Dynamics of transcendental functions [1st ed.] 9781351454032, 135145403X, 90-5699-161-2

Citation preview

DYNAMICS OF TRANSCENDENTAL FUNCTIONS

Asian Mathematics Series A series edited by Chung-Chun Yang, Department ofMathematics, The Hong Kong

University of Science and Technology, Hong Kong

Volume 1 Dynamics of Transcendental Functions Xin-Hou Hua and Chung-Chun Yang

Volumes in Preparation Approximate Methods and Numerical Analysis for Elliptic Complex Equations

Guo-Chun Wen Introduction to Statistical Methods in Modem Genetics

Mark C.K. Yang

This book is part of a series. The publisher will accept continuation orders which may be cancelled at any time and which provide for automatic billing and shipping of each title in the series upon publication. Please write for details.

DYNAMICS OF TRANSCENDENTAL FUNCTIONS

Xin-Hou Hua Institute of Mathematics, Nanjing University, China

and

Chung-Chun Yang Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong

(S;C? CRC Press

Taylor &Francis Group Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Croup, an informa business

Copyright © 1998 OPA (Overseas Publishers Association) N.Y. Published by license under the Gordon and Breach Science Publishers imprint.

All rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system, without permission in writing from the publisher. Printed at Ajanta Offset, New Delhi, India. Amsteldijk 166 1st Floor 1079 LH Amsterdam The Netherlands British Library Cataloguing in Publication Data Hua, Xin-Hou Dynamics of transcendental functions. - (Asian mathematics series; 1) I. Transcendental functions I. Title II. Yang, Chung-Chun 515.9 ISBN: 90-5699-161-2 ISSN: 1028-1428

Contents Introduction to the Series

ix

Preface

xi

1

Preliminaries 1.1 Some notations 1.2 Normal families . . . 1.3 Conformal mappings 1.4 The Poincare metric 1.5 Extremal lengths and prime ends 1.6 Ramified coverings . . . . . . . 1.7 Quasiconformal mappings .... 1.8 Nevanlinna's value distribution theory 1.9 Covering surfaces . . . . . . . . 1.10 Growth of composite functions 1.11 Wiman-Valiron theory ..... ~

2

3

1

1 2 6 9 11

15 15 19 23 25 27

The 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

Fixed Point Theory Classification of meromorphic functions Classification of fixed points . . . . . Fixed points of iterated functions . . Fixed points of composite functions . Conjugacy and the central problem . Attracting and repelling fixed points Rationally indifferent fixed points . Irrationally indifferent fixed points

33

The 3.1 3.2 3.3

Fatou and Julia Sets Definitions of the Fatou and Julia sets Completely invariant sets .... Some properties of the Julia set .

47 47 48 49

v

33 34 35 37 38 39 41 44

vi

CONTENTS

3.4 Baker's theorem . . . . . . 3.5 Expansivity of the Julia set

4 The 4.1 4.2 4.3 4.4

4.5 4.6 4. 7 4.8 4.9 4.10 4.11

4.12 4.13 4.14 4.15 4.16 4.17

Components of the Fatou Set Types of the components . . . . Multiply connected components .. Simply connected components . . . Cl~ification of periodic components . 4.4.1 The classification theorem . . 4.4.2 Constant limit functions . . . . 4.4.3 Nonconstant limit functions . . 4.4.4 Herman rings and Baker domains . Singular points . . . . . . . . . . FUrther results on limit functions Growth of functions . . . . . . Images of the components . . . . Prime ends and Baker domains . Functions without Baker domains . Functions with wandering domains 4.11.1 Functions for the class E .. 4.11.2 Functions for the class M . Functions without wandering domains Completely invariant components .. . The number of the components . . . . Connectivity of pre-periodic components . Bounded Fatou components . The Fatou exceptional value.

5 Geometry of Julia Sets 5.1 Cantor sets . . . . . . 5.1.1 Symbolic dynamics .. 5.1.2 Schwarzian derivative 5.1.3 Cantor sets 5.2 Cantor bouquets 5.3 Jordan arcs . . . . 5.4 Explosion . . . . . 5.5 Points that tend to infinity 5.6 Expanding functions . . . . 5. 7 Relations between E and 'P 5.8 Connectivity of Julia sets 5.9 Distribution of the Julia set

54 56

. . . . .

57 57 58 58 60 60 61 63 65 66 73 76 80 81 83 85 85 91 98 104 108 113 115 129

. . . . . . . . . . . .

131 131 131 133 134 136 139 141 143 146 147 149 150

vii

CONTENTS

6 Hausdorff Dimension of Julia Sets 6.1 General measure theory . . . 6.2 Caratheodory's construction . 6.3 Hausdorff measure . 6.4 Hausdorff dimension 6.4.1 Definition . . 6.4.2 Cantor sets . 6.4.3 Self-similar sets . 6.4.4 Capacity and dimension 6.4.5 Limiting sets and density 6.5 Dimension for the classes £ and P 6.5.1 For general functions . . . . 6.5.2 Combinations of exponential functions 6.5.3 The function in the class B . . . . . 6.6 Dimension of the Julia set for the class M .

. . . . . . . . . . . . . .

153 153 155 156 157 157 159 160 162 164 165 165 166 169 189

7 Miscellaneous 7.1 Measurable dynamics . 7.1.1 Recurrence 7.1.2 Ergodicity .. . 7.1.3 Stability .. . . 7.2 Permutable functions . 7.3 Convergence of Julia sets 7.4 Newton's method . 7.5 Random iterations

. . . . . . . .

193 193 193 195 200 215 219 221 222

Bibfiography

225

List of Special Symbols

239

Index

240

Introduction to the Series The Asian Mathematics Series provides a forum to promote and reflect timely mathematical research and development from the Asian region, and to provide suitable and pertinent reference or text books for researchers, academics and graduate students in Asian universities and research institutes, as well as in the West. With the growing strength of Asian economic, scientific and technological development, there is a need more than ever before for teaching and research materials written by leading Asian researchers, or those who have worked in or visited the Asian region, particularly tailored to meet the growing demands of students and researchers in that region. Many leading mathematicians in Asia were themselves trained in the West, and their experience with Western methods will make these books suitable not only for an Asian audience but also for the international mathematics community. The Asian Mathematics Series is founded with the aim to present significant contributions from mathematicians, written with an Asian audience in mind, to the mathematics community. The series will cover all mathematical fields and their applications, with volumes contributed to by international experts who have taught or performed research in Asia. The level of the material will be at graduate level or above. The book series will consist mainly of monographs and lecture notes but conference proceedings of meetings and workshops held in the Asian region will also be considered.

ix

Preface Complex dynamics as a subject is currently very much a focus of interest in terms of theoretical and applicable considerations. The subject originated in the 1920s when Fatou and Julia based their investigations on the Koebe-Poincare Uniformization Theorem, Montel's Normality Criterion and the work on functional equations by BOttcher, Koenigs, Leau, Poincare and SchrOder in the last century. Fatou and Julia independently discovered the dichotomy of the Riemann sphere into the sets now bearing their names, by considering the sequence of iterates of arbitrary nonlinear entire or meromorphic functions. The development of the theory of complex dynamics is divided into two principle aspects: the iteration of rational functions and the hereafter of transcendental functions. The dynamics of transcendental functions are quite different from the dynamics of rational functions, mainly because of essential singularities. Sullivan's so-called No Wandering Theorem, for instance, conjectured by Fatou in 1926 for rational functions does not hold for transcendental functions. Nowadays, several fine books on rational iteration, by A.F. Beardon, L. Carleson and T.W. Gamelin, J. Milnor, N. Steinmetz, are available. There are also some fragmental results concerning iterations of transcendental functions in Devaney's book on Chaotic Dynamical Systems. Until recently there has been no systematic book on the dynamics of transcendental functions. This book is intended to fill this gap and give a self-contained and quite comprehensive coverage of the most up-dated developments on the dynamics of transcendental functions. Many results in this book, including the authors and their co-workers, have not previously been published. The book may serve as a textbook for a course following a one-year introduction to analytic function theory. It will also be useful for mathematicians who want to become acquainted with this field . The early drafts of this manuscript were developed from lecture notes for a seminar course taught at the Hong Kong University of Science and Technology (HKUST). The authors thank the participants in the course for their useful comments and constructive feedback. The authors also thank I.N. Baker for his encouragement to write such a monograph. We also thank W. Bergweiler, A.E. Eremenko, G.M. Stallard, C. McMullen and R. Vaillancourt for sending us much material relating to the topics of the book. Additionally, we have benefited a great deal from discussions with M. Taniguchi, F . Maita.ni, P. Mattila and D.F. Shea. In addition, we want to express our gratitude to the referees for valuable suggestions. Finally we thank all those people, too numerous to list here, who have contributed in various ways to the writing of the book. The first author expresses a special word of thanks to the Croucher Foundation for awarding him a visiting fellowship to visit HKUST. That visit facilitated the xi

writing and completion of this joint manU8Crlpt, and the warm hospitality and excellent research environment provided by the HKUST made the work pleasant.

Xin-Hou Hua and Chung-Chun Yang

Chapter 1

Preliminaries In this chapter, we present topics and notations that are needed in this book. Part of the results are of interests themselves. Some are not included in the standard introductory gr~uate texts of complex analysis.

1.1

Some notations

The real line and the complex plane are denoted by lR and C respectively. The Riemann sphere and the punctured plane will be denoted respectively by:

= Cu{oo}, c· = c- {o}; C

the upper half-plane, unit disk and punctured disk by: lHl

= {z: Im(z) > 0},

tl. = {z:lzl-one correspondence between the points on the unit circle { and the prime ends P({). The impression /(P({}} is a subset of 8D which may be defined to be /(P({)) = C(h,{) ={wE

C:

thereexistz,. E A,z,.- {,h(z,.)- w}.

In the general case the cluster set of a prime end consists of all the limits of convergent fundamental sequences in the equivalence classes defined by the prime ends. It is easy to show that a cluster set is closed and connected. Distinct prime ends may have the same cluster sets.

15

1.6. RAMIFIED COVERINGS

1.6

Ramified coverings

The following notations and results are standard and can be found in many places such as Ahlfors-Sario [4]. Let V and W be two dimensional surfaces. A map f : V -+ W is called a d-sheeted ramified covering (1 ~ d ~ oo) if every a E W has a neighborhood D such that 1. /- 1(a) = {b1, ... , bd} and neighborhoods of the b;.

r 1(D) = l];'=1 B;, where B; are mutually disjoint

2. There are homeomorphisms t/1; : B; -+ ~ and 1/J; : D 1/J(a) = 0 such that 1/J; of o t/lj 1 (z) = z"•.

-+ ~

with q,,(b,) =

If k; > 1 for some i, then b; is called a branch point off, k; is called its branch index, and a is called the projection of the branch point.

Proposition 1.4 Let f: V-+ W be ad-sheeted ramified covering (I ~ d ~ oo). Then the following hold: {i) The inverse image of any point y E W consists of d points taking account of multiplicities. {ii} If W is connected and V = UV; is a decomposition of V into connected components, then f : V; -+ W is a ramified covering. In particular, flv, is onto. {iii) For any domain DC W, f: /- 1 (D)-+ D is a ramified covering. {iv} For any simply connected domain D C W, if B is a connected component of /- 1 (D) and B has no branch points, then f: B-+ D is a homeomorphism. Theorem 1.11 Let

f: V-+ W

be an analytic map of Riemann surface. Then

1. f is a finite-sheeted ramified covering if and only iff is proper {that is, the inverse image of any compact set is compact). 2. Suppose that V and W are domains on C and that f is continuous on V. Then I is a finite-sheeted ramified covering if and only if f(8V) caw.

1. 7

Quasiconformal mappings

Given a domain B, consider all quadrilaterals Q(z1 , ••• , z4 ) with Q C B. Let -+ B' be a sense-preserving homeomorphism. The number

f :B

16

CHAPTER 1. PRELIMINARIES

is called the maximal dilatation of f. It is

always~

1 by the fact:

M(Q(zt.z2,za,z4)) = 1/M(Q(z2,za,z4,zl)). Definition 1.4 A sense-prueroing homeomorphism with maximal dilatation K < +oo is said to be K -quasicon/0f"f114l (K -q.c.). Remark. This geometric definition was suggested by Pftuger in 1951, and systematically used by Ahlfors in 1953. By this definition, f is 1-quasiconformal if and only iff is conformal. From some characteristic on modulus in the section 1.5 it follows that a mapping f and its inverse are simultaneously K -quasiconformal, and if f : A - B is K 1quasiconformal and g: B- Cis K2-quasiconformal, then go f is K 1 K 2-q.c .. K -quasiconformal mappings possess the same compactness property as conformal mappings, i.e., the limit function of a sequence of K-quasiconformal mappings of a domain A, locally uniformly convergent in A, is either a constant or a K -quasiconformal mapping. One way to generalize the concept of conformal mappings is to consider diffeomorphisms. This local approach using derivatives is often much more convenient than the definition using modules. Suppose f has a continuous first derivative, that is, f E 0 1 • We will use the standard notations as follows: dz = dx + idy,

cJz = dx - idy,

/z

=

~(/.,- i/,),

h = ~(/., + i/,), d/ = f,dz + f-;4%.

The Jacobian J1 of I is given by

l/zl 2 -l/al 2 • Thus/ preserves orientation if and only if 1/al < 1/,1. The complex dilatation of J1 =

/ is defined by

/JJ(z) = P.t(z) = ,: ·

This dilatation p. is also called the Beltrami coefficient of f, and the equation

h = P./z

(1.15)

is called the Beltrami equation. Note that p. = 0 if and only iff is conformal, and for any conformal mapping ¢J we have JJ• R. Consider the CauchyGreen formula

1.

l(w) l(z) = -1. - dw- -1 271'1 1•1-r w - z 11'

hl

l,(w) - d ( d71, l•l R, the 0(1/z) term does not contribute to the first integral. We can replace l(w) by win the first integral, and this leads to

l(z)-z=-~ ff 11' }

f,(w)d(d'7=Tf,(z),

(1.16)

}I•I ~lz1- z2l on Da-3 , the integral over D;

is bounded by

The intgral over Do is bounded by

These estimations yield our conclusion.

1.8

•

Nevanlinna's value distribution theory

As a quantitative generalization of Picard's theorem, the theory of the distribution of values of meromorphic functions, developed by R. Nevanlinna and then L. Ahlfors was one of the most outstanding achievements of mathematics in this century. We here present a brief account. For a more comprehensive treatment, the reader is referred to the excellent monograph of Hayman[109] or Nevanlinna[163]. Let f(z) be meromorphic in lzl ~ R < oo. The starting point of the theory is the Poisson-Jensen formula which can be proved by the Poisson formula on harmonic functions. Poisson-Jensen formula Let f has zeros a 1 , ... , an and poles b1, ... , b,. in repe4ted according to multiplicity. If f(z) =f= 0, oo and z = pei6 , 0 ~ p < r < R, then

lzl < R,

log 1/(z)l

20

CHAPTER 1. PRELIMINARIES

If f(z) ha& a zero or a pole at z modification of f(z).

=

0, then we need only to make a suitable

Denote by n(r, f) the number of the poles of f(z) in lzl < r, counted according to multiplicities. Setting tog+ x = max(logx,O). We define

N(r,f) m(r,f)

T(r,f)

Lr n(t,

f)

~ n(O, f) dt + n(O, f) log r,

2~ L2fr tog+ 1/(re")ldD,

m(r, f)+ N(r, f),

w:here N(r, /), m(r, f) and T(r, f) are called counting function, proximity function and Nevanlinna characteristic respectively. The order >.(/) and the lower order p(/) are defined respectively as follows:

'(f) _ li

"'

-

msup

r-+oo

log T( r, f) , 1ogr

p(/) = liminf logT(r,f). r~ 1 (i :1: j). Now for any positive integer n we can find a number M > 0 such that

ll(z)- al < Mlz- a; I, z E {lz- a; I< 1}(j = 1, ... ,n). Consequently, log

+

and so, m(r, /

~

1

+

1

ll(z)- al ~~log Jz- a·l -log J

J•l

1 ( )

og z - a

)~

+

M,

1

L m(r, g(z ) - a; ) -log+ M. i=l n

On the other hand, we see easily that

!'f(r, I

1 o g(z)-

a)

~

f; N(r, g(z)-1 a;). n

Combining these two inequalities with Theorem 1.14 we get

T(r,l o g)~ nT(r,g)- 0(1). Since n is arbitrary, we have (1.22). Remark. The corresponding theorem for log M(r, ) is false unless g is of finite order.

1.11

Wiman-Valiron theory

In this section, we discuss Wiman-Valiron theory, which was initiated by Wiman [209, 210] and deepened by Valiron [204] and others such as [110, 111, 31, 220] and

28

CHAPTER 1. PRELIMINARIES

so on. The theory is important in the study of value distribution theory and its

applications such as the growth estimates of solutiens of differential equations in the complex domain. Suppose that n=O

is a transcendental entire function. We denote by J.&(r,g) and v(r,g}, respectively, the maximum term and the central index, i.e.,

J.&(r, g) :=sup lanlr" n

and

v(r,g) := max{n: lanlr"

= J.&(r,g)}.

When the function is clear, we simply write the central index and the maximum term as v(r) and J.&(r) respectively. Obviously, logJ.&(r) is a convex increasing function of log r, and v( r) is monotone increasing. Throughout all this section, we denote by A any suitable set with finite logarithmic measure. Wiman-Valiron theory describes the local behavior of g(z}, near a point where jg(z)l is large.

Theorem 1.27 Let g(z) be a transcendental entire function. Assume that c > 0, E > 0, 'f/ > 0 and 'Y > 1/2. Suppoae that lzol = r, jg(zo)l ~ 'IJM(r,g) and lrl < cv(r, g)--r. Then fork = 0, 1, ... , g 1/2. Suppose that lzol = r ¢A, lg(zo)l ~ TJM(r, g). Then there exits a function T(z) defined and analytic on lz- zol ~ crv(r, g)-'Y satisfying IT(z)v(r, g) - 21ril = o(1),

g(ze*>)

=

g(z).

PROOF. From the assumption we have 11- z/Zol ~ cfv(r), so that z = e"'Zo for some w satisfying lwl ~ 2c/v(r), provided r is large enough. It follows from (1.23) that g(z) = g(Zo)eO' for some u satisfying lu - wv(r)l = o(1). In particular, for larger, we have lui ~ 3c. Let

/l(u) = g(Zoe")- g(z),

h(u) = g{ZQ)e"v(r) - g(z).

From (1.23) we deduce that, if lu+21ri-uv(r)l = E with 0 1 and B > 1, the set S {r: T(Dr,g) ~ BT(r,g)}

:=

has a lower logarithmic density

logD logdensS S A(g)logB"

(1.27)

Lemma 1.3 (Bergweiler /3!/} Assume that 1(2:) and k(2:) are non-negative, nondecrealing and cont1e.1:, and that 1(2:) S k(2:) for 2: ~ 0. Then for arbitrary D > 1, we have 1'(2:) S Dk'(2:) on a set I satisfving (D -1)/D S fknl.l S logdensl. Now we can derive the following

Theorem 1.29 Let g(z) and P(z) be two transcendental entire function~. If there uists a set G having a positive lower logarithmic density such that for a positive number a, aT(r,g) S T(r,P), reG, (1.28) then for arbitrary £ > 0, there uists a subset H of G with a po6itive lower logarithmic density such that

v(r,g) S v(r,p)l+•,r PROOF.

For t

e H.

(1.29)

> 0, we can choose a positive number t' such that 1+£

=

(1 + £')2

1- £' •

From Borel's monotone lemma we have

logM(r,g) < T(r,g)l+•'/ 2 , r ¢A, and from (1.25), (1.26) and (1.28)

v(r,g)

S

(logM(r,g)) 1+-'/ 2 S T(r,g)

S

(v{r,P)logr) ..... where the composition is understood to be the multiplication in the semi-group ~ 0}. There should be no confusion with the ordinary power, which will be explicitly written as (/(z))" if necessary. is well defined for 1 when all z E C except for a countable set which consists of the poles of/, ... ,

{r : n E Z, n

r.1 < c 1/(z)l ~ clzl for z E ~(r). Thus for any starting point Zo E .1.(r), the orbit o+(Zo) converges to the origin and l.z:..l = lr(Zo)l S r.1 < 1, then we may choose a

and a neighborhood ~(r) of the origin so that

1/(z) -

>.zl S dlzl 2 z E .1.(r)

for some constant d. Thus 1/(z)l S (1>.1 + dr)lzl. We choose r so small that h = (1>.1 + dr )2 /1>.1 < 1. By induction we have lr(z)l

S (1>.1 + dr)"lzl,

z e .1.(r).

Define IPn(z) = r(z)/1>.1"· Then

h"dlzl 2

IIPn+t(Z) -~Pn(z)l S ~·

z E .1.(r).

Hence ~P,.(z) converges to IP(z) uniformly for z E .1.(r). Now by the definition of ~Pn, IPn o / = >.~Pn· Thus 1P o / = >.~P; if 1>.1 > 1, then the inverse map /- 1 is locally well defined and holomorphic having z = 0 as an attractive fixed point with multiplier 1/>.. Applying the above argument to /- 1 we can derive the same conclusion of conjugation. Now we prove the uniqueness. If there are two such maps IP and'¢, then '1/J o IP- 1 (>.w)

=

= =

'1/J o IP- 1 o 1P o 1 o ~P- 1 (w)

'1/J o I o

w-• o '1/J o ~P- (w) 1

>.'f/JoiP- (w). 1

Set '1/J o~P- 1 (w) = b1 w + ~w2 + · · ·· Substituting this into the above equation and comparing coefficients, we see that >.b,. = b,.>." for all n. Since >. is neither zero nor a root of unity, this implies that~= b:l = · · · = 0. Thus '¢(z) = b11P(z). This completes the proof. • Remark. The 1P in the above theorem is the so called Koenigs map.

Theorem 2.6 If Zo is an attractive fixed point, then there exists a disk D = < r} in Which the sequence r(z) (n = 1,2, ... ) CORVefYeS unifonnl7J to%().

{lz-Zol

PROOF. By hypothesis, we can choose a constant c such that 1/'(Zo)l a _positive constant r such that

/(z)- Zo

I Z-Zo

< c < 1 and

I= 1/(z) -/(Zo) I< c %-%()

for 0 < lz- Zol < r. Thus for lz- Zol < r, 1/(z)- Zol S clz- Zol· By induction, lr(z) - Zol S (z) of a neighborhood of 0 onto a neighborhood of 0 that conjugates f(z) tow". The conjugating function is unique up to multiplication by a (p- 1)- th root of unity. PROOF. The proof of the uniqueness is similar to the one in Theorem 2.6, we here prove only the existence. For small r, there is a constant c > 1 satisfying 1/(z)l ~ clzl" for all lzl ~ r. By induction,

lr(z)l ~ (clzl)"" , lzl ~ r. Now we choose r so small that cr < 1. By the conjugation we may assume a = 1. Let tl>n(z) = (r(z))"-" , n ~ 1, which are well defined in a neighborhood of 0 and satisfy

tPn of= ~+1• Thus if tl>n- 1/>, then

t1> of=

t/>;:1

1/>", and so we find a solution. Note that

( tl>1;nr)"-" (1 + O(lrln"_,.

= 1 + o(p-")

for lzl ~ r. Thus the product f1:'= 1tPn+l/tl>n converges uniformly on Hence the required t1> exists. The proof is complete.

2. 7

lzl

~

r.

Rationally indifferent fixed points

Now we consider the case that z = 0 is a rationally indifferent fixed point of/, that is, the multiplier is a root of unity. Choose a neighborhood N of the origin that is small enough so that f maps N conformally onto some neighborhood No of the origin.

CHAPTER 2. THE FIXED POINT THEORY

42

Figure 2.1: The Leau-Fatou flower with three petals A connected open set U, with compact closure U C N attracting petal for f at the origin if

J(U> c u u {0},

n N 0 , will be called an

n ,,. = {0}.

lc~O

Similarly, V C N n N 0 is a repelling petal for

f

if V is an attracting petal for /- 1 •

Theorem 2.8 (Leau-Fatou Flower Theorem)

/(z)

= z + az"+l +higher terms

J-et (a::/: 0, n ~ 1)

be holomorphic in some neighborhood N of the origin. Then there exist n di6joint attracting petal& Ui and n di6joint repelling petal& ~ so that the union of these 2n petal6, together with the origin itself, ftmr&6 a neighborhood N 0 of the origin. These petal& alternate with each other, so that each ui intersects only ~ and ~-1 (where Vo is defined to be V,.).

PROOF. (cf. Milnor [159, 7.8]). Let w

= -1/(naz").

We take the inverse z

=

y'-1/(naw). Then the sector between two repelling directions in the z-plane will correspond to the entire w-plane with a slip along the negative real axis. In particular, a neighborhood of the origin in such a vector will correspond to a neighborhood of infinity in such a slit w-plane. Note that, as z -+ 0,

Thus the corresponding self-transformation in thew-plane to z ~-+f(z) is

w ...... w' = w + 1 + o(1)

2. 7. RATIONALLY INDIFFERENT FIXED POINTS

as lwl -+ oo. In other words, number r > 0 such that

43

given any small number e

lw'- w- 11 0,

we can choose a

lwl > r.

It follows that the slope of the vector from w to w' satisfies Islope I < tan e, as long Now we construct an attracting petal for the point oo in thew-plane as follows. Let

as lwl > r.

E={w=u+iv:

lwl>r,u>c-J!L }, tan 2e

where the constant c > 0 is large enough. Then E is mapped into E and every backward orbit starting in E must eventually leave E. The assertion follows. Remark. Let 8; = _ar~a + ~ and 8j = 8; + ~. j = 0, 1, ... ,n -1. We say 8; and 8j are the repelling directions and attracting directions at the fixed point z = 0 respectively. Evidently these n equally distributed attracting directions are separated by the n equally distributed repelling directions. Corollary 2.3 There is no periodic orbit other than the fixed point 0, which is completely contained within the neighborhood No of the origin.

Now we prove the existence of a local conjugation near rationally indifferent fixed point z = 0. Theorem 2.9 Let f(z)

= ..Xz + az"+ 1 +higher terms

(a ::/= 0, n ~ 1)

be holomorphic in some neighborhood of the origin, where A is a primitive q-th root of unity. Then there exists a local holomorphic change of coordinate w = 1. Thus, (2. 7) and (2.8) present a contradiction. Therefore c = 1 and so rp o g = rp + 1. This implies that 1 (rp 0 ¢) 0 0 (rp 0 ..p)- (w) = w + 1,

r

i.e., rp o '1/1 is the desired conjugation.

2.8

Irrationally indifferent fixed points

Once more we consider holomorphic maps of the form /(z)

= AZ + 02z2 + a3z3 + ··· ,

defined in some neighborhood of the origin, where the multiplier A is of the form

A= e2ft",

0 e R\Z.

45

2.8. IRRATIONALLY INDIFFERENT FIXED POINTS

Next we shall study whether or not this function conjugates to the linear map

w ~--+ .Xw. In 1912, Kasner conjectured that such a linearization is always possible. Five years later, Pfeiffer [166] disproved this conjecture by giving a rather complicated description of certain holomorphic functions for which no linearization is possible. In 1919 Julia claimed to settle the question completely for non-linear rational functions by showing that such a linearization is never possible. But Julia's proof is incorrect and his conclusion is also wrong. In fact, in 1938, Cremer [69] proved that if I-XI 1 and liminfi.X"- 11 11" = 0, then there is an analytic function f(z) = .XZ+· ··such that no linearization is possible. Later in 1942, Siegel [186, 187] gave the first example of a unimodular .X for which linearization is possible. In order to state this result clearly, some facts in number theory are needed. A real number 8 is Diophantine if it is badly approximable by rational numbers, in the sense that there exist c > 0 and p. < oo so that

=

, e-!!.1~~ q

q~-'

for all integers p and q, q ::/: 0. This occurs if and only if .X= e2 ~~'111 satisfies

I-X9 -11 ~ c'q1 -", for some constant

c.

q ~ 1,

In fact, as q8 - p -+ 0,

I.X9

-

11

= le2wi(qll-p) -

11 "' 211'ql8- !!.1. q

For fixed p. > 2, if E is the set of 8 E [0, 1] such that 18- ~I then the measure of E satisfies

lEI

L 2 · q_,. · q = O(n

< q_,. infinitely often,

OQ

~

2

-") -+

0.

q=n

Thus almost all real numbers are Diophantine.

Theorem 2.10 (Siegel) If 8 is Diophantine, and iff has fixed point at 0 with multiplier e 2 ~~'", then there exists a local change of coordinate z = h(w), which conjugate& f to the iTTational rotation w 1-+ .Xw. · PROOF. We give a sketch of the proof. The details can be found in either one of the references [186, 187] and [57). Our aim is to find h(z) such that

h(.Xz)

= f(h(z)).

For convenience, we can normalize h so that h(O) = 0 and h'(O) = 1. If we define i(f} =/(f) - .XZ and h{f) = h(z}- z, then the above equation can be written as

h(.Xz) - .Xh(z)

= j(h(z)).

46

CHAPTER 2. THE FIXED POINT THEORY

Now expanding both sides in the power series by using 00

h(z)

= :Ea..z",

00

j(z)

= Lbnz", n=2

n=2

we obtain equations of the form a,.(~n

-

~) =

An(a2, • • ·, 0..-t. ~. • • •, bn)•

Using the condition I~" - ~~ ~ cn 1-" we can get the estimates of the ans and prove the power series converges. •

Definition 2.1 We call an inutionallJI indifferent fixed point i6 a Siegel point or Cremer point depending whether a local linearization i6 po&sible or not. Similarly we can define the Siegel cycle and the Cremer cycle. By Theorems 2.10 and 2.4 we obtain

Corollary 2.4 Let Zo be a Siegel point. Then there e:Nt& a di&k D : { lz - Zo I < r} and a &equence of po&itive integer& n,. (k = 1, 2, ... ) tending to oo &uch that in D, the &equence (k = 1, 2, ... ) converge& uniformly to z.

r·

Remark. For the quadratic polynomial P(z) = e2fri1Jz + z2 (9 e R\Z), precise conditions are known for the existence of a conjugation. Such exists if and only if E~ 1 logfn+tfqn < oo, where {p../qn} is the sequence of rational approximations to 9 coming from its continued fraction expansion. The sufficiency of this condition was proved by Brjuno [52] in 1965, and the necessity was established by Yoccoz [218] in 1988. In cl06ing this chapter, we present the following open problem.

Question 2.1 Can one find a 9 8UCh that the Schroder equation

ha8 no &olution h at all for any tramcendental entire function f The answer is yes when

f

is a polynomial (cf. [57, p.42]).

~

Chapter 3

The Fatou and Julia sets

In this chapter, we shall deduce some elementary properties of the Fatou and Julia sets of transcendental functions. Some of them are different from those of rational cases, due to the existence of Picard exceptional values. The reader should keep in mind that I always denote a given transcendental function, unless specified otherwU!e.

3.1 · Definitions of the Fatou and Julia sets Let

u = c, c, c· F(f)

provided that IE£, IE M, IE 'P respectively. We define

= {z E U : the sequence {r} is well defined and normal at z}

and J(f)

= U\F(/),

they are called the Fatou set and the Julia set of I respectively. When the function I is clear, we briefly write F and J instead ofF(/) and J(/) respectively. According to the definition, it is easily verified that F is open (possibly empty) and J is closed.

47

CHAPTER 3. THE FATOU AND JULIA SETS

48

3.2

Completely invariant sets

Definition 3.1 A set S is called forward invariant (or invariant) under J if z E S impliu that /(z) E S or /(z) is undefined; A setS is called backward invariant under J if z E S impliu that w E S for all w satisf71ing f(w) = z. A set S is called completel71 invariant if it is both forward and backward invariant. As we know, for any nonlinear rational function R, the Julia set J(R) and the Fatou set F(R) have the property that R(J(R))

= J(R) = R- 1 (J(R))

R(F(R))

= F(R) = R- 1(F(R))

and

(see e.g., [91], (131)}. However, for transcendental function, we will see that the case is different. The reason is that J poesibly has finite Picard exceptional values. Recall that PV(J) is the set of all finite Picard exceptional values. H J is transcendental and entire, then PV(J) contains at most one point. We have the following invariant theorem (see e.g. [124, 213]).

Theorem 3.1 Let

J be

a transcendental meromorphic function. Then F

= =

rt(F) /(F) u {PV(J) n F}.

(3.1} (3.2}

PROOF. Let ZoE C. H /(Zo) E F, then there exists a component W of F(J) such that /(Zo) E W. In particular, all r(z} (n = 1, 2, ... ) are holomorphic in W and holomorphic at Zo· We take R > 0 such that D = {lz- /(Zo)l < R} c W. Then the family {r} is normal in D. Now we choose U = {lz- Zol < r} such that lf(z) - /(Zo)l < R/2 for z e U. Then the family {r+t} is normal in U, that is, Zo E F. This implies that

(3.3} On the other hand, for arbitrary Zo E F, there exists a disk Do= {lz-Zol < ro} such that the family {r} is holomorphic and normal in D 0 • We take a disk Dt = {lz -/(Zo)l < rt} such that Dt C /({lz- Zol < ro/2}). Then the family 1 } is normal in D11 that is, /(Zo) e F. This implies that

{r-

/(F)

c

F,

(3.4}

and so,

(3.5} This and (3.3} imply (3.1).

49

3.3. SOME PROPERTIES OF THE JULIA SET

=

Now we prove (3.2). If PV(f) n F 0, then for any Zo E F, there exists Z1 E C such that /(zl) = Zo· By (3.3), z1 E F, and so, Zo = /(z1) E /(F). Since Zo is arbitrary, we have F C /(F). Combining this with (3.4) we obtain (3.2). If PV(f) n F :/: 0, then for any b E PV(f) n F, it follows that b ~ /(F), and consequently bE F\f(F). Conversely, assume a E F\f(F), i.e., a E F and a ~ f(F). If a ~ PV(f), then there exists a point c E C such that /(c) = a. By (3.3), c E F and a =/(c) E /(F), which is a contradiction. Therefore a E PV(/). The two relations above imply that F\f(F) = PV(f) n F. This and (3.4) give (3.2). The proof is complete. • Corollary 3.1 Iff has no Picard exceptional value, then

r

1

(F) = F =/(F).

In particular, for f E 'P, we have f(F(f))

= F(f) = /- 1 (F(/)).

The following example describes that the case in the above theorem does occur. Example. Let T~(z) = .Hanz, -1 e- , then by Devaney [71], J(E~) = C. Since 0 is a Picard exceptional value of E~, we have 0 ~ E~(J(E~)). If .X< e- 1 , then by Devaney [71], J(E~) is a Cantor set, and 0 E F(E~). Since 0 is a Picard exceptional value of E~, we obtain 0 ~ E~(F(E~)). 1

3.3

Some properties of the J uli~ set

Next we shall give some basic properties (see [92], etc.). Theorem 3.3 For any positive integer q, iff E £ U 'P, then J(r)

= J(/).

CHAPTER 3. THE FATOU AND JULIA SETS

50

Since the family {19ft} is contained in the family {/"}, we thus have zo E J(/) and assume zo ~ J(/9 ). Then there is a disk D = {lz- Zol < r} in which the family {19ft : n = 1, 2, ... } is normal. Since the family{/": n = 1,2, ... } is not normal in D, by Corollary 2.1 and Theorem 1.3, there exists a point ( e D, a fixed point a of J2 and a function fP E {/": n = 1,2, ... } such that PROOF.

F(/9 ) :J F(f), and so J(/9 ) C J(f). Now let

!"(() =a = J2(o). For any integer n with nq ~ p, there exists a non-negative integer m such that nq = p + 2m or nq = p + 2m + 1. In the first case,

/"'(() =/2m 0 f"(() = /2m(a) = 0 and in the second case

/"'(() = / o pm of"(() =/(a). Hence the sequence{/"'((): n = 1, 2, ... } is bounded. By Theorem 1.1, {f9"(z): n = 1,2, ... } is locally uniformly bounded in D. Thus there exists a positive constant M > 0 such that, for z E U = {lz- Zol < r/2},

1/"'(z)l $ M

(n = 1, 2, ... ).

(3.10)

Now for any n ~ 1, we know that n = mq + k for non-negative integer m and 0 $ k $ q- 1. Thus /"(z) = /,.(r(z)). From this and (3.10) we deduce that {/"(z): n = 1,2, ... } is uniformly bounded in U. By Theorem 1.1, ZoE F, a contradiction. •

r

Remark. Here we have to exclude I E M, because then is not meromorphic in the entire C so that F(/") and J(f") are not completely defined.

Theorem 3.4 The Fatou set F(f) contains all attracting and super-attracting fixed point, and all Siegel point, of/; the Julia set J(f) contains all repelling fixed points, all rationally indifferent fixed points and all Cremer points. PROOF. By Theorems 2.6 and Theorem 2. 7 we see that F(/) contains all attracting fixed points and all super-attracting fixed points. By Theorem 2.4, Theorem 2.10 and Theorem 1.1 we know that F(/) contains all Siegel points. Now we prove that J(/) contains all rationally indifferent fixed points. Without loss of generality, let z = 0 be a rationally indifferent fixed point of f. Then near z = 0, we have /(z) = z + az" + ..., a :f:. O,p ~ 2.

By induction,

/"(z) and

so

= z + naz" + ...,

(/")(p)(O) = p!an- oo

(3.11)

51

3.3. SOME PROPERTIES OF THE JULIA SET

as n-+ oo. If the family{/"} is normal at z = 0, then by /"(0) = 0, there exists a subsequence in the family {/"} such that it converges to some holomorphic 4> in some neighborhood of the origin. Thus ¢>(0) = 0. On the other hand, by (3.11) we see that tf>

(O) = oo. This is a contradiction. Thus z = 0 E J(/). By Theorems 2.4 and 1.1 and the definition of Cremer points, we know that J(/) contains all Cremer points. At last we shall prove that J(/) contains all repelling fixed points. Let zo be one of them. We have 1/'(zo)l > 1. By the chain role, (/")'(zo) = (f'(zo))" (n = 1, 2, ... ). Thus 1(/")'(zo)l -+ oo. Assume, on the contrary that, Zo E F(/). Since /"(zo) = zo, by Theorem 1.1, there exist positive numbers r and M such that 1/"(z)l ~ M (n = 1,2, ... ) for all z E {lz- zol ~ r}. Then by Cauchy's inequality, we have rl(/")'(zo)l ~ M, this is impossible. • Combining Theorem 3.3 with the above result we arrive at

Theorem 3.5 The Fatou set F(f) contains all attracting cycles, super-attracting cycles and all Siegel cycles off; the Julia set J(f) contains all repelling cycles, all rationally indifferent cycles and all Cremer cycles. Definition 3.2 We call a E C to be a Fatou exceptional value of the meromorphic function I if o- (a) is finite. We denote by FV(f) all the Fatou exceptional values of Nevanlinna's second fundamental theorem, we have

f.

Proposition 3.1 FV(f) contains at most two points, and PV(f)

According to

c FV(f).

Theorem 3.6 For any bE U\FV(/), we have J(/)

c

(Q J-"(b));

Furthermore, if bE J(/)\FV(/), then J(/)

=

(90 J-"(b)).

PROOF. Since b is not a Fatou exeptional value, U~=o/-"(b) is an infinite set. Now for any ZoE J(/) and any neighborhood U of zo, by Montel criterion, U':=0/"(U) contains U~=O/-"(b) with at most one exception. Therefore there exists a point Zt E U with Z1 :F Zo such that /"(zt) E /-m(b), i.e., Z1 E f-(n+m)(b). Thus Zo is a limiting point of U~/-"(b). Since Zo is arbitrary, we obtain the first relation. The second relation follows from this and Theorem 3.2. • Now we define the set

Po= Po(/):= o-(oo) = {z E C: j(z) = oofor somen EN}. From the definition of the Julia set, we see that Po(/)

c J(/).

52

CHAPTER 3. THE FATOU AND JULIA SETS

Proposition 3.2 Iff EM, then Po contains infinitely many pointa. PROOF. H f has at least three poles w; (i = 1,2,3), then by Theorem 1.18, there exists 1 ~ i ~ 3 such that f - w; has infinitely many solutions. Obviously, these solutions belong to Po. H f has at most two poles, then N(r, f) = O(logr). From the definition of M and Proposition 2.1 we see that J has at least one pole w, which is not a Picard exceptional value. Thus there exists bE C\{w} such that /(b)= w. H f(z) = w has infinitely many solutions, then these solutions belong to Po, and so Po is infinite; if /(z) = w has only finitely many solutions, then N(r, w, /) = O(logr). Note that N(r, /) = O(logr). By Theorem 1.18 we see that /(z) = b has infinitely many solutions, which are poles of J3. Hence, in all cases, Po is infinite. •

Lemma 3.1 Let f EM. Then Po ha.s no isolated point. PROOF. Given any Zo E Po such that r(Zo) = oo and any sufficiently small neighborhood U of ZQ, since Po has infinitely many points, there exists a point b E P0 , which is not Picard exceptional value. By the so called big Picard's theorem, there exists an a E U\{Zo} such that r+ 1 (a) =b. Suppose that /"'(b)= oo, then r+n+ 1(a) = oo. Thus a E Po, and so, Zo is not an isolated point. • Now we prove the following result (see [21]).

Theorem 3.7 For IE M, J(J) =Po· PROOF. Since Po C J(/), thus Po C J(J). On the other hand, for any Zt E C\Po, by Lemma 3.1, there exists a neighborhood Ut of Zt SUch that rlu1 are well defined and r(Ut) C C\Po for all n EN. Since Po is infinite, by Montel's theorem, {r} is normal in U1 • Therefore z1 f/. J(J). The proof is complete. •

Theorem 3.8 J(/) contains an infinite number of pointa and is unbounded. PROOF. H J EM, then the conclusion follows from the above theorem. Next we let f E £or P. Let g(z) = J'l(z). Then J(g) = J(J). We only need to prove that, for any R > 0, J(g) nD ~ 0, where D = {lzl > R}. By Corollary 2.1, g(z) - z has infinitely many zeros. Thus by Picard's theorem, we can take three distinct points a 1 , a 2 and binD such that

g(a;) =a; (j Setting

Do= {R ~

= 1,2),

g(b)

=a 1•

lzl ~ latl + la2l + lbl + 1}.

H J(g) n Do = 0, i.e., g"(z) (n = 1, 2, ... ) is normal in Do, then by Theorem 3.4, a 1 and a2 are either attracting fixed points or Siegel points of g.

53

3.3. SOME PROPERTIES OF THE JULIA SET

If a1 is an attracting fixed point of g, then by Theorems 1.1, 3.4, and 2.6, there exists a disk C = {lz- a 1 l < r} with C C Do in which the sequence g"(z) (n = 1, 2, ...) uniformly converges to a 1 • On the other hand, since g"(z) (n = 1, 2, ... ) is normal in D 0 , we can get a subsequence g"•, which converges locally uniformly to a holomorphic function h(z). Since h(z) = 111 in C, hence h(z) = 111 in D, and so, h(a2) = 111. However, by g"(a2) = 112 for any positive n we see that h(a2) = a 2, which is a contradiction. Thus, a 1 is a Siegel point of g. By Corollary 2.4, we get a disk C = {lz- a 1 l < r} with C c Do and a sequence of positive integers n,. such that the sequence g"• converges uniformly to z in C. Further, since g"(z) (n = 1, 2, ... ) is normal in D 0 , there is a subsequence in g"•, which converges locally uniformly to a holomorphic function h(z) in D 0 • Thus h(z) =zinC, and so h(z) = z in D0 • In particular, h(b) =b. On the other hand, h(b) = 111 by the assumption g(b) = a 1, a contradiction. • Theorem 3.9 If Zo E J(/), then for each finite value a, there exist a sequence of points (k --+ Zo and a sequence of positive integers nk --+ oo such that

(3.12) except at most for two finite values. PROOF. Consider a finite value a. If for each positive integer N, there exist a positive integer n and a point ( such that

n;:::N,

1

/(- Zol < N'

/"((}=a,

then this value a has the required property. In fact, for each positive integer k, there exist a positive integer nk and a point (k such that

Therefore if a does not have the required property of the theorem, then there exists a positive integer N such that for n;::: N, the function f"(z) does not take the value a in lz- Zol < j;,. It follows that if there are three finite values a3 (j = 1, 2, 3) not having the required property, then we can find a positive integer N 1 such that for n;::: N 11 the function f"(z) does not take the three values a3 in /z- Zol < 1/N1 • By Mantel's theorem the family{!" : n = 1, 2, ... }is then normal in /z- Zo/ < 1/N1 • This contradicts the hypothesis that ZoE J(/). • Remark. If f is entire, then there is at most one such exceptional value. Theorem 3.10 J(/) is perfect, that is, J(f)

= {J(/)}'.

PROOF. For any point Zo E J(/}, by Theorem 3.8 and Theorem 3.9, we can choose a point a E J(/) such that there exist a sequence of points (k --+ Zo and a sequence of positive integers nk--+ oo with f"•((k) =a. From Theorem 3.2 we know that (k E J(/). Thus ZoE {J(f)}', and so, J(f) C {J(f)}'. The converse is obvious • since J(f) is closed.

CHAPTER 3. THE FATOU AND JULIA SETS

54

Theorem 3.11 If J(J) has an interior point, then J(J)

= U.

By the 888UIDption, there exists a disk D := {lz- .tol < r} belonging to a be a finite value satisfying the condition (3.12). Then we can get a positive integer k such that (A, ED. Hence (A, E J(l). Since r•((~c) =a::/: oo, by Theorem 3.1, we have a E J(J). It is then clear that every finite value a E J(J) except at most two values. It follows from Theorem 3.10 that J(J) = U. • PROOF.

J(l). Let

Corollary 3.2 If J(f) ::/: U, then F(J) is unbounded. PROOF. If F(f) is bounded, then J(l) has interior points. By the above theorem, J(J) = U, which is a contradiction. •

3.4

Baker's theorem

The following result is due to Baker [lOJ when IE C. His proof was based on a powerful result known as the Ahlfors Five Islands Theorem. It was extended to class Pin [44, Theorem 5.2J and to class Min [21J.

Theorem 3.12 J(J) is the clo8Ure of repelling periodic points of f. PROOF.

We here present a simpler proof due to Schiwick [184J. Let

A= {wE C: 9(w,/)

~

1 }, 2

where 9(w, I) was defined in Section 1.8. By Corollary 1.5, A consists of at most four elements. In the following we prove that the repelling periodic points are dense in J(I)\A. Then the assertion follows, since the Julia set is perfect. For woE J(J)\A, we apply Theorem 1.7 to the sequence {r: n = 1,2, ... } and o = 0. Then there exist a subsequence {r• : k = 1, 2, ... }, complex numbers z... - Wo, real numbers p... - 0 and a nonconstant meromorphic function h(z) in the plane such that

r•(z... + Pn.z) - h(z)

uniformly on all compact subset of C. This implies

r•+l(z... + Pn.z) - I 0 h(z), and so,

r•+ 1(z... + Pn.z)- (z... + Pn.z) -1 o h(z)- Wo.

Since wo ¢ A, there exists a sequence {ZJ : j that

l(z1 ) = WQ,

= 1, 2, ... }, Zj pairwise distinct, such

l'(z1) ::/: 0.

3.4. BAKER'S THEOREM

55

U his transcendental, then by Corollary 1.5, there exists j E {1, ... , 5} such that 9(z;, h) < 1/2. Thus there exists Zo with h(Zo) = z; and h'(Zo) ::/= 0; It is easy to see that such j and Zo also exist if h is rational. In both cases we have

f o h(Zo) = wo, (/ o h)'(Zo) ::/= 0. Since the limit function that the equation

f o h(z)- w0

is not constant, Hurwitz's theorem implies

r•+l(z... + Pn.z) = Zn. + Pn.z

has a solution in. for sufficiently large k, and in. -+ Zo· Therefore z... + Pn•in• is a fixed point of r•+l and z... + Pn. Zn. -+ Wo. It is repelling for large n,. because of

• By this theorem, we can derive the following useful criterion.

Theorem 3.13 The sequence {r} is normal at ZoE C if and only if some subsequence {In•} of {r} is normal at Zo. PROOF. We assume that the subsequence {In•} of {r} is normal at Zo and suppose on the contrary that {r} is not normal at Zo· Then there exists a subsequence {ri} in {r•} which converges uniformly in a neighborhood D of Zo to a nonconstant meromorphic function 1/>(z) or oo. Since Zo E J(/), by Theorem 3.12, there exist two different periodic points a, b with period p and q such that a ED, bED and the cycles Q+(a) and o+(b) are different. Thus ri(a) E Q+(a) for any j. This implies that {ri} converges uniformly in D to the nonconstant meromorphic function 1/>(z). Note that Zo is not a pole of any and so Zo is not a pole of 1/>(z). Now for any given e > 0, we take a neighborhood Do of Zo with Do C D such that E l4>(z) - 1/>(Zo)l < 2, z E Do.

r•,

For this D 0 , there exists a positive integer N such that

l/t1 (z)- 1/>(z)l < and so,

E:

2, z E Do, t; > N,

l/t1 (z)- 1/>(Zo)l < e,

z E Do, t; > N.

By Theorem 3.12, we may assume that a E Do and bE D 0 . Thus Now for a fixed j with t1 > N, there exist two positive integers m and n with ~ p and 0 ~ n ~ q such that fi(a) = f"'(a) and fi(b) = r(b). Then

0~ m

lf"'(a) -.f"'(b)l < 2e.

CHAPTER 3. THE FATOU AND JULIA SETS

56

Since e is arbitrarily small, we arrive at /"'(a)

3.5

= r(b), a contradiction.

•

Expansivity of the Julia set

The following result is called the local expansivity of the Julia set (see Fatou (92]). Theorem 3.14 Let f be a transcendental meromorphic function. For any Zo E J(l) and any neighborhood D of Zo, if A is a bounded and closed set of C and An FV(/) = 0, then there exists an integer N > 0 such that, for any n 2: N,

Acr(D). PROOF. We distinguish two cases. (i) Let f E £UP. Suppose, on the contrary, that the conclusion does not hold. Then there exists a sequence j = 1, 2, ... } such that A - r~ (D) ::/: 0 for any j. Take a; e A - r1 (D). Since A is bounded and closed, we may suppose that a; -+ a E A. Note that a is not a Fatou exceptional value, there exist two different points a' and a" in f- 1 (a). Therefore, in some neighborhood U of a, we have two branches /1 1 (z) and /i 1 (z) of 1 (z) such that

{rJ :

r

/1 1 (U) n /i 1 (U) = 0, /1 1 (a) =a', /2 1 (a) =a". Without loss of generality, we assume that E /} 1 (U) and a~ E /i 1 (U) such that

aj

a,

E U

for all j. Then there exist

f(aj) = f(a'j) = a1 , aj-+ a', a'J-+ a". Since rl(z) ::/:a; for all zeD, we have

A.

This implies that r(D) :>A for any integer n 2: m + 1. The proof is complete.

•

Chapter 4

The Components of the Fatou Set

For a transcendental entire or meromorphic function I, we focus on the behavior of the components of the Fatou set F(/) and the behavior of I on its Fatou set. We will see that there are some essential differences among rational functions, transcendental entire functions and transcendental meromorphic functions .

4.1

Types of the components

Let U be a maximum domain of normality of the iterates of I, that is, a component of F . This domain is also called a stable domain or a Fatou component. Here and subsequently, "component" always means "connected component". Consider a fixed component U of F. There are several possibilities for the orbit of U under 1.

1. If r(u) c u for some integer n ~ 1, then we call u a periodic component of F. The minimum n is the period of the component. In particular, if n = 1, then such a component U is said to be an invariant component er a fixed component. 2. If lm(U) is periodic for some integer m ~ 0, we call U a pre-periodic component of F. In particular, if U is pre-periodic but not periodic, then we call

U a really pre-periodic component.

57

CHAPTER 4. THE COMPONENTS OF THE FATOU SET

58

3. Otherwise, all {/"(U)} are disjointed, and we call U a wandering domain.

4.2

Multiply connected components

We denote by indo'Y the index of a curve 'Y C C with respect to a point a.

Theorem 4.1 Let f be a tmnscendental entire function, and let U be a multiply connected component of the Fatou set F(f). We denote by 'Y a Jordan cun1e that is not contractible in U. Then 1.

r ....

00 uniformly on oompact subsets of u, and so the distance between f"('Y) and 0 is l4rge;

!. indo/"('Y)

> 0 /or all sufficientlyl4rge n

and indo/"('Y) .... oo as n .... oo.

PROOF. 1. We only need to prove the conclusion for such kind of compact subsets of U that are not contractible in U. Suppose on the contrary that the conclusion does not hold. Then there exists some sequence {/"•} such that If"• I $ M on a curve r which is not contractible in U and hence l(f"•)'l $ C insider. But inside r there are points of J(f), and we obtain a contradiction with Theorem 3.12. 2. Assume that indof"•('Y) = 0 for some sequence {nA:}· Then /"• does not have zeros inside 'Y· By the minimum principle, /"• .... oo inside 'Y· This is imp08Sible, since inside 'Y there are points of J(/). It is easy to show that indo/"('Y) .... oo as n-+ oo. •

Recently, Bergweiler-Terglane [42] gave a relation between multiply connected components ofF(/) and fixed points of/:

Theorem 4.2 Let f be a tmnscendental meromorphic function. If f has no weakly repelling fixed point, then f has no multiply connected wandering domain. In addition, if / is a tmnscendental entire function and f has only finitely many weakly repelling fixed points, then the same conclusion holds. Here a fixed point a of/ is called weakly repulsive if /(a)= a and 1/'(a)l

(see Shishikura[185)).

4.3

Simply connected components

The following result is standard (see [164] or [170)).

~

1

4.3. SIMPLY CONNECTED COMPONENTS

59

Proposition 4.1 A domain D is simply connected 0 there emu a curve "f tending to oo 6UCh that 1/(z)l ~ M(lzlc, f) for z E "f· Then all componmu ·of F(f) are 6impl71 connected. In particular, this is the case if log 1/(z)l = O(log lzl) as z- oo through some path.

4.4

Classification of periodic components

r

The behavior of in periodic oomponents is well understood. In this section, we shall give the classification theorem that was first stated by Baker, Kotus and Lii [21]. The proof is essentially due to Cremer [68] and Fatou [91].

4.4.1

The classification theorem

Theorem 4.5 Let I be a transcendental meromorphic function, and let U be a periodic component of period m. Then we hove the following possibilitiu: 1. U contains an attracting periodic point Zo of period m. Then r'"(z) - %0 for z E U as n- oo, and U is called the immediate attractive basin of %0 • JiUrthermore, U is called a Bottcher domain or a Schroder domain prcwided that %0 is IUper-attracting or not.

f. lJU contains a periodic point Zo of period m and r'"(z)- %0 for Z E U as n - oo. Then (/'")'(z0 ) = 1. In this case, U is called a Leau domain {or parabolic domains). 3. There ezisU an analytic homeomorphim& t/J : U - a IUch that t/J o f'" 0 f/J- 1(z) = e2waiz for some o E R\Q. In this case U is called a Siegel disc. We hove the following commutative diagram

I'" _ u ___

A

u

A

4.4. CLASSIFICATION OF PERIODIC COMPONENTS

61

-'· U is doubly connected and 1m is conjugate to either a rotation on an annulus or to a rotation followed by an inversion. This U is called a Herman ring (or Arnold-Herman ring). The Siegel disc and Herman ring are referred to as rotation domains. 5. rm(z) -+ .zo E 8U for z E U as n -+ oo but fm is not holomorphic at .zo, and U is called a B.aker domain {or infinite Fatou component {115}, or essentially parabolic domain {21}, or domains at oo {7.4]). In particular, if m = 1, then the only possible case is .zo = oo. PROOF. Without loss of generality, we may assume that m = 1, i.e., I(U) C U. The conclusion follows from the following Theorem 4.6, Theorem 4.7 and Theorem

u

4.4.2

•

Constant limit functions

To prove the classification theorem, we need to study all poesible limits of subsequences of {r} in the Fatou component U (cf. [29]). A function tf>(z) is a limit function of {r} on a Fatou component U'ifthere is some subsequence of {r} that converges locally uniformly in U to tf>. We denote by .C{U) all such limit functions.

Lemma 4.2 Let U C F{f) be a forward invariant component. If there exists a constant limit function (, then either { is a fixed point of I or ( = oo. PROOF.

Let

(-:/= oo be a constant limit of r'(z),

then

•

Thus ( is a fixed point of I.

Lemma 4.3 Let U be a component of F(f). Then .C(U) does not contain any repelling fixed point of I. Suppose that there exists a repelling fixed point a of I such that a E .C(U). Since a is repelling, we can choose a number M > 1 such that 1/'(a)l > M, and a neighborhood V of a such that for z E V, PROOF.

ll(z)- al = ll(z)- l(a)l

~

Mlz- al.

(4.1)

On the other hand, since a E J{f), by Theorem 3.1, for any z e U and any positive integer n, we have r(z) -:/=a. Now by a e .C(U), we can take z E U and ni -+ +oo such that Zj = rJ(z)-+ a. Then there exist infinitely many j's such that Zj E v and

lzJ+t This and {4.1) give a contradiction.

ai < lzi -

aj.

•

62

CHAPTER 4. THE COMPONENTS OF THE FATOU SET

Theorem 4.6 Suppose that U is a forward invariant Fatou component, and that

euery limit junction in .C(U) is constant. Then .C(U) contains exactly one element, with value b, say, and r(z) converyes to b locally and uniformly in u.

PROOF. Suppose on the contrary that .C(U) contains at least two values. Then there exists a finite value bE .C{U). By Lemma 4.2, b is a fixed point of f. Given Zo E U and any small € > 0, set

A= {mEN: x(/"'(Zo),b) < €}. It is easy to see that if /"'(Zo) converges as m -+ oo through some sequence in A, then the limit is b. Since .C(U) contains at least two values, we obtain some subsequence {m;} of the poe~itive integers such that m; '1. A but m; - 1 E A and

(a may equal to oo). On the other hand, by extracting a subsequence from m; -1 if necessary, we have

•

which contradicts a #: b. This completes the proof of the theorem.

Theorem 4. '1 Suppose that U is a forward invariant Fatou component, and that every limit function in .C(U) is constant. Let b be the unique limit function in Theorem 4.6. Then ezactlJI one of the following holds: (i) b = oo; (ii) b is an attmcting jized point of J and b E U; (iii) b is a mtionally indifferent jized point of I and b E au. PROOF. Obviously, bED'. We assume that b #: oo. By Lemmas 4.2 and 4.3, b is a fixed point of J and is not repelling. Now we consider two cases. H bE U, then by the Weierstrass theorem, n(Zo) = 1, and 0 from 6 to V, t/J(O) = Zo· Then

Yn{()

'1.

f>n(V). Let t/J be the Riemann map

= f>n(t/J{())- 1

63

4.4. CLASSIFICATION OF PERIODIC COMPONENTS

# 0. Thus the function Yn/~(0) belongs to 8 0 , 1 • By Corollary 1.2 we know that So,l is a normal family, consequently {n} is normal on V. Let (z) be a limit function of {n}· From

is univalent on fl., 9n(O) = 0, g~(O) = ~(Zo)t/1'(0)

n(/(z))

=

= n(z/(~~~;~ ~(O)

we deduce that

(/) = /'(0). If is nonconstant, then I is conjugate to a rotation about 0, contradicting r(z) -+ 0. Thus is constant, and the above formula implies that 1'(0) = 1, as desrred. •

4.4.3

N onconstant limit functions

In this subsection, we shall study nonconstant limit functions. Theorem 4.8 Suppoae that U iiJ a fonJJard invariant Fatou component, and that .C(U) contairu some non-corutant limit function~. Then (i) I i1J conformal in U, (ii) the identity map I iiJ in .C(U), (iii) any non-corutant limit function is conformal in U. {iv) .C(U) does not contain any corutant limit function. PROOF. Let (z) be a non-constant limit function. Then there exists a sequence of positive integers ni such that ri(z)-+ (z) locally uniformly on U. For any w E U, it is clear that the zeros of (z)- (w) are isolated, so we can choose a disk D centered at w such that D C U and (z) =/: (w) on 8D. Thus for all sufficiently large j and all z E 8D,

lr1 (z)- (z)i < :rE8D inf 1(z)- (w)j. By Rouche's theorem, there exists a point zeD such that ri(z) I(U) C U, and so (w) E U. This implies that (U) cU.

= (w).

Since (4.2)

By passing to a subsequence of the ni and relabelling if necessary, we may assume that mi = ni -ni-l -+ +oo as j -+ oo. Since {rJ} is normal in U, we may suppose that r~ converges to some limit function"' locally uniformly in u. Thus by (4.2),

As is non-constant, ,P must be the identity map I, and so IE .C(U). This proves (ii).

CHAPTER 4. THE COMPONENTS OF THE FATOU SET

64

Now for any z,w E U, if /(z)

f"'i(z)

= f(w), then

= rr 1 (f(z)) = rr 1 (f(w)) = f"'i(w).

Note that r"J -I, we have I(z) = I(w), and so z = w. This proves (i). To prove (iii) and (iv), 'we suppose that 4> is an arbitrary limit function and that rJ - 4> locally uniformly on U. Let g : U - U be the Inverse of f. Then by passing to a subsequence of n; if necessary, we may assume that gnJ converges locally uniformly to some function 1/J on U. It follows that

1/J(t/>(z)) = limgnJ(ri(z)) = z, and hence 4> is not constant, which proves (iv). Further, 4> is injective on U, and 1/J is also not constant and 1/J(U) C U by the same proof for (4.2). With this, we can reverse the roles of 4> and 1/J and deduce that 4>(1/J(z)) = z on U. It follows that 4> is a bijection of U onto itself. This proves (iii). • Combining this theorem with Theorem 4.7, we obtain the following well known result. Denjoy-Wolff Theorem Let f : ~ - ~ be analytic, and assume that f i6 not an elliptic Mobius tramf~tion nor the identity. Then there i6 o e ~ such that r(z)- Q locally uniformlJI in~. Theorem 4.9 Suppose that U i6 a forward. invariant Fatou component, and that .C(U) contaim some non-COMtant limit function. Then U iS· either a Siegel due or a Herman ring. PROOF.

Suppose first that U is simply connected. Let 4> map U conformally onto

~.Then

.-t : -

1/J = 4> o I o ~ ~ is a Mobius transformation. If .P has fixed points on the unit circle, then I.Pnl - 1 on ~. which contradicts Theorem 4.8. If '1/J has a fix-point in the disk, we may assume it is at the origin. Then t/J is a rotation. Obviously, the fixed point can not be attracting, otherwise .C(U) does not contains non-constant limit function, which contradicts the assumption. Since the fixed point is in F(/), by Theorem 3.4, the fixed point can only be a Siegel point. Thus U is a Siegel disc. Now assume that U is multiply connected. Let .P : ~ - U be the universal covering map, and let g be the associated group of covering transformations, i.e., the group of conformal self-maps g of ~ satisfying '1/J o g = '1/J. The lift of f to the unit disk via .P is a Mobius transformation T satisfying .p o T = I o ¢. Let r be the group obtained by adjoining T to Q. By Theorem 4.8 (i), r is not discrete. Let be the closure of r in the (Lie) group of conformal self-maps of~. and let ro be the connected component ofT containing the identity. If g E Q then also . T(g(T- 1 )) e Q. Since

r

1/J o Togo T- 1 = I

o .pogo T- 1

= I o 1/J o T- 1 = I

o , - 1 o .p

= ¢,

65

4.4. CLASSIFICATION OF PERIODIC COMPONENTS

r,

it follows that and hence fo, also conjugates Q to itself. Therefore h(g(h- 1 )) = g for all h E fo and g E Q, and every g E Q commutes with every h E fo . Now choose hE f 0 , which is not the identity. We consider three cases. (i) h has a fixed point in~. We may assume the fixed point is 0, so that h(z) = ei8 z . Then ei 8 g(O}

= h(g(O)) = g(h(O)) = g{O}

for any g E Q. Thus g has the form eiq,z. (ii) h has two different fixed points on = 1}. We map~ to the right half-plane by A(z} with the fixed points going to 0 and oo, then AohoA- 1(z) = ..Xz, .X =F 1. As above, AogoA- 1(z) = JJ/z, or JJZ for some I' > 0. In the first case, A o h o A- 1 (z) does not commute with AogoA- 1 (z), thus AogoA- 1 (z) = JJZ. {ill} h has one fixed point on {z: lzl = 1}. As in (ii) we can deduce that A o h o A- 1 (z) = z +.Xi and A o go A- 1 (z) = z + JJi for some nonzero real number A and real number I'· Combining all the cases above we know that Q belongs to the one-parameter group generated by h. Thus Q has the form {g"}~00 • This means that the fundamental group of U is isomorphic to the integers, and U is doubly connected. Since U is hyperbolic, U cannot be a punctured plane, and U is either an annulus {z : 1 < izl < R} or a punctured disk. The proof is complete. •

{z: lzl

4.4.4

Herman rings and Baker domains

When Fatou and Cremer wrote their papers, it was not known yet that Siegel disc and Herman ring do actually exist. Topfer [202] constructed a function with Siegel disc. An example of Herman ring for rational functions was given by Herman [112, p. 138], extending earlier work of Arnol'd [5] . By Corollary 4.1 we have the following conclusion. Proposition 4.2 Any tronscendental entire function does not have Herman rings.

The term "Baker domain" seems to have been used first in [84] . Proposition 4.3 If f is a transcendental entire function, then the only case for the Baker domain in the classification theorem is Zo = oo.

The existence of the Baker domain can be seen from the following examples. Example. Let /(z) = z + 1 + e-z. Then the right half-plane is /-invariant and Re(/"(z))-+ +oo for z in the right half-plane. Example. Let /(z) = ~- ez. We have

J2(z)

= 1- zexp(-z) z

!1r

- exp

(~z -

ez) "' z + z 2

+ ···

as z-+ 0 in W(e) = {z : < argz < ~1r , izl < e}. There is a component U of F(f}, which contains (-e,O} for small positive e. Further, J2(U) C U, J2"-+ 0 in U. Obviously J2 is undefined at 0.

CHAPTER 4. THE COMPONENTS OF THE FATOU SET

66

Question 4.1 Can one construct a function

f

E M 8Uch that F(f) contain.! a

Hennan ring?

4.5

Singular points

Singular points play an important role in the study of dynamics of transcendental functions. Let f be a meromorphic function. A point a E C is said to be a nonsingular point (of the inverse function /- 1 ) if it has a neighborhood V such that f : f- 1 (V) -+ V is an unbranched cover. The set of singular points is denoted by nng(/- 1 ). Singular points (or singularities) are of the following types (see Nevanlinna [1631): 1. a is a critical value (or an algebraic singularity), i.e., there exists Zo e C such that /(Zo) = a, /'(Zo) = 0. Such a point Zo is called a critical point of f. We denote by CV(f) all these values;

2. a is an asymptotic value (or transcendental singularity), i.e., there exists a curve r going to oo such that /(z) -+ a as z -+ oo along r. All these values are denoted by AV(f). In particular, if a has a simply connected neighborhood V such that for some component U of the set f- 1 (V) the mapping f : U -+ V\ {a} is a universal covering, then a is called a logarithmic branch point, and U is called an exponential tract. In addition, if there exists a neighborhood Vo and a component Uo of /- 1 (V0 ) such that /(z) ¥a for z E Uo, then a is called direct, otherwise it is indirect. Obviously, a logarithmic branch point is direct; 3. limit points of types 1 and 2. For any transcendental meromorphic function /, we have

In fact, for any transcendental meromorphic function/, we let 3' be the Riemann surface defined by the inverse z = f- 1 (w). z = /- 1 (w) is single-valued function and maps 3' conformally onto C. By Iversen's Theorem [127], the asymptotic values of f(z) correspond to the boundary of 3' and vice versa A corollary of Iversen's Theroem is that any Picard value of a meromorphic function is an asymptotic value of the function. Based on singularities, for the family of meromorphic functions, '!Nf! introduce two subclasses Sand B.

S

= {I: f

has only finitely many critical and asymptotic values}.

67

4.5. SINGULAR POINTS

According to Eremenko and Lyubich [84], the letterS was choosen in honor of Speiser, who introduced this class (cf. [163]}.

8={/: sing(/- 1 ) is bounded}. We remark that 8\Si=

0. For example, let Ua(z)

2

= 11'2 -

asin..fZ/..fZ,

211'2

where 11' 51rrIN for all j' then from Corollary 1.4 we deduce that if lz I < 1rr IN' then

u

e;

pu(z,e;)

r

~}f.;

Pu(w}ldwl

r

~}f.;

2ldwl

d(w,BU)

N

r

e;'

ej

1

~ 2n }f.; ldwl < 2'

71

4.5. SINGULAR POINTS

where the integral is along the straight line from{; to z. Since pu({;,{~c) ~ 1 for j =/: k this implies that the disks A({;, 1rr/N), j = 1, ... , N, are disjoint. This is a contradiction. Hence there exists 1 $ j $ N such that 5 d({;.CJU) $ ; .

The conclusion follows from: Lemma 4.5 Let f be a transcendental entire function with an invariant Baker domain U. Suppoae that K C U i6 compact and that r > 1. Then there exi&t& no such that En A(/"(z), rd(f"(z),8U)) =F 0

for all z E K and n

~no.

PROOF. Suppose that the conclusion does not bold. Then there exist sequences {z;} C K and n;-+ oo such that

EnA(f"i(z;),rd(f"i(z;),8U))

= 0.

Restricting to a subsequence if necessary we may assume that z; -+ .zo E K as j-+ oo. This implies that pu(z;,Zo)-+ 0 and hence that p(f"i(z;),f"i(.zo))-+ 0 as j-+ oo. From Corollary 1.4 we deduce that

and

1/"i(z;) - f"i(.zo)i/d(/"i(z;),8U)-+ 0 as j-+ oo. It follows that if 1 < u

< r, then En A((;,u6;) = 0

for all large j, here(; = f"i (.zo), 6; = d((;, 8U). Hence there exist branches ¢; of (/"J)- 1 , which are defined (and univalent) in A((;,u6;) and satisfy¢;((;)= .zo. We consider B; = {z : lz- (;I = 6;} and C; = ¢;(B;). Then there is b; E B; n8U with Cj = ¢;(b;) E C; n au. It follows from Koebe's distortion theorem that lc;l is bounded. We may thus assume that Cj -+ c E au. FUrther from Koebe's distortion theorem we deduce that 1¢'-(b·)l > u -1 . d(.zo,8U). ) ) - u +1 6; Combining this with Koebe's one quarter theorem we obtain that A(c,6)

c

¢;(A((;,u6;))

for some 6 > 0 and all large j. This implies that f"i is univalent in A(c,6), which is impossible forcE au C J(/). • Remark. By Proposition 4.4, we can extend this result to periodic domains. Next we will give relations between connectivity and critical points.

72

CHAPTER 4. THE COMPONENTS OF THE FATOU SET

Theorem 4.15 For any fEE, any doubly-connected component of F(f) clou not

contain critical points of f. PROOF. This result was proved by Baker [11]. Suppose that Ut is a doublyconnected component of F(f). By Corollary 4.1, Ut is bounded and wandering. Denote by a 1 and {31 the outer and inner boundary components of U1 • Write Un+t = rCUt), a2 =/(crt), /32 = /(/Jt). Then 8U2 = /(8Ut) = a2 U /32, which has at most two components. We claim that 02 and /32 are distinct components. In fact, if 8U2 is connected, then 8Un+l is also connected for each n E N by the complete invariance of J(/), which contradicts Theorem 4.1. By the maximum principle, a 2 is the outer component and /32 is the inner component. Let Kj = {z : 1 < lzl < Rj}, and let¢; : KJ - U; be conformal maps U = 1,2). It is assumed that tPj approaches a; or /3; as lzl approaches R; or 1 respectively. Then g = ¢2 1 (/(1/Jt}) maps Kt onto K2 and g(z)- 8K2 as z - 8Kt. Thus g extends analytically to lit and lg(z)l = 1 on lzl = 1, lg(z)l = R2 on lzl = Rt· Repeated application of the reflection principle shows that g can be continued to give an analytic map C- C such that z = 0 is the only solution of g(z) = 0. Further, for w E K2, all solutions of g(z) = w are in Kt. Hence g(z) is a polynomial of the form czm, lei = 1, m is a positive integer. Hence !I has no zeros in K 1 , whence /' has no zeros in U1 • The proof is complete. •

Now we give a relation between asymptotic values and boundedness of stable domains ([216]}. Theorem 4.16 For the tran&cendental entire function/, if all stable domain& are bounded, then F(f) does not contain any asymptotic values of f. In partJcular, this is true iff has a multiply connected component.

r

Let D be a component ofF(/). We take a branch g(w) of 1 • By Gross Theorem, one can continue g(w) analytically in D. If there exists w0 E DnAV(f), then there is a branch h(w) of the inverse of /(z) with h(D) C D 1 such that h(w)- oo as w- wo, where Dt is a component ofF(/). This implies that D1 is unbounded, a contradiction. • PROOF.

The fo)Jpwing result was also proved in [216).

Theorem 4.17 For tmn&cendental entire function /, if Zo is a Cremer point of /, then ZoE E'(/). PROOF. Recall that Zo is a Cremer point if Zo is a.n irrational indifferent periodic point of f and Zo is not a center of any Siegel disk. Without loss of generality, we suppose that Zo is a fixed point of f. If Zo ¢ E'(f), then we can take a small disk V centered at Zo such that

{V\{Zo}} n E(/) = 0.

4.6. FURTHER RESULTS ON LIMIT FUNCTIONS

73

Note that l(r)'(Zo)l = 1, we can define single-valued holomorphic function 1-n: V -+ C such that rn(Zo) = Zo· By Lemma 4.4, the family {1-n} is normal in V. Thus we may suppose that 1-n locally uniformly convergent to a holomorphic function h : V -+ C, h(Zo) = Zo and lh'{Zo)l = 1. Thus there exist two neighborhoods Dt and D2 of Zo such that h : D 1 -+ D2 is conformal. Therefore converges on D 2 to h- 1 : D 2 -+ V. However, since Zo is a Cremer point, ZoE J(f). Thus D2 n J{f) :f. 0. By the expandicity of Julia sets, for any compact set K that does not contain Fatou exceptional value, there exists positive integer N such that r(D2) ::.> K for all n > N. This is a contradiction. Therefore, Zoe E'{f). The proof is complete. •

r

4.6

Further results on limit functions

From the proofs of Theorem 4.8 and Theorem 4.9 we immediately obtain the following result. Theorem 4.18 Let f be a transcendental meromorphic function. If in a component D of F(f}, some subsequence of {r} has a non-constant limit function, then there is a component D 1 of F(f) and a positive integer p such that f"(z) maps D1 univalently onto D 1 and for some increasing sequence of integers n,. one has f'l"'~(z)-+ z in D1. Moreover JN(D) C D 1 for some N. Remark. The result was given by Fatou [91], Cremer [68] and Bhattacharyya [44] for rational functions, entire functions and holomorphic self-maps of the punctured plane respectively. We now turn to the consideration of constant limit functions. Combining Theorem 4.5 and Theorem 4.13, we have the following result {cf. Milnor [159, p.317], Fatou [91]}. Theorem 4.19 Let f be a nonconstant entire function. Then all constant limit functions of in (pre)periodic components are in E'(f) U {co}, except possibly in (pre-images of) super-attracting components.

r

Lemma 4.6 If H is a compact set of which no point is a limit point of any sequence {r(a)}, a e F{f}, then for large n and each compact set K c F{f}, we have /-n{H) n K = 0. Theorem 4.20 (Baker [12]} Let f E £ U 'P. I{E(f) has an empty interior and a connected complement, then no sequence {r~} has a nonconstant limit function in any component of F(f) . PROOF. Suppose there is a component D ofF{/) in which a subsequence of {r} has a nonconstant limit function. By Theorem 4.18 there is a component Dt of

CHAPTER 4. THE COMPONENTS OF THE FATOU SET

74

F(/) and an integer p such that fP(z) maps D1 univalently onto Dlt while the /(D1),••• , D, p- 1 (Dl) are all different. There is a branch of g = 1-p univalent on D 1 and mapping D 1 onto itself. There are therefore other branches of which map D1 into another component D' of F(/). No point b of D' is a limit point of any sequence {r(a)}, a E F(l}, since fP(b} e D~t while for n > p, r(b} belongs to one of the domains D,. (k = 1, ••. ,p}, and D' =/: D,. (for JP(D') = D1 =/: D'). Since E(f) has an empty interior, the domains D' and D 1 contain points t' and tin the complement M of"'£(/). M is a domain and it is therefore possible to join t, t' by a simple polygon 1r in M, which will have positive distance from "E(I). Let Q be a simply-connected neighborhood of 1r which does not meet"£(/). Take a branch g that form the iterate g", which are branches of 1-"". They are regular in D1 and hence at t, and may be analytically continued throughout Q (cf. Proposition 4.4). By Lemma 4.4 {g"} is normal in Q and in D1. From Lemma 4.6 we see that in any compact neighborhood H of t' such that t' e H c Q n D' we have g"(H) n K = 0 for any compact subset K C F(/) and large n. Thus either g"-+ oo in H or, if for some z E H there is a subsequence g"•(z) which tends to a finite limit; then this limit must belong to J(/). Thus any finite limit function ~of g"•(z) in H must satisfy ~(H) C J(/). Since J(/) =/: C, we know that J(/) and hence ~(H) has an empty interior. Thus~ is a constant. However," by Theorem 4.18, there is a subsequence fP"• of fP" that converges to z in D 1 • Now we may choose n,. in such a way that the corresponding terms of the normal family g" converge locally uniformly in D 1 :

domains D1, D2

=

=

r"

We have

z = g"•(fP"•(z))-+ ,P(z)

in D1. Thus g"• is convergent to z in D1 n Q, hence is normal and convergent throughout Q. ·But we have seen that any limit function is constant in H c D'nQ, a contradiction. • Baker (12] further proved the following result on the location of constant limits. Theorem 4.21 For I E £ u 'P, anr comtant limit of a sequence r•(z) in a component 9f F(f), belongs to L = "E(I) U A, where A = {oo} or {0, oo} provided that I is entire or I E 'P respectivelr.

PROOF. Consider any a rt L. We may take a disk K, which contains a and does not (z) -+ a in the component D ofF(/). Then for any fixed meet L. Suppose that b E D the points b,. = (b} belong to K for large k and we may define a branch z = g"•(w) ofthe inverse of such that g"•(b,.) =b. This branch is analytic at b,. and indeed continues analytically to give a function analytic in the whole disk K. By Lemma 4.4 {g"•} is normal in K and we can extract a subsequence locally uniformly convergent inK to a limit ~. Since b,. -+a, ~(a} = b. However, by taking a different c =/:binD and forming c1c = r•(c) we see that r•(D1 ) C K

r• r•

r•

4.6. FURTHER RESULTS ON LIMIT FUNCTIONS

75

for large k and any compact subset D 1 of D containing b, c so that g"• (c~e) Since c,.-+ a, ~(a)= c, and so, b = c, a contradiction.

= c. •

For some special stable domains, we have better results. Theorem 4.22 Let f E C U 'P, and let U be a wandering domain of f. Iff E £, then all limit functions of {r} in U are constants and are contained in (E'(f) n J(/)) U {oo}; Iff E 'P and U is a bounded annule, then the only possible limit functions of {r} in U are constants 0 and oo. PROOF. H f E 'P, then the iterates of U under fare again annules and a su~ quence must nest about one of the essential singularities. The limit function must therefore be the constant 0 or oo. Now we suppose that f E C. H there exists a subsequence {r•} of {r} such that which has a nonconstant limit function in U, then by Theorem 4.18, f""•(U) c r•(U) for some positive integer p. Thus U is not wandering, which is a contradiction. Hence all limit functions in U are constants. Obviously, any constant limit is in J(/) U {oo}. Next we shall prove that all constant limit functions are in

E'(f) U {oo} (see Bergweiler-Haruta-Kriete-Meier-Terglane [40]}. Assume that there exists a E C\E'(f) such that -+ a in U. From Theorem 4.1 we see that U and all Un = r(U) are simply connected. By hypothesis, U n E = 0 and Un n E = 0 for all n E N so that /- 1 exists locally on all Un and can be continued analytically in Un to a univalent function. Hence rlu is univalent. We choose ~(Zo, R) C U. Without loss of generality, we shall assume that a = 0. We chooser> 0 such that ~(0, r) n 'l\{0} = 0, and assume r•(~(Zo, R)) c ~(O,r}\{0}. From the Koebe 1/4 theorem we obtain

r•

Denote Q = {z: Rez < logr} and g,.: branch of the logarithm. Then

~(Zo,R)-+

Q by 9k

= logr•(z}, for some

Since Q is simply connected, the inverse function of g,. can be continued analytically to a singl~valued function h,. in Q, that is, h,. : Q -+ C and h~c(g~c(z)) = z for z E ~(Zo, R). We claim that h~c is not univalent in Q. Otherwise, let = g~c(Zo). Then Wk E Q, i.e., Rew,. < logr. By the Koebe 1/4 theorem we have

w,.

CHAPTER 4. THE COMPONENTS OF THE FATOU SET

76

By Corollary 2.2, there exists a periodic cycle {p,q} with A(O,r) Then h,.(Q) n {p,q} = 0. It follows that

n {p,q}

= 0.

~lhl,(w~c)l(logr- Rew,.) ~ M,

where M = min{l.zo-pj, lzo-qj}. Since Rew,.-+ -oo we conclude that hj,(w,.)-+ 0. But h,.(g,.(z)) = z, so hj,(w,.)gj,(Zo) = 1 and this gives a contradiction. Thus h,. is not univalent in Q. As Nevanlinna [163, p. 283] did, we deduce that there exists E N such that h,. is periodic with period 21rl1ci and h,. is univalent in the half strip {z : Re z < log r: c < I M z < c + 2wl,.i} if c is real. If l,. -+ oo, we will obtain a contradiction as before. Hence we may assume that f+ oo and, restricting to a subsequence if necessary, we may suppose that = l for all k. Now we consider G1c = exp(g,.fl), r' = r 111 and the function

z,.

z,.

z,.

H,.: A(O,r')\{0}-+ C defined by !l~c(z) = h~c(llogz). Clearly, H~c(G,.(z)) the Koebe 1/4 theorem again we obtain

= z for z E A(Zo, R) and using

so that Gl,(zo)l -+ o. Since H,. is univalent, 0 is not an essential singularity of H,.. Suppose that o is a (simple) pole of H,.. Then H,.(A(O,r')\{0}) contains a neighborhood of oo. But every neighborhood of oo contains periodic cycles of/, which, as noted above, cannot be contained in H~c(A(O,r')\{0}) as soon as they have an empty intersection with A(O, r ), a contradiction. Hence H 1c has an analytic and univalent continuation to A(O, r'). Define z,. = G~c(zo). As before, we deduce from the Koebe 1/4 theorem that

IHHz~c)l $ r' ~~z~cl" Since lz~cl -+ 0, we have IH.(z~c)l ~ 8M/r' for sufficiently large k. This contradicts H.(z,.)GI,(Zo) = 1 and therefore completes the proof. •

4. 7

Growth of functions

In this section, we will study the growth of the function in its stable domains, especially in Baker domain. We first establish the following result (see Baker

(19]).

77

4. 7. GROWTH OF FUNCTIONS

Theorem 4.23 If transcendental entire function f has a Baker domain D, then loglr(z)l

Further, for any z, z'

E

= O(n),

(z

E

D,n-+ oo).

D,

= 0(1), exist a curve r c D

log lr(z')l-log lr(z)l

If D is invariant, then there constant, K, L such that and

(n-+ oo). that tend! to oo, and positive

t(r) c r

Klzl < 1/(z)l < Llzl,

z E r.

PROOF. We may supp011e that D is invariant. By Theorem 4.4, D is simply connected. Take a finite a E 8D. For any Zo E D, we put z,. = r(Zo), n E N. The hyperbolic distance PD(Zo, zl) of points %(), Z1 with respect to D is certainly positive since r(Zo) -+ oo. Because the map does not increase the hyperbolic distance, so that PD(Zn, Zn+l) ~ PD(Zo, z1) for all n. The function w = log(z- a) maps D onto a simply connected domain G which contains no vertical segment of length > 21r. Since any w E G has a distance at most 1r from 80, it follows from the Koebe's distortion theorem that the Poincare metric Paldwl of G satisfies

1

1

Pa(w) ~ 4d(w,8G) ~ 47r' Thus forb, c E D, PD(b, c) = 6, b' = log(b- a), c' = log{c- a), we have

6 = pa(b', c') which gives

~ 4~ Re(b' - c'),

lb- al ~ lc- alexp(47r6). In particular, we have IZn+l - al

< Klzn - al,

where K = exp(411'PD(Zo, z1)), and so lz..-al < K"IZo - al, which implies log lz..l = O(n) as n-+ oo. Further if z' is another point of D and z~ = r(z'), then L- 1
1 and lr(a)l < A for all n. Let "Y be a path in D that joins a to a point b E D, such that the minimum modulus satisfies m( bl, /) > lbl 2, and also lbl > 2A. Now /b) joins /(a) to some point at least beyond jzl = lbl and by induction we see that contains inside lzl < A and some other point Zt such that lztl = lbl > 2A. Then for any subsequence n~r, cannot have a constant limit on "Y and so the only possible limits in D are non-constant. By Theorem 4.18, there is a component D 1 of F(l) and a positive integer p such that fP(z) maps D 1 univalently onto D 1 and for some increasing sequence of integers n1r one has f""•(z)-+ z in D1. Moreover IN(D) C D1 for some N. By the minimum modulus theorem, D1 is unbounded. Now IN(-y) contains a'= /N(a) with la'l 2A and /N("Y) c D1. Let r be the part of JN('y) from a' to the first intersection with lzl = lbl. Then Ji(r) always contains points of mudulus at least lbl 2 > 2lbl, provided j ~ 1, and this is true in particular for j = pm, m = 1,2, .... Thus it is impossible that f""•(z)-+ z on r. This is a contradiction. •

rb>

r

r•

The following example in Bhattacharyya [44] shows that the' above result is best possible. Example. Let /(z) = oos{(~z + !w2) 112 }, 0 < f < (311')11 2 , which has order 1/2, type f and has an attractive fixed point at 0. Thus 0 belongs to a component D of F(f) in which r(z)-+ 0. It is easy to check that D contains [-1, oo) and is therefore unbounded and invariant. If the Fatou set has some special type, we can also get some estimates on the growth of the functions (see Qiao [175, Theorem 3], cf. Bhattacharyya [45)).

4. 7. GROWTH OF FUNCTIONS

79

Theorem 4.25 Let I be a transcendental entire function. If the angular domain O(Zo, 9,6) = {z: Iarg(z-ZQ)-91 < 6} is contained in F(f), then for any 6' E (0, 6) we have 1/(z)l = O(lzl,.16 ), z E O(Zo, 9, 6). PROOF. By hypothesis and Theorem 4.4, there exists an unbounded and simply connected component D of F(f) such that f!(Zo, 9, 6) CD. It is easy to see that

(e- 18 z- e-iBZo)" -1 iBZo)" + 1

11'

= h(z) = (e-iBz- e

w

(u

= 26)

maps f!(ZQ,9,6) conformally to .6.. Let a= h- 1 (0) E f!(ZQ,9,6). By the Riemann mapping theorem, there exists a conformal mapping w = g(z) from D to .6. such that g(f(a)) = 0, g'(!(a)) > 0. Thus by the Schwarz theorem, 19 o I o h- 1 (w)l ~ lwl,

(lwl

< 1).

Now Koebe's distortion theorem implies that 1

l(g- (w) -f(a))g'(f(a))l

~

(

1

~~~1) 2 '

(lwl

< 1),

and so,

1 ll(z)l ~ ll(a)l + lg'(/(a))l Denote z - Zo

= re"", .X = sin~ > 0. h

I

2

(z)l

1 ( 1 -lh(z)l) 2 ,

z E O(.zo, 9, 6).

(4.3)

Then for z E f!(Zo, 9, 6),

1- 2cos = h- 1 of o h and A = {{ E 8~ : oo 'f. C(h,{)}, where C(h,{) is the impression of the prime end COfT'e31J01lding to{. If /3 is a non-empty open arc of 8~ in A, then (i) tJ> is analytic on {3, and {ii) tJ> maps /3 bijectively to an open arc tl>(/3) C A. PROOF. Suppose that z,. E ~. z,. -+ { E 8~, ti>(Zn) -+ ~. We may assume that lim h(z,.) = w exists, otherwise we need only to consider a subsequence. We have w E 8D C J(/), w :f:. oo. It follows that f o h(zn) -+ f(w) :f:. oo. Since f o h(zn) ED while /(w) E J(/) we see that f(w) E 8D. But then ti>(Zn)-+ 8~. Thus every limiting value of is analytic on /3 with tl>(/3) c 8~. The reftection principle applies and shows that tJ> is univalent in a neighborhood of each { E /3. Hence tJ> : /3 -+ tJ>(/3) is a bijection. If ~ = tJ>({), { E /3, then h o : ~ -+ ~ is analytic, tJ>(O) = 0. If (/3) C {)~, then (z) = cz for some constant c of unit modulus. Lemma 4.9 Ify maps D conformally to~ so that as z -+ 0 along the positive axis we have g(z)-+ -1, then arg(l +g(z)) - ~ arg z-+ 0 as z-+ 0 in Iarg zl < ~ -6, for any 6 > 0. In particular, if z-+ 0, arg z-+ 0 in D then arg(l + g(z)) -+ 0.

CHAPTER 4. THE COMPONENTS OF THE FATOU SET

82

PROOF OF THEOREM 4.28. By Theorem 4.5, if Dis not a Baker domain, then

there are three cases as follows:

(a) Dis a SchrOder domain or BOttcher domain. D contains an attractive fixed o locally uniformly in D. point o of f, and (b) Dis a Leau domain. o locally uniformly in Dando E 8D, o "f: oo, /(o) = o, /'(o) = 1. (c) D is a Siegel disc. flo is analytically conjugate to a rotation of the unit disk about the origin by an irrational multiple "Y1r of 1r. Convergent subsequences of have non-constant limits. D contains a fixed point o such that /'(o) = ei"Y•. I is univalent in D. Let h : A - D be a conformal mapping. H D contains a fixed point o of f we choose h so that h(O) = o. The function ~ = h- 1 of o h satisfies ~A) C A, ~(0) = 0. We define A= {{ E 8A: oo;. C(h,{)}. Clearly, A is open and A :1: 8A since D is unbounded. We need to prove only that A is empty in cases (a), (b) and (c). In the case (a), if A :1: 0, then it contains an arc {3 of maximal length (S 2w), {3 :1: 8A. Lemma 4. 7 shows that ~fj) is also an arc of A. Since f is not univalent in Din the case (a) by Theorem 4.13, ~cannot have the form cz. We thus obtain from Lemmas 4.7 and 4.8 that ~(/J) has a length strictly greater than that of {J, which is impossible. Hence A = 0. In the case (c), since flo is a homeomorphism, it follows that ~z) = ze'"Y•. H A :1: 0, then there is some open arc fJ C A and by Lemma 4.7 1 ~"({J) CA. Clearly 1 ~"(/J) = 8A, since 'Y is irrational. This is impossible so A= 0. In the case (b), we may assume that o = 0, so that the Taylor expansion off at 0 has the form

r -

r-

r

LJ:.

U:.

f(z)

=z +

:E 00

o;rl,

Om+1 :1:0

(4.7)

j•m+l

for some m E N. We may assume in (4. 7) that Om+l to rotate J(/) and D about 0. Let l;={z:argz=

(2j- 3)w m

},

< 0, otherwise we only need .

lSJSm+l.

By Theorem 2.8 there are m different components D; of F(f) that satisfy f : D; - D;, 0 e 8D;, 0 in D;. Moreover, in the sector S; of angle 21rjm bounded by l;, l;+l• there is a Jordan curve B; which is tangent to l; and l.H 1 at 0. The interior of {0} U B; is a domain H; C S; such that /(H;) C H;. The D; are numbered in such a way that H 1 CD;, we may assume that D 1 =D. For any zeD;, r(z)- 0 in the direction argz = " which is the bisector of S; at

r -

i:

0.

Suppose that the strip domain A satisfies the following two conditions:

< 8 < 1rj2 there exists u(8) such that A(8) = {w = u + iv: u > u(9),1vl < 9} CA.

1. For every 8 with 0

4.10. FUNCTIONS WITHOUT BAKER DOMAINS

83

2. There exist sequences w,. = u,. + iv,. and w~ = u~ + iv~ in 8A such that Uo < ... < Un -+ oo, Un+l - u,. -+ 0, v,. -+ 7r /2, and ~ < ... < u~ -+ oo, u~+l - u~ -+ 0, v~ -+ -71"/2. Suppose that z(w) maps A one-to-one conformally into the strip {z = x+iy: IYI < 71"/2} so that limu..... oo Rez(u) = oo. Tlten lim (y{w}- v) = 0. Re w-+oo, wEA{II)

Now we take z = h(t) so that -1 corresponds to 0. The inverse map is t = g(z). If A :f: 0, then A contains some arc {3. By Lemma 4.7 {3,. = c/J"({3) C A and 4J is a.na.lytic on each {3,.. Let w(6, t) be the harmonic measure of 6. The maximum principle shows that w({3,., c/J"(t)) ~ w({3, t) for n E N and t E A. Take t 0 such that w({3, to}3/4. Thus t 0 lies on the circular arc 6({3) which lies in A, passes through the endpoints of {3 and makes an angle 71"/4 with {3. It follows that c/J"(t0 ) lies in the domain U,. bounded by {3,. and 6({3,.). Now Zo = h(to) E D and r(Zo) -+ 0 tangent to the positive real axis. Thus by Lemma 4.9, t,. = c/J"(t0 ) = h- 1 (r(Zo)) = g(r(Zo)) approaches t = -1 in the real direction. It follows from t,. E U,. that -1 E {3,. for large n. Thus there is a maximal arc u C A which contains -1. Since Dis unbounded, u :f: 8/l.. By Lemma 4.7 4J is analytic in u. It follows from cjJ( -1) = -1 that c/J( u) C u. Also 4J is bijective on u. By the reflection principle, we may continue 4J so that it is analytic in the domain V = uU{C\8/l.}. For t E A, c/J"(t) -+ -1 in a direction tangent to the positive real axis, and for t E Jnt(A}, c/J"(t)-+ -1 in the real direction, cjJ(Jnt(A)) C Jnt(A). By considering the local iteration of 4J near -1, the Taylor expansion of 4J near -1 is of the form cjJ(t)

=t +

.X(t + 1} 3 + ... ,

.X

< 0.

(4.8}

If 8/l. \u is a set of single point a, then 4J would be meromorphic in C\{a} and, for each 8 E u, cjJ(t) = 8 would have only one solution in u and hence in the plane. Then t/J would be a Mobius transformation, univalent in fl.. However, I is not univalent in Din the case (b) by Theorem 4.13, a contradiction. Thus 86\u contains a continuum. Since t/J(B) C B, the iterates {t/J"} form a normal family in B, and so c/J"(t) -+ -1 for all t E B. However, fortE u, t/J: u-+ u is one-to-one and lt/J(t) + 11 > It+ 11, a contradiction. Thus A is empty. The proof is complete. •

Remark. If 8D is a Jordan arc in the proceeding theorem, then by Gross Star Theorem we can prove that I is univalent in D.

4.10

Functions without Baker domains

In this section, we shall show that there are no Baker domains for some functions.

84

CHAPTER 4. THE COMPONENTS OF THE FATOU SET

Theorem 4.29 Let

I

be a trarucendental entire function. Iff E B, then F(/)

does not contain Baker domains. This theorem follows from the following proposition ([82]).

Proposition 4.5 Let I be a trarucendental entire function and F(l), then the orbit o+(z) cannot tend to oo.

I

E B. If z E

PROOF. We first describe a logarithmic change of variable that is one of the basic tools for investigating entire functions in the class under consideration in a neighborhood of oo. This device goes back to Teicbmiiller [201]. Let all the singular values of the function 1-1 be contained in the disk Dr. Denote by H the exterior of this disk, and by G the complete inverse image 1-'(H). Let V be a component of G. Then Vis simply connected and bounded by a simple analytic curve going to oo at both endpoints. By Corollary 4.2, all the components of F(/) are simply connected. Now The function t/1 = ln I maps V conformally and univalently onto the half-plane P = {z: Rez >lor}. Let r be large enough so that 11(0)1 < r. Then 0 ~ G, and the function exp(z) is univalent on each component of U = exp- 1 G. We consider the commutative diagram I

u ___ i _

p

G _ _I __

H

The function j maps each component W of U conformally onto the half-plane P. We consider the mapping 4): P- W inverse to j. Then for z E W, ll.(j(z), Rej(z)-lnr) C P. It follows from the Koebe 1/4 theorem that the set W contains the disk of radius W(i(z))I(Rej(z) -ln r)/4. On the other hand, since exp(z) is univalent on W, W does not contain vertical segments of length 21r. Consequently, A

ll'(z)l ~

1 7r (Re l(z) -ln r), 4 A

z

e W.

(4.9)

Suppose that there exists a point Zo e F(l) such that z,. = r(Zo) - oo. Then there exists a disk ll.(Zo, s) of radius s about Zo in which the sequence {r} tends uniformly to oo. Consequently, all the Dn = r(ll.(Zo, s)) from some point on are contained in G. We may assume that Dn C G for all n. Let A be a component

4.11. FUNCTIONS WITH WANDERING DOMAINS

85

of the set exp- 1 (A(:o,s)), and let An =/"{A). Then exp(An) = Dn so that An C U and oo uniformly in A. Suppose that ( E A, (n = f'(() E An, and R.. is the radius of the maximal disk centered at (n that is iiUICribed in An. By the Koebe 1/4 theorem, R..+1 ~ R..lf'((n)l/4. Since &j((n) -+ +oo, it follows from (4.9) that lf'((n)l -+ oo. Consequently, R.. -+ oo. But then some domain An C U contains a vertical segment of length 271", a contradiction. The proof is complete. •

&i"-+

Remark. Rippon and Stallard have recently made a partial extension of Theorem 4.29 to meromorphic functions: Let Bn be the class of meromorphic functions such that the finite singularities of 1-n form a bounded set, then f has no Baker domain of period n.

4.11

Functions with wandering domains

Wandering components of the Fatou set will be called wandering domains. By Sullivan's theorem, there is no wandering domain for any nonlinear rational function. But for traiU~Cendental functions, the cases are very different. Next we shall discuss the three classes respectively.

4.11.1

Functions for the class E

For the class of traiUICendental entire functions, we have seen that any multiply connected components must be wandering. The first example of a multiply connected wandering domain was constructed by Baker [14]. Baker [17] proved the following result. Theorem 4.30 For any p such that 0 $ p $ oo there is an entire function of order p, which has multiply-connected wandering domains. In particular, when p = 0, there exists a transcendental entire function which has wandering domain, of infinite connectivity. PROOF. Let k, (n E N) denote any increasing sequence of positive integers and C a constant, such that 1 {4.10) 0 < < 4e2'

c

Suppose further that no is a positive integer and

r1

a number such that

{4.11}

CHAPTER 4. THE COMPONENTS OF THE FATOU SET

86

Denote by r,., n E N, numbers such that

r,.+l > 2r,.,

1~ n
2r,.,

n

~

no.

(4.14}

For we may take any n ~ no and assume in the induction that r,. > 2r,._ 1 > 2'rn-2 > · · ·, so that (4.13} gives

Define the entire function g(z} by

where the product converges uniformly in any compact region of the plane, since r; > rt2i-l. Now (4.15} rn+l < g(r,.) < ern+lo n ~no. since

g(r,.) rn+l

=.

TI

]=n+1

(1+ (r"),.") < {1+!}(1+!)··· 2i-l, that

lg(z}l Since g(r) have

~ g(1} ~ 011'(1 +rj 1 ) < e2C < ~·

= M(r,g}, it follows that V(•) := logg(e•) is convex and for s > 0 we V(2•} - V(O}

Hence V(2s}

~

~

2V(•)- 2V(O).

2V(•) - V(O}, which gives, using (4.13), g(r2)

Putting r

(4.16)

~ (~~it

> 4(g(r))2.

= r,. and noting (4.15} gives 1

2

2

4g(r,.) > rn+li

(4.17}

4.11. FUNCTIONS WITH WANDERING DOMAINS

putting r

87

= r~12 gives (4.18)

We remark that there is an integer n 1 such that rn+l

An= {z: r~ ~ lzl ~ r;Z1 }. For z E An and r

n

> r! for n > n 1 • Set >

n1.

= lzl, .!(2_

n

J~n

J

J

For n > n2 both x = ~/r1 ~ r/rn+l < r~!{ , j > n, andy= r;/r < r; 1 , j ~ n, are so small that log{(1 + x)/(1 - x)} < 3x and log{(l + y)/(1 - y)} < 3x. Thus for n > n2 we have 2

log g(r) jg(z)l

~ 3L

..!.:_ + 3

j>n r1

L

r,

;~n r

< 6 (-r- + rn) < log4, rn+l

r

if n 2 is large enough. Thus jg(z)l

1

> 4g(lzl),

z

E

An.

(4.19)

For z E An, the maximum modulus theorem and (4.18) give lul

~ u < r;z2.

while the minimum modulus theorem and (4.17), (4.19) give

Ig (z)I

. { 49 1 ( 2) 1 1/2 } - 1 ( 2) 2 rn '4g(rn+l) - 49 rn > rn+l·

~ mm

Hence g(An) C An+l· This implies that gk(z)-+ oo uniformly in each An, n > n2, ask-+ oo. Thus An C F(g). Since J(g) is not empty, the bounded component of the complement of An meets J(g) for all large n. Hence the component Un of F(g), which contains such an An, is not simply-connected. By Corollary 4.1, Un is wandering. To complete the proof it remains to show that g can be made to have any prescribed order of growth. For a given positive constant p, take k.. = [r~]. Then the number of zeros of g(z) in lzl ~ t (rn ~ t < rn+l) is

n(t)

Thus for

Tn ~

r < Tn+l•

logM(r,g)

=

logg(r)

= k1 + · · · kn = O(r~).

CHAPTER 4. THE COMPONENTS OF THE FATOU SET

88

On the other band,

logg(2r,.)

> k,.log2 > [r!Jlog2.

Thus g is of order p. The case p = oo is similar. For the case p = 0, we modify the above construction. Take C, to satisfy (4.10), (4.11) with k 1 = 1, (4.12). Define r,.+l = (;2 ( 1 +

for

~:)

2 ••• (

~=)

1+

no. r1t ... , rno

2

(4.20)

n;:: no. By induction it follows from (4.10)-(4.12) and (4.20) that > 2r,., n E N,

(4.21)

> 4r! > (n + l}r.. -1. n;:: no.

(4.22)

rn+1

and indeed rn+1 Define

/(z)

= C2

fr (1 + r;~)

2

(4.23)

t

j•1

where the product converges uniformly in any compact region of the plane. Set s,. h(z)

n+1 =-rn+1• n+ 2

= //'((z)) = 2 z

E ;. z+r;

1 -.

1

It is easily seen that all zeros of h are real and negative, and h is decreasing on each interval (-r;+l.-r;). Now h(-1,.-t)

> 2(n + 1) + 2 r,.

L

1

J"

< exp( _.!._) r,.

= 1 + 0(2-")

as n-+ oo, by (4.12). Thus /(•.. - 1 )

a,.

__!..2 + O(n-3 ) < 1

= 1-

for large n. Let

p1 = n (1- _n_ • r,.)2 n+1 0

1 0. Then there exists a mtional function R(z) with poles in P such that 1/(z) - R(z)i < e, z E K. Lemma 4.11 ([94, p.131}) Suppose that A is a closed set in C and that f is a function defined on A. Then f can be uniformly approximated on A by meromorphic functions without poles in A if and only if f can be uniformly approximated by mtional functions on each compact subset of A. Lemma 4.12 ([94, p.137}) Suppose that A is a closed set inC and that z1, Z2 lie in the same component ofC\A. Then, for each function h(z) meromorphic inC with a pole at z1 and for each e > 0, there exists a function h* meromorphic inC that is analytic at z1, has a pole at z2 , has no other poles except those of h, and for which Jh{z)- h*{z)i < e, z EA. Lemma 4.13 ([94, p.140/) Suppose that A is a closed set inC such that C\A is locally connected at oo. If the meromorphic function h has no poles on A, then for each e > 0, there exists a mtional function R(z) with poles outside A and an entire function g( z) such that

Jh{z)- (R(z)

+ g(z))J < e, z EA.

Theorem 4.31 Let kEN. Then there are four meromorphic functiom f, (i 1, ... , 4) such that

1. F(ft) has a k-connected bounded wandering domain,

2. F(h) has a k-connected unbounded wandering domain, 3. F(/3) has a bounded wandering domain of infinite connectivity,

=

CHAPTER 4. THE COMPONENTS OF THE FATOU SET

92

4· F(/4 )

ha8 a unbounded wandering domain of infinite connectivity.

PROOF. Define

'71

Let

1/>(z)

n-1

= 0,

'7n

= E Em, n > 1,

Bo = ~(-5, 1), Ba

= ~(-5, ~>·

1

- 8 0,

= 2(z + 5) 2 -

5 : Bo

x(z)

= -5.

We distinguish the proof into four steps. 1. The construction of ft. We describe the construction in the case when k = 3 (cf. figure 4.1). It will be clear that it can be adapted to any other value of k. Let 1 a,= 10n- 7, b,. = lOn- 3,6n = '1n + 2E'n•

Set

Bn =

~(0,

Gn

{ z : lz - IOn+ 51

Gn,l Gn,2 Ln

= = =

lOn}, 1

< 4- f1n, lz- bnl, lz- a.. I > 2 +'In},

{z:% E Gn,d(z,8Gn} > £~}, {z:% E Gn,d(z,8Gn} >En}, {z: z E Gn,d(z,8Gn} > 10-n-l}

for n E N. ThUB Gn, Gn,lt Gn,2 are triply-connected domains such that, for

m :f: n,

Gn,2 C Gn,l C Gn, Gm () Gn Write

C1n

= IOn -

9 + 6n, C2n

= a, +

= 0.

I

I

2 + 6n, C3n = bn + 2 + 6n,

so that Ctn 1 02n, Can lie in each of the components of Gn \Gn,l· Denote 1 a-= {z: z E Gt,d(z,8Gt) > 3}.

Note that "(z} = z + IO maps G,.,2 univalently onto Gn+l and

f/Jn(G*)

c

Gn+l,2•

n E N.

By Lemma 4.10, applied to K = Bo U 801 U ~. there is a rational function Rt (z) such that IRt(z) -1/>(z)l < z E Bo,

ef,

4.11. FUNCTIONS WITH WANDERING DOMAINS

93

.~

Figure 4.1: The construction for ft.

< e~, z E GW, IR1(z)- 4>(z)l < e~, z E 8G1,

IRt(z) -1/J(z)l

and R1 has poles in {at.b1,cn,c2t,C31,oo}. For n > 1 there is a rational function Rn such that IR..(z)l
5 on Gn,1· Thus ft(z) defines a transcendental meromorphic function inC and has poles in each Gn. Note that Bo is It-invariant, and so Bo C F(ft). Now /f(G•) C Gn+l,2• hence by Mantel's theorem, G• C F(ft), fr(z) -+ oo as n -+ oo for z E H which is a component of F(ft) containing G•. Since ft(aGn+l) C Bo, /j"(8Gn+1) C Bo for all m, it follows that Jr(H) C Gn+t. and His wandering. Next we prove that H is triply connected. Let g(z) = /(z) -1/J(z). By (4.28) and (4.29) and Cauchy's inequality we have lg'(z)l
~r" > r,.e'" on {z : lzl = 1 + r,.} C N,.. This growth also implies that Is fl. 'P. Thus N,. is wandering and obviously bounded. By the size of 1/sl in U1 and the maximum modUlus theorem, we know that h has poles. Since Ia: Uo-+ A(0,3/4), Ia has an attractive fixed point p e A(0,3/4) and Uo is part of a component No of F(/a) in which If-+ p. Each V(n,k) is mapped by fa into Uo and so belongs to a component N(n, k) in which If -+ p. Thus the connectivity of Nn is at least n + 2. Since Nn+l is bounded, fa has no poles in N;.. Thus Is : Nn-+ N,.+l is a branched map and 8/a(N,.) = fa(8N,.). H N,. has connectivity d < oo then 8Nn consists of d continua 'Yt.···• 'YtJ· Then fa(8N,.) has at most d components so that Nn+l has connectivity at most d, and hence the connectivity of all N,.. for m ~ n must also be at most d, a contradiction. Thus the connectivity of each Nn is infinite.

fa

4. The construction of (n ~ 2} by induction,

Yt

f•·

=

We construct sequences ·

0,

Vt = 2,

Jln, V,. (n

~

1), and

kn

4.11. FUNCTIONS WITH WANDERING DOMAINS

2Yn-1 + k..-

1

, 10 1 1 2Yn-1 + k.. + lO'

Yn I

Yn

k..

97

>

exp(y~-1)

> exp(k.,_l).

Thus 112 > 7, (Jin) and (y'n) are increasing. We have 11 1 1 n+l 1 1 tin- Jln = 102n - 5' Yn+l- Yn ~ k..+1- lO > 2 - 10 . Thus Jln+l > J/n. Now set 1nn

= (Jin + tl~)/2 =2m..- I + k.,

and a(1,p) = 2p+i, a(n+ 1,p) = 2a(n,p) +ik.., pEN, so that Ima(n,p) = m.,. Write

U(n,p) = A(a(n,p), ~), V(n,p) = A(a(n,p),

1 ), Vo = A(4i, 1), 20

CXl

Dn = {z: Yn ~ lmz ~ y~}\

U U{n,p). p=l

We define g(z) on the closed set A= Dn U Vo U Un,p V(n,p) by

g(z)=

{

2z + ik..+t. : (z-4i) 2 /2+4~,:

z E Dn, zEVo, 4a, : z E Un,p V(n,p),

which is analytic on A. By Lemma 4.11 there is a meromorphic function h,(z) which has no poles in A such that 1 lf.e(z)- g(z)l < , z EA. 10 We may assume that / 4 has infinitely many poles. By our constructions, h. : Dn -+ Dn+l• so that Dn belongs to a component Nn of F(/4) in which /"' -+ oo. If some Nn meets Nn+l then Nn is a Baker domain. For zo = im.. E Dn and E(z) = e•, the iterate z. = J:(zo) satisfies

Imz. > k.. + s > E•(y~) > E•(lzol). This rate of growth contradicts Theorem 4.23. Thus each Nn is an unbounded wandering domain of F(/4)· Similarly as in the proof for [3, we can deduce that Nn has infinite connectivity. The proof is complete. •

CHAPTER 4. THE COMPONENTS OF THE FATOU SET

98

Question 4.3 Are there transcendental meromorphic functions which po3sess at least two properties described in the abotJe theorem?

Based on the above two subsections, the following question arises naturally. Question 4.4 Can one find a function in P with wandering domains 'I

4.12

Functions without wandering domains

One of the most important results in the iteration theory of rational functions is Sullivan's No Wandering Domain Theorem: any nonlinear rational functions do not have wandering domains. This solved the known Fatou-Julia problem. It is natural to seek some transcendental functions which do not have wandering domains. A basic observation is to extend Sullivan's theorem to the class of transcendental meromorphic functions of finite type. Theorem 4.32 Jihnctions in S do not have wandering domains.

Remark. This result wa.s proved by Ba.ker-Kotus-Lii [23). It ha.d been obtained earlier by Ereme~Lyubich (82, 85) and Goldberg-Keen (98) for the class t n S and by Keen (134), Kotus (140], and Ma.kienko [153) for P n S {and, in fact, for the corresponding class of ana.lytic self-maps of c -). For some other subclasses of S, this ha.d been proved by Baker [16) and Deva.ney-Keen (74). To prove this theorem, we need the following lemmas. Lemma 4.14 Iff E

tnB,

then all the components of F(f) are simply connected.

This lemma follows from Theorem 4.1 and Proposition 4.5. Lemma 4.15 Let f E 1', then F(f) has at most one multiply connected component. Jihrthermore, if the multiply connected component exi.Jts, then it should be

doubly connected and it separatu 0 and oo.

PROOF. This is proved by Baker [18) and Keen [134). Sup~ that -y1 , "Y2 are disjoint Jordan curves in F(f). To prove the conclusion, we need only to prove that the region U bounded by "Y1t "Y2 contain no points in J(/). Suppose on the contrary that U n J(/) :/; 0. Then for arbitrarilly small positive number e, r(U) covers c• except for an e- neighborhood of 0 and oo for n > no( e). Now if some has a. non-constant limit function t/1 in the component Dt of F(f) which contains "Yt, it follows from Theorem 4.18 that for large n, rbt) is close to the compact set ~~ jJ (/N ( -y1 )) C c•, where p is the smallest positive integer such that p maps

r•

4.12. FUNCTIONS WITHOUT WANDERING DOMAINS

99

itself to the component ofF(/) which contains IN (D1 ). By the covering property of r(u), it follows that for large n, rb2) contains points near both 0 and oo, has no constant limit functions in the component D 2 of F(/) which so that contains 'Y'l· But then rb2) must also approximate a certain compact subset of c•, as is the case for rb1 }. This again contradicts the covering property of r(U}, as n -+ oo. Thus, for sufficiently large n, we have either llnl < E on 'Yt. 1r1 > 1/e on 'Y2 or lrl 0 such that X1 E F(f,..). Since all the components of the Fatou set are pr&-periodic, and F(/p) has only one cycle of periodic domains {D;}j~. there exists a non-negative integer m such that J;:'(xl) and /,..(-1) both belong to some domain D;. Therefore,

J;+"P(xt)- xo S 0 ( n - oo). On the other band, it is obvious that J;(x) > x for all x > 0 so that J:'p+"{x1 ) > x 1 > O(n = 1,2... ), a contradiction. Therefore, R+ C J(J,..). It is easy to see that the curve x = -11 ctg 11 consists of infinitely many components (this can be seen from the figure of the curve x =ctg y), and there is one and only one such a component in each horizontal strip {z : k1r < Imz < (k + 1)1r} (k = 0,±1,±2... ). We denote by f 1 and f2 the components which lie in {zl1r < Imz < 211'} and {zl - 211' < Imz < -11'} respectively. Obviously, one endpoint of r 1 tends to 00 towards the right asymptotic line 11 = 211'' and another endpoint of

111

4.14. THE NUMBER OF THE COMPONENTS

r 1 tends to 00 towards the left asymptotic line y = 7r' r 2 is the symmetric curve of r1 with respect to the real axis. Hence r 1 and r 2 bound an unbound region H. It is easy to verify that /"(rl) = /"(r2) = JR+. On the other hand, JR+ c J(/"). By the complete invariance of J(/") we know that r 1, r2 c J(f,.). Thus F(f,.) has at least two components. It follows from Theorem 4.42 that F(f,.) has infinitely many components. Since F(f,.) has no wandering domain, from Corollary 4.1 we deduce that all the components of F(f,.) are simply connected. If J.l E (0, 2), then it is easy to see that -JJ is attractive. Thus J(f,.) #0. The conclusion (ii) is proved. (iii). If J.l E [0, oo), then /,.(x) > x for x > 0, and so f;:(x) --+ oo as n--+ oo. Thus by Proposition 4.5, x E J(f,.). To prove (iv) and (v), we .need the following lemma. Lemma 4.17 There exists a sequence of unbounded positive numbers {lin} and a corruponding sequence of positive integers {mn} such that Sm,. (lin)= -1. PROOF. By the definition of Bn(JJ) we have n-1

sn(l-')

= -exp{L s~c(l-') + n1-1}

(n ~ 2).

(4.35)

k=O

Note that, for any positive constant c,

s1(1-1) + so(JJ) + CJ.l

= c1-1- 1- e"- 1 --+ -oo

asJJ--+ oo,

(4.36)

this and (4.35) imply that s 2 (JJ) --+ 0 as 1-' --+ oo. By induction we deduce from (4.35) that (4.37) for any n ~ 2. Suppose on the contrary that the assertion of the lemma does not hold. Then there exists M > 2 such that, for any 1-' > M and any n ~ 2, sn(JJ) > -1. In fact, if for any M > 2 we can find JJ1 > M and n ~ 2 such that Bn(l-'1) ~ -1, then by (4.37) and the continuity of sn(l-'), we can get a JJ" > JJ' > M such that sn(JJ") = -1, a contradiction. According to (4.37), we may take a fixed 1-'o > M such that (4.38) From (4.35) we see that sn(l-') < 0 for all n ~ 2 and sa(JJo) 1-'o}· Combining these and (4.38) we get

sa(JJo) < s2(1-'o) and

= B2(J.lo)exp{s2(/.l0) +

CHAPTER 4. THE COMPONENTS OF THE FATOU SET

112

By induction we deduce that

-1

< Sn(J.&o) < · · • < B2(J.&o) < 0

for any n ~ 2. Thuslimn-oo Bn(J.&o) exists. Let a be the limit. Then by the above inequality,

-1 . of the transcendental entire function is less than 1/2 and ---'log=-M..,...,(~2r~,/~) -+Cy-00 4 -:(4.39} logM(r,/)

f

as r

-+

oo, then evei'y component of F(/) is bounded.

Definition 4.1 Suppose that Ct and C2 are constants satisfying C 1 A positive numberr is called to be normal (Ct.C2) for f(z) if

> 1, C 2 > 1. (4.40)

We denote by N(Ct.C2) the set of the numbers r which are normal (Ct.C2) for

/(z).

Remark. From (4.39} we see that, for any small number ro such that [ro,oo) C N(2,c+E).

E

> 0, there exists a positive

Theorem 4.47 Suppose that f(z) is a transcendental entire function of order Assume that there exist constants C2 > 1 and C 1 > ~ such that, for sufficiently large ro, [ro,oo) C N(Ct.C2)· Then any component of F(f) is bounded.

>. < 1/2.

This result was proved by Hua and Yang [~25]. To this end, we need to estimate the minimum modulus of the functions. Lemma 4.19 The assumptions are the same as in Theorem

-1.-11.

exist sequences R.., t" -+ oo such that, for all sufficiently large n, (i) M(R.., f) = R..+t.

(ii) R!+r.'!n < t" < ~+:' (iii} logm•(t",/}

> (1- (:m~t)S}logM(Jl!+r.'!:rr,f}.

Then there

4.16. BOUNDED FATOU COMPONENTS

117

PROOF. Since the order of f(z) is less than one, by Weierstr888 theorem we may write f(z) in the form /(z) = cz"g(z) (c :F 0), where

g(z)

=2

rr(l- -=-) 00

n=l

Zn

and k is a nonnegative integer. We define

= 2 IT (1 +-=._ ), 00

G(z)

n=l

rn

(rn

= lz..l).

It is easy to see that

A(G) = A(g) =A(/). From Boas [47, p. 39, formula (3.1.4}1 we have

m*(r,G)

= IG(-r)l $; m*(r,g) $; M(r,g) $

M(r,G)

= G(r).

Since C 1 > C1, we can take a constant a E (A, 1/2) such that

b :=a -logC2/logC1 > 0. Let

~(

r

)=

1,.

We denote k(a) = have

00

log 11- tl- coa(wo) log(1 + t) dt ~~ '

l-"':'(,..a).

(r ;::: 0).

Then from Kjellberg [1381 (cf. Denjoy [70J) we

k(a) log(~+ r} :5 ~(r) :5 10log(~+ r).

Note that

(4.42)

00

logiG(-r)l and

= L:togl1-; n=l

=L

n

I

00

log IG(r)l and

(4.41}

n=l

log 11 +..!:_I rn

E r;;a convergent, we thus obtain

1 ,.

oo

logG(t) dt tl+a

=

+...!...) L 100 log(1tl+a,.,. dt 00

n=l "

= ~

-a

L...., r n

n=l

Joe log(l + t) dt ...1:..

rn

tl+a

I

CHAPTER 4. THE COMPONENTS OF THE FATOU SET

118

the interchange of limiting operations being justified because everything is positive;

"" f

and

f

log IG( -t)l dt

=

tl+•

f"

L

n=l

logl1- ..LI

tl+• '"" dt

'"

= "

-a

roo logl1tldt tl+a '

L.J r,. } .L.

n=l

-.

because the integral is dominated by the convergent integral

oo 1

=

llogiG(-t)lldt tl+•

0

~1oo IJog It- tlldt L.J

0

-a1

n•l

= ~ L.J r" n=l

Therefore by (4.42), for any r

tl+a 00

llogll-tlldt tl+o

0

.

< R,

l =[- r"

log IG( -t)l- oo.9(•o) logG(t) dt

R

tl+a

r

JR

r

logiG(-t)l-coe(•o)logG(t)dt tl+a

= f)rn)-"'{111( .:._) n=l

111( R)}

rn

~ k(a)

L""

log(1 + L) r" r,.

rn

-

10

n=l

= k(a)log~(r)

L

log(1 + li) R." r,.

n=l

_ 10 log;!R).

Thus, for any positive integer n, we have

l l

R

log!G(-t)l- (1- ~)logG(r)

R

logiG(-t)l- coe(•a)logG(t) dt fl+a

r

+ ;:::

l

R

coe(•o) logG(t)- [1- ~] logG(r) dt

tl+•

r

k(a)logG(r) _ 10 togG(R) ,... R." 1

+ ( -k(a) + a(2n + 1)3 ;:::

dt

tl+e

f"

)

r1. Let

> 0, there exists r 1 > r 0 > 0 such that M (r, f) >

rK for all

(4.44)

and take We define

Rn+1,

M(R,., f)=

Note that log Bn+t/ log Bn

-+

1 as n

-+

n ~ 1.

oo. Thus when K is large, we have (4.45)

Define

h(n) = Note that, for any r

~

cS(n) =

.R!+2/(2n+l)'

1 4n(n + 1)

(4.46)

r 1 , by the assumptions we know that

This implies that logM(h(n) 1H(n),J) ~ C:(n)logh(n)/logC1 logM(h(n),J). Applying (4.43) for r

= h(n) and R = h{n)l+cS(n)

(4.47)

we get

f"(n) 1+• logiG(-t)l- (1- ~]logG{h(n)) dt tl+a

JMnl

> logG(h{n)) J(n) h(n) 0

where J(n)

1

= { a(2n +

1) 3 -

'

c!(n)Iogh(n)/logCl} 2 lO h(n)acS(n) ·

(4 .48 )

(4.49)

Now from {4.41), (4.43)-(4.47) we deduce that M{n) log h(n) ~ log{10a{2n + 1) 3 ).

From this we can easily check that J(n)

~

0.

(.4.50)

CHAPTER 4. THE COMPONENTS OF THE FATOU SET

120

Since

logm•(t,f)

=

log M(t, f)

= S

~

and so, for t

~

log lei+ klogt + logm•(t,g) log lei + k log t +log IG( -t)l, log lei+ k log t +log M(t, g) log lei+ klogt +logG(t),

h(n), we have logm•(t, f)- [1- ( n ~ )3 }logM(h(n), f) 1 2 1

~ logiG(-t)l- [1- ( n+ )3 ]logG(h(n)). 2 1

Combining this with (4.48)-(4.50) we have

1

h(n,••• "'

qm•(t,/)- (1- ~)locM(h(n),f) tl+•

la(n)

dt ~ 0.

•

Thus there exists tn such that (ii) and (iii) holds. PROOF OF THEOREM

4.47. Define 1

c(n) = 2 + n + • n ~ 1. 1 For the sequence Rn and tn in Lemma 4.19 we know that

Rn )ld4> ~log M(2R, /), 21r lo Jl"'l

Now for any R 1 < R2, by {4.42), we have

{R2 u2(-t, R)- C08{tra)u2{t, R) dt }Rt tl+o

Let Q(t, R)

= tn;(t, R) + m3{t, R)- M2(t, R)- M3(t, R) + k(a)t(M~(t, R) + M~(t, R)).

Take R 1 = r and R2 and (4.58) that

= R/2.

l

r

It follows from the above inequality, {4.55), (4.57)

2 R/ Q(t,R) dt

tl+o

>

-C( )logM(2R,f) a

na

'

CHAPTER 4. THE COMPONENTS OF THE FATOU SET

124

where C(a) is a positive constant. Set H( R)

r,

and

= C(a) los M(2R, /) ,.- + ro lll/2 Q(t, R) dt JlO r tl+o

G(r, R)

= H(r, R) + lO(M2(r, R) + M3(r, R)).

Then for 0 < r < R/2, we know that H(r, R) > 0 and rG'(r,R)

= aH(r,R) + M2(r,R)- tn;(r,R) +M3(r,R)- m;(r,R).

Now by (4.56), G(ro, R) > C(a)(M2(ro, R) + M3(ro, R)) 2: C(a)v(ro) log 2. Hence

L

ll/2 G'(t, R) dt ,.., G(t,R)

< l -


R 1 ,

< {M-t(R,/'")}X(o).

Thus for all sufficiently large r with rl/coa(fto}

= M - l(R,/'")

(4.69)

CHAPTER 4. THE COMPONENTS OF THE FATOU SET

128 we have

It follows that there exists

such that

M(t.r>

=•·

Now by the induction hypothesis there is a simple closed curve r m in the annulus

t S lzl < tk = •.

r m• from (4.69) we get

IJ"'+l(z)l

~

m•(a, f)

M(a,f)- 0, we chooee n so that 2-"- 1 .! < 1, then conformally conjugate to T112·

Theorem 5.4 If 0

J(T~)

is a Cantor set and each

T~

PROOF. Since 0 is an attracting fixed point, there is a holomorphic map t!i to a neighborhood U of the origin so that t/J~ = f/1~ o T~ o f/1~ 1 (()

is quasif/1~

from

= .>.(.

Therefore we can find an annulus A~ = {( : >.r < 1.r) 2 diam(Ka).

141

5.4. EXPLOSION However, by Theorem 4.18 we conclude that any convergent sequence constant limit function in F{/), which is a contradiction. The proof is complete.

r• has only •

Remark. Theorem 5.4 shows that the type (ii) exists. The existence of the other two types can be seen from the following two examples. By Theorem 5.7, the type (iii) does not occur for the function of the form (5.10). Example ((74]). If A E a, IAI > 1 or A = 1. Then J(T>.) = R, here T>. is of the form (5.4). In fact, if A > 1, then T>. maps the upper half-plane into itself. Now T>. also preserves the imaginary axis and we have T>.(iy) = iA tanh(y). The graph of Atanh y shows that T>. has a pair of attracting fixed points located symmetrically about 0. By the Schwarz lemma, all points in the upper (resp. lower) halfplane tend, under the iteration, to one of these two points. Hence neither the upper nor the lower half-plane is in J(T>.)· From the facts that T; 1(JR) C R, T>.(R) = JRU{oo} and T.{(x) > 1 for all x E JR it follows that the real line is in J(T>.), and each interval ofthe form ( 2k; 1 1r, ¥1r) is expanded over all of JR. If U is an open interval in R, then there is an integer k such that 7'f+l(U) covers U. Thus there exist repelling fixed points and poles of r:H in U. Therefore T>. =JR. If A = 1 or A < -1, we can similarly deduce that T>. = JR. Example ([21]). Let l(z) = (tan v'%) 2 • Then J(/) = JR+. In fact, for the function I, z E JR+ if and only if l(z) E JR+ U{oo}. Thus C\R+ is invariant and C\R+ C F(/). Near z = 0 we have

l(z)

= z + ~z2 + ... 3

and by the local iteration of 1 near z = o we have r(x) -. o as n -. oo for negative X near 0. Thus -+ 0 locally uniformly in C\R+. Since l(x) >X for X E (0,6), 6 > 0, we see that, for X E a+, we can have r(x)-+ 0 only if r(x) = 0 for some x, which holds only for a countable subset of a+. Thus J(/) = R+. Example ((74). By the same reasoning above we can prove that J(Ae" /(e" e-•)) = a- U {0}, A > 0. Remark. For IE M, if J(/) contains an isolated Jordan arc, then J(/) must be a Jordan arc or a Jordan curve. Furthermore, if the isolated Jordan arc is analytic, then J{/) is a straight line, circle, segment of a straight line or an arc of a circle (see. (194]).

r

5.4

Explosion

In this section we describe an "explosion" in the Julia set (cf. (71]).

CHAPTER 5. GEOMETRY OF JULIA SETS

142

Definition

s.a

An e:tplo6ion occurs at a parameter value for a family of functiom

whenever the Julia sets of the functions in the family change suddenly, when the parameter is reached, from a nowhere dense subset to all the plane. We first consider the family of E>. of the form (5.5).

Proposition 5.2 If ~(0) -+ oo or ~(0) is pre-periodic, then we have J(E>.) =

c.

PROOF. Since E>. e S, thus there is no wandering domain. It is easy to see that E>. has only one singular value 0 (asymptotic value). The conclusion follows from the assumptions and the classification theorem. •

Corollary 5.1

If~>

1/e

or~=

k1ri, k E Z, then J(E>.) =C.

Combining Theorem 5.5 and this corollary we see that~= 1/e is an explosive point of E>..

Corollary 5.2 There exists a sequence

~(j)-+

1 such that J(E>.w) =C.

PROOF. From the above corollary we see that the family {Gn(~) = ~(0)} is not normal in any neighborhood of 1. By Montel's theorem, there exist ~-values arbitrarily close to 1 such that GJ(~) = 2k1ri for any k E Z. But then GJ+l(~) = ~ = E>.(O) so that 0 is pre-periodic. Thus the proposition implies that J(E>.) =C.

•

By a similar argument as in the above proposition, we have the following result.

Theorem 5.9 Let f E 8 be an entire junction. Suppose that the orbits of all critirol points and asymptotic valuu off tend to oo. Then J(/) =C. For the function F>.(Z) = 1/(>.. + e-2a), we know by the Theorem above that J(Fo) = C. On the other hand, if~ > 0, we consider the equation F>.(Z) = z, which is equivalent to /(z) = ~+ze-2a -1 = 0. Since /(0) = -1 < 0 and /(1/ ~) > 0, thus there exists a point zo with 0 < zo < 1/>.. such that /(zo) = 0, i.e., zo is a fixed point of F>.. It is easy to verify that 0 < F'(zo) < 1 so that zo is attractive and J(F>.) :/:C. This implies that

Proposition 5.3 ~ = 0 is an e:tplolive point of 1/(~ + e- 2"). Remark. Deveney and Keen [74] proved that J(F>.) is a Cantor set for~> 0.

5.5. POINTS THAT TEND TO INFINITY

5.5

143

Points that tend to infinity

Eremenko [80] considered the set

I(/)= {z: r(z)--+ ooasn--+ oo}. If f is a polynomial, then I(/) is the immediate attractive basin of the superattracting fixed point oo. In this case we easily find that J(/)

= 8I(f).

Eremenko [80] proved that this result also holds for transcendental entire functions.

Theorem 5.10 Iff E £, then J(/)

= 8I(J).

PROOF. AB in Theorem 1.27, we denote by A a set in Ill+ with finite logarithmic measure, which need not to be the same at each occurrence. The central index is denoted by v(r, /). Choose r 1 > 2, r 1 ~ A, so large that

M(r, /)

> 4r, r

~

r1;

lm(An[r11 oo)) < 1; v(r1, /) > 104. Take

w1.

r2,

w2

such that 1/(wl)l

= M(r1,f),

lw1l

= r1;

M(r1, /)/2 < r2 < 2M(rlt /), l/(w2)1

= M(r2,/),

lw2l

r2

r/. A;

= r2.

Consider the sector

The function

g(z)

= 11(r1, /)(log z -log wl) +log /(wl),

which is nearly logf(z) by (1.23) with k the square

= 0, maps the sector C~ univalently onto

{( = e+ i'1: le -log 1/(wl)ll < 5, 1'1 By R.ouche's theorem there exists a domain C 1 onto the square

by log I

c

arg /(wl)l

C~

< 5}.

which is mapped univalently

{( = e+ i'1: le -log 1/(wl)ll < 4, ,,_ arg /(wl)l < 4}.

144

CHAPTER 5. GEOMETRY OF JULIA SETS

Thus the image /(01 ) contains the annulus

Let

c~ = {z : Ilog lz/1021! < v(r:, f), Iarg(z/102)1 < v(r:, /)} · Then C2 C Q. Repeating the above construction, we obtain a sequence of domains C; -+ oo, C;+l c /(C;) sucll that there exists a uniform branch of in C;+l for which J-i(C;+l) c C;. Let B; = ri(C;+l)· We have Bi+ 1 c B;, thus there exists Zo such that

r•

00

ZoE

nB;.

This implies that Ji(Zo) e C;+l· Hence Zo E 1(/). For an arbitrary z e J(/), take a neighborhood U of z. Since the family {r} is not normal in U, there exists a pre-image z E U of /"'(Zo) for some m ~ 0. Hence z E 1(/). On the other hand, it is easy to see that Int(1(/)) C F(/). Thus J(/) C 81(/). The opposite inclusion is obvious. This completes the proof. • Combining this and Theorem 4.29 we obtain the following

Corollary 5.3

Iff E En 8, then J(/) =I(/).

For polynomials, it is evident that J(/) n /(/) = 0. Next we shall show that this is not true for all transcendental entire functions.

Theorem 5.11 (Eremenko (80)) For any I

e E, I(J) n J(/) :F 0.

We distinguish the proof into two cases. At first we suppose that F(/) has a multiply connected component D. Then by Corollary 4.1, D is bounded. Let Km be the unbounded component of the complement of Dm = /"'(D), and let 'Y be a Jordan curve in D non-homotopic to a point in D, 'Ym = /"'('Y)· By Theorem 4.1, we have 'Ym-+ oo, ind,('Y,;.) > 0 and 'Ym C Dm. Thus 'Ym separates 0 and Km, and so Km -+ oo. Fix a large number m, we have -+ oo in D, Dm n Dn = 0 for n > m. Hence 15" c Km for sufficiently large n > m. Co118eQuently, oD c I(/) and J(/) n 1(/) ':/: 0. Now we suppose that all components of F(/) are simply connected. Let B; be the same as in ~he proof of Theorem 5.10. By Theorem 1.27, for z E C,, PROOF.

r

(5.11)

5.5. POINTS THAT TEND TO INFINITY

145

By definition, diamC;+l ~const·r;+l/v(r;+t. /). Putting k

= j in (5.11) we obtain

diam/-l(C;+1 ) :5 const ( /)r;2ee v r;, v (r;+t. /). Applying (5.11) repeatedly fork= j - 1, ... , 1, we get

as j

--+

oo. Thus

n 00

Bj

= {Zo}·

j=l

Obviously, Zo E /(/). H Zo E F(/}, then there exists a neighborhood V of Zo such that V C F{/). By (5.12), Bn C V for n ~ no. We have /"(Bn) = Cn+l• consequently, f"+l(Bn) contains the annulus {z: e- 4 M(rn+t./) < lzl < e4 M(rn+l,J)}. Thus the set F(/) contains an arbitrarily large annulus. This contradicts to the assumption. Thus Zo E J(/), and /(/) n J(/) '# 0. The proof is complete. •

Set

. loglf"+l(z)l Io(/) = {z: n~~ logl/n(z)l = oo}.

The above proof shows that there exists z E /(/)such that 1/"+l(z)l "'M(l/"(z)I,J) as n --+ oo. Since logM(r,/)/logr function f it follows that

--+

oo as r

. logl/"+ 1 (z)l lim I n-+oo log /"(z)l

--+

oo for transcendental entire

= oo.

Thus / 0 (/) '# 0. In addtion, we remark that the arguments of Eremenko do not require that f is transcendental entire but only that f is analytic in a neighborhood of oo and that oo is an essential singularity of f. Therefore the same reasoning yields the following result. Theorem 5.12 Iff is a tronscendental entire function or an analytic aelf-map of C•. Then Io(/} '# 0 and J(f) = 8/o(/). The following question was proposed by Eremenko [80]. Question 5.1 /a every component of I(!) unbounded? Eremenko himself proved that the closure/(/) of/(/) has no bounded component.

CHAPTER 5. GEOMETRY OF JULIA SETS

146

5.6

Expanding functions

In this section, we shall discuss the expaosion and distortion properties of certain entire functions. A meromorphic function/ is called hyperbolic if En J(/) = 0 . The function f is called to be sub-expanding if d(E, J(f)) > 0. Furthermore, we say f is expanding if E is compact and "En J(/) = 0. Obviously, any expanding function is sub-expanding. However, the following example shows that the converse is not true. Example. Let f(z) = z+ 1-e". It is easy to see that f bas no finite asymptotic values, and all the critical values of/ consist of 2mri (n E Z), each of which is a fixed point of f. Thus E = E is not compact. Note that r(z+21ri) = r(z)+21fi, so that F(/) and J(f) are invariant under T(z) = z+211'i. Since 1/(z)l $ e- 2 for lzl $ 1, we have /(A) C A. Therefore A(2n7fi, I) C F(/), so that d(E, J(f)) ~ 1. This implies that f is sub-expanding but n6t expanding.

Proposition 5.4 If/ i3 an expanding tranacendental entire function, then {i} l(r)'(z)l ~ Clr(z)llog lr(z)l for some conatant C; {ii) lr'(z)l -+ oo for all z e J(f). PROOF. The proof was given by McMullen [158] which is similar to the one for rational functions [77]. Let V denote the unbounded component of the complement of E. Then on any bounded component the boundary values of lie in the compact set E and so the Iterates form a normal family by the maximum principle. This implies that J(/) C V. Let U = /- 1(V). By the definition of E, U C V and f : U -+ V is a covepng. If U = V, then f conformally conjugate to z-+ zm for some mEN. The· oonclusion follows. Next we suppose that U is a proper subset of V . Let ,\(z)ldzl be the hyperbolic metric on V. Since E is bounded, we have

r

(5.13)

Now by Schwarz lemma, 11/'(z)ll where

> 1, z E U,

II · II is the norm of the derivative of / l(r)'(z)l

(5.14)

in the ,\ metric. Obviously, we have

= ll(r)'(z)ll-\(z)/-'(r(z)).

(5.15)

The conclusion (i) follows from (5.13)-(5.15). Next we prove (ii). If r(z) tends to oo, then (ii) follows from (i); if the iterates r(z) return infinitely many often to some fixed compact subset K of J(/), then the derivative of in the -\ metric tends to oo since 1 expands by a definite amount on K. Thus l(r)'(z)l -+ oo by (5.15). The proof is complete. •

r

5. 7. RELATIONS BETWEEN E AND 'P

147

For sulrexpanding entire function, Stallard [190] showed that the above conclusion (ii) still holds.

Proposition 5.5 Iff is a sub-expanding transcendental entire function, then lr'(z)l --+ oo for all z E J(f). PROOF. Let d = d(E, J(f)). Suppose on the contrary that there exists a point J(/) such that l(r)'(z)l does not tend to oo. Then we may suppose that l(r)'(z)l < K for some K > 0 and all n (in fact, this holds for a sequence of n --+ oo). Let g be the branch of 1-n such that g(/"(z)) = z. Then g is holomorphic and univalent in the disk t~o(r(z),d). Moreover lg'(f"(z))l > 1/K. By Koebe's 1/4 theorem, we have z E

g(t~o(r(z),d))

Thus r(D)

c

::) t!.(z,dlg'(r(z))l/4) ::) t!.(z,d/(4K)) = D(say).

t!.(/"(z),d). This contradicts Theorem 3.14.

•

Proposition 5.6 Let f be an expanding entire function. Then there is a countable family of balls {D;}iei whose diameters tend to oo, with the following property: every z E J(/) has a sequence of neighborhoods Un whose diameter tend to 0, such that maps Un onto a member of the family {D;};ei with distortion bounded by a constant L independent of z and f.

r

PROOF. Take {D;hei = {t!.(z,d(z,E)/2): z E J(f)}. Since E is bounded and J(J) is unbounded, the diameter of the balls tends to oo. Now fix z E J(J). For each n let Dn be a member of {D;};ei containing r(z). We take Un = /-"(Dn)· By Theorem 1.10 this map has distortion bounded by a universal constant, and

Note that d(Dn) is comparable to lr(z)l, it follows from Proposition 5.6 (i) that

Thus d(Un)--+ 0. This completes the proof.

5. 7

•

Relations between £ and P

The dynamical properties between the class E and the class 'P are closely related. In fact, consider the following commutative diagram

148

CHAPTER 5. GEOMETRY OF JULIA SETS

J _ c ___

c

c· __9_ _ c·

Here we regard C as a covering space of entire defined by the diagram. That is,

c•, 1r(z)

= e•. If g is given, then

J is

(5.16) and J is uniquely determined by g up to a choice of a branch of the logarithm. We call J a lift of g. Any function g(z) mapping c· to itself has the form

g(z)

= z1 exp(t(z) + h(1/z)),

where t(z) and h(z) are entire which fix the origin, l is an integer. The function J(w) has the form: Obviously,

f(z + 21ri) = /(z) + l21ri, r 0) by (5.17), it follows from Corollary 4.3 that all components ofF(/) are simply connected. Thus by Lemma 4.1, there exists a constant c > 0 such that (5.20) for all large n. Since yl max{1, Ill} and n -+ oo, we have 1/"t(zi)I

= o (M"t).

Combining this with {5.20) we find that l/"k(z2)l

= o (M"k) .

(5.21)

On the other hand, from the choice of w2 and (5.16) we deduce that Ref"(z2) -+

oo and Ref"+ 1 (z2 )/Ref"(z2)-+ oo as n-+ oo. Thus

l/"(z2)1 2:: Ref"(z2) 2:: M" for all large n, contradicting (5.21). The proof is complete.

•

Remark. This result was stated in Keen [134, Lemma 2.2] and Fang (89, Lemma 1.2]. Bergweiler (38] gave a complete proof.

5.8

Connectivity of Julia sets

In this section, we shall consider the connectivity of Julia sets.

150

CHAPTER 5. GEOMETRY OF JULIA SETS

If J is a non-linear polynomial, then J(J) is connected if and only if no finite

critical values tend to oo under the iterates off (see e.g. [29, Theorem 9.5.1] or [159, Theorem 17.3)). However, if J is transcendental, this criterion does not hold. For example, for the exponential family b.(z) = ~e•, when 0 < ~ < 1/e, f>,(z) has a unique finite singular value z = 0 which is attracted to some fixed point of f>,(z). But J(f>,(z)) is disconnected. The following result is standard (see [164] or [170)).

Proposition 5. 7 A compact set K C C is disconnected if and only if there exists a Jordan curoe "Y which separates K. Now we can prove the following basic result.

Proposition 5.8 Let K C C be compact. Then K is connected if and only if each component of the complement Kc is 6imply connected. PROOF. • 0 such that the distance from all these end-points to C(b) is at least 6. Choosing k so large that 6 > b" = d(I~o,;), it follows that every interval I~o,i is contained in some / 1 {1 $ j $ n). Now for any I; and any fixed l such that there are some intervals lt,i inside 11 , let n be the smallest integer for which / 1 contains some In,i· Then n $ l. Let In,i 1 , ... , In,i,. be

160

CHAPTER 6. HAUSDORFF DIMENSION OF JULIA SETS

all the n-th generation intervals which meet 11 • Then p S 4, since otherwise 13

would contain some Ifl-l,i• Thus p

4d(I3 )• ~

p

L d(I,.,,.,.)• = L L

m=l

m a li,

cr.. ...

d(Il,a)• ~

L

r, ,,clj

d(Il,a)• ,

which implies that 4

~

L d(I, )• ~ L L J

r•.,cr1

,

d(I.,,,)• ~

L d(I.,,,)• = 1. a=l

Therefore H•(C(b)) ~ 1/4. Combining the above discussions we get Proposition 6.6 dimC(b)

= ~·

Remark. Actually the precise value H•(C(b)) = 1 for s = dimC(b), see Falconer [87, Theorem 1.14). Note that dimC(b) takes all the values between 0 and 1 when b runs from 0 to 1/2. Cantor sets in C. We can use the same ideas as above to construct Cantortype sets in C having a given Hausdorff dimension s. Suppose for k = 1, 2, ... we have compact sets Eat. .. ··••• iJ = 1, ... , m;, such that

d,.

.... ..

= max d(E,,

.•• )-+ Oask-+ oo,

m•+l

L

J•l

d(E,,, ...,,. , j)• = d(Ea, ,...,•• )•,

L

BnE, 1

for any ball B with d(B)

~d., ,

d(E,,, ...•••

>· s cd(B)•

·'•~·

where cis a positive constant. Then 00

0 < H•(n

U E,, ...... ) < oo.

A:z l& t .. ·••

See Beardon [28).

6.4.3

Self-similar sets

Roughly speaking, a set is self-similar if it can be split into parts which are geometrically similar to the whole set. The Cantor set is a simple example. Another standard example is von Koch's "snowflake" curve. In the construction one replaces at each stage a segment of length d by four segments of length d/3. The

161

6.4. HAUSDORFF DIMENSION

von Koch curve K is a limit of the polygonal curves thus obtained. It is a nonrectifiable curve having tangents at none of its points. We now state the basic ideas of Hutchinson's general theory (see [126]). Let D be a closed subset of the plane. A map S : D --+ D is called a contraction on D if there is a number r, 0 < r < 1, such that IS(x)- S(y)l ~ rlx- Yl

forx,y ED.

Clearly any contraction is continuous. If equality holds, then we callS a similarity. Suppose that S = {81, ... , SN}, N ~ 2, is a finite sequence of similarities with contraction ratios rb···• rN. We say that a non-empty compact set K is invariant under S if N

K= US;(K). i=1

It is somewhat surprising that the invariant set K is determined by S. In fact, for gicen S, there exists a unique compact set K invariant with respect to S. We define an invariant set K under S to be self-similar if with s = dim K, This definition is rather awkward to use, but the following somewhat stronger separation condition, called the open set condition, is very convenient: There is a non-empty open set 0 such that N

US;(O) c OandS;(O) n S (0) = 0 1

fori# j.

•=1

This is satisfied if the different parts S;(K) are disjoint. Theorem 6.4 If S satisfies the open set condition, then the invariant set K is self-similar and 0 < H 6 (K) < oo, whence s = dimK, where s is the unique number for which N

I>:= 1. •=1

Moreover, there are positive and finite numbers a and b such that ar•

~

H 6 (KnB(x,r))

~·br•

forx E K,O

r. It follows that Kn~(:z:,r)

and so IJ(Kn~(:z:,r)) ~

J.&(K,,,.

c

,J

Ki, ....... = (b,, ... b,,.)• ~ (r/d)•.

If U intersects K, then U C ~(x, r) for some x E K with r = mu(U). Thus IJ(U) ~ (mea(U)/d)•. This implies that for any cover {U,} of K,

0 < J.&(K) ~ p(U;U,) ~ c

L mu(U;)•, J

where cis a positive constant. Taking infima, we deduce that Hl(K) ~ J.&(K)/c if 6 > 0 is small enough, so H•(K) ~ IJ(K)fc. The conclusion follows from Proposition 6.5. •

6.4.4

Capacity and dimension

For Radon measure J.&, the t-energy is defined to be (6.1)

For A C C, let M(A)

= {J.&:

J,&isaRadonmeasurewithcompactsupport, 6pt(IJ) c A,O 0}

163

6.4. HAUSDORFF DIMENSION

By Proposition 6.1 we can deduce that the two least upper bounds in the above definition agree {cf. [156, pp.109-110]}.

Definition 6.5 Let s > 0. The {Riesz) a-capacity of a set A

cC

is defined by

= sup{I.(~-£)- : 1-' E M{A),~-£(C) = 1} with the interpretation c.(0) = 0. Proposition 6.7 For s > 0 and A c C, C.(A)

1

dim, A= sup{ s : C.(A) > 0} = inf{ s : C,(A) = 0}.

Remarks. By an approximation we could drop the requirement that the measures 1-' have compact support in the definitions of dim, A and C.(A). More generally we could use Borel measures instead of Radon measures.

Theorem 6.6 Let A C C. {1) If s > 0 and H•(A) < oo, then C.(A) (2) dime A ~ dim A.

= 0.

PROOF. (2) follows immediately from (1), thus we only need to prove (1). Suppose C,(A) > 0. Then there is 1-£ E M(A) with ~-L(A) = 1 and I.(~-£) < oo. Thus fix-

Yi-'d~-LY < oo

for almost all x E C, whence for such x, lim

r-+O+

JI.O.(z,r) ix -

Consequently, given c > 0 there are B for x E B and 0 < r ~ !J,

C A

~-L(Ll(x,r)) ~ r• I

J.o.(z,r)

Choose sets

E~t

Ez, ...

Yi-•di-£Y

= 0.

and lJ > 0 such that ~-£(B) > 1/2, and

ix- Yi -'d~-LY

~ cr'.

such that B

cUE,,

B

n E, :/:0,

d(E;) ~ !J,

Picking x; E B n E; and setting r, = d(E;), we have

Letting c-+ o+ we conclude that H•(A)

= oo, which proves (1).

Next we give a powerful tool which is so called Frostman's lemma ([93]).

•

164

CHAPTER 6. HAUSDORFF DIMENSION OF JULIA SETS

Theorem 6.7 Let B c C be a Borel set. Then H"(B) > 0 if and only exists J.1. E M(B) such that JJ.(~(x, r)) S r- for x E C and r > 0.

if there

The sufficient part can be proved similarly as in Theorem 6.6. The proof for the necessary part can be found in [156, pp.113-114]. Using Frostman's lemma we can give more complete information about the relations between Hausdorff measures and capacities of Borel sets. Theorem 6.8 Let A C C be a Borel set. (1) If s > 0 and c.(A) = 0, then H 1(A) = 0 fort > 3. {2) dim, A= dim A. PROOF. If H'(A) > 0, then Frostman's lemma implies that there exists a p. E M(A) for which J.~.(A(z, r)) ~ r. Then for 0 < s < t, I.(p.) < oo by Proposition 6.1. Hence C.(A) > 0 and (1) follows. (2) is an immediate consequence of (1) and Theorem 6.6. •

6.4.5

Limiting sets and density

In this subeection, we will give a basic result on the Hausdorff dimension of some limiting sets (see McMullen [158]). Let V and U be two sets in C. The (Lebesgue) density of VatU is defined to be

-'- -(V. U) ·= mu(V n U) '""'"'" ' · mu(U) · For k = 1, 2, ..., let £ 1c denote a finite collection of disjoint compact subsets of C, and let u,. denote the compact set obtained as the union of the elements of E~c. We make the following three assumptions: (a) Every E1c+1 E £1c+1 is contained in a unique E., E £.,; (b) Every E,. E £ 1c contains at least one element of £ lc+l; (c) For all k and all E., E E~c, dens(U~c+t. E,.)

;::: 6,.

and

d(E~c) ~

d,.,

where, 0 < 6,. ~ 1 and d,. decreasing ask-+ oo.

> 0 for all k. By (a) and (b), we may assume that d,. is

Proposition 6.8 Let V

= nr._ 1 U,..

Then

II 6~r. 00

dens(V,Ut);:::

A:=l

PROOF. By the assumptions (a) and (b), U;+l C U1 for all j. Thus dens(Um Ut)

Letting n

-+

=

mes(Un) mes(Ut)

=IT

oo, we obtain the conclusion.

J=l

mu(U;+l) ;::: me3(U;)

IT j=l

6;.

•

6.5. DIMENSION FOR THE CLASSES E AND P

Proposition 6.9 Let V

= nf:1 Uk.

165

Then

k+l logcS-:- 1 dim{V) 2:8 := 2 - lim sup ~ . k-oo j=l 11og k 1

TI -

PROOF. We may suppose that 8 > 0. For simplicity we assume that me8(Ut) = 1. Construct a sequence of probability measures 1-'k as follows: Let 1-'l be the restriction of Lebesgue measure to U1 • Construct 1-'k+t inductively so that within each Ek E Ek, 1-'k+l is proportional to Lebesgue measure on Uk+l• scaled so that 1-'k+l(Uk+l n Ek) = IJk(Ek)· For these measures, if m 2: k, then 1-'m(Ek) is the same for Ek E E~c. Let IJ denote a weak limit of the 1-'k· Then IJ is unique and supported on V, and for Ek E Ek,

~-' (

E ) < mes(Ek). k

nk u,

-

j=l

£.

(6.2)

Now for arbitrary 0 < c < s, it is easy to see from the assumption that lim sup k-oo

d2-a+t:

~+1

nj=l

6;

= 0.

(6.3)

Let B be a ball of small radius r . Choose k so that dk+ 1 $ r $ dk, and let B denote the union of all sets Ek+l E Ek+l which meet B . The d(B) $ 2d(B). It follows from (6.2) and (6.3) that

where C is a constant depending only on 8 and e. By Theorem 6.7 (Frostman's lemma), dim(V) 2: s - c. The conclusion follows by letting c-+ 0. •

6.5 6.5.1

Dimension for the classes£ and P For general functions

By Theorem 4.4 and Lemma 4.15 we have the following result.

Theorem 6.9 Iff E £U'P, then 1 :S dimJ(/) $ 2. We propose the following conjecture:

166

CHAPTER 6. HAUSDORFF DIMENSION OF JULIA SETS

Conjecture 6.1 For IE ELfP, dimJ{f) > 1. H this conjecture is true, then the following result proved by Stallard [192] shows that the lower bound is sharp. Let L be the boundary of the region G = {z: Re(z) > 0, llm(z)l < 1r} described in a clockwise direction. Consider the function defined by

E(z) = - 1 . 21rt

1 L

1

exp(e ) -t--dt, -z

z E C\G.

As shown in [168, Part III, problems 158-160], E(z) can be analytically continued to a transcendental entire function which will also be denoted by E(z). Theorem 6.10 Given any E > 0, there exists a positive constant K 0 (e) such that, for any K > K 0 (e), dim J(E(z)- K) $ 1 +e.

Next we consider some special classes.

6.5.2

Combinations of exponential functions

In this subeection we will study the family

{!(z) = ae" +be-" :a, bE C}.

(6.4)

This family is topologically complete in the sense that any entire function topologically conjugate to a member of the family is a member itself. We say a subset V of C is thin at oo if there exists an R and an e such that for all z, dens(V, ~(z, R)) < 1 -e. The following result was proved by McMullen [158].

Theorem 6.11 Let f be a member of the family {6.-1). (i) If ab =F 0, then J{f) alway3 hall positive area, and dim J(f) = 2. {ii) If a =F 0 but b = 0, then dim J(/) = 2. {iii} If a =F 0 but b = 0 and if J(/) is thin at oo, then the area of J(f) is zero. PROOF.

= fi%. Consider the sequence h,. = 2g/t k = 0, 1, 2, ••••

We first prove (i). Let g(x)

1

If we take sufficiently large x

> 0, then h,.(x)--+ oo ask--+ oo and the sum

E h,.+l exp(-h,.> 00

k=O

is sufficiently small. Consider the region D(h,.) = {z : IR.e(z)l > h,.}. Then for z E D(h,.), 1/'1 > O(exp(h,.)) and If" //'1 ~ 1. Let B be a box in D(h~c), we see that /(B)nD(hk+ 1 ) looks very nearly like a square of side at least O(exp(h,.)), possibly with a strip

6.5. DIMENSION FOR THE CLASSES E AND P

167

of width 2h,.+1 deleted. Thus the set /(B) n D(h~c+ 1 ) can be packed with disjoint boxes whose total density in /(B) exceeds 1- O(hk+t exp( -h,.)). We refer these boxes in such a packing as pack(f(B)). Let Eo= {Bo: Bois a box in D{ho)}

and define inductively

£,. = {G: G c V E E~c-t, f"(G) E pack{f"{V))}, which consists ofpreimages ofthe boxes packing f"(V)nD(h,.) for each V E £,._ 1 • We denote by u,. the union of all G E £,. (k = 0, 1, ... ). Since dens(L.Jpack{f"{V), f"(V)) ;::: 1 - O(h~c+t exp( -h,.)), we have, for any G E £,., dens{U~c+t, G)

;::: 1 -

O(h~c+ 1

exp( -h~c)).

Note that n,.u,. consists of points for which f"(z) E D(h~c) for all k. We deduce from Proposition 6.8 that the set of z in B0 such that f"(z) E D(h~c) for all k has density at least

IT (1 - O(hk+t exp( -h,.))). 00

lc=O

Thus there is always a set of positive area which tends to infinity at the rate of iterated exponentiation. By Corollary 5.3, this set must lie in J{f). The proof of (i) is thus complete. Next we prove (ii). Let I = [-11/4- arg)t.,11/4- arg)t.j. Then argf(z) E [-11/4,11/4] whenever Im(z) E /(mod211). Consider the sector D = {z: Im(z) E /(mod211)}. For z E D, we have Re/(z) > O(exp(Rez)). Thus as long as the iterates of a point remain in D, the point moves to the right at the rate of iterated exponentiation. Now choose x 0 so that 1/'(z)i » 1 when Re(z) > x 0 . Set T ={zED: Re(z) > x 0 }. We define a box B to be a square region in T of side length r, where r is chosen so small that f is injective on B and the image /(B) looks much like a square. This is possible by f" / f' = 1. Thus the set /(B) n T can be packed with disjoint boxes whose total density in /(B) is nearly 1/4. As before, we refer these boxes in such a packing as pack(f(B)). Let Eo= {B0 } and define inductively E,. = {G: G c V E E~c-t,f"(G) E pack{flc(V))}, which consists of preimages of the boxes packing f"(V) n T for each V E £,._ 1 • Since

.:~ens(upack{f"{V), '" O(exp(Re(z)) for z E T, and that /'(z) = /(z), we deduce

that, for z EVE t~e and g(x) Re{f"(z))

=~.

> O(gk{1)) and d(V) < d~; = 0{1/gk{1)).

This implies that for any z e n~: U~e, Jk(z) tend to oo at the rate of iterated exponentiation. By Corollary 5.3, n~: U~: must lie in J(f). On the other hand, by Proposition 6.9, the Hausdorff dimension of the set n~: U~: is 2. This completes the proof of (ii). To prove (iii), we need only to prove the following lemma.

Lemma 6.1 Let f(z) be an expanding entire function and suppose that V is a measumble completely invariant subset of J(f) . Then if V is thin at oo, its area is zero. PROOF. The proof is parallel to Sullivan's argument for rational functions (198]. We choose a family of balls {Dihei as in Proposition 5.6. The diameters of these balls tend to oo and V is thin at oo, so there is a uniform upper bound to the density of V in these balls. By the complete invarianty of V, we deduce from Proposition 5.6 that every point in V has arbitrarily small neighborhoods in which the density is less than one by a definite amount. Furthermore, these neighborhoods are images of round balls under maps of uniformly bounded distortion. Therefore, V has no points of positive Lebesguee density and consequently its area is zero. • Remarks. We can take some suitable a =F 0 and b =F 0 such that /(z) ae• +be-• is expanding. Thus there exists transcendental entire function such that the Julia set has positive area. But the corresponding result for rational function is not true. In fact, the Julia set of any expanding rational function always has measure zero. An example for the conclusion (iii) is the situation 0 < a < 1/e. This is included in the following general proposition.

Proposition 6.10 Let /(z)

= ae•

with a =F 0. Then the following relations hold:

f has an attmcting periodic cycle ~ is expanding ~ J(f) is thin at oo.

I

PROOF. Assume that f has an attracting cycle. Then by Theorem 4.13, this cycle contains at least one singular value. Note that f has only one singular value z = 0. Since the cycle is attracting, r(o) tends to a point in the cycle. We conclude that J(f) n E = 0. Thus f is expanding. Now we suppose that / is expanding. Then the singular value z = 0 is in F(f). Thus we can find a neighborhood U of z = 0 such that U c F(/). The preimage of U contains some left half.pJane H, and the preimage of H consists

6.5. DIMENSION FOR THE CLASSES

e AND P

169

of periodic horizontal strips S. Any sufficiently large ball must either lie in H or meet a definite amount of S. This means that J(f) is thin at oo. The proof is complete. •

6.5.3

The function in the class B

In this subsection we shall consider the case proved by Stallard [196].

Theorem 6.12 Iff E B, then dim J(f)

f E

B. The following result was

> 1.

The proof is divided into several steps as follows. Step 1. Some properties off when fEB.

PROOF.

Lemma 6.2 If sing(f- 1 ) C ~(O,ro/2), lwl = r ~ ro and !lwl < lzl < ~lwl and i/ we take a brunch g of f- 1 that is defined at w and continue g analytically along a curoe 'Y which lies within {t : lwl < it I < ~ lwl} such that 'Y winds at most once around 0, then

!

where L

= 34 •

PROOF.

For 8m

= 2m7r/25

(m

Bm

= 0, 1, ... , 24)

and cp

= argz, let

= ~(rexp[i(8m + ¢)], r/4).

Then the circle {lzl = r} can be covered by the union of the disks Bm. For each 0 ~ m ~ 24, we take a point Zm E Bm n Bm+l, where B2s := Bo. As r ~ ro, each branch g of /- 1 is univalent in ~(rexp[i(Bm + ¢)], r/2). It follows from Theorem 1.10 (Koebe's distortion theorem) that, for z E B 0 and wE Bn (0 ~ n ~ 24), lg'(z)i ~ Ljg'(Zo)l ~ L 2 lg'(zl)l ~ · · · ~ L 26 jg'(w)j.

The other inequality can be sinillarly derived. Lemma 6.3 Iff E B, then there exists Rt > 0 such that {i) if lf(z)i > Rt, then lf'(z)i > 1/(z)l(log lf(z)l-logro)/(47rjzl). {ii) if izi > R1, then, for each brunch g of /- 1,

lg'(z)l PROOF.

< 87rlg(z)l/(lzllog lzl).

Assume that

sing(r 1 )

c

~(0, ro/2), 1/(0)1

< ro, 1/(z)l

From the proof of Proposition 4.5 we see that, for w E ln(/- 1 (C\~(0, ro))),

~ ro.

•

170

CHAPTER 6. HAUSDORFF DIMENSION OF JULIA SETS

there exists a map g such that /(e"') =

Thus

e6(w)

·

and lg'(w)l ~ [Re(g(z)}-logro)/(411').

l/'(z)l > 1/(z)l(log 1/(z)l-logro). 4•lzl

The two conclusions follows.

•

For any function /(z), we set

/*(z) = (1 + lzl 2 )/*(z), where /*(z) is the spherical derivative of/ at

(6.5)

z.

Lemma 6.4 If/ E 8, then there exists a number R2 > Rt such that, if 1/"'(z)l ~ R 2 for each 0 ~ k ~ n, the branch g of /-n that maps r(z) to z satisfies: (i) for each 0 ~ k < n and each K ~ 4,

!"' o g(b.(r(z), lr(z)l/K)) c b.(/"'(z), l/"'(z)I/(4K)), (II} g Is univalent In b.(r(z), lr(z)l/4), (Iii) if we b.(r(z), lr(z)l/8), then L- 1 19'(r(z))l ~ lg'(w)l ~ Ll9'(r(z))l, (iv) ifw e b.(r(z), lr(z)l/8), then (3L)- 1g*(r(z)) ~ g*(w) ~ 3Lg*(r(z)). (v) if z e J(/) and lr(z)l ~ R2 for each n e N then, given M > 0, there exists n eN such that l(r)'(z)l ~ Mlr(z)l. Bing(/- 1 ) C b.(O,R2/2). Then for each 1 ~ k ~ n, the branch g,. of that maps J"'(z) to l"'- 1 (z) is univalent in b.(/"(z), 1/"'(z)l/2) and so, from Theorem 1.10 (Koebe's distortion theorem), PROOF. Let

r•

g,.(b.(f"(z), 1/ll:(z)l/K)) C b.(/"'- 1 (z), LgA:(f"'(z))l/"'(z)l/ K) for each K

If R2

~

4. It follows from Lemma 6.3 that, for R 2

(6.6)

> R1

> exp(321rL), then

and so, from (6.6),

9~r(b.(/"'(z), 1/ll:(z)l/K)) C b.(/"'- 1 (z), lf"'- 1 (z)l/(4K)). This proves (i). Note that g = Yt o · · · o Yn is univalent in b.(r(z), lr(z)l/4), (ii) follows. Also (iii) follows from Koebe's distortion theorem. Combining (i) and (iii) we get (iv).

171

6.5. DIMENSION FOR THE CLASSES & AND P

< Mlf"(z)i for

At last we prove (v}. Suppose on the contrary that l{f")'(z)i each n EN. By (ii) and Theorem 1.9 (Koebe's 1/4 theorem),

n 1 1/"(z)l 1 g(a(f (z), :tir(z)l) :> a(z, 161(/")'(z)i) :::> a(z, 16M).

Since a(r(z), ~ lr(z)l) n a(o, R2/2) r(a(z,

= 0, we have

1 M)) n a(O, R2/2) 16

=0

•

for each n EN, which contradicts Theorem 3.14. From the proof of Proposition 4.5 we can derive the following:

Lemma 6.5 Iff E B, then there exists' R3 > R2 such that, for each R ~ R3, there exists an analytic curve r joining a point ZR to oo such that lf(z)i = R for each z E r.

Step 2. Construction of some measures. Suppose that J(/) ::f:. C. We take a value R' satisfying

where R3 is as defined in Lemma 6.5, L we can choose a point

= 34

Zo E (J(/) n /(/)

and C > 4800L2 • By Corollary 5.3,

n {izl ~ 4R'})

such that lr(Zo)i ~ 4R' for each n E N and a(Zo, IZol/4) does not contain any Fatou exceptional values. We put

R

= IZol.

A = a(Zo. R/C),

B

= a(Zo. 2R/C).

Lemma 6.6 There exist w E C and r > 0 such that (i) U = a(w, r) c F(/) n A, {ii) if 9i (i = 1, 2) are branches of rn(i) satisfying f,. 0 g;{U} each 0 $ k $ n(i), and 9tlv ::f:. 92iu, then g, (U) n 92(U) = 0.

c

{izl ~ R'} for

PROOF. Let N be a component ofF(/). We claim that there exists an open set = 0 for each n EN. In fact, if N is wandering, then we may take V = N; If N is pre-periodic, then we can get V by Theorem 4.5. Let z E V\FV(f). By Theorem 3.6, there exist wE A and n' EN such that r'(w) = z. It follows that there exists r > 0 such that U = a(w,r) c A and

V c F(f) such that r(V) n V

r' c v.

CHAPTER 6. HAUSDORFF DIMENSION OF JULIA SETS

172

H f-(M+Ir)(U) n /-lr(U) ::/: 0 for some m, keN, then r ' (U) n r'+m(U) ::/: 0 and hence V n f"'(V) :1: 0 which contradicts our choice. Thus, for all m, k E N,

r(u) n r•(u) = 0. Suppose now that, for some n eN, there are two branches 91 and 92 of 1-n such that, fori= 1,2 and 0 ~ k ~ n,

Obviously, r < lwl/4. It follows from Lemma 6.4 {ii) that both g1 and 92 are univalent in U. Thus

•

This completes the proof of the lemma.

Now we define Rl = 2R' and R:.+l = M(R:., /). From the proof of Proposition 4.4 and the fact that nn9(/- 1 ) C ~(0, Rl) we deduce that ftn9(r">

·

Set ~ = max{R:., Rt Tn = lr(Zo)l. where Rt is as defined in Lemma 6.3 for By Lemma 6.5 we can take an analytic curve r joining a point ZR to oo such that 1/(z)l = R = IZol for each z e r.

r.

Lemma 6. 7 There exist m, p E N 6Uch that (i) r n {lzl = rp} ::/; 0 and G c {lzl ~ 2R:;.}, where G

{ii) There exist& a branch 90 of G' = {

eG = z;

(iii) For each z

F(wt) = F(W2)

(iv) For each r

9

10

= {10rp ~ lzl ~ 9rp};

rl 6Uch that Wo = 9o(Zo) e go(B) c G', where 17 19 rp ~ lzl ~ rp}i 18 18

there exist two distinct point& w 1 , w2

e

G' 6Uch that

> 0, the length of r n ~(0, r) is finite.

PROOF. (i) By the definition of Zo, there exist p e N such that rp = lfP(Zo)l > max{20R:;.,IzRI} and two distinct points Zt, Z2 e o-(Zo) n A. Since Zo e J(/), we have Zt,Z2 e J(f). Thus, by Theorem 3.14, there exist r' > 0 and meN such that a{zt,r') n a{z2,r') Thus (i) is proved.

= 0,a(zt,r') c A c

r(a(zt,r'))

(i = 1,2).

6.5. DIMENSION FOR THE CLASSES E AND 'P

173

(ii) Now we take a point r E rn{lzl = r,} so that lw'l ~ R, where w' = /(z'). Let l(R) be the segment joining w' to Zo and let y0 be the branch of /- 1 satisfying 9o(w') = z', then we can continue y0 uoivalently almig l(R) to Band so, by Lemma 6.2, for w E l(R) U B we have

Since each w E B can be joined to w' by a curve of length less than 21r R which lies in l(R) U B we deduce that

IYo(w)- z'l < 21TRL 26 Iy~(w')l. Now by Lemma 6.3,

r, = 811" Rlog R'

IYo(w')l lz'l IYo' (w')I < 811" lw'llog lw'l = 811" Rlog R and so, for each wEB, we deduce from (6.7) that

'( ')I < 1611"2RL2sr, / IYow RlogR < r, 36.

{6.8)

Thus wo = Yo(Zo) E Yo(B) C ~(z', r,/36) C G' which proves {ii). (iii) Let Yi be the branch of 1-m that maps A uoivalently into ~(zi, r') C A and let hi =Yo o Yi of (i = 1, 2). From (6.8) we see that hi (i = 1, 2) are branches of satisfying (6.9) hi(wo) E Yo(A) C ~(z',r,/36) .

rm

If z E G, then there exists a simple curve 'Y joins w 0 to z. For each w E 'Y we have

4

5lwol < (9/10) 2 lwol ~

c

G of length less than 21rr, which

lwl ~ (10/9)

5

lwol < 4lwol·

2

Note that IWol > 2R~ > R1 and siny(f-m) C ~{O,R~) C ~(O,R~), by Lemmas 6.2 and 6.3, we can continue hi uoivalently along 'Y and, for each wE -y,

It follows from (6.7) that 2

lhi(z)- hi(w)l ~ 1611" 9

L 26

r, 910 r,

(9 ) Iiir, 1og lOr,

< r,/36,

which and (6.9) yield that hi(z) E ~(z',r,/18) C G' {i

= 1,2).

174 As

CHAPTER 6. HAUSDORFF DIMENSION OF JULIA SETS

= z, this proves (iii). (iv) It is clear that there are only a finite number of branches 91, !J2, .•. , 9n of

f'"(~(z))

satisfying 9i(Zo) E ~{0, 2r). We cut the circle {lzl = R} at .z:o and, for 1 ~ i ~ n, continue 9i univalently in an anti-clockwise direction around the cut curve {lzl = R}. Suppose that z E rn~{O, r). There exists w' e {lzl < R} and a branch g of /- 1 such that z = g(w'). We continue g analytically in a clockwise direction from w' to .z:o along {zl = R}. As R ~ 4R' ~ R 1 and aing(/- 1 ) C ~{O,R'/2}, we deduce from Lemmas 6.2 and 6.3 that, for each won that arc of {zl = R}, /-1

26

26

'( )I < L26 l '( ')I < 811'L Ig(w')l < 87TL r I9 w 9 w - lw'lloglw'l - RlogR and so, as log R > log R' > 1611'2 L26 , lg(w')- g(.z:o)l ~ . Obviously g(.z:o) E ~{0, 2r} and so g r n ~{O,r)

1611'2 L 26 Rr Rl R og

< r.

= g, for some 1 ~ i ~ n. c

Thus

n

UYi({lzl = R}) •=1

•

and hence has finite length. This completes the proof.

Let g be a branch of 1-n satisfying 1/"'(g(z))l ~ I( for each 0 ~ k ~ n and each z E B. We define In

= {g(.z:o) E AUG}

= U In. 00

and

I

n=1

Lemma 6.8 {i) For each n, m EN, I,. n In+m = 0; {ii) If lf'""'{z)l ~ R:,. for each 0 ~ k ~ n, then 1/"'(z)l ~ 2R' for each 0 ~ k ~

mn;

(iii) If, in addition, z E AUG and f'""+l(z) = .z:o, then z e J,.,.+l·

PROOF. (i) If for some n, m E N, there exists z E J,. n ln+m• then r(z) = r+"'(z) .z:o, and hence /'"(.z:o) = f"'+"(z) = .z:o. Thus fP"'(.z:o) = .z:o for each pEN, which contradicts the fact that fP(.z:o) -too asp--. oo. (ii) Suppose on the contrary that there exist 0 ~ k < n and 0 < p < m such that lf"""+P(z)l < 2!(. Then

=

lf"'"'+"'(z)l


2R', we have 1/"'(z)l ~ 2!( for 0 ~ k ~ mn. We thus get a contradiction.

6.5. DIMENSION FOR THE CLASSES£ AND P

J75

(iii) Taking g to be the branch of /-(mn+l) that maps Zo to z and noting that B

for 0

~

k

~

c

~(Zo, l.zol/4),

1/k(z)l ;::: 21( ;::: R2

nm + 1, it follows from Lemma 6.4 (i) that fk(g(B))

c

~(fk(g(.zo)), 1/k(g(.zo))l/16) C {z:

lzl ;::: J(}

and hence z E Inm+I as claimed.

•

Now for each z E In, let

= 1/(r)*(z),

cS(z)

where /*(z) is the same as in (6.5). By Lemma 6.8 (i), cS(z) is a single-valued function. Set (6.10) s = inf{t : LcS(z)t < oo}. zEI

For any set H, we denote

D(H) where

=

sup x(z, w),

(6.11)

z ,wEH

x is the spherical metric.

Lemma 6.9 0< s

~

2.

Take a set U which satisfies Lemma 6.6. For each zEIn, there is a branch As R' ;::: R2 and U C B C ~(Zo, l.zol/8), we deduce from Lemma 6.4 (iv) that PROOF.

g

= 9z of /-n such that z = 9z(Zo)· D(g(U))

~

3Lg*(.zo)D(U)

~

3LcS(z)D(U)

and so there exists K > 0 such that, for each z E I , the spherical area of g(U) is at most K(cS(z)) 2 • If two different points z 1 ,z2 E /,then 9z 1 (U) ng,. 2 (U) = 0. Thus K Ezei cS(z) 2 < oo and so s ~ 2. Now by Lemma 6.7 (ii), there exists w0 E G' with f(wo) = Zo· We let Jn = {z : rn(z) = Wo, rk(z) E G', k = 0, 1, ... , n- 1}. It follows from G' C {lzl;::: ](,:_} C {lzl ;::: ~} and Lemma 6.8 that Jn C Jnm+l· Put K' = f*(wo),K > max{1, sup(/m)*(z)}. zEG'

For z E Jn we have cS(z)

=

1 umn+I )•(z)

1

= /*(wo)

n-l

II

k=O

1

1 (/m)•(fmk(z)) ;::: K' K".

CHAPTER 6. HAUSDORFF DIMENSION OF JULIA SETS

176

By Lemma 6.7 (iii), we see that Jn contains at least 2n points and so

This implies that Eaer 6( z)' The proof is complete. For 0

0. •

H C C we define

= c,

L

6(z)',

zeinG

where c, is chosen to be 1 or 1/ Eaer6(z)' provided that t < 8 or t > 8 respectively. .Then fort > s, 1St is a measure supported on AUG. Let ~-'• be a weak limit of the measures 1-'t as t '\. s. Then spt~-'• C AUG and ~S.(A U G) = 1. Lemma 6.10 (i) There emts an E > 0 6Uch that 1-'c(A) ~ E~St(G) for 0 (ii) For a ~ t ~ 2, 1-'t(A) ~ E/(1 +E).

1611'2 L 26 C, we obtain

A ( rn-1 ( Zo,) 1611'2 £2flrp-1rp) A( n-1( Zo ),rp-1 / C. ) g (z ) E ~ log C ~ r

r,

r,

(6.12)

Taking h to be the branch of /-c,- 1) satisfying h(JP- 1 (Zo)) = Zo· Ail 1/"(Zo)l ~ ~ R2 for 0 ~ q ~ p - 1, we deduce from Lemma 6.4 that

4R'

Combining this and (6.12) in the case q = 0 we see that there exists awE A such that JP(w) = z. Clearly 1/"(w)l > 2R' for 0 ~ q ~ p-1. Thus for each z E InnG, there exists awE A such that JP(w) = z and 1/"(w)l ~ 2R' for each 0 ~ q ~ p. Let g be the branch of r 0, 0 R/2 and so, as logR' > 81r, the result follows. (ii) Note that

E o·-..

2

+

zeinA

:E o·-.. . = 2

:E o·-..

zEinG 00

zEI

and by Lemma 6.10, /J•-o/2(G) ~ /J•-o/2(A)/c = /Ja-o/2(A)/c,

2

178

CHAPTER 6. HAUSDORFF DIMENSION OF JULIA SETS

we thus have

00

L L

l'•-o/2(.A) =

6(z)•-•/2 = oo.

(6.14)

n•l.rE/,.n.A

If there are only finitely many n such that

L

6(z)•-•12 ~ K,

.rE/,.nA

then there must exist some K'

~

K such that

L

6(z)•-o/2 ~ K'

.rEI,.n.A

for each n EN. It follows from (i) that, for each n EN,

L

6(z)•-•1 2 ~ sup 6(z)•l2 •

.rE/,.nA

and so,

.rE/,.nA

L L 00

L

6(z)•-• < K'(KIKi"] 0 12 ,

.rE/,.nA

6(z)•-•/2 < K'(KI)•/2

n=l .rEI,.nA

L K2no/2 < oo, 00

n=l

•

which contradicts (6.14). The proof is thus complete.

Now we put ~

= {g(.A): gisabranchofr"withg(Zo) E I,.},

~ .. =

{g(B): gisabranchofr"withg(~) E I,.}.

Lemma 6.12 There exists

no E N

such that, for n

~

no,

d(.A,.) < d(B,.) < 1

r"

PROOF. Let g denote the branch of which satisfies g(B) = B,. and g(.A) =.A,.. = z' E I,. we have IP'(g(Zo))l > [( for 0 ~ k ~ n and so, from Lemma

As g(Zo) 6.4,

D(.A,.)

< D(B,.)

~

supg•(z)D(B)

.rEB

~

3Lg•(Zo)D(B)

3U(z')D(B), here D() is the same as in (6.11). By Lemma 6.11 (i) we have 6(z') Combining this with the above inequality we get

< K 1Ki"·

6.5. DIMENSION FOR THE CLASSES & AND 'P As Bn

n In"# 0

and In

c

AUG

Bn

c {z: lzl :5

179

~r,}, the above inequality implies

11 {z: lzl :5 -gr,}

C

for large n. Thus for these n, there exists K

> 0 such that

The result follows.

Lemma 6.13 There exists a K3 > 0 such that, for each An E 2l., with n

~no,

•

K3#-'.(An) ~ (d(AnW· PROOF. For each z E Ip n A, there exists a w E An such that f"(w) = z. Let h and g be the branches of f-P and 1-n respectively such that h(Zo) = z E A and g(Zo) = z' E An. Then g(h) is the branch of r

£

" - 1 +t

(

D(A,.) ) 9L 2 D(A)

t

If n ~ no, then d(A,.) < 1 by Lemma 6.12. As A,. n I,. ~ 0 and I,. follows that, for some K' > 0, we have

D(A,.)

~

(6.16)

c AuG,

it

d(A,.)/K'.

Let t '\. 8 in (6.16), we imply from Theorem 6.1 that

~-'•

t ( D(A,.) )" (A" ) -> 1+£ 9L2K'D(A)

The proof is complete.

Step 3. The number 8 defined in {6.10} satisfi.es

8

> 1.

•

Lemma 6.14 Let m and p be the same as in Lemma 6.1. Then for each n ~ 0 there. exist 2" curoes 'Yn,j, j = 1, ... , 2", each of which join& {lzl = 19r,./18} to oo and lies in {lzl ~ 17r,./18} with (i) "Yo,1 c r, {ii} for each 0$ r $ n, f""('y,.,j) C 'Yn-l,j for some 1 $ j $ 2n-r, (iii) if i ~ j, then 'Yn,i n 'Yn,j = 0. At first we consider the case n = 0. By Lemma6.7 (i) we know that r joins By Lemma 6.7 (iv) we get a curve "Yo,I c rn{lzl ~ 19rp/18}. Now we assume that the result is true for n- 1 and, for some 1 $ j $ 2n-1, consider the curve 'Yn-l,j· Take z' with lz'l = 19rp/18. By Lemma 6.7 (ill), there exist two points w~o w 2 E G' such. that f'"(w 1) = /'"(!»2) = z'. Let h,. denote the branch of rm satisfying ~A:(z') = WA: (k = 1, 2). From Lemma 6.7 (i) we know that sing(rm> c ~(0, 1(,.) c ~(0, .~(:.) c ~(0, 9r,/10}, PROOF.

{lzl = 19r,/18} to oo.

thus we can continue h" univalently along 'Yn-l,j· Note that w1 E G', the curve r n,2j := hl('Yn-l,j) must join {lzl = 19r,/18} to oo. If r n,2j does not contain a curve 'Yn,2J C {lzl ~ 17r,./18} which joins {lzl = 19r,./18} to oo, then the length of r n,2j n {17r,/18 $ lzl $ 19rp/18}

= SUPJai:9Gr./18 rm(z)!. Since rm ~(g(Zo), jg'(Zo)IR/(4C)) ::> ~(g(Zo), jg'(w')IR/(4CL27 )) .

(6.22)

We now take a collection G,.,, of disjoint curves "Yk such that 'Yic C 'Y"J, 'Yic n-YnJ 0, /"'"+ 1 maps 'Yic univalently onto {lzl = R} and

u

,..ea.. ,

:f:

(6.23}

::>"'f',.J.

H "'fk = g,.( {lzl = R} ), then 'Yic C G by (6.21}, and so g,.(A) E .f),.,j. Combining (6.20}, (6.22) and (6.23) we obtain

"•({I•I•R})EG,. J

1

~ 87TLMC

L

d(g,.({lzl

= R}))

g•({ •I=R})eG,.,1

d("'.,,J} > r, ~ 87TL 54 C - 2881rL 54 C. This completes the proof of the lemma.

•

Now we prove that s > 1. H .4,.,.+1 = g(A) E ~n+lo then by Lemma 6.4, g is univalent in Band so all the closures of the sets in uj:1.f)nJ are disjoint. It therefore follows from Lemma 6.13 that, for n ~no. 2"

L L

J•l A,. ... -. e~ ...J

2"

d(Am,.+l)• ~ Ka

L L

J•l A.,.,.+l e~ .. ,

~.(A...n+l) ~ Ka.

(6.24)

For n ~no, 1 ~ j ~ 2" and each .4,.,.+1 E .f)nJ• we deduce from Lemma 6.12 that d(Am,.+t) < 1. Thus, if s ~ 1, then by (6.24) we have 2"

L

L

j=l A.,.,.+ I El),. J

d(Amra+t) ~ Ka.

(6.25)

6.5. DIMENSION FOR THE CLASSES t AND P

183

On the other hand, by Lemma 6.15 {ii) we have, for each n E N, 2n

L

L

d{Amn+l) 2:: 2" K4.

j=l Amn+l EJ)n ,j

Combining this with {6.25) we see that, for n 2:: no, 2" K 4 $ K 3 • Obviously, this is impossible, and so we must have s > 1 as claimed. Step 4. For the numbers defined in {6.10), we shall prove that dim.!(/) 2:: s. Lemma 6.16 Let a E {0, s). Then there exists a positive integer M such that

L

d(BM)•-a 2:: £2(•-a)d{B)'-a.

BME'BM,BMCB

PROOF. Take an integer n 2:: no and a set Bn = g(B) E ~n with g(.zo) EA. By Lemma 6.12 we see that d{Bn) < 1 and hence Bn C B. Note that g(.zo) E In, for 0 $ r $ n, we have

Thus, by Lemma 6.4,

It follows that there exists a constant K

> 1 such that

d(Bn) 2:: g*(.zo)d(B)/(3LK)

where z

= g(.zo) E In.

Hence, for n 2::

L

= 6(z)d(B)/(3LK),

no,

d(Bn)•-a 2:: d(B)(3LK) 4 -a

BnE'Bn,BnCB

L

h(A(I"-1-PM (g(.zo)), Rl 4800£ 3 , we deduce from Lemma 6.4, (6.29) and (6.31) that

f9-J-PM (BM)

= f"-2-PM (g(B)) = f"-2-PM (g(A(.zo, 2R/C)))

c A(/"-2-PM (g(.zo)), 2RLI(/"-2-PM (g))' (Zo)I/C)

c

A 0. We can find a disk in the plane, which we may assume to be the unit disk ~. such that mes(~ n X) > 0. For any positive integer k, there exist 2k disjoint sectors S;(i = 1, ... , 2k) in ~ such that mes(S; n X) > 0 for each j. Furthermore, for each S; and any!> 0, there is a sub-sector

193

194

CHAPTER 7. MISCELLANEOUS

such that

mes(O;(t) n X)> 0.

Next we construct a quasi-conformal homeomorphism as in the proof of Theorem 4.32. Suppose that the set aing(/- 1 ) has p points W = {a 1 , ••• , ap}. Take an integer N > 2p + 8. When k is large enough, we can choose N disjoint sectors such that each sector contains a sub-sector of the form

D;(6) = {z: argz E I1 =(a;- 6,a; + 6)} with the property that mu(D;(6) nX) > 0, where 0 < 6 < determined later, 6 < a1 < ... < aN < 271'- 6. Define

t/J ·(fJ)-{ 6exp{-f52/[f52-(9-a;) 2)} 1

0

-

,

,

;

is small and will be

iffJei,, if 9 fl. I1 •

(7.1)

Then 1/J; is C 00 ,lt/1;(9)1 S 6/e,ltPj(fJ)I S M6, where M > 1 is a constant independent of j and 6. Let

T= {t = (tll···•tN) ERN: For z

= re11 E ~and t E T,

it1 i S 6,1 Sj S N}.

put

Choose 6 so small that M N 6 < 1. Then

II#JO(t, z)lloo S c < 1. Let Xn = r o f(z) =go l/>(z).

198

CHAPTER 7. MISCELLANEOUS

Lemma 7.1 For any

E

> 0, let D

= {z:

cf>(z) -IRezll < e}. Then

dens(D, ~(z, 1)) - 1 tU

IRezl - oo.

PROOF. We will use a similar method as in the proof of Theorem 6.11. Taking x > jRezjlarge enough and h,. = 2tf(x- e). Consider the region D(h,.) = {z: IRe(z)l > h,.}. Then for z E D(h,.), 1/'1 > O(exp(h,.)) and If"/ /'I ~ 1. Let B be a box in D(h,.), we see that /(B) n D(hlc+l) looks very nearly like a square of side at least O(exp(h,.)), possibly with a strip of width 2hlc+ 1 deleted. Thus the set /(B) n D(h~c+d can be packed with disjoint boxes whose total density in f(B) exceeds 1 - O(hlc+ 1 exp( -h,.)). We refer these boxes in such a packing as pack(f(B)). Let

Eo= {Bo: Bo

= ~(z,1)}

and define inductively

E,. = {G: G c V E E~c-1./,.(G) E pack(f,.(V)) }. We denote by U,. the union of all G E E1c (k = 0,1, ... ). Since

we have, for any G E

Let V

= nr;, 1u,..

E~c,

Then

~

ll (1- O(h,.+l exp( -h,.)))

=

1- 0

00

dens(V, ~(z,1))

(f:

h,.+l exp( -h,.))

+ ...

lc=l

-

as x- oo. Note that for any wE V C 0

_

1

~(z,l),

there exists a y such that

lim IRer(w)l > lim g"(x- e) 2 g"(Jf) - n-++oo g"(Jf) ·

- n-++oo

It follows that cf>(w) ~ x- E > IRewl- e. On the other hand, for any z such that IRezl is large enough, we have cf>(z) $ IRezl +E. Thus, we can say w E D, i.e., V CD. The conclusion follows. •

7.1. MEASURABLE DYNAMICS

Lemma 7.2 For any intenJal I

c

199

[1,+oo}, f/J- 1 (1) has positive measure.

PROOF •• From Lemma 7.1 we see that the set {z : tjJ(z) e [x- 1,x + 1]} has positive measure when x is large enough. Now as n is large, gn(I) will contains [x- 1, x + 1]. Moreover,

f/J(r(f/J- 1 (/)))

= gn(I) :J [x- 1, x + 1].

Thus r(f/J- 1 (/)) has positive measure, and

r 1, it is easy to check that r(z)--+ oo as n--+ oo. Thus by Corollary 5.3, all these kinds of points are in J(f). For a fixed x > 1, we take two disjoint intervals 11 and / 2 such that i, C (g- 1 (x},x) for j = 1,2. Thus by Lemma 7.2, tj)- 1 (I,) has positive measure for j=1,2.Set 0 9 (I;) = {w = g"(z): n = 0, 1, ... , z E I;}

and

0J(f/J- 1 (I;)) = {w = r(z): n = 0, 1, ... , z E t/J- 1 (1;)}.

Then both 0 9 (1;} and

o,(4J- 1(I,)) have positive measure, and 0 9(/1) n 0 9(/2)

= 0.

(7.7}

Next we prove that n is also empty. In fact, if the intersection is not empty, then there exist positive integers n and m, and two points b; E / 3 (j = 1, 2) such that

o,(f/J- 1 (/1))

o,(f/J- 1 (/2 ))

r 0 and

IIm(E~+l(z})l

Aexp{Re(E~(z))}l sin(Im(E~(z)))l ~

Aexp{Re(E~(z))}jlm(E~(z))l

......

oo,

•

which is a contradiction.

PROOF OF THEOREM 7.5. Suppose the theorem is false, that is, there is a homeomorphism 4J : C --+ C such that

4J o EfJ

= E.., o l/J.

Since 0 is the unique exceptional value for both EfJ and E..,, ljJ(O) = 0. Furthermore, a+ is topologically distinguished as the only EfJ and E.., forward invariant curve which starts at 0 and tends to oo. Combining this and Corollary 7.2 we conclude that l/J(R+) =JR.+. Thus 4J restricts to a homeomorphism of the positive real axis. Among preimages of JR.+, IR- is distinguished for both EfJ and E..,. Hence lfJ(JR-) =a- . Let L; = {z: Imz = (2j+1)7r}, j E Z. Note that E~ 1 (JR-) consists of the lines L; for A> 1/e, hence the L,'s are permuted by l/J. Thus there exists an integer m such that l/J(L;) = l/J(Lj+m)· It follows from l/J(lR) = IR that m = 0, and so, ljJ(LJ)

= Lr

For all j E Z, let D; = {z : (2j- 1)7r < Imz < (2j + l)1r}. We cut the plane along the non-positive real axis and choose log~.; to be the branch of the inverse of E~(z} taking values in D;. Since 4J preserves each component of 8D; in an orientation preserving manner, it also preserves D;. Thus for all j,

4J o logp,; = log..,J olj). For any p, q E Z define

(7.8)

202

CHAPTER 7. MISCELLANEOUS

which lands on they axis at 2wqi. Note that 4>(/3) ="'(and A = {3, 'Y· It follows from (7.8) that, for all p, q,

log~,;(A)

= 2wqi for (7.9)

We claim that

1R = {Oit•4 : p,q E Z}

In fact, it suffices to consider x if Rez > K, then

(A= /3,"'f).

(7.10)

> K > 0. For convenience we take K = 2 80 that

l(log~,;)'(z)l S ~-

Fix x > K, large pandA= /3 or 'Y· Let q = q(p) be the smallest integer such that log(2wq) -log A 2: ~- 1 (x). 1f p is large enough, 1

log(2wq) -log A- ~- (x)


2 for j = 0, ... , p - 1. Hence

Therefore we find a sequence converging to x. Assume /3 < 'Y and fix a< 1 80 that /3 ))

> &E% (~,q(J>)) > M.

By induction, we deduce from (7.12) that &~-l(z"•q(J>)) "Y

"Y

> &E"-t(~'q(p)) {3 {3 •

Therefore IE"Y(~- 1 (z~· 9

))I

21rq

=

"'(exp(&~- 1 (z~· 9 U>>))

>

{3exp(&E~-t(~'q(p)))

=

21rq,

•

which is a contradiction.

We say that A is periodic if it is a periodic point of E>. and, A is escaped if £>:(0)-+ oo as n-+ oo. The following are some further results (see [177, 217]): Theorem 7.6 For the function E>., the set of structurally unstable A (J-unstable A) coincides with each of the following two sets: 1. the closure of the set of periodic A; 2. the boundary of the set of escaped A. Consequently, E>. is structurally stable if and only if it keeps the zero orbit bounded or unbounded by a perturbation of E>.. Now we consider the function f,..(z) can be found in [215]. Theorem 7.7 If ReJ.L

= zez+p.

The following three theorems

< 0 or J.l E {J.LIIJ.L-11 < 1}, then f,.. is structurally stable.

For the proof of this theorem, we need some preparations. We define a multivalued analytic function o:, : M = M1,. -+ C as the set of the solutions to the equation f"(o:) = o:, the function o:, has oJ?ly algebraic singularities. Let o:,,i(f) be a branch of o:,.. Set

N,

= {! E Ml(f")'(o:,,;{/)) = 1 for some i}, and

N=U!tN,, E=M -N. Lemma 7.4 Let 6 > 0 be fixed. Then the'following statements are equivalent: (a) The period of the longest attracting cycle off,.. is bounded uniformly in

O.s(J.Lo).

(b) For each J.l E 06(J.Lo), f,.. E E·

204

CHAPTER 7. MISCELLANEOUS (c) For any f.J E O.s(t.Jo), /,.

u J-stable in O.s(t.Jo).

{d) {Sn(l')} is normal in 06(J.&o).

Here O.s(t.Jo)

= {t.JIIt.J- t.Jol < 6}, ln(f.J) = f;( -1).

PROOF. We take M = {f,.lf.J E O.s(t.Jo)}, let 1/>(z) = ze"2 -~' 1 and tJ1 = id, hence t/1 o /,. 1 = /,. 2 o t/J. We see that /,. 1 " ' /,.2 (for any f.Jl, f.J2 E O.s(f.JO)), therefore, M is a submanifold of S2, here S9 denotes the class of the entire functions with q singularities. Note that aing(/; 1 ) = {0,/,.(-1)}. From r(c1 (/)) = J:(O) = 0 we know that /n(cl(/))';'= 1 is a normal family in O.s(t.Jo). Lemma 7.4 follows from r(C2(/)) = /;(-1) = an(l') and [217, Theorem 1]. • The following lemma is contained in [85, Theorem 10]. Lemma 7.5 Let f be an entire function of finite type. Iff has a unique attracting fixed point which attracts all points of F(f) and f u J-stable, then f u structurally stable.

PROOF OF 'fHEOREM 7.7. First we cosinder Ret.Jo < 0. By the proof of Proposition 4.8, for any R.ef.J < 0, we know that the function f,. has only one attractive fixed point z = 0 which attracts all points of F(f,.). Therefore, for any R.et.Jo < 0, there exists 6(> 0) such that O.s(t.Jo) C {ziRez < 0}, that is, for any I' E O.s(t.Jo), /,. has only one attractive fixed point Zo = 0. Suppose that there exists a I'' E O.s(t.Jo) and mEN such that am(#'')= -1, that is,/;( -1) = -1, and z1 = -1 is the critical point of /,.•. This contradicts the fact that the point Zo = 0 attracts all points of F(/,.•). Hence for any f.J E O.s(t.Jo) and n EN, an(l') :f: -1, and it is obvious that an(l') :f: 0 for any p. E O.s(t.Jo) and n E N. By Montel's Theorem, {an(P.)}:'= 1 is normal in O.s(t.Jo). From Lemma 7.4 we deduce that /,.., is J-stable in O.s(t.Jo), and so, by Lemma 7.5, /,.., is structurally stable in O.s(t.Jo). For any p. with IP. - 1 < 1, there exists an attractive fixed point Zo = -1'(1/'( -p.)l = IP. - 11 < 1). For any Rep. > 0, from 1/'(0)I = e'"" > 1 we see that 0 E J(J,.). It is easy to check that the point Zo = -1' attracts all points of F(f,.) for any p. E {p.: IP.- 11 < 1}, similar to the proof for R.ef.J < 0, we can obtain the last part of the theorem. • Theorem 7.8

!,. u not structurally stable when J::(-1) = -p. for some n EN.

.r:::,+l (

PROOF. Suppose that there exists f.JO with Ret.Jo > 2 and -1) = - f.JO such that /,..,(z) is structurally stable. Then there is a neighborhood O.s(t.Jo) of f.JO, such that, for any p. E O.s(t.Jo), there exists a homeomorphism hJ : C-+ C such that

(7.13) This yields

r:

0

h,(f,..,( -1))

= hJ 0 r,::,+ 1 ( -1) = hJ( -t.Jo).

It is easy to verify from (7.13) that if z is a repulsive fixed point or critical value or asymptotic value of/,.., then hJ(z) is a repulsive fixed point or critical value

7.1. MEASURABLE DYNAMICS

205

or asymptotic value of /1-1(z). Obviously, ht(-l'o) = -p., ht(/1-10(-1)) = /1-1{-1). Hence /;'+1(-1} = -p.. Keeping in mind that Bm+t(P.) = /;:'+ 1(-1), we deduce that Bm+1(P.)

= -p.

•

holds for any JJ E 06(1-'o), this is impossible. The proof of the theorem is complete .

Similarly we can prove the following result.

Theorem 7.9 If/;:( -1}

= -1 for some n EN,

then /1-1 is not structurally stable.

From Proposition 4.8 we know that the Fatou set F(/1-1) has only one component for ReJJ < 0 and has infinitely many components for JJ > 0. Therefore we have

Theorem 7.10 The function /1-1 is not structurally stable at the point JJ = 0. Next we shall study the conjugacy and density of the family

F = {T>.(z) =..X tanz, . X E C\{0}}. The functions in this family offer an easier domain to study than rational maps, because an exponential contraction occurs in the pre-image of a neighborhood of an omitted value. In the quadratic family {z2 + ..X, . X E C}, one can show that each hyperbolic component in the parameter plane contains a unique point (its "center") that has super-attractive periodic cycle. Since ~tan z has no superattractive cycles, we need another way to enumerate its hyperbolic components. A virtual center is a parameter value for which the asymptotic values eventually land on a pole. For each n E Z, let ln = (n + 1/2)7r +it, t E JR, and let

Ln

= {x + iy: (n- 1/2}7r ~ x < (n + 1/2}7r, 11 E IR}.

T;,i

We define (or T; 1 if~ is clear) the branch of the inverse map whose real part is in the strip Ln. For a given p E N we define a branch of r;" by

r-"

n,.,~

and set n,

= (n 1 , n 2 , ••• , n,). 1t

and

1 o . .. o r- 1 o r- 1 = rn,,~ n2,~ n1,A'

=

We call n, the itinerary of the map T;">.· Set P> {~ E

1-fl ={A E C\{0} : T>.

C\{0} : F(T>.) =f; 0} has an attracting periodic cycle}.

In any component of the Mandelbrot set, all the quadratic maps, except the one corresponding to the center are quasiconformally conjugate. We have an analogous situation for the hyperbolic components of 1t0 • The following conjugacy theorem was proved by Jiang [129], who also generated three computer pictures, figures 7.1, 7.2 and 7.3.

209

7.1. MEASURABLE DYNAMICS

Theorem 7.11 Each component 0 of 'H.0 corresponds to a quasiconformal subfamily of :F; that is, for any ~ 1 .~2 E 0, there exists a qutUiconformal map r/J : C -+ C such that T~, o t1> = tf> o T~,. PROOF. Let ~ E 0 c 1f.O and let Zo, z1 , ... , z,_ 1 be an attracting periodic cycle of T~ with multiplier mo. Then there exist a neighborhood U of Zo and a holomorphic homeomorphism ho: U-+ ~o. where ~o = {z: lzl < ro}, such that ho(Zo) = 0 and TPo = ho o Tfo o h0 1 has the form (,..... mo(. The annulus

Ao = {(: rolmol < 1(1 < ro}

is mapped one-to-one by tpo onto the annulus

AA = {(: rolmol 2 < 1(1 < rolmol}. Choose 1 > r > mo and pick m in ~r = {z : mo < lzl < r}. Let Am be the annulus defined by {( : rolml < 1(1 < ro} and set TPm(() = m(. Now choose a measurable structure on Ao as follows. Let f : Ao -+ Am be the affine stretch map f(r,8),..... (Kr,8) where K =1m/mol. Thus, •

fc

K -1 (

J.&(() = f, = K with

{l(()

+ 1 ('

being a measurable function satisfying

ll{l(()lloo = k =

K-1 K +1

< 1.

Extend f.t to ~ by setting {l( () = {l( f/J[) (()) for appropriate n. Pull f.t back to U by setting J.&(z) = f.t o ho(z). Extend J.&(z) to the inverse orbit of h0 1 (Ao) under Tfo by: ( np ) (r;:)'(z) J.&(z)- J.l T~o (z) (T;!')'(z)

for z E T~""(hQ 1 (Ao)). This defines the measurable structure in the whole attracting basin of Zo under 71'0 • This structure is transported to the basin of Z; by J10 • If T~0 has two attracting periodic cycles, then they are symmetric with respect to the origin, and the structure can be transported to the full basin of the second cycle by the symmetry. Consequently the structure J.l is defined everywhere in F(T~). Set J.&(z) = 0 for z E J(T~). By the measurable Riemann mapping theorem, there exists a quasiconformal map t1> : C ,..... C that is a solution of the Beltrami equation tf>f = J.I4Jz. The map t1> is uniquely determined up to an affine transformation. We assume that t1> fixes the origin and maps the two asymptotic values to a pair of points symmetric with respect to the origin. The map tf> o T"v o t/J- 1 is meromorphic by our construction, which fixes the origin and has exactly two symmetric asymptotic values and no critical points. By Theorem 5.2 there exists a

210

CHAPTER 7. MISCELLANEOUS

>.such that T~ = tjJoT>.o oq,- 1 E :F. It again follows from the measurable Riemann

mapping theorem that >. depends continuously on m with Am.ol = >.o. Therefore >.o and >. are in the same component n. of 1f!J. Since r can be arbitrarily cloee to 1, the multiplier m induces a covering map m: 0

~--+ ~. .

•

Now we consider the density in the family {>.tao z : >. E C\ {0}}. Let 0 be a component of 'lf!J\~. such that, for >. E 0, T~ has an attracting cycle of period p. Then it is easy to verify that T>. has one or two attracting periodic cycles of period p. From now on we reserve the notation Op for the components of 'lf!J\~. with two distinct attracting cycles. We use for the components with a single cycle of period 2p.

n;

Proposition 7.2 For any bounded hyperbolic component Op oro; of'lf!J\~. with > 1, the virtual center>.· is finite and 7f.'" 1(.>."i) = oo; that is, >.•i is a pre-pole (i.e., >.*i E Po) of order p- 1.

p

PROOF. For>. E Op, let Zo = .zo(.>.) be the attracting periodic point ofT>. of period p such that Zo belongs to the component Do of the regular set that contains the

asymptotic tract and such that >.i and z1 are both in D 1 = T>.(Do). Denote the preimage of .zo in the periodic cycle by z,- 1 and the preimage of Do by Dp- 1 . Then for some n, z,- 1 = T;,l(.zo). Since p > 1, the domains D,._ 1 and Do are different and so the map T~ : D,.-1 -+ Do is bijective. Since Do contains the asymptotic tract and the map is bijective, there must be a unique pole, say Bn, on 8Dp-1· To see this, note that there is a pre-asymptotic tract at Bn in D,.- 1 and so if 8Dp-I contained any other pole, there would be a pre-asymptotic tract in D,.- 1 at this pole and T>. would not he injective on Dp- 1. Suppoee that >. moves along the internal ray R( a) in Op to the virtual center >.• as r -+ 0, so that lim [Tf(.zo(.>.))]' = 0. >.~>.·

Since (taoz)' = sec2 z, and>.= Zi/tanZi- 1, the multiplier can be written as

The above two equ8Jities imply that sin 2z._ 1 -+ oo for some i, or equivalently, lmZi-1 -+ oo. Since Zo is in the asymptotic tract, we conclude Im.zo-+ oo. By hypothesis p > 1 and). " '::/: oo so Zp-2 '::/: z.- 1. We have lim .>.taoz,-1(.>.) >..!>.·

We conclude further that

=

lim Zo = oo.

>.~>.·

7.1. MEASURABLE DYNAMICS

211

and lim Z1(A)=.>.*i ~.!~·

so that .>."i is a pre-pole of ofT~· of order p -1. The other periodic orbit behaves symmetrically and hence -.>."i is also a pre-pole. If.>. E T~ has a single cycle containing the points Zo and Zp = -Zo, which lie in symmetric components Do and D, containing the asymptotic tracts. As above we deduce that there are unique poles Sn and -sn on their respective boundaries. If .>.• is the virtual center, we again conclude that .>.*i is a pre-pole of of/~· of order p-1. •

n;,

Set

C, = {.>.: Tf(>.i) = oo}, C =

' Uc,. 1

Keen-Kotus [135] were able to prove the following density theorem: Theorem 7.12 If >.o is such that J(T.x.,) = C, then for any e > 0 there exists a pEN and a .>.• such that l>.o- >.*I< e and T~· has either a parabolic point Z~· of period p with (71'.)'(z~·) = 1 or .>.• E Cp-1· PROOF. Choose a sequence cSn-+ 0, cSn < £. By Theorem 3.7 we can find pre-poles and repelling periodic points arbitrarily close to >.oi. Let v be a pre-pole such that lv- >.oil ~ !cSn; v = r:.,~.x.,(sm) for some odd q EN and some pole Sm. Define _1

1 >.o + iv i cSn i log(.>._ iv) ~ xk- 1og 1 >-o 1. 2 2 2

_

w = Tk,.x.,(v) =

where llog£1

Xk

>>

E [k7r·, (k + 1)7r), sow k2

Now consider T;~~(w). Since cSn > l.>.ol 2 ,

1j~~(w)~s,+

Xk-

c>.o Ao 16 I ~s,+-11 r I' 2log ii'o ogun i

Next choose l so large that T~.1.x.,(st) is very close to Sm and so that

T~1.x.,(~(s,, '

c>.o 1

Sj

+ nogT,;'l

))

has a small diameter. Note that

T,-1.\o 0 T;-1.\o (w ) ~ St + '

Thus

'

c.>.o 1

Sj

+ nogT,;'l

E

A( St.

L.1

Sj

c.>.o 1

+ nogT,;'l

)•

212

CHAPTER 7. MISCELLANEOUS Now

Vn(.\o)

=T:_~>... o T,;,1,... o T,~~(Bj)

is a pre-pole of order Pn = q + 3 landing on closes to ii. Thus,

Bj.

If we choose llarge enough, then

Vn(~}

so that

Tf;- (~(vn(~},fln}) ::> Tj~~ oTir~l,(~(vn(~},fln}). 1

Hence there must be a repelling periodic point z..,o(~} of period Pn + 1 such that if Z..,i(~} = r,...(z..,o}, i = 0, ... ,pn, then

Z..,o

R:

+ cS,.

(7.14}

Tj~~(Z..,p,.}

Z..,p,. -1 Z..,p,.

~i

= R:

R: Bj

+ 2:11:,n- ~1/iaol' 1

_1 1 (~ + iz..,o) T" ,...(z..,o} = + 2 . log ~ . • ' - •z...o i cSn I 2:A:,n - 2 log 1 ~ ' 2

(7.15) (7.16) (7.17)

where 2:1r,n E [kw, (k + 1}w}. Set C = 1(71.,}'(vn}l and set

The multiplier is m(z,.(~}} p,.-2

=

Cs1•j((kw} 2 we have

Since lm(z,.(~}}l

(c~·J.. ,y' j

~ (~ + z!,... -1) (~ + z!,p,.) (~ + z!,o) ·

R:

+ log2 cS,.}cS,.,

lm(z,.(~}}l

if we choose

1

= 0( 8ja" 2J: 1 !iJ: ). og a,.

s, so that l•1l >> (7.18)

We deform the cycle z.. (~} by varying,\. Note that, p,. + 1 is odd, and so the corresponding cycle is not symmetric. We define Tin as the radius of the maximal

213

7.1. MEASURABLE DYNAMICS

ball Bn = ~(Ao, f1n) on which Zn(.X) can be analytically continued. In particular then, the functions hn{.X) = Zn,p,.-1(-X) are holomorphic maps from Bn to C. Since a pole can never belong to a periodic cycle, hn(Bn) contains no poles. It follows that if B = n~ 1 Bn contains non-empty interior B, the family {hn} is normal on

B.

Set 9n(.X) = ~(.X). Then 9n(.X) =tan Zn,p-2(-X) +

+

p-2

k

k=1

•=1

L .X" tan Zn,p-2-k(..X) II sec2 Zn,p-1-k(..X)

,p-1

A

p-2

II sec i=O

2

Zn,1 (')dzn,O A d,X ,

where dzn,o(.X) tan Zn,p(.X) + Et,1 .X" tan Zn,p-k(.X)ll~,;: sec2 Zn,p-i(.X) d.X 1-m(..X)

The following three lemmas will complete the proof of the theorem.

Lemma 7.6 Let Zn,i(.X), i = 0, ... ,p be a repelling periodic cycle. If its multiplier m(.X) has large enough absolute value, then

.X2 sec2 Zn,p-1 (.X) sec2 Zn,p(.X)

tan Zn,p-1 (.X) .Xsec2 Zn,p-1(-X)

(7.19)

PROOF. In the equation for dzn,o/d.X we may replace the denominator by -m(.X) provided lm(.X)I is large enough and obtain the following expression for dzn,o/d>.:

Substituting this expression into the formula for 9n(>.) will lead to (7.19).

•

Lemma 7.7

PROOF. Applying Lemma 7.6 to the sequence Zn,i(Ao) (i = 0, ... ,pn) of periodic cycles we get

>.2o sec2 Zn,p,. -1 (>.o) sec2 Zn,p,. (>.o) AoZn,O

= -l..xg + z~.p.. (>.o)][~ + z~. 0 (>.o)J

tan Zn,p,. -dAn)

>.n sec2 Zn,p,. -1 (>.o) Zn,p,.(Ao)

214

CHAPTER 7. MISCELLANEOUS

Substituting equations (7.14)- (7.17) into the above we get

Since 6,. _. 0, 6,. > -log 1/tol >> ~ >> 1~ 2 1,

l(~i +6,.)( 2~")- $Jo,n + ~ logj:~ II~ 0 (~). Moreover,

Consequently

Since

x.,,,. E [k1r, (k + 1)11') and k is fixed we have

•

Lemma 7.8 B = {~}

PROOF • If B is not the singleton {~}.then there exists a ball i:J C B such that the family {h,.: i:J--. C, n EN} is normal on i:J. Thus, there is a subsequence h... which converges locally uniformly on i:J to some meromorphic function h : i:J --. C. By construction h(~) = l.imn-.oo h,.(~) = s1 , soh is analytic at~. It is therefore holomorphic on some small disc D = ~(~. r) C i:J. Thus

dh

d~ (~) =

where we may take "'(

1

1 g(~)d,\ 21Ti .., (~- ~)2'

= ~ + reil, 8 E [0, 211" ), we obtain

215

7.2. PERMUTABLE FUNCTIONS

Since {h...} converges uniformly on D, 9nk(.X) = d~x· also converges uniformly on D to~· The Cauchy formula implies that g(Ao) is bounded, while by Lemma 7.7, 9n.(Aa) = 0((5nlog2 5n)- 1 )-+ oo if 5n-+ 0. Thus we arrive at a contradiction. The boundary of Bn contains either a singularity of the cycle or the origin. By the argument above, ~(Ao, lAo I) cannot be contained in infinitely many Bn, so for all but finitely many periodic cycles there must be a singularity of the boundary on Bn. • Corollary 7.3 Let the asymptotic values of T~0 belong to J(T~). Then for any e > 0 there is a hyperbolic component Op of1t0 and a A E Op such that lA- Aoi < 2e.

7.2

Permutable functions

Fatou in 1918 proved that for two rational functions f and g, if they are permutable, i.e., f(g) = g(f), then F(/) = F(g). He then asked the following: Question 7.1 For two given permutable tmnscendental entire functions f and g, does it follows that F(f) = F(g) 'I This question is still open. However, for some particular permutable functionS, we can get an affirmative answer. The argument used to prove the following result is essentially due to Baker (16). Theorem 7.13 Suppose f and g are permutable tmnscendental entire functions. If g(z) = af(z) + b, where a, b are constants and a :f: 0. Then F(f) = F(g). PROOF. Obviously, it is enough to show that F(/) C F(g). To this end, we need only to prove that g(F(/)) c F(/). In fact, if g(F(/)) C F(/), then for every component U ofF(/), g(U) c F(/), hence {g"} omits at least two values in U. By Monte! theorem, {g"} is normal in U, this implies F(/) c F(g). Now suppose g(F(/)) C F(/) is false. Take o E F(/), g(o) ¢ F(/) and consider the neighborhood Uo C F(/) which contains o. {/"}either converges to an entire function, say h(z) in Uo or converges to infinity in U0 • In the former case, {/"} converges togo h in g(Uo)· By the fact that /" o g = go/", we have g(Uo) C F(/) which contradicts our assumption. From now on, we assume {/"} -+ oo in U0 • It follows that there exists n 0 such that for all n > n 0 , 1/"1 > A for all z E U0 , where A > lb/~ 1 • Thus, for z E /"(U0 ), where n > n 0 , 1/(z)i >A. Since g(o) ¢ F(/), by Theorem 3.14, /m takes all values with at most one exception for arbitrarily large m in g(U0 ). Therefore, there exists t = g({3), {j E Uo such that for m > n0

ir(t)i = lfm(g(t3))1 = lg(fm(t3))i


af('Y)- g('Y)I > laiA- 1,

lbf+1 >A lal '

which contradicts the choice of A. Hence g(F(/)) C F(/), and the proof is completed. • Remark. If g =a/ +b is permutable with/, where a and bare constants, then by the Hadamard three circles theorem, it is easy to conclude that f(az +b) = af(z) - b with lal = 1. Recently, Poon and Yang [173] proved the following result. Theorem 7.14 Suppose that I and g are two permutable transcendental entire functions. If both sing(f- 1 ) and sing(g- 1 ) are isolated in the finite complex plane, then F(J) = F(g).

As in the proof of Theorem 7.13, it is enough to show that g(F(/)) c F(f). Assume on the contrary that g(F(I)) ¢. F(J). Then there exists ZoE g(F(/)) and W 0 E F(/) such that Z0 E J(/) and g(w0 ) = Z0 • We consider two cases: (i) z0 f/. .ting(g- 1 ). Then there exists a disc D = A(z0 , r) such that the analytic branch g- 1 of g is defined on D with g- 1 (z0 ) = W 0 • Since z0 E J(/), in the disk D, there exists a sequent.'e of repelling periodic points {On} of I with the period tin such that On- Z 0 as n - 00. For each OnE D. we can find a disc Dn = A(on,cSn) such that Dn c D. Note that l' . (on)= On. By the continuity of 'there exists a disc Un = A(on,tn) such that, for all z E Un, r"(z) E Dn. Thus for all z E Un, and by the permutability of I and g, we have PROOF.

r·

g- 1 o !' · (z)

=f

· o g- 1 (z).

By taking the derivative on both sides, we have (g- 1 )' (onHr" )'(on)

= (r" >' (g- 1 (on))(g- 1 )'(on)·

Thus g- 1 (on) is a repelling periodic point of I and 9- 1 (on)- g- 1 {zo) =

Wo

W 0 E J(/), which is a contradiction. (ii) Z0 E sing(g- 1 ). Since the set •ing(g- 1 ) is isolated, there exists a disc D = A(z0 , r) such that Z0 is the only singularity of g"'" 1 in D. Moreover, since g(w0 ) = Zo and W 0 E F(/), we can find a disc U0 containing W 0 such that Uo C F(/) and g(Uo) c D.

i.e.

7.2. PERMUTABLE FUNCTIONS

217

E g(Uo)\{zo} such that z 1 E J{f). By our choice, Wt E F{f) such that g(wt) = z 1 • This is impossible by the case (i). Hence in the nighborhood g(U0 ), only Zo is a point of J{f), which contradicts the fact that J{f) is perfect. The proof is complete. •

Assume there exists

Zt

Zt

r/. sing(g-l ), and there exists a

Corollary 7.4 Let f and g be two tronscendental entire functions of finite order. Iff and g are permutable, then F{f) = F(g) . PROOF. Since f and g are of finite order, it follows from Theorem 4.11 that both f and g have only finitely many asymptotic values. The conclusion follows. •

Remark. In a recent paper, Wang and Hua [207] proved that F{f) are permutable and sing u-t) is bounded.

1 and g

= F(g)

if

Next we study some other dynamical properties for permutable functions. For two permutable rational functions, Schmidt [183] proved that they have the same attractive domains, Leau domains, Sigel disks and Herman rings. For transcendental entire functions, we have the following natural questions. Question 7.2 Suppose f and g are permutable tronscendental entire functions and U is a common component of F(f) and F(g). Does U have the same/similar dynamical property for f and g 'f Question 7.3 Suppose f and g are permutable tronscendental entire functions. If f has no wandering domain, does g also have no wandering domain'( Next we construct a counter-example for Questions 7.2 and 7.3. Example. Let /(z) = -sinz + z, g(z) = f(z) + 2rr. It is easy to check that g(f) = f(g). By Theorem 7.13, F(f) = F(g). Obviously, for any k E N, 2k7r is an attractive fixed point of f. Let u2k be the attractive stable domain containing the attractive fixed point 2krr. Then it is a subset of F(g). Note that, U2,. n U2k' = ¢ for k f k' and g(2k7r) = 2k7r + 2rr = 2(k + 1)rr. Thus

g(U2,.)

= u2(k+l>·

Hence (U2k)keN are wandering domains of g. Note that /(z) = -sinz + z has finitely many multiple zeros, the superattracting fixed points of f are zeros of f' with finitely many exceptions. By Bergweiler [40], f has no wandering domain. At last we prove the following result. Theorem 7.15 For two non-linear entire functions f and g, /(g) has no wandering domain if and only if g(f) has no wandering domain.

218

CHAPTER 7. MISCELLANEOUS

PROOF. Let K

= f(g)

and H

= g(/). H(g)

Then we have

= g(K).

(7.20)

We assume that H = g(/) bas no wandering domain and suppose that K has wandering domains {Ui}, where K(U;) = Ui+l and Um n Un = 0 for m :/: n. Obviously, g(Um) n g(Un) = 0 (m :/: n). Let Vm = g(Um)· Then {Vm} are pairwise disjoint and H(Vm)

= H o g(Um) =go K(Um) = g(Um+d = Vm+l·

By Montel theorem, we imply that Vm C F(H) for all m. Next we prove that each Vm is a component of F(H), and then we conclude that {Vm} are wandering domains of H, which is a contradiction. . Suppose there exists a point a E 8Vm n F(H) . Then a can not be of the form g(b) for some b E 8Um. In fact, if a = g(b) for some b E 8Um C J(K), then b is a limit point of the repelling periodic points z,. of K, say Ki"(Zn) = z,.. From (7.20) we deduce that g(z,.) are repelling periodic points of H, and so a E J(H), which is a contradiction. Hence a is not a limit point of g(8Um)· There exists a disk D =~(a, r) which contains no points of g(8Um\{oo}). Since a E 8Vm, we can take w' E D and r E Um such that w' = g(r). By iversen theorem, there exists a path joining w' and a in D except perhaps for one point in D. H a is not the exception, then the inverse branch of g is continuous on the path which lies in Um but never meets 8Um. This implies that a E g(Um), which is a contradiction. H a is the exception, then there exists an asymptotic path r tending to oo in Um such that g(f) -+a. In this case, Um is unbounded. Let Wm be the component of F(H) which contains Vm . By Theorem 4.3 and Corollary 4.1, Um and Wm are simply connected. Hence there exist conformal mappings 4>: ~-+ Um and tp: ~-+ Wm, so that h := t/J- 1 o go 4> maps ~ to ~. By a result of Beurling (see (169, Theorems 11.5 and 11.9]) there exists a set A C (0, 211'] of capacity zero with the property that if 8 rl. A, then there exists a9 E 8Um \ {oo} such that 4>( reif) -+ a8 as r -+ 1. It follows that g(tf>(reil))-+ g(aB) E 8Wm\{oo} C J(H). Hence lg(reil)l -+ 1 as r -+ 1, provided 8 rl. A. A result of Lohwater (see (67, Theorem 5.14]) now implies that ~ \h(~) contains at most one point. Therefore, Vm = Wm except for at most one point. This also completes the proof. • Remark. The above result was proved by Poon and Yang (173]. Baker and Singh [24] proved the same result under an additional condition that f and g are of finite order. Example. We know that eP has no wandering domain for any non-constant polynomial P(z). The above theorem implies that P(e"') also has not wandering domain. Obviously, P(e•) is not of finite type.

219

7.3. CONVERGENCE OF JULIA SETS

7.3

Convergence of Julia sets

Let /n and/ be meromorphic functions in the plane such that /n converges to f locally uniformly on C, that is, for each compact set K and e > 0, there exists a positive integer N such that 11/n- /IlK:= sup 1/n(z)- /(z)l < e zEK

for alln ~ N .

The problem is what we can deduce about the dynamics off from those of fn· Not so many things can be expected, it is often the case that the dynamics are completely different even iff and /n are very close. However, some information could be obtained in special cases. For example, Devaney, Goldberg and Hubbard [73] studied the exponential family E~(z) by approximating with the sequence of polynomials P~.d(z) := .X(1 + z/d)d (d = 1, 2... ). Krauskopf [144] showed that J(P~.d) converges to J(E~) under certain conditions. More generally, Kisaka [136] proved the following result.

Theorem 7.16 Let In and f be entire functions in the plane such that fn converges to f locally uniformly on C. Then the following two conclusions hold: (1) If F(f) consists only of attractive basin, then J(/n) converges to J(f) in the Hausdorff metric. {2) If J(f) = C, then J(/n) -+ J(f) in the Hausdorff metric. PROOF. We prove only (1) here, the proof for (2) is the same. Let

J(/n) = J(/n) U { 00 }, J(/) = J(/) U { 00 }. Then it is enough to prove J(/n)-+ J(/). For any w E F(/), by the assumption, we see that there exists an attractive basin A(Zo) of an attracting periodic cycle o+(Zo) such that w E A(Zo)

c F(f).

We may suppose that Zo is a fixed point of f. By Hurwitz theorem, there exists a positive integer No such that, for n ~ No, there is an attracting fixed point Zo,n of In such that Zo,n converges to Zo as n -+ oo. We denote by An the attracting basin of Zo,n · Fixing a number a with 1/'(Zo)l < a < 1 - e, where 0 < e < 1 -1/'(Zo)l. Then we can find a small disk ~(Zo, r) containing Zo and an integer N 1 ~No such that 1/'(z)J~a

and

l/~(z)l~a+e N1. This implies that ~(Zo, r)

C An C F(/n)

220

CHAPTER 7. MISCELLANEOUS

for all n

> Nt.

For any e > 0 and any set B, let Ut(B) be the e-neigbborhood of B. Now we

take sufficiently small e < r/4 such that U,,(w) C A(ZQ). Since Zo is attracting, there exists a q = q( w) E N such that

f9(Uc(w)) C li(Zo,r/2) C F(/n)• Thus there exists a N 2 E N such that

Uc(w) C F(/n) for n ~ N'J.. Therefore, for each w E "C\Uc(J{f)) C F{f), there exist a neighborhood Uc(w) and a positive integer N(w) such that, for n ~ N(w),

Uc(w) C F(/n)• Since C\Uc(J(J)) is compact, we can find finite number of points w; (2 in C\Uc(J(f)) such that

C\Uc(J(f)) C

= 1, ... , s)

•

U Uc(w;) C F(/n) i=l

for n ~ N

= maxtSiS• N(w;).

Hence

Uc(J(/)) :> J(/n)

(n ~ N).

Note that ](/) is compact, there exist finitely many points fJ; ](f) such that

(7.21)

U = 1, ..., m) in

m

J(f) C

U Uc 2([JJ). J=l

Now take a repelling periodic point zJ E Uc;2(f3;). By Hurwitz theorem, there exists N([J;) E N_such that /n has a repelling periodic point z;,n E Uc 2([3;). Obviously, z1 ,n E J(/n)· Thus

J(f) = J(f) n Uc 2([3;) C Uc(J(f")) for n ~ maxtSJSm N(fJ;). The conclusion follows from this and (7.21).

•

Remark. From the proof, it is easily seen that the above conclusion also holds for the class P.

7.4. NEWTON'S METHOD

7.4

221

Newton's method

Let g be a meromorphic function. One of the most effective algorithms for finding the zeros of g is the Newton iteration or Newton method, which iterates the Newton function f defined by g(z) (7.22} f(z) = z- g'(z). It is easy to see that ( is a zero of g if and only if ( is an attracting fixed point of In particular, any zero of g is a point of F(f). From

f.

f'(z)

= g(z)g"(z) g'(z)2

(7.23}

we see that the simple zeros of g correspond to the super-attracting fixed points of f. It is also clear that fn(z) --. (as n--. oo whenever z--. ( . However, if z E J(/}, then r(z) cannot tend to any zero of g, since J(/) is completely invariant. A natural problem is under what circumstances r(z} fails to converge to zeros of g. For more d!'tails, see [165]. We shall •ucentrate on transcendental function g. Obviously, f is also transcendental except when g = R( z) exp(p( z)) for rational function R( z} and polynomial p(z). This case was studied in detail by Haruta [106]. It is of interest to find classes for which Newton's method behaves similarly to that for polynomials. For example, Bergweiler [36] considered the following function g(z)

=

1 2

p(t)eq(tldt + c,

(7.24}

where p and q are polynomials and c is a constant. Obviously, g' and g" has only finitely many zeros. Theorem 7.17 (/36}} Let g be of the form {7.24) and let f be defined by {7.22}. Suppose that g(z) is not of the form g(z) = eaz+b for two constants a and b. Denote by z 1 ,. •• , Zm the zeros of g" that are not zeros of g'. If fn(z3 ) converges for all j, then r(z) converges to zeros of g for all z E F(/). Remark. The case g = eaz+b has to be excluded because then f is linear and the conclusion is false. Remark. For the typical meromorphic function tan z. Howland-Vaillancourt [116] and Howland-Thompson-Vaillancourt [117] applied Newton's method to the equation tan z - c = 0, where c is a constant. In particular, they got some numerical results on the dynamics of the family and investigated the structure of the corresponding Mandelbrot sets.

222

CHAPTER 7. MISCELLANEOUS Remark. As a generalization of Newton's method, one may consider the itera-

tion of the function

g(z)

= z- cg'(z), where cis a constant and satisfies lc-11 < 1. This is the so-called relaxed Newton /c(z)

method.

7.5

Random iterations

In this section, we discuss random dynamical systems formed by a set of finite meromorphic functions. This is somewhat like Barnsley's Iterated FUnction Systems (IFS) (see (26]), who use IFS, a set of finite contract affine mappings, to generate fractal modeling images of the world. Let F = {/; : i E I} be a set of non-constant meromorphic functions. We denote ~)I)= {(j1,.f2, ... ): j, E /for all i EN}.

For each u

= (j1.h, ... ) E ~)I), we define iteration sequence as follows: w;(z)

= /]1 (z),

W:+l(z)

= /1..+

1

(z) o W;'(z), n 2: 1.

For any u E E(I), set

F(u,F)

= {z: {w;'(z)}

are well defined and normal at z},

the complement is denoted by J(u, F). The two sets F( u, F) and J( u, F) are called Fatou set and Julia set of the sequence {/,, : j, E u} respectively. FUrthermore, we define F(F) = {z: z E F(u,F)for allu E ~)I)}. The complement is denoted by J(F). We call F(F) and J(F) the Fatou set and the Julia set of the system F. The inverse of W;'(z) is defined to be

w;n(z)

= /j~l 0 ••• 0 /;: 1 (z).

For these kinds of iterations, there are many different properties. Next we list some of them (see Biiger (54], Maa.louf (152], Ren (180], Zhou-Ren (223], etc.):

• J(F) is not necessarily forward invariant, • J(u,F) is not necessarily perfect set, • J(F) is not necessarily the whole plane when J(F) contains interior points. Example. Let F Example. Let F

= {z2 ,4z2 }. Then 1/4 E J(F), but (1/4) 2 rt J(F). = {bne" fz: n = 1, 2, ... } and u = {1, 2, ... },where bn

= n/

min lew;-t(a) /W:- 1 (z)l. aE{l/nSI•ISn}

7.5. RANDOM ITERATIONS

223

Then J(u,F) = {0, oo}. Example. Let F = {/I, ... , /m} , where fi, ... , fm are meromorphic and have a common super-attractive fixed point. Suppose that there exist two diffEmrent i and j such that f, has a super-attractive periodic point b and J(/1 ) contains a Jordan arc 'Y with bE 'Y· Then J(F) contains interior points but J(F) =I= C.

Bibliography [1] Aarts, J.M. and Oversteegen, L.: The geometry of Julia sets, 'frans. Amer. Math. Soc. 338(1993}, 897-918.

[2] Ahlfors, L. V.: Vber die asymptotischen Werte der meromorphen Funktionen endlicher Ordnung, Acta Acad. Aboensis Math. et Phys. 6(1932}. [3] Ablfors, L. V.: Zur Theorie der Uberlagerungsftachen, Acta Math. 65(1935}, 157-194.

[4] Ahlfors, L. V. and Sarlo, L.: Riemann Surfaces, Princeton Univ. Press 1960. [5] Amol'd, V.I.: Small denominators 1: On the mappings of the circumference onto itself, Amer. Math. Soc. 'fransl. (2}46(1965}, 213-284; translation from Izv. A.kad. Nauk. SSSR Ser. Mat. 25(1961}, 21-86. [6] Baker, I. N.: Zusammensetzungen ganzer Funktionen, Math. Z. 69(1958}, 121-163. [7] Baker, I. N.: Fixpoints and iterates of entire functions, Math. Z. 71(1959}, 146-153. [8] Baker, I. N.: The existence of fix-points of entire functions, Math. Z. 73(1960}, 280-284.

[9] Baker, I. N.: Multiply connected domains of normality in iteration theory. Math. Z. 81(1963}, 206-214. [10] Baker, I. N.: Repulsive fixpoints of entire functions, Math. Z. 104(1968}, 252-256.

[11] Baker, I. N.: Completely invariant domains of entire functions, Math. essays dedicated to A.J.Macintyre, Ohio Univ. Press, Athens, Ohio 1970. [12] Baker, I. N.: Limit functions and sets of non-normality in iteration theory, Ann. Acad. Sci. Fenn. Ser. A I Math. 467(1970}, 1-11. [13] Baker, I. N .: The domains of normality of an entire function, Ann. A cad. Sci. Fenn. Ser. A I Math. 1(1975), 277-283. 225

BIBLIOGRAPHY

226

[14] Baker, I. N.: An entire function which has wandering domains, J. Austral.

Matb. Soc. SerA. 22(1976}, 173-176.

[15] Baker, I. N.: The iteration of polynomials and transcendental entire functions, J. Austral. Math. soc. Ser. A 30{1981), 483-495. [16] Baker, I. N.: Wandering domains in the iteration of entire functions, Proc. London Math. Soc. 49{1984), 563-576. [17] Baker, I. N.: Some entire .functions with multiply-connected wandering domains, Ergod. Th. Dynam. Sys. 5{1985), 163-169. [18] Baker, I.N.: Wandering domains for maps of the punctured plane, Ann. Acad. Sci. Fenn. Ser. A I Math. 12{1987), 191-198. [19] Baker, I. N.: Infinite limits in the iteration of entire functions, Ergod. Th. Dynam. Sys. 8{1988), 503-507. [20] Baker, I.N., Kotus, J. and Lii, Y.: Iterates of meromorphic functions II, J. London Math. Soc. 42{1990), 267-278. [21] Baker, I.N., Kotus, J. and Lii, Y.: Iterates ofmeromorphic functions I, Ergod. Th. Dynam. Sys~ 11(1991), 241-248. · [22] Baker, I.N., Kotus, J. and Lii, Y.: Iterates of meromorphic functions III, Ergod. Th. Dynam. Sys. 11{1991), 603-618. [23] Baker, I.N., Kotus, J. and Lii, Y.: Iterates of meromorphic functions IV: Critical finite functions, Results in Math. 22{1992), 651-656. [24] Baker, I.N. and Singh, A.P.: Wandering domains in the iteration of compositions of entire functions, Ann. Acad. Sci. Fenn. Ser.A' I Math. 20(1995), 149-153. [25] Baker, I.N. and Weinreich, J. Boundaries which arise in the dynamics of entire functions, Rev. Romaine Math. Pures Appl. 36(.1991), 413-420. [26] Barry, P.D.: The minimum modulus of small integral and subharmonic functions, Proc. London Math. Soc. 12(1962), 445-495. [27] Barry, P.D.: On a theorem of Kjellberg, Quart. 179-191.

J~

Math. Oxford 15(1964),

[28] Beardon, A.F.: On the Hausdorff dimension of general Cantor sets, Proc. Cambridge Philos. Soc. 61(1965), 679-694. [29] Beardon, A.F.: Iteration of Rationall'Unctions, New York, Berlin, and Heidelberg 1991. [30] Beardon, A.F.: The Geometry of Discrete Groups, Springer-Verlag 1983.

BIBLIOGRAPHY

227

[31) Bergweiler, W.: On proof of a conjecture of Gross concerning fix-points, Math. 204(1990), 381-390.

z.

[32] Bergweiler, W.: On the number of fix-points of iterated entire functions, Arch. Math. 55(1990), 558-563. [33] Bergweiler, W.: Periodic points of entire functions: Proof of a conjecture of Baker, Complex Variables 17(1991), 57-72. (34] Bergweiler, W.: On the existence of fixpoints of composite meromorphic functions, Proc. Amer. Math. Soc. 114(1992), 879-880. [35) Bergweiler, W.: Iteration of meromorphic functions, Bull. Amer. Math. Soc. 29(1993), 151-188. [36] Bergweiler, W.: Newton's method and a class of meromorphic functions without wandering domains, Ergod. Th. Dynam. Sys. 13(1993), 231-247. [37] Bergweiler, W.: Invariant domains and singularities, Math. Proc. Camb. Phil. Soc. 117(1995), 525-532.

[38] Bergweiler, W.: On the Julia set of analytic self-maps of the punctured plane, Analysis 15(1995), 252-256. [39] Bergweiler, W. and Eremenko, A.: On the singularities of the inverse to a meromorphic function of finite order, Revista Matematica Iberoamericana 11(1995), 355-373. [40] Bergweiler, W., Haruta, M., Kriete, H., Meier, H.G. and Terglane, N.: On the limit functions of iterates in wandering domains, Ann. Acad. Sci. Fenn. Ser. A.I. Math. 18(1993),. 369-375. [41) Bergweiler, W. and Rohde, St.: Omitted values in domains of normality, Proc. Amer. Math. Soc. 123(1995), 1857-1858. [42] Bergweiler, W. and Terglane, N.: Weakly repelling fixpoints and the connectivity of wandering domains, TI-aos. Amer. Math. Soc. 348(1996), 1-12. [43] Bergweiler, W. and Yang, C. C.: On the value distribution of composite meromorphic functions, Bull. London Math. Soc. 25(1993), 357-361.

[44) Bhattacharyya, P.: Iteration of Analytic Functions, Ph.D. thesis, University of London, 1969. [45) Bhattacharyya, P.: On the domain of normality of an attractive fixpoint, TI-aos. Amer. Math. Soc. 153(1971), 89-98. [46] Bieberbach, L.: Uber die Koeffizienten Potenzreihen, wejche eine schlichte Abbilduog des Eioheitskreises vermitteln, S. -B. Preuss. Akad. Wiss. 1916, S. 940-955.

BIBLIOGRAPHY

228

[47) Boas, R. P.: Entire l'imctions, Academic Press, New York 1954.

[48] Bock, H.: On the dynamics of entire functions on the Julia set, Results in Math. 30(1996), 16-20.

[49] Bohr, H.: Uber einen Satz von Edmund Landau, Scripta Univ. Hierosolymitanarum 1(1923), or. 2, 5 pp. [50] BOttcher, L. E.:The principle laws of convergence of iterates and their applications to analysis, Izv. Kazan. Fiz.-Mat. Obshch. 14(1904), 155-234. [51] Branges, L. de: A proof of the Bieberbach conjecture, Acta Math. 154(1985), 137-152. [52] Brjuno, A. D.: Convergence of transformations of differential equations to normal forms, Dokl. Alcad. Nauk USSR 165(1965), 987-989. [53] Brolin, H.: Invariant sets under iteration of rational functions, Aik. Mat. 6(1965), 103-144. [54] Biiger, M.: On the Julia set of the composition of meromorphic functions, Analysis 16(1996), 385-397. [55] Bula, W.D. and Oversteegen, L.: A characterization of smooth Cantor bouquets, Proc. Amer. Math. Soc. 108(1990), 529-534. [56] Carat~ory, C.: Uber das lineare Mass von Punkmengen, eine Verallgemeinerung des L&ngenbegriffs, Nach. Ges. W~. GOttingen (1914), 406-426. [57] Carleson, L. and Gamelin, W .: Complex Dynamics, Springer-Verlag New York, Inc. 1993. [58] Cebotarev, N.G.: Uber die Realitat von Nullstellen ganzer transzendenten F\mktionen, Math. Ann. 99(1928), 660-686. [59] Chen, H.H. and Gu, Y.X.: An improvement of Marty's criterion and its applications, Sci. in China 36(1993), 674-681. [60] Chen, H.H. and Hua, X.H.: Normality criterion and singular directions, Proc. of the Inter. Con£. on Complex, The Inter. Press,(1994)183-189. [61] i1heo, H.H. and Hua, X.H.: Normal families of holomorphic functions, J. Austral. Math. Soc. 58(1995), 1-6. [62] Chuang, C.T.: A simple proof of a theorem of Fatou on the iteration and fix-points of transcendental entire functions, Contemporary Math. (edited by C.C.Yang and C.T.Chuang)48(1985), 65-70. [63] Chuang, C.T.: Normal Families of Meromorphic l'imctions, Singapore; Hong Kong: World Scientific, 1993.

BIBLIOGRAPHY

229

[64] Chuang, C.T. and Yang, C. C.: Fix-points and Factorization of meromorpbic functions, Singapore; Hong Kong: World Scientific, 1990. [65] Clunie, J.: The composition of entire and meromorphic functions, Math. essays dedicated to A.J.Macintyre, Ohio Univ. Press, Athens, Ohio 1970.

[66] Cohn, C.H.E.: Two primary factor inequalities, Pacific J. Math. 44(1973), 81-92. [67] Collingwood, E.F. and Lohwater, A.J.: The theory of cluster sets, Cambridge Univ. Press, London and New York 1966. [68] Cremer, H.: Uber die Schrooersche Funktionalgleichung und das Schwarzsche Eckenabbildungsproblem, Ber. Verb. Siicbs. Akad. Wiss. Leipzig, Math. Phys. Kl. 84(1932), 291-324. [69] Cremer, H.: Uber der Hii.ufigkeit der Nichtzentren, Math. Ann. 115(1938), 573-580. (70] Denjoy, A.: Sur un theoreme de Wiman, C. R. Acad. Sci. Paris 193(1931}, 828-830.

[71] Devaney R.: Julia sets and bifurcation diagrams for exponential maps, Bull. Amer. Math. Soc. 11(1984), 167-171.

[72] Devaney, R.: The structural instability of exp(z), Proc. Amer. Math. Soc. 94(1985), 544-548. [73] Devaney, L.R., Goldberg, L.R. and Hubbard, J.: Dynamical approximation to the exponential map by polynomials, Preprint. [74] Devaney, R. and Keen, L.: Dynamics of meromorphic maps with polynomial Schwarzian derivative, Ann. Sci. Ecole Norm. Sup. 22(1989), 55-81. [75] Devaney, R.L. and Tangerman, F.: Dynamics of entire functions near the essential singularity, Ergod. Th. Dynam. Sys. 6(1986), 489-503. [76] Dong, X.Y.: On iteration of a function in the sine family, J. Math. Anal. Appl. 165(1992), 575-586. [77] Douady, A.: Systemes dynamiques holomorphes, Asterisque 105(1983), 3963. [78] Douady, A.: Systemes dynamiques holomorphes, Ergod. Th. Dynam. Sys. 4(1984), 35-52. [79] Douady, A. and Goldberg, L.R.: The nonconjugacy of certain exponential functions, Holomorpbic functions and modulus I, Edit. by Drasin, D., Earle, C.J., Gehring, F.W., Kra, I. and Marden, A., Math. Sci. Research lnst. Publ. 10, Springer-Verlag New York Inc. 1988.

230

BIBLIOGRAPHY

[80] Eremenko, A.E. On the iterates of entire functions, Dynamical Systems and Ergodic Theory, Banach Center Publ., Vol. 23, Polish Scientific Publishers, Warsaw, 1989, 339-345.

[81] Eremenko, A.E. and Lyubich M.Yu.: Iterates of entire functions, Dokl. A.lcad. Nauk SSSR 279(1984), 25-27; English transl. in Soviet Math. Dokl. 30(1984), 592-594. [82] Eremenko, A.E. and Lyubich M.Yu.: Iterates of entire functions, Preprint No. ~84, Fiz.-Tekhn. Inst. Nizkikh Temperatur Akad. Nauk Ukr. SSR. Khar'kov 1984.

[83] Eremenko, A.E. and Lyubich M. Yu.: Examples of entire functions with pathological dynamics, J . London Math. Soc. 36(1987), 458-468.

[84] Eremenko, A.E. and Lyubich M.Yu.: The dynamics of analytic transforms, · Leningrad Math. J. 1(1990), 563-634.

[85] Eremenko, A.E. and Lyubich M.Yu.: Dynamical properties of some classes of entire functions, Ann. Inst. Fourier 42(1992), 989-1020.

[86] Essen, M.: The cos d Theorem, Lecture Notes in Math. 467, Springer, Berlin 1975. [87] Falconer, K.J.: Geometry of Fractal Sets, Cambridge Univ. Press 1985.

[88] Falconer, K.J.: Fractal Geometry, John Wiley and Sons 1990. (89] Fang, L.P.: Complex dynamical systems on 34(1991), 611-621.

c· (Chinese), Acta Math. Sinica

[90] Fang, L.P.: Measurable dynamics of some holomorphic maps, Proc. Int. Coni. on Complex Anal. at the Nankai Inst. of Math. (1992), International Press 1994, 79-89. (91] Fatou, P.: Sur les equations fonctionelles, Bull. Soc. Math . France 47(1919), 161-271; 48(1920), 33-94 and 208-314. (92] Fatou, P.: Sur I' iteration des fonctions transcendentes entieres, Acta Math. 47(1926), 337-370. [93] Frostman, 0.: Potentiel d'equilibre et capacite des ensembles avec quelquet~ applications a Ia thoorie des fonctions, Meddel. Lunds. Univ. Mat. Sem. 3(1935), 1-118. (94] Gaier, D.: Lectures on Complex Apprwcimation, Birkhauser, Boston 1985. [95] Garber, V.: On the iteration of rational functions, Math. Proc. Cambridge Pbilos. Soc. 84(1978), 497-505.

BIBLIOGRAPHY

231

[96] Gehring, F.: Injectivity of local quasi-isometries, Comm. Math. Helv. 57(1982), 202-220. [97] Ghys, E., Goldberg, L. and Sullivan, D.: On the measurable dynamics of z --+ exp(z), Ergod. Th. Dynam. Sys. 5(1985), 329-335. [98] Goldberg L.R. and Keen L.: A finiteness theorem for a dynamical class of entire functions, Ergodic Th. Dynam. Sys. 6(1986), 183-192. [99] Goldstein, R.: On factorization of certain entire functions, II, Proc. J. London Math. Soc. 22(1971), 485-506.

[100] Gross, F., Factorization of Meromorphic Functions, U. S. Government Printing Office, Washington D. C., 1972. [101] Gross, F. and Osgood, C.F.: On fixed points of composite entire functions, J. London Math. Soc. (2)28(1983)), 57-61. [102] Gross, F. and Yang, C.C.: The fix-points and factorization of meromorphic functions, 'ITans. Amer. Math. Soc. 168(1972), 211-219. [103] Gross, F. and Yang, C.C.: Further results on prime entire functions, 'ITans. Amer. Math. Soc. 142(1974), 347-355. [104] Gross, W.: Uber die singularitaten analytischer Funktionen, Monat. Math. Phys. 29(1918), 3-47. [105] Harada, T. and Taniguchi 1 M.: On Teichmiiller spaces of complex dynamics by entire functions, to appear in the Special Issue of the Bull. HK Math. Soc .. [106] Haruta, M.: The dynamics of Newton's method on the exponential function in the complex domain, Ph.D. thesis, Boston Univ. 1992. [107] Hausdorff, F.: Dimension und ausseres Mass, Math. Ann. 79(1919), 157-179. [108] Hayman, W.K.: Some applications of the transfinite diameter to the theory of functions, J. Analyse Math. 1(1951), 155-179.

[109] Hayman, W.K.: Meromorphic Functions, Oxford University Press, London, 1964.

[110] Hayman, W.K.: On the characteristic of functions meromorphic in the plane and of their integrals, Proc. London Math. Soc. 14(1965), 93-128.

[111] Hayman, W.K.: The local growth of power series: A survey of the WimanValiron method, Canada Math. Bull. 17(1974), 317-358. [112] Herman, M.: Exemples de fractions rationelles ayant une orbite dense sur la sphere de Riemann, Bull. Soc. France 112(1984), 93-142.

BIBLIOGRAPHY

232

[113] Herman, M.: Are there critical points on the boundary of singular domains?,

Comm. Math. Phys. 99(1985), 593-612. [114] Herman, M.: Conjugaison quasi-s~trique des diffoomorphismes du circle 8. des rotations et applications aux disques singuliers de Siegel!, Manuscript. [115] Herring, M.: Mapping of Fatou components, Preprint 1993. [116] Howland, J.L. and Vaillancourt, R.: Attractive cycles in the iteration of meromorphic functions, Numer. Math. 46(1985), 323-337. [117] Howland, J.L., Thompson, A. and Vaillancourt, R.: On the dynamics of a meromorphic function, Appl. Math. Notes 15{1990), 7 37. [118] Hua, X.H.: Julia set and its applications, J. Nanjing Univ. 3{1993), 1-7. [119] Hua, X.H.: Normality criterion and its applications, Ann. Sci. Report-Suppl. JNU(N.S.) English Series I 29(1993),34-37. [120] Hua, X.H.: The number of fix-points of composite entire functions and a problem of Baker, Complex Analysis and its Applications, edit. by Yang C.C., Wen, G. C., Li, K. Y. and Jiang, Y. M., Pitnam Research Notes in Math., 305(1994), 60-66.

[121] Hua, X.H.: A new approach to normality criterion, Manuscripta Math. 86(1995), 467-478. [122] . Hua, X. H. and Lappan, P.: Normal families and normal functions, Preprint. [123] Hua, X.H. and Mattila, P.: Hausdorff dimension of transcendental entire functions, in preparation.

[124] Hua, X.H. and Yang, C. C.: Fatou components and a problem of Bergweiler, to appear in Int. Journal of Bifurcation and Chaos. [125] Hua, X.H. and Yang, C.C.: Fatou components of entire functions of small growth, to appear in Ergodic Th. Dynam. Sys .. [126] Hutchinson, J.E.: Fractals and self similarity, Indiana Univ. Math. J. 30(1981 ). 713-747. [127] Iversen, F., Recherches sur les fonctions inverses des fonctions miomorphes, These. Helsingfors 1914. [128] Jang, C. M.: Julia set of the function zexp(z+J.t), Tohoku Math. J. 44(1992), 271. [129] Jiang, W.H.: Dynamics of..\ tan z, Ph.D. thesis, CUNY 1991. [130] John, F.: On quasi-isometries. I, Comm. Pure Appl. Math. 21(1968), 77-110.

BIBLIOGRAPHY

233

[131] Julia, G.: Memoire sur 1' iteration des fractions rationnelles, J. Math. Pures Appl. (8)1(1918), 47-245. [132] Katajamii.ki, K., Kinnunen, L. and Laine, 1.: On the value distribution of composite entire functions, Complex Variables 20(1990), 63-69. [133] Katajamii.ki, K., Kinnunen, L. and Laine, 1.: On the value distribution of some composite meromorpbic functions, Bull. London Math. Soc. 25(1993), 445-452. [134J Keen, L.: Dynamics of holomorphic self-maps of c·, Holomorphic functions

and modulus I, Edit. by Drasin, D., Earle, C.J., Gehring, F.W., Kra, I. and Marden, A., Math. Sci. Research Inst. Pub!. 10, Springer-Verlag New York Inc. 1988.

[135] Keen, L. and Kotus, J.: Density in the family A tan z, Preprint, Suny at Stony Brook 1995. [136] Kisaka, M.: Local uniform convergence and convergence of Julia sets, Nonlinearity 8(1995), 273-281. [137] Kisaka, M.: On the connectivity of Julia sets of transcendental entire func-

tions, Preprint.

[138] Kjellberg, B.: On the minimum modulus of entire functions of lower order less than one, Math. Scand. 6(1960}, 189-197. [139J Krenigs, G.: Recherches sur les integrals de certains equations fonctionelles, Ann. Sci. Ec. Norm. Sup. 1(1884) supplem, 1-41. [140] Kotus, J.: Iterated holomorphic maps of the punctured plane, Dynamical

systems, (Kurzhanski and Sigmund eds.), Lecture Notes Econom. and Math. Systems, Vol. 287, Springer, Berlin, Heidelberg, and New York 1987, 10-29. [141] Kotus, J.: On the Hausdorff dimension of Julia sets of meromorphic functions. I., Bull. Soc. Math. France 122(1994), 305-331. [142] Kotus, J.: On the Hausdorff dimension of Julia sets of meromorphic functions. II., Bull. Soc. Math. France 123(1995), 33-46. [143J Kotus, J., On ergodicity of Julia sets of meromorphic functions, Preprint.

[144] Krauskopf, B.: Convergence of Julia sets in the approximation of Aez by A[1 + (z/d)]d, Int. J. Bif. Chaos 3(1993), 257-270. [145] Kuroda, T. and Jang, C.M.: Julia sets of the function z exp(z+Jl) II, Tohoko Math. J., 49(1997}, 577-584. [146] Langley, J.K.: On the fix-points of composite entire functions of finite order, Proc. Royal Soc. Edinburgh, 124(1994}, 995-1001.

234

BIBLIOGRAPHY

[147) Lappan, P.: A criterion for a meromorphic function to be normal, Comment.

Math. Helv., 49(1974), 492-495. [148) Lehto, 0.: Univalent FUnctions and Teichmiiller Space. Springer-Verlag New York Inc. 1987.

[149) Liverpool, L.S.O.: On entire functions with infinite domains of normality, Aequationes Math. 10(1974), 189-200.

[150) LOwner, K.: Untersuchungen iiber schlichte konforme Abbildungen des Einheitskreises I, Math. Ann. 89(1923), 103-121.

[151) Lyubich, M. Yu.: The measurable dynamics of the exponential map, Siberian J. Math. B. (28)5(1987}, 111-127. [152) Maalouf, R.N.: Julia sets of inner compositions, Arch. Math. 67(1996}, 138141.

c·, Soviet Math. Dokl. 36(1988}, 418-420; translation from Dokl. Akad. Nauk. SSSR 297(1987}.

[153) Makienko, P.M.: Iterates of analytic functions of

[154) Mandelbrot, B., The Fractal Geometry of Nature, W.H.Freeman, New York 1983.

[155) Marty, F.:

Researches sur la repartition des valeurs d'une fonction meromorphe, Ann. F&c. Sci. Univ. Toulouse 23(1931}, 183-196.

[156) Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Adv. Math. 44, 1995.

[157) Mayer, J.C.: An explosion point for the set of endpoints of the Julia set of .Xexp(z}, Ergod. Th. Dynam. Sys. 10(1990}, 177-183.

[158) McMullen, C.: Area and Hausdorff dimension of Julia sets of entire functions, TI-ans. Amer. Math. Soc. 300(1987}, 329-342.

[159) Milnor, J., Dynamics in One Complex Variable: Introductory Lectures, Stony Brook lost. for Math. Sci., Preprint 1990. [160) Montel, P.: Families Normales. Gauthiers-Villars, 1927. [161) Moser, J.K.: Stable and Random Motions in Dynamical Systems, Princeton University Press 1973. [162) Nevanlinna, R.: Uber ruemannsche Facben mit endlich vielen Windungspunkten, Acta Math. 58, 1932. [163) Nevanlinna, R.: Analytic FUnctions Springer-Verlag, New York, 1970. [164) Newman, M.: Elements of the Thpology of Plane Sets, Cambridge Univ. Press 1939.

235

BIBLIOGRAPHY

[165] Peitgen, H.O.: Newton's Method and Dynamical Systems, Kluwer Academic Publishers 1989. [166] Pfeifer, G. A.: On the conformal mapping of curvilineaar angles. The functional equation t/>[/(x)] = attf>(x), 'frans. Amer. Math. Soc. 18(1917), 185-198. [167] P6lya, G.: On an integral function of an integral function, J. London Math. Soc. 1(1926), 12-15. [168] P6lya, G. and Szego, G.: Problems and Theorems in Analysis I, Springer, New York 1972. [169] Pommerenke, Ch.: Univalent functions, Vandenhock and Ruprecht, 1975. [170] Pommerenke, Ch.: Boundary behavior of conformal maps, Springer-Verlag Berlin, 1992. [171] Poon, K.K. and Yang, C.C.: Relations between the Fatou exceptional value and component of the Fatou set, Preprint. [172] Poon, K.K. and Yang, C.C.: Dynamical behavior of two permutable entire functions, to appear in Ann. Polon. Math. [173] Poon, K.K. and Yang, C.C.: On the Fatou set of two permutable entire functions, Preprint. [174] Prokopovich, G.S.: On superposition of some entire functions, Ukrain Mat. Zh. 26(1974), 188-195. (English translation). [175] Qiao, J. Y.: Stable domains in the iteration of entire functions (Chinese), Acta Math. Sinica 37(1994), 702-708. [176] Qiao, J. Y.: The Julia set of the mapping z 39(1994), 529-533.

--+

z exp(z + J.L), Chn. Sci. Bull.

[177] Qiu, W.Y. and Yin, Y.C.: Structural stability of exponential functions described by orbit of zero, Complex Variables 27(1995), 299-307. [178] Rees, M.: The exponential map is not recurrent, Math. Z. 191(1986), 593598. [179] Reitz, St.: On connectivity numbers of stable domains, Analysis 15(1995), 247-250. [180] Ren, F.Y.: Advances and problems in random dynamical systems, Preprint. [181] Rudin, W.: Real and Complex Analysis, McGraw-Hill, Inc., 1966. [182] Schiff, J. L.: Normal Families, Springer-Verlag, New York Inc., 1993. [183] Schmidt, W.: On the periodic stable domains of permutable rational functions, Complex variables 17(1992), 149-152.

236

BIBLIOGRAPHY

[184) Schwick, W.: Repelling periodic points in the Julia set, Bull. London Math.

Soc, to appear.

[185) Shishikura, M.: The connectivity of the Julia set and fixed points, Preprint IHES-M-~37(1990).

[186) Siegel, C. L.: Iteration of analytic functions, Ann. of Math. 43(1942), 607612. [187) Siegel, C. L. and Moeer, J. K.: Lectures on Celestial Mechanics, SpringerVerlag 1971. [188) Smale, S.: Diffeomorphisms with infinitely many periodic points, Differential and Combinatorial Topology, Princeton Univ. Pre88, Princeton, N.J. (1965), 63-80. [189) Smyth, W.F.: Constant limit of a sequence of iterates, Siam J. Math. Anal. 10(1979), 463-471. [190) Stallard, G.M.: Entire functions with Julia sets of zero measure, Math. Proc. Camb. Phil. Soc. 108(1990), 551-557. [191) Stallard, G.M.: A class of meromorphic functions with no wandering d~ mains, Ann. Acad. Sci. .Fenn. 16(1991), 211-226. [192) Stallard, G.M.: The Hausdorff dimension of Julia sets of entire functions, Ergod. Th. Dynam. Sys. 11(1991), 769-777. [193) Stallard, G.M.: The iteration of entire functions of small growth, Math. Proc. Camb. Phil. Soc. 114(1993), 43-55. [194) Stallard, G.M.: Meromorphic functions whoee Julia sets contain a free Jordan arc, Ann. Acad. Sci. .Fenn. 18(1993), 273-298. [195) Stallard, G.M.: The Hausdorff dimension of Julia sets of meromorphic functions, J. London Math, Soc. 49(1994), 281-295. [196) Stallard, G.M.: The Hausdorff dimension of Julia sets of entire functions II, •Math. Proc. Camb. Phil. Soc. 119(1996), 513-536. [197) Steinmetz, N.: Rational Iteration, de Gruyter Studies in Math. 16, Walter de Gruyter 1993. [198) Sullivan, D.: Conformal dynamical systems, Geometric Dynamics, Lecture Notes in Math., Springer-Verlag 1007(1983), 725-752. [199) Sullivan, D.: Quasiconformal homeomorphisms and dynamics 1: Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. 122(1985), 402-418.

BIBLIOGRAPHY

237

[200] Sullivan, D.: Quasiconformal homeomorphisms and dynamics II, Acta Math. 155(1985), 243-260.

[201] Teichmiiller, 0.: Eine Umkehrung des zweitten Hauptsatzes der Wertverteilungslehre, Deutsch. Math. 2{1937), 96-107. [202] Topfer, H.: Uber die Iteration der ganzen transzendenten F\mktionen, insbesondere von sin z und cos z, Math. Ann. 177(1939), 65-84. [203] Tsuji, M.: Potential Theory in Modern Function Theory, Maruzen, Tokyo 1959. [204] Valiron, G.: Lectures on the General Theory of Integral Functions, Toulouse: Edona.rd Privat 1923.

[205] Viana da Silva, M.: The differentiability of the hairs of exp(z), Proc. Amer. Math. Soc. 103{1988), 1179-1184. [206] Wang, X.L. and Hua X.H.: Components of the Fatou sets of transcendental entire functions, J. Nanjing Univ. Math. Biquart. 15(1998).

[207] Wang, X.L. and Hua X.H.: Dynamics of two permutable transcendental entire functions, Preprint. [208] Whyburn, G.T.: Analytic Topology, Amer. Math. Soc. Colloquiun Publ. Province, U.S.A., 1942. [209] Wiman, A.: fiber den Zusammenhang zwischen dem Maximalbetrage einer analytischen Funktion und dem grssten Gliede der zugehorigen Taylorschen Reihe, Act& Math. 37(1914), 305-326.

[210] Wiman, A.: fiber den Zusammenhang zwischen dem Maximalbetrage einer analytischen F\mktion und dem grssten Betrage bei gegebenem Argumente der F\mktion, Act& Math. 41{1916), 1-28. [211] Xue, G.F. and Pang, X.C.: Normality criterion of a family of meromorphic functions, J. East Cbn. Norm. Univ. 2{1988), 15-22. [212] Yang, C.C.: On the zeros of f(g(z))- a(z), Proc. Int. Conf. on Complex at Ne.nkai Inst., International Press, 1994, 261-270. [213] Yang, C. C. and Hua, X. H.: Dynamics of transcendental entire functions, J. Nanjing Univ. Math. Biquart. 14(1997), 1-4. [214] Yang, C.C. and Zheng, J.H.: On the fix-points of compo8ite meromorphic functions and generalizations, J. d'Analyse Math. 68{1996), 59-93. [215] Yang, D.G., Hao, Z.F. and Hua, X.H.: Dynamics of the function zexp(z+J.t), Preprint.

238

BIBLIOGRAPHY

[216] Yang, G.X.: Dynamics of entire functions and nonuniformlly quasiconformal mapp~,

Ph. D. Thesis. Peking Univ. 1991.

[217] Ye. Z.: Structural instability of exponential functions, 'frans. Amer. Math. Soc. 344(1994}, 379-389. [218] Yoccoz, J. C.: Linearisation des germes de diffeomorphimes holomorphes de (C. 0}, C. R. Acad. Sci. Paris 306(1988}, 55-58. [219] Zalcman, L.: A heuristic principle in complex function theory, Amer. Math. Monthly 82(1975}, 813-817. [220] Zheng, J.H. and Yang, C.C.: Further results on fix-points and zeros of entire functions, 'frans. Amer. Math. Soc. 347(1995), 37-50. [221] Zheng, J .H.: A quantitative estimate on fix-points of composite meromorphic functions, Canad. Math. Bull. 38(1995), 1-6.

[222] Zhou, J. and Li, Z.: Structural instability of the mapping z-+ Aexp(z)(A > e- 1 ), Sci. in China Ser. A 30(1989}, 1153-1161. [223] Zhou, W.M. and Ren, F.Y.: The Julia sets of the random iteration of transcendental functions (Chinese), Chn. Sci. Bull. 38(1993), 289-290.

List of Special Symbols

AV(f), 66

B, 67 Co(X), 154 x. 2

CV(f), 66 d(x,y), 2

dimE, 158 dimt s, 158

£, 34 E(f), 68 F(f), 47

ind0 -y, 58 J(f), 47 .C(U), 61

M,34 M(r,f), 2 m*(r,f), 2

o+(Zo), 34 o-(Zo), 34 'P, 34

PV(f), 33

s, 66

sing(f- 1 ), 66

>.(!), 20

p(f), 20

239

Index density theorem, 154 dilatation, 16 Diophantine, 45 direction repelling, 43 attracting, 43 distortion, 7 distortion lemma, 9 ergodic on J(/), 193 expanding, 146 sub-expanding, 146

Ahlfo~Shimizu characteristic, 24 Arnold-Herman ring, 61 asymptotic value, 66 Baker domain, 61 Beltrami coefficient, 16 Beltrami equation, 16 Beurling transformation, 18 Bottcher domain, 60 bounded distortion, 7 bounded type, 67 Cartan formula, 20 central problem, 39 chordal metric, 2 component, 57 periodic, 57 fixed, 57 invariant, 57 pre-periodic, 57 really pre-periodic, 57 conformal mapping, 6 conjugate, 38 counting function, 20 Cremer point, 45 critical point, 66 critical value, 66 cycle, 35 attracting, 35 indifferent, 35 rationally indifferent, 35 irrationally indifferent, 35 repelling, 35 super-attracting, 35 parabolic, 35 Denjoy-Carleman-Ahlfors Theorem, 67 Denjoy-Wolff Theorem, 64

pseud~ding, 195 exponential tract, 66 Fatou component, 57 Fatou set, 47 finite type, 67 first fundamental theorem, 20 fixed point, 34 attracting, 35 indifferent, 35 rationally indifferent, 35 irrationally indifferent, 35 repelling, 35 super-attracting, 35 fractal, 159 Hausdorff dimension, 158 Hausdorff distance, 3 Hausdorff metric, 3 Gross Star Theorem, 67 Herman ring, 61 Holder estimation, 18 hyperbolic, 10 hyperbolic function, 146 hyperbolic metric, 10 immediate attractive basin, 60 invariant Lemma of Schwarz, 11

240

241

INDEX J-stable, 200 Julia set, 47 k-quasiconformal, 16 Koebe 1/4 theorem, 7 Koebe distortion, 7 Koebe function, 7 Koenigs Linearization Theorem, 39 Koenigs map, 40 Leau domain, 60 Lebesgue density theorem, 155 limit function, 61 maximum domain of normality, 57 multiplier, 35 Nevanlinna characteristic, 20 nonlinearity, 9 normal, 3 open set condition, 161 orbit backward orbit, 34 forward orbit, 34 parabolic domains, 60 petal attracting petal, 42 repelling petal, 42 Poincare metric, 10 Poisson-Jensen formula, 19 post-singular set, 68 proximity function, 20 recurrent, 193 rotation doinain, 61 s-set, 158 SchrOder domain, 60 SchrOder equation, 39 Schwarz-Pick Lemma, 10 Second Fundamental Theorem, 21 self-similar, 161 Siegel disc, 60 similarity, 161 singularity, 66 algebraic singularity, 66 direct, 66 indirect, 66

transcendental singularity, 66 small distortion, 7 spherical metric, 2

stable domain, 57 strictly pre-periodic, 34 structurally stable, 200 thin at oo, 166 topological dimension, 158 topologically equivalent, 200 uniform expansion, 9 virtual center, 205 wandering domain, 58