∀n ≥ 0.

(31.61)

By (31.56), we know that μT (V c \T −1 (V )) < ∞. We infer from this, (31.59) and (31.61) that ∞

∫ τV c dμT − μT (V c \T −1 (V )) ≍ ∑ ∑ (n + 2) ω∈Ω(T) n=0

Vc

1−

p(ω)+1 hT p(ω)

,

where ≍ is the usual symbol of multiplicative comparability (see (25.34)). But the double series converges if and only if 1 − ppT +1 hT < −1, which is equivalent to T

hT >

2pT . pT + 1

So, by Corollary 10.1.14 and Kac’s lemma (Lemma 10.2.6), the measure μT is finite if and only if hT > 2pT /(pT + 1). As a consequence of this theorem and Bowen’s formula (Theorem 31.2.1(i); see also Theorem 31.3.11), we get the following two results. ̂→ℂ ̂ is a parabolic rational function with deg(T) ≥ 2 and for Corollary 31.4.3. If T : ℂ which HD(𝒥 (T)) ≤ 1, then the measure μT is infinite.

31.5 Hausdorff and packing measures

� 1275

̂→ℂ ̂ is a parabolic rational function with deg(T) ≥ 2 and for Corollary 31.4.4. If T : ℂ which HD(𝒥 (T)) > 1 and pT = 1, then the measure μT is finite. ̂→ℂ ̂ be a finite Blaschke product. (For more information on Example 31.4.5. Let B : ℂ Blaschke products, see [53].) By the Denjoy–Wolff theorem (see [31]), there exists a fixed point ω ∈ B(0, 1) such that lim Bn (z) = ω

n→∞

Hence,

∀z ∈ B(0, 1),

while

lim Bn (z) = 1/ω,

n→∞

̂ \ B(0, 1). ∀z ∈ ℂ

1

𝒥 (B) ⊆ 𝕊 .

If ω ∈ 𝕊1 and B′ (ω) = 1, then ω = 1/ω ∈ Ω0 (B). If additionally Crit(B) ∩ 𝕊1 = 0, then B is a parabolic rational function. So, Corollary 31.4.3 applies. ̂→ℂ ̂ be the quadratic polynomial Example 31.4.6. Let Q1/4 : ℂ Q1/4 (z) := z2 + 1/4 for which 1/2 is a parabolic fixed point since Q1/4 (1/2) = 1/2

′ and Q1/4 (1/2) = 1.

Apart from the superattracting point ∞, Q1/4 has only one critical point, namely 0. By Theorem 24.2.22, this point must belong to the immediate basin of attraction of 1/2. Therefore, Q1/4 is a parabolic rational function. A straightforward calculation shows that p(1/2) = 1. As HD(𝒥 (Q1/4 )) > 1 by [151] or [137] (the proof in [137] is quite similar to that of Theorem 30.2.2), Corollary 31.4.4 applies.

31.5 Hausdorff and packing measures In this section, we give a precise geometric description of the fractal structure of the Julia set of parabolic rational functions in terms of the Hausdorff and packing measures. This is the counterpart of Theorem 30.1.1(f). ̂→ℂ ̂ is a parabolic rational function with deg(T) ≥ 2, then the Theorem 31.5.1. If T : ℂ following statements hold: (a) If hT < 1, then HhT (𝒥 (T)) = 0 and there exists a constant c ∈ (0, ∞) such that mhT = cPhT . In particular, 0 < PhT (𝒥 (T)) < ∞. (b) If hT = 1, then there exist constants c, d ∈ (0, ∞) such that mhT = cPhT = dHhT . In particular, 0 < HhT (𝒥 (T))PhT (𝒥 (T)) < ∞. (c) If hT > 1, then PhT (𝒥 (T)) = ∞ and there exists a constant d ∈ (0, ∞) such that mhT = dHhT . In particular, 0 < HhT (𝒥 (T)) < ∞. (d) HD(𝒥 (T)) ≤ PD(𝒥 (T)) < 2.

1276 � 31 Conformal measures, invariant measures and fractal geometry The proof requires several intermediate results. Recall that 󵄨 Ω0 (T) := {ω ∈ Ω(T) 󵄨󵄨󵄨 T(ω) = ω and T ′ (ω) = 1}. The following result is a consequence of Lemma 24.2.11 and Fatou’s flower theorem (Theorem 24.2.24). ̂→ℂ ̂ is a parabolic rational function with deg(T) ≥ 2, then there Lemma 31.5.2. If T : ℂ exists κ > 0 such that T admits holomorphic inverse branches on all balls of radius κ centred on points of 𝒥 (T) and for every ω ∈ Ω0 (T) there is a unique analytic inverse ̂ of T such that Tω−1 (ω) = ω and branch Tω−1 : B(ω, κ) → ℂ 󵄨󵄨 −1 󵄨 󵄨󵄨Tω (z) − ω󵄨󵄨󵄨 < |z − ω|,

∀z ∈ B(ω, κ) ∩ 𝒥 (T) \ {ω}.

In particular, Tω−1 (B(ω, κ) ∩ 𝒥 (T)) ⊆ B(ω, κ) ∩ 𝒥 (T), and thus all iterates Tω−n : B(ω, κ) ∩ 𝒥 (T) → B(ω, κ) ∩ 𝒥 (T) of Tω−1 |B(ω,κ)∩𝒥 (T) are welldefined. One may further choose κ so small that 0 < κ < θT and [B(ω, κ) ∪ T(B(ω, κ))] ∩ B(ω0 , κ) = 0,

∀ω, ω0 ∈ Ω0 (T) with ω ≠ ω0 .

̂→ℂ ̂ be a parabolic rational function with deg(T) ≥ 2 and let Lemma 31.5.3. Let T : ℂ κ > 0 fulfill Lemma 31.5.2. There exists a constant C1 > 0 such that if T n (z) ∈ B(ω, κ/2) for some ω ∈ Ω0 (T) and T n−1 (z) ∉ B(Ω(T), κ/2), then for every r > 0 satisfying ′ 󵄨 󵄨 󵄨 󵄨 2K 󵄨󵄨󵄨T n (z) − ω󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨r ≤ (2K)−1 κ,

(31.62)

we have ′ ′ 󵄨 󵄨p(ω)(hT −1) mT (B(z, r)) 󵄨 󵄨p(ω)(hT −1) C1−1 󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 ≤ ≤ C1 󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 , αhT (ω) r

where K is a Koebe distortion constant, and p(ω) and αhT (ω) were respectively defined in (24.24) and (31.14). Proof. By the right inequality in (31.62), by Lemma 31.5.2 and by Koebe’s distortion theorem (Theorem 23.1.10), we get B(z, r) ⊆ Tz−n ( B(T n (z), K|(T n )′ (z)|r) ).

(31.63)

By the left inequality in (31.62), we have |T n (z) − ω| ≤ K|(T n )′ (z)|r. Using this and the right inequality in (31.62) successively, we obtain that B(T n (z), K|(T n )′ (z)|r) ⊆ B(ω, 2K|(T n )′ (z)|r) ⊆ B(ω, κ).

(31.64)

31.5 Hausdorff and packing measures

� 1277

Relations (31.63)–(31.64), the hT -conformality of mT in the form (31.9), the Koebe distortion theorem and Theorem 31.3.7 yield that mT (B(z, r)) ≤ mT (Tz−n ( B(ω, 2K|(T n )′ (z)|r) )) ′ 󵄨 󵄨h ≤ K hT 󵄨󵄨󵄨(Tz−n ) (T n (z))󵄨󵄨󵄨 T mT (B(ω, 2K|(T n )′ (z)|r)) α (ω) ′ 󵄨−h 󵄨 ≤ K hT 󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 T c(2K|(T n )′ (z)|r) hT ′ 󵄨 󵄨p(ω)(hT −1) αhT (ω) = cK hT (2K)αhT (ω) ⋅ 󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 ⋅r .

On the other hand (though in a similar way), by the right inequality in (31.62), by Lemma 31.5.2 and by the Koebe distortion theorem, we get Tz−n ( B(T n (z), K −1 |(T n )′ (z)|r) ) ⊆ B(z, r)

(31.65)

while, by the left inequality in (31.62) and (31.64), we have B(ω, (2K)−1 |(T n )′ (z)|r) ⊆ B(T n (z), K −1 |(T n )′ (z)|r) ⊆ B(ω, κ).

(31.66)

Relations (31.65)–(31.66), the hT -conformality of mT in the form (31.9), the Koebe distortion theorem and Theorem 31.3.7 give mT (B(z, r)) ≥ mT (Tz−n ( B(ω, (2K)−1 |(T n )′ (z)|r) )) ′ 󵄨 󵄨−h ≥ K −hT 󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 T mT (B(ω, (2K)−1 |(T n )′ (z)|r)) ′ α (ω) 󵄨 󵄨−h ≥ K −hT 󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 T c−1 ((2K)−1 |(T n )′ (z)|r) hT ′ 󵄨 󵄨p(ω)(hT −1) αhT (ω) = c−1 K −hT (2K)−αhT (ω) ⋅ 󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 ⋅r .

Take C1 = cK hT (2K)αhT (ω) . ̂→ℂ ̂ is a parabolic rational function with deg(T) ≥ 2, then Proposition 31.5.4. If T : ℂ there exists a constant C > 0 such that: (a) HhT ≤ CmT , and thus HhT (𝒥 (T)) < ∞. (b) mT ≤ CPhT , and hence PhT (𝒥 (T)) > 0. Proof. (a) Let A ⊆ 𝒥 (T) be any Borel set. Let Ac := A \ Ω−∞ (T). Since Ω−∞ (T) is countable, we know that HhT (Ac ) = HhT (A). Fix ε, ξ > 0. Per a construction similar to that in the proof of Lemma 31.3.4 (replace A by Ac and ε by ξ in (31.34)), the family {B(z, r(z))}z∈Ac forms an open cover of Ac such that mhT ( ⋃ B(z, r(z))) ≤ mhT (Ac ) + ξ z∈Ac

and

0 < r(z) ≤ ε, ∀z ∈ Ac .

1278 � 31 Conformal measures, invariant measures and fractal geometry By Besicovitch’s covering theorem (Theorem 15.4.3), there exists a countable subset c ∞ c {zk }∞ k=1 of A such that the collection {B(zk , r(zk ))}k=1 covers A and can be decomc posed into C = c(2) packings of A , where C is a universal constant, i. e., C is independent of the cover and of the subcover. Applying Lemma 31.3.3 to the measure mhT , we deduce that ∞

hT

∑ (r(zk ))

k=1

∞

≤ chT ,θT ∑ mhT (B(zk , r(zk ))) k=1

∞

≤ chT ,θT CmhT ( ⋃ B(zk , r(zk ))) k=1

≤ CchT ,θT (mhT (Ac ) + ξ). Letting ξ ↘ 0 and then ε ↘ 0, we conclude that HhT (A) = HhT (Ac ) ≤ CchT ,θT mhT (Ac ) ≤ CchT ,θT mhT (A). (b) Given z ∈ 𝒥 (T) \ Ω−∞ (T), let (nj (z))∞ j=1 be the sequence of all n ∈ ℕ such that ′ 󵄨 󵄨 n T (z) ∉ B(Ω(T), δT ) (cf. (31.24)). Recall that limj→∞ 󵄨󵄨󵄨(T nj (z) ) (z)󵄨󵄨󵄨 = ∞ according to Lemma 26.2.5(b). Moreover, by Lemma 31.3.3, ch−1T ,θT ≤

mT (B(z, rj (z))) (rj (z))hT

≤ chT ,θT ,

∀j ∈ ℕ,

′ 󵄨 󵄨−1 where rj (z) = 41 θT 󵄨󵄨󵄨(T nj (z) ) (z)󵄨󵄨󵄨 . So, limj→∞ rj (z) = 0. As the set Ω−∞ (T) is countable and the measure mT is nonatomic per Theorem 31.3.11, the result ensues from Theorem 15.5.4(b) with A = X = 𝒥 (T).

Recall that 𝒪+ (z) = {T n (z) : n ≥ 0} is the forward T-orbit of z. We now establish that the set of Julia points whose forward orbit’s closure does not intersect Ω(T), is negligible. Define 󵄨 S(T) := {z ∈ 𝒥 (T) 󵄨󵄨󵄨 𝒪+ (z) ∩ Ω(T) = 0}.

(31.67)

̂ → ℂ ̂ is a parabolic rational function with deg(T) ≥ 2, then Lemma 31.5.5. If T : ℂ HhT (S(T)) = PhT (S(T)) = mT (S(T)) = 0. Proof. Observe that S(T) = ⋃∞ k=1 Sk (T), where 󵄨 Sk (T) := {z ∈ 𝒥 (T) 󵄨󵄨󵄨 𝒪+ (z) ∩ B(Ω(T), 1/k) = 0} ⊆ 𝒥 (T) \ Ω−∞ (T). Since T(Sk (T)) ⊆ Sk (T), since 𝒥 (T)\Sk (T) is nonempty and relatively open and since the hT -conformal measure mT is conservative, ergodic and positive on nonempty relatively open sets per Theorem 31.3.9, we know by Theorem 10.1.11 that

31.5 Hausdorff and packing measures

mT (𝒥 (T) \ (𝒥 (T)\Sk (T))∞ ) = 0,

and thus

mT (Sk (T)) = 0,

� 1279

∀k ∈ ℕ.

Because S(T) = ⋃∞ k=1 Sk (T), we deduce that mT (S(T)) = 0 and the conclusion that HhT (S(T)) = 0 follows from Proposition 31.5.4(a). To corroborate that PhT (S(T)) = 0, we will use Lemma 31.3.3. Fix any k ∈ ℕ such n that 1/k < δT . Let z ∈ Sk (T). Then the sequence (nj (z))∞ j=1 of all n ∈ ℕ such that T (z) ∉ ∞ B(Ω(T), 1/k) is simply the full sequence (j)j=1 . Note that 1/k is an expansive constant for T|𝒥 (T) . This is all that is required about δT from relation (31.24) onwards. Hence, all the concepts and results introduced and proved from that point on remain true with δT replaced by 1/k. Only resulting constants are affected. In particular, θT will be replaced by θ1/k in the rest of this proof. Per Lemma 31.3.3, mT (B(z, rj (z)))

ch−1T ,θ1/k ≤

hT

(rj (z))

≤ chT ,θ1/k ,

∀j ∈ ℕ,

′ ′ 󵄨 󵄨−1 󵄨 󵄨 where rj (z) = 41 θ1/k 󵄨󵄨󵄨(T j ) (z)󵄨󵄨󵄨 . Per Lemma 26.2.5(b), limj→∞ 󵄨󵄨󵄨(T j ) (z)󵄨󵄨󵄨 = ∞ and thus limj→∞ rj (z) = 0. Observe that

󵄨󵄨 j+1 ′ 󵄨󵄨 󵄨(T ) (z)󵄨󵄨 󵄨󵄨 ′ j 󵄨 = 󵄨󵄨 = 󵄨󵄨T (T (z))󵄨󵄨󵄨, rj+1 (z) 󵄨󵄨(T j )′ (z)󵄨󵄨󵄨 󵄨 󵄨 rj (z)

∀j ∈ ℕ.

Since 𝒥 (T) is compact and 𝒥 (T) ∩ Crit(T) = 0 due to the parabolicity of T, there exists a constant C ≥ 1 such that 󵄨 󵄨 󵄨 󵄨 C −1 ≤ inf 󵄨󵄨󵄨T ′ (ζ )󵄨󵄨󵄨 ≤ sup 󵄨󵄨󵄨T ′ (ζ )󵄨󵄨󵄨 ≤ C. ζ ∈𝒥 (T) ζ ∈𝒥 (T)

Therefore, C −1 ≤

rj (z)

rj+1 (z)

≤ C,

∀j ∈ ℕ.

Let 0 < r ≤ 41 θ1/k and set j = j(r, z) ∈ ℕ so that rj+1 (z) < r ≤ rj (z). It ensues that mT (B(z, rj+1 (z))) hT

≤

mT (B(z, r)) mT (B(z, rj (z))) ≤ h r hT (rj+1 (z)) T

≤

mT (B(z, r)) mT (B(z, rj (z))) hT ≤ C h r hT (rj (z)) T

(rj (z)) C −hT

mT (B(z, rj+1 (z))) hT

(rj+1 (z))

C −hT ch−1T ,θ1/k ≤

mT (B(z, r)) ≤ chT ,θ1/k C hT . r hT

1280 � 31 Conformal measures, invariant measures and fractal geometry It follows from Theorems 15.5.3–15.5.4(a) that HhT (Sk (T)) = PhT (Sk (T)) = mT (Sk (T)) = 0 for all k such that 1/k < δT , and in fact for all k ∈ ℕ since the sequence (Sk (T))∞ k=1 is ascending. As S(T) = ⋃∞ S (T), the result holds. k=1 k The last ingredient needed to complete the proof of Theorem 31.5.1 is next. ̂→ℂ ̂ be a parabolic rational function with deg(T) ≥ 2. Proposition 31.5.6. Let T : ℂ (a) If hT < 1, then HhT (𝒥 (T)) = 0. (b) If hT ≥ 1, then HhT (𝒥 (T)) > 0. (c) If hT ≤ 1, then PhT (𝒥 (T)) < ∞. (d) If hT > 1, then PhT (𝒥 (T)) = ∞. Proof. (a) As #Ω(T) < ∞, there is q ∈ ℕ with Ω(T q ) = Ω0 (T q ) = Ω(T). As 𝒥 (T q ) = 𝒥 (T), we may without loss of generality assume that Ω(T) = Ω0 (T). So, pick ω ∈ Ω(T) = Ω0 (T) and let X(ω) := {z ∈ 𝒥 (T) \ Ω−∞ (T) : ω ∈ 𝒪+ (z)}. Let 0 < κ < δT satisfy Lemma 31.5.2, and thereby Lemma 31.5.3 as well. As the diameter of B(ω, κ/2) does not exceed an expansive constant of T : 𝒥 (T) → 𝒥 (T), there exists for each z ∈ X(ω) a strictly increasing sequence (nj (z))∞ j=1 of positive integers such

that limj→∞ |T nj (z) (z) − ω| = 0 and T nj (z) (z) ∈ B(ω, κ/2) while T nj (z)−1 (z) ∉ B(ω, κ/2) for every j ∈ ℕ (or equivalently T nj (z)−1 (z) ∉ B(Ω(T), κ/2) because of the choice of κ in Lemma 31.5.2). Setting ′ 󵄨 󵄨󵄨 󵄨−1 rj (z) = 2K 󵄨󵄨󵄨T nj (z) (z) − ω󵄨󵄨󵄨󵄨󵄨󵄨(T nj (z) ) (z)󵄨󵄨󵄨 ,

we glean from Lemma 31.5.3 that for j large enough 󵄨 󵄨p(ω)(hT −1) mT (B(z, rj (z))) 󵄨 󵄨p(ω)(hT −1) C −1 󵄨󵄨󵄨T nj (z) (z) − ω󵄨󵄨󵄨 ≤ ≤ C 󵄨󵄨󵄨T nj (z) (z) − ω󵄨󵄨󵄨 hT (rj (z)) where C = C1 (2K)p(ω)(hT −1) . Therefore, lim sup r→0

mT (B(z, r)) = ∞ when hT < 1 r hT

whereas

lim inf r→0

mT (B(z, r)) = 0 when hT > 1. r hT

If hT < 1, then HhT (X(ω)) = 0 per Theorem 15.5.3(a). We also know that HhT (S(T)) = 0 by Lemma 31.5.5. As Ω−∞ (T) is countable, we further have that HhT (Ω−∞ (T)) = 0. Since 𝒥 (T) = S(T) ∪ ⋃ω∈Ω(T) X(ω) ∪ Ω−∞ (T), we conclude that HhT (𝒥 (T)) = 0. So, part (a) holds. (d) On the other hand, as there is an ergodic T-invariant measure equivalent to mT (namely, μT from Theorem 31.4.1) and as the conformal mT is positive on nonempty open sets, we deduce that mT (⋃ω∈Ω(T) X(ω)) = 1. This, Theorem 15.5.4(b) and the above show that PhT (⋃ω∈Ω(T) X(ω)) = ∞ when hT > 1. Thus (d) holds.

31.5 Hausdorff and packing measures

� 1281

(b) According to Theorem 15.5.3(b), part (b) of the current lemma will hold if we establish that whenever hT ≥ 1, there exists M ≥ 1 such that mT (B(z, r)) ≤ M, r hT

∀0 < r ≤ 1, ∀z ∈ 𝒥 (T).

(31.68)

For this, it suffices to show the existence of constants C, M ≥ 1 such that for each z ∈ 𝒥 (T) there is a sequence of radii 0 < ri (z) ≤ 1 satisfying lim ri (z) = 0,

i→∞

C −1 ≤

ri (z) ≤C ri+1 (z)

and

mT (B(z, ri (z))) hT

(ri (z))

≤ M,

∀i ∈ ℕ.

Fix z ∈ 𝒥 (T). According to Proposition 26.2.2, outside of B(Ω(T), κ) all the holomorphic inverse branches of T n , n ∈ ℕ, are well-defined on balls of radius 2θκ , where θκ ∈ (0, κ). −1 For every n ∈ ℕ, let rn (z) := θκ (K|(T n )′ (z)|) . Using Koebe’s distortion theorem, observe that Tz−n (B(T n (z), θκ )) ⊇ B(z, rn (z)) whenever T n (z) ∉ B(Ω(T), κ). In that case, it follows from the hT -conformality of mT and Koebe’s distortion theorem that mT (B(z, rn (z))) ≤ mT (Tz−n (B(T n (z), θκ ))) ′ 󵄨 󵄨h ≤ K hT 󵄨󵄨󵄨(Tz−n ) (T n (z))󵄨󵄨󵄨 T mT (B(T n (z), θκ )) ′ h 󵄨 󵄨−h ≤ K hT 󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 T = K hT [Kθκ−1 rn (z)] T h

= K hT (Kθκ−1 )hT ⋅ (rn (z)) T .

Let ℕ(z) = {n ∈ ℕ : ∃ω ∈ Ω(T) such that T n (z) ∈ B(ω, κ) while T n−1 (z) ∉ B(ω, κ)}. For n ∈ ℕ(z), let k(n) ≥ n + 1 be the smallest k ∈ ℕ (if it exists) such that T k (z) ∉ B(ω, κ) or equivalently (because of the choice of κ in Lemma 31.5.2) such that T k (z) ∉ B(Ω(T), κ). Otherwise, set k(n) = ∞. Define intervals In by either In = [rk(n) (z), rn (z)] when k(n) < ∞

or

In = (0, rn (z)] when k(n) = ∞.

As 󵄨 󵄨 0 < inf 󵄨󵄨󵄨T ′ (x)󵄨󵄨󵄨 ≤ x∈𝒥 (T)

󵄨󵄨 j+1 ′ 󵄨󵄨 󵄨󵄨(T ) (z)󵄨󵄨 󵄩󵄩 ′ 󵄩󵄩 󵄨󵄨 j ′ 󵄨󵄨 ≤ 󵄩󵄩T 󵄩󵄩∞ , 󵄨󵄨(T ) (z)󵄨󵄨

∀j ∈ ℕ,

it follows from the above that it suffices to prove (31.68) for r ∈ ⋃n∈ℕ(z) In . Fix n ∈ ℕ(z)

and suppose that k(n) < ∞. Consider r ∈ [cn , bn ], where bn = rn (z) = θκ (K|(T n )′ (z)|) 󵄨 󵄨󵄨 󵄨−1 is the right endpoint of In and cn = K 󵄨󵄨󵄨T n (z) − ω󵄨󵄨󵄨󵄨󵄨󵄨(T n )′ (z)󵄨󵄨󵄨 . Applying Lemma 31.5.3 and remembering that hT ≥ 1 as part (b)’s hypothesis, we get for every r ∈ [cn , bn ] that −1

1282 � 31 Conformal measures, invariant measures and fractal geometry ′ ′ 󵄨p(ω)(hT −1) hT +p(ω)(hT −1) 󵄨 󵄨p(ω)(hT −1) αhT (ω) 󵄨 r r = C1 󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 mT (B(z, r)) ≤ C1 󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 −1 ′ 󵄨p(ω)(hT −1) 󵄨 [θκ (K|(T n )′ (z)|) ] ≤ C1 󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨

p(ω)(hT −1)

r hT

p(ω)(hT −1) hT

= C1 (θκ K −1 )

r .

It is left to prove (31.68) for r ∈ [an , cn ], where an = rk(n) (z) = θκ (K|(T k(n) )′ (z)|) is the left endpoint of In . To this end, for every i = n, n + 1, . . . , k(n) consider the radius −1

′ 󵄨 󵄨󵄨 󵄨−1 󵄨 󵄨 r i := K −1 τ 󵄨󵄨󵄨T i (z) − ω󵄨󵄨󵄨󵄨󵄨󵄨(T i ) (z)󵄨󵄨󵄨 = 2τθκ−1 󵄨󵄨󵄨T i (z) − ω󵄨󵄨󵄨ri (z),

where τ arises from Lemma 24.2.25. Since 2τθκ−1 κ ≤ r k(n) /an ≤ 2τθκ−1 diam(𝒥 (T)) and r n /cn = (2K 2 )−1 τ, it is enough to prove (31.68) for r ∈ [r k(n) , r n ]. Moreover, for every ζ ∈ Ω(T), let Tζ−1 : B(ζ , 2θκ ) → ℂ be the holomorphic branch of T −1 that fixes ζ . Observe that 󵄩󵄩 ′ 󵄩󵄩 󵄩󵄩T 󵄩󵄩∞ r 󵄩󵄩 ′ 󵄩󵄩−1 −1 ′ −1 , 󵄩󵄩T 󵄩󵄩∞ min{‖(Tζ ) ‖∞ : ζ ∈ Ω(T)} ≤ i+1 ≤ ′ ri inf{|T (x)| : x ∈ 𝒥 (T)}

∀n ≤ i < k(n).

Consequently, it suffices to prove (31.68) for r = r k(n) , . . . , r n . Fix n ≤ i ≤ k(n). By Lemma 24.2.25, there is an analytic inverse branch Tz−i : B(T i (z), 2τ|T i (z) − ω|) → ℂ which sends T i (z) to z. Using Koebe’s distortion theorem, notice that T i (B(z, r i )) ⊆ B(T i (z), K|(T i )′ (z)|r i ) = B(T i (z), τ|T i (z) − ω|). Hence, by the hT -conformality of mT , the Koebe distortion theorem and Lemma 31.3.8(a) we gather that ′ 󵄨 󵄨h mT (B(z, r i )) ≤ K hT 󵄨󵄨󵄨(Tz−i ) (T i (z))󵄨󵄨󵄨 T mT (B(T i (z), τ|T i (z) − ω|)) ′ h 󵄨 󵄨h 󵄨 󵄨−h ≤ K hT 󵄨󵄨󵄨(T i ) (z)󵄨󵄨󵄨 T ⋅ C2 󵄨󵄨󵄨T i (z) − ω󵄨󵄨󵄨 T = C2 K 2hT τ −hT ⋅ r i T .

Thus, the proof of (31.68) is complete in the case k(n) < ∞. If k(n) = ∞, then T n (z) = ω and for every r ∈ In := (0, rn (z)] relation (31.68) immediately follows from Theorem 31.3.7, the conformality of mT and the Koebe distortion theorem. (c) By Theorem 15.5.4(a), it is sufficient to prove that whenever hT ≤ 1, there exists M ≥ 1 such that mT (B(z, r)) ≥ M, r hT

∀0 < r ≤ 1, ∀z ∈ 𝒥 (T).

(31.69)

Proceed in a similar way to the proof of part (b), using Lemma 31.3.8(b) rather than Lemma 31.3.8(a). Proof of Theorem 31.5.1. According to Lemma 15.2.25, whenever they are nonzero and finite the measures HhT |𝒥 (T) and PhT |𝒥 (T) are hT -conformal since P(hT ) = 0 (cf. Definition 29.2.10). Part (a) then follows from Propositions 31.5.6(a,c) and 31.5.4(b), as well as Theorem 31.3.11(b). Part (b) ensues from Propositions 31.5.6(b,c) and 31.5.4, as well

31.6 Exercises

� 1283

as Theorem 31.3.11(b). Part (c) is a consequence of Propositions 31.5.6(b,d) and 31.5.4(a), ̂ per as well as Theorem 31.3.11(b). Finally, we can easily prove part (d). Since 𝒥 (T) ≠ ℂ Exercise 31.6.1, it follows from Theorem 29.2.13 and relation (31.13) that Leb2 (𝒥 (T)) = 0. As both measures H2 and P2 are constant multiples of Leb2 , part (c) does not hold, which firstly means that HD(𝒥 (T)) < 2. Secondly, it gives that PhT (𝒥 (T)) < ∞. Thus, PD(𝒥 (T)) ≤ hT . Since we just proved that hT = HD(𝒥 (T)) < 2, we are done. Remark 31.5.7. In fact, as for expanding rational functions, we have HD(𝒥 (T)) = PD(𝒥 (T)) = BD(𝒥 (T)) for all parabolic T but the proof of these equalities is much subtler and much more involved than for expanding functions. It was for the first time provided in [41] (cf. [124]). Remark 31.5.8. For the simplest parabolic quadratic polynomial Q1/4 (z) = z2 + 1/4, we know more than just the inequality HD(𝒥 (T)) < 2 established in Theorem 31.5.1(d). Namely, hQ1/4 = HD(𝒥 (Q1/4 )) < 1.295 per [55]. It also follows from Theorem 31.5.1 that 0 < HhQ (𝒥 (Q1/4 )) < ∞ 1/4

and PhQ (𝒥 (Q1/4 )) = ∞. 1/4

̂ →ℂ ̂ is a Blaschke product as in Example 31.4.5 and if On the other hand, if B : ℂ 1 some open arc in 𝕊 containing the parabolic fixed point ω is contained in the Fatou set of B (as a matter of fact in the immediate basin of attraction to ω), then hB = HD(𝒥 (B)) < 1 (see Exercise 31.6.2). So, it follows from Theorem 31.5.1 that 0 < PhB (𝒥 (B)) < ∞

and

HhB (𝒥 (B)) = 0.

31.6 Exercises ̂ for every parabolic rational function T : ℂ ̂ → ℂ. ̂ Exercise 31.6.1. Prove that 𝒥 (T) ≠ ℂ ̂ → ℂ ̂ be a Blaschke product as in Remark 31.5.8 (see also Exercise 31.6.2. Let B : ℂ Example 31.4.5). Show that HD(𝒥 (B)) < 1. Hint: Use the method from the proof of Theorem 31.5.1(d). ̂ → ℂ ̂ be a rational function with deg(T) ≥ 2 and such that Exercise 31.6.3. Let T : ℂ Ω(T) ⊈ PC(T). Show that for every t > δ(T) there exists a t-conformal measure supported on Ω−∞ (T).

32 Conformal measures, invariant measures and fractal geometry of subexpanding rational functions In this chapter, we continue the study of Hausdorff, packing, conformal, and invariant measures that are absolutely continuous with respect to conformal ones, this time for subexpanding rational functions. ̂ → ℂ ̂ is subexpanding if the Recall from Section 26.3 that a rational map R : ℂ restriction of R to the intersection of the Julia set 𝒥 (R) with the ω-limit set of the critical points of R, is expanding with respect to the spherical metric on 𝒥 (R). In the previous two chapters, we dealt with fractal geometry, Sullivan conformal measures, and invariant measures for expanding and parabolic rational functions. We observed a number of similarities and differences between them. In this chapter, we examine subexpanding maps from the same perspective. Overall, their properties turn out to be somewhat more similar to expanding mappings than to parabolic ones, though they still share some important features with these latter. We prove for subexpanding maps essentially the same properties as for expanding systems in Chapter 30 except that, as in the parabolic case in Chapter 31, there are t-conformal measures for all t > hR and all of them are purely atomic supported this time on the inverse orbits of critical points (see Theorem 32.1.13). The structure of this chapter is generally the same as that of Chapter 31. Briefly summarizing its content, Subsection 32.1.1 comprises general properties of subexpanding maps as well as properties of their Sullivan conformal measures, including relationships between conformal measures with different exponents, and culminates with Bowen’s formula. Subsection 32.1.2, especially Theorem 32.1.9, pertains to the existence, uniqueness, ergodicity and conservativity of an atomless hR -conformal measure mR . We also show in that theorem that mR is a geometric measure per (32.34). In Subsection 32.1.3, a full description of all Sullivan conformal measures for subexpanding rational functions is provided in Theorem 32.1.13. That description is quite different from that for expanding rational functions (cf. Theorem 30.1.1) and is more similar to the case of parabolic rational functions (cf. Theorems 31.2.1 and 31.3.11). In Section 32.2, we establish the existence and uniqueness of an R-invariant Borel probability measure μR which is absolutely continuous with respect to the hR -conformal measure mR . We further show that μR is ergodic and equivalent to mR . Finally, in Section 32.3, we provide in Theorem 32.3.1 an exact geometric description of the fractal structure of the Julia set of a subexpanding rational function in terms of the Hausdorff and packing measures. It is actually the same as that for expanding rational functions (cf. Theorem 30.1.1) and is much simpler than that for parabolic rational functions (cf. Theorem 31.5.1). All possible dimensions, geometrical and dynamical, are equal and both the Hausdorff and packing measures are finite, positive and geometric. https://doi.org/10.1515/9783110769876-032

1286 � 32 Conformal measures, invariant measures and fractal geometry Our presentation of subexpanding maps given in Section 26.3 and especially in Chapter 32, partly stems from [39] but substantially improves it and goes beyond it. In turn, the research in [39] was prompted by the paper [54] where it was proved that ̂ admits an any subexpanding rational function whose Julia set is the whole sphere ℂ invariant Borel probability measure absolutely continuous with respect to the Lebesgue measure.

32.1 Sullivan conformal measures for subexpanding rational functions 32.1.1 Technical preparations and Bowen’s formula We start this section with the following general result. ̂ → ℂ ̂ be a rational function with deg(Q) ≥ 2. Let ξ0 > 0 be Lemma 32.1.1. Let Q : ℂ ̂ there exist at least three different periodic points so small that for every point x ∈ ℂ ̂ a, b, c ∈ ℂ whose forward trajectories do not intersect the ball B(x, ξ0 ). Let z ∈ 𝒥 (Q). (a) If there is a strictly increasing sequence (nk = nk (z))∞ k=1 and a number ξ1 (z) ∈ (0, ξ0 ] such that for every k ∈ ℕ there exists a unique local holomorphic inverse branch −n ̂ of Qnk such that Qz−nk (Qnk (z)) = z, then Qz k : B(Qnk (z), ξ1 (z)) → ℂ ′ 󵄨 󵄨 lim sup󵄨󵄨󵄨(Qnk ) (z)󵄨󵄨󵄨 = ∞. k→∞

(32.1)

Moreover, for every t ≥ 0 and every t-conformal measure m for Q there exist a constant C4 ≥ 1 (depending only on ξ1 (z) with respect to z and depending in a monotonically increasing way on t) and a sequence (rk (z))∞ k=1 of positive numbers such that limk→∞ rk (z) = 0 and C4−1 ≤

m(B(z, rk (z))) t

(rk (z))

≤ C4 ,

∀k ∈ ℕ.

(b) Suppose that there exist a constant C5 ≥ 1 (potentially depending on z) and a number ξ2 (z) ∈ (0, ξ0 ] such that for every r ∈ (0, ξ2 (z)) there is n = n(r) ∈ ℕ for which ′ 󵄨 󵄨 C5−1 ≤ r 󵄨󵄨󵄨(Qn ) (z)󵄨󵄨󵄨 ≤ C5

̂ of Qn satisfying and a unique holomorphic inverse branch Qz−n : B(Qn (z), ξ2 (z)) → ℂ −n n Qz (Q (z)) = z. Then for every t ≥ 0 and every t-conformal measure m for Q there exists a constant C6 ≥ 1 (depending on C5 and ξ2 (z)) such that C6−1 ≤

m(B(z, r)) ≤ C6 , (r(z))t

∀0 < r ≤ 1.

32.1 Sullivan conformal measures for subexpanding rational functions

� 1287

Proof. (a) Passing to a subsequence if necessary, we may assume without loss of generality that there exists y ∈ 𝒥 (Q) such that lim Qnk (z) = y

k→∞

󵄨 󵄨 and 󵄨󵄨󵄨Qnk (z) − y󵄨󵄨󵄨 ≤ ι,

∀k ∈ ℕ,

where ι := ξ1 (z)/2. Hence, B(y, ι) ⊆ B(Qnk (z), 2ι) for all k ∈ ℕ and the family of restric−n ̂ ∞ is well-defined. It follows from Lemma 24.1.13 that tions {Qz k : B(y, ι) → ℂ} k=1 ′ 󵄨 󵄨 lim 󵄨󵄨󵄨(Qnk ) (z)󵄨󵄨󵄨 = ∞.

k→∞

So, relation (32.1) is proved. In order to prove the other part of (a), take ζ := γ/K, where K is the constant defined in Lemma 23.1.11 (a consequence of Koebe’s distortion theorems; set K = k2R,s (1/2) with R = ξ1 (z)/2 and s any number in (0, π)). Let ′ ′ 󵄨 󵄨−1 󵄨 󵄨 rk = rk (z) := 󵄨󵄨󵄨(Qnk ) (z)󵄨󵄨󵄨 ζ = 󵄨󵄨󵄨(Qz−nk ) (Qnk (z))󵄨󵄨󵄨ζ ,

∀k ∈ ℕ.

By Lemma 24.1.13, there is l ∈ ℕ such that diam(Qz k (B(Qnk (z), ξ1 (z)))) ≤ π − s for every −n ̂ we get k ≥ l. Fix any such k. Applying Lemma 23.1.11 to Qz k : B(Qnk (z), 2ι) → ℂ, −n

Qz−nk (B(Qnk (z), K −1 ζ )) ⊆ B(z, rk ) ⊆ Qz−nk (B(Qnk (z), Kζ )). Using the t-conformality of m, we obtain m(B(z, rk )) ≥ m(Qz−nk (B(Qnk (z), K −1 ζ ))) =

∫ B(Qnk (z),K −1 ζ )

󵄨󵄨 −nk ′ 󵄨󵄨t 󵄨󵄨(Qz ) 󵄨󵄨 dm

′ 󵄨 󵄨t ≥ K −t 󵄨󵄨󵄨(Qz−nk ) (Qnk (z))󵄨󵄨󵄨 m(B(Qnk (z), K −1 ζ ))

≥ K −t (rk /ζ )t Mm (K −1 ζ ) = (Kζ )−t Mm (K −1 ζ ) ⋅ rkt , where the compactness of supp(m) = 𝒥 (T) guarantees that for any r > 0, Mm (r) := inf{m(B(w, r)) : w ∈ 𝒥 (Q)} > 0. Analogously, ′ 󵄨 󵄨t m(B(z, rk )) ≤ K t 󵄨󵄨󵄨(Qz−nk ) (Qnk (z))󵄨󵄨󵄨 m(B(Qnk (z), Kζ )) ≤ K t ζ −t ⋅ rkt .

(32.2)

The proof of part (a) for k ≥ l is complete. For k < l (if any), part (a) follows by increasing C4 if necessary, since we are then dealing with “large” balls and “large” measures. (b) Up to small modifications, the proof is the same as the proof of (a). Recall that subexpanding rational functions were defined in Section 26.3. We keep here the notation used therein. In particular, R, ω(R), s, T, η, λ, p, γ, ε, A, τ and ΓR originate

1288 � 32 Conformal measures, invariant measures and fractal geometry from that section whereas θ = θ(ε, λ) comes from Lemma 25.1.11. Note that, from time to time, the letter s will be used to denote a parameter for the pressure or the exponent of a conformal measure. The context will determine the meaning of s. Fix any 1 1 󵄩 󵄩−1 δ ∈ (0, min{ θ, η, γ min{γ, θ}τK −2 A−2 󵄩󵄩󵄩T ′ 󵄩󵄩󵄩∞ }) 2 3

(32.3)

so small that if z ∉ B(Crit(T p ), ε) then there exists a unique holomorphic inverse branch −p ̂ of T p such that Tz−p (T p (z)) = z and Tz : B(T p (z), 2δ) → ℂ diam(Tz−p (B(T p (z), 2δ))) < γ.

(32.4)

Let ∞

Crit−∞ (T) := ⋃ T −n (Crit(T)). n=0

We prove the following simple result which nevertheless has significant consequences. ̂ →ℂ ̂ be a subexpanding rational function with deg(R) ≥ 2. If Lemma 32.1.2. Let R : ℂ − z ∈ 𝒥 (T) \ Crit∞ (T), then ′ 󵄨 󵄨 lim sup󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 = ∞.

(32.5)

n→∞

Moreover, there is a sequence (rk (z))∞ k=1 of positive numbers such that limk→∞ rk (z) = 0 and for which any t-conformal measure m for R admits a constant C7 = C7 (m) ≥ 1 (independent of z) such that m(B(z, rk (z)))

C7−1 ≤

t

(rk (z))

≤ C7 ,

(32.6)

∀k ∈ ℕ.

In consequence, if 0 ≤ s ≤ t and ν is an s-conformal measure for R, then t−s

C8−1 (rk (z))

≤

m(B(z, rk (z))) ν(B(z, rk (z)))

≤ C8 (rk (z))

t−s

,

(32.7)

where C8 = C8 (m, ν) := C7 (m)C7 (ν). Proof. As T is an iterate of R, it suffices to check that the hypotheses of Lemma 32.1.1(a) are satisfied for the point z and that inf{ξ1 (x) : x ∈ 𝒥 (T) \ Crit−∞ (T)} > 0. To this end, let 󵄨 ℕ(z) := {n ≥ p 󵄨󵄨󵄨 T n (z) ∉ B(ω(T), η)}. Suppose first that #ℕ(z) = ∞. Let (jk + p − 1)∞ k=1 be the strictly increasing sequence of all integers in ℕ(z). In view of Lemma 26.3.7, there exists for every k ∈ ℕ a unique

32.1 Sullivan conformal measures for subexpanding rational functions

� 1289

−j ̂ of T jk such that Tz−jk (T jk (z)) = z. holomorphic inverse branch Tz k : B(T jk (z), 2γ) → ℂ Setting ξ1 (z) := min{2γ, ξ0 }, we see that the hypotheses of Lemma 32.1.1(a) are satisfied for the point z in this case. Suppose now that #ℕ(z) < ∞. Let k be the largest integer in ℕ(z). As z ∉ Crit−∞ (T), ̂ of T k there is Δ > 0 and a unique holomorphic inverse branch Tz−k : B(T k (z), Δ) → ℂ such that Tz−k (T k (z)) = z. By (26.15), (26.18) and Lemma 25.1.11, there exists for every −j ̂ of T j such that j ∈ ℕ a unique holomorphic inverse branch Tk : B(T j+k (z), 2θ) → ℂ

Tk (T j+k (z)) = T k (z). In light of Lemma 32.1.1(a) (applied to the point T k (z)), of (26.18) and of Lemma 23.1.11, we have −j

diam(Tk (B(T j+k (z), θ))) < Δ −j

for all j sufficiently large. Therefore, for each such j the inverse branch composition −j ̂ is well-defined. Setting ξ1 (z) := min{θ, ξ0 }, we conclude that Tz−k ∘ Tk : B(T k+j (z), θ) → ℂ the hypotheses of Lemma 32.1.1(a) are satisfied for the point z in this case, too. Finally, observe that inf{ξ1 (x) : x ∈ 𝒥 (T) \ Crit−∞ (T)} = min{2γ, θ, ξ0 } > 0. Remark 32.1.3. In the proof of Lemma 32.1.2, we have also shown that for every point z ∈ 𝒥 (T) \ Crit−∞ (T) there exists a strictly increasing sequence (nk = nk (z))∞ k=1 such that ′ 󵄨 󵄨 lim sup󵄨󵄨󵄨(T nk ) (z)󵄨󵄨󵄨 = ∞ k→∞

and for every k ∈ ℕ there exists a unique holomorphic inverse branch ̂ Tz−nk : B(T nk (z), min{2γ, θ, ξ0 }) → ℂ of T nk such that Tz k (T nk (z)) = z. Furthermore, thanks to Lemma 32.1.1(a), we may and we do assume that {nk (z) : k ∈ ℕ} is the set of all n ∈ ℕ such that there exists a unique holomorphic inverse branch −n

̂ Tz−n : B(T n (z), min{2γ, θ, ξ0 }) → ℂ of T n sending T n (z) to z. As an immediate consequence of this remark and of the fact that 𝒥 (R) = 𝒥 (T) and Crit−∞ (R) = Crit−∞ (T), we get the following result. ̂→ℂ ̂ is a subexpanding rational function with deg(R) ≥ 2, then Corollary 32.1.4. If R : ℂ −

𝒥r (R) = 𝒥 (R) \ Crit∞ (R).

We now prove an analog of Lemma 31.3.4.

1290 � 32 Conformal measures, invariant measures and fractal geometry ̂ →ℂ ̂ be a subexpanding rational function with deg(R) ≥ 2. If Lemma 32.1.5. Let R : ℂ 0 ≤ s ≤ t and ν and m are respectively s-conformal and t-conformal measures for R, then m|𝒥 (R)\Crit− (R) is absolutely continuous with respect to ν|𝒥 (R)\Crit− (R) . In fact, ∞

∞

3

m(A) ≤ 4s C7 (m)(C7 (ν)) ⋅ ν(A) for every Borel set A ⊆ 𝒥 (R) \ Crit−∞ (R). If additionally s < t, then m(𝒥 (R) \ Crit−∞ (R)) = 0. Proof. Let A be any Borel subset of 𝒥 (R) \ Crit−∞ (R). Fix ε > 0. Since the measure ν is ̂ such that outer regular, there exists an open set A ⊆ G ⊆ ℂ ν(G) ≤ ν(A) + ε.

(32.8)

For every z ∈ A, take k ∈ ℕ so large that r(z) ≤ ε

and

B(z, r(z)) ⊆ G,

(32.9)

where r(z) := rk (z) is a radius arising from Lemma 32.1.2. It follows from the 4r-covering theorem (Theorem 15.4.1 and Remark 15.4.2(e)) that there exists a countable set A∗ ⊆ A such that the family {B(z, r(z)/4)}z∈A∗ consists of mutually disjoint sets and the family {B(z, r(z))}z∈A∗ covers A. Using Lemma 32.1.2, we obtain that t−s

m(A) ≤ ∑ m(B(z, r(z))) ≤ C8 (m, ν) ∑ (r(z)) z∈A∗

z∈A∗

ν(B(z, r(z))) 2

≤ C7 (m)C7 (ν)εt−s ∑ ν(B(z, r(z))) ≤ C7 (m)(C7 (ν)) εt−s ∑ (r(z)) z∈A∗

2

s

3

s

z∈A∗

= 4s C7 (m)(C7 (ν)) εt−s ∑ (r(z)/4) ≤ 4s C7 (m)(C7 (ν)) εt−s ∑ ν(B(z, r(z)/4)) z∈A∗

z∈A∗

3

3

= 4s C7 (m)(C7 (ν)) εt−s ν( ⋃ B(z, r(z)/4)) ≤ 4s C7 (m)(C7 (ν)) εt−s ν(G) s

3 t−s

≤ 4 C7 (m)(C7 (ν)) ε

z∈A∗

(ν(A) + ε).

Letting ε → 0 leads to the result. Denote hR := δ(R), where δ(R) is the minimal exponent of Sullivan conformal measures for the map R per Definition 29.2.10. We prefer to use the symbol hR rather than δ(R) since later on, as for parabolic rational functions, we will show that δ(R) is equal to several other significant

32.1 Sullivan conformal measures for subexpanding rational functions

� 1291

quantities ascribed to the function R and we do not want to prioritize any of them above any other. We end this subsection by establishing a variation of Bowen’s formula. ̂ → ℂ ̂ is a subexpanding rational function Theorem 32.1.6 (Bowen’s formula). If R : ℂ with deg(R) ≥ 2, then hR := δ(R) = ExpD∗∗ (𝒥 (R)) = ExpD∗ (𝒥 (R)) = ExpD(𝒥 (R)) = DD(𝒥 (R)) = HD(𝒥uer (R)) = HD(𝒥er (R)) = HD(𝒥r (R)) = HD(𝒥 (R)) ∈ (

pR , 2]. pR + 1

Proof. The first two lines in this formula follow from Theorem 29.2.12 and the definition hR := δ(R) while the last equality sign follows from Corollary 32.1.4 and the countability of the set Crit−∞ (R). The inequality δ(R) > pR /(pR + 1) is simply Theorem 29.2.14. 32.1.2 The atomless hR -conformal measure mR : existence, uniqueness, ergodicity and conservativity We first show that any Sullivan conformal measure that does not charge the critical set of T p is a geometric measure. ̂→ℂ ̂ be a subexpanding rational function with deg(R) ≥ 2. Proposition 32.1.7. Let R : ℂ If m is a t-conformal measure for R such that m(Crit(T p )) = 0, then there exists a constant C9 ≥ 1 such that for every z ∈ 𝒥 (R), C9−1 ≤

m(B(z, r)) ≤ C9 , rt

∀0 < r ≤ 1.

(32.10)

In particular, the measure m is atomless. Proof. First of all, keep in mind that 𝒥 (R) = 𝒥 (T) and that m is also t-conformal for the map T. In this proof, we will always be dealing with the map T. For every z ∈ 𝒥 (T), let C9 (z) ≥ 1 be the minimal constant such that (32.10) is satisfied (if no such constant exists, define C9 (z) = ∞). We need to prove that 󵄨 sup{C9 (z) 󵄨󵄨󵄨 z ∈ 𝒥 (T)} < ∞. The proof consists of several steps. Step 1: Using relations (26.15) and (26.18), Lemma 25.1.11, and Lemma 32.1.1(b) (with Q = T, C5 = ‖T ′ ‖∞ and ξ2 (z) = min{2θ(ε, λ), ξ0 }), we conclude that (32.10) is uniformly satisfied for all points z for which T

n

󵄨

𝒪+ (z) := {T (z) 󵄨󵄨󵄨 n ≥ 0} ⊆ B(ω(T), 2η).

1292 � 32 Conformal measures, invariant measures and fractal geometry By saying this, we mean here and in the sequel that sup{C9 (z)} taken over the set of points z under consideration is finite. Step 2: Consider z ∈ Crit(T p ). In view of Theorem 25.1.1, there exist ρ ∈ (0, 1/2) and conformal homeomorphisms ̂ (on their respective open connected image hi (B(0, 2ρ)); h2 being h1 , h2 : B(0, 2ρ) → ℂ even a translation) such that h1 (0) = z,

h2 (0) = T 2p (z),

and

T 2p (h1 (w)) = h2 (wq ),

∀w ∈ B(0, 2ρ),

where q := degz (T 2p ) ≥ 2. For each i = 1, 2, define a finite Borel measure mi on the Borel σ-algebra of B(0, ρ) by setting for every Borel set A ⊆ B(0, ρ), ′ 󵄨t 󵄨 mi (A) := ∫ 󵄨󵄨󵄨(hi−1 ) 󵄨󵄨󵄨 dm.

(32.11)

hi (A)

By (26.16), we know that 𝒪+ (T 2p (z)) ∈ B(ω(T), η/2). Applying Step 1 to T 2p (z), we get 󵄨 󵄨−t m2 (B(0, r)) ≍ 󵄨󵄨󵄨h2′ (0)󵄨󵄨󵄨 m(h2 (B(0, r))) ≍ m(h2 (B(0, r)))

{ { {

≤ ≥ ≤ ≥ ≤ ≥

m(B(h2 (0), ‖h2′ ‖∞ r))

m(B(h2 (0), 41 |h2′ (0)|r))

m(B(T 2p (z), ‖h2′ ‖∞ r))

m(B(T 2p (z), 41 |h2′ (0)|r)) t

C9 (T 2p (z))[‖h2′ ‖∞ r] −1

t

[C9 (T 2p (z))] [ 41 |h2′ (0)|r] ,

where ≍ is the usual symbol of multiplicative comparability (see (25.34)). So −1 C10 ≤

m2 (B(0, r)) ≤ C10 rt

(32.12)

for every r ∈ (0, ρ] and some constant C10 ≥ 1. For every Borel set F ⊆ hi (B(0, ρ)), we deduce from (32.11) that 󵄨 󵄨t m(F) = ∫ 󵄨󵄨󵄨hi′ 󵄨󵄨󵄨 dmi ,

i = 1, 2.

(32.13)

hi−1 (F)

Let A ⊆ B(0, ρ) be any Borel set on which the map w 󳨃→ wq is injective and let Aq denote the image of A under w 󳨃→ wq . Using (32.11), (32.13), the t-conformality of m and the change-of-variables formula several times, we get

32.1 Sullivan conformal measures for subexpanding rational functions

′ 󵄨t 󵄨 m2 (Aq ) = ∫ 󵄨󵄨󵄨(h2−1 ) 󵄨󵄨󵄨 dm = h2 (Aq )

� 1293

󵄨󵄨 −1 ′ 󵄨󵄨t 󵄨󵄨(h2 ) 󵄨󵄨 dm

∫ T 2p (h1 (A))

′ 󵄨t ′ 󵄨t ′ 󵄨 󵄨 󵄨t 󵄨 = ∫ 󵄨󵄨󵄨(h2−1 ) ∘ T 2p 󵄨󵄨󵄨 󵄨󵄨󵄨(T 2p ) 󵄨󵄨󵄨 dm = ∫ 󵄨󵄨󵄨(h2−1 ∘ T 2p ) 󵄨󵄨󵄨 dm h1 (A)

h1 (A)

=

󵄨󵄨 −1 󵄨t 󵄨 ′ 󵄨t 2p ′ 󵄨󵄨(h2 ∘ T ) ∘ h1 󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨h1 󵄨󵄨󵄨 dm1

∫ h1−1 (h1 (A))

′ 󵄨t ′ 󵄨t 󵄨 󵄨 = ∫󵄨󵄨󵄨(h2−1 ∘ T 2p ∘ h1 ) 󵄨󵄨󵄨 dm1 = ∫󵄨󵄨󵄨(wq ) 󵄨󵄨󵄨 dm1 (w) A

A

󵄨 󵄨t = q ∫󵄨󵄨󵄨wq−1 󵄨󵄨󵄨 dm1 (w) = qt ∫ |w|(q−1)t dm1 (w). t

A

(32.14)

A

Fix any ν ∈ (0, 1). For every r ∈ (0, ρ], set A(0; νr, r) := {w ∈ ℂ : νr ≤ |w| < r}. For every j = 0, 1, . . . , q − 1, let 󵄨󵄨 2π(j + 1) 2πj 󵄨 Aj (r) := {beiθ ∈ ℂ 󵄨󵄨󵄨 √q ν r ≤ b < r and ≤θ< }. 󵄨󵄨 q q q−1

The sets {Aj (r)}j=0 partition the annulus A(0; √q ν r, r). Moreover, the map w 󳨃→ wq is injective on each set Aj (r) and q

(Aj (r)) = A(0; νr q , r q ). Therefore, using (32.12)–(32.14) we obtain that q−1

qC10 r qt ≥ qm2 (A(0; νr q , r q )) = ∑ qt ∫ |w|(q−1)t dm1 (w) j=0

q−1

≥ ∑ qt [√q νr]

(q−1)t

j=0

= qt ν

(q−1)t q

Aj (r)

m1 (Aj (r))

r (q−1)t m1 (A(0; √q ν r, r)).

Similarly, set A(0; 0, r) := B(0, r) \ {0}. For every j = 0, 1, . . . , q − 1, let 󵄨󵄨 2πj 2π(j + 1) 󵄨 A0j (r) := {beiθ ∈ ℂ 󵄨󵄨󵄨 0 < b < r and ≤θ< }. 󵄨󵄨 q q

(32.15)

1294 � 32 Conformal measures, invariant measures and fractal geometry From (32.11) and the t-conformality of m, we readily deduce that ′ 󵄨t 󵄨 󵄨−t 󵄨 󵄨 󵄨−t m2 ({0}) = 󵄨󵄨󵄨h2′ (0)󵄨󵄨󵄨 m({T 2p (z)}) = 󵄨󵄨󵄨h2′ (0)󵄨󵄨󵄨 󵄨󵄨󵄨(T 2p ) (z)󵄨󵄨󵄨 m({z}) = 0

because z ∈ Crit(T 2p ). So, by (32.12)–(32.14), q−1

−1 qt qC10 r ≤ qm2 (B(0, r q )) = qm2 (A(0; 0, r q )) = ∑ qt ∫ |w|(q−1)t dm1 (w) j=0

A0j (r)

≤ qt ∫ |w|(q−1)t dm1 (w) ≤ qt r (q−1)t m1 (B(0, r)). B(0,r)

Hence, m1 (B(0, r)) −1 1−t ≥ C10 q rt

(32.16)

while we infer from (32.15) that m1 (A(0; √q ν r, r)) ≤ C10 q1−t ν

−

(q−1)t q

rt .

(32.17)

Set C11 := C10 q1−t ν q . The hypothesis that m(Crit(T p )) = 0 implies that m1 ({0}) = 0. Using this and (32.17), we obtain that −

(q−1)t

∞

m1 (B(0, r)) = m1 ( ⋃ A(0; (√q ν) ∞

n+1

n=0

∞

n

n+1

r, (√q ν) r)) = ∑ m1 (A(0; (√q ν)

1 rt . ≤ ∑ C11 [(√q ν)n r] = C11 q t √ 1 − ( ν) n=0

n=0

t

(32.18)

Using (32.13), (32.16) and (32.18), we deduce that 󵄨 󵄨t m(B(z, r)) ≍ 󵄨󵄨󵄨h1′ (0)󵄨󵄨󵄨 m1 (h1−1 (B(z, r))) ≍ m1 (h1−1 (B(z, r)))

{ { { So there is C12 ≥ 1 such that

≤ ≥ ≤

m1 (B(h1−1 (z), ‖(h1−1 )′ ‖∞ r))

m1 (B(h1−1 (z), 41 |(h1−1 )′ (z)|r)) m1 (B(0, ‖(h1−1 )′ ‖∞ r))

≥

m1 (B(0, 41 |(h1−1 )′ (z)|r))

≤

C11 [1 − √νt ] [‖(h1−1 )′ ‖∞ r]

≥

q

n

r, (√q ν) r))

−1

t

C10 q1−t [ 41 |(h1−1 )′ (z)|r] .

t

32.1 Sullivan conformal measures for subexpanding rational functions

−1 C12 ≤

m(B(z, r)) ≤ C12 , rt

� 1295

∀r ∈ (0, ρ].

Note that the constant C12 can be adjusted upwards so that these estimates become valid for all r ∈ (0, 1]. As #Crit(T p ) < ∞, this confirms that (32.10) is satisfied for all z ∈ Crit(T p ). Step 3: Suppose that z ∉ Crit(T p ), that 0 < r < 2−6 K −2 ‖T ′ ‖−p+1 ∞ min{γ, θ}, and that ′ 󵄨 󵄨 󵄩 󵄩−1 sup r 󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 ≤ K −1 󵄩󵄩󵄩T ′ 󵄩󵄩󵄩∞ min{γ, θ}. n≥0

Then it follows from (32.5) in Lemma 32.1.2 that z ∈ Crit−∞ (T). Let k ≥ 0 be the least p−1 integer such that z ∈ T −k (Crit(T)). Since z ∉ Crit(T p ) = ⋃j=0 T −j (Crit(T)), this means

that k ≥ p and there is l ∈ ℕ such that T p−1 (T l (z)) = T k (z) ∈ Crit(T). Since, moreover, T k (z) ∉ B(ω(T), 2η) by (26.18), it ensues from Lemma 26.3.7 that there exists a unique ̂ of T l such that Tz−l (T l (z)) = z. But holomorphic inverse branch Tz−l : B(T l (z), 2γ) → ℂ (T p ) (T l (z)) = (T p−1 ) (T l (z)) ⋅ T ′ (T k (z)) = 0. ′

′

That is, T l (z) ∈ Crit(T p ). Applying Lemma 23.1.11, we know that B(T l (z), K −1 |(T l )′ (z)|r) ⊆ T l (B(z, r)) ⊆ B(T l (z), K|(T l )′ (z)|r). Using this, the t-conformality of m, Theorem 23.1.9 (Koebe’s distortion theorem, spherical version) and applying Step 2 to T l (z) (note that K|(T l )′ (z)|r ≤ 1), we deduce that m(B(z, r)) = m(Tz−l (T l (B(z, r)))) = ≍

′ 󵄨 󵄨t K ±t 󵄨󵄨󵄨(Tz−l ) (T l (z))󵄨󵄨󵄨 󵄨−t ±t 󵄨 l ′

∫

󵄨󵄨 −l ′ 󵄨󵄨t 󵄨󵄨(Tz ) 󵄨󵄨 dm

T l (B(z,r))

⋅ m(T l (B(z, r)))

≍ K 󵄨󵄨󵄨(T ) (z)󵄨󵄨󵄨 ⋅ m(B(T l (z), K ±1 |(T l )′ (z)|r)) ′ t 󵄨 󵄨−t ±1 ±1 l ′ ≍ K ±t 󵄨󵄨󵄨(T l ) (z)󵄨󵄨󵄨 ⋅ C12 [K |(T ) (z)|r] ±1 ±2t = C12 K ⋅ rt . In conclusion, (32.10) is uniformly satisfied for all points z under consideration in Step 3. Step 4: Suppose that z ∉ Crit(T p ), that 0 < r < 2−6 K −2 ‖T ′ ‖−p+1 ∞ min{γ, θ}, and that ′ 󵄨 󵄨 󵄩 󵄩−1 r 󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 > K −1 󵄩󵄩󵄩T ′ 󵄩󵄩󵄩∞ min{γ, θ}

(32.19)

for some n ∈ ℕ. Fix a minimal such n. Then n ≥ p and, remembering that p ≥ 2, we have ′ 󵄩 󵄩−p 󵄨 󵄨 K −1 󵄩󵄩󵄩T ′ 󵄩󵄩󵄩∞ min{γ, θ} ≤ r 󵄨󵄨󵄨(T n−p+1 ) (z)󵄨󵄨󵄨 ≤ K −1 min{γ, θ}.

(32.20)

1296 � 32 Conformal measures, invariant measures and fractal geometry Case 1: Assume first that T n (z) ∉ B(ω(T), 2η). −(n−p+1)

In view of Lemma 26.3.7, there exists a unique holomorphic inverse branch Tz : ̂ of T n−p+1 such that Tz−(n−p+1) (T n−p+1 (z)) = z. Therefore, per B(T n−p+1 (z), 2γ) → ℂ Lemma 23.1.11 we know that B(T n−p+1 (z), K −1 |(T n−p+1 )′ (z)|r) ⊆ T n−p+1 (B(z, r)) ⊆ B(T n−p+1 (z), K|(T n−p+1 )′ (z)|r). Using this, the t-conformality of m, Theorem 23.1.9 (Koebe’s distortion theorem, spherical version) and (32.20), we obtain that m(B(z, r)) = m(Tz−(n−p+1) (T n−p+1 (B(z, r)))) =

󵄨󵄨 −(n−p+1) ′ 󵄨󵄨t ) 󵄨󵄨 dm ≥ 󵄨󵄨(Tz

∫ T n−p+1 (B(z,r))

≥ ≥ ≥

∫

󵄨󵄨 −(n−p+1) ′ 󵄨󵄨t ) 󵄨󵄨 dm 󵄨󵄨(Tz

B(T n−p+1 (z),K −1 |(T n−p+1 )′ (z)|r)

′ 󵄨t K 󵄨󵄨(Tz−(n−p+1) ) (T n−p+1 (z))󵄨󵄨󵄨 ⋅ m(B(T n−p+1 (z), K −1 |(T n−p+1 )′ (z)|r)) ′ 󵄨 󵄨−t K −t 󵄨󵄨󵄨(T n−p+1 ) (z)󵄨󵄨󵄨 ⋅ m(B(T n−p+1 (z), K −2 ‖T ′ ‖−p ∞ min{γ, θ})) −2 ′ −p −t t Mm (K ‖T ‖∞ min{γ, θ}) ⋅ γ r , −t 󵄨󵄨

where we recall that 󵄨 Mm (u) := inf{m(B(w, u)) 󵄨󵄨󵄨 w ∈ 𝒥 (T)} > 0 for every u > 0. Analogously, ′ 󵄨 󵄨t m(B(z, r)) ≤ K t 󵄨󵄨󵄨(Tz−(n−p+1) ) (T n−p+1 (z))󵄨󵄨󵄨 ⋅ m(B(T n−p+1 (z), K|(T n−p+1 )′ (z)|r)) ′ 󵄨 󵄨−t ≤ K t 󵄨󵄨󵄨(T n−p+1 ) (z)󵄨󵄨󵄨 −t 󵄩 󵄩pt ≤ K 2t 󵄩󵄩󵄩T ′ 󵄩󵄩󵄩∞ (min{γ, θ}) r t .

Consequently, (32.10) is uniformly satisfied for all points z under consideration in Case 1. Case 2: Assume that T n (z) ∈ B(ω(T), 2η) and let k ≥ 0 be the least integer such that {T k (z), T k+1 (z), . . . , T n (z)} ⊆ B(ω(T), 2η). Subcase 2a: If k = 0, by virtue of (26.15) and (26.18), proceed as in Case 1 using ̂ be the unique Lemma 25.1.11 instead of Lemma 26.3.7. Indeed, let Tz−n : B(T n (z), θ) → ℂ n −n n holomorphic inverse branch of T such that Tz (T (z)) = z per Lemma 25.1.11. By the definition of n (cf. (32.19)), we know that ′ 󵄩 󵄩−1 󵄨 󵄨 K −1 󵄩󵄩󵄩T ′ 󵄩󵄩󵄩∞ min{γ, θ} < r 󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 ≤ K −1 min{γ, θ}.

(32.21)

32.1 Sullivan conformal measures for subexpanding rational functions

� 1297

In a similar way to Case 1, we have ′ 󵄨 󵄨t m(B(z, r)) ≥ K −t 󵄨󵄨󵄨(Tz−n ) (T n (z))󵄨󵄨󵄨 m(B(T n (z), K −2 ‖T ′ ‖−1 ∞ min{γ, θ})) −t t ≥ Mm (K −2 ‖T ′ ‖−1 ∞ min{γ, θ}) ⋅ γ r .

and ′ ′ 󵄨t 󵄨 󵄨 󵄨t m(B(z, r)) ≤ K t 󵄨󵄨󵄨(Tz−n ) (T n (z))󵄨󵄨󵄨 m(B(T n (z), min{γ, θ})) ≤ K t 󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 −t 󵄩 󵄩t ≤ K 2t 󵄩󵄩󵄩T ′ 󵄩󵄩󵄩∞ (min{γ, θ}) r t .

In conclusion, (32.10) is uniformly satisfied for all points z under consideration in Subcase 2a. Subcase 2b: If k > 0, let l := max{0, k − p} Again by (26.15), (26.18) and Lemma 25.1.11 as well as (32.3), there is a unique holomor−(n−p−l) ̂ of T n−p−l such that T −(n−p−l) (T n (z)) = phic inverse branch T1 : B(T n (z), 2δ) → ℂ 1 p+l T (z). Sub-subcase 2b1: Assume that T l (z) ∉ B(Crit(T p ), ε). ̂ By (32.3)–(32.4), there is a unique holomorphic inverse branch T2 : B(T p+l (z), 2δ) → ℂ −p p+l p l of T such that T2 (T (z)) = T (z). −p

–

–

−p −(n−p−l) ̂ (which If l = 0, then the well-defined composition T2 ∘ T1 : B(T n (z), 2δ) → ℂ n l sends T (z) to T (z) = z) brings this sub-subcase to a conclusion in the same way as in Case 1 and Subcase 2a. If l > 0, then l = k − p. Since T k−1 (z) ∉ B(ω(T), 2η), it follows from Lemma 26.3.7 that ̂ of T l such there exists a unique holomorphic inverse branch Tz−l : B(T l (z), 2γ) → ℂ −p −(n−p−l) −l l −l n ̂ that Tz (T (z)) = z. By (32.4), the composition Tz ∘ T ∘ T : B(T (z), 2δ) → ℂ 2

1

makes sense and sends T n (z) to z. So, we are done in the same way as in Case 1 and Subcase 2a.

Therefore, (32.10) is uniformly satisfied for all points z under consideration in Subsubcase 2b1. Sub-subcase 2b2: Assume now that T l (z) ∈ B(c, ε) Set

for some c ∈ Crit(T p ).

1298 � 32 Conformal measures, invariant measures and fractal geometry 󵄨 󵄨 ρc := 󵄨󵄨󵄨T p+l (z) − T p (c)󵄨󵄨󵄨. Part 2b2(i): Assume that ′ 󵄨 󵄨 K −1 󵄨󵄨󵄨(T n−p−l ) (T p+l (z))󵄨󵄨󵄨τρc ≥ δ,

where τ ∈ (0, 1) emerges from Lemma 26.3.8. By Lemma 23.1.11 and Part 2b2(i)’s assumption, −(n−p−l)

T1

(B(T n (z), δ)) ⊆ B(T p+l (z), τρc ).

Moreover, in view of Lemma 26.3.8, there exists a unique holomorphic inverse branch −p T2 : B(T p+l (z), τρc ) → B(T l (z), γ) of T p sending T p+l (z) to T l (z). Hence, the composition −p

−(n−p−l)

T2 ∘ T1

̂ : B(T n (z), δ) → ℂ

is well-defined and maps T n (z) to T l (z). – If l = 0, then we are done as in Case 1 and Subcase 2a. – If l > 0, then l = k − p. Since T k−1 (z) ∉ B(ω(T), 2η), it follows from Lemma 26.3.7 ̂ that there exists a unique holomorphic inverse branch Tz−l : B(T l (z), 2γ) → ℂ of T l such that Tz−l (T l (z)) = z. The possibility of making up the composition −p −(n−p−l) ̂ finishes the proof in this situation as well. : B(T n (z), δ) → ℂ Tz−l ∘ T2 ∘ T1

Thus, (32.10) is uniformly satisfied for all points z under consideration in Part 2b2(i). Part 2b2(ii): Assume that ′ 󵄨 󵄨 K −1 󵄨󵄨󵄨(T n−p−l ) (T p+l (z))󵄨󵄨󵄨τρc < δ.

Set 󵄨 󵄨 ι := 󵄨󵄨󵄨T l (z) − c󵄨󵄨󵄨

and qc := degc (T p ).

Using (26.19) (with z replaced by T l (z)) and the definition of ρc , we have that ′ 󵄨 󵄨 K −1 󵄨󵄨󵄨(T n−p−l ) (T p+l (z))󵄨󵄨󵄨τA−1 ιqc < δ.

(32.22)

But by virtue of (26.20) (with z replaced by T l (z)) and (32.19), we obtain that ′ ′ 󵄨 󵄨 󵄨 󵄨 󵄩 󵄩−1 r 󵄨󵄨󵄨(T n−p−l ) (T p+l (z))󵄨󵄨󵄨 ⋅ Aιqc −1 ⋅ 󵄨󵄨󵄨(T l ) (z)󵄨󵄨󵄨 > K −1 󵄩󵄩󵄩T ′ 󵄩󵄩󵄩∞ min{γ, θ}.

Multiplying both sides of this inequality by ι and using (32.22) and (32.3), we get that −1 ′ ′ 󵄩 󵄩 󵄨 󵄨 1 󵄨 󵄨 ι < K 2 󵄩󵄩󵄩T ′ 󵄩󵄩󵄩∞ (min{γ, θ}) τ −1 δA2 r 󵄨󵄨󵄨(T l ) (z)󵄨󵄨󵄨 < γr 󵄨󵄨󵄨(T l ) (z)󵄨󵄨󵄨. 3

32.1 Sullivan conformal measures for subexpanding rational functions

� 1299

2 󵄨 1 󵄨 ′ ′ ′ 󵄨 󵄨 󵄨 󵄨 B(T l (z), γr 󵄨󵄨󵄨(T l ) (z)󵄨󵄨󵄨) ⊆ B(c, γr 󵄨󵄨󵄨(T l ) (z)󵄨󵄨󵄨) ⊆ B(T l (z), γr 󵄨󵄨󵄨(T l ) (z)󵄨󵄨󵄨). 3 3

(32.23)

This implies that

Since l < n, the definition of n (relation (32.19) and its minimality) yields ′ 󵄨 󵄨 γr 󵄨󵄨󵄨(T l ) (z)󵄨󵄨󵄨 ≤ γ.

–

If l = 0, then it follows from (32.23) and Step 2 (used at c) that m(B(z, γr/3)) ≤ C12 (2γ/3)t ⋅ r t

–

(32.24)

and

−1 m(B(z, γr)) ≥ C12 (2γ/3)t ⋅ r t .

In light of Lemma 25.6.2, these inequalities are sufficient to complete the proof of Part 2b2(ii) when l = 0. If l > 0, then l = k − p. Since T k−1 (z) ∉ B(ω(T), 2η), it follows from Lemma 26.3.7 ̂ of that there exists a unique holomorphic inverse branch Tz−l : B(T l (z), 2γ) → ℂ l −l l T such that Tz (T (z)) = z. Using Koebe’s distortion theorem, the t-conformality of m, (32.23)–(32.24) and Step 2 (at c), we conclude that m(B(z, K −1 γr/3)) ≤ C12 (2Kγ/3)t ⋅ r t

t

−1 and m(B(z, Kγr)) ≥ C12 (2K −1 γ/3) ⋅ r t .

In light of Lemma 25.6.2, these inequalities suffice to complete the proof of Part 2b2(ii) when l > 0. Thus, (32.10) is uniformly satisfied for all points z under consideration in Part 2b2(ii). Having successfully considered Parts 2b2(i)+(ii), relation (32.10) is uniformly satisfied for all points z under consideration in Sub-subcase 2b2. Both Sub-subcases 2b1 and 2b2 having been tackled, the proof in Subcase 2b is finished. Having established both Subcases 2a and 2b, we conclude that the proof in Case 2 is over. Cases 1 and 2 being done, the proof of Step 4 is complete. Steps 1–4 cover all possibilities and therefore the result holds. We can now easily prove the first main result of this section, which asserts the existence of an atomless hR -conformal measure that behaves like a geometric measure with exponent hR . ̂→ℂ ̂ is a subexpanding rational function with deg(R) ≥ 2, then Theorem 32.1.8. If R : ℂ there exists an atomless hR -conformal measure mR for R. In addition, C9−1 ≤

mR (B(z, r)) ≤ C9 , r hR

∀r ∈ (0, 1], ∀z ∈ 𝒥 (R).

(32.25)

1300 � 32 Conformal measures, invariant measures and fractal geometry Proof. In fact, we will show that for an appropriate choice of ξ ∈ 𝒥 (R) each associated PS limit measure m in the sense of Definition 29.1.12 and as constructed at the beginning ∞ −n k of Section 29.2, is atomless. As the critical set’s grand orbit ⋃∞ n=0 R (⋃k=0 R (Crit(R))), −n which coincides with the postcritical set’s backward orbit PC−∞ (R) := ⋃∞ n=0 R (PC(R)), is countable and as HD(Safe(R) ∩ 𝒥uer (R)) > 0 (by Theorem 28.5.3 and Lemma 29.2.3), there exists a point w ∈ [Safe(R) ∩ 𝒥uer (R)] \ PC−∞ (R). Clearly, PC−∞ (R) is completely R-invariant. Moreover, by its very Definition 29.2.2, the set Safe(R) is backward R-invariant, i.e. R−1 (Safe(R)) ⊆ Safe(R). Though 𝒥uer (R) is generally not backward R-invariant, the set 𝒥uer (R) \ PC−∞ (R) is. We easily deduce that [Safe(R) ∩ 𝒥uer (R)] \ PC−∞ (R) is backward R-invariant and hence ∞

⋃ R−j (w) ⊆ [Safe(R) ∩ 𝒥uer (R)] \ PC−∞ (R). j=0

Since R|𝒥 (R) is topologically exact, it is strongly transitive, i. e., the backward orbit of any Julia point is dense in the Julia set. Given that ω(R) is nowhere dense in 𝒥 (R) (by Proposition 26.3.4), by decreasing η > 0 if necessary, there is a point ξ ∈ [Safe(R) ∩ 𝒥uer (R)] \ [PC−∞ (R) ∪ B(ω(R), 4η)].

(32.26)

Let m be a PS limit measure in the sense of Definition 29.1.12 as performed at the beginning of Section 29.2 for the function φhR := −hR log |R′ |. Recall that hR := δ(R) = sξ (R) := inf{t ≥ 0 | cξ (t) ≤ 0} by Theorem 29.2.12 and cξ (sξ (R)) = 0 by (29.49). So, let (sj )∞ j=1 be a sequence strictly decreasing to 0 such that m = lim msj j→∞

per Definition 29.1.12. Fix σ ∈ (0, 1) sufficiently close to 1 as will be specified later in the course of the proof. By redefining finitely many terms of the sequence (bn )∞ n=1 from Lemma 29.1.10, we may assume that bn ≥ σ, bn+1

∀n ∈ ℕ.

As En+1 = R−1 (En ) for every n ∈ ℕ, it follows from (29.26) that msj (R(A)) =

1 ∞ ∑ Msj n=1

∑

x∈A∩En+1

′ 󵄨 󵄨−h 󵄨 󵄨h bn 󵄨󵄨󵄨(Rn+1 ) (x)󵄨󵄨󵄨 R e−sj (n+1) esj 󵄨󵄨󵄨R′ (x)󵄨󵄨󵄨 R

for every j ∈ ℕ and every special set A ⊆ 𝒥 (R). Using (32.27), we get

(32.27)

32.1 Sullivan conformal measures for subexpanding rational functions

msj (R(A)) ≥ σ

1 ∞ ∑ Msj n=1

∑

x∈A∩En+1

� 1301

′ 󵄨h 󵄨−h 󵄨 󵄨 bn+1 󵄨󵄨󵄨(Rn+1 ) (x)󵄨󵄨󵄨 R e−sj (n+1) esj 󵄨󵄨󵄨R′ (x)󵄨󵄨󵄨 R

1 󵄨 󵄨h ≥ σ(esj ∫󵄨󵄨󵄨R′ 󵄨󵄨󵄨 R dmsj − Msj A

∑ b1 )

x∈A∩E1

󵄨 󵄨h ≥ σ(e ∫󵄨󵄨󵄨R′ 󵄨󵄨󵄨 R dmsj − b1 #(A ∩ E1 )) sj

A

for all j ∈ ℕ large enough since limj→∞ Msj = ∞. If, in addition, ξ ∉ A,

i. e.,

A ∩ E1 = 0,

(32.28)

then 󵄨 󵄨h 󵄨 󵄨h msj (R(A)) ≥ σesj ∫󵄨󵄨󵄨R′ 󵄨󵄨󵄨 R dmsj ≥ σ ∫󵄨󵄨󵄨R′ 󵄨󵄨󵄨 R dmsj . A

(32.29)

A

Now, let z0 ∈ Crit(T p ). Set q := degz (T 2p ) ≥ 2. Choose a ∈ (0, 1) and b > 0 such that (1 − q(a + b) + b) ≥ 1/2. In light of (26.15), choosing σ ≥ λ−ahR /s ensures that 󵄨 󵄨−ah σ s ≥ 󵄨󵄨󵄨T ′ (z)󵄨󵄨󵄨 R ,

∀z ∈ B(ω(T), 2η).

If, additionally, a special set A for T is such that s−1

⋃ Ri (A) ⊆ B(ω(T), 2η), i=0

then using (32.29) s times and the chain rule yields ′ 󵄨h 󵄨 󵄨 󵄨(1−a)hR dmsj . msj (T(A)) = msj (Rs (A)) ≥ σ s ∫󵄨󵄨󵄨(Rs ) 󵄨󵄨󵄨 R dmsj ≥ ∫󵄨󵄨󵄨T ′ 󵄨󵄨󵄨 A

(32.30)

A

Having this and (32.26), we deduce in the same way as in Step 1 of the proof of Proposition 32.1.7 (via Lemma 32.1.1; see particularly (32.2)) that msj (B(z, r)) ≤ K (1−a)hR ζ −(1−a)hR r (1−a)hR ,

∀r ∈ (0, 1]

(32.31)

whenever {T n (z) : n ≥ 0} ⊆ B(ω(T), 2η). It follows from (32.26) that there exists u ∈ (0, 1) such that 2sp−1

ξ ∉ ⋃ Ri (B(Crit(R), u)). i=0

(32.32)

Let ρ > 0 be so small that hi (B(0, ρ)) ⊆ B(Crit(R), u), where the hi ’s are from Step 2 of the proof of Proposition 32.1.7. From msj define two Borel measures msj ,i as in (32.11).

1302 � 32 Conformal measures, invariant measures and fractal geometry Let A ⊆ B(0, ρ) be any Borel set on which the map w 󳨃→ wq is one-to-one. Using (32.32) and 2sp times relation (32.29), we obtain by proceeding as in (32.14) that msj ,2 (Aq ) =

󵄨󵄨 −1 ′ 󵄨󵄨hR 󵄨󵄨(h2 ) 󵄨󵄨 dmsj

∫ T 2p (h1 (A))

󵄨 󵄨h 󵄨h 󵄨 ≥ σ 2sp ∫ 󵄨󵄨󵄨(h2−1 )′ ∘ T 2p 󵄨󵄨󵄨 R 󵄨󵄨󵄨(T 2p )′ 󵄨󵄨󵄨 R dmsj h1 (A)

hR

= σ̃ q ∫ |w|(q−1)hR dmsj ,1 (w),

(32.33)

A

where σ̃ := σ 2ps ∈ (0, 1) is independent of ρ and A and satisfies limσ↗1 σ̃ = 1. Reducing ρ > 0 if necessary, we have σ̃ ≥ ρ(q−1)bhR . Therefore, σ̃ ≥ |w|(q−1)bhR for all w ∈ B(0, ρ) and it follows from (32.33) that msj ,2 (Aq ) ≥ qhR ∫ |w|(q−1)(1+b)hR dmsj ,1 (w) A

for every Borel set A ⊆ B(0, ρ) on which the map w 󳨃→ wq is one-to-one. Proceeding in the same way as in Step 2 of the proof of Proposition 32.1.7, relation (32.15) takes the form qC̃10 r q(1−a)hR ≥ qhR ν

q−1 (1+b)hR q

r (q−1)(1+b)hR msj ,1 (A(0; √q ν r, r)),

where the constant C̃10 arises from (32.31). As consequences corresponding to (32.17) and (32.18), we have that − msj ,1 (A(0; √q ν r, r)) ≤ C̃10 q1−hR ν

q−1 (1+b)hR q

− r (1−q(a+b)+b)hR ≤ C̃10 q1−hR ν

q−1 (1+b)hR q

hR

r2,

where the rightmost inequality follows from the choice of a, b. Hence, msj ,1 (B(0, r)) ≤ C̃11 q−1

1 1−ν

hR

hR 2q

r2,

R where C̃11 := C̃10 q1−hR ν q . Therefore, by means of a similar argument to that given in the analogous context of Step 2 in the proof of Proposition 32.1.7, we infer that

−

(1+b)h

hR

msj (B(z0 , r)) ≤ C 11 r 2 ,

∀r ∈ (0, 1],

for some constant C 11 ≥ 1. The set 󵄨󵄨 ∞ 󵄨󵄨 Nz0 := {r ∈ (0, 1] 󵄨󵄨󵄨 ∑ msj (𝜕B(z0 , r)) > 0} 󵄨󵄨 󵄨 j=1

32.1 Sullivan conformal measures for subexpanding rational functions

� 1303

is countable. Using the Portmanteau theorem (Theorem A.1.23), for every r ∈ (0, 1] \ Nz0 we obtain that hR

m({z0 }) ≤ m(B(z0 , r)) = lim msj (B(z0 , r)) ≤ C 11 r 2 . j→∞

Thus, m({z0 }) = 0. Since z0 was arbitrarily chosen among Crit(T p ), we deduce that m(Crit(T p )) = 0. The proof of Theorem 32.1.8 is complete by an application of Proposition 32.1.7. Beyond its existence and geometric nature, the main result about mR (its uniqueness, ergodicity and conservativity) is next. This is the counterpart of Theorem 31.3.9 for subexpanding maps. ̂→ℂ ̂ is a subexpanding rational function with deg(R) ≥ 2, then Theorem 32.1.9. If R : ℂ the dynamical system R : 𝒥 (R) → 𝒥 (R) is conservative and ergodic with respect to the hR -conformal measure mR . Moreover, supp(mR ) = 𝒥 (R) and C9−1 ≤

mR (B(z, r)) ≤ C9 , r hR

∀r ∈ (0, 1]

(32.34)

for every z ∈ 𝒥 (R). Furthermore, mR is the unique hR -conformal measure for R. Proof. The topological exactness of R (per Corollary 24.3.5) and the conformality of mR guarantee that supp(mR ) = 𝒥 (R). Relation (32.34) holds because of Theorem 32.1.8 and Proposition 32.1.7. We now prove the following. Claim 1. Any hR -conformal measure m′ for R is equivalent to mR . In particular, m′ (Crit−∞ (R)) = 0. Proof of Claim 1. The proof goes along the same lines as that of Claim 1 in Theorem 31.3.9 by replacing Ω−∞ (T) with Crit−∞ (R) and using Theorem 32.1.8 and Lemma 32.1.5 instead of Theorem 31.3.7 and Lemma 31.3.4. ◼ Claim 2. The measure mR is ergodic. Consequently, it is totally ergodic, i. e., it is ergodic with respect to every positive iterate Rk of R. Indeed, note that 𝒥 (Rk ) = 𝒥 (R), Rk is subexpanding, and the measure mR fulfills all the assertions of Theorem 32.1.8 for Rk . Proof of Claim 2. The proof is identical to that of Claim 2 in Theorem 31.3.9.

◼

Claim 3. If m′ is an hR -conformal measure for R and ρ := dm′ /dmR , then ρ is mR -a. e. T-invariant, i. e., ρ∘T =ρ

mR -a. e.

Proof of Claim 3. By virtue of Claim 1, the function ρ is well-defined mR -almost everywhere. As mR is atomless (per Theorem 32.1.8) and Crit−∞ (R) is countable, we know that

1304 � 32 Conformal measures, invariant measures and fractal geometry mR (Crit−∞ (R)) = 0. Recall further that 𝒥 (R) = 𝒥 (T) and Crit−∞ (R) = Crit−∞ (T). Fix z ∈ 𝒥 (R) \ Crit−∞ (R) = 𝒥 (T) \ Crit−∞ (T). Denote ι :=

1 min{2γ, θ, ξ0 }. 2

(32.35)

For all k ∈ ℕ, let Vk (z) := Tz−nk (z) ( B(T nk (z) (z), (3K 2 )−1 ι) ), where the sequence (nk (z))∞ k=1 and the corresponding holomorphic inverse branches −nk (z) nk (z) ̂ that send T nk (z) (z) to z, come from Remark 32.1.3. Since Tz : B(T (z), 2ι) → ℂ infk∈ℕ diam(Vk (z)) = 0, the family V = {Vk (z) : z ∈ 𝒥 (T) \ Crit−∞ (T), k ∈ ℕ} is (in the terminology of Section 2.8 in Federer [50]) fine at every z ∈ 𝒥 (T) \ Crit−∞ (T). Like (31.45), set ̂k (z) := { ⋃ W : W ∩ Vk (z) ≠ 0 and diam(W ) ≤ 2diam(Vk (z))}. V W ∈V

(32.36)

Relations similar to (31.46)–(31.49) hold: Tz−nk (z) (B(T nk (z) (z), ι)) ⊇ B(z, K −1 ι|(T nk (z) )′ (z)|−1 ), ′ 󵄨 󵄨−h mR (Tz−nk (z) (B(T nk (z) (z), ι))) ≤ K hR 󵄨󵄨󵄨(T nk (z) ) (z)󵄨󵄨󵄨 R , ′ 󵄨 󵄨−h mR (Vk (z)) ≥ MmR ((3K 2 )−1 ι)K −hR 󵄨󵄨󵄨(T nk (z) ) (z)󵄨󵄨󵄨 R , ′ 󵄨 󵄨−1 diam(Vk (z)) ≤ (3K)−1 ι󵄨󵄨󵄨(T nk (z) ) (z)󵄨󵄨󵄨 .

(32.37) (32.38) (32.39) (32.40)

Using (32.36), (32.40) and (32.37) successively, we get ̂k (z) ⊆ B(z, 3diam(Vk (z))) ⊆ B(z, K −1 ι|(T nk (z) )′ (z)|−1 ) ⊆ T −nk (z) (B(T nk (z) (z), ι)). V z Therefore, by (32.38), ̂k (z)) ≤ K hR 󵄨󵄨󵄨(T nk (z) )′ (z)󵄨󵄨󵄨−hR . mR (V 󵄨 󵄨 Along with (32.39), this yields ̂k (z)) mR (V

mR (Vk (z))

−1

≤ K 2hR [MmR ((3K 2 )−1 ι)] .

Thus, lim sup(diam(Vk (z)) + k→∞

̂k (z)) mR (V mR (Vk (z))

−1

) ≤ K 2hR [MmR ((3K 2 )−1 ι)]

< ∞.

32.1 Sullivan conformal measures for subexpanding rational functions

� 1305

So, the hypotheses of Theorem 2.8.17 in Federer [50] are satisfied. That theorem along with Theorem 2.9.8 in [50] imply that ρ(z) :=

m′ (Vk (z)) dm′ (z) = lim k→∞ mR (Vk (z)) dmR

for mR -a. e. z ∈ 𝒥 (R) \ Crit−∞ (R).

Since Vk (z) = Tz−1 (Vk−1 (T(z))) for all z ∈ 𝒥 (T) \ Crit−∞ (T) = 𝒥 (R) \ Crit−∞ (R) and all k ∈ ℕ large enough, it ensues that 󵄨󵄨 −1 ′ 󵄨󵄨hR ′ 󵄨󵄨(Tz ) 󵄨󵄨 dm = lim . ρ(z) = lim ′ 󵄨h 󵄨 k→∞ k→∞ mR (T −1 (Vk−1 (T(z)))) ∫V (T(z)) 󵄨󵄨󵄨(Tz−1 ) 󵄨󵄨󵄨 R dmR z k−1 ∫V

m′ (Tz−1 (Vk−1 (T(z))))

k−1 (T(z))

(32.41)

As ′ ′ 󵄨 󵄨 󵄨 󵄨 inf{󵄨󵄨󵄨(Tz−1 ) (w)󵄨󵄨󵄨 : w ∈ Vk−1 (T(z))} sup{󵄨󵄨󵄨(Tz−1 ) (w)󵄨󵄨󵄨 : w ∈ Vk−1 (T(z))} = 1 = lim , 󵄨󵄨 −1 ′ 󵄨 󵄨󵄨 −1 ′ 󵄨 k→∞ k→∞ 󵄨󵄨(Tz ) (T(z))󵄨󵄨󵄨 󵄨󵄨(Tz ) (T(z))󵄨󵄨󵄨

lim

we can continue (32.41) as follows: ρ(z) = lim

k→∞

m′ (Vk−1 (T(z)))

mR (Vk−1 (T(z)))

= lim

k→∞

m′ (Vk (T(z)))

mR (Vk (T(z)))

= ρ(T(z))

for mR -a. e. z ∈ 𝒥 (R) \ Crit−∞ (R). As mR (𝒥 (R) \ Crit−∞ (R)) = 0 according to Claim 1, we conclude that ρ ∘ T = ρ mR -almost everywhere. Claim 3 is thus proved. ◼ Claim 4. mR is the unique hR -conformal measure for R. Proof of Claim 4. Let m′ be an hR -conformal measure for R. As mR is quasi-R-invariant, it follows from Claims 2–3 that the Radon–Nikodym derivative ρ = dm′ /dmR is constant mR -almost everywhere. Since both measures m′ and mR are probabilistic, the derivative ρ must be identically equal to 1. So m′ = mR and Claim 4 is proved. ◼ Our next goal is to show that the measure mR is conservative. This will easily ensue from the following claim. Claim 5. If E ⊆ 𝒥 (R) is a forward or backward R-invariant Borel set, then mR (E) ∈ {0, 1}. Proof of Claim 5. By passing to the complement, it suffices to prove the statement for a forward R-invariant set E. Since mR (Crit−∞ (R)) = 0 and the set Crit−∞ (R) is completely R-invariant, we may assume that E ⊆ 𝒥 (R) \ Crit−∞ (R) = 𝒥 (T) \ Crit−∞ (T). Suppose for a contradiction that 0 < mR (E) < 1. By Lebesgue’s density theorem, there exists at least one point z ∈ E (in fact, the set of such points has mR -measure equal to mR (E)) such that mR (E ∩ B(z, r)) = 1. r→0 mR (B(z, r)) lim

(32.42)

1306 � 32 Conformal measures, invariant measures and fractal geometry ∞ Let (si (z))∞ i=1 be a subsequence of the sequence (nk (z))k=1 produced in Remark 32.1.3, si (z) ∞ subsequence such that the sequence (T (z))i=1 converges to a point ̂z. Seeking a contradiction, suppose that mR (B(̂z, (2K 2 )−1 ι) \ E) = 0, where ι > 0 is the number defined in (32.35). Given that R|𝒥 (R) is topologically exact according to Corollary 24.3.5, let N ∈ ℕ be such that RN (B(̂z, (2K 2 )−1 ι)) ⊇ 𝒥 (R). By partitioning the ball B(̂z, (2K 2 )−1 ι) into finitely many special Borel sets for RN (cf. Lemma 29.1.15), the hR -conformality of mR entails that mR (RN (B(̂z, (2K 2 )−1 ι) \ E)) = 0. From the forward R-invariance of E, we deduce that

0 = mR (RN (B(̂z, (2K 2 )−1 ι) \ E)) ≥ mR (RN (B(̂z, (2K 2 )−1 ι)) \ RN (E))

≥ mR (RN (B(̂z, (2K 2 )−1 ι)) \ E) ≥ mR (RN (B(̂z, (2K 2 )−1 ι))) − mR (E)

= 1 − mR (E).

Hence, mR (E) ≥ 1. This contradiction shows that u := mR (B(̂z, (2K 2 )−1 ι) \ E) > 0. As |T si (z) (z) − ̂z| < (2K 2 )−1 ι for all i ∈ ℕ large enough, say i ≥ N(z), we infer that B(̂z, (2K 2 )−1 ι) ⊆ B(T si (z) (z), K −2 ι),

∀i ≥ N(z).

Let ′ 󵄨 󵄨−1 ri (z) := K −1 󵄨󵄨󵄨(T si (z) ) (z)󵄨󵄨󵄨 ι.

It follows from Lemma 23.1.11 and the forward R-invariance of E that Tz−si (z) (B(̂z, (2K 2 )−1 ι) \ E) ⊆ Tz−si (z) (B(T si (z) (z), K −2 ι) \ E) ⊆ B(z, ri (z)) \ E,

∀i ≥ N(z).

By virtue of the hR -conformality of mR and of the Koebe distortion theorem (Theorem 23.1.9), we thereby get that mR (B(z, ri (z)) \ E) ≥ mR (Tz−si (z) (B(̂z, (2K 2 )−1 ι) \ E)) ′ 󵄨 󵄨h ≥ K −hR 󵄨󵄨󵄨(Tz−si (z) ) (T si (z) (z))󵄨󵄨󵄨 R mR (B(̂z, (2K 2 )−1 ι) \ E) ′ 󵄨 󵄨−h = uK −hR 󵄨󵄨󵄨(T si (z) ) (z)󵄨󵄨󵄨 R , ∀i ≥ N(z). On the other hand, it follows from Lemma 23.1.11 that B(z, ri (z)) ⊆ Tz−si (z) (B(T si (z) (z), ι)),

∀i ∈ ℕ.

Thus, by conformality and Koebe’s distortion theorem, ′ 󵄨 󵄨−h mR (B(z, ri (z)) ≤ K hR 󵄨󵄨󵄨(T si (z) ) (z)󵄨󵄨󵄨 R ,

∀i ∈ ℕ.

32.1 Sullivan conformal measures for subexpanding rational functions

� 1307

Therefore, mR (B(z, ri (z)) \ E) mR (B(z, ri (z)))

≥

′ 󵄨−h 󵄨 uK −hR 󵄨󵄨󵄨(T si (z) ) (z)󵄨󵄨󵄨 R = uK −2hR > 0, ′ 󵄨−h 󵄨 K hR 󵄨󵄨󵄨(T si (z) ) (z)󵄨󵄨󵄨 R

∀i ≥ N(z).

Hence, lim sup i→∞

mR (E ∩ B(z, ri (z))) mR (B(z, ri (z)))

≤ 1 − uK −2hR < 1,

contrary to (32.42). So, mR (E) ∈ {0, 1}. This finishes the proof of Claim 5.

◼

Claim 6. mT is conservative. Proof of Claim 6. The proof is the same as its counterpart in Theorem 31.3.9.

◼

This finishes the proof of Theorem 32.1.9. This theorem entails the set of transitive points to be of full mR -measure. ̂→ℂ ̂ is a subexpanding rational function with deg(R) ≥ 2, then Corollary 32.1.10. If R : ℂ the mR -measure of the set of all transitive points of the dynamical system R : 𝒥 (R) → 𝒥 (R) is equal to 1. Proof. Replace T by R in the proof of Corollary 31.3.10 and rely on Theorem 32.1.9 rather than Theorem 31.3.9. ̂ → ℂ ̂ is a subexpanding rational function with deg(R) ≥ 2, Corollary 32.1.11. If R : ℂ then mR (PC(R)) = 0. Proof. Clearly, R(PC(R)) ⊆ PC(R). The continuity of R then imposes that R(PC(R)) ⊆ PC(R). Moreover, R(ω(R)) ⊆ ω(R). The ergodicity of mR (per Theorem 32.1.9) implies that {mR (PC(R)), mR (ω(R))} ⊆ {0, 1}. By Proposition 26.3.4 and the Baire category theorem, the set PC(R) is nowhere dense in 𝒥 (R). Corollary 32.1.11 thus follows from Corollary 32.1.10 and the ergodicity of mR . Alternatively, note that the nowhere dense set ω(R) does not intersect the set of transitive points of R, and hence mR (ω(R)) = 0 by Corollary 32.1.10. Given that PC(R) ⊆ PC(R) ∪ ω(R) with PC(R) countable and that mR is atomless, we conclude that mR (PC(R)) ≤ mR (PC(R)) + mR (ω(R)) = 0.

1308 � 32 Conformal measures, invariant measures and fractal geometry 32.1.3 The complete structure of Sullivan’s conformal measures for subexpanding rational functions: the atomless measure mR and purely atomic measures We can now collect together the results of the previous sections and provide a short concluding proof of the main theorem of this section. It gives a full description of the structure of the Sullivan conformal measures of a subexpanding rational function. Recall that these maps are characterized by the disjointness of their critical set with their ω-limit set: Crit(R) ∩ ω(R) = 0. Define Crit∗ (R) := [Crit(R) ∩ 𝒥 (R)] \ PC(R). By Proposition 26.3.3, the accumulation points of PC(R) that lie in 𝒥 (R) are in ω(R). Therefore, Crit∗ (R) ∩ PC(R) = 0 and there exists Δ > 0 such that B(Crit∗ (R), 4Δ) ∩ PC(R) = 0.

(32.43)

We therefore get the following. ̂→ℂ ̂ is a subexpanding rational function with deg(R) ≥ 2, then Lemma 32.1.12. If R : ℂ for every point c ∈ Crit∗ (R), every n ∈ ℕ, and every point z ∈ R−n (c) there exists a unique −n ̂ holomorphic branch R−n sending c to z. z : B(c, 4Δ) → ℂ of R We now state and prove a theorem which parallels Theorem 31.3.11; it is not however identical to it. ̂→ℂ ̂ be a subexpanding rational function with deg(R) ≥ 2. Theorem 32.1.13. Let R : ℂ Recall from Theorem 32.1.6 (Bowen’s formula) that hR := δ(R) = ExpD∗∗ (𝒥 (R)) = ExpD∗ (𝒥 (R)) = ExpD(𝒥 (R)) = DD(𝒥 (R)) = HD(𝒥uer (R)) = HD(𝒥er (R)) = HD(𝒥r (R)) = HD(𝒥 (R)) ∈ (

pR , 2]. pR + 1

The following situations prevail: (a) If t < hR , then R does not admit any Sullivan t-conformal measure. (b) R has exactly one hR -conformal measure, namely mR . This measure is atomless, ergodic, conservative and geometric with exponent hR . (c) For every t > hR and every c ∈ Crit∗ (R), it holds that ∞

Σt (c) := ∑

′ 󵄨 󵄨−t ∑ 󵄨󵄨󵄨(Rn ) (z)󵄨󵄨󵄨 < ∞.

n=0 z∈R−n (c)

Accordingly, let

(32.44)

32.1 Sullivan conformal measures for subexpanding rational functions

νt,c :=

1 ∞ ∑ Σt (c) n=0

� 1309

′ 󵄨−t 󵄨 ∑ 󵄨󵄨󵄨(Rn ) (z)󵄨󵄨󵄨 δz ,

z∈R−n (c)

(32.45)

where δz denotes the Dirac measure concentrated at the point z. Then the set of all t-conformal measures for R coincides with the set of all convex combinations of measures {νt,c }c∈Crit∗ (R) . In particular, these conformal measures are atomic. Proof. (a) Pick ξ ∈ Safe(R) ∩ 𝒥uer (R). Let mt be the PS limit measure (in the sense of Definition 29.1.12) produced in Lemma 29.2.1 for the function φt := −t log |R′ |. Recall that hR := δ(R) = sξ (R) := inf{t ≥ 0 | cξ (t) ≤ 0} by Theorem 29.2.12 and cξ (sξ (R)) = 0 by (29.49). Given that t ∈ [0, hR ), we know from (29.46) that cξ (t) > 0. By Lemma 29.2.4, we deduce that eP(t) ≥ ecξ (t) > 1. Lemmas 32.1.1, 32.1.2 and 32.1.5 can be generalized to (t, λt )-conformal measures like Lemmas 31.3.3–31.3.4 were extended to Lemmas 31.3.5–31.3.6. The proof is then essentially analogous to that of Theorem 31.3.11(a) with Ω(T) replaced by Crit(R). It is left to the reader as an exercise. (b) This follows from the combination of Theorems 32.1.8–32.1.9. (c) The proof is similar to that of Theorem 31.3.11(c). Clearly, Claims 1–4 remain valid with T replaced by R. In place of Claim 5, we prove the following. Claim 5′ . If Σt (x) < ∞ for some t ≥ 0 and x ∈ 𝒥 (T), then Σt (y) < ∞ for every −n y ∈ Crit−∗∞ (R) := ⋃∞ n=0 R (Crit∗ (R)). Proof of Claim 5′ . By Claim 3, it suffices to establish the statement for y ∈ Crit∗ (R). This follows from (32.43) and Claim 4. ◼ In lieu of Claims 6–7, we prove the next two claims. Claim 6′ . If t > hR = sξ (R) and cξ (t) < 0, then Σt (ξ) < ∞ and Σt (y) < ∞ for every y ∈ Crit−∗∞ (R). Proof of Claim 6′ . Fix s ∈ (cξ (t), 0). By definition of cξ (t), we obtain that ∞

Σt (ξ) = ∑

n=0

∞

′ 󵄨 󵄨−t ∑ 󵄨󵄨󵄨(Rn ) (z)󵄨󵄨󵄨 ≤ ∑

z∈R−n (ξ)

′ 󵄨 󵄨−t ∑ 󵄨󵄨󵄨(Rn ) (z)󵄨󵄨󵄨 e−sn < ∞.

n=0 z∈R−n (ξ)

Therefore, this claim follows from Claim 5′ .

◼

Claim 7′ . If t > hR and cξ (t) = 0, then Σt (y) < ∞ for every y ∈ Crit−∗∞ (R). Proof of Claim 7′ . Let mt be the measure produced in Lemma 29.2.1. Since cξ (t) = 0, it follows from Lemma 29.2.1 that mt is t-conformal. As hR < t and mR is hR -conformal, it ensues from Lemma 32.1.5 that mt (𝒥 (R) \ Crit−∞ (R)) = 0. So, there exists c ∈ Crit(R) ∩ 𝒥 (R) for which mt (𝒪− (c)) > 0, and thus mt ({c}) > 0. The conformality

1310 � 32 Conformal measures, invariant measures and fractal geometry of mt also entails that mt (Crit(R) ∩ PC(R)) = 0. Then c ∈ [Crit(R) ∩ 𝒥 (R)] \ PC(R) =: Crit∗ (R). Given that mt (𝒪− (c)) = mt ({c})Σt (c), we conclude that −1

−1

Σt (c) = (mt ({c})) mt (𝒪− (c)) ≤ (mt ({c}))

< ∞.

◼

Consequently, this claim ensues from Claim 5′ . Together, Claims 6′ –7′ conclude the proof of (32.44).

By definition (32.45), all measures {νt,c }c∈Crit∗ (R) are t-conformal. Obviously, so are all their convex combinations. On the other hand, let m be a t-conformal measure for R. It follows from Lemma 32.1.5 (applied with s = hR ) that m(Crit−∞ (R)) = 1. The conformality of m entails that m(Crit(R) ∩ PC(R)) = 0. Consequently, 1 = m(Crit−∗∞ (R)) = ∑c∈Crit∗ (R) m(𝒪− (c)). Using the t-conformality of m, we readily deduce that m is a convex combination of measures νt,c , c ∈ Crit∗ (R). Thus, (c) holds.

32.2 Invariant probability measure equivalent to mR : existence, uniqueness and ergodicity In this section, we establish the existence and uniqueness of an R-invariant Borel probability measure which is absolutely continuous with respect to the hR -conformal measure mR . This is the counterpart of Theorem 31.4.1. ̂→ℂ ̂ is a subexpanding rational function with deg(R) ≥ 2, then Theorem 32.2.1. If R : ℂ there exists a unique R-invariant Borel probability measure μR on 𝒥 (R) which is absolutely continuous with respect to the hR -conformal measure mR . In fact, μR is equivalent to mR . It is also ergodic. Proof. We first show the existence of an R-invariant σ-finite Borel measure μR which is absolutely continuous with respect to mR . For every z ∈ 𝒥 (R) \ PC(R), fix u(z) > 0 such that B(z, 2u(z)) ∩ PC(R) = 0

and mR (𝜕B(z, u(z))) = 0.

As 𝒥 (R) \ PC(R) is a Lindelöf space, there is a sequence (zj )∞ j=0 in 𝒥 (R) \ PC(R) such that ∞

⋃ B(zj , u(zj )) ⊇ 𝒥 (R) \ PC(R). j=0

Out of the cover {B(zj , u(zj ))}∞ j=0 of 𝒥 (R) \ PC(R), define inductively a Borel partition 𝒜 = (An )∞ n=0 of 𝒥 (R) \ PC(R) (partitioning up to a set of mR -measure zero) by setting A0 := B(z0 , u(z0 ))

and

n−1

An := B(zn , u(zn )) \ ⋃ B(zj , u(zj )), j=0

∀n ∈ ℕ

1311

32.2 Invariant probability measure equivalent to mR : existence, uniqueness and ergodicity �

while throwing away empty sets. Clearly, An ⊆ B(zn , u(zn )),

∀n ≥ 0.

Due to the definition of u(z), z ∈ 𝒥 (R) \ PC(R), due to Theorem 23.1.9 (Koebe’s distortion theorem), due to Theorem 32.1.9, due to Corollary 32.1.11, and due to the finiteness-to-one of the map R, the partition 𝒜 satisfies all the requirements of Theorem 10.4.5. Define μ′R to be the R-invariant σ-finite Borel measure resulting from the application of this theorem with mR as reference measure. We now demonstrate that μ′R (B(z, κΓR )) < ∞,

(32.46)

∀z ∈ 𝒥 (R),

where ΓR > 0 comes from Lemma 26.3.9 and Corollary 26.3.10 whereas κ is from ̂ and every n ≥ 0, let Lemma 26.3.11. Indeed, for every set D ⊆ ℂ n

Sn (mR )(D) := ∑ mR (R−n (D)). k=0

As mR (B(z, κΓR )) > 0 by Theorem 32.1.9 and as mR (⋃∞ n=0 An ) = 1 by Corollary 32.1.11, there exists i ≥ 0 such that m(B(z, κΓR ) ∩ Ai ) > 0. As B(z, κΓR ) ∩ Ai is an open set having nonempty intersection with 𝒥 (R), there exists an open ball G ⊆ B(z, κΓR ) ∩ Ai with center in 𝒥 (R). Of course, mR (G) > 0. Given n ≥ 0, let W be a connected component of R−n (B(z, κΓR )) and let V ⊆ W be a connected component of R−n (G) contained in W . Using Koebe’s distortion theorem (Theorem 23.1.9), the hR -conformality of mR and Theorem 32.1.9, we get hR

mR (V ) ≍ (diam(V ))

h

=(

h

diam(V ) R diam(V ) R h ) (diam(W )) R ⪰ mR (W )( ) , diam(W ) diam(W )

where ≍ and ⪰ are symbols of multiplicative comparisons (cf. (25.34)). Applying Lemma 26.3.11 to the connected sets G and B(z, κΓR ), we obtain mR (V ) ⪰ mR (W )(

h

R diam(G) ) . diam(B(z, κΓR ))

Thus, Sn (mR )(B(z, κΓR )) ⪯ Sn (mR )(G) ⪯ Sn (mR )(Ai ). So, with the notation in Theorem 10.4.5 (see (10.21)), we get that (mR )n (B(z, κΓR )) ⪯ (mR )n (Ai ),

∀n ≥ 0.

1312 � 32 Conformal measures, invariant measures and fractal geometry It therefore follows from the first paragraph and the last displayed formula in Theorem 10.4.5 that ∞

((mR )n (B(z, κΓR )))n=0 ∈ l∞ . It ensues from (10.23) in Theorem 10.4.5 that (32.46) holds. Since 𝒥 (R) is a compact set, it follows from (32.46) that μ′R (𝒥 (R)) < ∞. We then define −1

μR := (μ′R (𝒥 (R))) μ′R . Finally, ergodicity, equivalence with mR and uniqueness of the measure μR follow from the last paragraph of Theorem 10.4.5 and from Theorem 32.1.9. Remark 32.2.2. As an immediate consequence of Theorem 32.2.1, we get that μRn = μR for every n ∈ ℕ. So μR is totally ergodic.

32.3 Hausdorff and packing measures In this short section, we give a precise geometric description of the fractal structure of the Julia set of a subexpanding rational function in terms of the Hausdorff and packing measures. It is nearly the same as that for expanding rational functions (cf. Theorem 30.1.1(f)) and is much simpler than that for parabolic rational functions (cf. Theorem 31.5.1). Theorem 32.3.1 (Fundamental fractal geometry for subexpanding rational functions). Let ̂ → ℂ ̂ be a subexpanding rational function with deg(R) ≥ 2. Recall from TheoR : ℂ rem 32.1.6 (Bowen’s formula) that hR := δ(R) = ExpD∗∗ (𝒥 (R)) = ExpD∗ (𝒥 (R)) = ExpD(𝒥 (R)) = DD(𝒥 (R)) = HD(𝒥uer (R)) = HD(𝒥er (R)) = HD(𝒥r (R)) = HD(𝒥 (R)) ∈ (

pR , 2]. pR + 1

The following statements hold: (a) mR is the only Sullivan conformal measure for R and its exponent is equal to hR . (b) mR is a geometric measure with exponent hR . (c) The measures mR , Hausdorff HhR |𝒥 (R) , and packing PhR |𝒥 (R) , are finite and positive, and are equal up to a multiplicative constant. ̂ then hR < 2. (d) If 𝒥 (R) ≠ ℂ,

32.4 Exercises

� 1313

Proof. (a, b) are part of Theorem 32.1.13(b). (c) Finiteness and positivity of the measures mR , HhR |𝒥 (R) and PhR |𝒥 (R) is an immediate consequence of (b) and Theorem 15.6.14. Equality up to a multiplicative constant now follows from (a) and the fact that both measures HhR |𝒥 (R) and PhR |𝒥 (R) are hR -conformal. ̂ it ensues from Theorem 29.2.13, Corollary 32.1.4 and the countabil(d) As 𝒥 (R) ≠ ℂ, − ity of Crit∞ (R) that Leb2 (𝒥 (R)) = 0. Since the measures H2 and P2 are constant multiples of Leb2 , we conclude from (c) that hR < 2. The study of conformal measures, measurable dynamics, and fractal geometry of Julia sets of rational functions concurrently exhibiting parabolic and subexpanding features has been done in [138] and [140] for the class of all nonrecurrent rational functions, i. e., functions for which c ∉ ω(c) for all critical points c in the Julia set. This study has also been carried out for elliptic functions in [73]. An important extension of subexpanding rational functions is the class of topological Collet–Eckmann rational functions. It has been thoroughly studied from the point of view of this chapter in [108].

32.4 Exercises ̂ → ℂ ̂ is a rational function with deg(T) ≥ 2 and Exercise 32.4.1. Show that if T : ℂ Crit(T) ⊈ PC(T), then for every t > δ(T) there exists a t-conformal measure supported on Crit−∞ (T).

Appendix A – A selection of classical results This appendix lists classical definitions and results that will be used in this volume.

A.1 Measure theory Proofs and further explanations of the results can be found in many books on measure theory, for instance, Billingsley [12, 13] and Rudin [112]. Lemma A.1.1 (Borel–Cantelli lemma). Let (X, 𝒜, μ) be a measure space and (An )∞ n=1 a se∞ ∞ quence in 𝒜. If ∑∞ μ(A ) < ∞, then μ(⋂ A ) = 0. ⋃ n n n=1 k=1 n=k A.1.1 Convergence theorems In measure theory, there are fundamental theorems that are especially helpful for finding the integral of functions that are the pointwise limit of sequences of functions. The first of these results applies to monotone sequences of functions. A sequence of functions (fn )∞ n=1 is monotone if it is increasing pointwise (fn+1 (x) ≥ fn (x) for all x ∈ X and all n ∈ ℕ) or decreasing pointwise (fn+1 (x) ≤ fn (x) for all x ∈ X and all n ∈ ℕ). We state the theorem for increasing sequences, but its counterpart for decreasing sequences can be easily deduced from it. Theorem A.1.2 (Monotone convergence theorem). Let (X, 𝒜, μ) be a measure space. If (fn )∞ n=1 is an increasing sequence of nonnegative measurable functions, then the integral of their pointwise limit is equal to the limit of their integrals, that is, ∫ lim fn dμ = lim ∫ fn dμ. X

n→∞

n→∞

X

Note that this theorem holds for almost everywhere increasing sequences of almost everywhere nonnegative measurable functions with an almost everywhere pointwise limit. For general sequences of nonnegative functions, we have the following immediate consequence. Lemma A.1.3 (Fatou’s lemma). Let (X, 𝒜, μ) be a measure space. For any sequence (fn )∞ n=1 of nonnegative measurable functions, ∫ lim inf fn dμ ≤ lim inf ∫ fn dμ. X

n→∞

n→∞

X

The following lemma is another application of the monotone convergence theorem. It offers another way of integrating a nonnegative function. https://doi.org/10.1515/9783110769876-033

1316 � Appendix A – A selection of classical results Lemma A.1.4. Let (X, 𝒜, μ) be a measure space. Let f be a nonnegative measurable function and A ∈ 𝒜. Then ∞

∫ f dμ = ∫ μ({x ∈ A : f (x) > r}) dr. A

0

Pointwise convergence of a sequence of integrable functions does not guarantee convergence in L1 . However, under one relatively weak additional assumption, this becomes true. The second fundamental theorem of convergence applies to sequences of functions which have an almost everywhere pointwise limit and are dominated (i. e., uniformly bounded) almost everywhere by an integrable function. Theorem A.1.5 (Lebesgue’s dominated convergence theorem). If a sequence of measurable functions (fn )∞ n=1 on a measure space (X, 𝒜, μ) converges pointwise μ-a. e. to a function f and if there exists g ∈ L1 (μ) such that |fn (x)| ≤ g(x) for all n ∈ ℕ and μ-a. e. x ∈ X, then f ∈ L1 (μ) and lim ‖fn − f ‖1 = 0

n→∞

and

lim ∫ fn dμ = ∫ f dμ.

n→∞

X

X

Note that lim ‖fn − f ‖1 = 0

n→∞

󳨐⇒

lim ‖fn ‖1 = ‖f ‖1

n→∞

󵄨 󵄨 since 󵄨󵄨󵄨‖fn ‖1 − ‖f ‖1 󵄨󵄨󵄨 ≤ ‖fn − f ‖1 and lim ‖fn − f ‖1 = 0

n→∞

󳨐⇒

lim ∫ fn dμ = ∫ f dμ.

n→∞

X

X

The opposite implications do not hold in general. Nevertheless, the following lemma states that any sequence of integrable functions (fn )∞ n=1 that converges pointwise almost everywhere to an integrable function f will also converge to that function in L1 if and only if their L1 norms converge to the L1 norm of f . Lemma A.1.6 (Scheffé’s lemma). Let (X, 𝒜, μ) be a measure space. If a sequence (fn )∞ n=1 of functions in L1 (μ) converges pointwise μ-a. e. to a function f ∈ L1 (μ), then lim ‖fn − f ‖1 = 0

n→∞

⇐⇒

lim ‖fn ‖1 = ‖f ‖1 .

n→∞

In particular, if fn ≥ 0 μ-a. e. for all n ∈ ℕ, then f ≥ 0 μ-a. e. and lim ‖fn − f ‖1 = 0

n→∞

⇐⇒

lim ∫ fn dμ = ∫ f dμ.

n→∞

X

X

A.1 Measure theory

� 1317

If g is a L1 function on a general measure space (X, 𝒜, μ), then the sequence of nonnegative measurable functions (gM )∞ M=1 , where gM = |g| ⋅ 1{|g|≥M} , decreases to 0 pointwise and is dominated by |g|. Therefore, the monotone convergence theorem (or, alternatively, Lebesgue’s dominated convergence theorem) affirms that lim

M→∞

|g| dμ = 0.

∫ {|g|≥M}

This suggests introducing the following concept. Definition A.1.7. Let (X, 𝒜, μ) be a measure space. A sequence of measurable functions (fn )∞ n=1 is uniformly integrable if lim sup

M→∞ n∈ℕ

∫

|fn | dμ = 0.

{|fn |≥M}

On finite measure spaces, there exists a generalization of Lebesgue’s dominated convergence theorem (Theorem A.1.5). Theorem A.1.8. Let (X, 𝒜, μ) be a finite measure space and (fn )∞ n=1 a sequence of measurable functions that converges pointwise μ-a. e. to a function f . 1 1 (a) If (fn )∞ n=1 is uniformly integrable, then fn ∈ L (μ) for all n ∈ ℕ and f ∈ L (μ). Moreover, lim ‖fn − f ‖1 = 0

n→∞

and

lim ∫ fn dμ = ∫ f dμ.

n→∞

X

X

(b) If f , fn ∈ L1 (μ) and fn ≥ 0 μ-a. e. for all n ∈ ℕ, then lim ∫ fn dμ = ∫ f dμ

n→∞

X

X

󳨐⇒

(fn )∞ n=1 uniformly integrable.

1 Corollary A.1.9. Let (X, 𝒜, μ) be a finite measure space and (fn )∞ n=1 a sequence in L (μ) 1 that converges pointwise μ-a. e. to a function f ∈ L (μ). The following statements are equivalent: (a) The sequence (fn )∞ n=1 is uniformly integrable. (b) lim ‖fn − f ‖1 = 0. n→∞

(c) lim ‖fn ‖1 = ‖f ‖1 . n→∞

It is obvious that uniform convergence of a sequence of functions implies pointwise convergence of the sequence. The converse is generally not true but the following partial converse holds. Theorem A.1.10 (Dini’s theorem). If X is a compact topological space and (fn )∞ n=1 is an increasing sequence of real-valued continuous functions on X converging pointwise to a con-

1318 � Appendix A – A selection of classical results tinuous function f , then the convergence is uniform. The same conclusion holds if (fn )∞ n=1 is decreasing instead of increasing. It is also natural to ask whether, in some way, a pointwise convergent sequence converges “almost” uniformly. Definition A.1.11. Let (X, 𝒜, μ) be a measure space. A sequence (fn )∞ n=1 of measurable functions on X is said to converge μ-almost uniformly to a function f if for every ε > 0 there exists Y ∈ 𝒜 such that μ(Y ) < ε and (fn )∞ n=1 converges uniformly to f on X \ Y . It is clear that almost uniform convergence implies almost everywhere pointwise convergence. The converse is not true in general but these two types of convergence are one and the same on any finite measure space. Theorem A.1.12 (Egorov’s theorem). Let (X, 𝒜, μ) be a finite measure space. A sequence (fn )∞ n=1 of measurable functions on X converges pointwise μ-almost everywhere to a limit function f if and only if that sequence converges μ-almost uniformly to f . The reader ought to convince themself that this result does not generally hold on infinite measure spaces. Convergence in measure is another interesting type of convergence. Definition A.1.13. Let (X, 𝒜, μ) be a measure space. A sequence (fn )∞ n=1 of measurable functions converges in measure to a measurable function f provided that for each ε > 0, lim μ({x ∈ X : |fn (x) − f (x)| > ε}) = 0.

n→∞

Lemma A.1.14. Let (X, 𝒜, μ) be a measure space. If a sequence (fn )∞ n=1 of measurable functions converges in L1 (μ) to a measurable function f , then (fn )∞ converges in measure to f . n=1 When the measure is finite, there is a close relationship between pointwise convergence and convergence in measure. Theorem A.1.15. Let (X, 𝒜, μ) be a finite measure space and (fn )∞ n=1 a sequence of measurable functions. ∞ (a) If (fn )∞ n=1 converges pointwise μ-a. e. to a function f , then (fn )n=1 converges in measure to f . ∞ (b) If (fn )∞ n=1 converges in measure to a function f , then there exists a subsequence (fnk )k=1 which converges pointwise μ-a. e. to f . ∞ (c) (fn )∞ n=1 converges in measure to a function f if and only if each subsequence (fnk )k=1 ∞ admits a further subsequence (fnk )l=1 that converges pointwise μ-a. e. to f . l

The previous two results reveal that, on a finite measure space, a sequence of integrable functions that converges in L1 to an integrable function admits a subsequence which converges pointwise almost everywhere to that function. In general, the sequence itself might not converge pointwise almost everywhere.

A.1 Measure theory

�

1319

In some sense, the following result is a form of convergence theorem. It asserts that Borel measurable functions can be approximated by continuous functions on “arbitrarily large” portions of their domain. Theorem A.1.16 (Luzin’s theorem). Let (X, ℬ(X), μ) be a finite Borel measure space and let f : X → ℝ = [−∞, ∞] be a Borel function. Given any ε > 0, for every B ∈ ℬ(X) there is a closed set E with μ(B \ E) < ε such that f |E is continuous. If B is locally compact, the set E can be chosen to be compact and then there is a continuous function fε : X → ℝ with compact support that coincides with f on E and such that supx∈X |fε (x)| ≤ supx∈X |f (x)|. A.1.2 Mutual singularity, absolute continuity and equivalence of measures We now leave aside convergence of sequences of functions and recall the definitions of mutually singular, absolutely continuous and equivalent measures. Definition A.1.17. Let (X, 𝒜) be a measurable space, and μ, ν be two measures on (X, 𝒜). (a) μ and ν are said to be mutually singular, denoted by μ⊥ν, if there exist disjoint sets Xμ , Xν ∈ 𝒜 such that μ(X \ Xμ ) = 0 = ν(X \ Xν ). (b) μ is absolutely continuous with respect to ν, denoted μ ≺≺ ν, if ν(A) = 0 󳨐⇒ μ(A) = 0. (c) μ and ν are equivalent if μ ≺≺ ν and ν ≺≺ μ. The Radon–Nikodym theorem provides a characterization of absolute continuity. Though it is valid for σ-finite measures, the following version for finite measures is sufficient for our purposes. Theorem A.1.18 (Radon–Nikodym theorem). Let (X, 𝒜) be a measurable space and let μ and ν be two finite measures on (X, 𝒜). The following statements are equivalent: (a) μ ≺≺ ν. (b) For every ε > 0, there exists δ > 0 such that ν(A) < δ 󳨐⇒ μ(A) < ε. (c) There exists a ν-a. e. unique function f ∈ L1 (ν) such that f ≥ 0 and μ(A) = ∫ f dν,

∀A ∈ 𝒜.

A

Remark A.1.19. The function f is often denoted by derivative of μ with respect to ν.

dμ dν

and called the Radon–Nikodym

A.1.3 The space C(X), its dual C(X)∗ and the subspace M(X) Another important result is Riesz representation theorem. Before stating it, we first establish some notation. Let X be a compact metrizable space. Let C(X) be the set of all continuous real-valued functions on X. This set becomes a normed vector space when endowed with the supremum norm

1320 � Appendix A – A selection of classical results ‖f ‖∞ := sup{|f (x)| : x ∈ X}. This norm defines a metric on C(X) in the usual way: d∞ (f , g) := ‖f − g‖∞ . The topology induced by the metric d∞ on C(X) is called the topology of uniform convergence on X. Indeed, limn→∞ d∞ (fn , f ) = 0 if and only if the sequence (fn )∞ n=1 converges to f uniformly on X. It is not hard to see that C(X) is a separable Banach space (i. e., a separable and complete normed vector space). Let C(X)∗ denote the dual space of C(X), i. e., 󵄨 C(X)∗ := {F : C(X) → ℝ 󵄨󵄨󵄨 F is linear and continuous}. Recall that a real-valued function F defined on C(X) is called a functional on C(X). It is well known that a linear functional F is continuous if and only if it is bounded, i. e., if and only if its operator norm ‖F‖ (sometimes denoted ‖F‖∞ ) is finite, where ‖F‖ := sup{|F(f )| : ‖f ‖∞ ≤ 1}. So C(X)∗ can also be described as the normed vector space of all bounded linear functionals on C(X). The operator norm defines a metric on C(X)∗ in the usual manner: d(F, G) := ‖F − G‖. The topology induced by the metric d on C(X)∗ is called the operator norm topology, or strong topology, on C(X)∗ . It is not difficult to see that C(X)∗ is a separable Banach space. Furthermore, a linear functional F is said to be normalized if F(1) = 1 and is called positive if F(f ) ≥ 0 whenever f ≥ 0. Finally, we denote the set of all Borel probability measures on X by M(X). This set is clearly convex and can be characterized as follows. Theorem A.1.20 (Riesz representation theorem). Let X be a compact metrizable space, and let F be a normalized and positive linear functional on C(X). Then there exists a unique μ ∈ M(X) such that F(f ) = ∫ f dμ,

∀f ∈ C(X).

(A.1)

X

Conversely, any μ ∈ M(X) defines a normalized positive linear functional on C(X) via formula (A.1). This linear functional is bounded. It immediately follows from Riesz representation theorem that every Borel probability measure on a compact metrizable space is uniquely determined by the way it integrates continuous functions on that space.

A.1 Measure theory

� 1321

Corollary A.1.21. If μ and ν are two Borel probability measures on a compact metrizable space X, then μ=ν

⇐⇒

∫ f dμ = ∫ f dν, X

∀f ∈ C(X).

X

Let us now discuss the weak ∗ topology on the set M(X). Recall that if Z is a set and (Zα )α∈A is a family of topological spaces, then the weak topology induced on Z by a collection of maps {ψα : Z → Zα | α ∈ A} is the smallest topology on Z that makes each ψα continuous. Evidently, the sets ψ−1 α (Uα ), for Uα open in Zα , constitute a subbase for the ∗ weak topology. The weak topology on M(X) is the weak topology induced by C(X) on its dual space C(X)∗ , where measures in M(X) and normalized positive linear functionals in C(X)∗ are identified via Riesz representation theorem. Note that M(X) is metrizable, although C(X)∗ with the weak∗ topology usually is not. Indeed, both C(X) and its subspace C(X, [0, 1]) of continuous functions on X taking values in [0, 1], are separable since X is a compact metrizable space. Then for any dense subset {fn }∞ n=1 of C(X, [0, 1]), a metric on M(X) is ∞

1 n 2 n=1

d(μ, ν) = ∑

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 f dμ − f dν ∫ ∫ 󵄨󵄨 . n 󵄨󵄨󵄨 n 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨X X ∗

In this book, we will sometimes denote by μn → μ the convergence of a sequence of ∗ measures (μn )∞ n=1 to a measure μ in the weak topology of M(X). Remark A.1.22. This notion is often presented as “weak convergence” of measures. This can be slightly confusing at first sight, but it helps to bear in mind that, as we have seen above, the weak∗ topology is just one instance of a weak topology. The Portmanteau theorem gives several characterizations of weak∗ convergence of Borel probability measures. Theorem A.1.23 (Portmanteau theorem). Let (μn )∞ n=1 and μ be Borel probability measures on a compact metrizable space X. The following statements are equivalent: ∗ (a) μn → μ. (b) For all continuous functions f : X → ℝ, lim ∫ f dμn = ∫ f dμ.

n→∞

X

X

(c) For all closed sets F ⊆ X, lim sup μn (F) ≤ μ(F). n→∞

1322 � Appendix A – A selection of classical results (d) For all open sets G ⊆ X, lim inf μn (G) ≤ μ(G). n→∞

(e) For all Borel sets A ⊆ X such that μ(𝜕A) = 0, lim μ (A) n→∞ n

= μ(A).

For us, the most important result concerning weak∗ convergence of measures is that the set M(X) of all Borel probability measures on a compact metrizable space X is a compact and convex set in the weak∗ topology. Theorem A.1.24. Let X be a compact metrizable space. The set M(X) is compact and convex in the weak∗ topology of C(X)∗ . Finally, we introduce the concept of conditional measure and relate it to the concept of expected value. Definition A.1.25. Let (X, 𝒜, μ) be a probability space and let B ∈ 𝒜 be such that μ(B) > 0. The set function μB : 𝒜 → [0, 1] defined by setting μB (A) :=

μ(A ∩ B) , μ(B)

∀A ∈ 𝒜

is a probability measure on (X, 𝒜) called the conditional measure of μ on B. Note that for every φ ∈ L1 (X, 𝒜, μ), ∫ φ dμB = ∫ φ dμB + ∫ φ dμB = X

B

X\B

1 ∫ φ dμ. μ(B) B

A.2 Functional analysis Arzelà–Ascoli’s theorem (cf. Theorem IV.6.7 in Dunford and Schwartz [45]) is a fundamental theorem in functional analysis and topology. Before stating it, let us first recall the following two definitions. Definition A.2.1. Let X be a set and ℝX be the set of real-valued functions on X. A subset F of ℝX is said to be pointwise bounded if sup{|f (x)| : f ∈ F} < ∞,

∀x ∈ X.

Definition A.2.2. Let X be a topological space, and let C(X) be the space of real-valued continuous functions on X. A subset F of C(X) is said to be equicontinuous if for every x ∈ X and every ε > 0, there is a neighborhood U(x, ε) of x such that

A.3 Complex analysis in one variable

(y, f ) ∈ U(x, ε) × F

󳨐⇒

� 1323

|f (y) − f (x)| < ε.

The Arzelà–Ascoli theorem gives a characterization of the relative compactness of sets of continuous functions. Theorem A.2.3 (Arzelà–Ascoli theorem). Let X be a compact Hausdorff space. A subset F of C(X) is relatively compact (i. e., has compact closure in the topology of uniform convergence induced by the supremum norm ‖ ⋅ ‖∞ ) if and only if F is equicontinuous and pointwise bounded. There is another Riesz theorem that we will use in this book. This one is about the extension of a positive linear functional. Theorem A.2.4. Let E be a real vector space, F be a vector subspace of E, and K be a convex cone in E. A linear functional L : F → ℝ is called K-positive if L|F∩K ≥ 0. A linear functional L : E → ℝ is said to be a K-positive extension of L if L|F = L and L|K ≥ 0. If E ⊆ F + K, i. e., if for every y ∈ E there exists x ∈ F such that y − x ∈ K, then every K-positive linear functional on F can be extended to a K-positive linear functional on E. Note that, in general, a K-positive linear functional on F cannot be extended to a K-positive linear functional on E. Already in two dimensions there is a counterexample. Let E = ℝ2 , F = ℝ×{0}, and K = {(x, y) : y > 0}∪{(x, 0) : x > 0}. The K-positive functional L(x, 0) = x cannot be extended to a K-positive functional on E.

A.3 Complex analysis in one variable Theorem A.3.1 (Cauchy’s theorem). Let D ⊆ ℂ be a connected open set with piecewise smooth boundary 𝜕D. If f is an analytic function on D that extends smoothly to 𝜕D, then ∫ f (z) dz = 0. 𝜕D

Theorem A.3.2 (Cauchy’s integral formula). Let D ⊆ ℂ be a connected open set with piecewise smooth boundary 𝜕D. If f is an analytic function on D that extends smoothly to 𝜕D, then f has derivatives of all orders on D which are given by f (n) (z) =

f (w) n! dw, ∫ 2πi (w − z)n+1

∀z ∈ D,

∀n ≥ 0.

𝜕D

Theorem A.3.3 (Morera’s theorem). Let D ⊆ ℂ be a connected open set. If a continuous function f : D → ℂ is such that ∫𝜕R f (z) dz = 0 for every closed rectangle R ⊆ D with sides parallel to the coordinate axes, then f is analytic on D.

1324 � Appendix A – A selection of classical results Theorem A.3.4 (Monodromy theorem). Let D ⊆ ℂ be a connected open set. Let f : D → ℂ be holomorphic at a point z0 ∈ D, and let γ0 , γ1 : [0, 1] → D be two homotopic paths from z0 to a point w. If f can be continued holomorphically along any path in D, then the holomorphic continuations of f along γ0 and γ1 coincide at w. Lemma A.3.5 (Schwarz lemma). Let 𝔻 = {z ∈ ℂ : |z| < 1}. If f : 𝔻 → ℂ is an holomorphic function such that f (0) = 0 and f (𝔻) ⊆ 𝔻, then |f ′ (0)| ≤ 1

and |f (z)| ≤ |z|,

∀z ∈ 𝔻.

If |f ′ (0)| = 1 or |f (z)| = |z| for some z ≠ 0, then f (z) = az for some a ∈ ℂ with |a| = 1. Theorem A.3.6 (Schwarz reflection principle). Let D be a connected open subset of ℂ that is symmetric with respect to the real axis, and let D+ := D ∩ {Im(z) > 0}. Let f be an holomorphic function on D+ such that Im(f (z)) → 0 as z ∈ D+ tends to D ∩ ℝ. Then f has an analytic extension to D, and that extension satisfies f (z) = f (z),

∀z ∈ D.

The next two results are exhibits of the general fact that the number of zeros of an analytic function does not change under small analytic perturbations. Theorem A.3.7 (Rouché’s theorem). Let D ⊆ ℂ be a connected open set with piecewise smooth boundary 𝜕D. If f and g are analytic functions on D ∪ 𝜕D such that |g(z)| < |f (z)| for all z ∈ 𝜕D, then f and f + g have the same number of zeros in D, counting multiplicities. Theorem A.3.8 (Hurwitz’s theorem). Let D ⊆ ℂ be a connected open set. Suppose that (fn )∞ n=1 is a sequence of holomorphic functions on D that converges normally on D (i. e., converges uniformly on compact subsets of D) to a function f , and suppose that f has a zero of order N at z0 ∈ D. Then there exists δ > 0 such that for all sufficiently large n, the functions fn each have exactly N zeros in the disc B(z0 , δ), counting multiplicities, and these zeros converge to z0 as n → ∞. (Note: The normal convergence imposes that f be holomorphic.) This is far from the only theorem attributed to Hurwitz. Theorem A.3.9 (Hurwitz’s theorem II). Let D ⊆ ℂ be a connected open set. Suppose that (fn )∞ n=1 is a sequence of univalent (i. e., one-to-one) holomorphic functions on D that converges normally on D to a function f . Then f is either constant or univalent on D. A partial converse to this theorem is the following. Theorem A.3.10 (Converse to Hurwitz’s theorem II). Let D ⊆ ℂ be a connected open set. Suppose that a sequence (fn )∞ n=1 of holomorphic functions on D converges normally on D to a univalent function f . Then for any compact set K ⊆ D, all but finitely many of the functions fn are univalent on K.

A.3 Complex analysis in one variable

� 1325

̂∞ Theorem A.3.11. Let G ⊆ ℂ be an open connected set. If (fn : G → ℂ) n=1 is a sequence of ̂ then for meromorphic functions converging uniformly to a univalent function f : G → ℂ, every compact set K ⊆ G there exists NK ∈ ℕ such that fn (G) ⊇ f (K),

∀ n ≥ NK .

Theorem A.3.12 (Implicit function theorem). Let U, V ⊆ ℂ be open sets and F : U ×V → ℂ be a continuous function that depends analytically on z ∈ U for each fixed w ∈ V . Suppose that F(z0 , w0 ) = 0 and 𝜕F (z , w0 ) ≠ 0. Let δ > 0 be such that F(z, w0 ) ≠ 0 for all z ∈ 𝜕z 0 B(z0 , δ) \ {z0 }. Then there exists ε > 0 such that for every w ∈ B(w0 , ε), there is a unique z =: g(w) ∈ B(z0 , δ) such that F(z, w) = 0. In fact, 1 g(w) = 2πi

∫ |ζ −z0 |=δ

ζ 𝜕F (ζ , w) 𝜕z F(ζ , w)

dζ ,

∀w ∈ B(w0 , ε),

and g : B(w0 , ε) → B(z0 , δ) is thus continuous. If, additionally, F(z, w) is analytic in w for each fixed z, then g is analytic and 𝜕F

g ′ (w) = − 𝜕w 𝜕F 𝜕z

(g(w), w) (g(w), w)

.

The following theorem is a corollary to the previous one. Theorem A.3.13 (Inverse function theorem). If an holomorphic function f : U → ℂ is such that f ′ (z0 ) ≠ 0 at some point z0 ∈ U, then there exist δ, ε > 0 such that for every w ∈ B(f (z0 ), ε), there is a unique z ∈ B(z0 , δ) for which f (z) = w and, writing z = f −1 (w), the inverse function f −1 : B(f (z0 ), ε) → B(z0 , δ) satisfies f −1 (w) =

1 2πi

∫ |ζ −z0 |=δ

ζ f ′ (ζ ) dζ , f (ζ ) − w

∀w ∈ B(f (z0 ), ε),

and f −1 is thus holomorphic. Theorem A.3.14 (Riemann mapping theorem). If D is a simply connected domain in the complex plane but is not the entire complex plane, then there is a conformal map of D onto the open unit disk 𝔻 = B(0, 1). We end this appendix with an extension of the Riemann mapping theorem. Theorem A.3.15 (Carathéodory’s theorem). If f maps the open unit disk 𝔻 conformally onto a bounded domain D ⊆ ℂ, then f has a continuous one-to-one extension to the closed unit disk 𝔻 if and only if 𝜕D is a Jordan curve.

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Index A(T , c) 1048 almost uniform convergence 1318 Area Theorem 945 Arzelà–Ascoli theorem 1323 attracting sectors 995 basin of attraction 963 – immediate 963 bordered surface 928 Bowen’s Formula 1233 Carathéodory’s theorem 1325 Cauchy’s integral formula 1323 Cauchy’s theorem 1323 Classification Theorem 1025 closed bordered surface 928 completeness of F-spectrum 1237 conditional measure 1322 conformal measure 1200 Cremer periodic point 1030 Cremer point 974 critical – point 1202 – set 1202 critical set 1028 critically finite 1097 Crit(T ) 1202 cross ratio 1013 degree 927 directions – attracting 977 – repelling 977 discrete 926 dual space 1320 dynamical dimension 1219 edge 928 Egorov’s theorem 1318 Euler characteristic 928 exceptional set 958 expanding dimension 1219 expanding/hyperbolic rational function 1087 expansive rational function 1093 face 928 Fatou coordinates 990, 997 Fatou set 955 https://doi.org/10.1515/9783110769876-035

Fatou’s Flower Theorem 1004 Fatou’s lemma 1315 Herman ring 1017, 1026, 1029, 1069 Hurwitz’s theorem 1324 Hurwitz’s theorem (converse) 1324 Hurwitz’s theorem II 1324 hyperbolic/expanding 1068 hyperbolic/expanding rational function 1068, 1087 Implicit function theorem 1325 Inverse function theorem 1325 Julia set 955 Koebe’s 41 -Theorem 946 Koebe’s distortion theorem, spherical version 950 Koebe’s Distortion Theorem (analytic version) 948 Koebe’s Distortion Theorem (Analytic version) 949 Koebe’s Distortion Theorem I, Euclidean Version 949 Lebesgue’s dominated convergence theorem 1316 local dimension 1235 Lyapunov stable 1042 Mañé’s partition 1083 measure – absolutely continuous 1319 – equivalent 1319 – mutually singular 1319 Monodromy theorem 1323 Monotone convergence theorem 1315 Morera’s theorem 1323 multifractal decomposition 1235 multifractal spectrum 1235 μ − (n, ε, δ)-spanning set 1185 non-singular 1200 normal 951 open bordered surface 928 operator norm topology 1320 parabolic 976 partition – Mañé’s 1083

1334 � Index

periodic point – attracting 960 – indifferent 960 – neutral 960 – repelling 960 Pesin’s Formula 1227, 1229 pointwise dimension – regular part 1235 – singular part 1235 Portmanteau theorem 1321 postcritical set 1028 pre-periodic point 960 proper 925, 931 proper open bordered surface 928 PS limit measure 1208 radial (or conical) points 1179 Radon–Nikodym derivative 1319 Radon–Nikodym theorem 1319 regular subdomain 937 repelling sectors 996 Riemann mapping theorem 1325 Riesz representation theorem 1320 Rotation Theorem 949 Rouché’s theorem 1324 safe 1213 Safe(T ) 1213 Scheffé’s lemma 1316

Schröder equation 973 Schwarz lemma 1324 Schwarz reflection principle 1324 Siegel disk 1017, 1026, 1029, 1069 Siegel point 972 simple parabolic fixed point 976 simplex 928 Sing(T ) 1202 singular – point 1201 – set 1202 special set 1200 strict expanding dimension 1219 subcomplex 928 subexpanding 1097 Sullivan’s Non-Wandering Theorem 1027 t-conformal measure 1212, 1221 temperature function 1236 topology of uniform convergence 1320 transition parameter 1206 triangulation 928 vertex 928 weak∗ topology 1321 X∗ (T ) 1202

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