Nonuniformly Hyperbolic Attractors: Geometric and Probabilistic Aspects 3030628132, 9783030628130

Table of contents :
Preface
Contents
1 Introduction
1.1 Physical Measures
1.2 SRB Measures
1.3 Decay of Correlations
References
2 Preliminaries
2.1 Partitions
2.1.1 Generating Partitions
2.1.2 Bases
2.1.3 Measurable Partitions
2.2 Jacobians
2.3 Basins
References
3 Expanding Structures
3.1 Gibbs-Markov Maps
3.1.1 Bounded Distortion
3.1.2 A Space for the Densities
3.1.3 Equilibrium Measures
3.2 Induced Maps
3.3 Tower Maps
3.3.1 Tower Extension
3.3.2 Convergence to the Equilibrium
3.3.3 Decay of Correlations
3.4 Lifting Observables
3.5 Application: Intermittent Maps
3.5.1 Neutral Fixed Point
3.5.2 Interval Map
3.5.3 Circle Map
References
4 Hyperbolic Structures
4.1 Young Structures
4.1.1 Quotient Return Map
4.1.2 Bounded Distortion
4.2 SRB Measures
4.2.1 Return Map
4.2.2 Original Dynamics
4.3 Tower Extension
4.3.1 Quotient Tower
4.4 Decay of Correlations
4.4.1 Reducing to the Quotient Tower
4.4.2 Regularity of the Discretisation
4.4.3 Specific Rates
4.4.4 The Non-exact Case
4.5 Regularity of the Stable Holonomy
4.5.1 Absolute Continuity
4.5.2 The Density Formula
4.6 Application: A Solenoid with Intermittency
4.6.1 Partially Hyperbolicity
4.6.2 Positive Lyapunov Exponent
4.6.3 Young Structure
References
5 Inducing Schemes
5.1 A General Framework
5.1.1 Bounded Distortion
5.2 The Partition
5.2.1 Inductive Construction
5.2.2 Key Relations
5.2.3 Metric Estimates
5.3 Inducing Times
5.3.1 Integrability
5.3.2 Tail Estimates
References
6 Nonuniformly Expanding Attractors
6.1 Nonuniform Expansion and Slow Recurrence
6.1.1 Hyperbolic Times and Preballs
6.2 Attractors
6.2.1 Ergodic Components
6.2.2 Unshrinkable Sets
6.3 Gibbs-Markov Induced Maps
6.4 SRB Measures
6.5 Decay of Correlations
6.6 Applications
6.6.1 Derived from Expanding
6.6.2 Viana Maps
References
7 Partially Hyperbolic Attractors
7.1 Dominated Splitting
7.1.1 Hölder Control of Centre-Unstable Disks
7.1.2 Hyperbolic Times and Predisks
7.1.3 Partial Hyperbolicity
7.2 Attractors
7.2.1 Ergodic Components
7.2.2 Unshrinkable Sets
7.3 Young Structure
7.3.1 Inducing Domain
7.3.2 Product Structure
7.3.3 Recurrence Times
7.4 SRB Measures
7.5 Decay of Correlations
7.6 Application: Derived from Anosov
References
Index

Citation preview

Springer Monographs in Mathematics

José F. Alves

Nonuniformly Hyperbolic Attractors Geometric and Probabilistic Aspects

Springer Monographs in Mathematics Editors-in-Chief Minhyong Kim, School of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea; Mathematical Institute, University of Warwick, Coventry, UK Katrin Wendland, Research group for Mathematical Physics, Albert Ludwigs University of Freiburg, Freiburg, Germany Series Editors Sheldon Axler, Department of Mathematics, San Francisco State University, San Francisco, CA, USA Mark Braverman, Department of Mathematics, Princeton University, Princeton, NY, USA Maria Chudnovsky, Department of Mathematics, Princeton University, Princeton, NY, USA Tadahisa Funaki, Department of Mathematics, University of Tokyo, Tokyo, Japan Isabelle Gallagher, Département de Mathématiques et Applications, Ecole Normale Supérieure, Paris, France Sinan Güntürk, Courant Institute of Mathematical Sciences, New York University, New York, NY, USA Claude Le Bris, CERMICS, Ecole des Ponts ParisTech, Marne la Vallée, France Pascal Massart, Département de Mathématiques, Université de Paris-Sud, Orsay, France Alberto A. Pinto, Department of Mathematics, University of Porto, Porto, Portugal Gabriella Pinzari, Department of Mathematics, University of Padova, Padova, Italy Ken Ribet, Department of Mathematics, University of California, Berkeley, CA, USA René Schilling, Institute for Mathematical Stochastics, Technical University Dresden, Dresden, Germany Panagiotis Souganidis, Department of Mathematics, University of Chicago, Chicago, IL, USA Endre Süli, Mathematical Institute, University of Oxford, Oxford, UK Shmuel Weinberger, Department of Mathematics, University of Chicago, Chicago, IL, USA Boris Zilber, Mathematical Institute, University of Oxford, Oxford, UK

This series publishes advanced monographs giving well-written presentations of the “state-of-the-art” in fields of mathematical research that have acquired the maturity needed for such a treatment. They are sufficiently self-contained to be accessible to more than just the intimate specialists of the subject, and sufficiently comprehensive to remain valuable references for many years. Besides the current state of knowledge in its field, an SMM volume should ideally describe its relevance to and interaction with neighbouring fields of mathematics, and give pointers to future directions of research.

More information about this series at http://www.springer.com/series/3733

José F. Alves

Nonuniformly Hyperbolic Attractors Geometric and Probabilistic Aspects

123

José F. Alves Department of Mathematics University of Porto Porto, Portugal

ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-3-030-62813-0 ISBN 978-3-030-62814-7 (eBook) https://doi.org/10.1007/978-3-030-62814-7 Mathematics Subject Classification: 37A05, 37A25, 37C40, 37D30 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Romery, Lavínia & Leonardo

“And after all is said and done And the numbers all come home The four rolls into three The three turns into two And the two becomes a one” Paul Simon

Preface

I started writing this text without even suspecting that I was starting to write a textbook. I suppose, like many other scientific books, it started out as lecture notes, designed to support a short course I lectured at SUSTech, Shenzhen—China, in July 2016. Since then, several other short courses around the world have motivated many improvements. The initial lecture notes were essentially a compilation of results published over the past two decades in various research articles by several authors. This text goes far beyond a simple compilation of articles. I dare say that writing an article is the art of convincing, writing a book is the art of explaining. Articles are rarely self-contained and often not presented in great detail. This sometimes contributes to the appearance of a certain folklore which is hard to understand (and impossible to reach in the literature) for beginners in the field. To write a textbook like this it is mandatory to go beyond the content of articles, for otherwise, it makes little sense to write a new text. The price to pay is the number of pages. This text was initially scheduled for the SpringerBriefs series. Due to restrictions on the length of the texts in that series (up to 120 pages, supposedly) and to make the presentation of this theory self-contained, examples and applications were discarded in the end, making the first version of this text merely the presentation of an abstract theory. However, on the recommendation of anonymous referees (with which I unconditionally agreed from the first hour), the initial version was promoted to the normal Springer series and recommended examples and applications were included. This meant an increase of about one hundred pages. Of course, only a few of the possible examples and applications from the vast list of publications on these topics in recent years have been included. These represent only the author’s taste and view of the theory. Many others can be found in the bibliographic references left here, and in other references not left, by mere forgetfulness or ignorance. This textbook is the production of a single author, but it has a godfather: indelibly associated to its genesis, Wael Bahsoun convinced Springer’s publishing editor and myself (and himself, I presume) that turning the lecture notes available on my webpage into a book would be a good idea (for both the publishing house and the author), with the attractive side of being a quick and easy task: in a couple vii

viii

Preface

of months a new book would see daylight. In the end, this possible couple of months became my whole sabbatical leave at Loughborough University, from October 2018 to August 2019, and effectively, a couple more months after my return to the University of Porto. In addition to my co-authors in several scientific articles that much contributed to this text, I would like to thank M. Benedicks, M. Carvalho, J. M. Freitas, S. Luzzatto, D. Mesquita, I. Melbourne, F. J. Moreira, V. Pinheiro, P. Varandas and H. Vilarinho for many fruitful conversations and valuable exchanges of ideas. I particularly wish to thank Wael Bahsoun, not only for pushing me towards this project that brought me a lot of pleasure (and a little bit suffering as well), but also for the many helpful conversations and advice on these topics, especially after reading a preliminary version of the text. Special thanks also to M. Viana for his availability in frequent exchanges of ideas, carried out exclusively through online chats—this efficient way of communicating in modern times. A final acknowledgement to Loughborough University for the excellent conditions provided and The Leverhulme Trust for the financial support through the Visiting Professorship VP2-2017-004. Also, partial financial support from CMUP (UID/MAT/00144/2013 & UIDB/00144/2020) and the projects PTDC/MAT-CAL/ 3884/2014 and PTDC/MAT-PUR/28177/2017 funded by FCT with national and European structural funds through the program FEDER, under the partnership agreement PT2020, must be recognised. Loughborough, England Porto, Portugal August 2019/July 2020

José F. Alves

Contents

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

1 1 3 5 6

2 Preliminaries . . . . . . . . . . . . . . . . 2.1 Partitions . . . . . . . . . . . . . . . 2.1.1 Generating Partitions . 2.1.2 Bases . . . . . . . . . . . . 2.1.3 Measurable Partitions . 2.2 Jacobians . . . . . . . . . . . . . . . 2.3 Basins . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

9 9 10 13 15 17 18 22

3 Expanding Structures . . . . . . . . . . . . . . . . 3.1 Gibbs-Markov Maps . . . . . . . . . . . . . . 3.1.1 Bounded Distortion . . . . . . . . . 3.1.2 A Space for the Densities . . . . . 3.1.3 Equilibrium Measures . . . . . . . 3.2 Induced Maps . . . . . . . . . . . . . . . . . . 3.3 Tower Maps . . . . . . . . . . . . . . . . . . . . 3.3.1 Tower Extension . . . . . . . . . . . 3.3.2 Convergence to the Equilibrium 3.3.3 Decay of Correlations . . . . . . . 3.4 Lifting Observables . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

23 23 27 29 32 37 41 46 48 76 80

1 Introduction . . . . . . . . . . . 1.1 Physical Measures . . . 1.2 SRB Measures . . . . . . 1.3 Decay of Correlations . References . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

ix

x

Contents

3.5 Application: Intermittent Maps 3.5.1 Neutral Fixed Point . . . 3.5.2 Interval Map . . . . . . . . 3.5.3 Circle Map . . . . . . . . . References . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. 84 . 84 . 87 . 91 . 100

4 Hyperbolic Structures . . . . . . . . . . . . . . . . . . . 4.1 Young Structures . . . . . . . . . . . . . . . . . . . 4.1.1 Quotient Return Map . . . . . . . . . . . 4.1.2 Bounded Distortion . . . . . . . . . . . . 4.2 SRB Measures . . . . . . . . . . . . . . . . . . . . . 4.2.1 Return Map . . . . . . . . . . . . . . . . . . 4.2.2 Original Dynamics . . . . . . . . . . . . . 4.3 Tower Extension . . . . . . . . . . . . . . . . . . . 4.3.1 Quotient Tower . . . . . . . . . . . . . . . 4.4 Decay of Correlations . . . . . . . . . . . . . . . . 4.4.1 Reducing to the Quotient Tower . . . 4.4.2 Regularity of the Discretisation . . . . 4.4.3 Specific Rates . . . . . . . . . . . . . . . . 4.4.4 The Non-exact Case . . . . . . . . . . . . 4.5 Regularity of the Stable Holonomy . . . . . . 4.5.1 Absolute Continuity . . . . . . . . . . . . 4.5.2 The Density Formula . . . . . . . . . . . 4.6 Application: A Solenoid with Intermittency 4.6.1 Partially Hyperbolicity . . . . . . . . . . 4.6.2 Positive Lyapunov Exponent . . . . . 4.6.3 Young Structure . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

103 103 106 107 109 109 115 117 120 124 126 129 131 135 137 141 143 146 148 151 154 158

5 Inducing Schemes . . . . . . . . . . . . . 5.1 A General Framework . . . . . . 5.1.1 Bounded Distortion . . . 5.2 The Partition . . . . . . . . . . . . . 5.2.1 Inductive Construction . 5.2.2 Key Relations . . . . . . . 5.2.3 Metric Estimates . . . . . 5.3 Inducing Times . . . . . . . . . . . 5.3.1 Integrability . . . . . . . . . 5.3.2 Tail Estimates . . . . . . . References . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

161 161 163 166 167 169 172 173 175 177 187

. . . . .

. . . . . . . . . . .

. . . . .

. . . . . . . . . . .

. . . . .

. . . . . . . . . . .

. . . . .

. . . . . . . . . . .

. . . . .

. . . . . . . . . . .

. . . . .

. . . . . . . . . . .

. . . . .

. . . . . . . . . . .

. . . . .

. . . . . . . . . . .

. . . . . . . . . . .

6 Nonuniformly Expanding Attractors . . . . . . . . . . . . . . . . . . . . . . . . . 189 6.1 Nonuniform Expansion and Slow Recurrence . . . . . . . . . . . . . . . . 189 6.1.1 Hyperbolic Times and Preballs . . . . . . . . . . . . . . . . . . . . . 192

Contents

6.2 Attractors . . . . . . . . . . . . . . . . . 6.2.1 Ergodic Components . . . 6.2.2 Unshrinkable Sets . . . . . 6.3 Gibbs-Markov Induced Maps . . 6.4 SRB Measures . . . . . . . . . . . . . 6.5 Decay of Correlations . . . . . . . . 6.6 Applications . . . . . . . . . . . . . . . 6.6.1 Derived from Expanding 6.6.2 Viana Maps . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .

xi

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

198 199 200 202 204 207 212 212 216 222

7 Partially Hyperbolic Attractors . . . . . . . . . . . . . . . . 7.1 Dominated Splitting . . . . . . . . . . . . . . . . . . . . . 7.1.1 Hölder Control of Centre-Unstable Disks 7.1.2 Hyperbolic Times and Predisks . . . . . . . 7.1.3 Partial Hyperbolicity . . . . . . . . . . . . . . . 7.2 Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Ergodic Components . . . . . . . . . . . . . . . 7.2.2 Unshrinkable Sets . . . . . . . . . . . . . . . . . 7.3 Young Structure . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Inducing Domain . . . . . . . . . . . . . . . . . . 7.3.2 Product Structure . . . . . . . . . . . . . . . . . . 7.3.3 Recurrence Times . . . . . . . . . . . . . . . . . 7.4 SRB Measures . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Decay of Correlations . . . . . . . . . . . . . . . . . . . . 7.6 Application: Derived from Anosov . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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

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

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

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

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

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

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

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

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

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

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

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

225 226 226 229 232 233 234 235 238 238 242 245 246 249 252 256

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

Chapter 1

Introduction

A central problem in the theory of Dynamical Systems is Palis Conjecture [42], which states that typical systems in finite dimensional Riemannian manifolds possess a finite number of measures (physical measures) which describe the time averages of almost all orbits with respect to the Lebesgue (volume) measure. The concept of a typical system depends on the context, and can, for instance, be interpreted as the system belonging in an open and dense subset of the space of transformations in some topology, or as a subset of total measure in some generic parametrised family of transformations. In this text, we consider some special classes of dynamical systems for which Palis conjecture has already been proved, addressing not only the existence of physical measures, but also some of their statistical properties. The existence of physical measures is, to a certain extent, related to the existence of attractors for the dynamical system, that is, sets which are invariant under the dynamics and attract the orbits of a subset of initial states with positive Lebesgue measure.

1.1 Physical Measures Let M be a differentiable compact finite dimensional Riemannian manifold with Lebesgue (volume) measure on the Borel sets of M and f : M → M. A Borel probability measure μ on M is said to be a physical measure for f if there exists a positive Lebesgue measure set of initial states x ∈ M such that, for any continuous ϕ : M → R, we have  n−1 1 ϕ( f j (x)) = ϕ dμ. n→∞ n j=0 lim

(1.1)

A physical measure μ is necessarily f -invariant, that is, μ( f −1 (A)) = μ(A), for every Borel set A ⊂ M. We define the basin of an f -invariant Borel probability © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. F. Alves, Nonuniformly Hyperbolic Attractors, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-62814-7_1

1

2

1 Introduction

measure μ on M as the set of points x ∈ M such that (1.1) holds for every continuous ϕ : M → R. A point belonging in the basin of a probability measure μ means that the averages of Dirac measures along orbit of that point converge in the weak* topology to μ. Since the integrals of continuous functions characterise Borel probability measures on compact metric spaces, then the basins of distinct physical measures are necessarily disjoint. Birkhoff Ergodic Theorem [14] gives that if μ is an f -invariant probability measure, then the time averages in (1.1) converge for μ almost  every x ∈ M. In addition, these time averages converge to the spatial average ϕdμ, whenever μ is an ergodic measure, that is, there is no Borel set A with f −1 (A) = A such that both A and M \ A have positive μ measure. Palis conjecture is already proved for uniformly hyperbolic dynamical systems, which are characterised by the existence of an invariant splitting of the tangent bundle for which the derivative of the map contracts/expands uniformly on each one of the invariant directions of that splitting. It follows from the pioneering works of Sinai, Ruelle and Bowen [16, 17, 47, 51] that uniformly hyperbolic systems have a finite number of physical measures whose basins cover Lebesgue almost all points in the ambient manifold. The success of the approach in these works is based on the possibility of coding the system with a finite number of symbols, via finite Markov partitions, and conjugating it to a subshift of finite type. Many other statistical properties for the physical measures of uniformly hyperbolic attractors can be deduced using this coding, comprising the existence of equilibrium states, exponential decay of correlations or exponential large deviations; see [16, 54]. Extensions of the classical ideas to countable Markov shifts have been obtained in [18, 30, 31, 48–50]. The problem of existence and finiteness of physical measures for a typical dynamical systems remains open in general, even for nonuniformly hyperbolic dynamical systems, where asymptotic exponential rates of expansion/contraction prevail in certain directions assured by Oseledets Theorem [41]. For systems of this type, Young introduced in [56, 57] some Markov-like partitions with infinitely many symbols, which allowed her to use an inducing scheme to define a new dynamical system with uniformly hyperbolic behaviour, retrieving from this induced system relevant statistical properties of the original system. In broad terms, inducing schemes are characterised by the existence of some region of the phase space partitioned into a countable number of elements with certain recurrence times associated. This approach can be thought of as a generalisation of the classic ideas of Sinai, Ruelle and Bowen, allowing greater flexibility in three main aspects: (i) codification of only a region of the phase space; (ii) codification with a countable number of symbols; (iii) variable return times. As in the theory of uniformly hyperbolic systems, using these structures we can deduce many other statistical properties of these measures. The strategy of inducing has been particularly well succeeded in certain classes of nonuniform hyperbolic systems, including logistic maps [55, 56], intermittent maps [57], Viana maps [2, 8, 26, 52], billiards [21, 22, 33, 56] and Hénon-like attractors [13, 53]. We also refer to several results obtained in the last two decades that lead to what can already be considered a theory of nonuniformly expanding or partially hyperbolic attractors, including the existence of physical measures [3, 5, 15, 45] and some of their statistical properties [6, 7, 9, 10, 19, 20, 23, 26, 34, 36].

1.1 Physical Measures

3

On the other hand, it was proved in [4, 5] that the existence of inducing schemes is not only sufficient, but also a necessary condition for the existence of physical measures for nonuniformly expanding or partially hyperbolic attractors.

1.2 SRB Measures The physical measures introduced by Sinai, Ruelle and Bowen for uniformly hyperbolic attractors have some special geometric properties that we describe next. Let m denote the Lebesgue measure on the Borel sets of a compact Riemannian manifold M and d the distance in M, both induced by the Riemannian metric. Given a submanifold γ ⊂ M, we use m γ to denote the Lebesgue measure on γ induced by the Riemannian structure restricted to γ . Consider f : M → M a C 1+α diffeomorphism of the manifold M, that is, a C 1 diffeomorphism whose derivative is Hölder continuous. Given x ∈ M and v ∈ Tx M, define the Lyapunov exponent λ(x, v) =

1 log D f n (x)v, |n|→+∞ n lim

if this limit exists. A result due to Oseledets [41] establishes that if f preserves an invariant Borel probability measure μ, then there exist measurable functions λi and a D f -invariant splitting Tx M = ⊕i E i (x) such that λ(x, v) = λi (x), for μ almost every x ∈ M and every v ∈ E i (x). In addition, λi and dim(E i ) are constant μ almost everywhere whenever μ is ergodic. We define the regular set R as the set of those points in M for which the Lyapunov exponents are defined. A result due to Pesin [43] establishes that if x ∈ R has at least one positive Lyapunov exponent, then there is a smooth disk γ u (x) ⊂ M tangent to E u (x) = ⊕λi >0 E i (x) such that, for all y ∈ γ u (x), d( f −n (y), f −n (x)) ≤ Cλn d(y, x), for all n ≥ 1,

(1.2)

where 0 < λ < 1 and C > 0 depends on x in a measurable way. The set γ u (x) is called the unstable disk of the point x ∈ R. This unstable disk is unique in the following sense: for an appropriate choice of λ, it consists of all points y close enough to x for which there is a constant C > 0 such that (1.2) is satisfied; see for example [12, Sect. 8.1]. The disk γ u (x) can be obtained as the image under the exponential map of the graph of a function on a ball of radius ε > 0 around the origin in E u (x) taking values in a complement to E u (x) in the tangent space Tx M. Sometimes, we refer to this ε > 0 as the size of the unstable disk, which we also denote by γεu (x), to highlight this fact. A stable disk γεs (x), or simply γ s (x), for a point x ∈ R with at least one negative Lyapunov exponent, can obtained in the same way using f −1 in place of f . Stable and unstable disks are invariant, in the sense that f (γ s (x)) ⊂ γ s ( f (x)) and f −1 (γ u (x)) ⊂ γ u ( f −1 (x)),

4

1 Introduction

for all x ∈ R. The sets   f n (γ u ( f −n (x))) and W s (x) = W u (x) = n≥0

n≥0

f −n (γ s ( f n (x))) (1.3)

are called the unstable manifold and the stable manifold of the point x, respectively. These are actually immersed submanifolds of M. Before we describe the physical measures obtained by Sinai, Ruelle and Bowen for uniformly hyperbolic attractors, we start with some general measure theoretical considerations. Let (X, B, μ) be a probability measure space, where X ⊂ M is a compact set and B is the σ -algebra of Borel sets. Given a partition P of X into Borel sets, let π : X → ξ be the map assigning to each x ∈ X the element ω ∈ P such that x ∈ ω. Consider the probability measure space (P, π∗ P, π∗ μ), where π∗ P = {Q ⊂ P : π −1 (Q) ∈ B} and π∗ μ is the push-forward of μ, given by π∗ μ(Q) = μ(π −1 (Q)), for all Q ∈ π∗ P. We define a disintegration of μ with respect to the partition P as a family {μω }ω∈P of probability measures on X such that μω (ω) = 1, for π∗ μ, almost every ω ∈ P, and given any continuous ϕ : X → R, the function P  ω → ϕdμω is measurable and     ϕdμ = ϕdμω dπ∗ μ. (1.4) X

P

X

We refer to the measures μω as the conditional measures of μ with respect to P and to π∗ μ as the quotient measure of μ. Rohlin Disintegration Theorem [46] establishes that μ has a disintegration with respect to any measurable partition P of a compact set X . Roughly speaking, the measurability of a partition means the existence of a countable number of cuts in X that generates all elements of P, up to measure zero; see Sect. 2.1.3. Assume now that f : M → M preserves a Borel probability measure μ. A measurable partition P of X ⊂ M is said to be subordinate to unstable manifolds if f has at least one positive Lyapunov exponent μ almost everywhere, and for μ almost every x ∈ X , the element of P containing x is a subset of W u (x). The probability measure μ is called a Sinai-Ruelle-Bowen (SRB) measure if, for any measurable partition P subordinate to unstable manifolds, the conditionals {μω }ω∈P of μ with respect to P are μ almost everywhere absolutely continuous with respect to the conditionals {m ω }ω∈P of the Lebesgue measure m. To see that an SRB measure is a physical measure, we need to check an additional transversally condition. Given embedded disks D, D ⊂ M intersecting   a set of stable disks {γ s (x)}x , we define the stable holonomy h : x γ s (x) ∩ D → x γ s (x) ∩ D ,

as the map assigning to each z ∈ γ s (x) ∩ D the unique point in γ s (x) ∩ Ds . We say that the stable holonomy is absolutely continuous if, for any A ⊂ x γ (x) ∩ D, we have m D (A) > 0 if, and only if, m D (h(A)) > 0. A result due to Pesin [43] establishes that if f is a C 1+α diffeomorphism whose Lyapunov exponents are all non-zero with respect to an ergodic invariant probability measure μ, then the stable holonomy is absolutely continuous. Using Birkhoff Ergodic Theorem, it can be proved that the basin of an ergodic probability measure μ in a compact Riemannian manifold M contains μ almost every point in M; see Proposition 2.12. Since time

1.2 SRB Measures

5

averages with respect to continuous functions are constant on stable manifolds, it follows that every ergodic SRB measure with non-zero Lyapunov exponents is a physical measure, provided the stable holonomy is absolutely continuous. This is an efficient way of obtaining physical measures for certain classes of dynamical systems, especially those with chaotic attractors. In Sects. 6.4 and 7.4, we deduce the existence of physical measures for nonuniformly expanding attractors and partially hyperbolic attractors whose central direction is mostly expanding using this method. For additional properties on SRB measures, see [58].

1.3 Decay of Correlations We aim at describing the statistical properties of invariant measures for a given dynamical system, especially with respect to its SRB measures. In recent years, many statistical features of dynamics, such as Central Limit Theorem, decay of correlations or large deviations have been studied by several authors in many contexts. Here, we will be focused on the decay of correlations, not only for its intrinsic value in terms of the mixing properties of the dynamical system, but also because it is frequently used to deduce many other statistical properties of the system; see for example [6, 24, 25, 27–29, 32, 34–40, 56, 57]. We define the correlation function of observables ϕ, ψ : M → R with respect to an f -invariant probability measure μ as    Cor μ (ϕ, ψ ◦ f n ) = ϕ(ψ ◦ f n )dμ − ϕdμ ψdμ . It is sometimes possible to obtain specific rates of decay to 0 when n → ∞ for this correlation function, provided the observables ϕ and ψ are sufficiently regular. In most of our results we will take at least the observable ϕ satisfying some Hölder continuity condition and the observable ψ essentially bounded. In the case of invertible systems with contracting directions we actually need ψ satisfying the Hölder continuity condition as well. Notice that choosing these observables characteristic functions of measurable sets, the convergence of the correlation function to zero gives the definition of a mixing measure. As observed before, a physical measure μ is sometimes absolutely continuous, or even equivalent, to the Lebesgue measure m. Assuming that m is normalised and ϕ is the density of m with respect to μ, we have   n Cor μ (ϕ, ψ ◦ f ) = ψ f ∗ dm − ψdμ . n

In this case, the decay of Cor μ (ϕ, ψ ◦ f n ) gives information on the speed at which the push-forwards f ∗n m converge to the physical measure μ. As with the existence of SRB measures, the rates on the decay of correlations for nonuniformly expanding maps and partially hyperbolic diffeomorphisms are

6

1 Introduction

obtained here using inducing schemes. As shown in [6], the existence of inducing schemes is not only sufficient, but also necessary for certain rates on the decay of correlations of SRB measures for nonuniformly expanding attractors; see also [1] and [11, Appendix A].

References 1. R. Aimino, J.M. Freitas, Large deviations for dynamical systems with stretched exponential decay of correlations. Port. Math. 76(2), 143–152 (2019) 2. J. F. Alves, SRB measures for non-hyperbolic systems with multidimensional expansion. Ann. Sci. École Norm. Sup. (4) 33(1), 1–32 (2000) 3. J.F. Alves, C. Bonatti, M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140(2), 351–398 (2000) 4. J.F. Alves, C.L. Dias, S. Luzzatto, Geometry of expanding absolutely continuous invariant measures and the liftability problem. Ann. Inst. H. Poincaré Anal. Non Linéaire 30(1), 101– 120 (2013) 5. J.F. Alves, C.L. Dias, S. Luzzatto, V. Pinheiro, SRB measures for partially hyperbolic systems whose central direction is weakly expanding. J. Eur. Math. Soc. (JEMS) 19(10), 2911–2946 (2017) 6. J.F. Alves, J.M. Freitas, S. Luzzatto, S. Vaienti, From rates of mixing to recurrence times via large deviations. Adv. Math. 228(2), 1203–1236 (2011) 7. J.F. Alves, X. Li, Gibbs-Markov-Young structures with (stretched) exponential tail for partially hyperbolic attractors. Adv. Math. 279, 405–437 (2015) 8. J.F. Alves, S. Luzzatto, V. Pinheiro, Markov structures and decay of correlations for nonuniformly expanding dynamical systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 22(6), 817– 839 (2005) 9. J.F. Alves, V. Pinheiro, Slow rates of mixing for dynamical systems with hyperbolic structures. J. Stat. Phys. 131(3), 505–534 (2008) 10. J.F. Alves, V. Pinheiro, Topological structure of (partially) hyperbolic sets with positive volume. Trans. Amer. Math. Soc. 360(10), 5551–5569 (2008) 11. V. Araújo, I. Melbourne, Mixing properties and statistical limit theorems for singular hyperbolic flows without a smooth stable foliation. Adv. Math. 349, 212–245 (2019) 12. L. Barreira, Y. Pesin, Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents Encyclopedia of Mathematics and its Applications (Cambridge University Press, Cambridge, 2007) 13. M. Benedicks, L.-S. Young, Markov extensions and decay of correlations for certain Hénon maps. Astérisque 261, 13–56 (2000). Géométrie complexe et systémes dynamiques (Orsay, 1995) 14. G.D. Birkhoff, Proof of the ergodic theorem. Proc. Natl. Acad. Sci. 17(12), 656–660 (1931) 15. C. Bonatti, M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115, 157–193 (2000) 16. R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, vol. 470, Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1975) 17. R. Bowen, D. Ruelle, The ergodic theory of Axiom A flows. Invent. Math. 29(3), 181–202 (1975) 18. J. Buzzi, O. Sarig, Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps. Ergodic Theory Dynam. Syst. 23(5), 1383–1400 (2003) 19. A. Castro, Backward inducing and exponential decay of correlations for partially hyperbolic attractors. Israel J. Math. 130, 29–75 (2002)

References

7

20. A. Castro, Fast mixing for attractors with a mostly contracting central direction. Ergodic Theory Dynam. Syst. 24(1), 17–44 (2004) 21. N. Chernov, Decay of correlations and dispersing billiards. J. Statist. Phys. 94(3–4), 513–556 (1999) 22. N. Chernov, H.-K. Zhang, Billiards with polynomial mixing rates. Nonlinearity 18(4), 1527– 1553 (2005) 23. D. Dolgopyat, On dynamics of mostly contracting diffeomorphisms. Comm. Math. Phys. 213(1), 181–201 (2000) 24. S. Gouëzel, Central limit theorem and stable laws for intermittent maps. Probab. Theory Related Fields 128(1), 82–122 (2004) 25. S. Gouëzel, Berry-Esseen theorem and local limit theorem for non uniformly expanding maps. Ann. Inst. H. Poincaré Probab. Statist. 41(6), 997–1024 (2005) 26. S. Gouëzel, Decay of correlations for nonuniformly expanding systems. Bull. Soc. Math. France 134(1), 1–31 (2006) 27. A. Korepanov, Equidistribution for nonuniformly expanding dynamical systems, and application to the almost sure invariance principle. Comm. Math. Phys. 359(3), 1123–1138 (2018) 28. A. Korepanov, Rates in almost sure invariance principle for dynamical systems with some hyperbolicity. Comm. Math. Phys. 363(1), 173–190 (2018) 29. A. Korepanov, Z. Kosloff, I. Melbourne, Martingale-coboundary decomposition for families of dynamical systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 35(4), 859–885 (2018) 30. F. Ledrappier, Y. Lima, O. Sarig, Ergodic properties of equilibrium measures for smooth three dimensional flows. Comment. Math. Helv. 91(1), 65–106 (2016) 31. Y. Lima, O.M. Sarig, Symbolic dynamics for three-dimensional flows with positive topological entropy. J. Eur. Math. Soc. (JEMS) 21(1), 199–256 (2019) 32. C. Liverani, Central limit theorem for deterministic systems, in International Conference on Dynamical Systems (Montevideo, 1995), vol. 362 of Pitman Research Notes in Mathematics Series, pp. 56–75. Longman, Harlow (1996) 33. R. Markarian, Billiards with polynomial decay of correlations. Ergodic Theory Dynam. Syst. 24(1), 177–197 (2004) 34. I. Melbourne, Large and moderate deviations for slowly mixing dynamical systems. Proc. Amer. Math. Soc. 137(5), 1735–1741 (2009) 35. I. Melbourne, M. Nicol, Almost sure invariance principle for nonuniformly hyperbolic systems. Comm. Math. Phys. 260(1), 131–146 (2005) 36. I. Melbourne, M. Nicol, Large deviations for nonuniformly hyperbolic systems. Trans. Amer. Math. Soc. 360(12), 6661–6676 (2008) 37. I. Melbourne, M. Nicol, A vector-valued almost sure invariance principle for hyperbolic dynamical systems. Ann. Probab. 37(2), 478–505 (2009) 38. I. Melbourne, P. Varandas, A note on statistical properties for nonuniformly hyperbolic systems with slow contraction and expansion. Stoch. Dyn. 16(3), 1660012, 13 (2016) 39. I. Melbourne, P. Varandas, Convergence to a Lévy process in the skorohod M1 and M2 topologies for nonuniformly hyperbolic systems, including billiards with cusps. Commun. Math. Phys. to appear 40. I. Melbourne, R. Zweimüller, Weak convergence to stable Lévy processes for nonuniformly hyperbolic dynamical systems. Ann. Inst. Henri Poincaré Probab. Stat. 51(2), 545–556 (2015) 41. V.I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems. Trudy Moskov. Mat. Obšˇc 19, 179–210 (1968) 42. J. Palis, A global perspective for non-conservative dynamics. Ann. Inst. H. Poincaré Anal. Non Linéaire 22(4), 485–507 (2005) 43. J.B. Pesin, Families of invariant manifolds that correspond to nonzero characteristic exponents. Izv. Akad. Nauk SSSR Ser. Mat. 40(6), 1332–1379, 1440 (1976) 44. Y.B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory. Uspehi Mat. Nauk 32(4(196)), 55–112, 287 (1977) 45. V. Pinheiro, Sinai-Ruelle-Bowen measures for weakly expanding maps. Nonlinearity 19(5), 1185–1200 (2006)

8

1 Introduction

46. V.A. Rohlin, On the fundamental ideas of measure theory. Amer. Math. Soc. Translation 1952(71), 55 (1952) 47. D. Ruelle, A measure associated with Axiom A attractors. Amer. J. Math. 98(3), 619–654 (1976) 48. O.M. Sarig, Thermodynamic formalism for countable Markov shifts. Ergodic Theory Dynam. Syst. 19(6), 1565–1593 (1999) 49. O. Sarig, Existence of Gibbs measures for countable Markov shifts. Proc. Amer. Math. Soc. 131(6), 1751–1758 (2003) 50. O.M. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy. J. Amer. Math. Soc. 26(2), 341–426 (2013) 51. J.G. Sinai, Gibbs measures in ergodic theory. Uspehi Mat. Nauk 27(4), 21–64 (1972) 52. M. Viana, Multidimensional nonhyperbolic attractors. Inst. Hautes Études Sci. Publ. Math. 85, 63–96 (1997) 53. Q. Wang, L.-S. Young, Toward a theory of rank one attractors. Ann. Math. (2) 167(2), 349–480 (2008) 54. L.-S. Young, Large deviations in dynamical systems. Trans. Amer. Math. Soc. 318(2), 525–543 (1990) 55. L.-S. Young, Decay of correlations for certain quadratic maps. Commun. Math. Phys. 146(1), 123–138 (1992) 56. L.-S. Young. Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. (2) 147(3), 585–650 (1998) 57. L.-S. Young, Recurrence times and rates of mixing. Israel J. Math. 108, 733–754 (1999) 58. L.-S. Young, What are SRB measures, and which dynamical systems have them? J. Statist. Phys. 108(5)(6), 733–754 (2002)

Chapter 2

Preliminaries

In this chapter, we present some concepts and results that will be used throughout this text. Some of them are quite standard, but we address them here in any case, for the sake of clarity of the exposition. In Sect. 2.1, we introduce some key notions related to partitions in a measure space: generating partitions, bases and measurable partitions. The first two notions, especially designed for the definition of Gibbs-Markov that we present in Chap. 3, the third one for the SRB measures in Chap. 4. The notion of Jacobian addressed in Sect. 2.2 will also be useful in Chap. 3. In the final section, we present some results about physical measures and the basin of a measure that will be useful in several places along the text.

2.1 Partitions Let M be a set with a measure m defined on a σ -algebra A of subsets of M. Under these conditions, we refer to (M, A, m) as a measure space, M as a measurable space and the elements of A as measurable sets. We say that a set M0 ⊂ M has full m measure if M0 is a measurable set such that m(M \ M0 ) = 0. Two measurable sets A, B ⊂ M are said to be equal m mod 0 if there is a set M0 with full m measure such that A ∩ M0 = B ∩ M0 . We say that a family P of subsets of M is an m mod 0 partition of M if there exists a set M0 with full m measure such that the elements in the family {ω ∩ M0 : ω ∈ P} are pairwise disjoint and their union is equal to M0 . In case M0 can be equal to M, we will refer to P simply as a partition of M. Two families P and Q of subsets of M are said to be equal m mod 0 if there exists a set M0 with full m measure such that the families {ω ∩ M0 : ω ∈ P} and {ω ∩ M0 : ω ∈ Q} coincide. Equalities m mod 0 are often referred to only as mod 0, when the underlying measure is obvious. We say that a sequence (Pn )n of mod 0 partitions is increasing, and write P1 ≺ P2 ≺ · · · , if each partition is a refinement of the preceding one, meaning that the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. F. Alves, Nonuniformly Hyperbolic Attractors, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-62814-7_2

9

10

2 Preliminaries

elements in Pn are mod 0 union of elements in Pn+1 , for all n ≥ 1. Given any sequence (Pn )n of mod 0 partitions into measurable sets, set for each n ≥ 1 n 

  Pi = ω1 ∩ · · · ∩ ωn | ωi ∈ Pi , for all 1 ≤ i ≤ n

i=1

and

∞ 

  Pn = ω1 ∩ ω2 ∩ ω3 ∩ · · · | ωn ∈ Pn , for all n ≥ 1 .

n=1

 n Pi is a mod 0 partition M, and the same holds for and ∞ Note that each i=1 n=1 Pn . of n Pi )n is increasing and ∞ P In addition, the sequence of partitions ( i=1 n=1 n is as a common refinement of all partitions in this sequence. It is actually the largest common refinement of them all, in the sense that any other common refinement must necessarily contain ∞ n=1 Pn .

2.1.1 Generating Partitions Let (M, A, m) be a measure space and (Pn )n a sequence of mod 0 partitions of M. We say that (Pn )n generates A (mod 0) if the σ -algebra generated by the family  ∞ n=1 Pn coincides with A (mod 0). Below, we present some results about generating sequences. The symbol will be used to denote the symmetric difference of two sets, defined as A B = (A \ B) ∪ (B \ A). Proposition 2.1 Let (Pn )n be an increasing sequence of countable mod 0 partitions that generates A (mod 0). Given δ > 0 and there  are n,  ≥ 1 and pairwise  A ∈A,  ωi < δ. disjoint sets ω1 , . . . , ω ∈ Pn such that m A i=1 Proof Since the conclusion is stated in terms of measure, it is no restriction to assume that each Pn is a partition of M. Take an arbitrary δ > 0. Let B be the family of all sets A ∈ A for which the conclusion holds. We are going to show that B is a σ -algebra. First, we show that M ∈ B. Take arbitrary δ > 0 and n ≥ 1. Since Pn is a countable mod 0 partition of M, we have m(M) =



m(ω).

(2.1)

ω∈Pn

If Pn is a finite partition, nothing needs to be done. Otherwise, choosing a partial sum of the series in (2.1) for which the remainder is smaller that δ and taking the elements of Pn corresponding to the terms in that partial sum, we easily deduce that M ∈ B. Now we show that, if A ∈ B, then M \ A belongs in B. Take an arbitrary δ > 0.

2.1 Partitions

11

Since  are n,  ≥ 1 and pairwise disjoint sets ω1 , . . . , ω ∈ Pn such that  A ∈ B, there ωi < δ/2. Note that m A ∪i=1





    δ = m A ωi ωi < . m (M \ A) M \ 2 i=1 i=1

(2.2)

 ωi is a union If the partition Pn is finite, it is sufficient to observe that M \ i=1 of pairwise disjoint sets in Pn . Otherwise, consider the elements of Pn listed as ω1 , ω2 , . . . , ω , ω+1 , . . . Since m is a finite measure and



N ∞   lim m (M \ A) ωi = m (M \ A) ωi ,

N →∞

i=+1

i=+1

choosing N sufficiently large and using (2.2), we ensure that



N ∞   δ m (M \ A) ωi < m (M \ A) ωi + 2 i=+1 i=+1



  δ + < δ. = m (M \ A) M \ ωi 2 i=1 This shows that M \ A also belongs in B. Finally, we show that if A1 , A2 , . . . are pairwise disjoint sets in B, then their union belongs in B. Given an arbitrary δ > 0, for each k ≥ 1 we may find n k ≥ 1 and ωk,1 , . . . , ωk,k ∈ Pn k such that

k  δ ωk,i < k . m Ak 2 i=1 Since

 ∞

(2.3)

  k k ∞  ∞

 Ak Ak ωk,i ⊂ ωk,i ,

k=1

k=1 i=1

k=1

i=1

using (2.3) we obtain m

 ∞

 k ∞  < δ. Ak ωk,i

k=1

k=1 i=1

Thus, we may choose N ≥ 1 such that m

 ∞ k=1

 k N  < δ. Ak ωk,i k=1 i=1

(2.4)

12

2 Preliminaries

Now, as P1 ≺ P2 ≺ · · · , choosing n = max{n 1 , . . . n N }, we may find ω1 , ω2 , · · · ∈ Pn such that k N  ∞   ωk,i = ωi . k=1 i=1

i=1

Since lim m

→∞

 ∞



   ∞ ∞ =m Ak ωi Ak ωi

k=1

i=1

k=1

=m

 ∞

i=1

 k N  < δ, Ak ωk,i

k=1

k=1 i=1

there exists  ≥ 1 such that m

 ∞

  < δ. Ak ωi

k=1

i=1

 Recalling that ω1 , . . . , ω ∈ Pn , we deduce that k≥1 Ak ∈ B. This concludes the proof ∞ that B is a σ -algebra. Finally, observe that B trivially contains the union n=1 Pn . Since, by assumption, this union generates the σ -algebra A and B ⊂ A, then we have B = A, and the conclusion follows.  Corollary 2.2 Let (Pn )n be an increasing sequence of countable mod 0 partitions that generates A (mod 0). Given δ > 0 and A with m(A) > 0, there are n ≥ 1 and ω ∈ Pn such that m(ω \ A) < δm(ω). Proof By Proposition 2.1, we may find integers n,  ≥ 1 and pairwise disjoint sets ω1 , . . . , ω ∈ Pn such that

m A

  i=1

ωi


, 2 i=1

which together with (2.5) yields

 



  δ m ωi \ A < m(A) < δm ωi . 2 i=1 i=1

(2.5)

2.1 Partitions

13

Since ω1 , . . . , ω are pairwise disjoint sets, it follows that 

m (ωi \ A) < δ

i=1



m (ωi ) .

i=1

Arguing by contradiction, we easily see that there exists some 1 ≤ i ≤  for which  m(ωi \ A) < δm(ωi ). Consider now a measurable map f : M → M and a measure m (not necessarily f -invariant) on M such that f ∗ m m, meaning that the push-forward measure f ∗ m is absolutely continuous with respect to m: m(A) = 0 =⇒ m( f −1 (A)) = 0. Given a mod 0 partition P, set for n ≥ 0, f −n P = { f −n (ω) : ω ∈ P}. Since f ∗ m m, it follows that f −n P is still a mod 0 partition of M, for all n ≥ 1. In this case, we have for all n ≥ 1 n−1 

  f − j P = ω0 ∩ f −1 (ω1 ) ∩ · · · ∩ f −n+1 (ωn−1 ) : ω0 , . . . , ωn−1 ∈ P

(2.6)

j=0

and

∞ 

  f −n P = ω0 ∩ f −1 (ω1 ) ∩ · · · : ωn ∈ P for all n ≥ 0 .

(2.7)

n=0

 −j P)n generates A (mod 0), In case the increasing sequence of partitions ( n−1 j=0 f we simply say that P is a generating partition. Corollary 2.3 Let m be a measure for which f ∗ m m. If P is a countable generatingpartition, then for all δ > 0 and A ∈ A with m(A) > 0, there are n ≥ 1 and −j P such that m(ω \ A) < δm(ω). ω ∈ n−1 j=0 f Proof Take Pn =

n−1 j=0

f − j P and use Corollary 2.2.



2.1.2 Bases Here, we introduce the concept of basis of a measure space, in line with [2, 3]. Let (M, A, m) be a measure space. We say that an increasing sequence (Pn )n of countable m mod 0 partitions is a basis of the measure space if

14

2 Preliminaries

•  (Pn )n generates A (mod 0); • ∞ n=1 Pn is the partition into single points (mod 0). A measure space with a basis is called a separable measure space. Below, we present some results related to Borel measure spaces. Suppose M is a metric space with a measure m defined on the Borel σ -algebra of M. The diameter of an m mod 0 partition P of M is defined as diam(P) = sup {diam(ω) : ω ∈ P} , where diam(ω) stands for the diameter of ω ⊂ M. Lemma 2.4 Let (Pn )n an increasing sequence of countable mod 0 partitions by Borel sets of M. If diam(Pn ) → 0, when n → ∞, then (Pn )n is a basis of M. Proof Let B denote the σ -algebra of Borel sets. For each n ≥ 0, let Mn be a measurable set with full m measure such that {ω ∩ Mn : ω ∈ Pn } are pairwise disjoint sets whose union is equal to Mn . Set M0 =

Mn .

n

We have that M0 is a measurable set with m(M \ M0 ) = 0. In addition, since diam(Pn ) → 0, for any x, x  ∈ M0 with x = y,there are n ≥ 1 and ω, ω ∈ Pn such that x ∈ ω and x  ∈ ω . This shows that ∞ n=1 Pn is the partition into single points (mod 0). We are left to show that (Pn )n generates B (mod 0). Since each Pn isformed by Borel sets, we just need to show that the σ -algebra generated by ∞ n=1 Pn contains B (mod 0). Consider an arbitrary open set U ⊂ M. ∈ U ∩ M0 , there are n x ≥ 1 and Since diam(Pn ) → 0, when n → ∞, for each x  ωx ∈ Pn x such that ωx ∩ M0 ⊂ U ∩ M0 . Since ∞ n=1 Pn is a countable set, the function U  ∩ M0  x → ωx takes only countably many values.This implies that ∞ U ∩ M0 = x∈U ωx ∩ M0 belongs in the σ -algebra ∞ generated by n=1 Pn restricted to M0 . Therefore, the σ -algebra generated by n=1 Pn restricted to M0 contains all open sets in M0 , and so it contains the σ -algebra B restricted to M0 . Corollary 2.5 Let m be a Borel measure for which f ∗ m  m. If P is a countn−1 − j P → 0, when n → ∞, able mod 0 partition of M such that diam j=0 f n−1 − j  P n is a basis of the Borel measure space. then j=0 f  −j Proof Take Pn = n−1 P and use Lemma 2.4.  j=0 f Corollary 2.6 Let m be a Borel measure for which f ∗ m m. If P is a countable n−1 −j P → 0, when n → ∞, then for mod 0 partition of M for which diam j=0 f  −j P such that all δ > 0 and A ∈ A with m(A) > 0, there are n ≥ 1 and ω ∈ n−1 j=0 f m(ω \ A) < δm(ω).  −j Proof Take Pn = n−1 P, use Corollaries 2.3 and 2.5.  j=0 f

2.1 Partitions

15

2.1.3 Measurable Partitions Let M be a compact metric space and m a measure on the σ -algebra of Borel sets of M. We say that an m mod 0 partition P of M into Borel sets is measurable if there is a sequence (E n )n of Borel sets of M and M0 ⊂ M with full m measure such that, for all ω ∈ P, (2.8) ω ∩ M0 = M0 ∩ E 1∗ ∩ E 2∗ ∩ · · · , where each E n∗ is either E n or its complement M \ E n . Note that the way of writing each ω ∩ M0 in the form of an infinite intersection as in (2.8) is unique, and moreover, each intersection in (2.8) gives a set ω ∩ M0 , for some ω ∈ P, or the empty set. The result below provides an alternative definition of measurable partition, which is sometimes convenient for deducing properties of dynamical systems for which a measurable partition is known a priori. Lemma 2.7 Let M be a compact metric space and m a measure on the Borel sets of M. An m mod 0 partition P of M into Borel sets is measurable if, and only if, there is anincreasing sequence (Pn )n of finite partitions of M into Borel sets such that P = ∞ n=1 Pn (mod 0). Proof Assume first that P is an m mod 0 measurable partition. This means that there are a set M0 ⊂ M with full m measure and a sequence of Borel sets (E n )n in M such that, for all ω ∈ P, ω ∩ M0 = M0 ∩ E 1∗ ∩ E 2∗ ∩ · · · , where each E n∗ is either E n or its complement M \ E n . Set for each n ≥ 1 En = {E n , M \ E n } and Pn =

n 

Ei .

i=1

into account the Clearly, (Pn ) n is an increasing sequence of finite partitions. Taking ∞ definition of ∞ n=1 Pn , it is easily verified that P coincides with n=1 Pn (mod 0). Assume now that (Pn )n is an increasing sequence of finite partitions of M into Borel sets such that ∞  P= Pn (mod 0). (2.9) n=1

  Consider the elements of n≥1 Pn listed as E 1 , E 2 , . . . By definition of ∞ n=1 Pn and the equality mod 0 in (2.9), there is a set M0 ⊂ M with full m measure such that, for all ω ∈ P,

16

2 Preliminaries

ω ∩ M0 = M0 ∩ E 1 ∩ E 2 ∩ · · · This shows that P is a measurable partition.



Remark 2.8 It is easily verified in the proof above that a version of the result can be obtained without any measure involved: if P is a partition of M, then each Pn is a partition of M as well, and vice versa. It is straightforward to check that every finite partition into Borel sets of a compact metric space is measurable. In the next example, we show that measurable partitions with an infinite (possibly uncountable) number of elements are still admitted. Example 2.9 Let K be a compact metric space with a countable base of open sets. We claim that the partition of K into single points is measurable. In fact, let (E n )n be a countable base for the topology of K . Given any x ∈ K and n ≥ 1, set  E n∗

=

En , if x ∈ E n ; / En . K \ E n , if x ∈

Since K is a Hausdorff space, we have {x} =

E n∗ .

(2.10)

n≥1

This shows that the partition of K into single points is measurable. Note that this happens regardless the measure we consider on the Borel sets of K . The following is a simple example of a partition into measurable sets which is not measurable, so we can see that not every partition into measurable sets is measurable. This example can easily be generalised to any partition by orbits of a dynamical system on a measurable space with an ergodic measure that does not give full weight to single orbits. Example 2.10 Let R : S 1 → S 1 be an irrational rotation of the circle S 1 and m the Lebesgue measure on S 1 . Notice that m is an ergodic R-invariant measure. Consider the partition P of S 1 into orbits of R. We claim that P is not a measurable partition. Assume, by contradiction, that there is a set S0 ⊂ S 1 with full m measure such that, for all ω ∈ P, ω ∩ S0 = S0 ∩ E 1∗ ∩ E 2∗ ∩ · · · , where each E n∗ is either E n or S 1 \ E n . This means that the R orbit of m almost every orbit in S 1 is contained in E n or in S 1 \ E n . It follows that, up to zero measure, every E n is invariant under R. By the ergodicity of m, we have either m(E n ) = 0 or m(E n ) = 1, for all n ≥ 1. Set for each n ≥ 1  E n∗

=

En , if m(E n ) = 1; 1 S \ E n , if m(E n ) = 0.

2.1 Partitions

17

Consider the set

E = S0 ∩ E 1∗ ∩ E 2∗ ∩ · · ·

Since m(E) = 1 and any intersection of this type is equal to a set ω ∩ S0 , for some ω ∈ P, or the empty set, we conclude that there is an R orbit with full m measure. Clearly, this cannot happen.

2.2 Jacobians Let f : M → M be a measurable map on the measurable space M with a finite measure m. Assume that there is a countable mod 0 partition P of M into invertibility domains of f , meaning that f is a bijection from each ω ∈ P to f (ω) with a measurable inverse. We say that a measurable function J f : M → [0, ∞) is a Jacobian of f (with respect to m) if  m( f (A)) =

J f dm, A

for every measurable set A ⊂ ω. The Jacobian obviously depends on the reference measure, but we will not make this explicit in the notation for the sake of simplicity. Although it is not relevant to what comes next, it can be shown that the Jacobian does not depend on the choice of invertibility domains and it is essentially unique; see [4, Proposition 9.7.2]. Using standard arguments in integration theory, it can be verified that, for all nonnegative function ψ and measurable set A ⊂ ω ∈ P, we have 

 f (A)

ψdm =

(ψ ◦ f )J f dm.

(2.11)

A

Note that if f has a strictly positive Jacobian, then both f and f −1 preserve sets of m n−1 measure zero. This implies that j=0 f − j P is a mod 0 partition of M, for all n ≥ 1.  −j P is a partition into invertibility domains of f n . Moreover, each n−1 j=0 f Lemma 2.11 Let f : M → M be a measurable map with a mod 0 partition P into invertibility domains and J f > 0. For all n ≥ 1, 1. f n has a Jacobian and Jfn =

n−1 

Jf ◦ f j;

j=0

2. if λ m and ϕ = dλ/dm, then for all ω ∈

n−1 j=0

f −jP

d f ∗n (λ|ω) ϕ ◦ f n |ω = |ω . dm Jfn

18

2 Preliminaries

Proof We prove the first item by induction on n ≥ 1. The formula is trivial for n = 1. Assume now that the formula holds for some n ≥ 1. This means that, for  −j f P, we have every measurable set A ⊂ ω ∈ n−1 j=0 m( f n (A)) =

 n−1 

J f ◦ f j dm.

(2.12)

A j=0

  −j Consider now A ⊂ ω ∈ nj=0 f − j P. Thus, f (A) ⊂ ω , for some ω ∈ n−1 P. j=0 f n−1 j Applying (2.12) to the set f (A) and (2.11) to ψ = j=0 J f ◦ f , we get  m( f n+1 (A)) =

n−1  f (A) j=0

J f ◦ f j dm =

 n−1 

 (J f ◦ f

j+1

)J f dm =

A j=0

n  f (A) j=0

J f ◦ f j dm.

This shows that the formula also holds for n + 1, and so the inductive step is done. Now, we provethe second item. Let g : f n (ω) → ω be the inverse branch of f n −j restricted to ω ∈ n−1 P. Take an arbitrary measurable set A ⊂ f n (ω). Applyj=0 f n ing (2.11) to f on the set g(A) and ψ = ϕ ◦ g/J f n ◦ g, we obtain 

 

 ϕ ϕ◦g ϕ dm = ◦ gdm = ◦ g ◦ f n J f n dm = ϕdm = λ(B). A Jfn ◦ g f n (B) J f n B Jfn B

Since λ(B) = λ(g(A)) = f ∗n (λ|ω)(A), we have d f ∗n (λ|ω) ϕ◦g = . dm Jfn ◦ g Composing both sides with f n , we get the conclusion.



2.3 Basins In this section, we consider maps f : M → M, where M is a metric space. Recall that the basin of an invariant probability measure μ on the Borel sets of M is the set of points x such that (1.1) holds for every continuous ϕ : M → R. Therefore, distinct Borel probability measures on a compact metric space, must have disjoint basins, since these measures are uniquely determined by their integrals over all continuous functions. It is easily verified that if Bμ is the basin of a probability measure μ, then f n (Bμ ) ⊂ Bμ , for all n ∈ Z.

(2.13)

2.3 Basins

19

Proposition 2.12 Let M be a compact metric space and f : M → M a Borel measurable map. If μ is an ergodic f -invariant Borel probability measure, then the basin of μ covers μ almost all of M. Proof The proof is based on the fact that, if M is compact, then the space C 0 (M) with the sup norm  0 has a dense sequence (ϕk )k . By the ergodicity of μ and Birkhoff Ergodic Theorem, for each k ≥ 1 there exists a measurable set Bk ⊂ M with μ(Bk ) = 1 such that for each x ∈ Bk  n−1 1 j lim ϕk ( f (x)) = ϕk dμ. n→∞ n j=0  Taking B = k≥1 Bk , we clearly have μ(B) = 1. Now, given ϕ ∈ C 0 (M) and ε > 0 arbitrary, take k ∈ N such that ϕ − ϕk 0 < ε/2. For each x ∈ B, we have lim sup n→∞

 n−1 n−1 1 1 ε ε ϕ( f j (x)) ≤ lim ϕk ( f j (x)) + = ϕk dμ + n→∞ n j=0 n j=0 2 2

and lim inf n→∞

 n−1 n−1 1 1 ε ε ϕ( f j (x)) ≥ lim ϕk ( f j (x)) − = ϕk dμ − . n→∞ n n j=0 2 2 j=0

This yields lim sup n→∞

n−1 n−1 1 1 ϕ( f j (x)) ≤ lim inf ϕ( f j (x)) + ε. n→∞ n n j=0 j=0

Since ϕ and ε > 0 are arbitrary, we easily conclude that every x ∈ B belongs in the basin of μ.  In the rest of this section, we deduce some consequences of Proposition 2.12. In some of them, we need that the map f : M → M does not send Borel sets of zero measure to Borel sets with positive measure. We write f ∗ m m to mean that m(A) = 0 =⇒ m( f (A)) = 0.

(2.14)

With this, we implicitly assume that the image of a Borel set with zero measure is still a Borel set. Unlike the push-forward f ∗ m, we do not claim that f ∗ m is a measure. Now, we consider some topological notions. We say that • f is transitive if, for every nonempty open sets U, V ⊂ M, there is n ≥ 1 such that f −n (U ) ∩ V = ∅.

20

2 Preliminaries

• f is topologically mixing if, for every nonempty open sets U, V ⊂ M, there is N ≥ 1 such that f −n (U ) ∩ V = ∅, for all n ≥ N . • f is locally eventually onto if, for every nonempty open set U ⊂ M, there is n ≥ 1 such that f n (U ) = M. Clearly locally eventually onto =⇒ topologically mixing =⇒ transitive.

(2.15)

Unlike the transitivity, the fact that f is topologically mixing implies that all powers of f are topologically mixing. It is also clear that if f has a dense orbit in M, then f is transitive. The next result shows that the converse is true for continuous maps on compact metric spaces. Though not used in this section, this result will be useful in the future. Lemma 2.13 Let f : M → M be a continuous map on a compact metric space M. If f is transitive, then f has a dense orbit. Proof Since M is a compact metric space, there exists a countable basis {Un }n for the topology of M. Every point in the set X=



f −k (Un )

n≥1 k≥0

 has a dense orbit in M. Moreover, the continuity of f implies that k≥0 f −k (Un ) is an open set in M, for all n ≥ 1. On the other hand, the fact that f is transitive implies that each k≥0 f −k (Un ) is dense in M, for all n ≥ 1. Since M is a complete metric space, then X is dense in M, by Baire Category Theorem, and so a nonempty set. From the proof of Lemma 2.13, we easily see that the compactness of M can be replaced by its completeness together with the existence of a countable basis for the topology. Corollary 2.14 Let f : M → M be a locally eventually onto map on a compact metric space M with a Borel measure m such that f ∗ m m. If μ is an ergodic f -invariant probability measure which is equivalent to m on some nonempty open set, then the basin of μ covers m almost all of M. Proof Since μ is an ergodic f -invariant probability measure, it follows from Proposition 2.12 that its basin Bμ covers μ almost all of M. Assuming μ equivalent to m on a nonempty open set U , then m almost all of U is contained in Bμ . Since f is locally eventually onto, there is some n ≥ 1 such that f n (U ) = M. Hence, M \ f n (Bμ ) = f n (U ) \ f n (Bμ ) ⊂ f n (U \ Bμ ).

2.3 Basins

21

Using that f ∗ m m and recalling (2.13), we easily deduce that m almost all of M belongs in the basin of μ.  The support of a measure m on the Borel sets of a metric space M is the set of points x ∈ M such that m(U ) > 0 for every open set U ⊂ M such that x ∈ U . Note that the support of m is equal to M, if and only if, m is strictly positive on all nonempty open sets. Corollary 2.15 Let f : M → M be a continuous transitive map on a compact metric space M with a Borel measure m whose support coincides with M and f ∗ m m. If μ1 , μ2 are ergodic f -invariant probability measures for which there are nonempty open sets U1 , U2 such that μ1 |U1 is equivalent to m|U1 and μ2 |U2 is equivalent to m|U2 , then μ1 = μ2 . Proof Since μ1 , μ2 are ergodic f -invariant probability measures, then the basin B1 of μ1 covers μ1 almost all of M and the basin B2 of μ2 covers μ2 almost all of M, by Proposition 2.12. Assume that U1 , U2 are nonempty open sets such that μ1 |U1 is equivalent to m|U1 and μ2 |U2 is equivalent to m|U2 . This in particular implies that m almost every point in U1 belongs in B1 . Since f is transitive, there is some n ≥ 1 such that f −n (U1 ) ∩ U2 = ∅. In addition, the continuity of f implies that f −n (U1 ) ∩ U2 is an open set. It follows from (2.13) applied to B1 and the fact that f ∗ m m that m almost every point in f −n (U1 ) belongs in B1 . On the other hand, m almost every point in U2 belongs in B2 . Since f −n (U1 ) ∩ U2 is an open set and the support of m is equal to M, there must be some point in B1 ∩ B2 . This implies that μ1 = μ2 . In the particular case of M a Riemannian manifold, we have some interesting consequences of Proposition 2.12. In this case, we take the reference measure m as the Lebesgue measure in the Borel sets of M. Recall that a Borel probability measure μ on M is a physical measure for f : M → M if its basin has positive Lebesgue measure. Corollary 2.16 Let f : M → M be a map on a compact Riemannian manifold M. If μ is an ergodic f -invariant probability measure such that μ m, then μ is a physical measure for f . Proof By Proposition 2.12, the basin of μ has full μ measure. Since μ m, the basin of μ cannot have zero m measure, and so it is a physical measure. Corollary 2.17 Let f : M → M be a locally eventually onto map on a compact Riemannian manifold M for which f ∗ m m. If μ is an ergodic f -invariant probability measure which is equivalent to m on some nonempty open set, then μ is the unique physical measure for f and its basin covers m almost all of M. Proof Use Corollaries 2.14 and 2.16.



22

2 Preliminaries

References 1. R. Mañé, Ergodic Theory and Differentiable Dynamics, volume 8 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] (Springer-Verlag, Berlin, 1987). Translated from the Portuguese by Silvio Levy 2. V.A. Rohlin, On the fundamental ideas of measure theory. Amer. Math. Soc. Transl. 1952(71), 55 (1952) 3. V.A. Rohlin, Lectures on the entropy theory of transformations with invariant measure. Uspehi Mat. Nauk 22(5 (137)), 3–56 (1967) 4. M. Viana, K. Oliveira, Foundations of Ergodic Theory, vol. 151, Cambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge, 2016) 5. P. Walters, An Introduction to Ergodic Theory, vol. 79, Graduate Texts in Mathematics (SpringerVerlag, New York, 1982)

Chapter 3

Expanding Structures

In this chapter, we introduce Gibbs-Markov maps and describe Young’s construction of a tower extension associated with an induced Gibbs-Markov map in [19]. In particular, we derive the existence of ergodic invariant probability measures which are absolutely continuous with respect to a reference measure, and conclusions on the polynomial or (stretched) exponential decay of correlations with respect to these measures. In the stretched exponential case, we use ideas from [6], which optimise the estimates in [19]. For results on decay of correlations using tower extensions with rates beyond polynomial or (stretched) exponential, see [7, 10, 11]. For possible generalizations and other statistical properties of Gibbs-Markov maps, see [1–3]. At the end of this chapter, we illustrate the utility of induced Gibbs-Markov maps, with an application to a class of one-dimensional maps with neutral fixed points associated with intermittent phenomena. Several results that we present here will be applied to a class of multidimensional nonuniformly expanding maps that we introduce in Chap. 6. In Chap. 4, we also apply some results on tower extensions associated with Gibbs-Markov maps to another class of tower extensions related to induced maps for systems with contracting directions.

3.1 Gibbs-Markov Maps Throughout this section, we consider a set 0 with a reference measure m on a σ algebra A of 0 such that 0 < m(0 ) < ∞ and a map F : 0 → 0 . Assume that P is a countable mod 0 partition of 0 into pairwise disjoint subsets. We say that F is a weak Gibbs-Markov map (with respect to the partition P) if conditions (G1 )–(G5 ) below are satisfied. (G1 ) Markov: F maps each ω ∈ Pbijectively to a mod 0 union of elements of P. n−1 −i F P)n is a basis of 0 . (G2 ) Separability: the sequence ( i=0

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. F. Alves, Nonuniformly Hyperbolic Attractors, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-62814-7_3

23

24

3 Expanding Structures

In the last property, we require that the measure space 0 is separable, having a basis obviously related to the dynamics; see Sect. 2.1.2. It follows from (G2 ) that the separation time   s(x, y) = min n ≥ 0 : F n (x) and F n (y) lie in distinct elements of P

(3.1)

is well defined and finite for distinct points x, y in a full m measure subset of 0 . For definiteness, set the separation time equal to zero for all other points. In the next two conditions, we require in particular the equivalence of the measures F∗ m and m and some bounded distortion property for the Jacobian of F over the elements of P. (G3 ) Nonsingular: F has a strictly positive Jacobian JF . (G4 ) Gibbs: there are C > 0 and 0 < β < 1 such that, for all ω ∈ P and x, y ∈ ω, log

JF (x) ≤ Cβ s(F(x),F(y)) . JF (y)

(G5 ) Long branches: there is δ0 > 0 such that m(F(ω)) ≥ δ0 , for all ω ∈ P. The weak Gibbs-Markov map F is said to be Gibbs-Markov map if it satisfies (G5 ) below. Note that (G1 ) and (G5 ) are both implied by (G5 ). (G5 ) Full branches: F maps each ω ∈ P bijectively to 0 (mod 0). Examples of (weak) Gibbs-Markov maps are widespread in the specialised literature, ranging from very simple to sophisticated ones. In Sect. 3.5, Chaps. 6 and 7, we will build some non-trivial examples of (weak) Gibbs-Markov maps. We start by presenting a simple Gibbs-Markov map (probably the simplest beyond the uninteresting ones) that will be used at the end of this section to produce other weak Gibbs-Markov maps with interesting properties (Fig. 3.1).

Fig. 3.1 Doubling map

3.1 Gibbs-Markov Maps

25

Example 3.1 (Doubling map) Let 0 = [0, 1] and F : 0 → 0 be the map given by F(x) = 2x (mod 1). Consider m the Lebesgue (length) measure on 0 and the mod 0 partition P = {(0, 1/2), (1/2, 1)} of 0 . Clearly, F has the full branch property (G5 ) with respect to P. Since F is smooth on each interval of P with F  = 2, the nonsingularity property (G4 ) holds for F and JF = 2. The Gibbs property (G3 ) also holds for any choice of C > 0 and 0 < β < 1, since JF is constant. We are left to prove the separability property (G2 ). Note that, if x, y are points in 0 such that F n (x) n − F n (y)|/2n , and F n (y) always lie in the same elements of P, then |x − y| ≤ |F  (x) n−1 − j  P → 0, for all n ≥ 1. Since 0 has finite diameter, it follows that diam j=0 F when n → ∞. Using Corollary 2.5, we obtain the separability property (G2 ). Thus, F is a Gibbs-Markov map with respect to the partition P. The previous example can be easily generalised to a class of piecewise smooth maps with a finite number of smoothness domains, as long as we assume these maps on each smoothness domain with full branches, derivative bounded away from one and Hölder continuous derivatives. The ideas used in the study of Example 3.1, are also important ingredients in the proofs of Lemma 3.2 and Lemma 3.3, below. These two results provide useful ways of verifying some of the conditions in the definition of a Gibbs-Markov map for maps defined on a metric space or a Riemannian manifold. Lemma 3.2 Let F : 0 → 0 be defined on the bounded metric space 0 with a metric d and a Borel reference measure m such that F∗ m m. If P is an m mod 0 partition of 0 for which there is 0 < α < 1 such that d(x, y) ≤ αd(F(x), F(y)), for all ω ∈ P and x, y ∈ ω, then n−1 −i F P)n is a basis of 0 ; 1. ( i=0 2. d(F n (x), F n (y)) ≤ diam(0 )α s(x,y)−n , for all x, y ∈ 0 and n ≥ 0. Proof The idea is to use Corollary 2.5. For that, we need to see that the diameters of −j the partitions Pn = n−1 P converge to zero when n goes to infinity. Consider j=0 F x, y ∈ 0 whose forward iterates until time n ≥ 1 always lie in the same elements of the partition P. It follows that d(x, y) ≤ α n d(F n (x), F n (y)). This implies that diam(ωn ) ≤ α n diam(0 ), for all n ≥ 1 and ωn ∈ Pn . Therefore, diam(Pn ) → 0, when n → ∞. Then, using Corollary 2.5, we obtain the first conclusion. Let us now prove the second conclusion. If n ≥ s(x, y), then there is nothing to be proved. Otherwise, consider x, y ∈ 0 with s(x, y) ≥ 1 and take 0 ≤ n < s(x, y). Note that there are at least s(x, y) − n iterates of F n (x), F n (y) lying in common elements of P. Using the assumption s(x, y) − n times, we get

26

3 Expanding Structures

d(F n (x), F n (y)) ≤ α s(x,y)−1 d(F n+1 (x), F n+1 (y)) .. . ≤ α s(x,y)−n d(F s(x,y) (x), F s(x,y) (y)) ≤ α s(x,y)−n diam(0 ). This completes the proof of the lemma.



The second conclusion in Lemma 3.2, motivates the definition of expanding map that we adopt in Sects. 3.4 and 4.1. In the case of a smooth Riemannian manifold, we can also provide practical ways of checking the separability, nonsingularity and Gibbs properties. Lemma 3.3 Let 0 be a manifold (possibly with a smooth boundary) and m the Lebesgue measure on the Borel sets of 0 . Let P be a countable m mod 0 partition of 0 into open sets whose closures have smooth boundary, and F : 0 → 0 be such that, for all ω ∈ P, the restriction of F to ω has an extension to the boundary of ω which is a C 1 diffeomorphism onto its image. Then, 1. F is nonsingular and JF (x) = | det D F(x)|, for m almost all x ∈ 0 ; 2. if there is 0 < α < 1 such that (D F|ω )−1 (x) ≤ α, for all x ∈ F(ω), then F satisfies the separability property; 3. if there are C, ζ > 0 such that, for all ω ∈ P and x, y ∈ ω, log

det D F(x) ≤ Cd(F(x), F(y))ζ , det D F(y)

then F satisfies the Gibbs property. Proof The first item is an application of the change of variables formula for differentiable maps. To obtain the other two items, we will apply Lemma 3.2. Given any ω ∈ P and x, y ∈ ω, consider a C 1 curve γ : [0, 1] → F(ω) such that γ (0) = F(x), γ (1) = F(y) and length(γ ) = d(x, y). Note that such a C 1 curve exists, by [4]. Let Fω : ω → F(ω) be given by the restriction of F to ω. We have d(x, y) ≤ length(Fω−1 ◦ γ )  1 = |(Fω−1 ◦ γ ) (t))|dt 0  1 ≤

D Fω−1 (γ (t)) · |γ  (t))|dt 0  1 |γ  (t))|dt ≤α 0

= α length(γ ) = αd(F(x), F(y)).

(3.2)

3.1 Gibbs-Markov Maps

27

 −n Applying Lemma 3.2, we have that P is a generating partition and ∞ P is the n=0 F m mod 0 partition into single points. The final conclusion is a consequence of (3.2) and Lemma 3.2.  Our main objective in the rest of this section is to prove Theorem 3.13 in Sect. 3.1.3, where we deduce the existence of invariant probability measures for weak GibbsMarkov maps which are absolutely continuous with respect to the reference measure. In the stronger case of a Gibbs-Markov map, uniqueness and exactness (ergodic, in particular) of these measures will also be deduced. The strategy adopted to prove the existence of such measures is to consider averages of the push-forwards of the reference measure and to take an accumulation point of the respective sequence of densities. For that, we need some suitable space that guarantees the existence of accumulation points. This space will be introduced in Sect. 3.1.2. In Sect. 3.1.1, we start with some useful bounded distortion results that will help us to prove that the densities of the push-forwards of the reference measure lie in the space of Sect. 3.1.2.

3.1.1 Bounded Distortion In this subsection, we prove a uniform bounded distortion property in convenient domains for the Jacobians of the iterates of a weak Gibbs-Markov map F : 0 → 0 and deduce some consequences of this. Note that, by Lemma 2.11, the iterates of a weak Gibbs-Markov map still have strictly positive Jacobians. Set for each n ≥ 1 Pn =

n−1 

F − j P.

(3.3)

j=0

From the definition of separation time, we easily deduce that s(F i (x), F i (y)) = s(F n (x), F n (y)) + n − i.

(3.4)

for all x, y ∈ ω ∈ Pn and 0 ≤ i < n. Lemma 3.4 If F : 0 → 0 is a weak Gibbs-Markov map, then there is some C > 0 such that, for all n ≥ 1 and x, y ∈ ω ∈ Pn , log

JF n (x) n n ≤ Cβ s(F (x),F (y)) . JF n (y)

Proof It follows from Lemma 2.11, (G4 ) and (3.4) that, for all x, y ∈ ω ∈ Pn ,

28

3 Expanding Structures

JF n (x) JF (F i (x)) log = JF n (y) JF (F i (y)) i=0 n−1

log

≤

n−1

Cβ s(F (x),F (y)) i

i

i=0

=

n−1

Cβ n−i β s(F

n

(x),F n (y))

i=0

≤

C n n β s(F (x),F (y)) . 1−β

Take the constant equal to C/(1 − β).



Corollary 3.5 If F : 0 → 0 is a weak Gibbs-Markov map, then there is some C > 0 such that, for all n ≥ 1, ω ∈ Pn and x ∈ ω, 1 1 m(ω) ≤ ≤ Cm(ω). C JF n (x) Proof From Lemma 3.4, we easily get C0 > 0 such that, for all x, y ∈ ω, 1 JF n (y) ≤ C0 . ≤ C0 JF n (x)

(3.5)

We have  m(F n (ω)) =

ω

 JF n (y)dm(y) = JF n (x)

ω

JF n (y) dm(y). JF n (x)

(3.6)

Note that, as we assume the Markov property (G1 ), the long branches property (G5 ) still holds for each F n with same constant δ0 . It follows from (3.5) and (3.6) that δ0 ≤ JF n (x)C0 m(ω) and m(0 ) ≥ JF n (x) Take C = max{C0 /δ0 , C0 m(0 )}.

1 m(ω). C0 

Corollary 3.6 If F : 0 → 0 is a weak Gibbs-Markov map, then there is some C > 0 such that, for all n ≥ 1, ω ∈ Pn and measurable sets A1 , A2 ⊂ ω, m(A1 ) m(F n (A1 )) ≤C . n m(F (A2 )) m(A2 )

3.1 Gibbs-Markov Maps

29

Proof Let C0 > 0 be the constant given by Corollary 3.5. We have

n m(A1 ) C0 m(A1 )/m(ω) m(F n (A1 )) A J F dm = 1 ≤ C02 . ≤ n n m(F (A2 )) m(A2 )/(C0 m(ω)) m(A2 ) A2 J F dm Take C = C02 .



Corollary 3.7 If F : 0 → 0 is a weak Gibbs-Markov map, then there is some C > 0 such that, for all n ≥ 0, ω ∈ Pn and x, y ∈ F n (ω), log

ρn,ω (x) ≤ Cβ s(x,y) , ρn,ω (y)

where ρn,ω = d F∗n (m|ω)/dm. Proof Given ω ∈ Pn and x, y ∈ F n (ω), consider x  , y  ∈ ω such that F n (x  ) = x and F n (y  ) = y. From Lemmas 3.4 and 2.11 with λ = m, we obtain log

JF n (y  ) ρn,ω (x) n  n  = log ≤ Cβ s(F (x ),F (y )) = Cβ s(x,y) , ρn,ω (y) JF n (x  )

and so we have proved the result.



3.1.2 A Space for the Densities Here, we present a relatively compact subspace of L 1 (m) in which we will find the densities of absolutely continuous invariant probability measures for weak GibbsMarkov maps. Given 0 < β < 1, consider the linear space   Fβ (0 ) = ϕ : 0 → R | ∃C > 0 : |ϕ(x) − ϕ(y)| ≤ Cβ s(x,y) , ∀x, y ∈ 0 . (3.7) Since the functions in Fβ (0 ) are bounded and we assume m a finite measure, then Fβ (0 ) ⊂ L ∞ (m) ⊂ L 1 (m). Given ϕ ∈ Fβ (0 ), set |ϕ|β = sup x= y

|ϕ(x) − ϕ(y)| and ϕ β = |ϕ|β + ϕ ∞ , β s(x,y)

where ∞ stands for the usual norm on L ∞ (m). It is easily verified that | |β defines a seminorm on Fβ (0 ). Therefore, β defines a norm on Fβ (0 ). The next result

30

3 Expanding Structures

gives in particular that Fβ (0 ) endowed with the norm β is relatively compact in the space L 1 (m). Proposition 3.8 If (ϕn )n is a sequence in Fβ (0 ) such that ϕn β ≤ C, for all n ∈ N, then (ϕn )n has a subsequence converging pointwise and in L 1 (m) to a function ϕ with ϕ β ≤ C. Proof Consider a sequence (ϕn )n in Fβ (0 ) such that ϕn β ≤ C, for all n ≥ 1. We construct a sequence (xi )i in 0 , choosing one point in each element of the  countable family n≥1 Pn . Our assumption gives that for all i ≥ 1 the sequence (ϕn (xi ))n is bounded in R, and therefore it has a converging subsequence. Using a diagonal argument, we may find a sequence of natural numbers n 1 < n 2 < · · · such that the sequence (ϕn k (xi ))k converges in R, for all i ≥ 1. We claim that (ϕn k (x))k is a Cauchy sequence in R, for all x ∈ 0 . Take an arbitrary ε > 0 and choose N ≥ 1 such that Cβ N < ε/3. Consider the element ω ∈ P N that contains x. Picking the respective xi ∈ ω, we have s(x, xi ) ≥ N . Hence |ϕn k (x) − ϕn k (xi )| ≤ Cβ N
0 such that d F∗n m/dm β ≤ C, for all n ≥ 0. Moreover, if F is a Gibbs-Markov map, then d F∗n m/dm ≥ 1/C, for all n ≥ 0. Proof Assume first that F is a weak Gibbs-Markov map. Set for each n ∈ N ρn =

d F∗n m d F∗n (m|ω) and ρn,ω = , dm dm

3.1 Gibbs-Markov Maps

31

for each ω ∈ Pn . We have ρn =



ρn,ω 1 F n (ω) ,

(3.8)

ω∈Pn

By Lemma 2.11 applied to λ = m and Corollary 3.5, there is C0 > 0 such that, for all n ≥ 1 and ω ∈ Pn , 1 m(ω) ≤ ρn,ω | F n (ω) ≤ C0 m(ω). C0

(3.9)

From (3.8) and (3.9), we get for all n ≥ 1 ρn ≤ C0 m(0 ).

(3.10)

Given any n ≥ 1 and ω ∈ Pn , we distinguish two possible cases. If x, y ∈ 0 are such that, for some ω ∈ Pn , only one of the points belongs in F n (ω), then s(x, y) = 0. Thus, using (3.10), we get |ρn (x) − ρn (y)| ≤ 2C0 m(0 ) = 2C0 m(0 )β s(x,y) . Otherwise, we may write |ρn (x) − ρn (y)| ≤



|ρn,ω (x) − ρn,ω (y)|1 F n (ω) .

(3.11)

ω∈Pn

By Corollary 3.7, there is some uniform constant C1 > 0 such that log ρn,ω (x) ≤ C1 β s(x,y) . ρn,ω (y) Since this last expression is uniformly bounded, we also have some uniform constant C2 > 0 for which ρn,ω (x) ρn,ω (x) ρ (y) − 1 ≤ C2 log ρ (y) . n,ω

n,ω

Using (3.9) and the two inequalities above, we get ρn,ω (x) |ρn,ω (x) − ρn,ω (y)| ≤ C0 m(ω) − 1 ρn,ω (y) ρn,ω (x) ≤ C0 C2 m(ω) log ρn,ω (y) ≤ C0 C1 C2 m(ω)β s(x,y) .

32

3 Expanding Structures

It follows from (3.11) that |ρn (x) − ρn (y)| ≤ C0 C1 C2 β s(x,y) . The conclusions in both possible cases, together with (3.10), show that there is some uniform upper bound for ρn β . Finally, assuming that F is a Gibbs-Markov map, it easily follows from (3.8) and (3.9), that ρn is bounded from below by some uniform positive constant.  Remark 3.10 Some additional useful information can be extracted from the previous proof. In fact, given any set A ⊂ 0 which is a union of elements of P, set   Pn (A) = ω ∈ Pn : F n (ω) = A . It follows from (3.8) and (3.9) that there is some uniform constant C0 > 0 for which ρn | A =

ω∈Pn (A)

ρn,ω ≥

1 C0



m(ω).

ω∈Pn (A)

Therefore, if there is c0 > 0 such that ω∈Pn (A) m(ω) ≥ c0 , for all n ≥ 1, then there is c > 0 such that ρn | A ≥ c, for all n ≥ 1.

3.1.3 Equilibrium Measures Here, we prove the existence of absolutely continuous invariant probability measures for weak Gibbs-Markov maps. Before we address the main result of this section, we introduce some useful general notions and results. We say that an F-invariant measure ν defined on the σ -algebra A is exact if A∈



F −n (A) =⇒ ν(A) = 0 or ν(0 \ A) = 0.

n≥0

Clearly, every exact measure is an ergodic: for all A ∈ A, F −1 (A) = A =⇒ ν(A) = 0 or ν(0 \ A) = 0. Actually, it can be shown that every exact measure is mixing: for all A, B ∈ A, μ( f −n (A) ∩ B) → μ(A)μ(B), as n → ∞; see for example [12, Proposition 12.2]. The next result is a straightforward consequence of Birkhoff Ergodic Theorem.

3.1 Gibbs-Markov Maps

33

Lemma 3.11 If ν0 and ν1 are F-invariant probability measures such that ν0 is ergodic and ν1 ν0 , then ν1 = ν0 . Proof First of all, observe that, since ν1 ν0 and ν0 is ergodic, then ν1 is necessarily ergodic. Therefore, given any measurable set A, Birkhoff Ergodic Theorem applied to the characteristic function 1 A gives for i = 0, 1 n−1 1 1 A (F j (x)) = νi (A), for νi almost all x. n→∞ n j=0

lim

(3.12)

Using that ν1 ν0 , we obtain a point x for which (3.12) holds simultaneously for  the measures ν0 and ν1 . This implies that ν0 (A) = ν1 (A). A slight elaboration of the previous proof leads to Lemma 3.12, below. For reasons that will become clear in the next subsection, we consider the more general situation of a map F : M → M defined on a measurable set M containing 0 , still with a reference measure m. Lemma 3.12 If F : M → M has an ergodic F-invariant probability measure ν m and 0 ⊂ M is such that ν(0 ) > 0 and ν|0 is equivalent to m|0 , then ν is the unique ergodic F-invariant probability measure such that ν m and ν(0 ) > 0. Proof Let ν  m be an F-invariant probability measure on M such that ν  (0 ) > 0. Given any measurable set A ⊂ M, Birkhoff Ergodic Theorem provides ϕ ∈ L 1 (ν  ) such that n−1 1 lim 1 A (F j (x)) = ϕ(x), for ν  almost all x ∈ 0 , n→∞ n j=0



and



ϕdν =



1 A dν  = ν  (A).

(3.13)

(3.14)

Still by Birkhoff Ergodic Theorem and the ergodicity of ν, we also have n−1 1 1 A (F j (x)) = ν(A), for ν almost all x ∈ 0 . n→∞ n j=0

lim

(3.15)

Since ν  m and we assume ν|0 equivalent to m|0 , it follows that ν  |0 ν|0 . From (3.13) and (3.15) we obtain ϕ(x) = ν(A), for ν  almost all x ∈ 0 .

(3.16)

Since ν  (0 ) > 0, it follows that if ν  is ergodic, then ϕ(x) = ν  (A), for ν  almost all x ∈ 0 , and therefore, ν  (A) = ν(A). 

34

3 Expanding Structures

Finally, we present the theorem on the existence of invariant probability measures for weak Gibbs-Markov maps. In the Gibbs-Markov case, we also obtain uniqueness. Theorem 3.13 If F : 0 → 0 is weak Gibbs-Markov, then F has some invariant probability measure ν m with dν/dm ∈ Fβ (0 ). Moreover, if F is Gibbs-Markov, then ν is unique, exact and dν/dm is bounded from below by some positive constant. j

Proof Assume first that F is weak Gibbs-Markov. Set ρ j = d F∗ m/dm, for each j ≥ 0. By Lemma 3.9, there exists C > 0 such that, for all j ≥ 0 ρ j ≤ C and ρ j β ≤ C.

(3.17)

By Proposition 3.8, there exist ρ ∈ L 1 (m) and n k → ∞, when k → ∞, such that n k −1 1 ρ = lim ρj, k→∞ n k j=0

(3.18)

with pointwise convergence and in the L 1 (m) norm. Let μ be the finite measure such that dμ/dm = ρ. Since 

 ρ j dm =

d F∗j m = m(0 ), for all j ≥ 0,

the convergence in L 1 (m) implies μ(0 ) = m(0 ) > 0. The pointwise convergence of gives ρ ≤ C and |ρ|β ≤ C, and

so ρ ∈ Fβ (0 ). Let us now prove the invariance ∞ for all ϕ ∈ L (m). μ. By (3.17), we have that ( ϕρ j dm) j is a bounded sequence,

On the other hand, Hölder inequality gives that (ϕ, ψ) → ϕψdm is continuous in L ∞ (m) × L 1 (m). It follows that 



 ϕ ◦ Fρdm =

ϕ◦F

 n k −1 1 ρ j dm k→∞ n k j=0 lim

n k −1  n k −1  1 1 = lim ϕ ◦ Fρ j dm = lim ϕ ◦ Fd F∗j m k→∞ n k k→∞ n k j=0 j=0 n k −1  n k −1  1 1 j+1 = lim ϕd F∗ m = lim ϕρ j+1 dm k→∞ n k k→∞ n k j=0 j=0

 = lim

k→∞

   n k −1  1 1 1 ϕρ j dm − ϕρ0 dm + ϕρ j+1 dm n k j=0 nk nk

    n k −1  n k −1 1 1 ρ j dm = ϕρdm. ϕρ j dm = ϕ lim k→∞ n k k→∞ n k j=0 j=0

= lim

3.1 Gibbs-Markov Maps

35

This gives the F-invariance of the finite measure μ. Normalising it, we obtain a probability measure ν m whose density has the desired properties. Assume now that F is a Gibbs-Markov map. Then, Lemma 3.9, gives ρ j ≥ 1/C, for all j ≥ 0. This implies that ρ ≥ 1/C, by pointwise convergence. In particular, the measure ν is equivalent to m. Uniqueness is then a consequence of Lemma 3.12 applied  with M = 0 . Finally, we prove that ν is exact. We need to show that, if A ∈ n≥0 F −n (A) and ν(A) > 0, then ν(A) = 1. Since ν m, it is sufficient to show that m(0 \ A) = 0. Take an arbitrary ε > 0 and let C > 0 be the constant given by Corollary 3.6. Using the separability property (G2 ) and Corollary 2.3, we get n ∈ N and ω0 ∈ Pn for which m(ω0 \ A) ε < 2 . m(ω0 ) C m(0 )

(3.19)

Take B ∈ A such that A = F −n (B). Since F is Gibbs-Markov, we have F n (ω) = 0 , for all ω ∈ Pn . Then, 0 \ B = F n (ω \ A), for all ω ∈ Pn . It follows from Corollary 3.6 and (3.19) that, for all ω ∈ Pn , m(ω \ A) m(0 \ B) ε m(ω0 \ A) ≤C ≤ C2 < . m(ω) m(0 ) m(ω0 ) m(0 ) This implies that m(0 \ A) =



m(ω \ A)
0 is arbitrary, we obtain m(0 \ A) = 0, and therefore, the exactness of the measure ν.  Remark 3.14 In the weak Gibbs-Markov case, we can still provide a condition guaranteeing a positive lower bound for the density of the measure given by Theorem 3.13. Indeed, the proof gives ρ = dν/dm as the pointwise limit in (3.18). Given n ≥ 1 and A ⊂ 0 a union of elements of P, set   Pn (A) = ω ∈ Pn : F n (ω) = A . It follows from Remark 3.10 that, if there is c0 > 0 such that m(ω) ≥ c0 , ω∈Pn (A)

then there is c > 0 such that ρn | A ≥ c for all n ≥ 1. Therefore, ρ| A ≥ c. The measure provided by Theorem 3.13 is not necessarily unique or even ergodic in the weak Gibbs-Markov case, as the next simple elaboration of Example 3.1 illustrates (Fig. 3.2).

36

3 Expanding Structures

Fig. 3.2 Example 3.15

Fig. 3.3 Example 3.16

Example 3.15 Consider 0 = [−1, 1] with the Lebesgue measure m as reference measure. Let F : 0 → 0 be the map given by F(x) = 2x (mod 1). Notice that F|[0,1] is the doubling map in Example 3.1, and F|[−1,0] can easily be identified with the doubling map. It is straightforward to check that F is weak a Gibbs-Markov map with respect to the mod 0 partition P = {(−1, −1/2), (−1/2, 0), (0, 1/2), (1/2, 1)} of 0 . Moreover, ν = m/2 is an F-invariant probability measure, clearly absolutely continuous with respect to m. However, since F −1 ((0, 1)) = (0, 1) and ν(0, 1) = 1/2, the measure ν is not ergodic. Recalling Remark 3.14, we might be tempted to conjecture that the density of an absolutely continuous invariant probability measure of a weak Gibbs-Markov map F : 0 → 0 is bounded from below whenever there is some element ω in the partition associated with F for which F(ω) = 0 . This is false, as the next example illustrates (Fig. 3.3). Example 3.16 Consider 0 = [−1, 1] with the Lebesgue measure m as reference measure. Let F : 0 → 0 be given by  2x (mod 1), F(x) = 2x + 1,

if x ≥ 0; if x < 0;

It is easily verified that F is a weak Gibbs-Markov map with respect to the fmod 0 partition P = {(−1, 0), (0, 1/2), (1/2, 1)} of 0 . Notice that the restriction F|[0,1] is

3.1 Gibbs-Markov Maps

37

Fig. 3.4 Permuting subintervals

the doubling map in Example 3.1. Moreover, −1 is a repelling fixed point for F, and the orbit of any other point in (−1, 0) eventually hits [0, 1] in forward iterations. This implies that any invariant probability measure for F that is absolutely continuous with respect to m must necessarily be supported on [0, 1]. On the other hand, the absence of full branches in a weak Gibbs-Markov map is not sufficient for the non-uniqueness, non-ergodicity or non-existence of a lower bound for the density of absolutely continuous invariant probability measures. An interesting example illustrating this fact will appear in Sect. 3.5.3. However, we could easily create a simple example at this stage, taking two copies of the doubling map, as we did in Example 3.15, this time permuting the two subintervals, as in Fig. 3.4. These examples show that the uniqueness, ergodicity and lower bounds for the density of absolutely continuous invariant probability measures of weak GibbsMarkov maps in general are subtle issues, depending heavily on the transitions between the domains of the partition. Typically, symbolic dynamics and Markov chains are powerful tools for this purpose; see for example [1].

3.2 Induced Maps In this section, we make a brief introduction to induced maps associated with an initial transformation. Some of the results we present here will be used in the next section on tower maps, whose return to base is a good example of an induced map. Let f : M → M be a measurable map of a measurable space M with a reference measure m. Take 0 ⊂ M with m(0 ) > 0. For simplicity, the restriction of m to 0 will still be denoted by m. Consider • a countable mod 0 partition P of 0 into disjoint invertibility domains of f ; • a function R : 0 → N constant in the elements of P such that f R(ω) (ω) ⊂ 0 , for all ω ∈ P. We associate to these objects a new map f R : 0 → 0 , defined on each ω ∈ P by f R |ω = f R(ω) |ω .

38

3 Expanding Structures

We will refer to f R as an induced map and to R as the recurrence time associated with the induced map. We do not require that R(ω) is the first return time to 0 for the points in ω ⊂ 0 . Remark 3.17 The reference measure m is used only to introduce some flexibility in the induction scheme, allowing, for example, m mod 0 partitions in 0 . This reference measure plays no role in the first three items of Theorem 3.18 below; these conclusions remain valid as long as the induced map makes sense. In the next result, we provide sufficient conditions for the existence of invariant measures (not necessarily finite) which are absolutely continuous with respect to the reference measure, for transformations admitting weak Gibbs-Markov induced maps. For now, the expression for the measure ν in Theorem 3.18 may seem somewhat mysterious. In Proposition 3.23, we bring some light to the subject, using this formula applied to a tower map, which can be interpreted as a discrete-time suspension of the base map. In Lemma 3.26, we see that the tower system is an extension of the original system and this is precisely the formula for the push-forward of the measure in the tower. induced map for f : M → M and ν0 an Theorem 3.18 Let f R : 0 → 0 be an j f R -invariant probability measure. If ν = ∞ j=0 f ∗ (ν0 |{R > j}), then 1. 2. 3. 4.

ν is an f -invariant measure with ν|0 ≥ ν0 ; ν is finite if, and only if, R ∈ L 1 (ν0 ); if ν0 is ergodic, then ν is ergodic; if f ∗ m m and ν0 m, then ν m.

Proof First, we prove that ν|0 ≥ ν0 |0 . Given any measurable set A ⊂ 0 , we have ν(A) =

∞

ν0 ( f − j (A) ∩ {R > j}) ≥ ν0 (A ∩ {R > 0}) = ν0 (A ∩ 0 ) = ν0 (A).

j=0

Let us show that ν is f -invariant. For any measurable set A ⊂ M, we have ν( f −1 (A)) =

∞

  ν0 f − j ( f −1 (A)) ∩ {R > j}

j=0

=

∞

∞     ν0 f −( j+1) (A) ∩ {R = j + 1} + ν0 f −( j+1) (A) ∩ {R > j + 1} .

j=0

j=0

Since ∞

   ν0 f −( j+1) (A) ∩ {R = j + 1} = ν0

j=0

= ν0



fR

 j≥1

−1

f − j (A) ∩ {R = j}

 (A) = ν0 (A)



3.2 Induced Maps

39

and ∞

  ν0 f −( j+1) (A) ∩ {R > j + 1} = ν(A) − ν0 (A ∩ {R > 0}) = ν(A) − ν0 (A),

j=0

we obtain ν( f −1 (A)) = ν(A). Now, we prove the second item. It follows from the definition of ν that ν(M) =

∞

ν0 {R > j} =

j=0

∞

 jν0 {R = j} =

Rdν0 .

j=1

This shows that ν is finite if, and only if, R is integrable with respect to ν0 . Suppose now that ν0 is ergodic. Take a measurable set A with f −1 (A) = A. We have ν(A) =

∞



ν0 f

−j



(A) ∩ {R > j} =

j=0

∞

  ν0 A ∩ {R > j}

(3.20)

j=0

and 



ν M\A =

∞

  ν0 f − j (M \ A) ∩ {R > j}

j=0

=

∞

  ν0 (M \ A) ∩ {R > j}

j=0

= ν0 (0 \ A).

(3.21)

Also, 

fR

−1

(A ∩ 0 ) =

 

 j≥1

f − j (A ∩ 0 ) ∩ {R = j}





f − j (A) ∩ f − j (0 ) ∩ {R = j} j≥1    = A ∩ {R = j}

=



j≥1

= A ∩ 0 . Assuming ν(A) > 0, it follows from (3.20) that ν0 (A ∩ 0 ) >  0. Bythe ergodicity of ν0 , we have ν0 (0 \ A) = 0. Using (3.21), we obtain ν M \ A = 0. Finally, assume that f ∗ m m and ν0 m. Given any A measurable set with m(A) = 0, we have m( f − j (A)) = 0, for all j ≥ 0. Since ν0 m, we obtain ν0 ( f − j (A)) = 0, for all j ≥ 0. Hence,

40

3 Expanding Structures

ν(A) =

∞

  ν0 f − j (A) ∩ {R > j} = 0.

j=0

This shows that ν m.



In the next result, we give sufficient conditions for the uniqueness of the measures obtained in Theorem 3.18. Corollary 3.19 Let f R : 0 → 0 be an induced weak Gibbs-Markov map for f : M → M. If f ∗ m m and R ∈ L 1 (m), then f has some invariant probability measure ν m with ν(0 ) > 0. Moreover, if ν is ergodic and ν|0 is equivalent to m|0 , then ν is the unique ergodic f -invariant probability measure such that ν m and ν(0 ) > 0. Proof By Theorem 3.13, the map f R has an invariant probability measure ν0 m. Moreover, dν0 /dm is bounded from above by some positive constant. Assuming R ∈ L 1 (m), we obtain R ∈ L 1 (ν0 ). Then, f has some invariant finite measure ν  m with ν  (0 ) ≥ ν0 (0 ) = 1, by Theorem 3.18. Take ν = ν  /ν  (M). The second conclusion is a consequence of Lemma 3.12.  Remark 3.20 From the proof of Corollary 3.19, we easily see that if the weak Gibbs-Markov map f R has an invariant probability measure ν0 m with dν0 /dm bounded from below by some positive constant, then ν|0 is also equivalent to m|0 . The last item of Corollary 3.21 will be particularly useful for proving the uniqueness of an SRB measure for intermittent maps in Sect. 3.5. In the Gibbs-Markov case, the ergodicity and the lower bound on the density of the invariant probability measure allow us to draw stronger conclusions than those in Corollary 3.19. Corollary 3.21 Let f R : 0 → 0 be a Gibbs-Markov map and ν0 m its unique j f R -invariant probability measure. If f ∗ m m and ν = ∞ j=0 f ∗ (ν0 |{R > j}), then 1. 2. 3. 4.

ν is an f -invariant ergodic measure with ν m and ν|0 ≥ ν0 ; ν is finite if, and only if, R is integrable with respect to m; dν/dm|0 is bounded from below by some positive constant; if ν is finite, then μ = ν/ν(M) is the unique ergodic f -invariant probability measure such that μ m and μ(0 ) > 0.

Proof The first item is a consequence of the first and last conclusions of Theorem 3.18. By Theorem 3.18, ν is finite if, and only if, R is integrable with respect to ν0 . Since Theorem 3.13, gives that dν0 /dm it bounded from above and below by positive constants, it follows that ν is finite if, and only if, R is integrable with respect to m. Since Theorem 3.18, gives ν|0 ≥ ν0 , it also follows that dν/dm restricted to 0 is bounded from below by some positive constant. The last item is a consequence of Corollary 3.19, and the fact that, in the Gibbs-Markov case, ν is always ergodic, by  the first item; recall that ν(M) ≥ ν0 (0 ) = 1.

3.3 Tower Maps

41

3.3 Tower Maps Our main objective for the remainder of this chapter is to draw conclusions about the decay of correlations with respect to the measures obtained in Corollary 3.21 via induced maps. For this, we associate to the induced map a tower, which has two main features: (i) the recurrence time is the first return time to the inducing region, (ii) the tower dynamical system is an extension of the original system. The tower construction may be thought of as a discrete-time suspension of the induced map with roof function the recurrence time. Consider • a measurable map F : 0 → 0 of a space 0 with a reference measure m; • a countable mod 0 partition P of 0 into disjoint domains of 0 ; • a measurable function R : 0 → N which is constant on the elements of P. We define the tower    = (x, ) : x ∈ 0 and 0 ≤  < R(x) , and the tower map T :  →  of F with recurrence time R by  T (x, ) =

(x,  + 1), if  < R(x) − 1; (F(x), 0), if  = R(x) − 1.

The set {(x, 0) ∈ } is naturally identified with 0 and we will make no distinction between these two sets. This gives in particular that the induced map T R : 0 → 0 can be identified with F. Remark 3.22 As with the induced maps, the reference measure m is used only to introduce more flexibility in the tower construction, allowing, for example, m mod 0 partitions in the base space. In particular, this reference measure plays no role in Proposition 3.23 below; its conclusions are generally valid as long as the tower construction makes sense. For each  ≥ 0, define the th level of the tower  = {(x, ) : (x, ) ∈ }. To simplify the notation, we often omit the level coordinate of a point in the tower. We will refer to 0 as the base of the tower, and to each set {(x, ) :  = R(x) − 1} ⊂  as a roof of the tower. The th level of the tower is naturally identified with the subset {R > } of 0 . This allows us to extend the measure m on 0 to a measure on  that we still denote by m. In general, this measure m is not finite on . The integrability of R with respect to the measure m (on 0 ) is a necessary and sufficient condition for the finiteness of m on . In fact,

42

3 Expanding Structures

m() =

≥0

m( ) =



m{R > } =

≥0



 m{R = } =

≥1

Rdm.

(3.22)

0

The countable partition P of 0 naturally induces a mod 0 countable partition on each level  . Collecting all these partitions, we obtain an m mod 0 partition Q of the tower . Given n ≥ 1, set Qn =

n−1 

T − j Q.

(3.23)

j=0

The next result gives, in particular, that any T -invariant probability measure is necessarily lifted from a probability measure invariant under the base map. Proposition 3.23 Let T :  →  be the tower map of F : 0 → 0 with recurrence time R. If ν is a T -invariant probability measure, then ν(0 ) > 0 and ν0 = (ν|0 )/ν(0 ) is an F-invariant probability measure. Moreover, ∞

1 1 . T∗j (ν0 |{R > j}) and ν(0 ) = ∞ ν {R > j} ν j=0 0 j=0 0 {R > j} j=0

ν = ∞

Proof First, we show that ν(0 ) > 0. Actually, since ν() = 1, there exists  ≥ 0 such that ν( ) > 0. Using the T -invariance of ν, we get ν(T − ( )) = ν( ) > 0. Since T − ( ) ⊂ 0 , it follows that ν(0 ) > 0. Now we prove the F-invariance of ν0 . Obviously, it is enough to show the F-invariance of the non-normalised measure ν|0 . Let A ⊂ 0 be a measurable set. Using the T -invariance of ν and the definition of the tower map, we obtain ∞       ν(A) = ν T −1 (A) = ν T − j (A) ∩ {R = j} = ν F −1 (A) , j=1

thus having ν0 is F-invariant. Let us now prove the two formulas in the last assertion. Observe that, except at the roof levels, T is an upward translation. So, using the T invariance of ν, we easily obtain for each j ≥ 0 and each measurable set A ⊂  j ν(A) = ν(T − j (A) ∩ {R > j}).

(3.24)

From (3.24) and the definition of ν0 , we get ν( j ) = ν{R > j} = ν0 {R > j}ν(0 ). Hence ∞ ∞ 1 1 , (3.25) ν0 {R > j} = ν( j ) = ν(0 ) j=0 ν(0 ) j=0 thus obtaining the final conclusion. Using (3.24) and (3.25), we have for any measurable set A ⊂ 

3.3 Tower Maps

43

ν(A) =

∞

ν(A ∩  j )

j=0

=

∞   ν T − j (A) ∩ {R > j} j=0

= ν(0 )

∞

  ν0 T − j (A) ∩ {R > j}

j=0 ∞

  1 ν0 T − j (A) ∩ {R > j} , j=0 ν0 {R > j} j=0

= ∞



which completes the proof.

We are particularly interested in towers associated with Gibbs-Markov maps. In those cases, we always take the partition to define the tower as that of the GibbsMarkov map. Consider the tower T :  →  associated with a Gibbs-Markov map F : 0 → 0 . By definition, F has a strictly positive Jacobian JF . It is straightforward to check that T also has a strictly positive Jacobian JT , given by  JT (x, ) =

1, if  < R(x) − 1; JF (x), if  = R(x) − 1.

(3.26)

We extend the separation time s associated with F to the tower map T , setting  



s((x, ), (x ,  )) =

s(x, x  ), if  =  ; 0, otherwise.

(3.27)

Consider 0 < β < 1 as in the Gibbs property (G3 ) for F. Similarly to (3.7), we introduce the linear space   Fβ () = ϕ :  → R | ∃C > 0 : |ϕ(x) − ϕ(y)| ≤ Cβ s(x,y) , ∀x, y ∈  , (3.28) and set for each ϕ ∈ Fβ () Cϕ = sup x= y

|ϕ(x) − ϕ(y)| . β s(x,y)

(3.29)

Assuming R ∈ L 1 (m), we have Fβ () ⊂ L ∞ (m) ⊂ L 1 (m); recall (3.22). Let also   Fβ+ () = ϕ ∈ Fβ () | ∃c > 0 : ϕ ≥ c . Note that 1/ϕ is bounded for all ϕ ∈ Fβ+ (). Given ϕ ∈ Fβ+ (), set

(3.30)

44

3 Expanding Structures

 Cϕ+ = max Cϕ , ϕ ∞ ,

   1   . ϕ  ∞

(3.31)

The next result provides, in particular, a unique absolutely continuous invariant probability measure for a tower map with integrable recurrence times. Theorem 3.24 Let T :  →  be the tower map of a Gibbs-Markov map F with recurrence time R ∈ L 1 (m). If ν0 is the unique F-invariant probability measure such that dν0 /dm ∈ Fβ+ (0 ), then ∞

1 T∗j (ν0 |{R > j}) ν {R > j} j=0 0 j=0

ν = ∞

is the unique T -invariant probability measure such that ν m. Moreover, ν is ergodic and dν/dm belongs in Fβ+ (). Proof Note that T R : 0 → 0 is naturally identified with the Gibbs-Markov map F. By Theorem 3.13, the measure ν0 is ergodic. Since R ∈ L 1 (m) and, clearly, T∗ m m, it follows from Corollary 3.21 that ν is an ergodic T -invariant probability measure and ν m. Let us now prove the uniqueness. If ν is a T -invariant probability measure, it follows from Proposition 3.23 that ν0 = (ν|0 )/ν(0 ) is a T R -invariant probability measure. Assuming ν m, we have ν0 m. Using that the measure ν0 is unique, by Theorem 3.13, and ν can be recovered from ν0 , by Proposition 3.23, we easily deduce the uniqueness of ν. Finally, we show that dν/dm belongs in Fβ+ (). By Theorem 3.13, there exist C, β > 0 such that dν0 dν0 1 ≤ C, ≤ ≤ C and C dm dm β

(3.32)

where ν0 m is the T R -invariant probability measure. On the other hand, we have ν|0 = ν(0 )ν0 , by Proposition 3.23. Hence, there must be C0 > 0 such that dν dν dν 1 ≤ (x0 ) ≤ C0 and (x0 ) − (y0 ) ≤ C0 β s(x0 ,y0 ) , C0 dm dm dm

(3.33)

for all x0 , y0 ∈ 0 . Now, recall that the reference measure m in the tower has been introduced considering copies of the reference measure restricted to certain subsets of the base naturally identified with the higher levels. Since T is an upward translation, except at the roof levels, we have that the measure ν in a higher level of the tower is a copy of the measure ν0 restricted to the subsets of the base. Since the separation time for points in higher levels coincides with the separation time of their natural representatives in the base, we deduce that the inequalities in (3.33) can be  extrapolated to all points in . Hence, dν/dm ∈ Fβ+ (). Let T be the tower map associated with a Gibbs-Markov map on a set 0 with recurrence time R. Observe that, if gcd(R) = d ≥ 2, then the measure ν given by

3.3 Tower Maps

45

Theorem 3.24, is not mixing. To justify this, it is enough to see that T −dn+1 (0 ) ∩ 0 = ∅, for all n ≥ 1, and recall that ν(0 ) > 0, by Proposition 3.23. Since every exact measure is mixing, the next result shows that the exactness of ν is in fact equivalent to gcd(R) = 1. In its proof we are going to use that if gcd(R) = 1, then there exist t0 ∈ N and R1 , . . . , Rk ∈ R(0 ) such that, for all t ≥ t0 , we can find integers a1 , . . . , ak ≥ 0 for which t = a1 R1 + · · · + ak Rk ; see for example [15, Corollary of Lemma A.3]. Theorem 3.25 Let T be the tower map of a Gibbs-Markov map with recurrence time R ∈ L 1 (m). If gcd(R) = 1, then the unique T -invariant probability measure ν m is exact. Proof Let A be the σ -algebra in . We need to show that ν(A) = 1 for every A ∈ ∩n≥0 T −n (A) with ν(A) > 0. Take an arbitrary ε > 0. Theorem 3.24 gives C0 > 0 such that dν/dm ≤ C0 . Since m() < ∞, we can choose  ≥ 0 such that    ε m \ j < . 2C0 j≤

(3.34)

Assuming gcd(R) = 1, there exist t0 ∈ N and R1 , . . . , Rk ∈ R(0 ) such that for each t ≥ t0 we may find integers a1 , . . . , ak ≥ 0 for which t = a1 R1 + · · · + ak Rk . This implies that T −t (0 ) ∩ 0 = ∅ for all t ≥ t0 . Hence, there exists t ≥ 0 such that  j. (3.35) T t (0 ) ⊃ j≤

Using that T is nonsingular with respect to m, and taking C > 0 the constant given by Corollary 3.6, we may find δ > 0 such that for all B ∈ A m(B) < Cm(0 )δ =⇒ m(T t (B))
0. Since ν m, we have m(A) > 0. Then, there exists s ≥ 0 such that m(T s (A) ∩ 0 ) > 0. Applying Corollary 2.3 to the Gibbs-Markov map F : 0 → 0 and the set T s (A) ∩ 0 , we find n ∈ N and ω ∈ Pn such that m(ω \ T s (A)) < δ. m(ω) Applying Corollary 3.6 to ω ∈ Pn , we obtain m (F n (ω \ T s (A))) m(ω \ T s (A)) ≤C < Cδ. n m (F (ω)) m(ω)

(3.37)

46

3 Expanding Structures

Now, note that points in ω ∈ Pn have the same first n recurrence times to the base of the tower. This means that there are R1 , . . . , Rn ∈ R(0 ) such that F n |ω = T R1 +···+Rn |ω . Set r = R1 + · · · + Rn . Since T r (ω) = 0 and 0 \ T r +s (A) ⊂ T r (0 ) \ T s (A), it follows from (3.37) that m(0 \ T r +s (A)) ≤ m(T r (0 ) \ T s (A)) < Cm(0 )δ.

(3.38)

Using (3.38) and (3.36) with B = 0 \ T r +s (A), we may write m(T t (0 ) \ T t+r +s (A)) ≤ m(T t (0 \ T r +s (A)))
1 − ε. Since ε > 0 is arbitrary, we have ν(A) = 1, and so ν is exact.



3.3.1 Tower Extension The construction of the tower above was carried out for general maps F, as long as we have some partition of its domain and associated recurrence times. A special case occurs when F is an induced map f R . In that situation, we always consider the partition in the construction of the tower as the one associated with f R and the recurrence time as R. The first item of the next result gives that the map π :  → M defined as (3.39) π(x, , ) = f  (x) is a measurable semiconjugacy between T and f . Since {R > j} ⊂ 0 and 0 may be thought of as a subset of both  and M, it makes sense to consider the pushj forwards f ∗ (ν|{R > j}), for any measure ν on . Lemma 3.26 Let f : M → M be a measurable map, T :  →  the tower map of an induced map f R : 0 → 0 and ν a T -invariant measure. Then

3.3 Tower Maps

47

1. π is measurable and π ◦ T = f ◦ π ;

j 2. π∗ ν is an f -invariant measure and π∗ ν = ∞ j=0 f ∗ (ν|{R > j}); 3. if ν is ergodic, then π∗ ν is ergodic. Proof For the measurability of π it is enough to show that, if A ⊂ M is a measurable set, then p1 (π −1 (A)) is a measurable set, where p1 :  → 0 is the projection in the first coordinate. Since p1 (π −1 (A)) = f − (A) ∩ {R > }, the measurability of π follows from the measurability of f and R. Now we prove that π ◦ T = f ◦ π . Given (x, ) ∈ , we consider two possible situations: if  < R(x) − 1, we have π(T (x, )) = π(x,  + 1) = f +1 (x) = f ( f  (x)) = f (π(x, )); if, on the other hand,  = R(x) − 1, we have π(T (x, R(x) − 1)) = π( f R (x), 0) = f R (x) = f ( f R(x)−1 (x)) = f (π(x, R(x) − 1)).

The f -invariance of π∗ ν is a consequence of the fact that ν is T -invariant and π is a measurable semiconjugacy between T and f . Also, for any measurable set A ⊂ M, we have π∗ ν(A) =

∞

ν(π −1 (A) ∩  ) =

∞

=0

=

∞

=0

ν(T

−

(π

−1

(A)) ∩ T

−

=0

=

∞

ν(T − (π −1 (A) ∩  ))

( )) =

∞

ν(π −1 ( f − (A)) ∩ T − ( ))

=0

ν( f − (A) ∩ {R > }).

=0

To justify the last equality, recall that T − ( ) = {R > } ⊂ 0 and π restricted to 0 is the identity. The ergodicity of π∗ ν is a consequence of the ergodicity of ν and the fact that π is a semiconjugacy. Given a measurable set A ⊂ M with π∗ ν(A) > 0, we have ν(π −1 (A)) > 0, which implies ν( \ π −1 (A)) = 0, by the ergodicity of ν. Hence π∗ ν(M \ A) = ν(π −1 (M \ A)) = ν(π −1 (M) \ π −1 (A)) = ν( \ π −1 (A)) = 0, and so π∗ ν is ergodic.



It follows from Theorem 3.24 that if T :  →  is the tower map of a GibbsMarkov map f R : 0 → 0 with R ∈ L 1 (m), then T has a unique invariant probability measure ν m, which moreover is ergodic. The next result gives in particular that π∗ ν coincides with the f -invariant measure μ given by Corollary 3.21. Proposition 3.27 Let T :  →  be the tower map of an induced Gibbs-Markov map f R : 0 → 0 with R ∈ L 1 (m), and ν the unique T -invariant probability measure such that ν m. Then μ = π∗ ν is the unique ergodic f -invariant probability

48

3 Expanding Structures

measure such that μ m and μ(0 ) > 0. Moreover, dμ/dm is bounded from below by some positive constant. Proof Let ν0 m be the unique ergodic f R -invariant probability measure. Since we assume R ∈ L 1 (m), Corollary 3.21 gives that ∞

1 f ∗j (ν0 |{R > j}) ν {R > j} j=0 0 j=0

μ = ∞

(3.40)

is the unique ergodic f -invariant probability measure with μ m and μ(0 ) > 0. Moreover, dν/dm is bounded from below by some positive constant. On the other hand, by Lemma 3.26, we have π∗ ν =

∞

f ∗j (ν|{R > j}).

(3.41)

j=0

Using the uniqueness of ν0 and Proposition 3.23, we have 1 ν0 . ν j=0 0 {R > j}

ν|0 = ∞

(3.42)

Now, since we have {R > j} ⊂ 0 for each j ≥ 0, it easily follows from (3.40),  (3.41) and (3.42) that π∗ ν = μ. Recall that π is a semiconjugacy between T and f , by Lemma 3.26. Proposition 3.27 now shows that we may think of the measure preserving dynamical system (T, ν) as an extension of the measure preserving dynamical system ( f, μ) or, alternatively, the measure preserving dynamical system ( f, μ) as a quotient of the measure preserving dynamical system (T, ν).

3.3.2 Convergence to the Equilibrium The purpose of this subsection is to prove Theorem 3.28 below, which establishes decay rates, in terms of the recurrence times for the convergence of signed measures to the equilibrium measure under push-forwards by a tower map. The utility of Theorem 3.28 will be clear in Sects. 3.3.3 and 4.4. First, we introduce some useful terminology. We say that a constant C > 0 depending on ϕ ∈ Fβ+ () has bounded dependence on Cϕ+ if, for every A > 0, there is B > 0 such that C ≤ B whenever Cϕ+ ≤ A. We say that a constant c > 0 depending on ϕ ∈ Fβ+ () has lower bounded dependence on Cϕ+ if, for every A > 0, there exists b > 0 such that c ≥ b whenever Cϕ+ ≤ A.

3.3 Tower Maps

49

Theorem 3.28 Let T :  →  be the tower map of a Gibbs-Markov map with recurrence times R ∈ L 1 (m) and gcd(R) = 1. If ν is the unique T -invariant probability measure such that dν/dm ∈ Fβ+ (), then 1. if m{R > n} ≤ Cn −a for some C > 0 and a > 1, then for any probability measure λ with ϕ = dλ/dm ∈ Fβ+ () there exists C  > 0 with bounded dependence on Cϕ+ such that |T∗n λ − ν| ≤ C  n −a+1 ; 2. if m{R > n} ≤ Ce−cn for some C, c > 0 and 0 < a ≤ 1, given 0 < β < 1, there is c > 0 such that, for any probability measure λ with ϕ = dλ/dm ∈ Fβ+ (), there exists C  > 0 with bounded dependence on Cϕ+ such that a

 a

|T∗n λ − ν| ≤ C  e−c n . The fact that the constant C  > 0 in Theorem 3.28 has bounded dependence on Cϕ+ plays a crucial role in Sect. 4.4, specifically in the conjunction of inequality (4.53) with Proposition 4.18. Before proceeding to the proof of Theorem 3.28, we present a simple result showing that the conclusions in Theorem 3.28 are optimal. Proposition 3.29 Let T :  →  be the tower map of a Gibbs-Markov map with recurrence time R ∈ L 1 (m). If ν is the unique T -invariant probability measure such that dν/dm ∈ Fβ+ (), then there are (infinitely many) probability measures λ on  with dλ/dm ∈ Fβ+ () and for some c > 0 |T∗n λ − ν| ≥ c



m{R > }.

>n

Proof For any sufficiently small constant c > 0, consider a probability measure λ on  such that dν dλ |(\0 ) = |(\0 ) + c. (3.43) dm dm Taking for example dλ/dm constant on 0 , we easily ensure that dλ/dm ∈ Fβ+ ();  recall that dν/dm ∈ Fβ+ (). Now, given any A ⊂ >n  , it easily follows from the way we have introduced the measure m on the tower  that m(T −n (A)) = m(A). Since T −n (A) ⊂  \ 0 , using (3.43) and the T -invariance of ν, we get  dλ n dm (T∗ λ)(A) = −n dm T (A)    dν ≥ + c dm T −n (A) dm = ν(T −n (A)) + cm(T −n (A)) = ν(A) + cm(A)    dν = + c dm A dm

50

3 Expanding Structures

This shows that

dT∗n λ  dν  |( >n  ) ≥ |(  ) + c. dm dm >n 

Hence |T∗n λ

   n  dT∗ λ dν dm ≥ cm − ν| =  = c m{R > }, − dm dm >n >n 

thus proving the result.

The remainder of this section is devoted to the proof of Theorem 3.28. We follow the strategy in [19] closely, with the only exception in the estimate of the stretched exponential decay, where we take into account an improvement in [6] that led to an optimal conclusion. Consider the tower map T :  →  of a Gibbs-Markov map F : 0 → 0 with recurrence time R ∈ L 1 (m). Take probability measures λ, λ on  whose densities with respect to m belong in Fβ+ (). Later, we will take λ = ν, but for the time being we consider λ a general probability measure on . Set ϕ=

dλ dλ and ϕ  = . dm dm

(3.44)

Let P = λ × λ be the product measure on  × . The strategy for proving Theorem 3.28 is based on a coupling argument that will appear in Sect. 3.3.2. For this, we consider in Sect. 3.3.2 a simultaneous return time S :  ×  → N. In Proposition 3.45below, we show that |T∗n λ − ν| decays essentially at the same speed of P{S > n} with n. Finally, in Sect. 3.3.2.1, we show that the decay of P{S > n} is comparable to the decay of m{R > n}, both in the polynomial and (stretched) exponential cases. Bounded Distortion Here we obtain bounded distortion results for the tower map T :  → . Let Q be the natural mod 0 partition of  introduced in Sect. 3.3 and Qn defined as in (3.23), for all n ≥ 1. Lemma 3.30 There exists C > 0 such that, for all n ≥ 1, ω ∈ Qn and x, y ∈ ω, log

JT n (x) n n ≤ Cβ s(T (x),T (y)) . JT n (y)

Proof It follows from Lemma 3.4 that there is some C > 0 such that for all k ≥ 1 and all x0 , y0 ∈ 0 belonging in the same element of Pk , we have log

JF k (x0 ) k k ≤ Cβ s(F (x0 ),F (y0 )) . JF k (y0 )

Consider now x, y ∈ Qn . It follows from Lemma 2.11 and (3.26) that

(3.45)

3.3 Tower Maps

51

JT n (x) = JF k (x0 ) and JT n (y) = JF k (y0 ),

(3.46)

where k is the number of visits of x, y to 0 prior to n and x0 , y0 are the elements in 0 corresponding to x, y, respectively. Since we have s(T n (x), T n (y)) = s(F k (x0 ), F k (y0 )),

(3.47)

using (3.45), (3.46) and (3.47), we get the desired conclusion.



Lemma 3.31 There exists C > 0 with bounded dependence on Cϕ+ such that, for all n ≥ 1, ω ∈ Qn with T n (ω) = 0 and x, y ∈ 0 , dT∗n (λ|ω) (x) dm



dT∗n (λ|ω) (y) ≤ C. dm

Moreover, the dependence of C on Cϕ+ can be removed if we assume the number of visits of ω to 0 up to time n sufficiently large. Proof Given x, y ∈ 0 , let x0 , y0 ∈ ω be such that T n (x0 ) = x and T n (y0 ) = y. Taking ϕ = dλ/dm and using Lemma 2.11, we may write dT∗n (λ|ω) (x) dm



dT∗n (λ|ω) JT n (y0 ) ϕ(x0 ) (y) = · . dm JT n (x0 ) ϕ(y0 )

(3.48)

By Lemma 3.30, there exists some constant C  > 0 such that JT n (y0 ) ≤ C . JT n (x0 )

(3.49)

On the other hand, considering i the number of visits of ω to 0 until n, we have s(x0 , y0 ) ≥ i. Using that ϕ ∈ Fβ+ (), we may write ϕ(x0 ) − ϕ(y0 ) ϕ(x0 ) ≤ 1 + (C + )2 β i . ≤ 1 + ϕ ϕ(y0 ) ϕ(y0 )

(3.50)

From (3.48), (3.49) and (3.50), we get dT∗n (λ|ω) (x) dm



  dT∗n (λ|ω) (y) ≤ C  1 + (Cϕ+ )2 β i . dm

Note that the constant C  does not depend on ϕ. Moreover, the dependence of the last expression on Cϕ+ is clearly bounded, and it can be removed if we take i sufficiently large.  Simultaneous Returns to the Base

52

3 Expanding Structures

Here, we introduce a simultaneous return time function on the product space  ×  that will be useful in the coupling argument of the next subsection. First of all, consider R ∗ :  → N the return time to the base, defined for x ∈  by   R ∗ (x) = min n ≥ 0 : T n (x) ∈ 0 . Note that

m{R ∗ > n} =

>n

m( ) =



m{R > }.

(3.51)

>n

Since we assume gcd(R) = 1, it follows from Theorem 3.24 that the measure ν is exact, and so mixing. Also, by Theorem 3.24, we have dν/dm bounded from above by some constant. Thus, there exist n 0 ∈ N and γ0 > 0 such that m(T −n (0 ) ∩ 0 ) ≥ γ0 , for all n ≥ n 0 .

(3.52)

Let p, p  :  ×  →  be the projections to the first and second spaces, respectively. We introduce a sequence of stopping times 0 = τ0 < τ1 < τ2 < · · · on  × , τ1 = n 0 + R ∗ ◦ T n 0 ◦ p τ2 = τ1 + n 0 + R ∗ ◦ T τ1 +n 0 ◦ p  τ3 = τ2 + n 0 + R ∗ ◦ T τ2 +n 0 ◦ p τ4 = τ3 + n 0 + R ∗ ◦ T τ3 +n 0 ◦ p  .. . with falls to 0 alternating between the two coordinates. Notice that τi+1 − τi ≥ n 0 , for all i ≥ 0. Consider the product map T × T :  ×  →  × . We define the simultaneous return to the base S :  ×  → N by     S(x, x  ) = min τi (x, x  ) : (T τi (x,x ) (x), T τi (x,x ) (x  )) ∈ 0 × 0 . i≥2

(3.53)

Note that, as ν is mixing, then ν × ν is ergodic. Since m × m is equivalent to ν × ν, by Theorem 3.24, we have S well defined m × m almost everywhere. Also, by construction, S ≥ 2n 0 . For all n ≥ 0, we have T n ◦ p = p ◦ (T × T )n and T n ◦ p  = p  ◦ (T × T )n .

(3.54)

We use Q × Q to denote the product partition of  × . Observe that T × T sends each element of Q × Q bijectively onto a union of elements of Q × Q. We introduce inductively a sequence of partitions ξ0 ≺ ξ1 ≺ ξ2 ≺ ξ3 ≺ · · · of  ×  in the following way. First, we take ξ0 = { × }. Assuming ξi−1 has been defined for some i ≥ 1, the definition of ξi depends on whether i is odd or even

3.3 Tower Maps

53

• if i is odd, ξi is the refinement of ξi−1 obtained partitioning each  ∈ ξi−1 in the ˜ and T τi sends ˜ such that τi is constant on each  first coordinate into subsets  ˜ bijectively to 0 . p() • if i is even, ξi is the refinement of ξi−1 obtained partitioning each  ∈ ξi−1 in the ˜ and T τi sends ˜ such that τi is constant on each  second coordinate into subsets   ˜ p () bijectively to 0 . From the construction of these objects we easily see that the following two properties hold for each i ≥ 1 • the functions τ1 , τ2 , . . . , τi are ξi -measurable; • the sets {S = τi−1 } and {S > τi−1 } are ξi -measurable. By ξi -measurable we mean that functions are constant on elements of ξi and sets can be written as union of elements of ξi . Before proceeding, let us understand a bit more the elements  ∈ ξi . For definiteness, assume that i ≥ 1 is odd. Observe that  = ω × ω with ω, ω ⊂ . Moreover, at time τi−1 we have T τi−1 (A) contained in some element of Q, and T τi−1 mapping B bijectively to 0 . Finally, at time τi , we have T τi mapping A bijectively to 0 , and T τi (B) spread over several levels  with  ≤ τi − τi−1 . In the remainder of this subsection we provide some useful estimates related to the functions τ0 < τ1 < τ2 < · · · and the product measure P = λ × λ on  × . Given measurable sets A, B with P(A) > 0, we consider the conditional probability P(B | A) =

P(A ∩ B) . P(A)

Observe that if ξ is a measurable partition, then for every measurable set B and every ξ -measurable set A with P(A) > 0, we have min {P(B | )} ≤ P(B | A) ≤ max {P(B | )} . ∈ξ ⊂A

∈ξ ⊂A

(3.55)

In fact, denoting ξ A the set of those  ∈ ξ such that  ⊂ A, we have P (B ∩ A) = P(B | A) = P(A) Since

P (B ∩ ) ≤ min ∈ξ A P()

∈ξ A



∈ξ A

P (B ∩ )

∈ξ A

P()

P (B ∩ )

∈ξ A

P()

≤ min

∈ξ A

P (B ∩ ) , P()

we obtain (3.55). This observation will be particularly useful in Sect. 3.3.2.1. Proposition 3.32 There exists ε0 > 0 with lower bounded dependence on Cϕ , Cϕ  such that, for all i ≥ 1 and  ∈ ξi with S| > τi−1 , we have

54

3 Expanding Structures

P(S = τi | ) ≥ ε0 . Moreover, the dependence of ε0 on Cϕ , Cϕ  can be removed if we take i large enough. Proof Assume first that i ≥ 1 is odd. We know that  = ω × ω . Moreover, {S = ˜ where ω˜ is the set of points in ω that T τi−1 maps to 0 τi } ∩  has the form ω × ω, τi −τi−1 and T makes them return to 0 . We have   τ T∗ i−1 (λ |ω ) T −(τi −τi−1 ) (0 ) ∩ 0 ˜ λ (ω) P(S = τi | ) =   = . τ λ (ω ) T∗ i−1 (λ |ω )(0 ) Using Lemma 3.31, we easily get C > 0 with bounded dependence on Cϕ+ such that P(S = τi | ) ≥ C

−2 m



 T −(τi −τi−1 ) (0 ) ∩ 0 . m(0 )

Now, observe that by construction we have τi − τi−1 ≥ n 0 , and (3.52) gives γ0 > 0 such that m(T −n (0 ) ∩ 0 ) ≥ γ0 , for all n ≥ n 0 . This implies that there exists ε0 > 0 depending on Cϕ+ for which the conclusion holds. The case i even can be proved similarly and yields the dependence of ε0 on Cϕ+ . The fact that C has bounded dependence on Cϕ , Cϕ  implies that ε0 > 0 has lower bounded dependence on Cϕ , Cϕ  . Moreover, this dependence can be removed if we take i large enough.  In the proof of the next result, we will use the fact that there exists some constant C1 > 0 such that dT∗n m ≤ C1 , for all n ≥ 1. (3.56) dm Indeed, by Theorem 3.24, there is C0 > 0 such that 1 dν ≤ C0 . ≤ C 0 dm This implies that for any measurable set A ⊂  and any n ≥ 1, we have m(T −n (A)) ≤ C0 ν(T −n (A)) = C0 ν(A) ≤ C02 m(A), and so (3.56) goes with C1 = C02 . Proposition 3.33 There exists C > 0 with bounded dependence on Cϕ+ , Cϕ+ such that, for all n, i ≥ 0 and  ∈ ξi , we have P(τi+1 − τi > n + n 0 | ) ≤ Cm{R ∗ > n}. Moreover, the dependence of C on Cϕ+ , Cϕ+ can be removed if we take i large enough.

3.3 Tower Maps

55

Proof Consider first the case i = 0. Using (3.56), we may write for all n ≥ 0 P{τ1 > n 0 + n} = (T∗n 0 λ){R ∗ > n} ≤ ϕ ∞ (T∗n 0 m){R ∗ > n} ≤ Cϕ+ C  m{R ∗ > n},

which gives the desired estimate for i = 0. Consider now i ≥ 1. Assume first that i is odd. Recall that for  ∈ ξi , we have  = ω × ω with T τi−1 (ω) contained in some element of Q and T τi−1 mapping ω bijectively to 0 . This implies that ω ∈ Qτi−1 and 1 1 T τi−1 (λ |ω ), T∗τi−1 p∗ (P|) = τi−1   P() T∗ (λ |ω )(0 ) ∗ which together with Lemma 3.31 yields C0 > 0 with bounded dependence on Cϕ+ for which 1 C02 T∗τi−1 p∗ (P|) ≤ m. (3.57) P() m(0 ) Lemma 3.31 also gives that the dependence of C0 on Cϕ+ , Cϕ+ can be removed if we take i large enough. Moreover, recalling that τi+1 (x, x  ) = τi (x, x  ) + n 0 + R ∗ (T τi (x  )), we may write P(τi+1 − τi > n 0 + n | ) = P(R ∗ ◦ T τi +n 0 ◦ p  > n | )   1 (P|) R ∗ ◦ T τi +n 0 ◦ p  > n = P()   1 T∗τi +n 0 p∗ (P|) R ∗ > n = P()   1 T∗τi−1 p∗ (P|) R ∗ > n . = T∗τi −τi−1 +n 0 P() Using (3.56) and (3.57), we easily find a constant C > 0 with bounded dependence on Cϕ+ for which the last expression is bounded by Cm{R ∗ > n}. The case i even  can be proved similarly and yields the bounded dependence of C on Cϕ+ . Coupling  = (T × T ) S :  ×  →  × , with Consider the induced dynamical system T S as in (3.53). We introduce functions 0 = S0 < S1 < S2 < · · · , defined for each n ≥ 1 as (3.58) Sn = Sn−1 + S ◦ (T × T ) Sn−1 .

56

3 Expanding Structures

Note that

n = (T × T ) Sn . T

(3.59)

Let ξ˜ be the partition of  ×  into rectangles  such that S is constant on  and  maps  bijectively to 0 × 0 . For each n ≥ 1, define T ξ˜n =

n−1 

− j ξ˜ . T

j=0

n maps  Each ξ˜n is a partition into sets  ⊂  ×  on which Sn is constant and T bijectively to 0 × 0 . We introduce a separation time in  × , defining for each w, z ∈  ×    n (z) lie in distinct elements of ξ˜ . n (w) and T s˜ (w, z) = min n ≥ 0 : T Let  be the density of the product measure P = λ × λ with respect to m × m. Note that for each (x, x  ) ∈  × , we have (x, x  ) = ϕ(x)ϕ  (x  ).

(3.60)

Bounded Distortion . We start with a Here we obtain some bounded distortion properties for the map T simple relation between the separation times on  and  × . Lemma 3.34 For all u = (x, x  ) and v = (y, y  ) in  ×  we have s(x, y) ≥ s˜ (u, v) and s(x  , y  ) ≥ s˜ (u, v). n (u) = (T × T )k (u). Proof Assume Consider k ≥0 such that T  that s˜ (u, v) > n. i Defining I = i ≤ k : (T × T ) (u) ∈ 0 × 0 , we have (T × T )i (v) ∈ 0 × 0 for each i ∈ I . Moreover, I has at least n elements. Now observe that the p or p  projection of any rectangle  ⊂ 0 × 0 belonging in the partition ξ˜ must necessarily be contained in some element of the initial partition P of 0 , for otherwise  would not map  bijectively to 0 × 0 . Hence, for each i there are ωi , ωi ∈ P T such that (T × T )i (u) ∈ ωi × ωi and (T × T )i (v) ∈ ωi × ωi . This obviously shows  that s(x, y) > n and s(x  , y  ) > n. It is straightforward to check that the product of two maps having Jacobians with respect to a certain reference measure still has a Jacobian with respect to the product  coinmeasure. Moreover, this Jacobian is the product of the two Jacobians. Since T cides with iterates of T × T on domains of injectivity, it follows from Lemma 2.11  has a Jacobian JT with respect to the product measure m × m on  × . that T Lemma 3.35 There exists C > 0 such that for all n ≥ 1 and u, v ∈  ∈ ξ˜n we have

3.3 Tower Maps

57

log

JTn (u) n n ≤ Cβ s˜(T (u),T (v)) . JTn (v)

Proof Take any u, v ∈  with  ∈ ξ˜n . Denoting u = (x, x  ) and v = (y, y  ), there exists some k ≥ 0 such that n (y, y  ) = (T × T )k (y, y  ). n (x, x  ) = (T × T )k (x, x  ) and T T

(3.61)

By Lemma 3.34, we have n (u), T n (v)) and s(T k (x  ), T k (y  )) ≥ s˜ (T n (u), T n (v)). s(T k (x), T k (y)) ≥ s˜ (T (3.62) We may write log

Jn (x, x  ) Jn (y, x  ) JTn (x, x  ) = log T + log T   JTn (y, y ) JTn (y, x ) JTn (y, y  ) J(T ×T )k (y, x  ) J(T ×T )k (x, x  ) + log = log J(T ×T )k (y, x  ) J(T ×T )k (y, y  )

JT k (y)JT k (x  ) JT k (x)JT k (x  ) + log JT k (y)JT k (x  ) JT k (y)JT k (y  )  JT k (x ) JT k (x) + log . = log JT k (y) JT k (y  ) = log

(3.63)

Notice that since u, v ∈  with  ∈ ξ˜n and (3.61) holds, then x, y belong in the same element of Qk and x  , y  belong in the same element of Qk . Let C0 > 0 be the constant given by Lemma 3.30. Using (3.63) and (3.62), we easily get log

JTn (x, x  ) k k k  k  n n ≤ C0 β s(T (x),T (y)) + C0 β s(T (x ),T (y )) ≤ 2C0 β s˜(T (u),T (v)) ,  JTn (y, y ) 

thus finishing the proof.

Using the previous lemmas we deduce the following bounded distortion property for the density  of the product measure P = λ × λ . Lemma 3.36 There exists C > 0 with bounded dependence on Cϕ+ , Cϕ+ such that for all u, v ∈  ×  we have log

(u) ≤ C β s˜(u,v) . (v)

58

3 Expanding Structures

Proof Since ϕ, ϕ  ∈ Fβ+ (), we have 1 Cϕ+ Cϕ+

≤  ≤ Cϕ+ Cϕ+ .

Applying the mean value theorem to the logarithm, we have for all u, v ∈  ×  | log (u) − log (v)| ≤ Cϕ+ Cϕ+ |(u) − (v)|.

(3.64)

Writing u = (x, x  ) and v = (y, y  ), we get |(u) − (v)| = |ϕ(x)ϕ  (x  ) − ϕ(y)ϕ  (y  )| ≤ |ϕ(x)| · |ϕ  (x  ) − ϕ  (y  )| + |ϕ  (y  )| · |ϕ(x) − ϕ(y)|   ≤ Cϕ+ Cϕ+ β s(x ,y ) + Cϕ+ Cϕ+ β s(x,y) . (3.65) It follows from (3.64), (3.65) and Lemma 3.34 that | log (u) − log (v)| ≤ 2(Cϕ+ )2 (Cϕ+ )2 β s(u,v) , which gives the result. Clearly, C has bounded dependence on Cϕ+ , Cϕ+ .



Lemma 3.37 There exists C > 0 with bounded dependence on Cϕ+ , Cϕ+ such that for all n ≥ 1,  ∈ ξ˜n and u, v ∈ 0 × 0 we have ∗n (P|) ∗n (P|)  d T dT (u) (v) ≤ C. d(m × m) d(m × m) n (u 0 ) = u and Proof Given u, v ∈ 0 × 0 , let u 0 , v0 ∈  be such that T n  T (v0 ) = v. Taking  = d P/d(m × m) and using Lemma 2.11, we may write ∗n (P|) dT (u) d(m × m)

 n d T∗ (P|) Jn (v0 ) (u 0 ) (v) = T · . d(m × m) JTn (u 0 ) (v0 )

(3.66)

Observing that for all u 0 , v0 ∈  ∈ ξ˜n , we have s˜ (u 0 , v0 ) ≥ n, it follows from Lemma 3.35, that there is C0 > 0 such that JTn (v0 ) ≤ C0 . JTn (u 0 )

(3.67)

On the other hand, Lemma 3.36, gives C > 0 such that (u 0 ) ≤ eC . (v0 )

(3.68)

3.3 Tower Maps

59

Using (3.66), (3.67) and (3.68), we get ∗n (P|) dT (u) d(m × m)

 n d T∗ (P|) (v) ≤ C0 eC . d(m × m)

By Lemma 3.36, this last expression has bounded dependence on Cϕ+ , Cϕ+ .



As a consequence of the bounded distortion property above, we obtain the following useful estimate. Proposition 3.38 There exists C > 0 with bounded dependence on Cϕ+ , Cϕ+ such that for all n, i ≥ 0, we have P{Si+1 − Si > n} ≤ C(m × m){S > n}. Proof The estimate is true for i = 0, for the density of P with respect to m × m is given by (x, x  ) = ϕ(x)ϕ  (x  ), which is bounded. Assume now i ≥ 1 and take i maps  bijectively to 0 × 0 . It easily follows from (3.58)  ∈ ξ˜i . Recall that T i . So, we may write and (3.59) that Si+1 − Si = S ◦ T P(Si+1 − Si > n | ) =

∗i (P|){S > n} T . ∗i (P|)(0 × 0 ) T

Using Lemma 3.37, we obtain C > 0 with bounded dependence on Cϕ+ , Cϕ+ such that (m × m){S > n} . P(Si+1 − Si > n | ) ≤ C 2 (m × m)(0 × 0 ) It follows that P{Si+1 − Si > n} =



P(Si+1 − Si > n | )P()

∈ξ˜i

≤ C2

(m × m){S > n} P() (m × m)(0 × 0 )

∈ξ˜i

= C2

(m × m){S > n} P( × ). (m × m)(0 × 0 )

Noting that d P/d(m × m) ≤ Cϕ+ Cϕ+ , the conclusion follows.



Densities in Induced Times ˜0 ≥ ˜1 ≥ ˜ 2 ≥ · · · , each  ˜ i corresponding We introduce a sequence of functions  . to the density of the total measure remaining after i iterations by the induced map T First of all, take an integer i 1 ≥ 0 such that

60

3 Expanding Structures

C β i1 < C T,

(3.69)

with C > 0 as in Lemma 3.36. Observe that since C has bounded dependence on Cϕ+ , Cϕ+ , the same is true for i 1 . Take a small constant ε > 0, to be chosen in Lemma 3.40. Given u ∈  × , let i (u) be the element of ξ˜i that contains u. Define  (u), if i ≤ i 1 ; ˜ i (u) = (3.70)  ˜ i−1 (v)  ˜ i−1 (u) − ε JTi (u) minv∈i (u) , if i > i 1 .  JTi (v)

A key feature of these densities is given in the next result. Lemma 3.39 For all i ≥ 1 and  ∈ ξ˜i , we have ∗i (( ∗i (( ˜ i−1 −  ˜ i )((m × m)|)) = p∗ T ˜ i−1 −  ˜ i )((m × m)|)). p∗ T ˜ i =  for all i ≤ i 1 , the conclusion is obvious for i ≤ i 1 . Assume Proof Since  now i > i 1 . Given any measurable set A ⊂ , we have ∗i (( ˜ i−1 −  ˜ i )((m × m)|))(A) p∗ T −i (A × )) ˜ i )((m × m)|))(T ˜ i−1 −  = ((  ˜ i−1 −  ˜ i )d(m × m) = (  = 

(3.71)

∩T −i (A×)

∩T −i (A×)

ε min

= A×

v∈

ε min v∈

˜ i−1 (v)  Ji (u)d(m × m) JTi (v) T

˜ i−1 (v)  d(m × m) JTi (v)

˜ i−1 (v)  = ε min m(A)m(). v∈ JT i (v)

(3.72)

Now, it is not difficult to see that starting with the p  instead of p in (3.71), we arrive to the same expression in (3.72).  In Proposition 3.41, we will see that a definite fraction of mass is deducted when ˜ i )i>i1 to the next one. To prove that, we pass from one element of the sequence ( ˜ i )i≥i1 and (i )i≥i1 in the following we introduce auxiliary sequences of densities ( way: given u ∈  × , set for each i ≥ i 1 ˜ i (u) = 

˜ i (u) ˜ i−1 (u)   , i (u) = and Ei (u) = ε · min i (v). v∈i (u) JTi (u) JTi (u)

It follows from Lemmas 2.11 and (3.70) that for i > i 1

(3.73)

3.3 Tower Maps

61

i (u) =

˜ i−1 (u)  ˜ i (u) = i (u) − Ei (u). and  i−1 (u)) JT(T

(3.74)

Note that, although not explicitly stated in the notation, for the sake of simplicity, all these functions depend on the choice of constant ε > 0. In the next result, we show that for small enough ε > 0 we have a uniform bounded distortion property for the ˜ i. functions  Lemma 3.40 There exist C > 0 and ε > 0 such that for all i ≥ i 1 and u, v in the same element of ξ˜i , we have log

˜ i (u)  i i ≤ Cβ s˜(T (u),T (v)) . ˜ i (v) 

Proof Fix δ > 0 small so that (1 + δ)β < 1. Denoting C0 > 0 the constant given by Lemma 3.35, take  C = max 2,

 (1 + δ)β C0 . 1 − (1 + δ)β

(3.75)

We are going to prove by induction on i ≥ i 1 that the result holds with this constant C. i1 (u), T i1 (v)) + i 1 . It Given u, v in a same element of ξ˜i1 , we have s˜ (u, v) = s˜ (T follows from Lemmas 3.35, 3.36 and (3.73) that log

˜ i1 (u) ˜ i (u)   J(v) = log 1 + log T ˜ i1 (v) ˜ i1 (v) JT(u)   JT(v) (u) + log = log (v) JT(u) i1 (u),T i1 (v))

≤ C β s˜(u,v) + C0 β s˜(T

i1 (u),T i1 (v))+i 1

≤ C β s˜(T

i1 (u),T i1 (v))

+ C0 β s˜(T

.

Then, by (3.69) and the choice of C, we have log

˜ i1 (u)  i1 i1 i1 i1 ≤ 2C0 β s˜(T (u),T (v)) ≤ Cβ s˜(T (u),T (v)) . ˜ i1 (v)

(3.76)

This gives the desired conclusion for i = i 1 . Assume now that the result holds for i − 1 with i > i 1 . Given u, v in the same element of ξ˜i , it follows from (3.74), Lemma 3.35 and the induction hypothesis that log

i−1 (v)) ˜ i−1 (u)  i (u) J( T = log + log T i−1  (u)) ˜ i−1 (v) i (v) JT(T  i−1 (u),T i−1 (v))

≤ (C + C0 )β s˜(T

.

(3.77)

62

3 Expanding Structures

This in particular gives e−C−C0 ≤

i (u) ≤ eC+C0 . i (v)

(3.78)

Now, setting εi = Ei (u) = Ei (v), by (3.74) we may write   ˜ i (u) i (u) − εi  i (u) i (v) − log · = log log ˜  (v)  (u)  (v) − ε i (v) i i i i   εi i − iε(u)  (v) = log 1 + i i 1 − iε(v) ε i − εi  (v) i (u) ≤ 2 i , i 1 − ε(v)

(3.79)

i

provided we assume ε < 1/2. Indeed, under this assumption, we have for all z ∈  εi minw∈ i (w) 1 =ε ≤ε< . i (z) i (z) 3 This implies

 1−

εi i (v)

−1


− . 2

(3.81)

Using the mean value theorem we easily get (3.79). Let us now obtain an estimate for the expression in (3.79). We have ε i − εi εi 1 i (v) i (u) · = εi i 1 −  (v) i (u) 1 − ε(v) i i

i (u) ε − 1 ≤ · i (v) 1−ε

i (u) · − 1 .(3.82) i (v)

Recalling (3.78) we may find a constant C1 (only depending on C and C0 ) such that i (u) ≤C1 log i (u) . − 1  (v) i (v) i

(3.83)

It follows from (3.77), (3.79), (3.82) and (3.83), that     ˜ i (u)  2 C1 ε i (u) 2 C1 ε i−1 i−1 log ≤ 1 + (C + C0 )β s˜ (T (u),T (v)) . ≤ 1+ log ˜ 1−ε i (v) 1−ε i (v)

3.3 Tower Maps

63

i−1 (u), T i−1 (v)) = s˜ (T i (u), T i (v)) + 1 and choosing Hence, using the relation s˜ (T ε > 0 sufficiently small (only depending on C1 and δ), we get ˜ i (u)  i i i i ≤ (1 + δ) β(C + C0 )β s˜(T (u),T (v)) ≤ Cβ s˜(T (u),T (v)) , log ˜ i (v)  thus completing the induction step. For the last inequality recall, the choice of the constant C in (3.75).  From now on, we take the constant ε > 0 in (3.70) as in Lemma 3.40. Proposition 3.41 There exists 0 < ε1 < 1 such that for all i > i 1 , we have ˜ i−1 . ˜ i ≤ (1 − ε1 )  Proof We need to show that there exists 0 < ε1 < 1 such that for all i > i 1 and all u ∈  × , we have ˜ i−1 (u) −  ˜ i−1 (u). ˜ i (u) ≥ ε1   (3.84) We claim that it is enough to show that there exists some uniform constant C > 0 such that for all i > i 1 and all  ∈ ξ˜i , we have max v∈

˜ i−1 (v)  ˜ i−1 (v)   min ≤ C. JTi (v) v∈ JTi (v)

(3.85)

Actually, if this holds, then using (3.70), we obtain for all u ∈  ˜ i−1 (u) −  ˜ i (u) = ε JTi (u) min  v∈

˜ i−1 (v)  ε ˜ i−1 (u), ≥  JTi (v) C

which obviously implies (3.84) with ε1 = ε/C. Now, for proving (3.85), observe that from (3.74), we easily get for all i > i 1 and u, v ∈  ∈ ξ˜i 



i−1 (v)) ˜ i−1 (u)  J( T i (u) = log . + log T i−1  (u)) ˜ i−1 (v) i (v) JT(T  (3.86) Hence, using Lemmas 3.35 and 3.40, we easily obtain a uniform constant C > 0 for which (3.85) holds.  ˜ i−1 (v) ˜ i−1 (u)    log JTi (u) JTi (v)

= log

Remark 3.42 It is worth noting that the constant ε > 0 in Lemma 3.40 can be chosen not depending on Cϕ , Cϕ  . However, ε depends on δ, which ultimately depends on β. The same is true for the constant ε1 in Proposition 3.41, since it is obtained quotienting ε by some uniform constant.

64

3 Expanding Structures

Densities in Real Time

  ˜ n to build a new sequence Here, we use the sequence densities in induced time  n of densities whose iterations under T × T will be very useful. We introduce functions 0 ≥ 1 ≥ 2 ≥ · · · , setting for each n ≥ 0 and u ∈  ×  ˜ i (u), n (u) = 

if

Si (u) ≤ n < Si+1 (u).

(3.87)

˜ i =  for all i ≤ i 1 , we have Since Si1 ≥ i 1 and (3.70) gives  i = , for all i ≤ i 1 .

(3.88)

It easily follows that for each n ≥ 1  = n +

n (k−1 − k ).

(3.89)

k=1

A key feature of these functions is obtained in the next equality. Lemma 3.43 For all 1 ≤ k ≤ n, we have p∗ (T × T )n∗ ((k−1 − k )(m × m)) = p∗ (T × T )n∗ ((k−1 − k )(m × m)) . Proof Consider for each 1 ≤ k ≤ n and i ≥ 1 the set Ak,i = {u ∈  ×  : k = Si (u)}. Note that Ak,i = Ak, j for i = j, and each Ak,i is a union of elements of ξ˜i . ˜ i−1 −  ˜ i on  ∈ ξ˜i |Ak,i , Define Ak = ∪i Ak,i . By (3.87), we have k−1 − k =  and k = k−1 on  ×  \ Ak . It follows from (3.54), (3.59) and Lemma 3.39 that for each 1 ≤ k ≤ n p∗ (T × T )n∗ ((k−1 − k )(m × m)) = T∗n−k p∗ (T × T )k∗ ((k−1 − k )(m × m)) ˜ i−1 −  ˜ i )(m × m)|) = T∗n−k p∗ (T × T )∗Si (( i

=

i

=

= =

∗i (( ˜ i−1 −  ˜ i )(m × m)|) T∗n−k p∗ T

∈ξ˜i |Ak,i

i

∗i (( ˜ i−1 −  ˜ i )(m × m)|) T∗n−k p∗ T

∈ξ˜i |Ak,i

i

=

∈ξ˜i |Ak,i



˜ i−1 −  ˜ i )(m × m)|) T∗n−k p∗ (T × T )∗Si ((

∈ξ˜i |Ak,i

T∗n−k p∗ (T × T )k∗ ((k−1 − k )(m × p∗ (T × T )n∗ ((k−1 − k )(m × m)).

m))

3.3 Tower Maps

65

Thus, we have proved the result.



The next corollary shows that n is actually the component of  in decomposition (3.89) that really matters for the total variation we need to estimate. Corollary 3.44 For all n ≥ 1, n T λ − T n λ ≤ ( p∗ − p  )(T × T )n (n (m × m)) . ∗ ∗ ∗ ∗ Proof Recall that P = λ × λ = (m × m). Using (3.54) and (3.89), we may write n T p∗ ((m × m)) − T n p  ((m × m)) ∗ ∗ ∗ p∗ (T × T )n ((m × m)) − p  (T × T )n ((m × m)) ∗ ∗ ∗ ≤ p∗ (T × T )n∗ (n (m × m)) − p∗ (T × T )n∗ (n (m × m)) n ( p − p  )∗ [(T × T )n ((k−1 − k )(m × m))] . + ∗

n T λ − T n λ = ∗ ∗ =

k=1



Lemma 3.43 gives that all terms in the last summation vanish.

Using Corollary 3.44, we finally obtain the relationship between |T∗n λ − T∗n λ | and P{S > n} mentioned in the comments after Theorem 3.28, which happens to be a fundamental step to prove that theorem. Fix ε1 > 0 as in Proposition 3.41. Proposition 3.45 There is C > 0 with bounded dependence on Cϕ+ , Cϕ+ such that, for all n ≥ 1, |T∗n λ − T∗n λ | ≤ 2P{S > n} + C

∞

 (1 − ε1 )i (i + 1)P S >

i=1

 n . i +1

Proof It follows from Corollary 3.44 that n T λ − T n λ ≤ 2 ∗



∗

n d(m × m).

(3.90)

We may write 

 n d(m × m) =

{Si1 >n}

n d(m × m) +

∞  i=i 1

{Si ≤nn}

n d(m × m) =

{Si1 >n}

d(m × m) = P{Si1 > n}.

Using (3.87), (3.88) and Proposition 3.41, we also have, for each i ≥ i 1 ,

(3.92)

66

3 Expanding Structures



 {Si ≤n n} = P{S1 > n} +

i 1 −1

P{Si ≤ n < Si+1 }

i=1

= P{S > n} + (1 − ε1 )−i1

i 1 −1

(1 − ε1 )i1 P{Si ≤ n < Si+1 }.

i=1

Together with (3.94), this yields ∞ n T λ − T n λ ≤ 2P{S > n} + C  (1 − ε1 )i P{Si ≤ n < Si+1 }, ∗ ∗

(3.95)

i=1

for some C  > 0 with bounded dependence on Cϕ+ , Cϕ+ . Recall that i 1 was chosen in (3.69) and has bounded dependence on Cϕ+ , Cϕ+ , and ε1 was introduced in Proposition 3.41 and is independent of Cϕ+ , Cϕ+ . We claim that for each i ≥ 1 P{Si ≤ n < Si+1 } ≤

i j=0

 n . P S j+1 − S j > i +1 

(3.96)

In fact, assuming Si+1 > n, there must be some 0 ≤ j ≤ i with S j+1 − S j > n/(i + 1), for otherwise, we would have Si+1 =

i i (S j+1 − S j ) ≤ j=0

j=0

n = n. i +1

3.3 Tower Maps

67

Hence, {Si ≤ n < Si+1 } ⊂

i  

S j+1 − S j >

j=0

 n , i +1

which clearly implies (3.96). Let now C  > 0 be the constant given by Proposition 3.38. Using (3.95) and (3.96), we get |T∗n λ − T∗n λ | ≤ 2P{S > n} + C 

∞ (1 − ε1 )i P {Si ≤ n < Si+1 } i=1

 ∞ i  i ≤ 2P{S > n} + C (1 − ε1 ) P S j+1 − S j > i=1

j=0

n i +1

 ∞   i ≤ 2P{S > n} + C C (1 − ε1 ) (i + 1)(m × m) S > i=1



 n . i +1

Since P = λ × λ , with dλ/dm and dλ /dm both belonging in Fβ+ (), we may obviously replace m × m by P in the last expression above. 

3.3.2.1

Specific Rates

Taking λ = ν in Proposition 3.45, we easily see that the conclusions of Theorem 3.28 can be drawn once we find appropriate estimates for P{S > n}. We will use the results in Sect. 3.3.2 for P = λ × ν, with λ m a probability measure such that ϕ = dλ/dm ∈ Fβ+ () and β > 0 such that ϕ  = dν/dm ∈ Fβ+ (). Note that the constant ε1 in Proposition 3.45 depends on β (recall Remark 3.42). However, this does not affect the exponent a in the polynomial case. On the contrary, this causes the exponent c to depend on β, in the (stretched) exponential case. The estimates for P{S > n} will be obtained below, considering separately the polynomial and (stretched) exponential cases. Polynomial Decay Assume that there are C > 0 and a > 1 such that for all n ≥ 1 m{R > n} ≤ Cn −a . It follows from (3.51) that there is C0 > 0 (depending only on C and a) such that m{R ∗ > n} =



m{R > } ≤ C0 n −a+1 .

(3.97)

>n

The polynomial case of Theorem 3.28 will be obtained as a consequence of the next result, which we present in general form, suitable for application in other situations. It will be particularly useful in Sect. 3.5.

68

3 Expanding Structures

Proposition 3.46 Let S and 0 = τ0 < τ1 < τ2 < · · · be measurable functions on a measurable space X and ξ1 ≺ ξ2 ≺ · · · be measurable partitions such that τ0 , . . . , τi and {S > τi−1 } are ξi -measurable, for all i ≥ 1. Assume that P is a probability measure on X for which (A1 ) there is ε0 > 0 such that, for all i ≥ 1 and  ∈ ξi with S| > τi−1 , P(S = τi | ) ≥ ε0 ; (A2 ) there are C, a > 0 and n 0 ≥ 0 such that, for all i ≥ 0 and  ∈ ξi , P(τi+1 − τi > n + n 0 | ) ≤ Cn −a . Then, there exists C  > 0 such that P{S > n} ≤ (1 − ε0 )

n 2n 0 +1

!

+ C

∞ (i + 1)a+1 (1 − ε0 )i−1 n −a . i=1

Moreover, if C has bounded dependence on some D, then C  also has bounded dependence on D. Proof We start by deducing some simple consequences of (A1 ) and (A2 ). It follows from (A1 ) that for any  ∈ ξi with S| > τi−1 , we have P (S > τi | ) = 1 − P (S = τi | ) ≥ 1 − ε0 .

(3.98)

Observing that {S > τi } ⊂ · · · ⊂ {S > τ1 } ⊂ {S > τ0 } = X , we have for each i ≥ 1 ⎛ P {S > τi } = P ⎝

i  

⎞ i &    S > τj ⎠ = P S > τ j | S > τ j−1 .

j=1

j=1

Since {S > τ j−1 } is ξ j -measurable, using (3.98) and (3.55) on each factor of the product above, we obtain for all i ≥ 1 P{S > τi } ≤ (1 − ε0 )i .

(3.99)

Assume from now on that n ≥ 2n 0 . It follows from (A2 ) and (3.55) that, for all j ≥ 0 and A a ξ j -measurable set, P(τ j+1 − τ j > n | A) ≤ C(n − n 0 )−a = C



n − n0 n

−a

n −a ≤ C2a n −a . (3.100)

3.3 Tower Maps

69

Set C1 = 2a C and, for n ≥ 2n 0 , ' q(n) =

( n . 2n 0 + 1

By (3.100), we have for all 1 ≤ i ≤ q(n),  n −a   n . P τ j+1 − τ j > | A ≤ C1 i i

(3.101)

Let us now estimate P{S > n}. We may write q(n)−1   P{S > n : τi ≤ n < τi+1 }. (3.102) P{S > n} = P S > n : τq(n) ≤ n + i=0

It follows from (3.99) that     P S > n : τq(n) ≤ n ≤ P S > τq(n) ≤ (1 − ε0 )q(n) .

(3.103)

Now, we analyse the terms of the summation in (3.102). For each 0 ≤ i < q(n), we have P {S > n : τi ≤ n < τi+1 } ≤ P {S > τi : n < τi+1 } .

(3.104)

We claim that P {S > τi : n < τi+1 } ≤

i j=0

 n . P S > τi : τ j+1 − τ j > i +1 

(3.105)

In fact, having τi+1 > n, there must be some 0 ≤ j ≤ i such that τ j+1 − τ j > n/ (i + 1), for otherwise, we would have τi+1

i i = (τ j+1 − τ j ) ≤ j=0

j=0

Hence, {S > τi : n < τi+1 } ⊂

i   j=0

n = n. i +1

S > τi : τ j+1 − τ j >

 n , i +1

which clearly implies (3.105). It follows from (3.104) and (3.105) that P {S > n : τi ≤ n < τi+1 } ≤

i j=0

 P S > τi : τ j+1 − τ j >

 n . i +1

(3.106)

70

3 Expanding Structures

Now, we estimate the terms in the summation above. Consider first j = 0. Using (3.101), we get    −a n n P τ1 − τ0 > . ≤ C1 i +1 i +1 This gives in particular for i = 0 P {S > τ0 : τ1 − τ0 > n} = P {τ1 − τ0 > n} ≤ C1 n −a .

(3.107)

Assume now 1 ≤ i < q(n). As {S > τi } ⊂ · · · ⊂ {S > τ0 }, we may write  i     n n P S > τi : τ1 − τ0 > S > τk : τ1 − τ0 > =P i +1 i +1 k=0 &    i n n . P S > τk | S > τk−1 : τ1 − τ0 > ≤ P τ1 − τ0 > i + 1 k=1 i +1 

(3.108)  Since the set A = S > τk−1 : τ1 − τ0 > using (3.98), we obtain i &

n i+1



is ξk -measurable and S| A > τk−1 ,



n P S > τk | S > τk−1 : τ1 − τ0 > i + 1 k=1

 ≤ (1 − ε0 )i .

(3.109)

It follows from (3.107), (3.108) and (3.109) that for all 0 ≤ i < q(n) 

n P S > τi : τ1 − τ0 > i +1



 ≤ C1

n i +1

−a

(1 − ε0 )i ,

(3.110)

thus having an upper bound for the term corresponding to j = 0 in (3.106). Consider now 1 ≤ j ≤ i. We have   n P S > τ j : τ j+1 − τ j > i +1  = P S > τ j : τ j+1 − τ j >  ≤ P τ j+1 − τ j >

n i +1

(3.111)

 n | S > τ j−1 P{S > τ j−1 } i +1 (3.112)  | S > τ j−1 P{S > τ j−1 }. (3.113)

  Since S > τ j−1 is ξ j -measurable, using (3.101) we get

3.3 Tower Maps

71

 P τ j+1 − τ j >

n | S > τ j−1 i +1



 ≤ C1

n i +1

−a

.

(3.114)

By (3.99), we have P{S > τ j−1 } ≤ (1 − ε0 ) j−1 .

(3.115)

It follows from (3.113), (3.114) and (3.115) that 

n P S > τ j : τ j+1 − τ j > i +1



 ≤ C1

n i +1

−a

(1 − ε0 ) j−1 .

(3.116)

This gives in particular an upper bound for the term in summation (3.106) corresponding to j = i. Assume now 1 ≤ j < i. Since  S > τi : τ j+1 − τ j >

n i +1



 ⊂ · · · ⊂ S > τ j : τ j+1 − τ j >

 n , i +1

we have  P S > τi : τ j+1 − τ j >  =P S > τ j : τ j+1 − τ j >

n i +1



n i +1

(3.117)  & i

 P S > τk | S > τk−1 : τ j+1 − τ j >

k= j+1

 n . i +1

(3.118)   Since A = S > τk−1 : τ j+1 − τ j > n/(i + 1) is ξk -measurable and S| A > τk−1 , using (3.98) and (3.55), we obtain i &



n P S > τk | S > τk−1 : τ j+1 − τ j > i + 1 k= j+1

 ≤ (1 − ε0 )i− j ,

(3.119)

which together with (3.116) and (3.117) gives, for all 1 ≤ j < i, 

n P S > τi : τ j+1 − τ j > i +1



 ≤ C1

n i +1

−a

(1 − ε0 )i−1 .

(3.120)

Recall that (3.116) gives the previous inequality for j = i and (3.109) gives it for j = 0. Using (3.106) and (3.120), we get P {S > n : τi ≤ n < τi+1 } ≤

i

 P S > τi : τ j+1 − τ j >

j=0

≤ C1 (i + 1)a+1 (1 − ε0 )i−1 n −a .

n i +1



(3.121)

72

3 Expanding Structures

It follows from (3.102), (3.103) and (3.121) that P{S > n} ≤ (1 − ε0 )q(n) + 2C1

∞ (i + 1)a+1 (1 − ε0 )i−1 n −a . i=1

Take C  = 2C1 = 2a+1 C.



Recalling what was said at the beginning of Sect. 3.3.2.1, the polynomial case of Theorem 3.28 is now a consequence Proposition 3.46, along with Propositions 3.32 and 3.33; recall (3.97). It is worth noting that the term (1 − ε0 )[n/(2n 0 +1)] in the con clusion of Proposition 3.46 arises in (3.103) as an upper bound for P S > τq(n) . Moreover, the constant ε0 provided by Proposition 3.32 has lower bounded dependence on Cϕ+ , Cϕ+ , and this dependence can be removed for i = q(n) large enough. (Stretched) Exponential Decay Assume that there are C, c > 0 and 0 < a ≤ 1 such that m{R > n} ≤ Ce−cn , for all n ≥ 1. Then, there exists C0 > 0 such that   a m{R > } ≤ C0 e−cn . (3.122) m R∗ > n = a

>n

The (stretched) exponential case of Theorem 3.28 will be a consequence of Proposition 3.48 below. We first give an auxiliary result. Lemma 3.47 Given C, c > 0 and N sufficiently large, we have for all  ≥ 1  & n 1 ,...,n  ≥N

ni = p

Ce−cni ≤ Ce−cp . a

a

(3.123)

i=1

Proof We prove this by induction on  ≥ 1. Trivially, we have the equality for  = 1. Suppose now that the inequality holds for some  ≥ 1. Using the induction hypothesis in the first inequality below, we may write

+1 &

n 1 ,...,n +1 ≥N

ni = p

i=1

Ce−cni = a



a

k,n +1 ≥N k+n +1 = p

≤

 &

Ce−cn +1



n 1 ,...,n  ≥N

n i =k

C 2 e−cn +1 e−ck a

Ce−cni

a

i=1

a

k,n +1 ≥N k+n +1 = p

≤ 2C 2



e−c( j

a

+(n− j)a )

e−cn

(( j/n)a +(1− j/n)a )

N ≤ j≤ 2p

≤ 2C 2



N ≤ j≤

p 2

a

.

3.3 Tower Maps

73

Now, observe that x → (x a + (1 − x)a − 1) /x a defines a strictly positive continuous function on the interval [0, 1/2]. So, there is b > 0 such that x a + (1 − x)a ≥ 1 + bx a , for all 0 ≤ x ≤ 1/2. Applying this to x = j/ p ≤ 1/2, we obtain

+1 &

n 1 ,...,n +1 ≥N

ni = p

i=1



Ce−cni ≤ 2C 2 a

e−cn

a

(1+b( j/n)a )

≤ 2 C 2 e−cp

N ≤ j≤ 2p

Taking N large enough so that 2 C

j≥N e

a



e−cbj . a

j≥N

−cbj a

≤ 1, we finish the proof.



Proposition 3.48 Let S, κ and 0 = τ0 < τ1 < τ2 < · · · be measurable functions on a measurable space X such that S(x) = τκ(x) for all x ∈ X and ξ1 ≺ ξ2 ≺ · · · be measurable partitions such that τ0 , . . . , τi and {S > τi−1 } are ξi -measurable for all i ≥ 1. Assume that P is a probability measure on X for which (B1 ) there is ε0 > 0 such that for all i ≥ 1 and  ∈ ξi with S| > τi−1 P(S = τi | ) ≥ ε0 ; (B2 ) there are n 0 ≥ 0, C > 0 and 0 < a ≤ 1 such that for all i ≥ 0 and  ∈ ξi P(τi+1 − τi > n + n 0 | ) ≤ Ce−cn . a

Then, there exist C  > 0 and  > 0 such that  a

P{S > n} ≤ (1 − ε0 )[n ] + C  e−c n . a

Moreover,  only depends on a, c and if C has bounded dependence on some D, then C  also has bounded dependence on D. Proof Consider q(n) = [n a ], for some small  > 0, to be specified below. Given x ∈ X with S(x) > n, set n i = τi (x) − τi−1 (x), for each 1 ≤ i ≤ κ(x). Note that κ(x)

n i = S(x) > n.

i=1

If κ(x) > q(n), we do not do anything. Otherwise, we take N for which the conclusion of Lemma 3.47 holds and set J = {i ≤ κ(x) : n i ≥ N }. For n sufficiently large, we have i∈J

ni =

κ(x) i=1

ni −

κ(x) i=1 i ∈J /

n i > n − κ(x)N ≥ n − q(n)N ≥

n . 2

74

3 Expanding Structures

This shows that 





J ⊂{1,...,q(n)}

n ≥N

i i∈J n i ≥n/2

i∈J

{S > n} ⊂ {κ > q(n)} ∪

{τi − τi−1 = n i } .

(3.124)

Let us now find estimates for the sets in (3.124). Given  ∈ ξi with S| > τi−1 , it follows from (B1 ) that P (S > τi | ) = 1 − P (S = τi | ) ≥ 1 − ε0 .

(3.125)

  Recalling that S > τq(n) ⊂ · · · ⊂ {S > τ1 } ⊂ {S > τ0 } = X , we may write 

⎛



q(n)



P S > τq(n) = P ⎝



⎞

j=1

&

q(n)

S > τj ⎠ =

  P S > τ j | S > τ j−1 .

j=1

Since each {S > τ j−1 } is ξ j -measurable, using (3.125) and (3.55), we obtain   P{κ > q(n)} = P S > τq(n) = (1 − ε0 )q(n) .

(3.126)

Up to multiplying C in (B2 ) by some constant (just depending on n 0 , c and a), we may suppose that for all i ≥ 0, n ≥ 1 and  ∈ ξi , we have P(τi+1 − τi > n | ) ≤ Ce−c(n+1) . a

(3.127)

One can easily see that the multiplying constant can actually be taken any upper bound a a for the sequence e−c(n−n 0 ) /e−c(n+1) . Now, consider an arbitrary J ⊂ {1, . . . , q(n)} with its elements ordered as i 1 < · · · < i  . We have  P





⎛

{τi − τi−1 = n i } ≤ P ⎝

i∈J

  

τi j − τi j −1

⎞  > ni j − 1 ⎠

j=1    &  = P τi1 − τi1 −1 > n i1 − 1 P τi j − τi j −1 > n i j − 1 | A j , (3.128) j=2

  j−1  where A j = s=1 τis − τis −1 > n is − 1 , for all 2 ≤ j ≤ . Now, we estimate the factors in (3.128). Using (3.127), we deduce      P τi1 − τi1 −1 > n i1 − 1 ∩  P τi1 − τi1 −1 > n i1 − 1 = ∈ξi1 −1

=



P



  τi1 − τi1 −1 > n i1 − 1 |  P()

∈ξi1 −1

≤ Ce−cni1 . a

(3.129)

3.3 Tower Maps

75

On the other hand, since each A j is ξi j −1 -measurable, it follows from (3.55) and (3.127) that for all 2 ≤ j ≤  −cn ia

j

,

−cn ia

j

.

P(τi j − τi j −1 > n i j − 1 | A j ) ≤ Ce which together with (3.128) and (3.129) gives P

 

 {τi − τi−1 = n i } ≤

i∈J

 &

Ce

(3.130)

j=1

By (3.130) and Lemma 3.47, we have ⎛ ⎜ P⎝

⎞ 



J ⊂{1,...,q(n)}

n i ≥N i∈J n i ≥n/2



 &



⎟ {τi − τi−1 = n i }⎠ ≤

J ⊂{1,...,q(n)}

n i ≥N i∈J n i ≥n/2

i∈J

≤

Ce

=0

≤C

n 1 ,...,n  ≥N

n i ≥n/2

 q(n)  q(n) =0

= C2

q(n)



j

j=1

 & q(n)   q(n) 

−cn ia

Ce−cni

a

i=1

e−cp

a

p≥n/2

e

−cpa

.

(3.131)

p≥n/2

Recalling that q(n) = [n a ], for  > 0 sufficiently small (just depending on a and c), we easily find C0 , c > 0 so that 2q(n)



 a

e−cp ≤ C0 e−c n , a

(3.132)

p≥n/2

for all n ≥ 1. It follows from (3.124), (3.126), (3.131) and (3.132) that  a

P{S > n} ≤ (1 − ε0 )q(n) + CC0 e−c n . Take C  = CC0 .



Recalling what was said at the beginning of Sect. 3.3.2.1, the (stretched) exponential case of Theorem 3.28 is now a consequence Proposition 3.48 together with Propositions 3.32 and 3.33; recall (3.122). It is worth noting that the term a 3.48 arises in (3.126) as an upper (1 − ε0 )[n ] in the conclusion of Proposition  bound for P{κ > q(n)} = P S > τq(n) . Moreover, the constant ε0 provided by Proposition 3.32 has lower bounded dependence on Cϕ+ , Cϕ+ , and this dependence can be removed for i = q(n) large enough.

76

3 Expanding Structures

3.3.3 Decay of Correlations Here, we use Theorem 3.28 to obtain estimates on the decay of correlations in terms of the tail of the recurrence times for tower maps. Let T :  →  be the tower map of a Gibbs-Markov map F : 0 → 0 with recurrence time R ∈ L 1 (m). Theorem 3.24 gives a unique ergodic T -invariant probability measure ν m. Furthermore, ν is equivalent to m and its density is bounded above and below by positive constants. Given ϕ ∈ L ∞ (m) with ϕ = 0, set ϕ∗ =

1 (ϕ + 2 ϕ ∞ ). (ϕ + 2 ϕ ∞ )dν

(3.133)

Note that ϕ ∗ is strictly positive and its integral with respect to ν is equal to one. Lemma 3.49 Let T :  →  be the tower map associated with a Gibbs-Markov map with recurrence time R ∈ L 1 (m) and ν m the unique ergodic T -invariant probability measure. Given ϕ ∈ L ∞ (m) with ϕ = 0 and λ the probability measure on  such that dλ/dν = ϕ ∗ , then 1. 1/3 ≤ ϕ ∗ ≤ 3; 2. if ϕ ∈ Fβ (), then ϕ ∗ ∈ Fβ+ (); 3. Cor ν (ϕ, ψ ◦ T n ) ≤ 3 ϕ ∞ ψ ∞ |T∗n λ − ν|, for all ψ ∈ L ∞ (m). Proof Take ϕ ∈ L ∞ (m) such that ϕ = 0. We have

ϕ ∞ ≤ ϕ + 2 ϕ ∞ ≤ 3 ϕ ∞ .

(3.134)

Since ν is a probability measure, we get 1 1 1 ≤ ≤

. 3 ϕ ∞

ϕ ∞ (ϕ + 2 ϕ ∞ )dν

(3.135)

It follows from (3.134) and (3.135) that 1/3 ≤ ϕ ∗ ≤ 3. Assume now that ϕ ∈ Fβ (). For all x, y ∈ , we have ϕ(x) − ϕ(y) 1 ϕ ∗ (x) − ϕ ∗ (y) · =

. β s(x,y) β s(x,y) (ϕ + 2 ϕ ∞ )dν

(3.136)

This implies that ϕ ∗ ∈ Fβ () and Cϕ ∗ ≤ Cϕ / ϕ ∞ . Now take ψ ∈ L ∞ (m). Setting a = (ϕ + 2 ϕ ∞ )dν, we write

3.3 Tower Maps

77

   Cor μ (ϕ, ψ ◦ T n ) = ϕ(ψ ◦ T n )dν − ϕdν ψdν    = (ϕ + 2 ϕ ∞ )(ψ ◦ T n ) dν − (ϕ + 2 ϕ ∞ )dν ψ dν    = a ϕ ∗ (ψ ◦ T n ) dν − ϕ ∗ dν ψ dν   = a (ψ ◦ T n ) dλ − ψ dν   = a ψ dT∗n λ − ψ dν ≤ a ψ ∞ |T∗n λ − ν|. 

Since (3.134) gives a ≤ 3 ϕ ∞ , the conclusion follows.

Now, we present a simple result that we state in a general form, suitable for being applied in other situations. Lemma 3.50 Let T1 : M1 → M1 and T2 : M2 → M2 preserve measures μ1 and μ2 , respectively. Given a semiconjugacy S between T1 and T2 such that S∗ μ1 = μ2 , we have Cor μ2 (ϕ, ψ ◦ T2n ) = Cor μ1 (ϕ ◦ S, ψ ◦ S ◦ T1n ), for all ϕ, ψ : M → R such that the expressions make sense. Proof Since S∗ μ1 = μ2 , we have 



 ϕd S∗ μ1 ψd S∗ μ1    = (ϕ ◦ S)(ψ ◦ T2n ◦ S)dμ1 − ϕ ◦ Sdμ1 ψ ◦ Sdμ1    = (ϕ ◦ S)(ψ ◦ S ◦ T1n )dμ1 − ϕ ◦ Sdμ1 ψ ◦ Sdμ1

Cor μ2 (ϕ, ψ ◦ T2n ) =

ϕ(ψ ◦ T2n )d S∗ μ1 −

= Cor μ1 (ϕ ◦ S, ψ ◦ S ◦ T1n ), 

and so we have proved the result.

Now, we introduce a natural partition for towers whose recurrence times have a nontrivial common divisor. This partition will be introduced for general towers with no specific properties in the base map and will also be considered for the towers in Chap. 4. Let T :  →  be the tower map of F : 0 → 0 , with recurrence time R. Set q = gcd(R) and, for each 1 ≤ i ≤ q, ϒi =

 ≡i−1(mod q)

 .

(3.137)

78

3 Expanding Structures

We have that {ϒ1 , . . . , ϒq } is a partition of . Moreover, T (ϒ1 ) = ϒ2 , . . . , T (ϒq−1 ) = ϒq and T (ϒq ) = ϒ1 . Consider a tower map T  :  →  with the same base map F, now with recurrence time R  = R/q. Note that gcd(R  ) = 1. It is not difficult to see that Si (x, ) = (x, q + i − 1)

(3.138)

defines a map Si :  → ϒi , which is a measurable conjugacy between T  and T q |ϒi , for all 1 ≤ i ≤ q. Assume now that F : 0 → 0 is a Gibbs-Markov map. We claim that (3.139) ϕ ∈ Fβ () =⇒ ϕ ◦ Si ∈ Fβ ( ). It is enough to show that Cϕ◦Si ≤ Cϕ .

(3.140)

To see this, recall that the separation time on a tower is defined only in terms of the associated base map. Since F is the base map for the two towers T and T  , it follows that s(x, y) = s(Si (x), Si (y)) for all x, y ∈  . This clearly implies (3.140). There are situations in which we can ensure q = 1. In such cases, the T q -invariant measures given by the next theorem are all coincident with the T -invariant measure and therefore, the correlation decay is obtained with respect to this measure. Theorem 3.51 Let T :  →  be the tower map of a Gibbs-Markov map with recurrence time R ∈ L 1 (m) and ν the unique ergodic T -invariant probability measure such that ν m. If gcd(R) = q, then there are exact T q -invariant probability measures ν1 , . . . , νq such that T∗ ν1 = ν2 ,…,T∗ νq = ν1 and ν = (ν1 + · · · + νq )/q. Moreover, for all 1 ≤ i ≤ q, 1. if m{R > n} ≤ Cn −α for some C > 0 and α > 1, then for all ϕ ∈ Fβ () and ψ ∈ L ∞ (m), there is C  > 0 such that Cor νi (ϕ, ψ ◦ T qn ) ≤ C  n −a+1 ; 2. if m{R > n} ≤ Ce−cn for some C, c > 0 and 0 < a ≤ 1, given 0 < β < 1 there is c > 0 such that, for all ϕ ∈ Fβ () and ψ ∈ L ∞ (m), there is C  > 0 such that a

 a

Cor νi (ϕ, ψ ◦ T qn ) ≤ C  e−c n . Proof First, we prove the existence of the measures ν1 , . . . , νq . Consider the partition {ϒ1 , . . . , ϒq } of  as in (3.137), for which T (ϒ1 ) = ϒ2 , . . . , T (ϒq−1 ) = ϒq and T (ϒq ) = ϒ1 .

(3.141)

3.3 Tower Maps

79

Since ν is a T -invariant measure, it easily follows that ν|ϒi is an invariant measure for T q : ϒi → ϒi , for each 1 ≤ i ≤ q. Setting νi =

1 (ν|ϒi ), ν(ϒi )

(3.142)

we clearly have ν = (ν1 + · · · + νq )/q. And it follows from (3.141) that T∗ ν1 = ν2 , . . . , T∗ νq−1 = νq and T∗ νq = ν1 . Also, since ν m, we have νi m as well, for all 1 ≤ i ≤ q.  Let T  :  →  be the tower map of f R = ( f q ) R with recurrence time R  = R/q. We have R  ∈ L 1 ( ) and gcd(R  ) = 1. Note that T  can be seen as the tower map of an induced map Gibbs-Markov map for f q with recurrence time R  . By Theorem 3.24, T  has a unique invariant probability measure ν  m. Now, fix 1 ≤ i ≤ q and consider Si :  → ϒi the measurable conjugacy between T  and T q |ϒi given in (3.138). It follows that Si ∗ ν  is an exact T q -invariant probability measure. Moreover, since the measurable maps Si and Si −1 clearly preserve sets of m measure 0, it easily follows that Si ∗ ν  is the unique probability measure invariant under T q |ϒi : ϒi → ϒi and absolutely continuous with respect to m. Since νi also has this property, we must have Si ∗ ν  = νi . Let us finally prove the conclusions on the decay of correlations for νi . Since Si :  → ϒi is a measurable conjugacy between T  and T q |ϒi , it follows from Lemma 3.50 that Cor νi (ϕ, ψ ◦ T qn ) = Cor ν  (ϕ ◦ Si , ψ ◦ Si ◦ T  ). n

Observe that ϕ ◦ Si ∈ Fβ ( ), by (3.139). Assuming without loss of generality that ϕ = 0, it follows from Lemma 3.49, that Cor ν  (ϕ ◦ Si , ψ ◦ Si ◦ T  ) ≤ 3 ϕ ∞ ψ ∞ |T∗ λ − ν  |, n

n

where λ is the probability measure on  such that dλ/dν = (ϕ ◦ Si )∗ . The final conclusions follow from Theorem 3.28.  Remark 3.52 The proof of Theorem 3.51 gives that each νi is the normalisation of the restriction of ν to the union of the levels  with  ≡ i − 1(mod q), where ν the unique ergodic T -invariant probability measure such that ν m; recall (3.137) and (3.142). In particular, each measure νi is equivalent to m in the union of those levels, by Theorem 3.24.

80

3 Expanding Structures

3.4 Lifting Observables The main purpose of this section is to deduce some results on the decay of correlations with respect to the measure obtained in Corollary 3.21 (or Proposition 3.27) for systems with induced maps. Using Theorem 3.51 and the fact that the tower map is an extension of the original dynamics, we immediately have that for observables that lift to the appropriate space of observables in the tower. In order to make the space of observables independent of the tower, we need additional properties on the ambient space and the induced map. Assume that f : M → M is defined on a metric space M with a metric d and m is a reference measure on M, not necessarily a Borel measure. Let f R : 0 → 0 be an induced map and P be the mod 0 partition of 0 associated with f R . We say that the induced map f R is expanding if there are C > 0 and 0 < β < 1 such that, for all ω ∈ P and x, y ∈ ω, • d( f R (y), f R (x)) ≤ Cβ s(x,y) ; • d( f j (x), f j (y)) ≤ Cd( f R (x), f R (y)), for all 0 ≤ j ≤ R. This is actually a weaker version of the expanding condition (Y3 ) that we consider in Sect. 4.1. The reason this is sufficient in the current context is because we use it in conjunction with the fact that f R a Gibbs-Markov map. Given η > 0, we say that a function ϕ : M → R is η-Hölder continuous if |ϕ|η ≡ sup x= y

|ϕ(x) − ϕ(y)| < ∞. d(x, y)η

Set Hη = {ϕ : M → R | ϕ is η-H¨older continuous}.

(3.143)

Recall that π is the semiconjugacy between T and f introduced in (3.39). Lemma 3.53 If T :  →  is the tower map associated with a Gibbs-Markov expanding induced map f R : 0 → 0 , then ϕ ∈ Hη =⇒ ϕ ◦ π ∈ Fβ η (). Proof We need to show that there is K > 0 such that, for all x, y ∈ , |ϕ(π(x)) − ϕ(π(y))| ≤ K β ηs(x,y) ,

(3.144)

If s(x, y) = 0 (in particular, if x, y belong in distinct levels of ), then (3.144) is an easy consequence of the fact that ϕ is bounded. Consider now  ≥ 0 and x, y ∈  with s(x, y) ≥ 1. By definition, there are ω ∈ P and x0 , y0 ∈ ω such that s(x0 , y0 ) = s(x, y) ≥ 1. This gives in particular R(x0 ), R(y0 ) > . Since ϕ ∈ Hη , then

3.4 Lifting Observables

81

|ϕ(π(x)) − ϕ(π(y))| = |ϕ(π(x0 , )) − ϕ(π(y0 , ))| = |ϕ( f  (x0 )) − ϕ( f  (y0 ))| ≤ |ϕ|η d( f  (x0 ), f  (y0 ))η .

(3.145)

Since we assume f R a Gibbs-Markov expanding induced map, there is C > 0 such that d( f  (x0 ), f  (y0 )) ≤ Cd( f R (x0 ), f R (y0 )) ≤ C 2 β s( f

R

(x0 ), f R (y0 ))

= C 2 β s(x0 ,y0 )−1 = C 2 β s(x,y)−1 .

(3.146)

It follows from (3.145) and (3.146) that |ϕ(π(x)) − ϕ(π(y))| ≤ |ϕ|η C 2η β −η β ηs(x,y) . This yields (3.144), with K = |ϕ|η C 2η β −η .



In the next result, we obtain the aimed conclusion on the decay of correlations for transformations admitting induced Gibbs-Markov maps, with the decay rates depending on the tail of recurrence times. The existence of a unique f -invariant probability measure μ in the conditions of the next result was obtained in Corollary 3.21. Theorem 3.54 Let M be a metric space with a reference measure m, f : M → M a measurable map with f ∗ m m and f R : 0 → 0 a Gibbs-Markov expanding induced map such that R ∈ L 1 (m). If gcd(R) = q and μ is the unique ergodic f invariant probability measure μ m such that μ(0 ) > 0, then f q has p ≤ q exact probability measures μ1 , . . . , μ p with f ∗ μ1 = μ2 ,…, f ∗ μ p = μ1 and μ = (μ1 + · · · + μ p )/ p such that, for all 1 ≤ i ≤ p, 1. if m{R > n} ≤ Cn −a for some C > 0 and a > 1, then for all ϕ ∈ Hη and ψ ∈ L ∞ (m) there exists C  > 0 such that Cor μi (ϕ, ψ ◦ f qn ) ≤ C  n −a+1 ; 2. if m{R > n} ≤ Ce−cn for some C, c > 0 and a > 1, then given η > 0, there is c > 0 such that, for all ϕ ∈ Hη and ψ ∈ L ∞ (m), there is C  > 0 for which a

 a

Cor μi (ϕ, ψ ◦ f qn ) ≤ C  e−c n . Proof Let T :  →  be the tower map associated with f R : 0 → 0 and ν its unique invariant probability measure such that ν m; recall Theorem 3.24. By Proposition 3.27, we have μ = π∗ ν, with π :  → M as in (3.39). Set q = gcd(R). It follows from Theorem 3.51 that there are exact T q -invariant probability measures ν1 , . . . , νq with ν = (ν1 + · · · + νq )/q, such that T∗ ν1 = ν2 , . . . , T∗ νq−1 = νq

82

3 Expanding Structures

and T∗ νq = ν1 . Set μi = π∗ νi , for each 1 ≤ i ≤ q. Recall that π ◦ T = f ◦ π , by Lemma 3.26. Therefore, f ∗ μi = f ∗ π∗ νi = π∗ T∗ νi = π∗ ν j+1 = μ j+1 ,

j = i (mod q).

q

This, in particular, gives f ∗ μi = μi , for all 1 ≤ i ≤ q. Also, since each νi is exact with respect to T q , it easily follows that each μi is exact with respect to f q . Notice that as π is not injective, we may have f ∗ μ p = μ1 , for some p < q (with p necessarily dividing q). Consider the smallest p in these conditions. Taking k ∈ N such that q = pk, we have  μ = π∗

 1 1 k 1 (ν1 + · · · + νq ) = (μ1 + · · · + μq) = (μ1 + · · · + μ p ) = (μ1 + · · · + μ p ). q q q p

Now, we prove the conclusions on the decay of correlations. Since π ◦ T = f ◦ π , we also have π ◦ T q = f q ◦ π . Recalling that μi = π∗ νi , for all 1 ≤ i ≤ q, it follows from Lemma 3.50 that Cor μi (ϕ, ψ ◦ f qn ) = Cor νi (ϕ ◦ π, ψ ◦ π ◦ T qn ). Finally, taking ϕ ∈ Hη , Lemma 3.53 provides 0 < β < 1 such that ϕ ◦ π ∈ Fβ (). Using Theorem 3.51, we get the conclusions.  Remark 3.55 The proof of Theorem 3.54 provides additional useful information on the measures μ1 , . . . , μ p . It actually shows that μi = π∗ νi , for all 1 ≤ i ≤ p, where π is the map in (3.39), each νi is an exact T d -invariant probability measure as in Theorem 3.51, and T :  →  is the tower map associated with f R : 0 → 0 . In the next result, we show that if the measure μ in the previous result is ergodic for all powers of f , then we have q = 1 (and thus p = 1). Corollary 3.56 Let M be a metric space with a reference measure m, f : M → M a measurable map with f ∗ m m and f R : 0 → 0 a Gibbs-Markov expanding induced map such that R ∈ L 1 (m). If the unique ergodic f -invariant probability measure μ m such that μ(0 ) > 0 is ergodic for all f n with n ≥ 1, then 1. if m{R > n} ≤ Cn −a for some C > 0 and a > 1, then for all ϕ ∈ Hη and ψ ∈ L ∞ (m) there exists C  > 0 such that Cor μ (ϕ, ψ ◦ f n ) ≤ C  n −a+1 ; 2. if m{R > n} ≤ Ce−cn for some C, c > 0 and a > 1, then given η > 0, there is c > 0 such that, for all ϕ ∈ Hη and ψ ∈ L ∞ (m), there is C  > 0 for which a

 a

Cor μ (ϕ, ψ ◦ f n ) ≤ C  e−c n .

3.4 Lifting Observables

83

Proof Consider 1 ≤ p ≤ q and probability measures μ and μ1 , . . . , μ p as in Theorem 3.54. In particular, μi μ, for all 1 ≤ i ≤ p. Since we assume μ ergodic for f q , it follows from Lemma 3.11 that μi = μ, for all 1 ≤ i ≤ p. The expected conclusions for Cor μ (ϕ, ψ ◦ f qn ) then follow from Theorem 3.54. Applying those conclusions to the observables ψ ◦ f, · · · , ψ ◦ f q−1 in place of ψ, we finish the proof. Let f R be an induced map for f and P the mod 0 partition associated with f R . We say that f R is an open induced map if f  (ω) is an open set for all 0 ≤  < R|ω and ω ∈ P. For the next result, we need to assume that the reference measure m is a Borel measure. We use f ∗ m for the usual push-forward of m and f ∗ m as in (2.14). Corollary 3.57 Let M be a metric space with a Borel reference measure m whose support coincides with M, f : M → M a continuous map such that f ∗ m m and f ∗ m m, and f R : 0 → 0 a Gibbs-Markov expanding open induced map such that R ∈ L 1 (m). If f is topologically mixing and μ is the unique ergodic f -invariant probability measure μ m with μ(0 ) > 0, then 1. if m{R > n} ≤ Cn −a for some C > 0 and a > 1, then for all ϕ ∈ Hη and ψ ∈ L ∞ (m) there exists C  > 0 such that Cor μ (ϕ, ψ ◦ f n ) ≤ C  n −a+1 ; 2. if m{R > n} ≤ Ce−cn for some C, c > 0 and a > 1, then given η > 0, there is c > 0 such that, for all ϕ ∈ Hη and ψ ∈ L ∞ (m), there is C  > 0 for which a

 a

Cor μ (ϕ, ψ ◦ f n ) ≤ C  e−c n . Proof Consider 1 ≤ p ≤ q and probability measures μ and μ1 , . . . , μ p as in Theorem 3.54. Take an arbitrary 1 ≤ i ≤ p. Observe that μi μ m. According to Remark 3.55, we have μi = π∗ νi , where π is the map in (3.39) and νi is an exact T q invariant probability measure as in Theorem 3.51. By Remark 3.52, the probability measure νi is equivalent to m on the levels  with  ≡ i − 1 (mod q). Each of these levels  contains some element ω of the partition associated with the Gibbs-Markov map f R with  < R|ω . Since we assume f R an open induced map, then Ui = f  (ω) is an open set. We claim that μi |Ui is equivalent to m|Ui . In fact, since π |ω = f  |ω , we have μi ≥ ( f  |ω )∗ νi . Thus, if μi (A) = 0 for some A ⊂ Ui , then 0 = ( f  |ω )∗ νi (A) = νi ( f − (A) ∩ ω). Since νi |ω is equivalent to m|ω , we have m( f − (A) ∩ ω) = 0. Using that f ∗ m m, we obtain m(A) = 0, and therefore μi |Ui is equivalent to m|Ui . Since this holds for all 1 ≤ i ≤ p and f topologically mixing implies f q transitive, it follows from Corollary 2.15 that the measures μ1 , . . . , μ p are all equal. Necessarily, they are all equal to μ. Now, Theorem 3.54 gives the expected conclusions for Cor μ (ϕ, ψ ◦ f qn ).

84

3 Expanding Structures

Applying it to the observables ψ ◦ f, · · · , ψ ◦ f q−1 in place of ψ, we finish the proof.  Note that we have used the fact that f is topologically mixing only to deduce that the powers of f are all transitive.

3.5 Application: Intermittent Maps In this section, we exhibit two examples of transformations with indifferent fixed points, associated with phase transitions from stable periodic behaviour to a chaotic one, related by Pomeau and Manneville in [13] to an intermittent transition to turbulence in convective fluids. We consider a simpler first model on the interval, introduced by Liverani, Saussol and Vaienti in [9], where it is relatively easy to define a Gibbs-Markov induced map, an then deduce some interesting statistical properties of the original system. In the second example, introduced by Young in [19], the construction of a Gibbs-Markov induced map is no longer so straightforward. However, it has the advantage of allowing us to introduce a higher dimensional system with contracting directions in Sect. 4.6. The strategy we use to study the two examples is that of [19]. In both cases, we see that when the order of the tangency at the neutral fixed point is high, the Dirac measure at the fixed point is the unique physical measure of the dynamical system. In the proof of this last part, we use ideas from [17]. For more results on one-dimensional maps with neutral fixed points, see [8, 16]. See also [5] for a skew product map on the square. Here, we use m to denote the Lebesgue (length) measure on the interval or the circle. In this context, by an SRB measure we mean an invariant probability measure which is absolutely continuous with respect to m. Recall that an ergodic SRB measure is a physical measure, by Corollary 2.16.

3.5.1 Neutral Fixed Point Before presenting the two maps in the interval and in the circle, we get some information about the dynamics near a neutral fixed point that will be useful in both cases. Given real functions (or sequences) f (x) and g(x), we use f (x)  g(x) to say that there is some uniform constant C > 0 such that f (x) ≤ Cg(x) for all x. Also, f (x) ≈ g(x) means that both f (x)  g(x) and g(x)  f (x). Given z 0 > 0 and α > 0, let I = [0, z 0 ] and f : I → R be such that (i1 ) f (0) = 0 and f  (0) = 1; (i2 ) f  > 1 on I \ {0}; (i3 ) f is C 2 on I \ {0} and f  (x) ≈ x α−1 .

3.5 Application: Intermittent Maps

85

Note that f is a C 1+α map for each 0 < α < 1. Actually, it follows from (i1 ) and (i3 ) that (3.147) f  (x) − 1 ≈ x α and f (x) − x ≈ x α+1 . Let (z n )n be the sequence in I defined recursively for n ≥ 0 as f (z n+1 ) = z n . Set Jn = (z n , z n−1 ), for each n ≥ 1. We claim that (z n )n has the same asymptotics of the sequence (1/(n + n 0 )1/α )n , for some (hence all) n 0 ≥ 0. First of all, note that 1 1 1 − ≈ = 1/α 1/α (n + n 0 ) (n + n 0 + 1) (n + n 0 )1/α+1



1 (n + n 0 )1/α

α+1

.

On the other hand, using (3.147), we obtain α+1 . m(Jn ) = z n−1 − z n = z n−1 − f (z n−1 ) ≈ z n−1

It follows that ' zn ∈

1 1 , (n + n 0 + 1)1/α (n + n 0 )1/α

(

 =⇒ m(Jn ) ≈

1 (n + n 0 )1/α

α+1

.

This means that there is some uniform bound on the number of intervals Jn meeting an interval of the form [1/(n + n 0 + 1)1/α , 1/(n + n 0 )1/α ], and vice versa. Thus, z n ≈ n −1/α and m(Jn ) ≈ n −(α+1)/α .

(3.148)

Lemma 3.58 There is C > 0 such that, for all n ≥ 1 and x, y ∈ Jn , log

( f n ) (x) ≤ C f n (x) − f n (y) . n  ( f ) (y)

Proof Consider k ≥ 1 and x, y ∈ Jk . The mean value theorem provides u between x and y for which  log f  (x) − log f  (y) ≤ f (u) |x − y| ≈ u α−1 |x − y| ≤ z α−1 |x − y|. (3.149) k−1 f  (u)

Together with (3.148), this yields ∞ k=1

α−1 z k−1 m

(Jk ) 

∞  (α−1)/α  (α+1)/α 1 1 k=1

k

k



∞ 1 . 2 k k=1

(3.150)

86

3 Expanding Structures

It follows from (3.149) and (3.150) that, for all 1 ≤  ≤ n and x, y ∈ J , −1 ( f  ) (x) log   | log f  ( f k (x)) − log f  ( f k (y))| ≤ ( f ) (y) k=0



−1

α−1 z −k−1 | f k (x) − f k (y)|

k=0

≤

−1

α−1 z −k−1 m (J−k )

k=0 ∞ 1  . k2 k=1

(3.151)

This conclusion is still weaker than necessary, but we will use it to obtain the stronger version stated. Using the mean value theorem, for all 0 ≤ j < n and x, y ∈ Jn− j , we may find u, v ∈ Jn− j such that | f n (x) − f n (y)| = ( f n− j ) (u)| f j (x) − f j (y)| and

  m ( f (J1 )) = ( f n− j ) (v) m Jn− j ,

which together with (3.151) applied to  = n − j and u, v ∈ Jn− j gives | f j (x) − f j (y)| | f n (x) − f n (y)|   . ≈ m ( f (J0 )) m Jn− j

(3.152)

It follows from (3.149) and (3.152) that, for all n ≥ 1 and x, y ∈ Jn , log

n−1 ( f n ) (x) ≤ | log f  ( f j (x)) − log f  ( f j (y))| ( f n ) (y) j=0



n−1

α−1 j j z n− j−1 | f (x) − f (y)|

j=0



n−1

 | f n (x) − f n (y)|  α−1 . z n− m J n− j j−1 m ( f (J0 )) j=0

Using (3.150), we finish the proof.



3.5 Application: Intermittent Maps

87

3.5.2 Interval Map Given α > 0, consider I = [0, 1] and the map f : I → I , defined as  f (x) =

x + 2α x α+1 , if 0 ≤ x ≤ 1/2; 2x − 1, if 1/2 < x ≤ 1.

(3.153)

It is worth noting that these expressions are taken for simplicity. Actually, we could take the first branch defined on an interval [0, z 0 ] by a map f satisfying (i1 )–(i3 ) as in Sect. 3.5.1 with f (z 0 ) = 1. For the second branch, any map with strictly greater than one Hölder continuous derivative on [z 0 , 1], mapping [z 0 , 1] diffeomorphically to [0, 1], would be fine as well. Theorem 3.59 Let f : I → I be defined as in (3.153). Then 1. for α < 1, the map f has a unique SRB measure μ. Moreover, μ is exact, the support of μ coincides with I , its basin covers m almost all of I and, for each Hölder continuous ϕ : I → R and ψ ∈ L ∞ (m), Cor μ (ϕ, ψ ◦ f n )  1/n 1/α−1 ; 2. for α ≥ 1, the Dirac measure at 0 is a physical measure for f and its basin covers m almost all of I . The proof of Theorem 3.59 will be given in the rest of this subsection. It is easy to check that the restriction of f to [0, 1/2] satisfies conditions (i1 )–(i3 ) Sect. 3.5.1, with z 0 = 1/2. We are going to build an induced map for f defined on the whole interval I . Consider sequences (z n )n≥0 and (Jn )n≥1 as before. Set J0 = (1/2, 1) and R| Jn = n + 1, for all n ≥ 0.

(3.154)

Since f n+1 (Jn ) = (0, 1) for each n ≥ 0, we have an induced map f R : I → I . Moreover, by (3.148), we have m{R > n} ≈

k≥n

m(Jk ) ≈

 1 1+1/α k≥n

k

≈ n −1/α .

(3.155)

We are going to see that f R is Gibbs-Markov with respect to the m mod 0 partition of I P = {J0 , J1 , . . . }. Lemma 3.60 f R : I → I is a Gibbs-Markov map and gcd(R) = 1.

88

3 Expanding Structures

Proof The full branch property (G5 ) is obvious in this case. Moreover, since f R restricted to each interval in the partition P is a diffeomorphism onto its image, the nonsingular property (G3 ) for f R with respect to m holds and J f R = ( f R ) . Let us verify the separability property (G2 ). We claim that ( f R ) | Jn ≥ 2, for all n ≥ 0.

(3.156)

This is obvious for n =+0, since R| J0 = 1 and f  | J0 = 2. Take now n ≥ 1. By (3.154), n f  ( f i (x)), for each x ∈ Jn . In addition, f  ( f i (x)) > 1 we have ( f R ) (x) = i=0 n for all 0 ≤ i < n. Since f (x) ∈ J0 , then f  ( f n (x)) = 2, and therefore, we have the claim. The separability property is then a consequence of (3.156) and Lemma 3.3. Now, we prove the Gibbs property (G4 ). Given n ≥ 0 and x, y ∈ Jn , using (3.154) and the fact that f  ( f n (x)) = 2 = f  ( f n (y)), we get log

( f n ) (x) f  ( f n (x)) ( f n ) (x) ( f R ) (x) = log + log = log . ( f R ) (y) ( f n ) (y) f  ( f n (y)) ( f n ) (y)

(3.157)

In addition, Lemma 3.58, provides a uniform constant C > 0 for which log

( f n ) (x) ≤ C| f n (x) − f n (y)| ( f n ) (y) C ≤ | f n+1 (x) − f n+1 (y)| 2 C R = | f (x) − f R (y)|. 2

(3.158)

Using (3.157), (3.158) and Lemma 3.3, we deduce the Gibbs property for f R . Finally,  since R| J0 = 1, we obviously have gcd(R) = 1. Note that so far everything works for any value of α > 0. Let us prove the first item of Theorem 3.62. Assume that 0 < α < 1. In this case, (3.155) implies that R ∈ L 1 (m). Therefore, it follows from Lemma 3.60 and Corollary 3.21 that f has a unique SRB μ. Recall that the domain of f R is the whole interval I . By Corollary 3.21, we also have that μ is equivalent to m. This in particular implies that the support of μ coincides with I and its basin covers m almost all of I , by Proposition 2.12. Since gcd(R) = 1, Theorem 3.54 gives the exactness of μ and the conclusion on the decay of correlations. The second item of Theorem 3.59 will be derived from the next result that we present in an abstract setting, suitable for application in more situations. Proposition 3.61 Let M be a compact metric space and m a finite measure on the Borel sets of M. Assume that f : M → M has an induced map f R : M → M such that 1. there is an ergodic f R -invariant probability measure ν which is equivalent to m and R ∈ / L 1 (ν);

3.5 Application: Intermittent Maps

89

2. there are x0 ∈  M with f (x0 ) = x0 and neighbourhoods M = U0 ⊃ U1 ⊃ · · · of x0 such that n≥0 Un = {x0 } and, for all n ≥ 0 and An = Un \ Un+1 , f (An+1 ) ⊂ An and R| An = n + 1. Then, for m almost every x ∈ M and every continuous function ϕ : M → R, n−1 1 ϕ( f j (x)) = ϕ(x0 ). n→∞ n j=0

lim

Proof We claim that it is enough to prove that, for m almost all x ∈ M and all k ≥ 1,  1  # 0 ≤ j < n : f j (x) ∈ M \ Uk → 0, when n → ∞. n

(3.159)

In fact, consider any continuous function ϕ : M → R. Given an arbitrary ε > 0, choose k ≥ 1 sufficiently large so that x ∈ Uk =⇒ ϕ(x) ∈ (ϕ(x0 ) − ε, ϕ(x0 ) + ε). Given any x ∈ M and n ≥ 1, set     and Jkn (x) = 0 ≤ j < n : f j (x) ∈ / Uk . Ikn (x) = 0 ≤ j < n : f j (x) ∈ Uk We may write n−1

ϕ( f j (x)) =

j=0



ϕ( f j (x)) +

j∈Ikn



ϕ( f j (x))

j∈Jkn

Note that #Ikn (x)(ϕ(x0 ) − ε)
0 is arbitrary, it follows from the last four displayed formulas that, for m almost all x ∈ M

90

3 Expanding Structures n−1 1 ϕ( f j (x)) = ϕ(x0 ). n→∞ n j=0

lim

This proves the claim. Therefore, let us show that, for m almost all x ∈ M and all k ≥ 1, we have (3.159). Since ν is an ergodic f R -invariant probability measure which is equivalent to m, by Birkhoff Ergodic Theorem, we have lim

n→∞

 1  # 0 ≤ j ≤ n : ( f R ) j (x) ∈ A = ν(A ), n

(3.160)

for all  ≥ 0 and m almost all x ∈ M. Take a point x ∈ M in these conditions. Set for each  ≥ 0 and i ≥ 0   Ni () = # 0 ≤ j ≤ i : ( f R ) j (x) ∈ A . Consider functions 0 = τ0 < τ1 < τ2 < · · · , defined inductively for i ≥ 1 by τi = τi−1 + R ◦ f τi−1 (x).

(3.161)

Note that f τi (x) = ( f R )i (x), for each i ≥ 1. Given n ≥ 1, take an integer s ≥ 0 such that τs ≤ n < τs+1 . Since we assume R| A =  + 1, corresponding to each 0 ≤ j ≤ i for which ( f R ) j (x) ∈ A , there are  + 1 iterations under f for the point x. Therefore τs =



( + 1)Ns ().

(3.162)

≥0

Given any 0 ≤ i ≤ s, consider  ≥ 0 such that f τi (x) ∈ A . If 0 ≤  < k, then for all j ∈ [τi , τi+1 ), f j (x) ∈ A−( j−τi ) = U−( j−τi ) \ U−( j−τi )+1 ⊂ M \ U−( j−τi )+1 ⊂ M \ Uk . Moreover, τi+1 − τi = R| A =  + 1 ≤ k. If, on the other hand,  ≥ k, since R| A =  + 1 ≥ k + 1 = R| Ak , then we have exactly k values j ∈ [τi , τi+1 ) for which f j (x) ∈ M \ Uk . In particular, there are at most k values j ∈ [τi , n) for which f j (x) ∈ M \ Uk . Altogether, this means that   Ns () + k. # 0 ≤ j < n : f j (x) ∈ M \ Uk ≤ k ≥0

Since

≥0

Ns ()/s = 1, using (3.162), we get

3.5 Application: Intermittent Maps

91

 k ≥0 Ns () + k 1  j

# 0 ≤ j < n : f (x) ∈ M \ Uk ≤ n n − τs + ≥0 ( + 1)Ns ()

k ≥0 Ns ()/s + k/s ≤ ≥0 ( + 1)Ns ()/s 2k . ≤ ≥0 ( + 1)Ns ()/s

By (3.160), the denominator converges to ≥0 ( + 1)ν(A ) = Rdν = ∞, when s → ∞. Observing that n → ∞ implies s → ∞, we have proved (3.159).  Finally, we prove the second item of Theorem 3.59. Assume that α ≥ 1. From (3.155), we get R ∈ / L 1 (m). Note that f R is a Gibbs-Markov map, regardless of the value of α > 0. It follows from Theorem 3.13 that f R has a unique ergodic SRB measure ν which is in fact equivalent to m. Moreover, dν/dm is bounded from above and below by positive constants. This implies R ∈ / L 1 (ν). Applying Proposition 3.61 with x0 = 0 and Un = [0, z n−1 ], for each n ≥ 1, we obtain the second item of Theorem 3.59.

3.5.3 Circle Map Given α > 0, let f : S 1 → S 1 be a degree d ≥ 2 map of the circle S 1 = R/Z so that (c1 ) f (0) = 0 and f  (0) = 1; (c2 ) f  > 1 on S 1 \ {0}; (c3 ) f is C 2 on S 1 \ {0} and x f  (x) ≈ |x|α , for x close to 0. The closeness in the last item means essentially a local chart at 0 not containing the point 1, to avoid ambiguity in the quotient. Note that f is a C 1+α local diffeomorphism. Actually, it follows from (c1 ) and (c3 ) that, for x near 0, f  (x) − 1 ≈ |x|α and f (x) − x ≈ sgn(x)|x|α+1 ,

(3.163)

where sgn denotes the sign function. Theorem 3.62 Let f : S 1 → S 1 satisfy (c1 )–(c3 ). Then 1. for α < 1, the map f has a unique SRB measure μ. Moreover, μ is exact, equivalent to m, its basin covers m almost all of S 1 and, for every Hölder continuous ϕ : S 1 → R and ψ ∈ L ∞ (m), Cor μ (ϕ, ψ ◦ f n )  1/n 1/α−1 ; 2. for α ≥ 1, the Dirac measure at 0 is a physical measure for f and its basin covers m almost all of S 1 .

92

3 Expanding Structures

In the remainder of this subsection, we prove Theorem 3.62. Unlike the interval map in the previous subsection, here it will not be so simple to construct a GibbsMarkov map induced map for f . The only exception is the case d = 2, where we could adopt a procedure simpler than in the general case. This will be illustrated in Section 4.6, where we use f with degree d = 2 to build a solenoid with intermittency in the solid torus. Let us now explain how we build, in general, a Gibbs-Markov induced map for f . To begin with, we introduce an auxiliary weak Gibbs-Markov induced map. Consider {I1 , . . . , Id } an m mod 0 partition of S 1 into open intervals such that f | Ii : Ii → S 1 \ {0} is a diffeomorphism, for each 1 ≤ i ≤ d. Take these intervals arranged in the natural order, with 0 being simultaneously the infimum of I1 and the supremum of Id . Let z 0 be the supremum of I1 and z 0 the infimum of Id . We further partition I1 and Id into countably many subintervals as follows: consider (z n )n and (z n )n sequences in I1 and Id , respectively, defined recursively for n ≥ 0 as  ) = z n . f (z n+1 ) = z n and f (z n+1

Set for each n ≥ 1  , z n ). Jn = (z n , z n−1 ) and Jn = (z n−1

Notice that

 ) = Jn . f (Jn+1 ) = Jn and f (Jn+1

(3.164)

Remark 3.63 The restriction of f to [0, z 0 ] clearly satisfies conditions (i1 )–(i3 ) in Sect. 3.5.1. Therefore, the conclusions therein hold for this restriction. Taking into account the expressions in (3.163), it is not hard to see that similar results can be drawn for the restriction of f to [z 0 , 0]. In particular, the conclusions in (3.148) and in Lemma 3.58 are valid for the intervals Jn and Jn . For each 2 ≤ i ≤ d − 1 (if d ≥ 3) and n ≥ 1, set R| Ii = 1 and R| Jn = R| Jn = n.

(3.165)

It follows from Remark 3.63 that     1 1+1/α  m(Jk ) + m(Jk ) ≈ ≈ n −1/α . m{R > n} ≈ k k≥n k≥n

(3.166)

Consider the m mod 0 partition of S 1 P = {I2 , . . . , Id−1 } ∪ {Jn , Jn : n ≥ 1},

(3.167)

ignoring, of course, the first part of the union above for d = 2. Up to mod 0 sets, we have for all ω ∈ P

3.5 Application: Intermittent Maps

  f R (ω) ∈ S 1 , I1 ∪ · · · ∪ Id−1 , I2 ∪ · · · ∪ Id .

93

(3.168)

Now we see that f R is weak Gibbs-Markov with respect to the partition P. The proof of the next result is similar to that of Lemma 3.60. Lemma 3.64 f R : S 1 → S 1 is a weak Gibbs-Markov map. Proof The Markov property (G1 ) and the long branches property (G5 ) follow from (3.168). Also, since f R restricted to each elements of P is a diffeomorphism onto its image, f R has the nonsingularity property (G3 ) with respect to m and J f R = ( f R ) . Now we check the separability property (G2 ). We claim that there exists σ > 1 such that (3.169) ( f R ) |ω ≥ σ, for all ω ∈ P. In fact, since f  > 1 on S 1 \ {0}, there exists σ > 1 such that f  |ω ≥ σ for all ω ∈ {I2 , . . . , Id−1 , J1 , J1 }. Recalling that R|ω = 1 in this case, we have proved the claim for ω ∈ {I2 , . . . , Id−1 , J1 , J1 }. Assume now that ω = Jn , for some n ≥ 2 (the +n−1  i case ω = Jn is similar). We have ( f R ) (x) = i=0 f ( f (x)) for all x ∈ Jn . In addition, f  ( f i (x)) > 1 for all 0 ≤ i ≤ n − 2. Since f n−1 (x) ∈ f n−1 (Jn ) = J1 and f  | J1 ≥ σ , we obtain the claim also for ω ∈ {Jn , Jn : n ≥ 1}. Now, the generating property is a consequence (3.169) together with Lemma 3.3. Finally, we check the Gibbs property (G4 ). Assume first ω = Ii , for some 2 ≤ i < d. Since f is a C 2 local diffeomorphism on S 1 \ {0}, there is Ci > 0 such that, for all x, y ∈ Ii , | log f  (x) − log f  (y) | ≤ Ci |x − y|. Since R|ω = 1, it follows from (3.169) that, for all x, y ∈ Ii , log

( f R ) (x) ≤ Ci |x − y| ≤ Ci | f R (x) − f R (y)|. ( f R ) (y)

(3.170)

Assume now that ω ∈ {Jn , Jn : n ≥ 1}. Since R|ω = n, it follows from Lemma 3.58 that there is some uniform constant C0 > 0 such that, for all x, y ∈ ω, log

( f R ) (x) ≤ C0 | f R (x) − f R (y)|; ( f R ) (y)

(3.171)

recall Remark 3.63. The Gibbs property is then a consequence of (3.169), (3.171) and Lemma 3.3.  Note that f R is not Gibbs-Markov, since the full-branch property fails on the intervals Jn and Jn . Therefore, the ergodicity or uniqueness of an SRB measure for f R are not directly guaranteed by any of our previous results. Lemma 3.65 f R : S 1 → S 1 has a unique SRB measure ν. Moreover, ν is ergodic and dν/dm is bounded from above and below by positive constants.

94

3 Expanding Structures

Proof It follows from Lemma 3.64 and Theorem 3.13 that f R has an SRB measure ν such that dν/dm is bounded from above by a positive constant. To prove that ν is ergodic, we need to show that, for any A ⊂ S 1 with ν(A) > 0 and ( f R )−1 (A) = A, we have ν(A) = 1. Given any ε > 0, Corollary 2.3 provides n ≥ 1 and ω ∈ Pn such that m(ω \ A) < ε. (3.172) m(ω) Although the ideas to prove that ν is ergodic and dν/dm is bounded from below by some positive constant are similar in the cases d = 2 and d ≥ 3, we consider each case separately. Assume first that d = 2. By (3.172) and Corollary 3.6, there is C > 0 such that m(ω \ A) m(( f R )n (ω) \ A) ≤C < Cε. m(( f R )n (ω)) m(ω) It follows from (3.168) that ( f R )n (ω) is equal to I1 or I2 , depending on whether ω ⊂ I2 or ω ⊂ I1 . Since ε > 0 is arbitrary, we have m(Ii \ A) = 0, for some i = 1, 2. Assume for definiteness that m(I1 \ A) = 0 (the other case is similar). This implies R −1 that ν(I1 \ A) = 0, by the absolute continuity  R −1 of ν. Since  ( f ) (A) =1 A and ν is invariant, we also have ν(I2 \ A) = ν ( f ) (I1 \ A) = 0. Hence, ν(S \ A) = 0, and therefore ν is ergodic. Now we prove that ρ = dν/dm is bounded from below by some positive constant, still under the assumption that d = 2. Since I1 and I2 are both unions of elements in P, by Remark 3.14, it is enough to show that there is c0 > 0 such that, for all n ≥ 1 and i = 1, 2, we have ω∈Pn (Ii ) m(ω) ≥ c0 , where   Pn (Ii ) = ω ∈ Pn : ( f R )n (ω) = Ii . Indeed, since f R (I1 ) = I2 and f R (I2 ) = I1 , it follows that

m(ω) ≥ min{m(I1 ), m(I2 )},

ω∈Pn (Ii )

for all n ≥ 1 and i = 1, 2. Assume now that d ≥ 3. Since ν m, for the ergodicity of ν it is enough to show that m(S 1 \ A) = 0. Consider ω as in (3.172). It follows from (3.168) that ( f R )n (ω) is equal to one of the sets S 1 , I2 ∪ · · · ∪ Id or I1 ∪ · · · ∪ Id−1 . We discuss the first two possibilities (the last one is similar to the second). If ( f R )n (ω) = S 1 , then S 1 \ A = ( f R )n (ω \ A). Using (3.172) and Corollary 3.6, we obtain C > 0 such that m(ω \ A) m(S 1 \ A) ≤C < Cε. m(S 1 ) m(ω) Since ε > 0 is arbitrary, we have m(S 1 \ A) = 0 and therefore ν is ergodic. If ( f R )n (ω) = I2 ∪ · · · ∪ Id , then (I2 ∪ · · · ∪ Id ) \ A = ( f R )n (ω \ A). Observe that

3.5 Application: Intermittent Maps

95

there exists some constant C  > 0 such that m(I2 ∪ · · · ∪ Id ) ≤ C  m(I2 ). Using (3.172) and Corollary 3.6, we have for some C > 0 m(I2 \ A) m((I2 ∪ · · · ∪ Id ) \ A) m(ω \ A) ≤ C ≤ C C < C  Cε. m(I2 ) m(I2 ∪ · · · ∪ Id ) m(ω) Now, recalling that I2 ∈ P and f R (I2 ) = S 1 , using again Corollary 3.6, we get m(I2 \ A) m(S 1 \ A) ≤C < C  C 2 ε. m(S 1 ) m(I2 ) Since ε > 0 is arbitrary, we have m(S 1 \ A) = 0, and so the ergodicity of ν. Now, we prove that dν/dm is bounded from below by a positive constant, under the assumption d ≥ 3. Set   P0n = ω ∈ Pn : ( f R )n (ω) = S 1 . By Remark 3.14, it is enough to show that there is c0 > 0 such that, for all n ≥ 1,

m(ω) ≥ c0 .

(3.173)

ω∈P0n

For convenience, take P0 = {I1 , I2 ∪ · · · ∪ Id−1 , Id }. Given n ≥ 1 and ω ∈ Pn−1 , set   P0n (ω ) = ω ∈ P0n : ω ⊂ ω . Note that, by (3.168), we have for all ω ∈ Pn−1   ( f R )n−1 (ω ) ∈ S 1 , I1 ∪ · · · ∪ Id−1 , I2 ∪ · · · ∪ Id . Set for each ω ∈ Pn−1   ω0 = x ∈ ω : ( f R )n−1 (x) ∈ I2 ∪ · · · ∪ Id−1 . Since ω0 is equal to the disjoint union of the elements ω ∈ P0n (ω ), we have ω∈P0n

m(ω) =





ω ∈Pn−1

ω∈P0n (ω )

m(ω) =



m(ω0 ).

(3.174)

ω ∈Pn−1

Since f R is a weak Gibbs-Markov map, Corollary 3.6 provides C > 0 for which    1 m ( f R )n−1 (ω0 ) 1 m(ω0 ) 1    ≥ m ( f R )n−1 (ω0 ) = m(I2 ∪ · · · ∪ Id−1 ). ≥  R n−1  m(ω ) C m ( f ) (ω ) C C (3.175)

96

3 Expanding Structures

Take c0 = m(I2 ∪ · · · ∪ Id−1 )/C. It follows from (3.174) and (3.175) that ω∈P0n

m(ω) ≥



m(ω0 ) ≥

ω ∈Pn−1



c0 m(ω ) = c0 ,

ω ∈Pn−1

thus having shown (3.173). Finally, since f R is defined on S 1 , the uniqueness of the SRB measure ν is a consequence of the fact that dν/dm is bounded from below by a positive constant  and Lemma 3.12, with M = S 1 = 0 . Note that so far everything works for all α > 0. Let us first prove the second item of Theorem 3.62. Assume that α ≥ 1. In this case, (3.166) implies that R ∈ / L 1 (m). R Moreover, f has a unique ergodic SRB measure and dν/dm is bounded from above and below by positive constants, by Lemma 3.65. Therefore, the non-integrability of R with respect to ν is a consequence of R ∈ / L 1 (m). Applying Proposition 3.61 with x0 = 0 and Un the interval in S 1 containing 0 delimited by the points z n−1  and z n−1 , we obtain the second item of Theorem 3.62. Assume now that 0 < α < 1. In this case, (3.166) implies that R ∈ L 1 (m). So, the existence of a unique SRB μ for f follows from Lemmas 3.64, 3.65 and Corollary 3.19; recall also Remark 3.20. It also follows from these results that μ is equivalent to m. Therefore, the support of μ coincides with S 1 and its basin covers m almost all of S 1 , by Proposition 2.12. We are going to use the map f R to build an induced Gibbs-Markov map for f with an upper bound for the measure of tail of the recurrence times of the same order of m{R > n}. This will provide us with the expected estimate on the decay of correlations. Consider functions 0 = τ0 < τ1 < τ2 < · · · , defined inductively for i ≥ 1 by (3.176) τi = τi−1 + R ◦ f τi−1 . Notice that

( f R )i = f τi , for all i ≥ 1.

(3.177)

Now, consider I0 the union of at least two consecutive intervals in the sequence (Jn )n . For the time being, we could simply take I0 = I1 , but it will be convenient in Sect. 4.6 to have an induced map on an interval with arbitrarily small length. The reason for taking at least two consecutive intervals is to ensure that the greatest common divisor of the recurrence times is equal to one; see Proposition 3.67 below. Set for each x ∈ S 1   S(x) = min τi (x) : f τi (x) (x) ∈ I0 , i≥1

and for each i ≥ 1 Pi =

i−1  =0

( f R )− P,

(3.178)

3.5 Application: Intermittent Maps

97

where P is the partition in (3.167). Using (3.168) and (3.177), we easily see that, for all i ≥ 1 and ω ∈ Pi , f τi (ω) ∈ {S 1 , I2 ∪ · · · ∪ Id , I1 ∪ · · · ∪ Id−1 }.

(3.179)

We will see that f S induces a Gibbs-Markov map on the interval I0 . We also need to estimate m{S > n}. For this purpose, we will use Proposition 3.46. The assumptions (A1 ) and (A2 ) are checked in the next result. Lemma 3.66 There are C, ε0 > 0 such that 1. for all i > 1 and ω ∈ Pi with S|ω > τi−1 , m (S = τi | ω) ≥ ε0 ; 2. for all i ≥ 1 and ω ∈ Pi , m (τi+1 − τi > n | ω) ≤ Cm{R > n}. Proof Assume that S|ω > τi−1 , for ω ∈ Pi with i > 1. By construction, the set f τi−1 (ω) belongs in P. In addition, f τi−1 (ω) must be contained in S 1 \ I1 , otherwise we would not have S|ω > τi−1 . It follows from (3.168) that f R ( f τi−1 (ω)) = f τi (ω) necessarily contains I0 . This means that f τi ({S = τi } ∩ ω) = I0 .

(3.180)

Since f R is a weak Gibbs-Markov map, using (3.179), (3.180) and Corollary 3.6, we have, for some uniform constant C  > 0 m ({S = τi } ∩ ω) m(ω) 1 m ( f τi ({S = τi } ∩ ω)) ≥  C m ( f τi (ω)) 1 ≥  m(I0 ). C

m (S = τi | ω) =

This gives the first item with ε0 = m(I0 )/C  . Now we prove the second item. Given i ≥ 0 and ω ∈ Pi , it follows from (3.179) that, for some uniform constant δ > 0, m ( f τi (ω)) ≥ δ. Using Corollary 3.6 and (3.181), we get

(3.181)

98

3 Expanding Structures

m (τi+1 − τi > n | ω) = m (R ◦ f τi > n | ω) 1 = (m|ω) {R ◦ f τi > n} m(ω) 1 = f τi (m|ω) {R > n} m(ω) ∗ m ({R > n} ∩ f τi (ω)) ≤ C m ( f τi (ω)) C m{R > n}. ≤ δ This gives the second item with C = C  /δ.



Note that each Pi is a mod 0 partition into intervals of S on which τi is constant. Moreover, {S > τi−1 } and τ0 , . . . , τi are Pi -measurable, for all i ≥ 1. Therefore, using (3.166), Lemma 3.66 and Proposition 3.46 with P = m and ξi = Pi , we get 1

m{S > n}  n −1/α .

(3.182)

Our next step is to build a mod 0 partition Q into intervals of I0 such that f S maps each element in Q diffeomorphically to I0 . We define inductively families (Fn )n≥2 and (En )n≥2 of pairwise disjoint subintervals of I0 , gathering in Fn all the intervals ω ⊂ I0 returning to I0 in time n, with n ≥ R|ω . Consider for each n ≥ 1 P0n = {ω ∈ Pn : ω ⊂ I0 } . Let us now start the inductive construction with n = 2. Set   P12 = ω ∈ P02 : f 2 (ω) ⊃ I0 . Since f R is weak Gibbs-Markov, given any ω ∈ P12 , there are intervals ω1 ⊂ ω which are mapped diffeomorphically to I0 under f 2 . Let F2 be the family of all these intervals ω1 ⊂ ω, with ω ∈ P12 . Let also E2 be the family of all connected components of ω \ ω1 , with ω ∈ P12 , together with the intervals ω ∈ P02 \ P12 . Observe that each ω0 ∈ E2 is P3 -measurable. The procedure in the inductive step is similar to the first step. Assuming that Fn−1 and En−1 have been defined for some n ≥ 3, set P1n = {ω ∈ En−1 : f n (ω) ⊃ I0 }. Again, given any ω ∈ P1n , there are subintervals ω1 ⊂ ω mapped diffeomorphically to I0 under f n . Let Fn be the family of these intervals ω1 ⊂ ω, with ω ∈ P1n , and En the family of connected components of ω \ ω1 , with ω ∈ P1n , together with the intervals ω ∈ P0n \ P1n . Observe that each ω0 ∈ En is Pn+1 -measurable. Finally, set Q=

 n≥2

Fn .

3.5 Application: Intermittent Maps

99

Note that, by construction, Q is a family of pairwise disjoint intervals ω ⊂ I0 such that f S maps each ω ∈ Q diffeomorphically to I0 . If follows from (3.182) and the saturation argument in the construction of the sequence (Fn )n≥2 that Q is a mod 0 partition of I0 . In the next result, we see that f S : I0 → I0 is Gibbs-Markov with respect to Q. Recall that I0 is the union of at least two consecutive intervals in the sequence (Jn )n . Proposition 3.67 f S : I0 → I0 is a Gibbs-Markov map and gcd(S) = 1. Proof The full branch property (G5 ) follows immediately from the construction of Q. Recalling (3.177) and (3.178), the separability property (G2 ) is a consequence of (3.169), the chain rule and Lemma 3.3. Since f S restricted to each interval in Q is a diffeomorphism onto its image, we also have that the nonsigular property (G3 ) holds for f S . Now, we check the Gibbs property (G4 ). Observe that the sequence of times corresponding to iterations under f S is a subsequence of the functions (τi )i introduced in (3.176). Consider 0 = i 0 < i 1 < i 2 < · · · such that for each k ≥ 1, we have τik = τik−1 + S ◦ f τik−1 . In this way

( f S )k = f τik , for all k ≥ 0.

Assume that x, y ∈ ω are such that ( f S )k (x) = f τik (x) and ( f S )k (x) = f τik (y) lie in the same elements of Q, for all 0 ≤ k ≤ n. Since f R is Markov and f S maps each ω ∈ Q diffeomorphically to I0 , we necessarily have that ( f R )i (x) = f τi (x) and ( f R )i (y) = f τi (y) lie in the same elements of P, for all 0 ≤ i ≤ i n . This shows that the separation time s(x, y) with respect to f R is greater or equal to the separation time s  (x, y) with respect to f S . Since f R is a weak Gibbs-Markov map, Lemma 3.4 provides C > 0 and 0 < β < 1 such that, for all x, y ∈ ω ∈ Pτi1 log

J f τi1 (x) J( f R )i1 (x) J f S (x) S S = log = log ≤ Cβ s( f (x), f (y)) . J f S (y) J f τi1 (y) J( f R )i1 (y)

(3.183)

Recalling that s( f S (x), f S (y)) ≥ s  ( f S (x), f S (y)) and each element of Q is contained in some element of Pτi1 , we have the Gibbs property for f S . Finally, we show that gcd(S) = 1. Recalling that we I0 is the union of at least two consecutive intervals in (Jn )n , set n 0 = min {n ≥ 1 : Jn ⊂ I0 } . Observe that there is some interval ω ⊂ I2 with f (ω) = I0 . Since f n 0 (Jn 0 ) ⊃ ω, there is some interval ωn 0 ⊂ Jn 0 such that f n 0 +1 (ωn 0 ) = I1 . Taking ωn 0 +1 ⊂ Jn 0 +1 ⊂ I0 with f (ωn 0 +1 ) = ωn 0 , we have f n 0 +2 (ωn 0 +1 ) = I0 . Since S| I0 ≥ n 0 + 1, we have  S(ωn 0 ) = n 0 + 1 and S(ωn 0 +1 ) = n 0 + 2, and therefore, gcd(S) = 1.

100

3 Expanding Structures

Remark 3.68 Recall that in (3.171) we obtained some uniform constant C > 0 such that, for all x, y belonging in an element of the partition associated with the weak Gibbs-Markov map f R , log

( f R ) (x) ≤ C| f R (x) − f R (y)|. ( f R ) (y)

Using this, and reasoning as in the proof of Lemma 3.4, we can improve the estimate in (3.183), obtaining log

( f S ) (x) ≤ C| f S (x) − f S (y)|. ( f S ) (y)

This will be useful in the proof of Proposition 4.33. The exactness of the unique SRB measure for f : S 1 → S 1 and the conclusion about the decay of correlations in the first item of Theorem 3.62 are finally a consequence of Theorem 3.54, together with Proposition 3.67.

References 1. J. Aaronson, An introduction to infinite ergodic theory, vol. 50, Mathematical Surveys and Monographs (American Mathematical Society, Providence, RI, 1997) 2. J. Aaronson, M. Denker, Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps. Stoch. Dyn. 1(2), 193–237 (2001) 3. J. Aaronson, M. Denker, M. Urba´nski, Ergodic theory for Markov fibred systems and parabolic rational maps. Trans. Amer. Math. Soc. 337(2), 495–548 (1993) 4. R. Alexander, S. Alexander, Geodesics in Riemannian manifolds-with-boundary. Indiana Univ. Math. J. 30(4), 481–488 (1981) 5. W. Bahsoun, C. Bose, Y. Duan, Decay of correlation for random intermittent maps. Nonlinearity 27(7), 1543–1554 (2014) 6. S. Gouëzel, Decay of correlations for nonuniformly expanding systems. Bull. Soc. Math. France 134(1), 1–31 (2006) 7. M. Holland, Slowly mixing systems and intermittency maps. Ergodic Theory Dynam. Syst. 25(1), 133–159 (2005) 8. H. Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points. Ergodic Theory Dynam. Syst. 24(2), 495–524 (2004) 9. C. Liverani, B. Saussol, S. Vaienti, A probabilistic approach to intermittency. Ergodic Theory Dynam. Syst. 19(3), 671–685 (1999) 10. V. Maume-Deschamps, Projective metrics and mixing properties on towers. Trans. Amer. Math. Soc. 353(8), 3371–3389 (2001) 11. I. Melbourne, D. Terhesiu, Decay of correlations for nonuniformly expanding systems with general return times. Ergodic Theory Dynam. Syst. 34(3), 893–918 (2014) 12. M. Pollicott, M. Yuri, Dynamical Systems and Ergodic Theory, vol. 40, London Mathematical Society Student Texts (Cambridge University Press, Cambridge, 1998) 13. Y. Pomeau, P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems. Comm. Math. Phys. 74(2), 189–197 (1980)

References

101

14. V.A. Rohlin, On the fundamental ideas of measure theory. Amer. Math. Soc. Transl. 1952(71), 55 (1952) 15. E. Seneta, Non-negative Matrices and Markov Chains. Springer Series in Statistics (Springer, New York, 2006). Revised reprint of the second (1981) edition (Springer-Verlag, New York) 16. M. Thaler, Estimates of the invariant densities of endomorphisms with indifferent fixed points. Israel J. Math. 37(4), 303–314 (1980) 17. M. Viana, Stochastic dynamics of deterministic systems, in 22th Brazilian Mathematics Colloquium (IMPA Mathematical Publications, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1997) 18. L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. (2) 147(3), 585–650 (1998) 19. L.-S. Young, Recurrence times and rates of mixing. Israel J. Math. 110, 153–188 (1999)

Chapter 4

Hyperbolic Structures

In this chapter, we introduce Young structures and the corresponding tower extensions. Associated to each Young structure there is a return map, which can be seen as the analog of a Gibbs-Markov map for systems with contracting directions. For systems with Young structures, we deduce the existence of SRB measures and provide estimates on the decay of correlations with respect to these measures. Our presentation is ultimately based on [20], whose conclusions hold only for exponential rates. We extend the conclusions to the polynomial and stretched exponential cases. In addition, we present some simplifications in the set of conditions that define Young structures. The exponential decay of correlations in [20] is obtained by means of a spectral gap argument for the transfer operator of the tower map; see also [6, 7]. Our strategy here to study the decay of correlations is based on a mixture of techniques from [20, 21], reducing it to a problem with respect to a system in the conditions of Theorem 3.28 and applying this result. We follow the approach in [3], complemented by ideas of Gouëzel developed by Melbourne et al. in [11, 14].

4.1 Young Structures Let M be a finite dimensional compact Riemannian manifold M. Consider d the distance on M and m the Lebesgue (volume) measure on the Borel sets of M, both induced by the Riemannian metric. Given a submanifold γ ⊂ M, we use dγ to denote the distance on γ and m γ to denote the Lebesgue measure on γ , both induced by the restriction of the Riemannian metric to γ . Consider f : M → M a piecewise C 1+η diffeomorphism, possibly having critical, singular or discontinuity points that are not reached by the iterations of the domains that we describe below in the times associated with them. First, we intro-

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. F. Alves, Nonuniformly Hyperbolic Attractors, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-62814-7_4

103

104

4 Hyperbolic Structures

duce continuous families  s and  u of C 1 disks in M. Typically,  s is a family of Pesin stable disks and  u a family of Pesin unstable disks, but it does not have to be that way in general. Anyhow, we will impose some weak forward contraction on the disks of  s and weak backward contraction on those of  u . For this reason, we will continue to refer to the disks in  s as stable disks and to those in  u as unstable disks. We will also refer to invariant probability measures whose conditional measures are absolutely continuous with respect to the conditionals of Lebesgue measure on the disks of  u as SRB measures. We say that  is a continuous family of C 1 disks in M if there are a compact metric space K , a unit disk D in some Rk and an injective continuous function  : K × D → M such that •  = {({x} × D) : x ∈ K }; •  maps K × D homeomorphically onto its image; • x → |{x}×D defines a continuous map from K into Emb1 (D, M), where Emb1 (D, M) denotes the space of C 1 embeddings of D into M. Note that the disks in  have all the same dimension dim D, that we also denote dim . We say that a compact set  ⊂ M has a product structure if there are continuous families of C 1 disks  s and  u such that •  = (∪γ ∈s γ ) ∩ (∪γ ∈u γ ); • dim  s + dim  u = dim M; • each γ ∈  s meets each γ ∈  u in exactly one point. We say that  has a full product structure if it has a product structure and every disk in  u is contained in . We say that 0 ⊂  is an s-subset, if 0 has a product structure with respect to families 0s and 0u such that 0s ⊂  s and 0u =  u ; u-subsets are defined similarly. Let γ ∗ (x) denote the disk in  ∗ containing x ∈ , for ∗ = s, u. Consider the holonomy map γ ,γ : γ ∩  → γ ∩ , defined for each x ∈ γ ∩  by (4.1) γ ,γ (x) = γ s (x) ∩ γ . We say that a compact set  has a Young structure (with respect to f ), if  has a product structure given by continuous families of C 1 disks  s and  u such that m γ ( ∩ γ ) > 0, for all γ ∈ γ u , and the conditions (Y1 )–(Y5 ) below are satisfied. A Young structure is called a full Young structure if the product structure associated with it is a full product structure. (Y1 ) Markov: there are pairwise disjoint s-subsets 1 , 2 , · · · ⊂  such that • m γ ( \ ∪i i ) ∩ γ ) = 0 for all γ ∈  u ; • for each i ≥ 1, there is Ri ∈ N such that f Ri (i ) is a u-subset and, moreover, for all x ∈ i , f Ri (γ s (x)) ⊂ γ s ( f Ri (x)) and f Ri (γ u (x)) ⊃ γ u ( f Ri (x)).

4.1 Young Structures

105

This Markov property allows us to introduce a recurrence time R :  → N and a return map f R :  → , defined for each i ≥ 1 by R|i = Ri and f R |i = f Ri |i .

(4.2)

Note that R and f R are defined on a full m γ measure subset of  ∩ γ , for each γ ∈  u . Thus, there is a set  ⊂  intersecting each γ ∈  u in a full m γ measure subset, such that ( f R )n (x) belongs in some i for all n ≥ 0 and x ∈  . For points x, y ∈  , we define the separation time   s(x, y) = min n ≥ 0 : ( f R )n (x) and ( f R )n (y) lie in distinct i ’s , with the convention that min(∅) = ∞. For definiteness, we set the separation time equal to zero for all other points. For the remaining conditions, we consider constants C > 0 and 0 < β < 1 only depending on f and . (Y2 ) Contraction on stable disks: for all i ≥ 1, γ ∈  s and x, y ∈ γ ∩ i , • d(( f R )n (y), ( f R )n (x)) ≤ Cβ n , for all n ≥ 0; • d( f j (y), f j (x)) ≤ Cd(x, y), for all 1 ≤ j ≤ Ri . (Y3 ) Expansion on unstable disks: for all i ≥ 1, γ ∈  u and x, y ∈ γ ∩ i , • d(( f R )n (y), ( f R )n (x)) ≤ Cβ s(x,y)−n , for all n ≥ 0; • d( f j (y), f j (x)) ≤ Cd( f R (x), f R (y)), for all 1 ≤ j ≤ Ri . (Y4 ) Gibbs: for all i ≥ 1, γ ∈  u and x, y ∈ γ ∩ i , log

det D f R |Tx γ R R ≤ Cβ s( f (x), f (y)) . R det D f |Ty γ

(Y5 ) Regularity of the stable holonomy: for all γ , γ ∈  u , the measure (γ ,γ )∗ m γ is absolutely continuous with respect to m γ and its density ργ ,γ satisfies 1 ≤ C

 γ ∩

ργ ,γ dm γ ≤ C and log

ργ ,γ (x) ≤ Cβ s(x,y) , ργ ,γ (y)

for all x, y ∈ γ ∩ . We say that a Young structure has integrable recurrence times if R is integrable with respect to m γ , for some γ ∈  u . By (Y5 ), this is equivalent to say that R is integrable with respect to m γ , for all γ ∈  u . In fact, the measure (γ ,γ )∗ m γ is equivalent to m γ , since (Y5 ) holds for all γ , γ ∈  u . In addition, the inequalities in (Y5 ) involving the integral mean that there exists some uniform constant C > 0 such that, for all γ ∈  u , 1 (4.3) ≤ m γ (γ ∩ ) ≤ C. C

106

4 Hyperbolic Structures

In Sect. 4.5, we will show that, under reasonable additional conditions on the set with a Young structure, property (Y5 ) can be deduced from (Y1 )–(Y4 ). Remark 4.1 As in the proof of Lemma 3.2, it is easily verified that the first part of (Y3 ) holds whenever there is some α > 0 such that d(x, y) ≤ αd( f R (x), f R (y)), for all i ≥ 1, γ ∈  u and x, y ∈ γ ∩ i .

4.1.1 Quotient Return Map Here, we introduce a quotient map associated to the return map of a Young structure. We also show that this quotient is a Gibbs-Markov map. This map will be used in several instances throughout this chapter and will help us to bring useful information in Chap. 3 into the current context. Given γ ∈  u , consider γ :  → γ ∩ , defined for each x ∈  by (4.4) γ (x) = γ s (x) ∩ γ . Fixing some γ0 ∈  u , we define the quotient map of f R F : γ0 ∩  −→ γ0 ∩  x −→ γ0 ◦ f R (x).

(4.5)

To simplify notation, the restriction of each m γ to γ ∩  will still be denoted by m γ . Proposition 4.2 F : γ0 ∩  → γ0 ∩  is a Gibbs-Markov map with respect to the m γ0 mod 0 partition P = {γ0 ∩ 1 , γ0 ∩ 2 , . . . } of γ0 ∩ . Proof By definition of a set with a Young structure, we have m γ0 (γ0 ∩ ) > 0. Using the Markov property (Y1 ), we see that P is an m γ0 mod 0 partition of γ0 ∩ . Now, we need to check properties (G2 )–(G4 ) and (G5 ) in Sect. 3.1 for the map F. It follows from the regularity of the stable holonomy (Y5 ) and the fact that f is a diffeomorphism that F maps each ω ∈ P bijectively to a full m γ0 measure subset of γ0 ∩ . Thus, the full branch property (G5 ) holds. Now, we check the separability 2.5. To apply it, we need to see that property (G2 ) for F. We will use Corollary  −j P converge to zero when n goes to the diameters of the partitions Pn = n−1 j=0 F infinity. First of all, note that, since R is constant on stable leaves, we have  n F n (x) = γ0 ◦ f R (x),

(4.6)

for all x ∈ γ0 ∩  and n ≥ 1. Assume that x, y are points in γ0 ∩  whose forward iterates under F until time n always lie in the same elements of P. Together with (4.6), this gives that the forward iterates under f R until time n always lie in the same

4.1 Young Structures

107

elements of the partition associated with the Young structure. This implies s(x, y) ≥ n. It follows from (Y3 ) that d(x, y) ≤ Cβ s(x,y) ≤ Cβ n . Hence, diam(ωn ) ≤ Cβ n , for all n ≥ 1 and ωn ∈ Pn , and therefore, diam(Pn ) → 0, when n → ∞. The separability property (G2 ) is then a consequence of Corollary 2.5. Now, we show that F satisfies the nonsingularity condition (G3 ). Given ω ∈ P and a measurable set A ⊂ ω, consider γ1 ∈  u such that f R (ω) ⊂ γ1 . Denoting by f 0R the restriction of f R to γ0 , we may write   R m γ0 (F(A)) = m γ0 −1 γ1 ,γ0 ( f (A) = (γ1 ,γ0 )∗ m γ0 ( f R (A))  = ργ1 ,γ0 dm γ1 f R (A)  = (ργ1 ,γ0 ◦ f R ) det D f 0R dm γ0 . A

This shows that JF |ω = (ργ1 ,γ0 ◦ f R ) det D f 0R |ω .

(4.7)

Since ργ1 ,γ0 > 0 and f is a diffeomorphism, then JF |ω is strictly positive, and therefore, F has the nonsingularity property. Finally, we check the Gibbs condition (G4 ). It follows from (Y4 ), (Y5 ) and (4.7) that, for all ω ∈ P and x, y ∈ ω log

ργ ,γ ( f R (x)) det D f 0R (x) JF (x) = log 1 0 R + log JF (y) ργ1 ,γ0 ( f (y)) det D f 0R (y) ≤ 2Cβ s( f

R

(x), f R (y))

.

Now, note that (4.6) gives in particular that, for points in γ0 ∩ , the separation time with respect to F or f R coincide. Since s( f R (x), f R (y)) = s(x, y) − 1, then F has the Gibbs property. 

4.1.2 Bounded Distortion In this subsection, we obtain some useful bounded distortion results for the return map f R associated with a set  having a Young structure. Let Q be the family of pairwise disjoint s-subsets 1 , 2 , . . . provided by (Y1 ). For each n ≥ 1, set Qn =

n−1  i=0

( f R )−i Q.

(4.8)

108

4 Hyperbolic Structures

We introduce functions 0 = τ0 < τ1 < · · · , recursively defined for n ≥ 1 by τn = τn−1 + R ◦ f τn−1 .

(4.9)

Note that τn is constant on each Q ∈ Qn and f τn | Q = ( f R )n | Q , for each n ≥ 1. Lemma 4.3 There is C > 0 such that, for all n ≥ 1, Q ∈ Qn , γ ∈  u and x, y ∈ γ ∩ Q, det D f τn |Tx γ τn τn ≤ Cβ s( f (x), f (y)) . log det D f τn |Ty γ Proof It follows from (Y3 ) that, for all x, y ∈ γ and 0 ≤ i < n, s( f τi (x), f τi (y)) = s( f τn (x), f τn (y)) + n − i. Using (Y4 ), we get det D f R |T f τi (x) γ det D f τn |Tx γ = log τ det D f n |Ty γ det D f R |T f τi (y) γ i=0 n−1

log

≤

n−1

Cβ s( f

τi (x), f τi (y))

i=0

=

n−1

Cβ n−i β s( f

τn

(x), f τn (y))

i=0

≤

C τn τn β s( f (x), f (y)) . 1−β

Since C > 0 and 0 < β < 1 are uniform constants, we have proved the result.



The following consequence of the previous lemma will be useful in the proof of Theorem 4.7 below. Corollary 4.4 There is some constant C > 0 such that, for all n ≥ 1 and γ0 , γ ∈  u γ with f ∗τn m γ0 (γ ∩ ) > 0, the density ζn = d f ∗τn m γ0 /dm γ satisfies γ

1 ζn (x) ≤ γ ≤ C. C ζn (y) for all x, y ∈ γ ∩ . Proof Given any γ , γ0 ∈  u and n ≥ 1, set Pn = {γ0 ∩ Q : Q ∈ Qn } and γ = ζn,ω

d f ∗τn (m γ0 |ω) , dm γ

4.1 Young Structures

109

for each ω ∈ Pn . We have ζnγ =



γ ζn,ω .

ω∈Pn f τn (ω)=γ ∩

This implies that

min

ω∈Pn f τn (ω)=γ ∩

γ

ζn,ω (x) γ ζn,ω (y)





γ

ζn (x) ≤ γ ≤ ζn (y)

max

ω∈Pn f τn (ω)=γ ∩

 γ ζn,ω (x) . γ ζn,ω (y)

(4.10)

for all x, y ∈ γ ∩ . Given x ∈ γ ∩  and ω ∈ Pn , let xω,n denote the unique point γ in ω such that f τn (xω,n ) = x. Noting that ζn,ω (x) = (det D f τn |Txω,n γ )−1 , the conclusion follows from (4.10) and Lemma 4.3. 

4.2 SRB Measures The main goal of this section is to prove that the return map associated with a Young structure has a unique SRB measure, and use it to build an SRB measure for the original dynamical system. This will be done in the next two subsections.

4.2.1 Return Map The first result of this subsection shows, in particular, that any SRB measure for the return map associated with a Young structure is necessarily ergodic. As we consider the Lebesgue measure as the reference measure on the unstable disks, it is legitimate to consider the notion of SRB measure for the return map as well. Lemma 4.5 Let  be a set with a Young structure and F : γ0 ∩  → γ0 ∩  a quotient of the return map f R :  → . If ν is an SRB measure for f R , then 1. ν0 = (γ0 )∗ ν is the unique F-invariant probability measure such that ν0  m γ0 ; 2. ν is ergodic. Proof It is easily verified that F ◦ γ0 = γ0 ◦ f R . The F-invariance of ν0 is then an easy consequence of the f R -invariance of ν and γ0 being a measurable semiconjugacy between the maps f R and f . It follows from Proposition 4.2 that F is a Gibbs-Markov map. By Theorem 3.13, the map F has a unique invariant probability measure, which is absolutely continuous with respect to the reference measure m γ0 . Therefore, it is enough to check that (γ0 )∗ ν  m γ0 . Let A ⊂ γ0 be a Borel set with m γ0 (A) = 0. We need to see that ν((γ0 )−1 (A)) = 0. For each γ ∈  u , let νγ be the conditional measure of ν on γ . By Rohlin Disintegration Theorem, there exists a measure ν¯ on  u such that

110

4 Hyperbolic Structures

  ν (γ0 )−1 (A) =

 γ0 ,γ (A)dνγ d ν¯ .

Since ν is an SRB measure for f , we have νγ  m γ , for ν¯ almost all γ ∈  u . As we assume γ0 ,γ absolutely continuous, we easily get ν((γ0 )−1 (A)) = 0. Now we prove that ν is ergodic. Observe, first of all, that the F-invariant probability measure ν0 = (γ0 )∗ ν is ergodic, by Theorem 3.13. Given ϕ ∈ L 1 (ν), the limit ∞ 1 ϕ(( f R ) j (x)) n→∞ n j=0

ϕ ∗ (x) = lim

(4.11)

exists for ν almost every x ∈ , by Birkhoff Ergodic Theorem. It is enough to show that ϕ ∗ is constant ν almost everywhere. Assume first ϕ that is continuous. Since Birkhoff averages for continuous functions are constant on stable disks, we may also think of ϕ ∗ as a function on γ0 ∩ . Moreover, as ϕ ∗ ◦ f R (x) = ϕ ∗ (x), for ν almost every x ∈ , we easily get ϕ ∗ ◦ F(x) = ϕ ∗ (x) for ν0 almost every x ∈ γ0 ∩ . The ergodicity of ν0 then gives that ϕ∗ is constant ν0 almost everywhere. This implies that ϕ ∗ is constant ν almost everywhere. For the general case, consider the bounded linear operator  : L 1 (ν) → L 1 (ν), defined as (ϕ) = ϕ ∗ . Since  maps the dense space of continuous functions into the closed subspace of constant functions, it follows  that (ϕ) is a constant function for all ϕ ∈ L 1 (ν). In the next result, we show that the partition induced by the unstable disks on a set with a product structure is measurable. As there is still no measure involved, the partitions here are understood in the sense of families of pairwise disjoint sets whose union is the whole set. Lemma 4.6 If  is a compact set with a product structure, then the partition U on  induced by the family of unstable disks is measurable. Moreover, there is an increasing sequence of finite partitions (Un )n into u-subsets of  such that U = ∞ n=1 Un . Proof By Lemma 2.7, it is sufficient to prove the second conclusion. Let  u be the set of unstable disks associated with . By definition,  u is a continuous family of C 1 disks in M. This implies that there exist a compact metric space K , a unit disk D ⊂ Rk and an injective continuous function  : K × D → M such that •  u = {({x} × D) : x ∈ K }; •  maps K × D homeomorphically onto its image. Let P be the partition of K into single points. In Example 2.9, we have seen that P is a measurable partition. By Lemma 2.7, there exists an increasing sequence (Pn )n of finite partitions into Borel sets of K such that P = ∞ n=1 Pn ; recall also Remark 2.8. Given any n ≥ 1 and ω ∈ Pn , set Uω =  ∩ (ω × D). It is easily verified that each Uω is a u-subset of . In addition,

4.2 SRB Measures

111

Un = {Pω : ω ∈ Pn } is a partition of , for all n ≥ 1. Since P =

∞ n=1

Pn , then U =

∞ n=1

Un .



Since there is no measure involved in Lemma 4.6, the partition U is measurable with respect to any measure on . For here on, we use ν¯ to denote the quotient measure given by the Rohlin Disintegration of a Borel measure ν on  with respect to the partition U; recall (1.4). For simplicity, we sometimes think of this quotient measure as a measure on the set  u . In the proof of the next result, we use the fact that a sequence (νn )n of probability measures on the Borel sets of a compact metric space converges to a probability measure ν in the weak* topology if, and only if, νn (A) → ν(A) as n → ∞, for every Borel set A with ν(∂ A) = 0; see for example [15, Theorem 6.1]. Theorem 4.7 If f R is the return map of a set  with a Young structure, then f R has a unique SRB measure ν. Moreover, ν is ergodic, the basin of ν covers m almost all of  and the densities dνγ /dm γ are bounded from above and below by uniform positive constants on γ ∩ , for ν¯ almost all γ ∈  u . Proof Fixing some γ0 ∈  u , let (νn )n be the sequence of probability measures on  given for each n ≥ 1 by n−1 1 R j νn = ( f ) ∗ m γ0 . n j=0 Since  is a compact set, we can extract a subsequence (νn k )k converging to a probability measure ν in the weak* topology. First, we prove that ν is an f R -invariant measure. It is worth noting that, as f R is not necessarily continuous, there is no reason for the operator ( f R )∗ to be continuous, and therefore, the invariance of ν is not immediate. Given any Borel set A ⊂  with ν(∂ A) = 0, we have n k −1   1 m γ0 ( f R )− j (A) . k→∞ n k j=0

ν(A) = lim

Since the conditional measures of ν on unstable disks are equivalent to the conditional measures of m and f R preserves sets of zero measure on unstable disks, we obtain ν(∂( f R )−1 (A)) = 0. It follows that n k −1    1 m γ0 ( f R )− j ( f R )−1 (A) . k→∞ n k j=0

ν(( f R )−1 (A)) = lim

This means that the sequence (νn∗k )k given by νn∗k =

n k −1 1 ( f R )∗j+1 m γ0 n k j=0

112

4 Hyperbolic Structures

converges to f ∗R ν in the weak* topology. On the other hand, given any continuous function ϕ :  → R,  lim

k→∞

ϕdνn∗k = lim



 ϕd

k→∞

j=0

1 k→∞ n k

= lim

1 k→∞ n k

= lim

1 k→∞ n k

= lim

n k −1 

= lim

j=0

 k→∞

 ϕ ◦ ( f R ) j dm γ0 −

 ϕdm γ0 +

ϕ ◦ ( f R )n k dm γ0

j=0 n k −1 

ϕ ◦ ( f R ) j dm γ0

j=0

ϕd

k→∞

ϕ ◦ ( f R ) j+1 dm γ0

n k −1 



= lim

n k −1 1 j+1 ( f R )∗ m γ0 nk

n k −1 1 j ( f R )∗ m γ0 nk j=0

ϕdνn k .

This shows that the sequences (νn k )k and (νn∗k )k have a common weak* limit. Since (νn k )k converges to ν and (νn∗k )k converges to f ∗R ν in the weak* topology, it follows that ν is an f R -invariant probability measure. Now, we prove the absolute continuity and the uniform bounds on the densities of the conditional measures of ν. By Lemma there are finite partitions U1 ≺ U2 ≺ 4.6, ∞ Ui . Given any γ ∈  u , take for each · · · into u-subsets of  such that U = i=1 i ≥ 1 the element Uγ ,i ∈ Ui such that γ ∩=



Uγ ,i .

i≥1

Slightly enlarging the sets Uγ ,i , if necessary, we may assume that ν(∂Uγ ,i ) = 0. Assuming that γ ∈  u and n ≥ 1 are fixed, set for each i ≥ 1 and 0 ≤ j < n   i, j = γ ∈  u : ( f R )∗j m γ0 (γ ∩ ) > 0 . Given a Borel set A ⊂ γ ∩  with m γ (∂ A) = 0, let S A be the s-subset of  formed by the union of the stable disks passing through the points in A. For any γ ∈ i, j we have  γ R j ζ j dm γ . ( f )∗ m γ0 (γ ∩ A) = γ ∩S A

Choosing any point z ∈ γ ∩ , it follows from Corollary 4.4 that there is some uniform constant C0 > 0 such that

4.2 SRB Measures

113

1 γ γ ζ (z )m γ (γ ∩ S A ) ≤ ( f R )∗j m γ0 (γ ∩ S A ) ≤ C0 ζ j (z )m γ (γ ∩ S A ). C0 j In particular, for A = γ ∩ , we have S A = , and therefore 1 γ γ ζ (z )m γ (γ ∩ ) ≤ ( f R )∗j m γ0 (γ ∩ ) ≤ C0 ζ j (z )m γ (γ ∩ ). C0 j It follows from the last two displayed formulas that ( f R )∗ m γ0 (γ ∩ S A ) m γ (γ ∩ S A ) 1 m γ (γ ∩ S A ) ≤ . ≤ C02 2 m (γ ∩ ) j m γ (γ ∩ ) C0 γ ( f R )∗ m γ0 (γ ∩ ) j

By (Y5 ) and (4.3), there is some uniform constant C1 > 0 such that m γ (γ ∩ S A ) 1 1 ≤ C1 and ≤ ≤ m γ (γ ∩ ) ≤ C1 C1 m γ (A) C1 It follows from the last two displayed formulas that ( f R )∗ m γ0 (γ ∩ S A ) 1 m (A) ≤ ≤ C02 C12 m γ (A). γ j C02 C12 ( f R )∗ m γ0 (γ ∩ ) j

(4.12)

Now, observing that for all i ≥ 1 either we have νn (Uγ ,i ) = 0 or i, j = ∅. In the latter case, we may write νn (Uγ ,i ∩ S A ) =

n−1 1 ( f R )∗j m γ0 (γ ∩ S A ). n j=0 γ ∈ i, j

We have in particular for A = γ νn (Uγ ,i ) =

n−1 1 ( f R )∗j m γ0 (γ ∩ ). n j=0 γ ∈ i, j

It follows that ( f R )∗ m γ0 (γ ∩ S A ) j

min

γ ∈i, j

j

( f R )∗ m γ0 (γ ∩ )

( f R )∗ m γ0 (γ ∩ S A ) νn (Uγ ,i ∩ S A ) ≤ max γ ∈i, j ( f R ) j m (γ ∩ ) νn (Uγ ,i ) ∗ γ0 j

≤

114

4 Hyperbolic Structures

Thus, using (4.12) and taking C = C02 C12 , we have νn (Uγ ,i ∩ S A ) 1 m γ (A) ≤ ≤ Cm γ (A). C νn (Uγ ,i ) Taking limits in n, we obtain ν(Uγ ,i ∩ S A ) 1 m γ (A) ≤ ≤ Cm γ (A). C ν(Uγ ,i ) Then, by the Martingale Convergence Theorem, 1 m γ (A) ≤ νγ (A) ≤ Cm γ (A). C This shows that νγ |γ ∩  m γ |γ ∩ , for ν¯ almost all γ ∈  u , with the densities bounded from above and below by uniform positive constants. It remains to verify that ν is ergodic, the basin of ν covers m almost all of  and ν is the unique SRB measure for f R . The ergodicity of ν is an immediate consequence of Lemma 4.5. It follows from Proposition 2.12 that the basin of ν covers ν almost all of . This is equivalent to say that νγ almost all of γ ∩  belongs in the basin of ν, for ν¯ almost all γ ∈  u . Therefore, by the equivalence of the conditional measures, there is some γ0 ∈  u such that m γ0 almost every point in γ0 ∩  belongs in the basin of ν. By the property (Y2 ) of contraction on stable disks, we have that Birkhoff averages of continuous functions are constant on each γ ∈  s . It follows from (Y5 ) that m γ almost every point in γ belongs in the basin of ν, for all γ ∈  u . Thus, m almost all of  belongs in the basin of ν. Finally, assume that ν is another SRB measure. It follows from Lemma 4.5 that ν is necessarily ergodic. Moreover, since ν is an SRB measure, the absolute continuity of its conditional measures gives that, for some γ ∈  u , there is a subset of γ with positive m γ measure whose points belong in the basin of ν . Using once more the contracting property on stable disks we deduce that the basin of ν contains a subset of  with positive m measure. Since the basin of ν contains m almost all of , it follows that the basins of the SRB measures ν  and ν are not disjoint, and therefore ν = ν . Remark 4.8 In the case of a full Young structure, the construction of the measure ν in the previous theorem gives that if a point z belongs in the support of ν, then the intersection of  with the unstable disk containing z is contained in the support of ν. This is an immediate consequence of Rohlin Disintegration Theorem and the fact that the densities of the conditional measures on unstable disks are bounded from below by a positive constant.

4.2 SRB Measures

115

4.2.2 Original Dynamics The next result ensures, in particular, the existence of ergodic SRB measures for diffeomorphisms admitting some set with a Young structure and integrable recurrence times. Theorem 4.9 If f : M → M has a set  with a Young structure with integrable recurrence time R, then f has a unique ergodic SRB measure μ with μ() > 0. Moreover, the measure μ is given by ∞

1 f ∗j (ν|{R > j}), ν{R > j} j=0 j=0

μ = ∞

where ν is the unique SRB measure for f R , and R ∈ L 1 (ν). Proof We will prove that the measure μ given by the formula has the desired properties. The f -invariance and the ergodicity of μ are a consequence Theorem 3.18; recall Remark 3.17. In addition, by Theorem 4.7, the densities dνγ /dm γ are uniformly bounded from above and below by positive constants. Since f is a diffeomorphism, j then f ∗ ν has also absolutely continuous conditional measures on the set of unstable disks { f j (γ ) : γ ∈  u }, for all j ≥ 0. This gives that μ is an SRB measure for f . Now, assuming R integrable with respect to some m γ , for some γ ∈  u , it follows from the regularity property (Y5 ) and Rohlin Disintegration Theorem that ∞

 ν{R > j} =

Rdν < ∞.

j=0

From the definition of μ we easily get μ ≥ ν, and therefore ν() > 0. j=0 ν{R > j}

μ() ≥ ∞

(4.13)

We are left to prove the uniqueness. Let μ be another ergodic SRB measure for f such that μ () > 0. Take an arbitrary continuous ϕ : M → R. Since μ () > 0, it follows from Birkhoff Ergodic Theorem and the absolute continuity of the conditional measures of μ that there exists γ ∈  u and A ⊂  with m γ (A) > 0 such that  n−1 1 j lim ϕ( f (x)) = ϕdμ , for every x ∈ A. n→∞ n j=0

(4.14)

On the other hand, since (4.13) holds and ν has conditional measures on unstable disks whose densities with respect to the conditional measures of m are bounded from below by some positive constant, the same holds for the conditionals of μ. Then,

116

4 Hyperbolic Structures

using that Birkhoff time averages of continuous functions are constant on stables disks, it follows from the ergodicity of μ and (Y5 ) that, for all γ ∈  u and m γ almost all x ∈ γ , we have  n−1 1 j ϕ( f (x)) = ϕdμ. (4.15) lim n→∞ n j=0   From (4.14) and (4.15), we get ϕdμ = ϕdμ. Since ϕ is an arbitrary continuous  function, we must have μ = μ. In the next result, we give a description of the support of the measure given by Theorem 4.9. Proposition 4.10 Let f : M → M have a set with a full Young structure  contained in some compact transitive set  ⊂ M such that f | is continuous. If ν0 is the unique SRB measure for the return map f R associated with , then the support of the j measure ν = ∞ j=0 f ∗ (ν0 |{R > j}) coincides with . Proof First, we show that the support of ν is contained in . It is enough to show that any open set with positive ν measure necessarily intersects . Consider an open set U ⊂ M with ν(U ) > 0. Taking into the account the expression of ν, there must be some j ≥ 0 for which ν0 ( f − j (U ) ∩ ) > 0. This implies that f − j (U ) intersects  ⊂ . Since we assume  a transitive compact set and f | continuous, then  is a forward invariant set, and therefore, U intersects . This gives that the support of ν is contained in . Now, we show that the support of ν contains . It is enough to prove that any open set intersecting  has positive ν measure. Let U be an open set in M such that U ∩  = ∅. Consider  s and  u the continuous families of C 1 disks associated with the full Young structure. It follows from Remark 4.8, that the support of ν0 contains some unstable disk γ0u ∈  u . Since ν ≥ ν0 , the support of ν contains γ0u as well. Then, the union U0 of all stable disks in  s is a set with nonempty interior in M that intersects . Since  is a transitive set and f | is continuous, then  is f -invariant. Moreover, Lemma 2.13, gives that there is some point z ∈  whose forward orbit is dense in . Without loss of generality, we may assume that z ∈ U0 . Let γ0s be the disk in  s containing z 0 and x0 be the intersection of γ0s with γ0u . Since z 0 has dense orbit in  and z 0 belongs in the same stable disk of x0 , then there is some n 0 ≥ 1 such that f n 0 (x0 ) belongs in the open set U . By the continuity of f n 0 in x0 ∈ , the set f −n 0 (U ) is a neighbourhood of x0 . Recalling that x0 belongs in the support of the measure ν which is f -invariant, we have ν(U ) = ν( f −n 0 (U )) > 0. This shows that every point in  belongs in the support of ν.  The main objective of the upcoming sections is to provide rates for the decay of correlations with respect to the SRB measure given by Theorem 4.9. This will be achieved for both polynomial and (stretched) exponential rates, depending on the rates for the recurrence times.

4.3 Tower Extension

117

4.3 Tower Extension In this section, we consider a tower map associated with the return map f R :  →  of a set  with a Young structure. In fact, we will consider two tower maps: one associated with the return map f R , another one associated with a quotient of f R . We will se that the latter is actually a quotient of the first one. As in Section 3.3, consider the tower associated with f R :  →    ˆ = (x, ) : x ∈  and 0 ≤  < R(x) ,  : ˆ → , ˆ given by and the tower map T  (x, ) = (x,  + 1), if  < R(x) − 1; T ( f R (x), 0), if  = R(x) − 1. It is worth mentioning that the current situation is not exactly as in Sect. 3.3, as we do not have a mod 0 partition of the set  with respect to a reference measure. Nevertheless, the Markov property (Y1 ) provides a countable collection of pairwise disjoint subsets of  that induces an m γ mod 0 partition on each γ ∩  with γ ∈  u ; recall Remark 3.22. All objects can therefore be defined as in Sect. 3.3, with the role of the reference measure being played by m γ on each γ . ˆ 0 of the tower  ˆ is naturally identified with the set , and As before, the base  ˆ each level  with the set {R > } ⊂ . This allows us to refer to stable and unstable disks through points in the tower, naturally considering the corresponding disks of their representatives in the base. To simplify the notation, we often omit the level coordinate in the representation of a point in the tower. ˆ 0 whose union Since {1 , 2 , . . . } is a family of pairwise disjoint subsets of  u intersects each unstable disk γ ∈  on a full m γ measure subset of points in γ ∩ , ˆ  with similar properties. Collecting we may consider subfamilies on each level  ˆ of pairwise disjoint subsets of all these subfamilies, we obtain a countable family Q u ˆ  whose union intersects each unstable disk γ ∈  on a full m γ measure subset of points. Given n ≥ 1, set n−1  ˆn = ˆ − j Q. Q (4.16) T j=0

We have seen in Theorem 4.7 that the return map f R has a unique SRB measure. In the next result, we show that this SRB measure gives rise to a unique ergodic SRB . Note that, as each level of the tower is identified with a subset of measure for T the set with a product structure, it still makes sense to talk about SRB measures for tower maps. Considering, as before, the map π :

ˆ −→ M  (x, ) −→ f  (x),

(4.17)

118

4 Hyperbolic Structures

Lemma 3.26 gives that

. f ◦π =π ◦T

(4.18)

The next result gives in particular that the push-forward under π of the unique ergodic SRB measure in the tower coincides with the ergodic SRB measure given by Theorem 4.9. This means that, once again, we can interpret the tower system as an extension of the initial dynamical system.  the tower map of the return map f R of a set  with a Young Theorem 4.11 Let T structure  and integrable recurrence times. If ν is the unique SRB measure for f R , then ∞ 1 ∗j (ν|{R > j}) νˆ = ∞ T j=0 ν{R > j} j=0 . Moreover, νˆ is ergodic and μ = π∗ νˆ is the unique is the unique SRB measure for T SRB measure for f with μ() > 0. R :  ˆ 0 is naturally identified with  and T ˆ0 → ˆ 0 is identified Proof Note that  R R has a unique ergodic with the return map f . It follows from Theorem 4.7 that T SRB measure ν. Moreover, as we assume the Young structure with integrable recurrence times, Theorem 4.9 gives that R ∈ L 1 (ν). Then, by Theorem 3.18, ∞

1 ∗j (ν|{R > j}) T ν{R > j} j=0 j=0

νˆ = ∞

-invariant probability measure; recall Remark 3.17. Since T  is an is an ergodic T ˆ j coincides upward translation in all but the roof levels of the tower, we have that νˆ | with ν|{R > j}, for all j ≥ 0, up to a multiplicative constant. Therefore, the absolute continuity (the equivalence, in fact) of the conditional measures of νˆ follows easily from that property for the measure ν. -invariant probability measure, then Let us now prove the uniqueness. If νˆ is a T R  ˆ ˆ ν = (ˆν |0 )/ˆν (0 ) is a T -invariant probability measure, by Proposition 3.23. More, we clearly have that ν is an SRB over, assuming that νˆ is an SRB measure for T R  measure for T . Since this SRB measure ν is unique, by Theorem 4.7, and νˆ can be recovered from ν, by Proposition 3.23, we obtain the uniqueness of νˆ . Finally, we show that μ = π∗ νˆ is the unique ergodic SRB measure for f with μ() > 0. Since we assume the Young structure with integrable recurrence times, it follows from Theorem 4.9 that ∞

1 f ∗j (ν|{R > j}) j=0 ν{R > j} j=0

μ = ∞

(4.19)

4.3 Tower Extension

119

is the unique SRB measure of f with μ() > 0, where ν is the unique ergodic SRB measure for f R , given by Theorem 4.7. On the other hand, letting νˆ be the SRB , it follows from Lemma 3.26, that measure for T π∗ νˆ =

∞

f ∗j (ˆν |{R > j}).

(4.20)

j=0

Then, the uniqueness of ν and Proposition 3.23, imply that 1 ν. ν{R > j} j=0

νˆ |0 = ∞

(4.21)

Since {R > j} ⊂ 0 , for each j ≥ 0, it follows from (4.19), (4.20) and (4.21) that . the measure π∗ νˆ coincides with μ. The next result will be useful in the proof of Lemma 4.17, to deduce that the lift of a Hölder continuous observable to the tower still has some regularity properties. ˆ and n ≥ 1, set Given x ∈     j (x) ∈  ˆ0 . (4.22) bn (x) = # 1 ≤ j ≤ n : T Note that bn is constant on the elements of Qn . For the next result, consider a constant 0 < β < 1 as in the conditions that define the Young structure on the set . ˆ 2n and x, y ∈ Q, Proposition 4.12 There is C > 0 such that, for all n ≥ 1, Q ∈ Q   n (x)), π(T n (y))) ≤ C β bn (x) + β bn (Tn (x)) . d(π(T ˆ 2n and x, y ∈ Q. It follows that b2n (y) = b2n (x) and in parProof Consider Q ∈ Q ticular, bn (y) = bn (x). Take z ∈ Q such that z ∈ γ u (x) ∩ γ s (y). We have n (y))) ≤ d(π(T n (x)), π(T n (z))) + d(π(T n (z)), π(T n (y))). n (x)), π(T d(π(T (4.23) Let us estimate the two terms on the right hand side of (4.23). For the first one, note ˆ n , the points T n (x), T n (z) belong in the same unstable disk n (Q) ∈ Q that, since T ˆ n . Consider  ≥ 0 such that Q ⊂  ˆ  and take x0 , z 0 ∈  ˆ0 of some element Q ∈ Q  n  (x0 ) = T  (x) and T  (z 0 ) = T n (z). Note that such that T n (x)). bn (z 0 ) = bn (x0 ) = bn (T It follows from (4.18), (4.24) and (Y3 ) that

(4.24)

120

4 Hyperbolic Structures

n (x)), π(T n (z))) = d(π(T  (x0 )), π(T  (z 0 ))) d(π(T = d( f  (π(x0 )), f  (π(z 0 ))) = d( f  (x0 ), f  (z 0 )) ≤ Cd( f R (x0 ), f R (z 0 )) n (x))−1

≤ C 2 β bn (T

.

(4.25)

In this last inequality, we have used (4.24), and the fact that the points x0 , z 0 have the ˆ 0 until time n, always falling into the same elements in same number of returns to  ˆ 0 . This is because x0 , z 0 ∈ Q ∈ Q2n . Now, we estimate the second term partition of  ˆ0 ˆ  and y0 , z 0 ∈  on the right hand side of (4.23). Consider  ≥ 0 such that Q ⊂   (y0 ) = T n (y) and T  (z 0 ) = T n (z). Note that y0 , z 0 belong in a same such that T ˆ 0 . It follows from (4.18) that stable disk in  n (y))) = d(π(T n+ (z 0 )), π(T n+ (y0 ))) n (z)), π(T d(π(T = d( f n+ (π(z 0 )), f n+ (π(y0 ))) = d( f n+ (z 0 ), f n+ (y0 )).

(4.26)

Consider b = bn+ (y0 ) = bn (y) = bn (x) ≥ 0.  R j Assume first that b ≥ 1. Setting Rb = b−1 j=0 R(( f ) (y0 )), we have n +  = Rb + Rb j, for some 0 ≤ j < R( f (y0 )). It follows from (Y2 ) that d( f n+ (z 0 ), f n+ (y0 )) = d( f j ( f Rb (z 0 )), f j ( f Rb (y0 ))) ≤ Cd( f Rb (z 0 ), f Rb (y0 )) = Cd(( f R )b (z 0 ), ( f R )b (y0 )).

(4.27)

If on the other hand b = 0, then we have n +  < R(y0 ), and it follows from (Y2 ) that d( f n+ (z 0 ), f n+ (y0 )) ≤ Cd(y0 , z 0 ), thus extending (4.27) also to the case b = 0. Recalling that b = bn (x), it follows from (4.26), (4.27) and (Y2 ) that n (y))) ≤ C 2 β bn (x) . n (z)), π(T d(π(T

(4.28)

From (4.23), (4.25) and (4.28), we easily get the conclusion.

.

4.3.1 Quotient Tower Here, we analyse the relationship between the tower associated with the return map f R of a set with a Young structure and the tower associated with the quotient of f R . Fix some γ0 ∈  u and consider the quotient map

4.3 Tower Extension

121

F : γ0 ∩  → γ 0 ∩  as in (4.5). It follows from Proposition 4.2 that F is Gibbs-Markov with respect to the m γ0 mod 0 partition P = {γ0 ∩ 1 , γ0 ∩ 2 , . . . } of γ0 ∩ . We can therefore consider the tower map T :→ of this Gibbs-Markov map F with recurrence time R. As before, we keep denoting the reference measure on  by m γ0 . Notice that R|γ0 ∩i = R|i = Ri , for all i ≥ 1. Since γ0 ∩  ⊂ , it follows that, for all  ≥ 0 | , ˆ  and T | = T  ⊂ 

(4.29)

 as in the beginning of Section 4.3. Hence, it makes sense to consider ˆ  and T with  the map ˆ −→   :    (4.30) (x, ) −→ γ0 (x),  . It is straightforward to check that  ◦  =  ◦ T, T

(4.31)

 and T . In the next result thus  being a semiconjugacy between the tower maps T , ν) we see that this semiconjugacy gives in fact that (T ˆ is an extension of (T, ν). , then ∗ νˆ is the unique Proposition 4.13 If νˆ is the ergodic SRB measure for T ergodic T -invariant probability measure absolutely continuous with respect to m γ0 . Proof By Theorem 4.11, we have ∞

1 ∗j (ν|{R > j}), T j=0 ν{R > j} j=0

νˆ = ∞

(4.32)

where ν is the unique SRB measure for f R . On the other hand, by Theorem 3.13, there is a unique F-invariant probability measure ν0  m γ0 . Under the identification ˆ 0 and , we clearly have | ˆ 0 = γ0 . It follows from Lemma 4.5 that of  ∗ ν = (γ0 )∗ ν = ν0 .

(4.33)

Since R is constant on stable disks, we have ∞ j=0

ν{R > j} =

∞ j=0

ν{R ◦ γ0 > j} =

∞ j=0

(γ0 )∗ ν{R > j} =

∞

ν0 {R > j}.

j=0

(4.34)

122

4 Hyperbolic Structures

Using (4.32), (4.33), (4.34) and the fact that  is a semiconjugacy, we obtain ∞

1 ∗j (ν|{R > j}) ∗ T ν {R > j} 0 j=0 j=0

∗ νˆ = ∞

∞

1 T∗j ∗ (ν|{R > j}) ν {R > j} 0 j=0 j=0

= ∞

∞

1 T∗j (ν0 |{R > j}). j=0 ν0 {R > j} j=0

= ∞

Now, Theorem 3.24 gives that ∗ νˆ is the unique ergodic T -invariant probability . measure absolutely continuous with respect to m γ0 . We finish this subsection with a result relating the function bn in (4.22) with the tail of recurrence times. Since (4.29) holds, each bn can be thought of as defined in  as well. Set for each k ≥ 1 k−1 R ◦ T j. (4.35) Rk = j=0

The next result will be particularly useful in conjunction with Proposition 4.12 for the proof of Lemma 4.17 in Sect. 4.4.1. Proposition 4.14 Given any 0 < σ < 1, there is C > 0 such that, for all n ≥ 1,  σ bn dν ≤ C



m γ0 {R ≥ k} + Cn

k≥n/3

 n . σ k m γ0 R k > 3 k≥1



Proof First of all, note that, by Theorem 3.13, it is sufficient to get the conclusion with the measure ν0 instead of m γ0 , where ν0 is the unique F-invariant measure such that ν0  m γ0 . For each n ≥ 1, we have  σ dν = bn

∞

σ k ν{bn = k}.

k=0

Given x with bn (x) ≥ 1, set rn (x) = min{1 ≤ j ≤ n : T j (x) ∈ 0 } and sn (x) = max{1 ≤ j ≤ n : T j (x) ∈ 0 }.

(4.36)

4.3 Tower Extension

123

For each k ≥ 1, we may write {bn = k} =



{bn = k, rn = i, sn = j}

1≤i≤ j≤n

⊂

 1≤i j} = ν(0 )

j≥n



ν0 {R > j} ≤ ν(0 )

j≥n



ν0 {R ≥ j}.

j>n/3

Together with (4.36) and (4.41), this gives the desired conclusion.

.

4.4 Decay of Correlations The goal of this section is to prove Theorem 4.15 below, that provides estimates on the decay of correlations for the SRB measures given by Theorem 4.9 for diffeomorphisms admitting Young structures. We draw conclusions for polynomial and (stretched) exponential decay, depending on the tail of recurrence times. Theorem 4.15 Let f : M → M have a Young structure  with integrable recurrence time R and μ be the unique ergodic SRB measure for f with μ() > 0. If gcd(R) = q, then f q has p ≤ q exact invariant probability measures μ1 , . . . , μ p with f ∗ μ1 = μ2 , . . . , f ∗ μ p = μ1 and μ = (μ1 + · · · + μ p )/ p. Moreover, for all 1≤i ≤ p 1. if m γ {R > n} ≤ Cn −a for some γ ∈  u , C > 0 and a > 1, then for all ϕ, ψ ∈ Hη there is C > 0 such that Cor μi (ϕ, ψ ◦ f qn ) ≤ C n −a+1 ; 2. if m γ {R > n} ≤ Ce−cn for some γ ∈  u , C, c > 0 and 0 < a ≤ 1, then given η > 0, there is c > 0 such that, for all ϕ, ψ ∈ Hη , there is C > 0 for which a

a

Cor μi (ϕ, ψ ◦ f qn ) ≤ C e−c n . The remainder of this section is devoted to the proof of Theorem 4.15. Consider a compact set  having a Young structure with integrable recurrence time R, given by families of disks  s and  u . Fixing some γ0 ∈  u , let F : γ0 ∩  → γ 0 ∩ 

(4.42)

4.4 Decay of Correlations

125

be the quotient map of f R as in (4.5). By (Y5 ), we may assume without loss of generality that the polynomial or (stretched) exponential decay for the measure of {R > n} holds with respect to the measure m γ0 on the unstable disk γ0 . Consider the tower maps introduced in Sect. 4.3 : ˆ → ˆ and T :  → , T associated with f R and F, respectively, the latter tower map with recurrence time R. Notice that R is constant on the elements of the partition associated with F as well. Consider also ˆ → M and  :  ˆ → π : as in (4.17) and (4.30), respectively. By (4.18) and (4.31), we have  = f ◦ π and  ◦ T  = T ◦ . π ◦T

(4.43)

, it follows from Theorem 4.11 Considering νˆ the unique ergodic SRB measure for T and Proposition 4.13 that μ = π∗ νˆ and ν = ∗ νˆ ,

(4.44)

where ν is the unique ergodic T -invariant measure such that ν  m γ0 . Given ϕ, ψ ∈ Hη , set ϕˆ = ϕ ◦ π and ψˆ = ψ ◦ π. (4.45) Once here, it might be a good idea to keep in mind the following commuting diagram: 

T

 ˆ 

  T

π M



ˆ  π

f

M

ϕˆ ϕ

R.

 and f , it follows from Since μ = π∗ νˆ and π is a semiconjugacy between T Lemma 3.50, that n ). ˆ ψˆ ◦ T (4.46) Cor μ (ϕ, ψ ◦ f n ) = Cor νˆ (ϕ,

126

4 Hyperbolic Structures

Therefore, the conclusions of Theorem 4.15 follow once obtained the respective n ). In the next subsection, we reduce this to a problem ˆ ψˆ ◦ T estimates for Cor ν (ϕ, on the quotient tower T :  → . The proof of Theorem 4.15 will be completed in Sect. 4.4.2 and Sect. 4.4.3, first in the case gcd(R) = 1. In this case, we have necessarily p = 1 and μ1 = μ. The general case will deduced from the particular case in Sect. 4.4.4.

4.4.1 Reducing to the Quotient Tower ˆ → R as in (4.45), let ϕk :  ˆ →R Take an arbitrary n ≥ 1 and k ≈ n/4. Given ϕˆ :  ˆ 2k by be the discretisation of ϕ, defined on each Q ∈ Q   k (x) : x ∈ Q , ϕk | Q = inf ϕˆ ◦ T

(4.47)

ˆ 2k as in (4.16). Note that this discretization ϕk can be as well interpreted as a with Q ˆ → R is defined in function on the quotient tower . A discretization ψk for ψˆ :  the same way. We use  0 to denote the sup norm of continuous functions on M and  1 to ˆ Since ∗ νˆ = ν, denote the L 1 -norm with respect to the probability measure νˆ on . 1 this last norm coincides with the L -norm with respect to the probability measure ν on  for functions which are constant on stable disks. Proposition 4.16 For all 1 ≤ k ≤ n, we have n ) ≤ Cor ν (ϕk , ψk ◦ T n ) + 2ϕ0 ψˆ ◦ T k − ψk 1 + 2ψ0 ϕˆ ◦ T k − ϕk 1 . Cor νˆ (ϕ, ˆ ψˆ ◦ T

-invariant, we may write Proof Since the probability measure νˆ is T        n )ϕd n ) =  (ψˆ ◦ T ˆ νˆ ϕd ψd ˆ ν ˆ Cor νˆ (ϕ, ˆ ψˆ ◦ T ˆ ν ˆ −         n−k ϕd ≤  ψk ◦ T ˆ νˆ − ψk d νˆ ϕd ˆ νˆ           k − ψk ) ◦ T n−k ϕd k )d νˆ ϕd +  (ψˆ ◦ T ˆ νˆ  +  (ψk − ψˆ ◦ T ˆ νˆ  n−k ) + 2ψˆ ◦ T k − ψk 1 ϕ0 .. ≤ Cor νˆ (ϕ, ˆ ψk ◦ T

(4.48)

-invariant, then T ∗ preserves absolute continuity with respect to Note that as νˆ is T ∗k (ϕk νˆ ) with respect to νˆ , we have νˆ . Defining ζk the density of the signed measure T

4.4 Decay of Correlations

127

       n−k ) =  ψk ◦ T n−k ϕd Cor νˆ (ϕ, ˆ ψk ◦ T ˆ ν ˆ − ψ d ν ˆ ϕd ˆ ν ˆ k         n−k ζk d νˆ − ψk d νˆ ζk d νˆ  ≤  ψk ◦ T           n−k )(ϕˆ − ζk )d νˆ  +  ψk d νˆ (ϕˆ − ζk )d νˆ  +  (ψk ◦ T        n−k ) + 2ψ0  (ϕˆ − ζk )d νˆ  . (4.49) ≤ Cor νˆ (ζk , ψk ◦ T   Observing that

∗k ((ϕˆ ◦ T ∗k (ϕk νˆ ) k )ˆν ) dT dT = ϕˆ and = ζk , d νˆ d νˆ

we obtain            (ϕˆ − ζk )d νˆ  =  d T ∗k ((ϕˆ ◦ T k − ϕk |d νˆ , k )ˆν ) − d T ∗k (ϕk νˆ ) ≤ |ϕˆ ◦ T     which together with (4.49) gives n−k ) ≤ Cor νˆ (ζk , ψk ◦ T n−k ) + 2ψ0 ϕˆ ◦ T k − ϕk 1 . ˆ ψk ◦ T Cor νˆ (ϕ, Recalling (4.48), it remains to obtain n−k ) = Cor ν (ϕk , ψk ◦ T n ). Cor νˆ (ζk , ψk ◦ T

(4.50)

First of all, observe that    n−k n−k   ∗n (ϕk νˆ )). ˆ = ψk d(T (ψk ◦ T )ζk d νˆ = ψk d(T∗ (ζk ν)) Since ψk and ϕk are constant on stable disks and (4.43), (4.44) hold, we may write 

∗n (ϕk νˆ )) = ψk d(T

It follows that





∗n (ϕk νˆ )) = ψk d(∗ T

n−k )ζk d νˆ = (ψk ◦ T



 ψk d(T∗n (ϕk ν)) =

(ψk ◦ T n )ϕk dν.

 (ψk ◦ T n )ϕk dν.

On the other hand,       k  ψk d νˆ ζk d νˆ = ψk dν d(T∗ (ϕk νˆ )) = ψk dν ϕk dν. The last two formulas yield (4.50).



128

4 Hyperbolic Structures

In the next result, we give an estimate for the L 1 -norm terms in Proposition 4.16. Since ϕ and ψ play symmetric roles in that part, it is enough to consider the case of ϕ. As in (4.35), set R =

−1

R ◦ F j , for all  ≥ 0,

j=0

where F : γ0 ∩  → γ0 ∩  is the Gibbs-Markov quotient map in (4.42) associated with the return map f R of the set  with a Young structure. The next result is essentially a corollary of Proposition 4.12 and Proposition 4.13. Lemma 4.17 There is C > 0 such that, for all k ≥ 1, k − ϕk 1 ≤ C ϕˆ ◦ T



m γ0 {R ≥ k} + Ck

≥k/3



η



β m γ0

≥1

 k R > . 3

ˆ 2k . By definition of ϕˆ and ϕk , for each x ∈ Q, there Proof Take any k ≥ 1 and Q ∈ Q n (y); recall (4.45) and (4.47). Hence, is y ∈ Q such that ϕk (x) = ϕ ◦ π ◦ T k (x) − ϕ ◦ π ◦ T k (y)| k (x) − ϕk (x)| = |ϕ ◦ π ◦ T |ϕˆ ◦ T k (x)), π(T k (y)))η . ≤ |ϕ|η d(π(T By Proposition 4.12, there is C0 > 0 such that, for all x ∈ Q,   k (y))) ≤ C0 β bk (x) + β bk (Tk (x)) . k (x)), π(T d(π(T Since 0 < η < 1, we have C1 > 0 (depending on |ϕ|η ) such that     ϕ ◦ π ◦ T n (x) − ϕk (x) ≤ C1 β ηbn (x) + β ηbn (Tn (x)) . ˆ 2k and using the T -invariance of ν, ˆ we obtain Integrating over the sets Q ∈ Q k − ϕk 1 ≤ C1 ϕˆ ◦ T

    k ˆ σ bk + σ bk ◦T d νˆ = 2C1 σ bk d ν.

Since bk is constant on stable disks, it follows from Proposition 4.13 and (4.44) that 

 σ d νˆ = bk

 σ

bk

◦ d νˆ =

 σ d∗ νˆ =

Using Proposition 4.14, we conclude the proof.

bk

σ bk dν. 

4.4 Decay of Correlations

129

In the next two subsections, we give the final estimates on the correlation term in Proposition 4.16, and on the upper bound for the L 1 -norms in Proposition 4.16, given by Lemma 4.17.

4.4.2 Regularity of the Discretisation Here, we control the correlation term in Proposition 4.16. For this, we use the assumption gcd(R) = 1. In fact, this is the only place where this assumption plays a role. The control of the correlation term is essentially a consequence of Theorem 3.28. First, we need to verify that the density of a certain push-forward of the measure whose density is the discretisation in (4.47), is regular enough. This will be done in Proposition 4.18. Suppose, without loss of generality, that ϕk = 0. Consider ϕk∗ associated with ϕk as in (3.133) and λ∗k the probability measure on  whose density with respect to ν is ϕk∗ . Since ϕk ∞ ≤ ϕ0 and ψk ∞ ≤ ψ0 , it follows from Lemma 3.49 that Cor ν (ϕk , ψk ◦ T n ) ≤ 3ϕ0 ψ0 |T∗n λ∗k − ν|.

(4.51)

Set λk = T∗2k λ∗k and φk the density of λk with respect to m γ0 . We have φk =

dT∗2k λ∗k dλ∗k dλk = and = ϕk∗ ρ. dm γ0 dm γ0 dm γ0

(4.52)

where ρ = dν/dm γ0 . Since T∗n λ∗k = T∗n−2k λk , it follows from (4.51) that Cor ν (ϕk , ψk ◦ T n ) ≤ 3ϕ0 ψ0 |T∗n−2k λk − ν|.

(4.53)

Comparing (4.51) with (4.53), we have used 2k iterates to move from the measure λ∗k to the measure λk . This was done for the sole purpose of making the density of the measure more regular, and is fundamental in the proof of the next result. We consider the space Fβ+η () as in (3.30), with 0 < β < 1 as in the Gibbs property (G3 ) for the Gibbs-Markov map F; recall also (3.31). Proposition 4.18 There is C > 0 such that φk ∈ Fβ+ () and Cφ+k ≤ C, for all k ≥ 1. Proof Given k ≥ 1, consider Q2k as in (3.23). Consider the elements of Q2k listed as Q 1 , Q 2 , . . . and set Ti2k = T 2k | Q i , for all i ≥ 1. Observe that each Q i is a domain of injectivity of T 2k . Denote by Ti−2k the inverse of Ti2k on Ti2k (Q i ). From the second item of Lemma 2.11 applied to the measures ν and λk and (4.52), we obtain ρ=

ρ ◦ T −2k i

1 2k and φk = −2k Ti (Q i )

i≥1

JT 2k ◦ Ti

(ϕ ∗ ρ) ◦ T −2k k

i≥1

i

JT 2k ◦ Ti−2k

1Ti2k (Q i ) .

(4.54)

130

4 Hyperbolic Structures

Note that 1/3 ≤ ϕk∗ ≤ 3, by Lemma 3.49, and ρ = dν/dm γ0 belongs in Fβ+ (), by Theorem 3.24. Thus, there is some constant C0 > 0 not depending on k such that 1 ≤ φk ≤ C 0 . C0

(4.55)

It remains to check that φk ∈ Fβ (), with Cφk uniformly bounded. By (4.54) we have         (ϕk∗ ρ) Ti−2k (x) (ϕk∗ ρ) Ti−2k (y)  |φk (x) − φk (y)| =  −2k  1Ti2k (Q i ) (x) −  −2k  1Ti2k (Q i ) (y)  .   JT 2k Ti (x)  2k J (y) T T i i≥1 If x, y are points in  such that, for some i ≥ 1, only one of them belongs in Ti2k (Q i ), then necessarily s(x, y) = 0. From (4.55), we get |φk (x) − φk (y)| ≤ 2C0 = 2C0 β s(x,y) .

(4.56)

Otherwise, we have x ∈ Ti2k (Q i ) if, and only if, y ∈ Ti2k (Q i ). For those values of i ≥ 1 such that x, y ∈ Ti2k (Q i ), let xi , yi ∈ Q i be such T 2k (xi ) = x and T 2k (yi ) = y. We have  ϕ ∗ (xi )ρ(xi ) ϕ ∗ (yi )ρ(yi )  k   k (4.57) |φk (x) − φk (y)| =  J 2k (x ) − J 2k (y )  1Ti2k (Q i ) (x). i i T T i≥1 Since 1/3 ≤ ϕk∗ ≤ 3 and ϕk∗ is constant on each Q i , we have for each i ≥ 1   ∗  ϕk (xi )ρ(xi ) ϕk∗ (yi )ρ(yi )     J 2k (x ) − J 2k (y )  ≤ i i T T

   ρ(xi ) ρ(yi )   3 . − JT 2k (xi ) JT 2k (yi ) 

(4.58)

It follows from (4.54) that

Therefore,

ρ(z i ) ≤ ρ∞ , for z i ∈ {xi , yi }. JT 2k (z i )

(4.59)

     ρ(xi )  ρ(yi )JT 2k (xi )  ρ(yi )    ≤ ρ . 1 − − ∞  J 2k (x ) JT 2k (yi )  ρ(xi )JT 2k (yi )  i T

(4.60)

By Theorem 3.24 and Lemma 3.30, there is some uniform constant C1 > 0 such that log

JT 2k (yi ) ρ(yi ) 2k 2k ≤ C1 β s(xi ,yi ) and log ≤ C1 β s(T (xi ),T (yi )) . ρ(xi ) JT 2k (xi )

Since (4.59) holds, we also have some uniform constant C2 > 0 such that

(4.61)

4.4 Decay of Correlations

131

    1 − ρ(yi )JT 2k (xi )  ≤ C2 log ρ(yi )JT 2k (xi ) .  ρ(xi )JT 2k (yi )  ρ(xi )JT 2k (yi ) Since s(xi , yi ) ≥ s(T 2k (xi ), T 2k (yi )) = s(x, y), it follows from (4.60) and (4.61), that    ρ(xi ) ρ(yi )  ρ(yi )JT 2k (xi ) s(x,y)  ,  J 2k (x ) − J 2k (y )  ≤ C2 ρ∞ log ρ(x )J 2k (y ) ≤ 2C1 C2 ρ∞ β i i i T i T T (4.62) which together with (4.54), (4.57) and (4.58) yields   ρ Ti−2k (x) |φk (x) − φk (y)| ≤ 2C1 C2 ρ∞  −2k  1Ti2k (Q i ) ≤ 2C1 C2 ρ2∞ . β s(x,y) 2k T J (x) T i i≥1 Recalling also (4.56), we clearly have φk ∈ Fβ+ () with Cφ+k bounded from above by some uniform constant.  The conclusion of Proposition 4.18, together with the assumption gcd(R) = 1, makes it possible to apply Theorem 3.28. We obtain in this way the desired conclusions for the decay of |T∗n−2k λk − ν|, and therefore for the term Cor ν (ϕk , ψk ◦ T n ) in Proposition 4.16, by (4.53). Note that, since we assume k ≈ n/4, then n − 2k ≈ n/2.

4.4.3 Specific Rates Here, we find the appropriate estimates for the upper bound of the L 1 -norm terms in Proposition 4.16 given by Lemma 4.17. For that, we no longer need to assume gcd(R) = 1. We consider the polynomial and (stretched) exponential cases separately. Polynomial Decay Assume that there exist constants C > 0 and a > 1 such that m γ0 {R > k} ≤ Ck −a .

(4.63)

For each  ≥ 1, we have

R−1 >

k 3

 =

⎧ −1 ⎨ ⎩

j=0

R ◦ Fj >

⎫ k⎬ 3⎭

⊂

−1

 j=0

R ◦ Fj >

 k . 3

(4.64)

Since F is Gibbs-Markov, by Theorem 3.13 it has a unique invariant probability measure ν0  m γ0 . Moreover, there exists C0 > 0 such that

132

4 Hyperbolic Structures

1 dν0 ≤ ≤ C0 . C0 dm γ0

(4.65)

Using the F-invariance of ν0 , we easily get ν0

 −1

j=0

k R◦F > 3



≤ ν0

j

 k . R> 3

Together with (4.63), (4.64) and (4.65), this yields

m γ0 R−1

k > 3

Hence, k



 ≤

C02 m γ0

k R> 3



≤ C02 C3a a+1 k −a .

 k σ  m γ0 R−1 > ≤ C k −a+1 , 3 ≥1



 where C = C02 C3a ≥1 σ  a+1 . Since k ≈ n/4, we obtain the desired estimate in the polynomial case for the L 1 norm terms of Proposition 4.16; recall Lemma 4.17. (Stretched) Exponential Decay Assume there exist constants C, c > 0 and 0 < a ≤ 1 such that m γ0 {R > k} ≤ Ce−ck . a

(4.66)

We start with a general probabilistic result. Proposition 4.19 Let X 0 , . . . , X  be nonnegative random variables on a space with a probability measure P. If there are C, c > 0 and 0 < a ≤ 1 such that, for all k ≥ 1, 1 ≤ j ≤  and k0 , . . . , k j−1 ≥ 0,

P{X 0 > k} ≤ C exp (−ck a ) , P{X j > k | X 0 = k0 , . . . , X j−1 = k j−1 } ≤ C exp (−ck a ) ,

then there is C > 0 such that, for all 0 < b ≤ c/2 and k ≥ 1,   P{X 0 + · · · + X  > k} ≤ (1 + bC )+1 exp −bk a . Proof Markov’s Inequality gives P{X 0 + · · · + X  > k} ≤

E(b(X 0 + · · · + X  )a ) . exp(bk a )

(4.67)

4.4 Decay of Correlations

133

Observe that  E(exp(bX 0a )) =

∞ 0

 P{exp(bX 0a ) ≥ t}dt = 1 +

∞

1

P{exp(bX 0a ) ≥ t}dt.

Making the substitution t = exp(cs a ), we obtain  E(exp(bX 0a )) = 1 + ba

∞



0

≤ 1 + Cba ≤ 1 + bC , where C = Cb



∞

s a−1 exp(bs a )P{X 0 ≥ s}ds ∞

s a−1 exp(−(b − a)s a )ds

0

s a−1 exp(−cs a /2)ds.

0

In a similar way we prove that E(exp(bX aj | X 0 , . . . , X j−1 )) ≤ 1 + bC . Hence, E(exp(b(X 0 + · · · +X  )a )) ≤ E(exp(b(X 0a + · · · + X a ))) = E(E(exp(b(X 0a + · · · + X a ))) | X 0 , . . . , X −1 ) a = E(exp(b(X 0a + · · · + X −1 ))E(exp(bX a ) | X 0 , . . . , X −1 )) a ≤ (1 + bC )E(exp(b(X 0a + · · · + X −1 ))) .. . ≤ (1 + bC )+1 .



Using (4.67), we conclude the proof. Now, defining X i = R ◦ F i for all i ≥ 0, we have for each  ≥ 1

m γ0

k R > 3



= m γ0 X 0 + · · · + X −1

 k . > 3

Our goal now is to show that the random variables X 0 , . . . , X  verify the assumptions of Proposition 4.19, with P = m γ0 . By (4.66), we have m γ0 {X 0 > k} = m γ0 {R > k} ≤ Ce−ck . a

(4.68)

For all 1 ≤ j ≤ , set ρ j = 1{X 0 =k0 ,...,X j−1 =k j−1 } . Since 1{X j =k j } = 1{R◦F j =k j } = 1{R=k j } ◦ F j , letting λ j be the measure whose density with respect to m γ0 is ϕ j , we may write

134

4 Hyperbolic Structures

 m γ0 {X 0 = k0 , . . . , X j = k j } =

1{R=k j } ◦ F j ϕ j dm γ0  = 1{R=k j } d(F∗j (λ j |ω)) ω∈P j

 =

1{R=k j }

d(F∗j (λ j |ω)) dm γ0 . dm γ0 ω∈P

(4.69)

j

It follows from Lemma 2.11 and Corollary 3.5 that there exists C0 > 0 such that for all x ∈ γ0 ∩  and ω ∈ P j , we have d(F∗ (λ j |ω)) ϕ j (x ) ≤ C0 m γ0 (ω)ϕ j (x ), (x) = dm γ0 JF  (x ) j

where x ∈ ω is such that F  (x ) = x. Since ϕ j is constant on each ω ∈ P j , we get  d(F∗j (λ j |ω)) ≤ C0 ϕ j dm γ0 = C0 m γ0 {X 0 = k0 , . . . , X j−1 = k j−1 }, dm γ0 ω∈P j

which together with (4.69) yields m γ0 {X 0 = k0 , . . . , X j = k j } ≤ C0 m γ0 {R = k j }m γ0 {X 0 = k0 , . . . , X j−1 = k j−1 }. Using (4.66), we obtain

m γ0 {X 0 = k0 , . . . , X j = k j } m {X 0 = k0 , . . . , X j−1 = k j−1 } γ 0 k j >k ≤ C0 m γ0 {R = k j }

m γ0 {X j > k | X 0 = k0 , . . . , X j−1 = k j−1 } =

k j >k

= C0 m γ0 {R > k} ≤ C0 Ce−ck . a

By Proposition 4.19, there is C > 0 such that for all 0 < b ≤ c/2 and k ≥ 1, we have



  k k ka = m γ0 X 0 + · · · + X −1 > ≤ C (1 + bC ) exp −b a . m γ0 R  > 3 3 3 where C is a normalising factor which is needed because m γ0 is not necessarily a probability measure. Hence,

4.4 Decay of Correlations

135



 k ka ≤ C σ  m γ0 R  > σ  (1 + bC ) exp −b a . 3 3 ≥1 ≥1



Choosing b > 0 such that σ (1 + bC ) < 1, we obtain the desired estimate for the L 1 terms in Proposition 4.16, also in the (stretched) exponential case. Recall that k ≈ n/4 and Lemma 4.17.

4.4.4 The Non-exact Case In this subsection, we show how to deduce the proof of Theorem 4.15 in the general case from the case already seen. Let q = gcd(R) ≥ 1. First, we prove the existence of exact SRB measures for f q . Afterwards, we derive the conclusions on the decay of correlations. SRB Measures The proof of the existence of SRB measures follows ideas similar to those in the proofs of Theorem 3.51 and Theorem 3.54. As in (3.137), consider the parˆ of the Young structure , given by ϒi = tition {ϒ1 , . . . , ϒq } of the tower  # ˆ ≡i−1(mod q)  , for each 1 ≤ i ≤ q. We have (ϒ1 ) = ϒ2 , . . . , T (ϒq−1 ) = ϒq and T (ϒq ) = ϒ1 . T , we have ν| Letting νˆ be the unique SRB measure for T ˆ ϒi an invariant measure for q  T : ϒi → ϒi , for all 1 ≤ i ≤ q. Setting νˆ i = (ˆν |ϒi )/ν(ϒi ) for each 1 ≤ i ≤ q, we have νˆ = (ˆν1 + · · · + νˆ q )/q. Moreover, ∗ νˆ q−1 = νˆ q and T ∗ νˆ q = νˆ 1 . ∗ νˆ 1 = νˆ 2 , . . . , T T Since νˆ i is the normalised restriction of an SRB measure, it easily follows that νˆ i is an q , for each 1 ≤ i ≤ q. Note that  is also a Young structure for f q , SRB measure for T  :  ˆ → ˆ , with recurrence time R = R/q. Consider the respective tower map T now with recurrence time R satisfying gcd(R ) = 1, and νˆ the unique SRB measure . Now fix 1 ≤ i ≤ q, and consider Si :  ˆ → ϒi the measurable conjugacy for T q   between T and T |ϒi given by Si (x, ) = (x, q + i − 1). q -invariant probability measure. Since the maps It follows that Si ∗ νˆ is an exact T −1 Si and Si preserve sets of m γ measure 0 on each unstable disk γ and νˆ is the unique SRB measure for T , it easily follows that Si ∗ νˆ is the unique SRB measure q |ϒi : ϒi → ϒi . Hence, Si ∗ νˆ = νˆ i and so νˆ i is exact. for T

136

4 Hyperbolic Structures

ˆ → Now, we obtain the SRB measures for f q . Consider the semiconjugacy π :  M as in (4.43), and define μi = π∗ νˆ i , for each 1 ≤ i ≤ q. We have ∗ νˆ i = π∗ νˆ j+1 = μ j+1 , f ∗ μi = f ∗ π∗ νˆ i = π∗ T

j = i (mod q).

q

anq this gives f ∗ μi = μi , for all 1 ≤ i ≤ q. Since π is only a semiconjugacy, we may have f ∗ μ p = μ1 , for some p = kq with k ∈ N. Considering the smallest p in these conditions, we have μ1 , . . . , μ p pairwise distinct exact f q -invariant probability measures. We have μ = π∗

1 1 k 1 (ˆν1 + · · · + νˆ q ) = (μ1 + · · · + μq ) = (μ1 + · · · + μ p ) = (μ1 + · · · + μ p ). q q q p

Recall that μ = π∗ νˆ is the unique ergodic SRB measure for f with μ() > 0, by Theorem 4.11. Since μi / p ≤ μ, it easily follows that μi has conditional measures on unstable disks γ absolutely continuous withe respect to m γ . Hence, μi is an exact SRB measure for f q , for each 1 ≤ i ≤ p. Decay of Correlations Take νi an exact SRB measure for f q as above. Given a Hölder continuous function ϕ : M → R, set (4.70) ϕˆ = ϕˆ ◦ Si . k−1

− j Qˆ , where Qˆ is the family of pairwise disjoint sets T ˆ → R be defined for each Q ∈ Qˆ 2k ˆ . Let also ϕk :  as in Sect. 4.3 for the tower  Given k ≥ 1, let Qˆ k =

j=0

   k (x) : x ∈ Q . ϕk | Q = inf ϕˆ ◦ T  = T q ◦ Si , it easily follows from (4.70) that Since Si ◦ T  k = ϕ ◦ π ◦ T qk ◦ Si ϕˆ ◦ T

(4.71)

for all k ≥ 1. Recalling that μi = π∗ νˆ i = π∗ Si ∗ νˆ , Lemma 3.50 gives qn ) = Cor νˆ (ϕˆ , ψˆ ◦ Tˆ n ). Cor μi (ϕ, ψ ◦ f qn ) = Cor νˆi (ϕ, ˆ ψˆ ◦ T

Now let F : γ0 ∩  → γ0 ∩  be as in (4.5), the quotient map of ( f q ) R , and T : ˆ →  as in (4.30), we have  →  the tower map of F . Considering  :    ◦ T = T ◦  . It follows from Proposition 4.16 that n

 k − ψk 1 + 2ψ0 ϕˆ ◦ T  k − ϕk 1 , Cor νˆ (ϕˆ , ψˆ ◦ Tˆ ) ≤ Cor ν (ϕk , ψk ◦ T n ) + 2ϕ0 ψˆ ◦ T

ˆ . where  1 denotes the L 1 -norm with respect to the probability measure νˆ on  Using (4.71), for each Q ∈ Qˆ 2k we have

4.4 Decay of Correlations

137

     k (x) : x ∈ Q = inf ϕ ◦ π ◦ T qk (x) : x ∈ Si (Q) . ϕk | Q = inf ϕˆ ◦ T This shows that ϕk = ϕqk ◦ Si . Now, recalling (3.140) and Proposition 4.18, we use Theorem 3.28 to estimate the term Cor ν (ϕk , ψk ◦ T n ) as in (4.53). The estimates for L 1 norms above can be easily deduced from the case already seen. Actually, since ϕk = ϕqk ◦ Si and (4.71) holds, we may write  k − ϕk 1 = ϕˆ ◦ T = = ≤

   

 k − ϕk |dν |ϕˆ ◦ T qk ◦ Si − ϕqk ◦ Si |dν |ϕ ◦ π ◦ T qk − ϕqk |dνi |ϕ ◦ π ◦ T qk − ϕqk |dν. |ϕ ◦ π ◦ T

qk − ϕqk 1 (now, At this point, we can use the estimates in Sect. 4.4.3 for ϕ ◦ π ◦ T 1 the L norm with respect to the measure ν). Recall that in that part we have not used that gcd(R) = 1.

4.5 Regularity of the Stable Holonomy In this section, we show that the property (Y5 ) can be deduced from (Y1 )–(Y4 ), under quite general additional assumptions. Since the pioneering works of Anosov and Sinai [4, 5] for Anosov diffeomorphisms, the absolute continuity of the stable holonomy has been obtained in several contexts, with general results in [16] for the so-called Pesin blocks; see also [17]. Here, we use some ideas from [20], to present a relatively simple and self-contained proof of the regularity condition (Y5 ), taking advantage of specific properties of Young structures. In the final part of the argument, we also use ideas from [13], where a proof of the absolute continuity of the stable holonomy for Anosov diffeomorphisms is provided. Let  be a compact set with a product structure given by continuous families of C 1 disks  s and  u for which (Y1 ) holds. Let Q = {1 , 2 , . . . } be the family of pairwise disjoint s-subsets provided by (Y1 ) and f R :  →  the associated return map. Consider (Qn )n as in (4.8), and 0 = τ0 < τ1 < · · · functions as in (4.9). Recall that τn (x) = τn (x ), for every n ≥ 1, γ ∈  s and x, x ∈ γ ∩ . Assuming that C, ζ > 0 and 0 < β < 1 are uniform constants, consider (Y0 ) for all n ≥ 1, γ , γ ∈  u , x ∈ γ ∩  and x = γ ,γ (x), ∞ i=n

log

det D f R |T f τi (x ) f τi (γ ) ≤ Cβ n ; det D f R |T f τi (x) f τi (γ )

138

4 Hyperbolic Structures

(Y4 ) for all n ≥ 1, Q ∈ Qn , γ ∈  u and x, y ∈ γ ∩ Q, log

ζ  det D f τn |Tx γ ≤ Cd f τn (x), f τn (y) . τ det D f n |Ty γ

It is easily verified that (Y4 ) is a consequence of (Y3 ) and (Y4 ). In Sects. 4.6.3 and 7.3.2, we obtain sets with full product structures satisfying (Y0Y0 ) and (Y4 ). In the next result, we obtain the absolute continuity part in (Y5 ) and a formula for the density. We write simply (Y0 )–(Y4 ) to mean all the conditions from (Y1 ) to (Y3 ), together with (Y0 ) and (Y4 ). Theorem 4.20 Let  be a compact set with a full product structure for which (Y0 )– (Y4 ) hold. Then, (γ ,γ )∗ m γ is absolutely continuous with respect to m γ , for all γ , γ ∈  u , and ∞

$ det D f R |T f τi (x ) f τi (γ ) d(γ ,γ )∗ m γ , (x) = dm γ det D f R |T f τi (x) f τi (γ ) i=0 for all x ∈ γ and y = γ ,γ (x). We postpone the proof of Theorem 4.20 to Sects. 4.5.1 and 4.5.2. It follows from (Y0 ) that the infinite product above converges uniformly on the compact set . In line with the notation in (Y5 ), let ργ ,γ be the density in Theorem 4.20. In the next corollary, we deduce the inequalities required in (Y5 ) for ργ ,γ . Corollary 4.21 Let  be a compact set with a full product structure for which (Y0 )– (Y4 ) hold. Then, there is C > 0 such that 1 ≤ C

 γ ∩

ργ ,γ dm γ ≤ C and log

ργ ,γ (x) ≤ Cβ s(x,y)/2 . ργ ,γ (y)

for all γ , γ ∈  u and x, y ∈ γ . Proof The first part is a consequence of the fact that we have a continuous family of C 1 disks and a full product structure. Given any x, y ∈ γ , take n ≈ s(x, y)/2. Let w, z ∈ γ be such that γ ,γ (w) = x and γ ,γ (z) = y. By Theorem 4.20, we have log =

∞ $ ργ ,γ (x) det D f R |T f τi (x) f τi (γ ) det D f R |T f τi (z) f τi (γ ) = log · ργ ,γ (y) det D f R |T f τi (w) f τi (γ ) det D f R |T f τi (y) f τi (γ ) i=0 n−1 i=0

det D f R |T f τi (x) f τi (γ ) det D f R |T f τi (z) f τi (γ ) + log det D f R |T f τi (y) f τi (γ ) i=0 det D f R |T f τi (w) f τi (γ ) n−1

log

(4.72)

4.5 Regularity of the Stable Holonomy

+

∞ i=n

139 ∞

log

det D f R |T f τi (x) f τi (γ ) det D f R |T f τi (z) f τi (γ ) + . log det D f R |T f τi (w) f τi (γ ) det D f R |T f τi (y) f τi (γ ) i=n

(4.73)

By (Y4 ), there is C > 0 such that n−1 i=0

log

n−1 det D f R |T f τi (y) f τi (γ ) s( f τi+1 (x), f τi+1 (y)) Cβ ≤ det D f R |T f τi (x) f τi (γ ) i=0

=

n−1

Cβ s(x,y)−i−1

i=0

=

n−1

Cβ n−i−1 β s(x,y)−n

i=0

≤

C β s(x,y)−n . 1−β

Since s(x, y) − n ≈ s(x, y)/2, there is some uniform constant C0 > 0 n−1 i=0

log

det D f R |T f τi (y) f τi (γ ) ≤ C0 β s(x,y)/2 . det D f R |T f τi (x) f τi (γ )

Noting that s(w, z) = s(x, y), the same conclusion can be drawn for the summation in (4.72) involving the points f τi (w) and f τi (z). Using (Y0 ), we get the adequate estimates for the summations in (4.73) as well.  Before addressing the proof of Theorem 4.20, we provide practical ways of obtaining conditions (Y0 ) and (Y4 ), which will be useful in Sects. 4.6.3 and 7.3.2. The first condition in Lemma 4.22, requires contraction on stable disks at all large moments, not only at the recurrence times, as in (Y2 ). This actually coincides with the condition imposed in [20] for the stable disks. The second condition in Lemma 4.22 requires the Hölder continuity of the Jacobian of f in the unstable direction. This occurs in great generality for partially hyperbolic sets of C 1+η diffeomorphisms; see [8]. Lemma 4.22 Assume that there are C, ζ > 0 and 0 < α < 1 such that, for all n ≥ 1, γ , γ ∈  u , x ∈ γ ∩  and x = γ ,γ (x), 1. d( f n (x), f n (x )) ≤ Cα n ; det D f |Tx γ ≤ Cd(x, x )ζ . 2. log det D f |Tx γ Then (Y0 ) holds. Proof By the chain rule, we may write

140

4 Hyperbolic Structures ∞

log

i=n

∞ det D f |T f i (x ) f i (γ ) det D f R |T f τi (x ) f τi (γ ) = log det D f R |T f τi (x) f τi (γ ) det D f |T f i (x) f i (γ ) i=τ n

≤C

∞

d( f i (x), f i (x ))ζ

i=τn

≤ C2

∞

αζ i

i=τn

≤

C α ζ τn . 1 − αζ

Since τn ≥ n, we get the conclusion. The proof of the next lemma essentially consists of mimicking the proof of Lemma 4.3, under the adequate assumptions. In the first condition we impose a form of expansion along unstable disks related to the first part of (Y3 ), but stronger. The second condition points to the Hölder continuity of the Jacobian of f R in the unstable direction. Lemma 4.23 Assume that there are C, ζ > 0 and 0 < α < 1 such that, for all i ≥ 1, γ ∈  u and x, y ∈ γ ∩ i , 1. d(x, y) ≤ αd( f R (y), f R (x)); det D f R |Tx γ ≤ Cd( f R (x), f R (y))ζ . 2. log det D f R |Ty γ Then (Y4 ) holds. Proof Consider Q ∈ Qn and γ ∈  u . It follows from the first assumption that for all 0 ≤ j < n and x, y ∈ γ ∩ Q, d( f τ j (x), f τ j (y)) = α n− j d( f τn (x), f τn (y)). Using the second assumption, we get det D f R |T f τ j (x) γ det D f τn |Tx γ = log det D f τn |Ty γ det D f R |T f τ j (y) γ j=0 n−1

log

≤

n−1

Cd( f τ j (x), f τ j (y))ζ

j=0

=

n−1

Cα η(n− j) d( f τn (x), f τn (y))ζ

j=0

≤ This gives (Y4 ).

C d( f τn (x), f τn (y))ζ . 1 − αη 

4.5 Regularity of the Stable Holonomy

141

4.5.1 Absolute Continuity Here, we prove the absolute continuity of the push-forward measure by the holonomy map in Theorem 4.20. Let  be a compact set with a full product structure given by continuous families of C 1 disks  s and  u for which (Y0 )–(Y4 ) hold, and (Qn )n be as in (4.8). Recall that τn is constant on each Q ∈ Qn and f τn | Q = ( f R )n | Q , for all n ≥ 1. Given γ , γ ∈  u , set   Pn = {γ ∩ Q : Q ∈ Qn } and P n = γ ∩ Q : Q ∈ Qn ,

(4.74)

for each n ≥ 1. Since  has a full product structure, it follows from (Y1 ) that (Pn )n is a sequence of m γ mod 0 partitions of γ by pairwise disjoint sets and (P n )n is a sequence of m γ mod 0 partitions of γ by pairwise disjoint sets. Moreover, γ ,γ maps each γ ∩ Q ∈ Pn bjiectively to γ ∩ Q ∈ P n . We denote ω the element in P n associated with an element ω ∈ Pn in this way, and vice versa. Lemma 4.24 There is C > 0 such that, for all n ≥ 1 and ω ∈ Pn , m γ (ω) ≤ Cm γ (ω ). Proof It follows from Lemma 4.3 that there is a uniform constant C0 > 0 such that, for all ω ∈ Pn and x, y ∈ ω ∈ Pn , 1 det D f τn |Tx γ ≤ C0 . ≤ C0 det D f τn |Ty γ Fixing points z ∈ ω and z = γ ,γ (z) ∈ ω , we may write m γ ( f τn (ω)) =



| det D f τn |Tx γ |dm γ     det D f τn |Tx γ  τn  dm γ  ≥ | det D f |Tz γ |   τn ω det D f |Tz γ 1 ≥ | det D f τn |Tz γ | m γ (ω). C0 ω

(4.75)

Similarly, we get m γ ( f τn (ω )) ≤ C0 | det D f τn |Tz γ | m γ (ω ).

(4.76)

On the other hand, since  u is a continuous family of C 1 disks and both f τn (ω) and f τn (ω ) belong in  u , there is some uniform constant C2 > 0 such that m γ ( f τn (ω)) ≤ C2 m γ ( f τn (ω )). It follows from (4.75), (4.76) and (4.77) that

(4.77)

142

4 Hyperbolic Structures

   det D f τn |Tz γ   m γ (ω ). m γ (ω) ≤ C0 2 C2  det D f τn |Tz γ 

(4.78)

We have det D f R |T f τi (z) f τi (γ ) det D f τn |Tz γ = log . τ det D f n |Tz γ det D f R |T f τi (z ) f τi (γ ) i=0 n−1

log

Since  s and  u are continuous families of C 1 disks, the expression above defines a continuous function ϕn on . It follows from (Y0 ) that (ϕn )n is a uniformly converging sequence of continuous functions on , and therefore, a sequence of uniformly bounded functions. The same goes for (eϕn )n . Together with (4.78), this gives the conclusion.  Consider for each n ≥ 1 n = Consider also γˆ =



ω and  n =



ω∈Pn

ω ∈P n





n≥1

n and γˆ =

ω .

 n .

(4.79)

(4.80)

n≥1

Note that γˆ ⊂ γ ∩  and each P n is a cover of γˆ by pairwise disjoint sets. It follows from (Y1 ) that m γ (γˆ ) = m γ (γ ), m γ (γˆ ) = m γ (γ ) and γ ,γ (γˆ ) = γˆ .

(4.81)

The next proposition gives the absolute continuity of the push-forward measure in Theorem 4.20. Proposition 4.25 For all γ , γ ∈  u , the measure (γ ,γ )∗ m γ is absolutely continuous with respect to m γ . Proof Let  : γˆ → γˆ be defined by the restriction of γ ,γ to γˆ . Set m = m γ |γˆ and m = m γ |γˆ . By (4.81), it is enough to prove that ∗ m  m . Since ∗ m and m are regular measures, we just need to show that, for all ε > 0, there is δ > 0 such that, for any compact set K ⊂ γˆ , (4.82) m (K ) < δ =⇒ ∗ m(K ) < ε. Take arbitrarily an ε > 0 and a compact set K ⊂ γˆ . Letting C > 0 be the constant given by Lemma 4.24, choose δ > 0 such that 2Cδ < ε. Since m is a regular measure and diam(P n ) → 0, there exist n ≥ 1 and a family Fn ⊂ P n such that

4.5 Regularity of the Stable Holonomy

K ⊂



143

ω and

ω ∈Fn



m (ω ) < m (K ) + δ.

(4.83)

ω ∈Fn

Consider the family Fn formed by the elements in Pn corresponding to those in Fn . Using Lemma 4.24 and (4.83), we get ∗ m(K ) ≤ ∗ m



ω



=m

ω ∈Fn

 ω∈Fn

ω ≤C



m (ω ) < C(m (K ) + δ).

ω ∈Fn

Assuming m (K ) < δ, we obtain ∗ m(K ) < 2Cδ < ε, thus proving (4.82).



4.5.2 The Density Formula Here, we obtain the density formula in Theorem 4.20. Assume that  is a compact set with a product structure given by continuous families of C 1 disks  s and  u for which conditions (Y0 )–(Y4 ) are satisfied. Since  has a full product structure, the holonomy map γ ,γ is a bijection from γ to γ . Note that, for all ≥ 1 and ω ∈ Pn , we have (4.84) γ ,γ |ω = f −τn ◦  f τn (ω), f τn (ω ) ◦ f τn |ω . In the next result, we essentially show that −1 γ ,γ = γ ,γ can be uniformly approximated by the inverses of the maps that we obtain replacing  f τn (ω), f τn (ω ) in the expression above by a C 1 diffeomorphism. Lemma 4.26 For all n ≥ 1 and ω ∈ Pn , there is ψω,n : f τn (ω) → f τn (ω ) a C 1 diffeomorphism such that  −1  1. lim sup sup d ψω,n ◦ f τn (x), f τn ◦ γ ,γ (x) = 0; n→+∞ ω∈P x∈ω n   −1 ( f τn (x)) = 0. 2. lim sup sup 1 − det Dψω,n n→+∞ ω∈P x∈ω n

Proof Since  u is a continuous family of C 1 disks, there are a compact set K , a unit disk D in some Rk and an injective continuous function  : K × D → M such that  u = {({x} × D) : x ∈ K } . Moreover,  maps K × D homeomorphically onto its image, and x → |{x}×D defines a continuous map from K into Emb1 (D, M). Since  has a full product structure, the disks f τn (ω ) and f τn (ω ) necessarily belong in  u . Thus, we may consider points xω,n , xω ,n ∈ K such that ({xω,n } × D) = f τn (ω) and ({xω ,n } × D) = f τn (ω ).

144

4 Hyperbolic Structures

This naturally induces C 1 diffeomorphisms ω,n : D → f τn (ω) and ω ,n : D → f τn (ω ). Set

ψω,n = ω ,n ◦ −1 ω,n .

The two conclusions are now consequence of the contraction property (Y2 ), together with the fact that x → |{x}×D defines a continuous map from K into Emb1 (D, M). Note that f τn (ω) and f τn (ω ) are disks in  u close to each other in the C 1 topology for large n, uniformly on ω ∈ Pn . This gives in particular the second item. For the first one, note also that f τn ◦ γ ,γ (x) is the point in the unstable disk f τn (ω ) resulting from the intersection of f τn (ω ) with γ s (( f τn )(x)). This point is necessarily close to f τn (x) for large n, since also we assume  s a continuous family of C 1 disks.  Given n ≥ 1, consider n and  n as in (4.79). Let n : n →  n be the C 1 diffeomorphism defined for each ω ∈ Pn by n |ω = f −τn ◦ ψω,n ◦ f τn |ω .

(4.85)

Since each ψω,n is a C 1 diffeomorphism, it is easily verified that (n )∗ m γ is absolutely continuous with respect to m γ and, for all ω ∈ P n and x ∈ ω d(n )∗ m γ det D f τn |Tx γ −1 (x) = ( f τn (x)). · det Dψω,n dm γ det D f τn |T−1 γ n (x) Set for each n ≥ 1 ρn =

(4.86)

d(n )∗ m γ . dm γ

Note that −1 n and ρn are not defined all over γ , but their common domain contains the full m γ measure set γˆ introduced in (4.80). Set

ρ(x) =

∞ $ i=0

det D f R |T f τi (x) f τi (γ ) . det D f R |T f τi (γ ,γ (x)) f τi (γ )

By (Y0 ), the function ρ is well defined in γ and bounded. Lemma 4.27 (−1 n |γˆ )n converges uniformly to γ ,γ |γˆ and (ρn |γˆ )n converges uniformly to ρ|γˆ . Proof For all n ≥ 1 and ω ∈ P n , we have −τn −τn −1 ◦ f τn ◦ −1 ◦ ψω,n ◦ f τn |ω −1 n |ω = f n |ω = f

4.5 Regularity of the Stable Holonomy

and

145

γ ,γ |ω = f −τn ◦ f τn ◦ γ ,γ |ω

By (Y3 ), there is some uniform constant C > 0 such that, for all x ∈ ω , −1 −1 ◦ f τn (x), f −τn ◦ f τn ◦ γ ,γ (x)) ≤ Cd(ψω,n ◦ f τn (x), f τn ◦ γ ,γ (x)). d( f −τn ◦ ψω,n

Using Lemma 4.26, we easily get the first of the two conclusions. Let us now prove the second conclusion. Multiplying and dividing the right hand side of (4.86) by det D f τn |Tγ ,γ (x) γ , we get ρn (x) =

det D f τn |Tγ ,γ (x) γ det D f τn |Tx γ −1 · · det Dψω,n ( f τn (x)). det D f τn |Tγ ,γ (x) γ det D f τn |T−1 γ (x) n

(4.87)

It follows from the chain rule and (Y0 ) that the first factor in (4.87) converges uniformly to ρ. We are left to show that the other two factors in (4.87) converge uniformly to one. By (Y4 ), there are C0 , ζ > 0 such that    det D f τn |Tγ ,γ (x) γ   ζ  τn ◦ γ ,γ (x) .  ≤ C0 d f τn ◦ −1 log n (x), f τ n   det D f |T−1 γ n (x) Since this last quantity is uniformly bounded, there is some uniform constant C1 > 0 such that        det D f τn |T (x) γ det D f τn |Tγ ,γ (x) γ     γ ,γ − 1 ≤ C1 log      det D f τn |T−1 γ det D f τn |T−1 γ  n (x) n (x)  ζ τn ≤ C0 C1 d f τn ◦ −1 ◦ γ ,γ (x) . n (x), f Together with (4.85), this gives for all x ∈ ω,     det D f τn |T (x) γ  −1 ζ   γ ,γ = C0 C1 d ψω,n − 1 ◦ f τn (x), f τn ◦ γ ,γ (x) .   τ   det D f n |T−1 γ n (x) By Lemma 4.26, this last quantity converges uniformly to zero on the set γˆ . This implies that the first factor of the product in (4.87) converges uniformly to one on γˆ Still by Lemma 4.26, the third factor also converges uniformly to one on γˆ .  With the information provided by the previous lemma, we can now conclude the proof of Theorem 4.20. We need to show that ργ ,γ = ρ. Take an arbitrary ε > 0. Given any Borel set A ⊂ γˆ , take a compact set K ⊂ A such that 

 ρdm γ ≤ A

ρdm γ + ε. K

(4.88)

146

4 Hyperbolic Structures

Set for each δ > 0

  Uδ = x ∈ γ : d(x, γ ,γ (K )) < δ .

Since the map γ ,γ is continuous, the set γ ,γ (K ) is compact. Then, there exists some δ > 0 such that m γ (Uδ ) ≤ m γ (γ ,γ (K )) + ε. By Lemma 4.27, the sequence (−1 n |γˆ )n converges uniformly to γ ,γ |γˆ . Thus, there (K ) ⊂ Uδ , for all n ≥ n 0 . It follows that, for all n ≥ n 0 , exists n 0 ∈ N such that −1 n (n )∗ m γ (K ) = m γ (−1 n (K )) ≤ m γ (Uδ ) ≤ m γ (γ ,γ (K )) + ε. By Lemma 4.27, we also have that (ρn |γˆ )n converges uniformly to ρ|γˆ . Recalling that the set γˆ has full m γ measure in γ , we may write 

 ρdm γ = lim

n→+∞

K

ρn dm γ = lim (n )∗ m γ (K ) ≤ m γ (γ ,γ (K )) + ε. n→+∞

K

Together with (4.88), this yields  (γ ,γ )∗ m γ (A) = m γ (γ ,γ (A)) ≥ m γ (γ ,γ (K )) ≥

ρdm γ − 2ε. A

Since ε > 0 is arbitrary, we obtain ργ ,γ ≥ ρ. Now, by dominated convergence theorem, we have   ρdm γ = lim ρn dm γ = (n )∗ m γ (γ ) = m γ (γ ) γ

n→+∞ γ

Together with the inequality already proved, this gives for any Borel set A ⊂ γ (γ ,γ )∗ m γ (A) = (γ ,γ )∗ m γ (γ ) − (γ ,γ )∗ m γ (γ \ A)   ρdm γ − ρdm γ ≤ γ γ \A  = ρdm γ , A

which implies that ργ ,γ ≤ ρ.

4.6 Application: A Solenoid with Intermittency In this section, we present a diffeomorphism to which we can apply Theorem 4.15 and obtain an SRB measure with polynomial decay of correlations. This diffeomorphism was introduced in [3] and obtained by replacing the dynamics in the unstable

4.6 Application: A Solenoid with Intermittency

147

direction near a fixed point of the classical solenoid map by a map with a neutral fixed point as in Sect. 3.5.1. Similar conclusions were drawn in [1] for diffeomorphisms arising from an Anosov map by transforming one of its fixed points into a neutral fixed point, in both the stable and unstable directions. See also [2, 9, 10, 12] for more general examples of “almost hyperbolic” diffeomorphisms with conclusions only on the existence of physical measures (either SRB measures or not) for those diffeomorphisms. We are going to use an intermittent map f : S 1 → S 1 as in Sect. 3.5.3 to define a new map on the three-dimensional solid torus. Recall that f is C 2 on S 1 \ {0} and depends on a parameter α > 0 such that, for x near 0, x f (x) ≈ |x|α . Therefore, f is a C 1+α map on S 1 . As in Sect. 3.5.3, consider d be the degree of f and {I1 , . . . , Id } an m mod 0 partition of S 1 into open intervals such that f | Ii : Ii → S 1 \ {0} is a diffeomorphism for all 1 ≤ i ≤ d. Consider I1 partitioned into subintervals (Jn )n , as in (3.164). For simplicity, here we assume f | J1 ≥ 5.

(4.89)

Consider the solid torus M = S 1 × D 2 , where D 2 is the unit disk in R2 . Let F : M → M be defined by F(x, y, z) =

f (x),

1 1 1 1 cos(2π x) + y, sin(2π x) + z . 2 5 2 5

(4.90)

Since f is a C 1+α local diffeomorphism, then F is a C 1+α diffeomorphism. It is not hard to check that M is a trapping region for F, meaning that F(M) is contained in the interior of M. Thus,  F n (M) = n≥0

is a compact attractor for F. It is easily verified that p0 = (0, 5/8, 0) is a fixed point of F, naturally belonging in , and 1 is an eigenvalue of D F( p0 ). Therefore,  is not a hyperbolic set for F. On the other hand, since f is topologically conjugate to x → d x (mod 1), we can easily build a topological conjugacy between F and a classical hyperbolic solenoid diffeomorphism. Therefore, the attractor  is locally the product of an interval by a Cantor set; see for example [18, Sect. 7.7]. In particular, for each p ∈ , there exists a continuous curve in  passing through p. Let m denote the Lebesgue (volume) measure on M and d the distance in M. Theorem 4.28 Let F : M → M be defined as in (4.90). Then 1. for α < 1, the map F has a unique ergodic SRB measure μ. Moreover, μ is exact, the support of μ coincides with , its basin covers m almost all of M and, for all Hölder continuous ϕ, ψ : M → R,

148

4 Hyperbolic Structures

Cor μ (ϕ, ψ ◦ F n )  1/n 1/α−1 ; 2. for α ≥ 1, the Dirac measure at p0 is a physical measure for F and its basin covers m almost all of M. We first prove the second item of Theorem 4.28. Assume that α ≥ 1. In this case, Theorem 3.62 gives that the Dirac measure δ0 at zero is a physical measure for f and its basin B covers m 1 almost all of S 1 . It follows that π −1 (B) covers m almost all of M. For each n ≥ 1 and p ∈ π −1 (B), set μ p,n =

n−1 1 δ F j ( p) . n j=0

It is enough to show that, for each p ∈ π −1 (B), any weak* accumulation point of the sequence (μ p,n )n coincides with δ p0 . Assume that (μ p,n k )k converges to a probability measure μ on M. Since the push-forward π∗ is continuous, we have that π∗ μ p,n k → π∗ μ, when k → ∞. Using the linearity of π∗ and (4.91), we get π∗ μ p,n k =

n k −1 n k −1 n k −1 1 1 1 π∗ δ F j ( p) = δπ(F j ( p)) = δ f j (π( p)) . n k j=0 n k j=0 n k j=0

Since π( p) ∈ B, this last sequence converges to δ0 , when k → ∞. Therefore, π∗ μ = δ0 and so μ is an F-invariant measure with its support contained in the stable disk γ s ( p0 ). Since F is a contraction on γ s ( p0 ), with p0 its unique fixed point, it must be μ = δ p0 . This gives the second item of Theorem 4.28. The first item of Theorem 4.28 will be given in the remainder of this section.

4.6.1 Partially Hyperbolicity Let π : M → S 1 be given by π(x, y, z) = x. Recalling (4.90), we easily see that f ◦ π = π ◦ F.

(4.91)

Consider I0 ⊂ S 1 a union of at least two consecutive intervals in the sequence (Jn )n and f S : I0 → I0 the induced Gibbs-Markov map given by Proposition 3.67. In (4.105) below, we will specify the choice of I0 . Let m 1 denote the Lebesgue (length) measure on S 1 and Q be the m 1 mod 0 partition of I0 associated with f S . Considering the elements of Q listed as ω1 , ω2 , ω3 . . . , set for each i ≥ 1 i = { p ∈  : π( p) ∈ ωi } and Si = S(ωi ). Given p ∈ M, define the cone

(4.92)

4.6 Application: A Solenoid with Intermittency

  2 2 2 Ccu p = (v1 , v2 , v3 ) ∈ T p M : 15v1 ≥ v2 + v3 .

149

(4.93)

cu Lemma 4.29 For all p ∈  we have D F( p)Ccu p ⊂ C F( p) and the angle between any cu two nonzero vectors in D F(F −n ( p))C F −n ( p) converges to zero as n goes to infinity. Moreover

1. D F( p)v ≥ v/4, for all p ∈ M and v ∈ Ccu p ; 2. D F Si ( p)v ≥ 5v/4, for all p ∈ i , v ∈ Ccu p and i ≥ 1. Proof For each p = (x, y, z) ∈ , we have ⎛

⎞ f (x) 0 0 D F( p) = ⎝−π sin(2π x) 1/5 0 ⎠ . π cos(2π x) 0 1/5

(4.94)

, let (v¯1 , v¯2 , v¯3 ) = D F( p)(v1 , v2 , v3 ). Using that f ≥ 1 and Given (v1 , v2 , v3 ) ∈ Ccu √ p a sin t + b cos t ≤ a 2 + b2 , for all a, b, t ∈ R, we obtain v¯22

+

v¯32

2

2 1 1 = −π sin(2π x) v1 + v2 + π cos(2π x) v1 + v3 5 5     π 1 2 1/2 v2 + v32 ≤ π 2 v12 + |v1 | v22 + v32 + 5 25 % & √ 15 15 2 π+ ≤ π + v 2 < 15v12 ≤ 15v¯12 . 5 25 1

cu This proves that D F( p)Ccu p ⊂ C F( p) . For each n ≥ 1, set

K np = D F(F −n ( p))Ccu F −n ( p) . Note that for p ∈  we have F −n ( p) ∈ , for all n ≥ 1, and so this sets are well defined. Now we show that the angles between any two nonzero vectors in K np converge to zero when n → ∞. Given any (v1(1) , v2(1) , v3(1) ), (w1(1) , w2(1) , w3(1) ) ∈ K 1p , there are (v1(0) , v2(0) , v3(0) ), (w1(0) , w2(0) , w3(0) ) ∈ Ccu F −1 ( p) for which     (0) (0)  v (1) − 21 w1(0) sin x + 15 w2(0)  w2(1)   − 21 v1 sin x + 15 v2  2 −  (1) − (1)  =   v  w1   v1(0) f (x) w1(0) f (x) 1     1  v2(0) w2(0)  1  v2(0) w2(0)  ≤ = − −  .   (0) (0) (0) 5 f (x)  v1 w1  5  v1 w1(0)  Similarly,

     v (1) w3(1)  1  v3(0) w3(0)   3  (1) − (1)  ≤  (0) − (0)  . v w1  5  v1 w1  1

150

4 Hyperbolic Structures

By induction on n, for all vectors (v1(n) , v2(n) , v3(n) ), (w1(n) , w2(n) , w3(n) ) ∈ K np there are vectors (v1(0) , v2(0) , v3(0) ), (w1(0) , w2(0) , w3(0) ) ∈ Ccu F −n ( p) such that for i = 2, 3      v (n) wi(n)  wi(0)  1  vi(0)  i  (n) − (n)  ≤ n  (0) − (0)  . v w1  5  v1 w1  1 This gives that the angle between (v1(n) , v2(n) , v3(n) ) and (w1(n) , w2(n) , w3(n) ) goes to zero when n → ∞. In addition, for all (v1 , v2 , v3 ) ∈ Ccu p we have 1 2 1 (v1 + v22 + v32 ) = (v1 , v2 , v3 )2 . 16 16

D F( p)(v1 , v2 , v3 )2 ≥ v12 ≥

Finally, assume that p = (x, y, z) belongs in i , and take v ∈ Ccu p . Note that x ∈ ωi and Si ≥ R, with R as in (3.165); recall (3.178). Since f R−1 (x) ∈ J1 and f ≥ 1, it follows from (4.89) that ( f Si ) (x) =

S$ i −1

f ( f j (x)) ≥ 5.

j=0

Hence, D F Si ( p)(v1 , v2 , v3 )2 ≥ [( f Si ) (x)]2 v12 ≥

25 2 25 (v + v22 + v32 ) = (v1 , v2 , v3 )2 . 16 1 16

This concludes the proof of the lemma.



Corollary 4.30 There is a D F-invariant splitting T M = E cu ⊕ E s such that 1. D F( p)v ≤ v/5, for all p ∈  and v ∈ E sp ; 2. D F( p)v ≥ v/4, for all p ∈  and v ∈ E cu p . Proof Note that E sp = {0} × R2 is D F-invariant and contracted by the factor 1/5. By Lemma 4.29, the sets K np = D F(F −n ( p))(Ccu F −n ( p) ), n ≥ 1, form a nested sequence and the angle between any two nonzero vectors in K np converges to zero when n → ∞. This implies that E cu p =

∞ 

D F(F −n ( p))Ccu F −n ( p)

n=1 cu s cu is one-dimensional. Since E cu p ⊂ C p , we have T p M = E p ⊕ E p .



4.6 Application: A Solenoid with Intermittency

151

The previous result gives that  is a partially hyperbolic set for F. This topic will be further explored in Chapter 7. For now, we just need the existence of centreunstable disks: for each p ∈ , there exists an embedded C 1 disk Wεcu ( p) such that T p Wεcu ( p) = E cu p and F(Wεcu ( p)) ∩ Bε (F( p)) ⊂ Wεcu (F( p)),

(4.95)

where Bε (F( p)) denotes the ball of radius ε > 0 around the point F( p) ∈ . In addition, the disks Wεcu ( p) depend continuously on p ∈  in the C 1 topology; see [19, Theorem IV.1]. Note that these centre-unstable disks are not unique, in general.

4.6.2 Positive Lyapunov Exponent Assume from now on that 0 < α < 1. Let ν be the ergodic SRB measure for f provided by Theorem 3.62. In the next result, we show that there is a probability measure on M whose push-forward by the projection π in (4.91) is ν. Lemma 4.31 There exists an F-invariant Borel probability measure μ on M such that π∗ μ = ν. Moreover, the support of μ coincides with . Proof For any continuous ϕ : M → R, consider ϕ − : S 1 → R and ϕ + : S 1 → R defined for each x ∈ S 1 by ϕ − (x) =

inf {ϕ( p)} and ϕ + (x) =

p∈π −1 (x)

sup {ϕ( p)}.

p∈π −1 (x)

We are going to see that the limits  lim

n→+∞

(ϕ ◦ F n )− dν and

 lim

n→+∞

(ϕ ◦ F n )+ dν

(4.96)

exist and coincide. Let us prove the first case, the second is similar. It is enough to  verify that ( (ϕ ◦ F n )− dν)n is a Cauchy sequence in R. Given an arbitrary ε > 0, take δ > 0 such that d( p, q) < δ =⇒ |ϕ( p) − ϕ(q)| < ε.

(4.97)

Since F contracts by a factor 1/5 on each disk π −1 (x), there is n 0 ≥ 1 such that, for all n ≥ n 0 and x ∈ S 1 , (4.98) diam(F n (π −1 (x)) < δ. We have for all n ≥ n 0 and k ≥ 0     (ϕ ◦ F n+k )− (x) − (ϕ ◦ F n )− ( f k (x)) = inf ϕ ◦ F n+k |π −1 (x) − inf ϕ ◦ F n |π −1 ( f k (x)) .

152

4 Hyperbolic Structures

Since F n+k (π −1 (x)) ⊂ F n (π −1 ( f k (x))), it follows from (4.97) and (4.98) that the right hand side in the previous equality is bounded by     sup ϕ ◦ F n |π −1 ( f k (x)) − inf ϕ ◦ F n |π −1 ( f k (x)) < ε.

(4.99)

Together with the invariance of ν, this yields 

(ϕ ◦ F n+k )− dν −



(ϕ ◦ F n )− dν =



(ϕ ◦ F n+k )− − (ϕ ◦ F n )− ◦ f k dν < ε,

 and so have proved that ( (ϕ ◦ F n )− dν)n is a Cauchy sequence. It also follows from (4.97) and (4.98) that, for all n ≥ n 0 and x ∈ S1 ,     0 ≤ (ϕ ◦ F n )+ (x) − (ϕ ◦ F n )− (x) = sup ϕ ◦ F n |π −1 (x) − inf ϕ ◦ F n |π −1 (x) < ε. This implies that the limits in (4.96) coincide. Now, define T : C 0 (M) → R by  T (ϕ) = lim

n→+∞



n −

(ϕ ◦ F ) dν = lim

n→+∞

(ϕ ◦ F n )+ dν.

Given ϕ, ψ ∈ C 0 (M), we have  T (ϕ + ψ) = lim

n→+∞



≥ lim

n→+∞

((ϕ + ψ) ◦ F n )− dν (ϕ ◦ F n )− dν + lim

n→+∞



(ψ ◦ F n )− dν

= T (ϕ) + T (ψ). Using the other limit in the definition of T , we also have T (ϕ + ψ) ≤ T (ϕ) + T (ψ). Note that T (cϕ) = cT (ϕ) is obvious for c ≥ 0. For c < 0, we have   n − T (cϕ) = lim (cϕ ◦ F ) dν = lim c(ϕ ◦ F n )+ dν = cT (ϕ). n→+∞

n→∞

Thus, T is a linear functional. Since T (1) = 1 and T (ϕ) ≥ 0, for all ϕ ≥ 0, it follows from Riesz-Markov Theorem that there exists a probability measure μ on the Borel sets of M such that T (ϕ) = ϕdμ, for all ϕ ∈ C 0 (M). Let us show that π∗ μ = ν. Given any ϕ ∈ C 0 (S 1 ), we have 

 ϕdπ∗ μ =

 ϕ ◦ π dμ = lim

n→+∞

(ϕ ◦ π ◦ F n )− dν = lim

n→+∞



(ϕ ◦ π ◦ F n )− dν. (4.100)

It follows from (4.91) that, for all n ≥ 1, (ϕ ◦ π ◦ F n )− = (ϕ ◦ f n ◦ π )− ≤ ϕ ◦ f n ≤ (ϕ ◦ f n ◦ π )+ = (ϕ ◦ π ◦ F n )+ .

4.6 Application: A Solenoid with Intermittency

153

Now, integrate these functions with respect to ν and  take limits. Recalling that ν is  f -invariant and (4.100) holds, we get ϕdπ∗ μ = ϕdν. This means that π∗ μ = ν. Using one more time (4.91), we obtain F∗ μ = F∗ π∗ ν = π∗ f ∗ ν = π∗ ν = μ, which provides the invariance of μ. We are left to show that the support of μ coincides with . Since all points in M \  are wandering points, the support of μ is necessarily contained in . Take U a neighbourhood of an arbitrary point p = (x, y, z) ∈ . Without loss of generality, we may assume that U = Br1 (x) × Br2 (y, z), where Br1 (x) and Br2 (y, z) are balls of radius r > 0 in S 1 and D 2 , respectively. Take n sufficiently large so that diam(F n (π −1 (t))) < r , for all t ∈ S 1 . Set pn = F −n ( p) and choose Vn a small neighbourhood of π( pn ) in S 1 such that f n (Vn ) ⊂ Br1 (x). We clearly have F n (Vn × D 2 ) ⊂ U . It follows from Theorem 3.62, that ν(Vn ) > 0. Using that μ is F-invariant and π∗ μ = ν, we get μ(U ) = μ(F −1 (U )) ≥ μ(Vn × D 2 ) = μ(π −1 (Vn )) = ν(Vn ) > 0. This shows that the support of μ coincides with .



Using the previous lemma, we prove in next one that there is a full μ measure set of points in  for which the local unstable disk is defined and contained in . Since the support of μ coincides with , this gives a dense set of points in  for which this holds. Recall that Wεcu ( p) is de centre-unstable disk through a point p ∈  and satisfies (4.95). Lemma 4.32 There is a set A ⊂  with μ(A) = 1 such that F has a positive Lyapunov exponent in the direction of E cu p , for all p ∈ A. Moreover, the local unstable disk γ u ( p) through each p ∈ A is contained in Wεcu ( p) ∩  and T p γ u ( p) = E cu p . Proof Let μ be the probability measure given by Lemma 4.31. We have π∗ μ = ν, where ν is the unique ergodic SRB measure ν for f provided by Theorem 3.62. By Oseledets Theorem, the Lyapunov exponent lim

n→±∞

1 log D F n ( p)u, n

is defined for μ almost every p = (x, y, z) ∈ M and every u = (u 1 , u 2 , u 3 ) ∈ T p M \ {0}. Recalling the expression of D F( p) in (4.94), we have D F n ( p)u ≥ ( f n ) (x)|u 1 |.

(4.101)

Since ν is equivalent to m 1 , it follows from Birkhoff Ergodic Theorem that, for m 1 almost every x ∈ S 1 ,

154

4 Hyperbolic Structures

 n−1 1 1 log f ( f j (x)) = log f dν > 0. log( f n ) (x) = lim n→∞ n n→∞ n j=0 lim

(4.102)

Let B be the set of points x ∈ S 1 for which (4.102) holds. Taking A = π −1 (B), it follows from Lemma 4.31 that μ(A) = ν(B) = 1. Moreover, from (4.101) and (4.102), we deduce that F has a positive Lyapunov exponent in the direction of E cu p , for all necessarily has u 1 = 0, p ∈ A; note that a nonzero vector u = (u 1 , u 2 , u 3 ) ∈ E cu p since E sp = {0} × R2 . This gives the first item. s From what we have seen above, we conclude that T p M = E cu p ⊕ E p is the Oseledets splitting for all p ∈ A. It follows from Pesin Theorem that a local unstable u disk γ u ( p) is defined for all p ∈ A with T p γ u ( p) = E cu p . Moreover, γ ( p) consists of the set of points q sufficiently close to p verifying (1.2), for some suitable choice of 0 < λ < 1. Since  is at a positive distance from the boundary of M and F −n ( p) ∈  for all n ≥ 0, choosing the size of the unstable disk γ u ( p) sufficiently small, we have F −n (q) ∈ M, for all q ∈ γ u ( p). This shows that γ u ( p) ⊂ , by definition of . Finally, we show that γ u ( p) ⊂ Wεcu ( p). Choosing ε > 0 sufficiently small, we have Tq Wεcu ( p) contained in the cone Cqcu , for all p ∈  and q ∈ Wεcu ( p). Given p ∈ , take γ u ( p) sufficiently small so that π(Wεcu (F −n ( p))) ⊃ F −n (γ u ( p)), for all p ∈ .

(4.103)

Let D p be the disk in Wεcu ( p) that π maps diffeomorphically to γ u ( p). Using (4.103), we deduce that F −n (D p ) ⊂ Wεcu (F −n ( p)) and π maps F −n (D p ) diffeomorphically to F −n (γ u ( p)), for all n ≥ 0; recall that π is constant on vertical (stable) disks. Since Tq F −n (D p ) ⊂ Cqcu , it follows that (1.2) is satisfied for some choice of C > 0.  Therefore, D p is a local unstable disk through p.

4.6.3 Young Structure Here, we introduce a compact set  with a Young structure given by continuous families of C 1 disks  s and  u . As a consequence, we deduce the first item of Theorem 4.28. We define  u by mean of an inductive process that we describe below. First of all, take n 0 sufficiently large so that, for p ∈  with π( p) ∈ Jn 0 ∪ Jn 0 +1 , we have π(Wεcu ( p)) ⊃ Jn 0 ∪ Jn 0 +1 . By Lemma 4.32, there exists some point p0 ∈  for which γ u ( p0 ) ⊂ Wεcu ( p0 ) ∩  u and T p0 γ u ( p0 ) = E cu p0 . Choosing the size of this local unstable disk γ ( p0 ) small enough, we may have u T p γ u ( p0 ) ⊂ Ccu p , for all p ∈ γ ( p0 ).

(4.104)

4.6 Application: A Solenoid with Intermittency

155

This implies that π maps γ u ( p0 ) diffeomorphically onto its image. In particular, π(γ cu ( p0 )) contains some open interval in S 1 . Since f is topologically conjugate to the map x → d x (mod 1), there exist an interval I ⊂ π(γ u ( p0 )) and n 1 ≥ 1 such that f n 1 maps I diffeomorphically to the interval I0 = Jn 0 ∪ Jn 0 +1 .

(4.105)

Take γ ⊂ γ u ( p0 ) such that π projects γ diffeomorphically to I , and set γ0 = F n 1 (γ ). Notice that γ0 is an unstable disk contained in  and π(γ0 ) = I0 . It follows from (4.104) and Lemma 4.29 that T p γ0 ⊂ Ccu p , for all p ∈ γ0 .

(4.106)

This implies that π projects γ0 diffeomorphically to I0 . This disk γ0 will be used as the first element of an inductive construction leading to  u . For that, we use some properties of f : S 1 → S 1 . Recall that, by Proposition 3.67, the map f has an induced Gibbs-Markov map f S : I0 → I0 . Let Q = {ω1 , ω2 , . . . } be the m 1 mod 0 partition of I0 associated with f S . Consider the sequence of sets (i )i in  and the respective sequence of times (Si )i as in (4.92). Now, we define inductively a sequence (n )n≥0 in the following way. We start with 0 = {γ0 } . Assuming that n−1 is defined for some n ≥ 1, set n =



 F Si (i ∩ γn−1 ) : γn−1 ∈ n−1 .

i≥1

Observe that π maps each γn ∈ n diffeomorphically to I0 and γn is the forward iterate of a subset of γ0 . It follows that γn ∈ n is an unstable disk contained in Wεcu ( pn ) ∩ , for some pn ∈ , by the invariance property in (4.95). Since the union of all disks in the families n with n ≥ 0 is not necessarily a compact set, we still need to take the accumulation points of that union. Set =

 

γn .

n≥0 γn ∈n

Given any q ∈ , there are n 1 < n 2 < · · · , disks γn k ∈ n k and points qk ∈ γn k converging to q, when k → ∞. As we have seen above, for each k ≥ 1, there are pk ∈  and a centre-unstable disk Wεcu ( pk ) ⊃ γn k . Taking a subsequence, if necessary, we may assume that ( pk )k converges to some point p ∈ . Since the centre-unstable disks Wεcu ( pk ) depend continuously on pk ∈  in the C 1 topology, then the disks γ jk converge in the C 1 topology to a disk γ∞ ⊂ Wεcu ( p) containing q, when k → ∞. We define  u as the set of all these disks γ∞ . Note that =

 γ ∈ u

γ.

156

4 Hyperbolic Structures

Finally, set

   s = {x} × D 2 : x ∈ S 1 .

(4.107)

The next result is a key step towards the conclusion of the proof of Theorem 4.28. Proposition 4.33 The set  has a full Young structure with recurrence time S. Moreover, gcd(S) = 1 and m γ0 {S > n}  1/n 1/α . Proof First of all, note that  has a full product structure given by the families of disks  u and  s constructed above. Moreover,  u is a continuous family of C 1 disks, by construction, and  s is a continuous family of C 1 disks, trivially. Since each γ ∈  u is contained in , we have m γ ( ∩ γ ) = m γ (γ ) > 0, for all γ ∈  u . For each i ≥ 1, set i = i ∩  and S|i = Si , with i and Si as in (4.92). With these choices, it is straightforward to verify that the Markov property (Y1 ) holds. The contraction (Y2 ) and expansion (Y3 ) are a consequence of the first item of Corollary 4.30 and the second item of Lemma 4.29, respectively. Indeed, from these results, we can easily see that, for all n ≥ 1, γ ∈  s and x, y ∈ γ , n 1 n n d(F ( p), F (q)) ≤ d( p, q) 5 and, for all i ≥ 1, γ ∈  u and x, y ∈ γ ∩ i , d( p, q) ≤

4 d(F S ( p), F S (q)). 5

(4.108)

Therefore, we have (Y2 ) and (Y3 ); recall Remark 4.1. Now, we check the Gibbs condition (Y4 ). It follows from (4.106) and Lemma 4.29 that u T p γ ⊂ Ccu p , for all γ ∈  and p ∈ γ .

(4.109)

Indeed, this property is obvious if γ ∈ n , for some n ≥ 0, and it passes to the limit disks, by continuity. It follows that, for each γ ∈  u , the projection π : M → S 1 induces a C 1 diffeomorphism πγ : γ → I0 . Moreover, there is some uniform constant C0 > 0 such that, for all γ ∈  u , p, q ∈ γ and x, y ∈ I0 , log

det πγ−1 (x) det πγ ( p) ≤ C0 d( p, q), log ≤ C0 |x − y| det πγ (q) det πγ−1 (y)

(4.110)

and |πγ ( p) − πγ (q)| ≤ C0 d( p, q). Since π is a semiconjugacy between F and f , we may write

(4.111)

4.6 Application: A Solenoid with Intermittency

157

F Si |γ ∩i = π F−1Si (γ ∩i ) ◦ f Si ◦ πγ |γ ∩i ,

(4.112)

for all i ≥ 1 and γ ∈  u . This implies that   D F Si |T p γ = Dπ F−1Si (γ ∩i ) f Si (πγ ( p)) ( f Si ) (πγ ( p))Dπγ ( p) for all p ∈ γ ∩ i . By (4.108), (4.110), (4.111) and Remark 3.68, we find some uniform constant C1 > 0 such that, for all i ≥ 1 and p ∈ γ ∩ i ,   det π F−1Si (γ ∩i ) f Si (πγ ( p)) det D F Si |T p γ   log = log det D F Si |Tq γ det π F−1Si (γ ∩ ) f Si (πγ (q)) i

det πγ ( p) ( f Si ) (πγ ( p)) + log + log S i ( f ) (πγ (q)) det πγ (q)  Si  Si ≤ C1 d f (πγ ( p)), f (πγ (q))   = C1 d πγ (F Si ( p)), πγ (F Si (q)) ≤ C0 C1 d(F Si ( p), F Si (q)).

(4.113)

This actually gives (Y4 ), which together with (Y3 ) implies (Y4 ). Finally, we verify the regularity of the stable holonomy (Y5 ) through Theorem 4.20 and Corollary 4.21. We will use Lemmas 4.22 and 4.23 to check (Y0 ) and (Y4 ). Note that  is contained in the compact invariant set  ⊂ M with a partially hyperbolic splitting T M = E s ⊕ E cu . Since F is a C 1+α diffeomorphism, the fibre bundles E s and E cu are Hölder continuous on ; see [10]. Since the unstable disks are contained in , the Hölder continuity of the E cu direction and the uniform contraction on stable disks enable us to apply Lemma 4.22, thus obtaining (Y0 ). Property (Y4 ) is a consequence of (4.108) and (4.113), together with Lemma 4.23. Applying Theorem 4.20 and Corollary 4.21, we get (Y5 ). Thus, the set  has a full Young structure with recurrence time S. Now, recall that the sequence (Si )i introduced in (4.92) coincides with the sequence of recurrence times for the induced map f S : I0 → I0 , which is a Gibbs-Markov induced map with gcd(S) = 1, by Proposition 3.67. In addition, m 1 {S > n}  n −1/α , by (3.182). Since πγ0 is a C 1 diffeomorphism, we finally get m γ0 {S > n}   n −1/α . Now, using Proposition 4.33, Theorems 4.9 and 4.15, we obtain an exact SRB measure μ such that, for all Hölder continuous ϕ, ψ : M → R, Cor μ (ϕ, ψ ◦ F n )  1/n 1/α−1 . Since F is topologically conjugate to the classical solenoid map, then  is a transitive set for F. It follows from Proposition 4.10 that the support of μ coincides with . Moreover, μ almost every point in M belongs in the basin of μ, by Proposition 2.12. On the other hand, it follows from Theorem 4.7 and the formula for μ in Theorem 4.9 that the densities dμγ /dm γ are bounded from below by some uniform constant, for

158

4 Hyperbolic Structures

almost all γ ∈  u . Therefore, there exists some unstable disk γ0 ∈  u such that m γ0 almost every point in γ0 belongs in the basin of μ. Set I0 = π(γ0 ). Since f is topologically conjugate to the map x → d x (mod 1), then there is n 0 ≥ 1 such that f n 0 (I0 ) = S 1 . Using that π is a semiconjugacy between F and f , we get π(F n 0 (γ0 )) = f n 0 (π(I0 )) = S 1 .

(4.114)

Since the basin of μ is an invariant set, we have that m F n0 (γ0 ) almost every point in the curve F n 0 (γ0 ) belongs in the basin of μ. Finally, it follows from (4.114) and the fact that M = S 1 × D 2 is foliated by stable disks that m almost every point in M belongs in the basin of μ. This gives, in particular, that μ is the unique ergodic SRB measure for F.

References 1. J.F. Alves, D. Azevedo, Statistical properties of diffeomorphisms with weak invariant manifolds. Discrete Contin. Dynam. Syst. 36(1), 1–41 (2016) 2. J.F. Alves, R. Leplaideur, SRB measures for almost Axiom A diffeomorphisms. Ergodic Theory Dynam. Syst. 36(7), 2015–2043 (2016) 3. J.F. Alves, V. Pinheiro, Slow rates of mixing for dynamical systems with hyperbolic structures. J. Stat. Phys. 131(3), 505–534 (2008) 4. D. V. Anosov. Geodesic flows on closed Riemann manifolds with negative curvature. Proceedings of the Steklov Institute of Mathematics, No. 90, Translated from the Russian by S, American Mathematical Society (Providence, R.I, Feder, 1967), p. 1969 5. D.V. Anosov, J.G. Sina˘ı, Certain smooth ergodic systems. Uspehi Mat. Nauk 22(5 (137)), 107–172 (1967) 6. V. Baladi, Positive Transfer Operators and Decay of Correlations, vol. 16, Advanced Series in Nonlinear Dynamics (World Scientific Publishing Co., Inc., River Edge, NJ, 2000) 7. M. Benedicks, L.-S. Young. Markov extensions and decay of correlations for certain Hénon maps. Astérisque 261, 13–56 (2000). Géométrie complexe et systèmes dynamiques (Orsay, 1995) 8. M. Brin, Hölder continuity of invariant distributions, in A. Katok, R. de la Llave, Y. Pesin, H. Weiss (eds.) Smooth Ergodic Theory and Its Applications, vol. 69, American Mathematical Society (2001) 9. H. Hu, Conditions for the existence of SBR measures for “almost Anosov” diffeomorphisms. Trans. Amer. Math. Soc. 352(5), 2331–2367 (2000) 10. H.Y. Hu, L.-S. Young, Nonexistence of SBR measures for some diffeomorphisms that are “almost Anosov”. Ergodic Theory Dynam. Syst. 15(1), 67–76 (1995) 11. A. Korepanov, Z. Kosloff, I. Melbourne, Explicit coupling argument for nonuniformly hyperbolic transformations. Proc. Roy. Soc. Edinb. Sect. A 149(1), 101–130 (2019) 12. R. Leplaideur, Existence of SRB-measures for some topologically hyperbolic diffeomorphisms. Ergodic Theory Dynam. Syst. 24(4), 1199–1225 (2004) 13. R. Mañé, Ergodic Theory and Differentiable Dynamics, volume 8 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] (Springer-Verlag, Berlin, 1987). Translated from the Portuguese by Silvio Levy 14. I. Melbourne, D. Terhesiu, Decay of correlations for nonuniformly expanding systems with general return times. Ergodic Theory Dynam. Syst. 34(3), 893–918 (2014) 15. K.R. Parthasarathy, Probability Measures on Metric Spaces. Probability and Mathematical Statistics, No. 3 (Academic Press, Inc., New York-London, 1967)

References

159

16. J.B. Pesin, Families of invariant manifolds that correspond to nonzero characteristic exponents. Izv. Akad. Nauk SSSR Ser. Mat. 40(6), 1332–1379, 1440 (1976) 17. C. Pugh, M. Shub, Ergodic attractors. Trans. Amer. Math. Soc. 312(1), 1–54 (1989) 18. C. Robinson, Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, Studies in Advanced Mathematics (CRC Press, Boca Raton, FL, 1995) 19. M. Shub, Global Stability of Dynamical Systems. With the collaboration of Albert Fathi and Rémi Langevin, Translated from the French by Joseph Christy (Springer-Verlag, New York, 1987) 20. L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. (2) 147(3), 585–650 (1998) 21. L.-S. Young, Recurrence times and rates of mixing. Israel J. Math. 110, 153–188 (1999)

Chapter 5

Inducing Schemes

In this chapter, we develop a general framework suitable for application to both nonuniformly expanding and to partially hyperbolic systems. The main result is Theorem 5.1 below, which is a key step towards the construction of the Gibbs-Markov induced maps or Young structures as in the two previous chapters, also providing information about the tail of recurrence times. The proof of the integrability of the inducing times is based on ideas which have been used in [4] to prove a onedimensional result originally from [6], successfully adapted to higher dimensions in [2, 7]. For the estimates on the tail of recurrence times, we follow the approach in [3], which is in turn an adaptation of arguments in [5].

5.1 A General Framework Let f : M → M be a map of a finite dimensional Riemannian manifold M, and  be a submanifold of M, possibly equal to M or having a boundary. Consider d the distance on  and m the Lebesgue (volume) measure on the Borel sets of , both induced by the Riemannian metric. For each n ≥ 0, denote dn = d f n () and m n = m f n () , where d f n () stands for the distance in the submanifold f n () and m f n () stands for the Lebesgue measure on the Borel sets of f n (), both induced by the Riemannian metric on M. Throughout this chapter, we assume that there exists a disk 0 ⊂  with the same dimension of  for which conditions (I1 )–(I3 ) below hold (I1 ) There is a sequence (Hn )n of compact sets in 0 such that m 0 almost every point in 0 belongs in infinitely many Hn ’s.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. F. Alves, Nonuniformly Hyperbolic Attractors, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-62814-7_5

161

162

5 Inducing Schemes

(I2 ) There is δ1 > 0 such that every x ∈ Hn has a neighbourhood Vn (x) of x in  that f n maps diffeomorphically to a disk of radius δ1 around f n (x). Moreover, there are C0 , η > 0 and 0 < σ < 1 such that, for all Vn (x) and y, z ∈ Vn (x), • dn−k ( f n−k (y), f n−k (z)) ≤ σ k dn ( f n (y), f n (z)), for all 1 ≤ k ≤ n. det D f n |Ty  • log ≤ C0 dn ( f n (y), f n (z))η . det D f n |Tz  n (x) ⊂ Vn (x) such that f n maps Wn (x) It will be useful to consider Wn (x) ⊂ W n (x) to disks of radius δ1 /9 and δ1 /3, respectively, both centred at f n (x). and W (I3 ) There exist L , δ0 > 0 such that, for each x ∈ Hn , there are 0 ≤  ≤ L and ωn, ⊂ Wn (x), with f n+ mapping ωn, and  ωn, diffeomorphically ωn, ⊂  to concentric disks of radius δ0 and 2δ0 , respectively. Moreover, there are ωn, and y, z ∈ f n ( ωn, ), C1 , η > 0 such that, for all  1 dn+ j ( f j (y), f j (z)) ≤ dn+ ( f  (y), f  (z)) ≤ C1 dn (y, z), for all 0 ≤ j ≤ ; C1 det D f  |Ty  ≤ C1 dn+ ( f  (y), f  (z))η . • log det D f  |Tz 

•

n (x) and Vn (x) are neighbourhoods of the reference point x Note that Wn (x), W in , but this is not necessarily true for  ωn, . For simplicity, we will often denote the ωn, as  ω. In such case, we also set sets ωn, as ω and  n (ω) = W n (x), Vn (ω) = Vn (x) and ω = . Wn (ω) = Wn (x), W

(5.1)

In Chaps. 6 and 7, the neighbourhoods Vn (x) will be defined using expanding properties of points in Hn . Then, using some transitivity of the system, a part of ωn, of Vn (x) returns in time n +  diffeomorphically onto 0 (Chap. 6) or onto an unstable disk of a Young structure as in Chap. 4 for which 0 is an unstable disk (Chap. 7). In order to make the framework sufficiently flexible for application in both situations, we do not impose any return property at this stage. The asymptotic frequency for which typical points in 0 belong to Hn plays an important role in some results that will follow, especially when this happens with uniform positive frequency: assume that there is 0 < θ ≤ 1 such that for m 0 almost every x ∈ 0 we have n 0 ∈ N for which n ≥ n 0 =⇒

 1  # 1 ≤ j ≤ n : x ∈ H j ≥ θ. n

(5.2)

We define h θ (x) as the minimum n 0 with this property. In (5.38) below, we consider a weaker form of asymptotic frequency for which typical points in 0 belong in Hn . Theorem 5.1 If (I1 )–(I3 ) hold, then there is an m 0 mod 0 partition P of 0 into domains ωn, as in (I3 ). Moreover, setting R(x) = n +  for each x ∈ ωn, ∈ P,

5.1 A General Framework

163

1. there are C > 0 and an arbitrarily small 0 < β < 1 such that, for all x, y ∈ ω ∈ P, a. d0 (x, y) ≤ βd R ( f R (x), f R (y)); b. d j ( f j (x), f j (y)) ≤ Cd R ( f R (x), f R (y)), for all 0 ≤ j ≤ R; det D f R |Tx  ≤ Cd R ( f R (x), f R (y))η ; c. log det D f R |Ty   2. there are S1 , S2 , · · · ⊂ 0 with n≥1 m 0 (Sn ) < ∞ such that, for all n ≥ 1, Hn ∩ {R > L + n} ⊂ Sn ; 3. if h θ is defined for some θ , then there are E 1 , E 2 , · · · ⊂ 0 with m 0 (E n ) converging to zero exponentially fast with n such that, for all n ≥ 1, {R > n + L} ⊂ {h θ > n} ∪ E n . The partition P will be obtained by mean of an inductive construction performed in Sect. 5.2. The first two items will be deduced in Sects. 5.2 and 5.3, whilst the third one will be deduced in Sect. 5.3.2. The first item of Theorem 5.1 is obviously pointing to the construction of expanding Gibbs-Markov induced maps. The second item constitutes an important step towards the integrability of the inducing times, as it will be clear in Proposition 5.12. The last item shows that the tail of inducing times can be known, once we know the tail of {h θ > n}.

5.1.1 Bounded Distortion Here, we derive some useful consequences of the bounded distortion properties in conditions (I2 ) and (I3 ). We start with a simple standard result that will be used several times. Lemma 5.2 There is C > 0 such that 1. for all Vn (ω) as in (I2 ) and Borel sets A1 , A2 ⊂ Vn (ω), m 0 (A1 ) m n ( f n (A1 )) ≤C . n m n ( f (A2 )) m 0 (A2 ) 2. for all ω˜ n, as in (I3 ) and Borel sets A1 , A2 ⊂ f n (ω˜ n, ), m n (A1 ) m n+ ( f  (A1 )) ≤C .  m n+ ( f (A2 )) m n (A2 ) Proof We prove only the first item, the second one can be proved similarly. By (I2 ), we can easily find some C0 > 0 such that, for all y, z ∈ Vn (x),

164

5 Inducing Schemes

det D f n |Ty  ≤ C0 . det D f n |Tz  Using a change of variables and fixing some z ∈ Vn (x), we may write  | det D f n |Ty |dm 0 (y) m n ( f n (A1 ))  A1 = n m n ( f n (A2 )) A2 | det D f |Ty |dm 0 (y)



| det D f n |Ty | dm 0 (y) A1 | det D f n |T | z =  | det D f n |Ty | dm 0 (y) A2 | det D f n |T | z m 0 (A1 ) . ≤ C02 m 0 (A2 ) Take C = C02 .



The conclusion of Proposition 5.3 below will be useful both in Chaps. 6 and 7. It is worth noticing that, for this result, the existence of a sequence (Hn )n as in (I1 ) is only needed for the points in A ∩ 0 , not necessarily almost everywhere in 0 . Recall that, for each x ∈ Hn , we have defined Wn (x) as the subset of Vn (x) that f n maps diffeomorphically to the disk of radius δ1 /9 centred at f n (x). Proposition 5.3 Let A ⊂ M be such that f (A) ⊂ A and m 0 (A ∩ 0 ) > 0. Assume that (Hn )n is a sequence of sets in 0 for which (I2 ) holds, and every point in A ∩ 0 belongs in infinitely many Hn ’s. Then, there are 1 ≤ n 1 < n 2 < · · · and, for each k ≥ 1, there are xk ∈ Hn k such that m n k (A ∩ f n k (Wn k (xk ))) = 1. k→∞ m n k ( f n k (Wn k (xk ))) lim

Proof It is enough to show that for any ε > 0 we may find an arbitrarily large n and x ∈ Hn such that m n (A ∩ f n (Wn (x))) ≥ (1 − ε)m n ( f n (Wn (x))). Set A0 = A ∩ 0 . Take a small number δ > 0. Let K be a compact subset of A0 and B an open neighbourhood of A0 in 0 such that m 0 (B \ K ) < δm 0 (A0 ) and m 0 (A0 ) < 2m 0 (K ).

(5.3)

Since every point in A0 belongs in infinitely many Hn ’s and (I2 ) holds, there exists an arbitrarily large n 0 ∈ N such that for all n ≥ n 0 and x ∈ K we have Vn (x) ⊂ B. Since K is a compact set, there are x1 , ..., xr ∈ K and n(x1 ), . . . , n(xr ) ≥ n 0 such that (5.4) K ⊂ Wn(x1 ) (x1 ) ∪ · · · ∪ Wn(xr ) (xr ).

5.1 A General Framework

165

For the sake of notational simplicity, set for each 1 ≤ i ≤ r Vi = Vn(xi ) (xi ), Wi = Wn(xi ) (xi ) and n i = n(xi ). Let n ∗1 < n ∗2 < · · · < n ∗s be such that {n 1 , . . . , n r } = {n ∗1 , . . . , n ∗s }. Let I1 ⊂ N be a maximal subset of {1, . . . , r } such that for each i ∈ I1 both n i = n ∗1 , and Wi ∩ W j = ∅ for every j ∈ I1 with j = i. Inductively, we define Ik for 2 ≤ k ≤ s in the following way: supposing that I1 , . . . , Ik−1 have already been defined, let Ik be a maximal set of {1, . . . , r } such that for each i ∈ Ik both n i = n ∗k , and Wi ∩ W j = ∅ for every j ∈ I1 ∪ ... ∪ Ik with i = j. Finally, define I = I1 ∪ · · · ∪ Is . By construction, we have {Wi }i∈I a family of pairwise disjoint sets. We claim that {Vi }i∈I is a cover of K . To see this, recall that given any W j with 1 ≤ j ≤ r , there is i ∈ I with n i ≤ n j such that W j ∩ Wi = ∅. Applying f ni we get f ni (W j ) ∩ Dδ1 /4 ( f ni (xi )) = ∅. It follows from (I2 ) that   δ1 diam f ni (W j ) ≤ 2δ1 σ (n j −n j ) /9 < . 4 Hence, f ni (W j ) ⊂ Dδ1 ( f ni (xi )) = f ni (Vi ). This gives W j ⊂ Vi . Thus, we have proved that given any W j with 1 ≤ j ≤ r , there is some i ∈ I such that W j ⊂ Vi . Recalling (5.4), this means that {Vi }i∈I is a cover of K . Now, using Lemma 5.2, we easily find τ > 0 such that m 0 (Wi ) ≥ τ m 0 (Vi ) for all i ∈ I . Together with (5.3), this yields m0



 Wi

≥τ



i∈I

m 0 (Vi ) ≥ τ m 0



i∈I

 Vi

≥ τ m 0 (K ) >

i∈I

τ m 0 (A0 ). 2

(5.5)

We claim that there is i 0 ∈ I for which 2δ m 0 (Wi0 \ A0 ) < . m 0 (Wi0 ) τ

(5.6)

Assuming by contradiction that (5.6) does not hold, we have m0



  2δ

 Wi \ A0 ≥ m 0 Wi . i∈I i∈I τ

(5.7)

Then, using (5.3), (5.5) and (5.7), we get δm 0 (A0 ) > m(B \ K ) ≥ m 0



  2δ

 Wi \ A0 ≥ m 0 Wi > δm 0 (A0 ), i∈I i∈I τ

166

5 Inducing Schemes

which clearly gives a contradiction. Now, taking D = f ni0 (Wi0 ), it follows from (5.6), the fact that f (A) ⊂ A and Lemma 5.2 that there is C > 0 such that m ni0 (D \ A0 ) m ni0 (D)

m ni0 ( f ni0 (Wi0 \ A0 ))

≤

m n i0 ( f

n i0

(Wi0 ))

≤C

  m ni0 Wi0 \ A0 m ni0 (Wi0 )

Since δ > 0 can be chosen arbitrarily small, the result follows.

=

2 Cδ . τ 

5.2 The Partition In this section, we describe an inductive process leading to an m 0 mod 0 partition of the disk 0 as in Theorem 5.1. The construction in this section is closely related to constructions performed in [1–3, 7], based on ideas from [5]. First, we introduce some auxiliary sets and constants. Given a domain ω = ωk, as in (I3 ), set for each n>k   (5.8) ω : dk+ ( f k+ (y), f k+ (ω)) ≤ δ0 σ n−k . An (ω) = y ∈  ω) contains a neighRemark 5.4 It follows from the definition of An (ω) that f k+ ( bourhood of the outer component of the boundary of f k+ (An (ω)) of size at least ω) 2δ0 − δ0 (1 + σ n−k ) = δ0 (1 − σ n−k ). So, using (I3 ), we easily deduce that f k ( contains a neighbourhood of the outer component of the boundary of f k (An (ω)) of size at least δ0 (1 − σ n−k )/C1 . Notice that a similar conclusion holds if in (I3 ) we ωn, mapped to a concentric disk require ωn, mapped to a disk of radius aδ0 and  of radius bδ0 , with a and b not necessarily constant, but still uniformly bounded from above and below by positive constants, and b − a bounded below by a uniform positive constant. The previous remark motivates the choice of some constants that we will make next. Considering δ1 > 0 and 0 < σ < 1 as in (I2 ) and C1 , δ0 > 0 as in (I3 ), we take δ2 = δ0 + N0 ∈ N large enough so that

δ1 C1 , 2

C 1 σ N0 < 1

(5.9)

(5.10)

and N1 ∈ N large enough so that δ2 σ

N1

≤ δ0

  δ0 1 − σ N 1 2δ1 N1 σ ≤ and . 9 C1

(5.11)

5.2 The Partition

167

5.2.1 Inductive Construction Here, we define inductively sequences (Pn )n , (n )n and (Sn )n , using the sets (Hn )n given by (I1 ). Each Pn will be the union of elements of the partition constructed at step n, and n the set of points which do not belong to any element of the partition defined up to time n. The set Sn contains domains which could have been chosen for the partition at time n but intersect other elements previously chosen. A key point in our argument is the conclusion of Proposition 5.5 below, which asserts that every point in Hn either belongs to an element of the partition constructed until that moment or to an Sn . First Step We start our inductive process at time N0 , for some N0 satisfying (5.10). Since HN0 is a compact set, there is a finite set FN0 ⊂ HN0 such that H N0 ⊂

x∈FN0

W N0 (x).

Consider x1 , . . . , x jN0 ∈ FN0 and, for each 1 ≤ i ≤ j N0 , a domain ω N0 ,i ⊂ W N0 (xi ) as in (I3 ), such that P N0 = {ω N0 ,1 , . . . , ω N0 , jN } is a maximal family of pairwise 0 disjoint sets contained in 0 . These are precisely the elements of the partition P constructed in our first step of the induction. Set

 N0 = 0 \ For each ω ∈ P N0 , set

ω∈P N0

ω.

N0 (ω), S N0 (ω) = W

N0 (ω) as in (5.1). Define also with W S N0 (0 ) =

ω∈P N0

S N0 (ω),

and for c0 =  \ 0     S N0 c0 = x ∈ 0 : d(x, ∂0 ) < 2δ1 σ N0 . Finally, define S N0 = S N0 (0 ) ∪ S N0 (c0 ). For definiteness, set n = Sn = 0 for each 1 ≤ n < N0 .

168

5 Inducing Schemes

General Step The induction step of the construction follows ideas of the first step with some adaptations. Given n > N0 , assume that Pk , k and Sk have already been defined for all k with N0 ≤ k ≤ n − 1. Let Fn be a finite subset of the compact set Hn such that

Wn (x). (5.12) Hn ⊂ x∈Fn

Consider x1 , . . . , x jn ∈ Fn and, for each1 ≤ i ≤ jn , a domain ωn,i ⊂ Wn (xi ) as in (I3 ) for which Pn = ωn,1 , . . . , ωn, jn is a maximal family of pairwise disjoint sets contained in n−1 , such that for each 1 ≤ i ≤ jn we also have ωn,i ∩

n−1

ω∈Pk

k=N0

 An (ω) = ∅.

(5.13)

The sets in Pn are the elements of the partition P obtained in the n-th step of the construction. Set

n

ω. (5.14) n = 0 \ ω∈Pk

k=N0

Given ωk, ∈ Pk , for some N0 ≤ k ≤ n, define for n − k < N1 k (ω) Sn (ω) = W and for n − k ≥ N1   Sn (ω) = y ∈  ω : 0 < dk+ ( f k+ (y), f k+ (ω)) ≤ δ2 σ n−k . Define Sn (0 ) = and

n



k=N0

ω∈Pk

Sn (ω)

(5.15)

(5.16)

    Sn c0 = x ∈ 0 : d(x, ∂0 ) < δ1 σ n .

Define also Sn = Sn (0 ) ∪ Sn (c0 ). Finally, set P=

n≥N0

Pn

By construction, the elements in P are pairwise disjoint and contained in 0 . However, there is still no evidence that the union of these elements covers a full m 0 measure subset of 0 . This will be obtained in Corollary 5.9 below.

5.2 The Partition

169

5.2.2 Key Relations Before we address the key property given by Proposition 5.5, we highlight some properties of the sets introduced in (I2 ) and (I3 ). First of all, observe that for each x ∈ Hk and y ∈ Hn with n ≥ k we have 

k (x); Wn (y) ∩ Wk (x) = ∅ =⇒ Wn (y) ⊂ W k (x) = ∅ =⇒ W n (y) ⊂ Vk (x). n (y) ∩ W W

(5.17)

To see this, recall that by (I2 ), we have diam( f k (Wn (y))) ≤

2δ1 n−k 2δ1 σ . ≤ 9 9

(5.18)

Then, assuming that Wn (y) intersects Wk (x), we necessarily have that f k (Wn (y)) intersects f k (Wk (x)), which by definition is a disk of radius δ1 /9 around f k (x). Together with (5.18), this implies that f k (Wn (y)) is contained in the disk of radius δ1 /3 centred at f k (x), and so, as Wn (y) and Wk (x) are both contained in , the first case of (5.17) follows. The second case can be proved similarly. By definition, for each ω ∈ Pk with k ≥ N0 , we have k (ω) ⊃ Sk (ω) ⊃ Sk+1 (ω) ⊃ · · · , W and for n ≥ k + N1 we have

Wk (ω) ⊃ Sn (ω).

(5.19)

(5.20)

It follows from the definitions and (5.17) that for all k2 ≥ k1 ≥ N0 and all ω1 ∈ Pk1 , ω2 ∈ Pk2 we have k1 (ω1 ) = ∅ =⇒ Sk2 (ω2 ) ∪ ω2 ⊂ Vk1 (ω1 ) Sk2 (ω2 ) ∩ W

(5.21)

and for all n ≥ N1 k1 (ω1 ). Sk2 +n (ω2 ) ∩ Wk1 (ω1 ) = ∅ =⇒ Sk2 +n (ω2 ) ∪ ω2 ⊂ W

(5.22)

Proposition 5.5 For all n ≥ N0 , we have Hn ∩ n ⊂ Sn . Proof Consider the finite set Fn ⊂ Hn in (5.12) used to define the elements in Pn . Given z ∈ Hn ∩ n , there must be some y ∈ Fn such that z ∈ Wn (y). Let ωn, be the domain associated to Wn (y) as in (I3 ). It is enough to show that Wn (y) ⊂ Sn (0 ) ∪ n (y) ⊃ Wn (y), and so we are done. If ωn, ∈ / Sn (c0 ). If ωn, ∈ Pn , then Sn (ωn, ) = W Pn , then at least one of the three cases below holds:

170

5 Inducing Schemes

n (ω) and 1. ωn, ∩ ω = ∅, for some ω ∈ Pn . In this case, we have Sn (ω) = W Wn (y) ∩ Wn (ω) = ∅. Hence, using (5.17) we get n (ω) = Sn (ω) ⊂ Sn (0 ). Wn (y) ⊂ W 2. ωn, ∩ An (ω) = ∅, for some N0 ≤ k < n and ω ∈ Pk . Observe that by definition, we have An (ω) ⊂  ω ⊂ Wk (ω). Assume first that n − k < N1 . Then, as in the previous situation, we have k (ω) = Sn (ω) ⊂ Sn (0 ). Wn (y) ⊂ W Assume now that n − k ≥ N1 . We claim that, in this case, we have ω. Wn (y) ⊂ 

(5.23)

ω) contains a neighbourIndeed, it follows from Remark 5.4 that the set f k ( k (An (ω)) of size at least hood of the outer component of the boundary of f   N1 δ0 1 − σ /C1 . On the other hand, by definition of Wn (y) and (I2 ) we have diam( f k (Wn (y))) ≤

2δ1 n−k 2δ1 N1 σ σ . ≤ 9 9

(5.24)

Recalling (5.11) and observing that in the situation we are considering the set ω). Then, since f k (Wn (y)) intersects f k (An (ω)), we have f k (Wn (y)) ⊂ f k ( ω = ∅ and f k maps  ω bijectively onto its image, we deduce (5.23). Wn (y) ∩  Now, letting 0 ≤ ω ≤ L be the integer associated to the domain ω, using (I3 ) and (5.24) we obtain diam( f k+ω (Wn (y))) ≤

2δ1 C1 σ n−k . 9

(5.25)

Since the set f k+ω (Wn (y)) intersects f k+ω (An (ω)), we have for each w ∈ f k+ω (Wn (y)) d0 (w, 0 ) ≤ δ0 σ n−k +

2δ1 C1 σ n−k = δ2 σ n−k . 9

This shows that Wn (y) ⊂ Sn (ω) ⊂ Sn (0 ). 3. ωn, ∩ c0 . This in particular implies that Wn (y) intersects ∂0 . By (I3 ), we get diam (Wn (y)) ≤

2δ1 n σ , 9

and so Wn (y) ⊂ Sn (c0 ). 

5.2 The Partition

171

We finish this subsection with a result that will only be used to estimate the tail of recurrence times in Sect. 5.3. Incidentally, this is the only place where we use the sets An (ω) introduced in (5.8). Fix an integer N2 ≥ N1 such that   δ0 (1 − σ ) δ2 1 + C12 σ N2 < δ0 and δ2 C1 σ N2 < . C1

(5.26)

Lemma 5.6 If k2 > k1 ≥ N0 , then for all ω1 ∈ Pk1 and ω2 ∈ Pk2 we have Sk2 +N2 (ω1 ) ∩ Sk2 +N2 (ω2 ) = ∅. Proof Assume by contradiction that there exists x ∈ 0 such that x ∈ Sk2 +N2 (ω1 ) ∩ Sk2 +N2 (ω2 ).

(5.27)

Consider 0 ≤ 1 , 2 ≤ L associated to the domains ω1 , ω2 as in (I3 ), respectively. By (5.15), we have dk2 +2 ( f k2 +2 (x), f k2 +2 (ω2 )) < δ2 σ N2 . It follows from (I3 ) that there must be some y ∈ ω2 such that dk2 ( f k2 (x), f k2 (y)) < δ2 C1 σ N2 .

(5.28)

Also, from (5.20) and (5.27) give in particular that Sk2 +N2 (ω2 ) intersects Wk1 (ω1 ). k1 (ω1 ). Then, using (I2 ) and (5.28) Hence, by (5.22), we have Sk2 +N2 (ω2 ) ∪ ω2 ⊂ W we obtain dk1 ( f k1 (x), f k1 (y)) ≤ σ k2 −k1 dk2 ( f k2 (x), f k2 (y)) < δ2 C1 σ N2 +k2 −k1 .

(5.29)

On the other hand, since x ∈ Sk2 +N2 (ω1 ) we have by definition dk1 +1 ( f k1 +1 (x), f k1 +1 (ω1 )) < δ2 σ N2 +k2 −k1 .

(5.30)

Recalling the first part of (5.26), we easily conclude that x ∈ Ak2 (ω1 ). Then, ω1 ) contains a neighbourhood of the outer component Remark 5.4 gives that f k1 ( of the boundary of f k1 (Ak2 (ω1 )) of size at least δ0 (1 − σ k2 −k1 2 ) δ0 (1 − σ ) ≥ . C1 C1 ω1 ). From the second part of (5.26) and (5.29), we easily deduce that f k1 (y) ∈ f k1 ( It follows from (I3 ) and (5.29) that dk1 +1 ( f k1 +1 (x), f k1 +1 (y)) < δ2 C12 σ N2 +k2 −k1 , which jointly with (5.30) and the first part of (5.26) yields

(5.31)

172

5 Inducing Schemes

dk1 +1 ( f k1 +1 (y), f k1 +1 (ω1 )) < δ2 (1 + C12 ) σ N2 +k2 −k1 < δ0 σ k2 −k1 . This implies that y ∈ Ak2 (ω1 ) with ω1 ∈ Pk1 . Since y ∈ ω2 and ω2 ∈ Pk2 , we have a contradiction with (5.13).

5.2.3 Metric Estimates In the proof of Proposition 5.5, we have deduced some relations about the sets Sn , just using properties of the distance on the sets introduced in (I2 ) and (I3 ). Here, we obtain additional properties in terms of measure. Lemma 5.7 There exists C > 0 such that for all n ≥ k ≥ N0 and ω ∈ Pk we have m 0 (Sn (ω)) ≤ Cσ n−k m 0 (ω). Proof Consider first the case n ≥ k + N1 . Letting  = ω , recall that   ω : 0 < dk+ ( f k+ (y), f k+ (ω)) ≤ δ2 σ n−k . Sn (ω) = y ∈  ω diffeomorphically to a disk of radius 2δ0 and ω to a disk Moreover, f k+ maps  ω). Then, there must be some uniform constant of radius δ0 concentric with f k+ ( D > 0 such that m k+ ( f k+ (Sn (ω))) ≤ Dσ n−k . Let C0 > 0 be the constant given by Lemma 5.2. Taking δ > 0 a uniform lower bound for the measure of disks of radius δ0 , we have m 0 (Sn (ω)) C02 D n−k m k+ ( f k+ (Sn (ω))) ≤ C02 ≤ σ . m 0 (ω) m k+ ( f k+ (ω)) δ

(5.32)

Consider now the case n < k + N1 . Since Sn (ω) ⊂ Vk (ω), by Lemma 5.2 we have m 0 (Sn (ω)) ≤ C0 m 0 (ω).

(5.33)

Finally, choose C ≥ C00 D/δ sufficiently large so that C0 ≤ Cσ N1 . Using (5.32), (5.33) and observing that σ N1 ≤ σ n−k for n < k + N1 , we easily get the desired conclusion.  Lemma 5.8

∞

m 0 (Sn ) < ∞.

n=N0

Proof Recall that Sn = Sn (0 ) ∪ Sn (c0 ). First, we consider the terms in Sn (0 ). It follows from (5.16) and Lemma 5.7 that for each n ≥ N0 , we have

5.2 The Partition

173

m 0 (Sn (0 )) ≤

n

m 0 (Sn (ω))

k=N0 ω∈Pk

≤

n

Cσ n−k m 0 (ω)

k=N0 ω∈Pk

=C

n

σ n−k m 0



k=N0

ω∈Pk

ω



Hence,

m 0 (Sn (0 )) =

n≥N0

n≥N0 k≥0

σ k m0

ω∈Pn

 ω =

1 m 0 (0 ). 1−σ

On the other hand, recalling that   Sn (c0 ) = x ∈ 0 : d(x, ∂0 ) < δ1 σ n ,

(5.34)

we may find C > 0 such that m 0 (Sn (c0 )) ≤ Cσ n . This obviously gives that the sum of the corresponding terms is finite.  Corollary 5.9 P is an m 0 mod 0 partition of 0 . Proof By definition of the sets n , it is enough to show that the intersection  of all these sets has zero m 0 measure. Assume by contradiction that m 0 ∩n≥N0 n > 0. Using (I1 ), we find some set B ⊂ 0 with m 0 (B) > 0 such that for every x ∈ B we have infinitely many times n 1 < n 2 < · · · (in principle depending on x) so that x ∈ Hn k ∩ n k for each k ∈ N. It follows from Proposition 5.5 that x ∈ Sn k for all k ∈ N. On the other hand, using Lemma 5.8 and Borel-Cantelli Lemma, we easily deduce that for m 0 almost every x ∈ 0 we cannot have x ∈ Sn for infinitely many values of n. Clearly, this gives a contradiction with the fact that m 0 (B) > 0 and  x ∈ Sn k for all k ∈ N and x ∈ B. The previous result gives the first conclusion of Theorem 5.1: there is an m 0 mod 0 partition P of 0 into domains ωn, as in (I3 ). The remaining conclusions in the itemised list of Theorem 5.1 will be obtained in the following sections.

5.3 Inducing Times Here, we prove the first two items of Theorem 5.1. In the previous section, we have  constructed an m mod 0 partition P = n≥N0 Pn of the disk 0 , where each element of Pn is a domain ωn, related to some point in Hn as in (I3 ); recall Sect. 5.2.1 and Corollary 5.9. Set

174

5 Inducing Schemes

R(x) = n + 

(5.35)

for each n ≥ N0 and x ∈ ωn, ∈ Pn . Remark 5.10 From the inductive construction in Sect. 5.2.1, we can easily see that each Pn has only a finite number of elements. This in particular implies that for each n ≥ 1, there is a finite number (possibly zero) of elements ω ∈ P for which R|ω = n. We claim that, for all n ≥ 1, Hn ∩ {R > L + n} ⊂ Sn .

(5.36)

To see this, observe first that for n < N0 there is nothing to prove, since Sn = 0 , by definition. Assume now n ≥ N0 . If R(x) > n + L for some x ∈ Hn , then by (5.35) we necessarily have x ∈ n . It follows from Proposition 5.5 that x ∈ Sn , and therefore (5.36) holds. The second item of Theorem 5.1 is then a consequence of Lemma 5.8 and (5.36). The conclusions of first item of Theorem 5.1 are given by the next result. Lemma 5.11 There are C > 0 and 0 < β < 1 arbitrarily small such that, for all n ≥ N0 and x, y ∈ ωn, ∈ Pn , 1. d0 (x, y) ≤ βd R ( f R (x), f R (y)); 2. d j ( f j (x), f j (y)) ≤ Cd R ( f R (x), f R (y)), for all 0 ≤ j ≤ R; det D f R |Tx  ≤ Cd R ( f R (x), f R (y))η . 3. log det D f R |Ty  Proof Recall that by construction, we have each ωn, ∈ Pn a subset of some Vn (z) with z ∈ Hn and f n+ mapping ωn, diffeomorphically to a disk of radius δ0 . Using (I2 ) and (I3 ), we obtain for all x, y ∈ ωn, d0 (x, y) ≤ σ n dn ( f n (x), f n (y)) ≤ C1 σ n dn+ ( f n+ (x), f n+ (y)). Since n ≥ N0 , it follows from (5.10) that C1 σ n ≤ C1 σ N0 < 1. Taking β = C1 σ N0 and recalling that R|ωn, = n + , we obtain the first item. Note that C1 is a uniform constant and, choosing N0 large we make β small. We split the proof of the second item into two cases. If n ≤ j ≤ n + , it follows from (I3 ) that, for all x, y ∈ ωn, , d j ( f j (x), f j (y)) ≤ C1 dn+ ( f n+ (x), f n+ (y)). If on the other hand 0 ≤ j < n, it follows from (I2 ) and (I3 ) that for all x, y ∈ ωn, d j ( f j (x), f j (y)) ≤ σ n− j dn ( f n (x), f n (y)) ≤ C1 σ n− j dn+ ( f n+ (x), f n+ (y)). In both cases, we obtain the conclusion of the second item with C = C1 .

5.3 Inducing Times

175

Finally, we prove the third item. Using (I2 )–(I3 ) and basic properties of the Jacobian, we have log

det D f  |T f n (x)  det D f n |Tx  det D f R |Tx  = log + log det D f R |Ty  det D f  |T f n (y)  det D f n |Ty  ≤ C1 dn+ ( f n+ (y), f n+ (z))η + C0 dn ( f n (y), f n (z))η η

≤ C1 dn+ ( f n+ (y), f n+ (z))η + C0 C1 dn+ ( f  ( f n (y)), f  ( f n (z)))η . η

Since R|ωn, = n + , we obtain the third item with C = C1 + C0 C1 .



The proof of the last item of Theorem 5.1 is left to Sect. 5.3.2 below. In the next subsection, we give a useful criterium for the integrability of R.

5.3.1 Integrability Theorem 5.1 will be useful to build Gibbs-Markov maps as in Chap. 3 or Young structures as in Chap. 4. In the first case, we choose each domain ω ∈ P in such a way that f R (ω) = 0 for all ω ∈ P, thus obtaining from Lemma 5.11 that f R : 0 → 0 is a Gibbs-Markov map. In the second case, we take the set 0 in a family of unstable disks  u defining a Young structure and each f R (ω) an unstable disk in  u . Then, using a quotient map as in (4.5) with domain 0 , we have again a Gibbs-Markov map F : 0 → 0 . Proposition 5.12 below gives a useful criterium for the integrability of the inducing time function R. For its statement, we need some  additionalj concepts. Consider (Hn )n as in (I1 )–(I3 ). Take R0 = 0 and Rk = k−1 j=0 R ◦ F , for each k ≥ 1. We say that a sequence (Hn∗ )n of sets in 0 is F-concatenated in (Hn )n if x ∈ Hn∗ =⇒ F i (x) ∈ Hn−Ri ,

(5.37)

whenever Ri (x) ≤ n < Ri+1 (x), for some i ≥ 0. In Sect. 6.3, we will actually take Hn∗ = Hn , for all n ≥ 1. The possibility of having this new sequence (Hn∗ )n will be particularly useful in Sect. 7.3. We say that (Hn∗ )n is a frequent sequence if there exists θ > 0 such that, for m 0 almost every x ∈ 0 ,  1  lim sup # 1 ≤ j ≤ n : x ∈ H j∗ > θ. n→∞ n

(5.38)

Proposition 5.12 Let F : 0 → 0 be a Gibbs-Markov map with respect to a partition P, and R : 0 → N be constant in the elements of P. Assume that there exist that is F-concatenated in (Hn )n ; 1. a frequent sequence (Hn∗ )n of sets in 0  2. a sequence (Sn )n of sets in 0 such that n≥1 m 0 (Sn ) < ∞; 3. L ∈ N such that Hn ∩ {R > L + n} ⊂ Sn , for all n ≥ 1. Then R is integrable with respect to m 0 .

176

5 Inducing Schemes

Proof Consider the ergodic F-invariant probability measure ν  m 0 given by Theorem 3.13. Since dν/dm 0 is bounded above and below by positive constants, it is enough to check the integrability of R with respect to ν. Assume by contradiction that R ∈ / L 1 (ν). Since R is a positive function, it follows from Birkhoff Ergodic Theorem that  k−1 1 lim R(F i (x)) → Rdν = ∞, (5.39) k→∞ k i=0  for ν almost every x ∈ 0 . Since n≥1 m 0 (Sn ) < ∞, it follows from Borel–Cantelli Lemma that ν almost every x ∈ 0 belongs in a finite number of sets Sn . Define s(x) = # {n ≥ 1 : x ∈ Sn } for x ∈ 0 . Using that dν/dm 0 is bounded above by a positive constant and Birkhoff Ergodic Theorem, we have for ν almost every x ∈ 0 1 s(F i (x)) → k i=0 k−1

 sdν =



ν(Sn ) < ∞.

(5.40)

n≥1

Since (Hn∗ )n is F-concatenated in (Hn )n , given i ≥ 0 and Ri ≤ j < Ri+1 , we have F i (x) ∈ H j−Ri , whenever x ∈ H j∗ . We cannot have R(F i (x)) < j − Ri , for otherwise we would have Ri+1 − Ri = R(F i (x)) < j − Ri ≤ Ri+1 − Ri . Set k = j − Ri . Since we assume Hk ∩ {R > k + L} ⊂ Sk , we have F i (x) ∈ Sk or R(F i (x)) = k +  for some 0 ≤  ≤ L. Thus, the number of integers j with Ri ≤ j < Ri+1 such that x ∈ H j∗ is bounded by the number of integers k such that F i (x) ∈ Sk or F i (x) ∈ {R = k + }, for some 0 ≤  ≤ L. This means that   # Ri ≤ j < Ri+1 : x ∈ H j∗ ≤ 1 + s(F i (x)). Given n ≥ 1, define r (n) = min{Ri : Ri > n}. For each n ≥ 1, we have #{1 ≤ j ≤ n : x ∈ H j∗ } ≤

r (n) r (n) (1 + s(F i (x))) ≤ r (n) + s(F i (x)). i=0

i=0

Therefore,  r (n)  r (n) 1  1 ∗ i # j ≤ n : x ∈ Hj ≤ s(F (x)) . 1+ n n r (n) i=0 Observe that if r (n) = k, then by definition we have Rk−1 ≤ n < Rk . Hence,   n Rk Rk 1 Rk−1 ≤ < = 1+ , k r (n) k k+1 k which together with (5.39) gives

(5.41)

5.3 Inducing Times

177

n Rk 1 R(F i (x)) = ∞. = lim = lim n→∞ r (n) k→∞ k k→∞ k i=0 k−1

lim

(5.42)

It follows from (5.40), (5.41) and (5.42) that lim

n→∞

 1  r (n) # 1 ≤ j ≤ n : x ∈ H j∗ = lim = 0, n→∞ n n

which clearly contradicts the fact that (Hn∗ )n is a frequent sequence.



5.3.2 Tail Estimates This subsection is entirely devoted to the proof of the last item of Theorem 5.1. Assume that there exists some 0 < θ ≤ 1 for which h θ is defined. We need to show that there is a sequence (E n )n of sets in 0 with m 0 (E n ) → 0 exponentially fast with n such that, for every n ≥ 1, {R > n + L} ⊂ {h θ > n} ∪ E n .

(5.43)

We start with a simple result, essentially a consequence of the definition of h θ and Proposition 5.5. Lemma 5.13 If d0 (x, ∂0 ) > δ1 σ θn/2 and n ≥ h θ (x), for some x ∈ n with n ≥ 2N0 /θ , then there are θ n/2 ≤ t1 < · · · < tk ≤ n with k ≥ [θ n/2] such that x ∈ Sti (0 ) for all 1 ≤ i ≤ k. Proof Consider n ≥ 2N0 /θ and x ∈ n with n ≥ h θ (x). If follows from (5.2) that #{1 ≤ j ≤ n : x ∈ H j } ≥ θ n. This implies that there are times t1 < · · · < tk ≤ n with k ≥ [θ n/2] and t1 ≥ θ n/2 ≥ N0 such that x ∈ Hti ∩ n , for all 1 ≤ i ≤ k.

(5.44)

Recalling that n ⊂ ti , it follows from Proposition 5.5 that for all 1 ≤ i ≤ k Hti ∩ n ⊂ Hti ∩ ti ⊂ Sti (0 ) ∪ Sti (c0 ).

(5.45)

Since d0 (x, ∂0 ) > δ1 σ θn/2 , we have d0 (x, ∂0 ) > δ1 σ ti for all 1 ≤ i ≤ k. Hence, x∈ / Sti (c0 ), for all 1 ≤ i ≤ k. Together with (5.44) and (5.45), this gives x ∈ Sti (0 ) for all 1 ≤ i ≤ k.  Given k, n ≥ 1, define  X n (k) = x ∈ n | ∃t1 < · · · < tk ≤ n : x ∈

k  i=1

 Sti (0 ) .

178

5 Inducing Schemes

It follows from Lemma 5.13 that   n ⊂ {h θ > n} ∪ x ∈ 0 | d(x, ∂0 ) ≤ δ1 σ θn/2 ∪ X n



θn 2

 .

Note that there exists C > 0 such that for all n ≥ 1   m 0 x ∈ 0 | d(x, ∂0 ) ≤ δ1 σ θn/2 ≤ Cσ θn/2 . So, taking 

E n = x ∈ 0 | d(x, ∂0 ) ≤ δ1 σ

θn/2



 ∪ Xn

θn 2

 ,

the proof of (5.43) is reduced to show that there exist C, c > 0 such that for all n ≥ k ≥ 1 we have (5.46) m 0 (X n (k)) ≤ Ce−cn . This will be proved in the sequel in several steps. Fix N2 ≥ N1 as in (5.26) and take an integer N3 ≥ N2 , to be specified in (5.57). Given x ∈ X n (k), consider • the moments u 1 < · · · < u p ≤ n for which x belongs to some Su i +ni (ωi ) with ωi ∈ Pu i and n i ≥ N3 ; • the moments v1 < · · · < vq ≤ n for which x belongs to some Svi +ni (ωi ) with ωi ∈ Pvi and n i < N3 . Observe that

p q (n i + 1) + (n i + 1) ≥ k. i=1

(5.47)

i=1

We distinguish two possible cases: 1.

p i=1

ni ≥

k . 2

Defining for each p ∈ N and n 1 , . . . , n p ≥ N3 the set 

 u1 < · · · < u p ≤ n :x∈ Su i +ni (ωi ) Y (n 1 , . . . , n p ) = x ∈ 0 | ∃ ω1 ∈ Pu 1 , . . . , ω p ∈ Pu p i=1 p

and Yk =

n 1 ,...,n p ≥N3  n i ≥ k2

we have in this case x ∈ Yk .

Y (n 1 , . . . , n p ).



5.3 Inducing Times

2.

p

179

k . Since we assume n 1 , . . . , n p ≥ N3 , we must have in this case p < 2 i=1 k/(2N3 ). Using (5.47) and the fact that n 1 , . . . , n q < N3 , we may write ni
0 such that for all n 1 , . . . , n p > N2 we have m 0 (Y (n 1 , . . . , n p )) ≤ D0 σ n 1 +···+n p . p

Proof Defining for each u 1 ≥ N0 and ω1 ∈ Pu 1 the set

Yuω11 (n 1 , . . . , n p ) =

⎧ ⎨

⎫ p ⎬  u2 < · · · < u p x |∃ : u2 > u1, x ∈ Su i +n i (ωi ) , ⎩ ⎭ ω2 ∈ Pu 2 , . . . , ω p ∈ Pu p i=1

we may write Y (n 1 , . . . , n p ) =





Yuω11 (n 1 , . . . , n p ).

u 1 ≥N0 ω1 ∈Pu 1

Noting that the elements ω1 ∈ Pu 1 with u 1 ≥ N0 are pairwise disjoint, it is enough to show that there is some constant D0 > 0 such that for all u 1 ≥ N0 and ω1 ∈ Pu 1 we have

180

5 Inducing Schemes

  p m 0 Yuω11 (n 1 , . . . , n p ) ≤ D0 σ n 1 +···+n p m(ω1 ).

(5.50)

We shall prove (5.50) by induction on p. Considering D0 > C, where C > 0 is the constant in Lemma 5.7, we get the result for p = 1. Now suppose that p > 1. We may write



Yuω11 (n 1 , . . . , n p ) = Yuω22 (n 2 , . . . , n p ) (5.51) u 2 >u 1 ω2 ∈Pu 2

By Lemma 5.6, for all ω2 ∈ Pu 2 with u 2 > u 1 we have Su 2 +N2 (ω1 ) ∩ Su 2 +N2 (ω2 ) = ∅.

(5.52)

Assuming, with no loss of generality, that Yuω11 (n 1 , . . . , n p ) is nonempty, we have in particular (5.53) Su 1 +n 1 (ω1 ) ∩ Su 2 +n 2 (ω2 ) = ∅. From (5.52) and (5.53), we deduce that u 1 + n 1 < u 2 + N2 , or equivalently u 2 − u 1 > n 1 − N2 . Then, using (5.19) and (I2 )–(I3 ), we easily get     diam f u 1 (Su 2 +n 2 (ω2 )) ≤ σ u 2 −u 1 diam f u 2 (Su 2 +n 2 (ω2 )) ≤ 2δ0 C1 σ n 1 −N2 . (5.54) Consider now β1 the outer component of the boundary of f u 1 (Su 1 +n 1 (ω1 )) in ω1 ) and N1 a neighbourhood of β1 in f u 1 ( ω1 ) of size 2δ0 C1 σ n 1 −N2 . Since we f ( assume n 1 ≥ N2 ≥ N1 , by definition of Su 1 +n 1 (ω1 ) there exists 0 ≤ 1 ≤ L such that f 1 (β1 ) coincides with the boundary of a disk of radius not exceeding 2δ0 concentric with f u 1 +1 (ω1 ). It follows from (I3 ) that the m u 1 measure of the submanifold β1 is uniformly bounded. Hence, there must be some constant D > 0 (not depending on u 1 ≥ N0 nor on ω1 ∈ Pu 1 ) such that u1

m u 1 (N1 ) ≤ Dσ n 1 .

(5.55)

Now, since f u 1 (Su 2 +n 2 (ω2 )) intersects β1 , it follows from (5.54) that f u 1 (Su 2 +n 2 (ω2 )) is contained in N1 . This gives (5.56) f u 1 (ω2 ) ⊂ N1 . On the other hand, by the induction hypothesis, we have m 0 (Yuω22 (n 2 , . . . , n p )) ≤ D0

p−1

σ n 2 +···+n p m 0 (ω2 ),

Using (5.19) and the definition of Yuω11 (n 1 , . . . , n p ), we easily deduce that Yuω11 (n 1 , . . . , n p ) ⊂ Su 1 +n 1 (ω1 ) ⊂ Vu 1 (ω1 ).

5.3 Inducing Times

181

Consider C0 > 0 the constant given by Lemma 5.2. We have m u 1 ( f u 1 (Yuω22 (n 2 , . . . , n p ))) ≤ C0 D0

p−1

  σ n 2 +···+n p m u 1 f u 1 (ω2 ) .

Observe that the sets ω2 ∈ Pu 2 with u 1 < u 2 are pairwise disjoint. Moreover, by (5.21) these sets are all contained in Vu 1 (ω1 ). As f u 1 is injective on Vu 1 (ω1 ), we easily get that the sets f u 1 (ω2 ) with ω2 ∈ Pu 2 and u 1 < u 2 are also pairwise disjoint. It follows from (5.51), (5.55) and (5.56) that m u 1 ( f u 1 (Yuω11 (n 1 , . . . , n p ))) ≤

n



m u 1 ( f u 1 (Yuω22 (n 2 , . . . , n p )))

u 2 =u 1 +1 ω2 ∈Pu 2 p−1

≤ C 0 D0

n

σ n 2 +···+n p



m u 1 ( f u 1 (ω2 ))

u 2 =u 1 +1 ω2 ∈Pu 2

≤ ≤

p−1 C0 D0 σ n 2 +···+n p m u 1 (N1 ) p−1 DC0 D0 σ n 1 +···+n p .

Taking D0 ≥ DC0 , we finish the proof.



Observe that the constant D0 given by Lemma 5.14 may depend on the integer N2 introduced in (5.26); however, it does not depend on the integer N3 . Thus, we are allowed to choose an integer N3 ≥ N2 sufficiently large so that σ + D0 σ N3 < 1.

(5.57)

The next result gives the expected estimate on the measure of the set Yk . Proposition 5.15 There are D1 > 0 and λ1 < 1 such that for all k ≥ 1 we have m 0 (Yk ) ≤ D1 λk1 . Proof By the definition of Yk and Lemma 5.14, we just need to show that

D0 σ n 1 +···+n p p

n 1 ,...,n p ≥N3  n i ≥ k2

decays exponentially fast with k. We use the generating series



n≥1

n 1 ,...,n p ≥N3  n i =n

p D0 σ n 1 +···+n p z n

=

∞ p=1

=

D0

∞

p σ z

n n

n=N3

D0 σ N 3 z N 3 . 1 − σ z − D0 σ N 3 z N 3

182

5 Inducing Schemes

Under condition (5.57), the function above has no pole in a neighbourhood of the unit disk in C. Thus, the coefficients of its power series decay exponentially fast: there are constants D1 > 0 and λ1 < 1 such that p D0 σ n 1 +···+n p ≤ D1 λn1 , n 1 ,...,n p ≥N3 ,  n i =n



and so we are done.

Our next goal is to show that m 0 (Z n (q)) decays exponentially fast to 0 with q. Now we introduce some useful notation. Consider T = {Sn (ω) ∪ ω | for some ω ∈ Pn and n ≥ N0 } . For each T = Sn (ω) ∪ ω ∈ T with ω ∈ Pn , define ωT = ω and t (T ) = n. We will refer to ωT as the core of T ∈ T . Notice that for any T1 , T2 ∈ T we have T1 = T2 =⇒ ωT1 ∩ ωT2 = ∅,

(5.58)

and from (5.21) it follows that T1 ∩ T2 = ∅ =⇒ T2 ⊂ Vt (T1 ) (ωT1 )

(5.59)

Finally, using Lemma 5.7, we easily deduce the existence of D > 0 such that for every T ∈ T we have (5.60) m 0 (T ) ≤ Dm 0 (ωT ) . For proving that m 0 (Z n (q)) decays exponentially fast with q, we need a couple of auxiliary lemmas that we prove in the sequel. We begin fixing some large integer N4 to be specified in (5.72). Given x ∈ Z n (q), consider the times v1 < · · · < vq ≤ n as in the definition of Z n (q) given by (5.48). Take the smallest positive integer u such that N4 u ≥ n and, for each 1 ≤ i ≤ u, pick in the interval ((i − 1)N4 , i N4 ] the first element in {v1 , . . . , vq }, if there exists at least one. Denote the subsequence of those elements by w1 < · · · < w p . We necessarily have p ≥ [q/N4 ] , and so ( p + 1)N4 ≥ q. Keeping only elements with odd indexes, we get a sequence t1 < · · · < t with 2 ≥ p. This implies that q − N4 ≥ . (5.61) 2N4 Moreover, by construction we have ti+1 − ti ≥ N4 , for each 1 ≤ i ≤ .

(5.62)

5.3 Inducing Times

183

According to the definition of Z n (q), for each 1 ≤ i ≤  there exists some Ti ∈ T such that t (Ti ) = ti . Set I = {1 ≤ i ≤  | Ti ⊂ T1 ∩ · · · ∩ Ti−1 }. Now we have two possible cases: 1. #I ≥ /2. Define for n, k ≥ 1 the set Z n0 (k) = {x ∈ n | ∃T1 ⊃ · · · ⊃ Tk with t (T1 ) < · · · < t (Tk ) ≤ n and x ∈ Tk } . Keeping only elements with indexes in I and recalling (5.61), we easily see that in this case we have x ∈ Z n0 ([(q − N4 )/(4N4 )]) . 2. #I < /2. Considering J = {1, . . . , } \ I , we necessarily have # J ≥ /2. We define   j0 = sup J and i 0 = inf i < j0 | T j0 ⊂Ti . Next we define   j1 = sup{ j ≤ i 0 | j ∈ J } and i 1 = inf i < j1 | T j1 ⊂Ti . Proceeding inductively, the process must necessarily stop at some ir0 . By construction, we have r0

{i s + 1, . . . , js }, J⊂ s=0

which then gives

r0  ( js − i s ) ≥ # J ≥ . 2 s=0

On the other hand, it follows from (5.62) that |t js − tis | ≥ N4 ( js − i s ) for all 0 ≤ s ≤ r0 . Altogether, this yields r0 r0 t (T js ) − t (Tis ) t js − tis q − N4  = ≥ ≥ . N N 2 4N4 4 4 s=0 s=0

Motivated by this second case, we consider for each T ∈ T the set FT of finite sequences (T0 , . . . , T2r ) ∈ T 2r +1 with r ≥ 0 and T0 = T such that t (T2i−2 ) ≤ t (T2i−1 ) ≤ t (T2i ) − N4 and T2i ⊂ T2i−1 for all 1 ≤ i ≤ r . Then, we define for each k ≥ 0 and T ∈ T the sets Z (k, T ) =

⎧ ⎨ ⎩

x ∈ 0 | ∃(T0 , . . . , T2r ) ∈ FT :

r t (T2i ) − t (T2i−1 ) i=1

N4

≥ k, x ∈

2r  i=0

⎫ ⎬ Ti

⎭

.

184

5 Inducing Schemes

and Z 1 (k) =



Z (k, T ).

(5.63)

T ∈T

The considerations in the case #I < /2 above show that x∈Z ([(q − N4 )/ (4N4 )] , T ) , with T = Tir0 and sequence (Tir0 , Tir0 , T jr0 , . . . , Ti0 , T j0 ) ∈ F (T ). Hence, we have     q − N4 q − N4 0 1 Z n (q) ⊂ Z n ∪Z . (5.64) 4N4 4N4 We are reduced to show that the measure of the sets Z n0 (k) and Z 1 (k) decays exponentially fast with k. Lemma 5.16 There exists λ2 < 1 such that for all k ≥ 1 we have m 0 (Z n0 (k)) ≤ λk2 m 0 (0 ). Proof We define T1 as the class of elements T ∈ T with t (T ) ≤ n not contained / T1 in any other element of T . Then, we define T2 as the class of elements T ∈ with t (T ) ≤ n contained in elements of T1 and not contained in any other elements of T . Proceeding inductively, this process must stop in a finite number k of steps. We say that the elements in Ti have rank i. Then, we define Ck = i=1 T ∈Ti ωT  0 ˜ ˜ and Z k = ( T ∈Tk T ) \ Ck . We claim that Z n (k) ⊂ Z k . Actually, given x ∈ Z n0 (k), we have T1 , . . . , Tk ∈ T with T1 ⊃ · · · ⊃ Tk and t (T1 ) < · · · < t (Bk ) ≤ n such that x ∈ Tk ∩ n . Clearly, Tk has rank r ≥ k. Take T1 ⊃ · · ·⊃ Tr with Ti ∈ Ti and Tr = Tk . In particular, x ∈ Ti for i = 1, . . . , k, and so x ∈ T ∈Tk T . On the other hand, as x ∈ n and Ck ∩ n = ∅, we get x ∈ / Ck . This gives x ∈ Z˜ k . Now, we show that for each k we have m 0 ( Z˜ k+1 ) ≤

D m 0 ( Z˜ k ), D+1

(5.65)

where D > 0 is the constant given by (5.60). To see this, we start by showing that for all T ∈ Tk+1 we have (5.66) ωT ⊂ Z˜ k \ Z˜ k+1 . Take any T ∈ Tk+1 and let T  ∈ T be an element of rank k containing T . As the cores are pairwise disjoint, we have ωT ∩ Ck = ∅, and so ωT ⊂ T  \ Ck ⊂ Z˜ k . As, by definition, we have ωT ⊂ Ck+1 , it follows that ωT ∩ Z˜ k+1 = ∅. This gives (5.66). Let us now prove (5.65). Using (5.60) and the fact that the cores of the elements in T are pairwise disjoint, we may write m 0 ( Z˜ k+1 ) ≤

T ∈Tk+1

m 0 (T ) ≤ D

T ∈Tk+1

m 0 (ωT ) ≤ Dm 0 ( Z˜ k \ Z˜ k+1 ).

5.3 Inducing Times

185

Therefore, (D + 1)m 0 ( Z˜ k+1 ) ≤ Dm 0 ( Z˜ k+1 ) + Dm 0 ( Z˜ k \ Z˜ k+1 ) = Dm 0 ( Z˜ k ). Inductively, this yields m 0 ( Z˜ k ) ≤



D D+1

k m 0 (0 ).

Since Z k ⊂ Z˜ k , the result follows with λ2 = D/(D + 1).



Lemma 5.17 There is D2 > 0 such that for all k ≥ 0 and all T ∈ T we have m 0 (Z (k, T )) ≤ D2 (D2 σ N4 )k m 0 (ωT ) . Proof We prove the result by induction on k ≥ 0. For k = 0, we have by (5.60) m 0 (Z (0, T )) ≤ m 0 (T ) ≤ Dm 0 (ωT ). So, in this case, it is enough to take D2 ≥ D. Given any k ≥ 1, we may write Z (k, T ) ⊂

k



r =1 T1 ∩T =∅



Z (k − r, T2 ).

T1 ∩T2 =∅,T2 ⊂T1 t (T2 )−t (T1 ) ≥r N4

Our goal now is to estimate the measure of the triple union above. Take T1 ∈ T intersecting T and T2 ∈ T intersecting T1 with T2 ⊂ T1 and t (T2 ) − t (T1 ) ≥ r N4 . Consider t1 = t (T1 ) and t2 = t (T2 ). Notice that for each ti (ωTi )) is, by definition, a disk of radius δ1 /3. Hence, i = 1, 2, the set f ti (Ti ) = f ti (W using (5.19) and (I2 ), we may write diam( f t1 (T2 )) ≤ σ t2 −t1 diam( f t2 (T2 )) ≤

2 2 δ1 σ t2 −t1 ≤ δ1 σ r N4 . 3 3

(5.67)

Now, let β1 be the boundary of the disk f t1 (T1 ) in f t1 ( ωT1 ), and consider N1 a ωT1 ) of size 2δ1 σ r N4 /3. Notice that N1 is necessarily neighbourhood of β1 in f t1 (  contained in f t1 Vt1 (ωT1 ) . Let U1 be the subset of Vt1 (ωT1 ) which is mapped by f ti diffeomorphically to N1 . One can easily see that there exists some constant C > 0 (not depending on t1 ≥ N0 or on T1 ) such that m t1 ( f t1 (U1 )) ≤ Cσ r N4 m t1 ( f t1 (T1 )).

(5.68)

Consider C0 > 0 the constant given by Lemma 5.2. Since T1 ⊂ Vt1 (ωT1 ), using (5.60) and (5.68) we obtain

186

5 Inducing Schemes

m 0 (U1 ) ≤ C0 Cσ r N4 m 0 (T1 ) ≤ C0 C Dσ r N4 m 0 (ωT1 ).

(5.69)

Now, as T2 ∩ T1 = ∅ with T2 ⊂ T1 , it follows that f t1 (T2 ) intersects β1 . It follows from (5.67) that f t1 (T2 ) is contained in N1 = f t1 (U1 ). This gives ωT2 ⊂ U1 . Since the cores of these possible T2 are pairwise disjoint, from (5.69) we get

    m 0 ωT2 ≤ C0 C Dσ r N4 m 0 ωT1 .

(5.70)

T1 ∩T2 =∅,T2 ⊂T1 t (T2 )−t (T1 ) ≥r N4

On the other hand, since T1 ∩ T = ∅, it follows from (5.59) that T1 ⊂ Vt (T ) (ωT ). Moreover, as the cores of these possible T1 are pairwise disjoint, by Lemma 5.2 we have   m 0 (ωT1 ) ≤ m 0 Vt (T ) (ωT ) ≤ C0 m 0 (ωT ) . (5.71) T1 ∩T =∅

Finally, using (5.70), (5.71) and the inductive hypothesis, we may write m 0 (Z (k, T )) ≤

k r =1 T1 ∩T =∅

≤

k r =1 T1 ∩T =∅

≤

k



m 0 (Z (k − r, T2 ))

T ∩T =∅,T2 ⊂T1  1 2 t (T2 )−t (T1 ) ≥r N4



D2 (D2 σ N4 )k−r m 0 (ωT2 )

T ∩T =∅,T2 ⊂T1 1 2  t (T2 )−t (T1 ) ≥r N4

D2 (D2 σ N4 )k−r C0 C Dσ r N4 m 0 (ωT1 )

r =1 T1 ∩T =∅

≤ D2 (D2 σ N4 )k C02 C D

k

D2−r m 0 (ωT ).

r =1

Hence, taking D2 > 0 large enough so that C02 C D D2−1 1 − D2−1

we finish the proof.

≤ 1, 

Finally, we are able to specify the choice of the integer N4 . We take N4 sufficiently large such that (5.72) D2 σ N4 < 1, with D2 > 0 as in Lemma 5.17. It follows from (5.58) and Lemma 5.17 that m 0 (Z 1 (k)) decays exponentially fast with k; recall (5.63).

References

187

References 1. J.F. Alves, C.L. Dias, S. Luzzatto. Geometry of expanding absolutely continuous invariant measures and the liftability problem. Ann. Inst. H. Poincaré Anal. Non Linéaire 30(1), 101–120 (2013) 2. J.F. Alves, C.L. Dias, S. Luzzatto, V. Pinheiro, SRB measures for partially hyperbolic systems whose central direction is weakly expanding. J. Eur. Math. Soc. (JEMS) 19(10), 2911–2946 (2017) 3. J.F. Alves, X. Li, Gibbs-Markov-Young structures with (stretched) exponential tail for partially hyperbolic attractors. Adv. Math. 279, 405–437 (2015) 4. W. de Melo, S. van Strien, One-dimensional dynamics, vol. 25, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] (Springer-Verlag, Berlin, 1993) 5. S. Gouëzel, Decay of correlations for nonuniformly expanding systems. Bull. Soc. Math. France 134(1), 1–31 (2006) 6. G. Keller, Exponents, attractors and Hopf decompositions for interval maps. Ergodic Theory Dynam. Syst. 10(4), 717–744 (1990) 7. V. Pinheiro, Sinai-Ruelle-Bowen measures for weakly expanding maps. Nonlinearity 19(5), 1185–1200 (2006)

Chapter 6

Nonuniformly Expanding Attractors

In this chapter, we obtain some results for a class of nonuniformly expanding maps introduced in [1]. In particular, we obtain SRB measures and rates for the decay of correlations with respect to these SRB measures, presenting results that contribute to an already quite general theory on this type of maps. Since the works of Krzy˙zewski, Szlenk [14] and Ruelle [23] that the statistical properties of (uniformly) expanding maps are well understood; see also [15] for some interesting topological properties of such maps. Some of the classical results will be obtained here from the theory that we develop for a broader class of nonuniformly expanding maps, possibly having sets of critical or singular points. In this chapter, we aggregate techniques from several works published in the last two decades, leading to the construction of SRB measures [1, 3, 4, 17] and the respective decay of correlations [5, 6, 16]. In Sect. 6.1, we introduce the class maps to be considered and define hyperbolic times, a key tool introduced in [1]. In Sect. 6.2, we use these hyperbolic times to obtain a finite number of attractors, following the strategy in [4, 18]. In Sect. 6.3, we construct Gibbs-Markov induced maps on each attractor. In the two subsequent sections, we use these induced maps, together with results from Chap. 3, to obtain an SRB measure on each attractor and estimates on the decay of correlations.

6.1 Nonuniform Expansion and Slow Recurrence Let M be a finite-dimensional compact Riemannian manifold, possibly with a boundary, m be the Lebesgue measure on M and d the distance induced by the Riemannian structure. Throughout this chapter, by an SRB measure, we mean an invariant probability measure which is absolutely continuous with respect to the Lebesgue measure. Consider f : M → M a C 1+η local diffeomorphism out of a set C ⊂ M, possibly the empty set. In practice, C can be a set of points where the derivative of f is not an isomorphism (critical set), a set of points where the derivative simply does not exist © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. F. Alves, Nonuniformly Hyperbolic Attractors, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-62814-7_6

189

190

6 Nonuniformly Expanding Attractors

(singular set), or even the boundary of M. We simply do not care about the image of the points in C and will only be concerned with points whose forward orbit do not hit C. We say that C is a nondegenerate set (with respect to f ) if m(C) = 0 and conditions (C1 )–(C3 ) below hold: (C1 ) If A is Borel set, then f (A) and f −1 (A) are both Borel sets and m(A) = 0 =⇒ m( f −1 (A)) = 0 and m( f (A)) = 0. This gives in particular that f ∗ m  m and f ∗ m  m, where f ∗ denotes the usual push-forward of the measure m and the meaning of f ∗ m  m is introduced in (2.14). Condition (C1 ) also gives that m( f j (C)) = 0, for all j ∈ Z. We also assume that there exist K , α > 0 such that (C2 ) D f (x) ≤ K d(x, C)−α , for every x ∈ M \ C; (C3 ) for every x, y ∈ M \ C with d(y, x) < d(x, C)/2, we have   • log D f (x)−1  − log D f (y)−1  ≤

K d(x, y)η ; d(x, C)α K d(x, y)η . • |log | det D f (x)| − log | det D f (y)|| ≤ d(x, C)α When f : M → M is a C 1+η local diffeomorphism, we take C = ∅ and, by definition, assume that d(x, ∅) = 1, for all x ∈ M. In such case, (C1 )–(C3 ) are trivially verified. Note that condition (C2 ) has nontrivial meaning only when the norm of the derivative of f is unbounded near C. The two subitems in (C3 ) mean that the functions log (D f )−1  and log | det D f | are locally Hölder in M \ C, with the Hölder constant possibly getting worse as we approach C. Observe that (C2 ) and (C3 ) become less restrictive when α increases or η decreases. A local diffeomorphism f : M → M of a differentiable manifold M is called expanding if, for some choice of a Riemannian metric on M, there exists λ > 0 such that for every x ∈ M (6.1) log D f (x)−1  < −λ. Note that this is equivalent to say that there is σ < 1 such that D f (x)−1  < σ , for every x ∈ M. Note that, in dimension greater than one, D f (x)−1  < 1 is not equivalent to D f (x) > 1. Indeed, condition D f (x)−1  < 1 ensures that D f (x) expands in every direction, whereas D f (x) > 1 means that D f (x) has some direction of expansion. Here, we consider maps for which (6.1) is weakened in two significant ways: i) reducing the set of points for which it holds, and ii) imposing it only in asymptotic time averages. We say that f is nonuniformly expanding on H if there exists λ > 0 such that, for some choice of a Riemannian metric on M and all x ∈ H , n−1 1 −1 log D f ( f j (x))  < −λ. lim inf n→∞ n j=0

(6.2)

6.1 Nonuniform Expansion and Slow Recurrence

191

−1

We implicitly assume that D f ( f j (x)) is defined for all x ∈ H and j ≥ 0. In Sect. 6.5, we consider a stronger version of nonuniform expansion, with the lim inf replaced by lim sup to obtain estimates on the decay of correlations of certain SRB measures. However, for the existence of those measures, this weaker condition with lim inf is enough. Having C = ∅, we need an extra assumption on the way orbits approach this set. We say that f has slow recurrence to C on the set H if, for every ε > 0, there exists r > 0 such that for all x ∈ H lim sup n→+∞

n−1 1 − log dr ( f j (x), C) < ε, n j=0

(6.3)

with dr (x, C) is the truncated distance, defined for x ∈ M \ C as  dr (x, C) =

1, if d(x, C) ≥ r ; d(x, C), otherwise.

Condition (6.3) essentially means that orbits in H do not come too often and too close to the set C. The usefulness of nonuniform expansion and slow recurrence will become clear in Proposition 6.3. In dimension one, condition (6.2) is equivalent to the existence of a positive Lyapunov exponent at x. In general, nonuniform expansion at a point x ∈ M implies the existence of dim(M) positive Lyapunov exponents at x ∈ M, that is lim sup n→∞

1 log D f n (x)v > λ > 0, n

∀v ∈ Tx M.

(6.4)

However, the converse is not necessarily true. Example 6.1 Let f : T 2 → T 2 be a local diffeomorphism of the flat torus T 2 = R2 /Z2 with the usual Riemannian metric. Consider a period two orbit { p, q} such that, with respect to the canonical bases of the tangent spaces, we have  D f ( p) =

1/2 0 0 3



 and D f (q) =

 3 0 . 0 1/2

It is straightforward to verify that the Lyapunov exponents at the points p or q are both log(3/2)/2 > 0. However, the limit in (6.2) with x = p or x = q equals log 2 > 0. Note that, unlike (6.2), condition (6.4) does not depend on the Riemannian metric on the manifold M. It remains an interesting open question to know whether (6.4), instead of the nonuniform expansion condition (6.2), is sufficient for the results on the existence of SRB measures that we obtain in Sect. 6.4. This is actually a particular case of Viana Conjecture [25]. The main problem in trying to implement the strategy

192

6 Nonuniformly Expanding Attractors

below under condition (6.4) lies precisely in obtaining hyperbolic times. Though situations as in Example 6.1 occur, it is not difficult to see that (6.2) holds with respect to f 2 for the period two orbit in Example 6.1. We ignore whether condition (6.4) on a set with positive Lebesgue measure implies (6.2) for some power of f on a set with positive Lebesgue measure.

6.1.1 Hyperbolic Times and Preballs Let f : M → M be a C 1+η local diffeomorphism out of a nondegenerate set C ⊂ M. We take once and for all constants K , α > 0 as in the definition of a nondegenerate set, and b > 0 such that   1 (6.5) b < min 1, α −1 η . 2 Given σ ∈ (0, 1) and r > 0, we say that n is a (σ, r )-hyperbolic time for x ∈ M if n−1

D f ( f j (x))−1  ≤ σ n−k and dr ( f k (x), C) ≥ σ b(n−k) ,

j=k

for all 0 ≤ k < n. Observe that the second condition above is trivially verified for every r > 0 when C = ∅. The next result is due to Pliss [19] and plays a crucial role in the proof of the existence of (infinitely many) hyperbolic times for points whose orbit is nonuniformly expanding and has slow recurrence to C.

Lemma 6.2 Let 0 < c ≤ A and a1 , . . . , an ≤ A be such that nj=1 a j ≥ cn. Then,

i a j ≥ 0 for all there exist 1 ≤ n 1 < · · · < n  ≤ n with  ≥ cn/A such that nj=k 1 ≤ k ≤ n i and 1 ≤ i ≤ .

Proof Set S0 = 0 and Sk = kj=1 a j for every 1 ≤ k ≤ n. Take 1 ≤ n 1 < · · · < n  ≤ n a maximal sequence such that Sni ≥ Sk for all 0 ≤ k ≤ n i and

1i ≤ i ≤ . a j ≥ 0, Observe that as Sn > 0 = S0 , we have necessarily  ≥ 1. Moreover, nj=k for all 1 ≤ i ≤  and 1 ≤ k ≤ n i . We are left to show that  > cn/A. Setting n 0 = 0, by the choice of the maximal sequence we obtain Sni −1 < Sni−1 , for all 1 ≤ i ≤ . Adding ani to both sides and recalling that ani ≤ A, we easily get Sni − Sni−1 ≤ A,

  for all 1 ≤ i ≤ . Observing that Sn  ≥ Sn ≥ cn, we obtain cn ≤ Sn  = i=1 Sni −  Sni−1 ≤ A. In the next result, we deduce the existence of hyperbolic times. In the case of a local diffeomorphism, this can be easily obtained as an application of the previous result.

6.1 Nonuniform Expansion and Slow Recurrence

193

Proposition 6.3 Let f : M → M be a C 1+η local diffeomorphism out of a nondegenerate set C. Given λ > 0, there exist (ε1 , r1 ), (ε2 , r2 ) and θ > 0 such that if, for x ∈ M and i = 1, 2, n−1 n−1 1 1 log D f ( f j (x))−1  < −λ and − log dri ( f j (x), C) < εi , n j=0 n j=0

then x has (e−λ/4 , r2 )-hyperbolic times 1 ≤ n 1 < · · · < n  ≤ n with  ≥ θ n. Proof By condition (C2 ) of a nondegenerate set, we may find K 0 > 0 such that − log D f (x)−1  ≤ −K 0 log d(x, C),

(6.6)

for every x in a sufficiently small neighbourhood V of C. Fix ε1 > 0 such that K 0 ε1 ≤ λ/2, and assume that for some 0 < r1 < 1 we have n−1 1 − log dr1 ( f j (x), C) < ε1 . n j=0

(6.7)

large so that K 1 is also an upper bound Take a constant K 1 ≥ −K 0 log r1 sufficiently   for − log (D f )−1  on M \ V . Set J = 0 ≤ j < n : − log D f ( f j (x))−1  > K 1 . Observe that for j ∈ J we have f j (x) ∈ V . Then, using (6.6) we get for each j ∈ J − K 0 log r1 < − log D f ( f j (x))−1  ≤ −K 0 log d( f j (x), C).

(6.8)

This shows that d( f j (x), C) < r1 , and so dr1 ( f j (x), C) = d( f j (x), C) < r1 for every j ∈ J . From (6.6), (6.7), (6.8) and the choice of ε1 and r1 , we easily deduce that  j∈J



λ − log D f ( f j (x))−1 ≤ K 0 − log dr1 ( f j (x), C) < K 0 ε1 n ≤ n. 2 j∈J

Now, define for each 0 ≤ j < n  bj =

/ J − log D f ( f j (x))−1 , if j ∈ 0, if j ∈ J.

We have b j ≤ K 1 for all 0 ≤ j < n and n−1  j=0

bj =





λ − log D f ( f j (x))−1 − − log D f ( f j (x))−1 > n. 2 j=0 j∈J

n−1 

194

6 Nonuniformly Expanding Attractors

n Setting a j = b j−1 − λ/4 for 1 ≤ j ≤ n, we have j=1 a j > λn/4. Applying K 1 − λ/4, we obtain θ1 > 0 and Lemma 6.2 to a1 , . . . , an with c = λ/4 and A =

pi a j ≥ 0 for every 1 ≤ k ≤ pi 1 ≤ p1 < · · · < p1 ≤ n, with 1 ≥ θ1 n, such that j=k and 1 ≤ i ≤ 1 . This gives pi −1 

−1

− log D f ( f (x))  ≥ j

j=k

pi −1  j=k

 pi   λ λ aj + ≥ ( pi − k) bj = 4 4 j=k+1

(6.9)

for every 0 ≤ k ≤ pi − 1 and 1 ≤ i ≤ 1 . Now, choose ε2 > 0 small enough so that ε2 < θ1 bλ/4, with b > 0 as in (6.5). Assume that for some r2 > 0, we have n−1  1 − log dr2 f j (x), C < ε2 . n j=0

Defining a j = − log dr2 ( f

j−1

(6.10)

(x), C) + bλ/4 for 1 ≤ j ≤ n, we have

n 

 aj ≥

j=1

 bλ − ε2 n . 4

Applying Lemma 6.2 to a1 , . . . , an with c = bλ/4 − ε2 and A = bλ/4, we get 2 ≥ θ2 n and 1 ≤ q1 < · · · < q2 ≤ n such that qi 

 log dr2 f

j−1

j=k+1

bλ (x), C ≥ − (qi − n), 4

(6.11)

for all 0 ≤ k ≤ qi − 1 and 1 ≤ i ≤ 2 . Moreover, θ2 =

4ε2 c =1− . A bλ

Our choice of ε2 gives that θ = θ1 + θ2 − 1 > 0. Hence, there exist  ≥ θ n and 1 ≤ n 1 < · · · < n  ≤ n for which n i −1

− log D f ( f j (x))−1  ≥

j=k

and

n i −1 j=k

log dr2 ( f j (x), C) ≥ −

λ (n i − k) 4

bλ (n i − k), 4

for all 0 ≤ k ≤ n i − 1 and 1 ≤ i ≤ . It easily follows that

6.1 Nonuniform Expansion and Slow Recurrence n i −1

195

λ

λ

D f ( f j (x))−1  ≤ e− 4 (ni −k) and dr2 ( f k (x), C) ≥ e− 4 b(ni −k) ,

j=k

for all 1 ≤ i ≤  and 0 ≤ k ≤ n i − 1.



In the next corollary, we deduce the existence of hyperbolic times for points in a set where the map is nonuniformly expanding and has slow recurrence to a nondegenerate set. Corollary 6.4 Let f : M → M be a C 1+η local diffeomorphism out of a nondegenerate set C. If f is nonuniformly expanding and has slow recurrence to C on a set H , then there exist σ ∈ (0, 1) and r, θ > 0 such for that all x ∈ H we have 1 lim sup # {1 ≤ k ≤ n : k is a (σ, r )-hyperbolic time for x} ≥ θ. n→∞ n Proof Consider λ > 0 such that the nonuniform expansion condition (6.2) holds for all x ∈ H . Take (ε1 , r1 ), (ε2 , r2 ) and θ > 0 as in Proposition 6.3. Since we assume the slow recurrence condition (6.3) with lim sup, it easily follows that for each x ∈ H there are infinitely many values of n such that for i = 1, 2 we have n−1 n−1 1 1 log D f ( f j (x))−1  < −λ and − log dri ( f j (x), C) < εi . n j=0 n j=0

It follows from Proposition 6.3 that for any such n the point x has (σ, r )-hyperbolic times 1 ≤ n 1 < · · · < n  ≤ n, with σ = e−λ/4 and r = r2 .  In Proposition 6.7 below, we give the main features of hyperbolic times. We start with a preliminary result. Lemma 6.5 Let f : M → M be a C 1+η local diffeomorphism out of a nondegenerate set C. Given σ ∈ (0, 1) and r > 0, there exists δ1 > 0 such that, for any x with a (σ, r )-hyperbolic time n, we have D f (y)−1  ≤ σ −1/2 D f (x)−1 , for all y ∈ M with d(y, x) < δ1 σ n/2 . Proof Let n be a (σ, r )-hyperbolic time for x. We have in particular dr (x, C) ≥ σ bn . This means that either d(x, C) ≥ r or d(x, C) = dr (x, C) ≥ σ bn , if d(x, C) < r . Take δ1 < min{r, 1}/2. If d(x, C) ≥ r , then δ1 σ n/2 < δ1
0 in the previous result can be easily obtained simply using the fact that x → D f (x)−1  is uniformly continuous on the compact manifold M. In such case, we can also choose δ1 > 0 uniform in a C 1 neighbourhood of f , as long as 0 < σ < 1 is uniform in that neighbourhood. This happens, for instance, when the value λ > 0 in Proposition 6.3 can be taken uniform. The sets Vn (x) obtained in the next proposition will be called hyperbolic preballs. Proposition 6.7 Let f : M → M be a C 1+η local diffeomorphism out of a nondegenerate set C. Given σ ∈ (0, 1) and r > 0, there exists δ1 > 0 such that for all x ∈ M with a (σ, r )-hyperbolic time n there is a neighbourhood Vn (x) of x such that 1. f n maps Vn (x) diffeomorphically to Bδ1 ( f n (x)); 2. for all y, z ∈ Vn (x), a. d( f n−k (y), f n−k (z)) ≤ σ k/2 d( f n (y), f n (z)), for all 1 ≤ k ≤ n; | det D f n (y)| ≤ Cd ( f n (y), f n (z))η . b. log | det D f n (z)| Proof We use δ1 > 0 given by Lemma 6.5 and prove inductively that if n ≥ 1 is a (σ, r )-hyperbolic time for x ∈ M, then f n maps a neighbourhood Vn (x) of x diffeomorphically to Bδ1 ( f n (x)); moreover, for all y ∈ Vn (x), we have D f n (y)−1  ≤ σ −n/2 .

(6.12)

Let n = 1 be a (σ, r )-hyperbolic time for x ∈ M. By Lemma 6.5, we have for y ∈ Bδ1 σ 1/2 (x) (6.13) D f (y)−1  ≤ σ −1/2 D f (x)−1  ≤ σ 1/2 . This means that D f expands all directions by a factor σ −1/2 on the ball Bδ1 σ 1/2 (x). Then, there must be some neighbourhood V1 (x) of x contained in Bδ1 σ 1/2 (x) which is mapped diffeomorphically onto the ball Bδ1 ( f (x)). Moreover, (6.13) gives (6.12) for n = 1 and y ∈ V1 (x).

6.1 Nonuniform Expansion and Slow Recurrence

197

Assume now that n > 1 is a (σ, r )-hyperbolic time for x ∈ M. Since n − 1 is a (σ, r )-hyperbolic time for f (x), we have by induction hypothesis that there exists a neighbourhood Vn−1 ( f (x)) of f (x) which is mapped diffeomorphically by f n−1 onto Bδ1 ( f n (x)); moreover, D f n−1 (z)−1  ≤ σ −(n−1)/2

(6.14)

for all z ∈ Vn−1 ( f (x)). We define an inverse branch of f n on the ball Bδ1 ( f n (x)) by lifting geodesics in the following way: given a point in the boundary of Bδ1 ( f n (x)) and a geodesic γ connecting f n (x) to that point, there is a lift γn of γ restricted to a small neighbourhood of f n (x) with initial point at x. Moreover, as long as γn does not leave the ball of radius δ1 σ n/2 around x, it follows from Lemma 6.5 that D f (y)−1  ≤ σ −1/2 D f (x)−1 ,

(6.15)

for any y in the curve γn . Observing that the curve f (γn ) is necessarily contained in Vn−1 ( f (x)), using (6.14) and (6.15) we get D f n (y)−1  ≤ σ −n/2

(6.16)

for any y in the curve γn . This means that the derivative of f n on γn is a σ −n/2 expansion, and this gives that the length of γn is less than δ1 σ n/2 . We conclude that the lift of γ is well defined on the whole geodesic γ , and we have a well-defined branch of f n on the ball of radius δ1 around f n (x). We call Vn (x) the image of that inverse branch. From (6.16) we obtain (6.12).  Now observe that, by construction, we have f k (Vn (x)) = Vn−k f k (x) for all 1 ≤ k ≤ n. On the other hand, (6.12) gives in particular that f n−k a σ −(n−k)/2 expansion on Vn−k ( f k (x)). This implies that  d( f n−k (y), f n−k (z)) ≤ σ k/2 d f n (y), f n (z) ,

(6.17)

for all y, z ∈ Vn (x) and all 1 ≤ k ≤ n. Finally, we prove the bounded distortion property. We assume without loss of generality that δ1 < 1/8. Since n is a (σ, r )-hyperbolic time for x and b < 1/2 by (6.5), we have for all 1 ≤ k ≤ n d( f k (y), C) ≥ d( f k (x), C) − d( f k (x), f k (y)) ≥ σ b(n−k) − 2δ1 σ (n−k)/2 1 ≥ σ b(n−k) 2 ≥ 4δ1 σ (n−k)/2 ≥ 2d( f k (y), f k (z)).

(6.18)

(6.19)

198

6 Nonuniformly Expanding Attractors

For the last inequality, recall that d( f n (y), f n (z)) ≤ δ1 for all y, z ∈ Vn (x). Since (6.19) holds, using that C is a nondegenerate set, we get log

K | det D f ( f k (y))| ≤ d( f k (y), f k (z))η . k k | det D f ( f (z))| d( f (y), C)α

Altogether with (6.17) and (6.18), this yields log

n−1  | det D f n (y)| | det D f ( f k (y))| = log n | det D f (z)| | det D f ( f k (z))| k=0

≤

n−1  k=0

2α K

σ η(n−k)/2 d( f n (y), f n (z))η . σ bα(n−k)

Recalling that bα < η/2 by (6.5), we are done.



Corollary 6.8 Let f : M → M be a C 1+η local diffeomorphism out of a nondegenerate set C. If f is nonuniformly expanding and has slow recurrence to C on H ⊂ M, then there exists δ > 0 such that for every set A ⊂ H with f (A) ⊂ A and m(A) > 0 there is a ball B of radius δ > 0 such that m(B \ A) = 0. Proof By Corollary 6.4, there exist σ, r > 0 such that every point in H has infinitely many (σ, r )-hyperbolic times. Take any ball 0 ⊂ M such that m(A ∩ 0 ) > 0. For each n ≥ 1, let Hn = {x ∈ H ∩ 0 : n is a (σ, r ) -hyperbolic time for x}. Clearly, every point in A ∩ 0 belongs in infinitely many Hn ’s. Proposition 6.7 shows that (I2 ) holds for the sequence of sets (Hn )n in 0 . Moreover, Proposition 5.3 provides δ = δ1 /9 > 0 (independent of A) such that, for every k ≥ 1, we have a ball Bk of radius δ on which the relative measure of A is bigger than 1 − 1/k. Take x an accumulation point of the centres of these balls Bk , and B the ball of radius δ around x. 

6.2 Attractors This section is entirely devoted to the proof of Theorem 6.9 below, where we obtain a finite number of compact sets attracting the orbits of most points in a set with positive Lebesgue measure on which a map is nonuniformly expanding and has slow recurrence to a nondegenerate set. We start by introducing some useful notions, which essentially correspond to an adaptation of standard notions to our context of maps possibly with discontinuity points. Let f : M → M be a C 1+η local diffeomorphism out of a nondegenerate set C ⊂ M. The ω-limit of a point x ∈ M, denoted ω(x), is

6.2 Attractors

199

the set of accumulation points of the orbit of x. Note that ω(x) is a nonempty compact set, since M is compact. If the map f is not continuous, there is no reason to expect f (ω(x)) ⊂ ω(x). However, it is still true that f (ω(x) \ C) ⊂ ω(x).

(6.20)

We say that ⊂ M is an elementary set if • is compact set for which f ( \ C) ⊂ ; • there is some point in whose forward orbit is dense in and does not hit C. Since C has nonempty interior (it actually has zero Lebesgue measure), it is easily verified that if is an elementary set and f | is continuous, then f ( ) = . We can therefore think of f | as map defined on and taking values in . Under these circumstances, we say that f if transitive, topologically mixing or locally eventually onto on the set , whenever these properties are valid for f | : → . In the next result, we see that if f is nonuniformly expanding and has slow recurrence to C on a set H ⊂ M with positive Lebesgue measure, then there is a finite number of elementary sets attracting almost all points in H . Theorem 6.9 Let f : M → M be a C 1+η local diffeomorphism out of a nondegenerate set C. If f is nonuniformly expanding and has slow recurrence to C on a set H ⊂ M with m(H ) > 0, then there are elementary sets 1 , . . . ,  ⊂ M such that, for m almost every x ∈ H , there is 1 ≤ j ≤  such that ω(x) = j . Moreover, for each 1 ≤ j ≤ , there is a ball  j ⊂ j such that f is nonuniformly expanding and has slow recurrence to C on m almost all of  j . The remainder of this section is devoted to the proof of Theorem 6.9. We assume that a C 1+η local diffeomorphism f : M → M out of a nondegenerate set C is nonuniformly expanding and has slow recurrence to C on H ⊂ M with m(H ) > 0.

6.2.1 Ergodic Components We say that ⊂ M is an invariant set for f if f −1 ( ) = . A Borel set E ⊂ M is called an ergodic component of f if E is an invariant set such that, for any invariant Borel set E  ⊂ E, we have m(E  ) = 0 or m(E  ) = m(E). Lemma 6.10 For every ergodic component E of f , there is a compact set ⊂ M such that ω(x) = , for m almost every x ∈ E. Proof Let E ⊂ M be an ergodic component of f . Given an open set A ⊂ M, set A∗ = {x ∈ E : ω(x) ∩ A = ∅}. Since A∗ ⊂ E is an invariant set and E is an ergodic component, we necessarily have m(A∗ ) = 0 or m(A∗ ) = m(E). Now, consider X 1 = M and let A1 be a finite cover

200

6 Nonuniformly Expanding Attractors

of X 1 by open balls of radius 1. By the previous considerations, for every A ∈ A1 we have m(A∗ ) = 0 or m(A∗ ) = m(E). Since A1 is a finite cover of X 1 , there exists at least one A ∈ A1 such that m(A∗ ) = m(E). Let    A1 = A ∈ A1 : m(A∗ ) = 0 and X 2 = X 1 \

A∈A1

A.

Then X 2 is a nonempty compact set and ω(x) ⊂ X 2 for m almost every x ∈ E. We can therefore repeat the procedure above with a finite cover A2 of X 2 by open balls of radius 1/2. Inductively, we construct sequences (An )n , (An )n and nonempty compact sets M = X 1 ⊃ X 2 ⊃ · · · and ω(x) ⊂ X j , for m almost every x ∈ E. Set

=



Xn.

n≥1

We have ω(x) ⊂ , for m almost every x ∈ E. We claim that ⊂ ω(x), for m almost every x ∈ E. In fact, given y ∈ , we have y ∈ X n for every n ≥ 1, and so there is someAn ∈ An \ An such that y ∈ An . Since diam(An ) → 0 as n → ∞, it follows that n An = {y}. Moreover, as An ∈ An \ An , we have m(An ) = m(E), for every n. Therefore, ω(x) ∩ An = ∅, for m almost all x ∈ E and all n. This implies that y ∈ ω(x), for m almost all x ∈ E. It follows that ⊂ ω(x), for m almost every x ∈ E. Since each ω(x) is a forward invariant compact set, we are done. 

6.2.2 Unshrinkable Sets We say that a Borel set U ⊂ M with m(U ) > 0 is unshrinkable if it is an invariant set for which there exists δ > 0 such that every invariant set S ⊂ U with m(S) > 0 necessarily satisfies m(S) > δ. Lemma 6.11 If U ⊂ M is unshrinkable, then U is (m mod 0) contained in the union of a finite number of ergodic components. Proof Take U1 = U and define F1 = {S ⊂ U1 : m(S) > 0 and f −1 (S) = S }. Note that F1 is nonempty because U1 ∈ F1 . Moreover, S, S  ∈ F1 and m(S \ S  ) > 0

⇒

S \ S  ∈ F1 .

(6.21)

Now consider the partial order  on F1 defined by strict inclusion in measure terms, meaning that S  S  if S ⊃ S  and m(S \ S  ) > 0. We claim that for this partial order, every totally ordered subset of F1 is finite, and in particular it has a lower bound. Indeed, arguing by contradiction, suppose that there is an infinite sequence S1  S2  · · · in F1 . This means that we have S1 ⊃ S2 ⊃ · · · and m(Sk \ Sk+1 ) > 0, for all k ≥ 1. Then  m(Sk \ Sk+1 ) = m(S1 ) < ∞, k≥1

6.2 Attractors

201

and therefore m(Sk \ Sk+1 ) → 0, as k → ∞. Since Sk \ Sk+1 ∈ F1 by (6.21), this contradicts our assumption that U1 = U is unshrinkable. This shows that every totally ordered subset of F1 has a lower bound. By Zorn Lemma, there exists at least one minimal element S1 ∈ F1 , which therefore must be an ergodic component. Now let U2 = U1 \ S1 , which is again invariant. If m(U2 ) = 0, then U = U1 is m mod 0 contained in S1 . Since S1 is an ergodic component, we are done. If, on the other hand, m(U2 ) > 0 we repeat the argument above to obtain an ergodic component S2 ⊂ U2 . Inductively, we construct a collection of disjoint ergodic components S1 , . . . , S and continue as long as m(U \ S1 ∪ · · · ∪ S ) > 0. However, as U is unshrinkable there is δ > 0 such that m(Si ) ≥ δ for all 1 ≤ j ≤ , and so this process must stop after a finite number of steps.  Corollary 6.12 If U ⊂ M is unshrinkable, then there is a finite number of compact sets 1 , . . . ,  ⊂ M such that, for m almost every x ∈ U , there is 1 ≤ j ≤  for which ω(x) = j . Proof A consequence of Lemma 6.10 and Lemma 6.11.



Let us now finish the proof of Theorem 6.9. Assume that f is nonuniformly expanding and has slow recurrence to C on a set H ⊂ M with m(H ) > 0. Since the nonuniform expansion and slow recurrence to C are defined by time averages, discarding, if necessary, the mod 0 set of points in H whose preimages intersect C, we may assume, without loss of generality, that f −1 (H ) = H . It follows from Corollary 6.8 that H is unshrinkable. By Corollary 6.12, there are compact sets 1 , . . . ,  ⊂ M such that for m almost every x ∈ H there exists some 1 ≤ j ≤  such that ω(x) = j . Before we see that each j above is an elementary set, we prove the second conclusion of Theorem 6.9. Take δ > 0 as in Corollary 6.8. We are going to prove that each j above contains a ball  j of radius δ > 0 such that f is nonuniformly expanding and has slow recurrence to C for m almost every point in  j . For 1 ≤ j ≤  and n ≥ 1, define   A j,n = x ∈ H : d( f k (x), j ) ≤ 1/n, for all k ≥ 0 . Since each A j,n is a subset of H with f (A j,n ) ⊂ A j,n and m(A j,n ) > 0, it follows from Corollary 6.8 that there is a ball B j,n of radius δ > 0 such that m(B j,n \ A j,n ) = 0. Let xn be the centre of the ball B j,n , for each n ≥ 1. Consider x an accumulation point of the sequence (xn )n , and let  j be the ball of radius δ around x. It is not difficult to see that f is nonuniformly expanding and has slow recurrence to C for m almost every x ∈  j . Since j is a compact set, we have  j ⊂ j . Since (6.20) holds, it remains to check that, for all 1 ≤ j ≤ , there is some point in j whose forward orbit is dense in j and does not hit the set C. Recall that, by construction, there is a positive m measure set of points in H whose ω-limit coincides with j . Moreover, the orbits of m almost all of these points do not hit C, since we assume C a nondegenerate set. Recalling that j contains some open ball, the orbit of any such point necessarily hits j and is dense in j .

202

6 Nonuniformly Expanding Attractors

6.3 Gibbs-Markov Induced Maps In this section, we show that for each elementary set given by Theorem 6.9 there is an induced Gibbs-Markov map defined on some ball 0 ⊂ . Despite its intrinsic interest, the next result will be particularly useful in Sect. 6.4 to obtain SRB measures supported on the attractors given by Theorem 6.9. Theorem 6.13 Let f : M → M be a C 1+η local diffeomorphism out of a nondegenerate set C and ⊂ M an elementary set. If f is nonuniformly expanding and has slow recurrence to C for m almost every point in a ball  ⊂ , then f has a Gibbs-Markov expanding open induced map defined on some ball 0 ⊂  with integrable recurrence times. This theorem will be proved in the remainder of this section. Let ⊂ M be an elementary set and  ⊂ a ball such that f is nonuniformly expanding and has slow recurrence to C for m almost every point in . The idea is to show that there exists a ball 0 ⊂  on which the conditions (I1 )–(I3 ) in Sect. 5.1 hold, and then apply Theorem 5.1. Consider H ⊂  with m(H ) = m() such that f is nonuniformly expanding and has slow recurrence to C on H . By Corollary 6.4, there exist σ, r > 0 such that every point in H has infinitely many (σ, r )-hyperbolic times. Define for each n ≥ 1 (6.22) Hn = {x ∈ H : n is a (σ, r ) -hyperbolic time for x} . It follows easily from the definition of hyperbolic time that for all 0 ≤ j < n x ∈ Hn =⇒ f j (x) ∈ Hn− j .

(6.23)

In addition, by Corollary 6.4, there exists θ > 0 such that for all x ∈ H  1  lim sup # 1 ≤ j ≤ n : x ∈ H j ≥ θ. n n→∞

(6.24)

This last condition plays a fundamental role in the proof of the integrability of the recurrence times. It gives in particular that m almost every point in  belongs in infinitely many Hn ’s, thus ensuring the validity of (I1 ) in any ball 0 ⊂ . We will also obtain (I2 ) in any ball 0 ⊂ , taking the hyperbolic preballs Vn (x) and the constant δ1 > 0 as in Proposition 6.7. Condition (I3 ) will be obtained as a consequence of the next lemma. Lemma 6.14 There exist C > 0, L ∈ N and p ∈  such that for δ0 > 0 small enough and any ball B ⊂ of radius δ1 /9 there exist U ⊂ B and an integer 0 ≤  ≤ L such that f  maps U diffeomorphically to B2δ0 ( p) ⊂ . Moreover, for all x, y ∈ U we have

6.3 Gibbs-Markov Induced Maps

203

1 d( f j (x), f j (y)) ≤ d( f  (x), f  (y)) ≤ Cd(x, y), for all 0 ≤ j ≤ ; C   det D f  (x)  ≤ Cd( f  (x), f  (y))η . 2. log  det D f  (y)

1.

Proof Since  is a ball contained in the elementary set , there exists some point q ∈  whose orbit is dense in and does not hit C. Thus, we may take L ∈ N such that p = f L (q) belongs in the interior of , the set {q, f (q), . . . , f L (q)} is disjoint from C and δ1 /27-dense in , meaning that any point in is at a distance of at most δ1 /27 from the set {q, f (q), . . . , f L (q)}. Therefore, choosing δ0 > 0 sufficiently small, we have B2δ0 ( p) ⊂  and the connected components of the preimages of B2δ0 ( p) up to time L disjoint from the critical set C and contained in a ball of radius δ1 /27. This implies that any ball B ⊂ M of radius δ1 /9 contains a preimage U of B2δ0 ( p) which is mapped diffeomorphically onto B2δ0 ( p) in at most L iterates. Since we have a finite number of iterations, we may assure those at most L iterates of U at a strictly positive distance from the set C. Using that f is C 1+η in subsets at a positive distance from C, the two inequalities in the first item follow.  Now take p ∈ , L ∈ N and δ0 > 0 small so that the conclusions of Lemma 6.14 n (x) ⊂ Vn (x) hold. Define 0 = Bδ0 ( p) ⊂ . Given x ∈ Hn , consider Wn (x) ⊂ W n  as in Sect. 5.1, with f mapping Wn (x) and Wn (x) diffeomorphically to disks centred at f n (x), of radius δ1 /9 and δ1 /3, respectively. By Lemma 6.14, for each Wn (x), there exist 0 ≤  ≤ L and U ⊂ f n (Wn (x)) such that f  maps U diffeomorphically ωn, to B2δ0 ( p). Let U  be the subset of U that f  maps to Bδ0 ( p). Finally, define  the subset of Wn (x) that f n maps diffeomorphically to U and ωn, the subset of  ωn, that f n maps to U  . By Lemma 6.14, the items of condition (I3 ) are clearly verified. At this point, we are in a situation where (I1 )–(I3 ) in Sect. 5.1 hold. Now, let P be m mod 0 partition into domains ωn, of 0 and R : 0 → N be the function given by Theorem 5.1. Notice that, by construction, we have f R (ωn, ) = 0 , for each ωn, ∈ P. Thus, we have an induced map f R : 0 → 0 . It follows from Theorem 5.1, Lemma 3.2 and Lemma 3.3 that f R is an Gibbs-Markov expanding induced map. Since f j |ω is a diffeomorphism onto its image, for all 0 ≤ j ≤ R(ω), then f R is an open induced map as well. We are left to prove that R ∈ L 1 (m).

For this, we will use Proposition 5.12. Taking (Sn )n as in Theorem 5.1, we have n≥1 m(Sn ) < ∞ and Hn ∩ {R > L + n} ⊂ Sn , for some L ∈ N. In this case, we take Hn∗ = Hn , for every n ≥ 1. Condition (6.24) gives precisely that (Hn )n is a frequent sequence of sets. It remains to check that (Hn )n is f R -concatenated in (Hn )n . Consider R0 = 0 and Rk =

k−1 

R ◦ ( f R ) j , for each k ≥ 1.

j=0

Given i ≥ 0 and Ri ≤ n < Ri+1 , we have ( f R )i = f Ri . It easily follows from (6.23) that if x ∈ Hn∗ , then F i (x) = f Ri ∈ Hn−Ri . Hence, (Hn )n is f R -concatenated in (Hn )n . It follows from Proposition 5.12 that R is integrable with respect to m.

204

6 Nonuniformly Expanding Attractors

Remark 6.15 In the proof of Theorem 6.13 we have not used the last item of Theorem 5.1, since we have no previous information on the sequence (Hn )n used to introduce a function h θ as in (5.2). However, if h θ can be defined for some 0 < θ ≤ 1, we may use the last item of Theorem 5.1 to get {R > n + L} ⊂ {h θ > n} ∪ E n , for a sequence (E n )n of sets in 0 such that m(E n ) decays exponentially fast with n.

6.4 SRB Measures In this section, we prove some results on the existence and uniqueness of ergodic SRB measures for local diffeomorphisms out of a nondegenerate set C. We start by an additional property for the measure in Corollary 3.21, under some topological assumptions. Lemma 6.16 Let f : M → M be a C 1+η local diffeomorphism out of a nondegenerate set C and ⊂ M an elementary set. If f R : 0 → 0 is a Gibbs-Markov map R defined on an open

∞set j 0 ⊂ M and ν0 is the unique SRB measure for f , then the support of ν = j=0 f ∗ (ν0 |{R > j}) coincides with . Proof First we show that the support of ν is contained in . Assume by contradiction that U is an open set in M such that U ∩ = ∅ and ν(U ) > 0. Taking into the account the expression of ν, there must be some j ≥ 0 for which ν0 ( f − j (U ) ∩ 0 ) > 0. Since is an elementary set, we have f ( \ C) ⊂ . Recalling that 0 ⊂ and U ∩ = ∅, it follows that f − j (U ) ⊂ C. But this implies m( f − j (U ) ∩ C) = 0, since we assume C a nondegenerate set. By the absolute continuity of ν0 , it must be ν0 ( f − j (U ) ∩ C) = 0, thus a contradiction. Now we show that the support of ν contains . Let U be an arbitrary open set in M such that U ∩ = ∅. Since is an elementary set, there is some point x ∈ whose forward orbit is dense in and disjoint from C. With no loss of generality, we may assume that x ∈ 0 . Choose n ≥ 0 such that f n (x) ∈ U . Since 0 is an open set in M, by the continuity of f out of C, there is some open set V ⊂ 0 such that f n (V ) ⊂ U . By Corollary 3.21, the measure ν is f -invariant and equivalent to m on 0 . It follows that ν(V ) > 0, for V is an open set contained in 0 . By the invariance of ν, we have ν(U ) = ν( f −n (U )) ≥ ν(V ) > 0. This shows that the support of ν contains .  Using the previous lemma, we can show that each elementary set given by Theorem 6.9 is the support of a unique ergodic SRB measure. Theorem 6.17 Let f : M → M be a C 1+η local diffeomorphism out of a nondegenerate set C and ⊂ M an elementary set such that f is nonuniformly expanding and has slow recurrence to C on a set H ⊂ with m(H ) > 0. Then, f has a unique ergodic SRB measure μ whose support coincides with . Moreover, the basin of μ contains m almost all points in some ball contained in .

6.4 SRB Measures

205

Proof Applying Theorem 6.9 to H , we get elementary sets 1 , . . . ,  ⊂ M such that, for m almost every x ∈ H , there is some 1 ≤ j ≤  for which ω(x) = j . Since is an elementary set and H ⊂ , then the sets 1 , . . . ,  must all be contained in . Choose any 1 ≤ i ≤ . Theorem 6.9 also gives that f is nonuniformly expanding and has slow recurrence to C for m almost every point in some ball i ⊂

i . By Theorem 6.13, there is a Gibbs-Markov map f R : 0 → 0 with integrable recurrence times defined on an open ball 0 ⊂ i . Let ν0 be the unique ergodic SRB that ν0 is equivalent measure for f R given by Theorem 3.13. This theorem

also gives j f (ν |{R > j}) is ergodic, to m on 0 . By Corollary 3.19, the measure ν = ∞ ∗ 0 j=0 f -invariant and absolute continuous with respect to m. Moreover, ν is finite, since we assume the induced map with integrable recurrence times. Therefore, μ = ν/ν(M) is an SRB measure for f . We are going to see that μ satisfies the desired conclusions. By Lemma 6.16, the support of μ coincides with . Assume that μ is another ergodic SRB measure for f whose support coincides with . Since 0 contains an open set in , then necessarily μ (0 ) > 0. It follows from Corollary 3.21 that μ = μ. Finally, Proposition 2.12 gives that μ almost all of is contained in the basin of μ. Therefore, m almost all of the ball 0 ⊂ is contained in the basin of μ, since μ is  equivalent to m on 0 . Using Theorem 6.17, we obtain the following measure theoretical version of the topological decomposition given by Theorem 6.9, with the elementary sets replaced by SRB measures. Corollary 6.18 Let f : M → M be a C 1+η local diffeomorphism out of a nondegenerate set C such that f is nonuniformly expanding and has slow recurrence to C on a set H ⊂ M with m(H ) > 0. Then, f has ergodic SRB measures μ1 , . . . , μ such that, for m almost every x ∈ H , there is 1 ≤ j ≤  for which x belongs in the basin of μi and ω(x) coincides with the support of μi . Moreover, the basin of each μ j contains m almost all points in some ball contained in the support of μ j . Proof By Theorem 6.9, there are elementary sets 1 , . . . ,  ⊂ M such that, for m almost every x ∈ H , there is 1 ≤ j ≤  such that ω(x) = j . Set H j = {x ∈ H : ω(x) = j }, for each 1 ≤ j ≤ . It follows from Theorem 6.17 that, for each 1 ≤ j ≤ , there is an ergodic SRB measure μ j whose support coincides with j . In addition, each j contains a ball B j such that m almost all of B j belongs in the basin of μ j . Clearly, Hj =

∞  n=0

f −n (B j ) and H =

 

Hj.

j=1

with the last equality being valid m mod 0. Since f ∗ m  m, by the nondegeneracy of C, then m almost every point in H j belongs in the basin of μ j . 

206

6 Nonuniformly Expanding Attractors

The proof of Corollary 6.18 gives as many ergodic SRB measures as the elementary sets provided by Theorem 6.9. The next result shows that requiring nonuniform expansion and slow recurrence to C Lebesgue almost everywhere in in Theorem 6.17, we obtain a stronger conclusion. Corollary 6.19 Let f : M → M be a C 1+η local diffeomorphism out of a nondegenerate set C and ⊂ M an elementary set with m( ) > 0 such that f is nonuniformly expanding and has slow recurrence to C on m almost all of . Then f has a unique ergodic SRB measure μ with support contained in . Moreover, the support of μ coincides with and its basin covers m almost all of . Proof Take H ⊂ with m(H ) = m( ) such that f is nonuniformly expanding and has slow recurrence to C on H . By Corollary 6.18, there are ergodic SRB measures μ1 , . . . , μ such that, for m almost every x ∈ H , there is 1 ≤ j ≤  for which x belongs in the basin of μi and ω(x) coincides with the support of μi . Moreover, the basin of each μ j contains m almost all points in some ball B j contained in the support of μ j . Now, we show that all these measures coincide. Take any 1 ≤ i, j ≤ . Since has a dense orbit not hitting C and f ∗ m  m by the nondegeneracy of C, then there is some n ≥ 1 such that f n (Bi ) intersects B j in a subset with positive m measure. Since the forward iterates of points in the basin of μi still belong in the basin of μi , then the basins of μi and μ j have some point in common, and therefore μi = μ j . This shows that μ1 , . . . , μ all coincide. By Theorem 6.17, the support of the unique ergodic SRB measure coincides with , and by Corollary 6.18, its basin covers m almost all of ; recall that in this case H has full m measure in . The fact that the basin of this measure covers m almost all of makes it impossible to have another ergodic SRB measure with support contained in .  The conclusion of the previous result can be strengthened under an extra topological assumption. Corollary 6.20 Let f : M → M be a C 1+η local diffeomorphism out of a nondegenerate set C and ⊂ M an elementary set with m( ) > 0 such that f is nonuniformly expanding and has slow recurrence to C on m almost all of . If f | is continuous and locally eventually onto on , then f has a unique SRB measure μ with support contained in . Moreover, μ is ergodic, the support of μ coincides with and its basin covers m almost all of . Proof By Corollary 6.19, it is enough to show that if μ is an SRB measure for f with support contained in , then μ is ergodic. Consider A ⊂ with f −1 (A) = A and μ(A) > 0. We need to show that μ( \ A) = 0. Since f is nonuniformly expanding and has slow recurrence to C on m almost all of , Corollary 6.8 provides some open set U such that m(U \ A) = 0. Since f | is locally eventually onto on , there exists n ≥ 1 such that f n (U ∩ ) = . We may write

\ A ⊂ \ f n (A) = f n (U ∩ ) \ f n (A) ⊂ f n (U \ A).

6.4 SRB Measures

207

Using the fact that m(U \ A) = 0 together with the nondegeneracy of C, we deduce that m( f n (U \ A)) = 0, and therefore m( \ A) = 0. It follows that μ( \ A) = 0, by the absolute continuity of μ.  From the previous result, we can easily derive the classical result on the existence and uniqueness of SRB measures for expanding maps of connected manifolds. Obviously, connectedness is necessary for the uniqueness. Lemma 6.21 If f : M → M is a C 1 expanding map of the connected manifold M, then f is locally eventually onto on M. Proof It is enough to show that, for any open ball Bε ( p) of radius ε > 0 around a point p ∈ M, we have f n (Bε ( p)) = M. Assume, by contradiction, that f n (Bε ( p)) = M for all n ≥ 1. Thus, for each n ≥ 1 we may find a point yn ∈ M \ f n (Bε ( p)). Let γn be a smooth curve joining f n ( p) to yn . Assume without loss of generality that length(γn ) ≤ diam(M). Since f is a local diffeomorphism, there is a unique curve γˆn joining p to some point x ∈ M \ Bε ( p) such that f n (γˆn ) = γn . Considering σ < 1 such that D f (x)−1  < σ for every x ∈ M, for each n ≥ 1 we have  length(γn ) = =



γn (t)dt

n

D f (γˆn (t))γˆ  (t) dt

≥ σ −n



n

γˆn (t)dt

= σ −n length(γˆn ). Since length(γn ) ≤ diam(M) and length(γˆn ) ≥ ε for all n ≥ 1, we arrive to a contradiction.  Corollary 6.22 If f : M → M is a C 1+η expanding map of a connected manifold, then f has a unique SRB measure μ. Moreover, μ is ergodic, the support of μ coincides with M and its basin covers m almost all of M. Proof Use Lemma 6.21 and Corollary 6.20 (with C = ∅). Observe that f is nonuniformly expanding on the whole M.  We point out that for the general conclusions in the last result, we have use in an important way the fact that the derivative of the expanding map is Hölder continuous. In the C 1 case, counterexamples in the circle were actually provided by several authors; see [9, 11, 12, 20–22].

6.5 Decay of Correlations In Theorem 6.9, we have seen that if H is a set with m(H ) > 0 on which f is nonuniformly expanding and has slow recurrence to C, then m almost all points in H are

208

6 Nonuniformly Expanding Attractors

attracted to an elementary set . In addition, there is a ball  ⊂ such that f is nonuniformly expanding and has slow recurrence to C on m almost all of . In Theorem 6.17, we have shown that each of these elementary sets supports a unique ergodic SRB measure. Here, we obtain estimates on the decay of correlations with respect to this SRB measure. The idea is to provide additional information on the tail of recurrence times for induced Gibbs-Markov maps as in Theorem 6.13. For this, we need a nonuniform expansion condition stronger than that in (6.2). Let f : M → M be a local diffeomorphism out of a nondegenerate set C ⊂ M. Given a set of points H ⊂ M whose forward orbit does not hit C, we say that f is strongly nonuniformly expanding on H , if there exists λ > 0 such that, for all x ∈ H , lim sup n→∞

n−1 1 −1 log D f ( f i (x))  < −λ. n i=0

(6.25)

Note that (6.25) implies the nonuniform expansion in (6.2). Fixing λ > 0 as in (6.25), we may define for each x ∈ H 

n−1 1 −1 E(x) = min N ≥ 1 : log D f ( f i (x))  < −λ, ∀n ≥ N n i=0

 .

Suppose, in addition, that f has slow recurrence to C on H . Considering (ε1 , r1 ), (ε2 , r2 ) as in Proposition 6.3, we may also define for x ∈ H and i = 1, 2 ⎧ ⎨

⎫ n−1 ⎬ 1 Ri (x) = min N ≥ 1 : − log dri ( f j (x), C) < εi , ∀n ≥ N . ⎩ ⎭ n j=0 Set for each x ∈ H

h H (x) = max {E(x), R1 (x), R2 (x)} .

(6.26)

Obviously, we have h H (x) = E(x) whenever C = ∅, that is, when f : M → M is a local diffeomorphism. Note that h H depends on choices of (ε1 , r1 ), (ε2 , r2 ) and λ. We will not make this dependency explicit in the notation, for the sake of simplicity. In the proof of Theorem 6.23 below, we show that h H is related to a function h θ as in (5.2), for a value of 0 < θ ≤ 1 given by Proposition 6.3. In fact, the strong nonuniform expansion introduced above can be seen as a useful criterium for introducing such a function h θ . Theorem 6.23 Let f : M → M be a C 1+η local diffeomorphism out of a nondegenerate set C. If ⊂ M is an elementary set and  ⊂ is a ball such that f is strongly nonuniformly expanding and has slow recurrence to C on a set H ⊂  with m(H ) = m(), then there are L > 0 and a Gibbs-Markov expanding open induced map f R defined on an open ball 0 ⊂  such that

6.5 Decay of Correlations

209

{R > n + L} ⊂ {h H > n} ∪ E n , for some sequence of sets (E n )n in 0 with m(E n ) → 0 exponentially fast, as n → ∞. Proof The Gibbs-Markov expanding open map f R : 0 → 0 has already been obtained in Theorem 6.13, under the weaker assumption of nonuniform expansion. Recall that for the definition of h H in (6.26), we have previously fixed λ > 0 and (ε1 , r1 ), (ε2 , r2 ) as in Proposition 6.3. Consider 0 < θ ≤ 1 given by Proposition 6.3. It easily follows from the definition of h H that, for all x ∈ H and n ≥ h H (x), there are  ≥ θ n and (σ, r )-hyperbolic times 1 ≤ n 1 < · · · < n  ≤ n for x, with σ = e−λ/4 and r = r2 . For each n ≥ 1, set Hn = {x ∈ H : n is a (σ, r ) -hyperbolic time for x} . Clearly, n ≥ h H (x) =⇒

 1  # 1 ≤ j ≤ n : x ∈ H j ≥ θ. n

This shows that a function h θ as in (5.2) can be defined and, moreover, h θ ≤ h H . By the third item of Theorem 5.1, we have {R > n + L} ⊂ {h θ > n} ∪ E n , for a sequence (E n )n of sets in 0 such that m(E n ) decays exponentially fast with n; recall Remark 6.15. Since {h θ > n} ⊂ {h H > n}, we are done. Corollary 6.24 Let f : M → M be a C 1+η local diffeomorphism out of a nondegenerate set C. If ⊂ M is an elementary set and  ⊂ is a ball such that f is strongly nonuniformly expanding and has slow recurrence to C on H ⊂  with m(H ) = m(), then there is a Gibbs-Markov expanding open induced map f R defined on an open ball 0 ⊂  such that 1. if m{h H > n} ≤ Cn −a for some C > 0 and a > 1, then there is C  > 0 such that m{R > n} ≤ C  n −a ; 2. if m{h H > n} ≤ Ce−cn for some C, c > 0 and a > 1, then there are C  , c > 0 such that  a m{R > n} ≤ C  e−c n . a

Proof By Theorem 6.23, there exist L > 0 and a Gibbs-Markov expanding open induced map f R defined on a ball 0 ⊂  ⊂ for which {R > n + L} ⊂ {h H > n} ∪ E n ,

(6.27)

210

6 Nonuniformly Expanding Attractors

where (E n )n is a sequence of sets in 0 such that m(E n ) → 0 exponentially fast with n. Since we consider {h H > n} decaying no faster than exponential, it follows  from (6.27) that m{R > n} decays (at least) at the same speed of m{h H > n}. Given ρ > 0, let Hρ be the space of ρ-Hölder continuous functions defined on M and taking values in R introduced in (3.143). Under the assumptions of the next corollaries, the existence of a unique ergodic SRB measure μ whose support coincides with has already been obtained in Theorem 6.17. Actually, under the weaker assumption of nonuniform expansion. Corollary 6.25 Let f : M → M be a C 1+η local diffeomorphism out of a nondegenerate set C and ⊂ M an elementary set containing a ball  such that f is strongly nonuniformly expanding and has slow recurrence to C on a set H ⊂  with full m measure in . If μ is the unique ergodic SRB measure for f whose support coincides with , then there are 1 ≤ p ≤ q and exact SRB measures μ1 , . . . , μ p for f q with f ∗ μ1 = μ2 ,…, f ∗ μ p = μ1 and μ = (μ1 + · · · + μ p )/ p such that, for all 1 ≤ i ≤ p, 1. if m{h H > n} ≤ Cn −a for some C > 0 and a > 1, for all ϕ ∈ Hρ and ψ ∈ L ∞ (m), there is C  > 0 such that Cor μi (ϕ, ψ ◦ f qn ) ≤ C  n −a+1 ; 2. if m{h H > n} ≤ Ce−cn for some C, c > 0 and a > 1, given ρ > 0, there is c > 0 such that, for all ϕ ∈ Hρ and ψ ∈ L ∞ (m), there is C  > 0 such that a

 a

Cor μi (ϕ, ψ ◦ f qn ) ≤ C  e−c n . Proof By Corollary 6.24, there is an expanding Gibbs-Markov map f R defined on an open ball 0 ⊂  with m{R > n} controlled in terms of m{h H > n} for the rates under consideration. Taking q = gcd(R), Theorem 3.54 provides exact SRB  measures μ1 , . . . , μ p for f q with the stated properties. In the next result, we improve the conclusions in Corollary 6.25, obtaining decay rates with respect to the original dynamics, under an additional assumption of ergodicity of the SRB measure with respect to the powers of the dynamics. Corollary 6.26 Let f : M → M be a C 1+η local diffeomorphism out of a nondegenerate set C and ⊂ M an elementary set containing a ball  such that f is strongly nonuniformly expanding and has slow recurrence to C on a set H ⊂  with full m measure in . If the unique ergodic SRB measure μ for f whose support coincides with is ergodic for all f n with n ≥ 1, then 1. if m{h H > n} ≤ Cn −a for some C > 0 and a > 1, then, for all ϕ ∈ Hρ and ψ ∈ L ∞ (m), there exists C  > 0 such that Cor μ (ϕ, ψ ◦ f n ) ≤ C  n −a+1 ;

6.5 Decay of Correlations

211

2. if m{h H > n} ≤ Ce−cn for some C, c > 0 and a > 1, given ρ > 0, there is c > 0 such that, for all ϕ ∈ Hρ and ψ ∈ L ∞ (m), there is C  > 0 such that a

 a

Cor μ (ϕ, ψ ◦ f n ) ≤ C  e−c n . Proof By Corollary 6.24, there is an expanding Gibbs-Markov map f R defined on an open ball 0 ⊂  ⊂ with the corresponding decays for m{R > n} in terms of  m{h H > n}. Using Corollary 3.56 we get the conclusions. The next result shows that assumption on the ergodicity of the SRB measure with respect to the powers of the dynamics can be replaced by a topologically mixing assumption on . Corollary 6.27 Let f : M → M be a C 1+η local diffeomorphism out of a nondegenerate set C and ⊂ M an elementary set for which there is  ⊂ a ball  such that f is strongly nonuniformly expanding and has slow recurrence to C on a set H ⊂  with m(H ) = m(). If f | is continuous and topologically mixing on

and μ is the unique ergodic SRB measure for f whose support coincides with , then 1. if m{h H > n} ≤ Cn −a for some C > 0 and a > 1, then, for all ϕ ∈ Hρ and ψ ∈ L ∞ (m), there exists C  > 0 such that Cor μ (ϕ, ψ ◦ f n ) ≤ C  n −a+1 ; 2. if m{h H > n} ≤ Ce−cn for some C, c > 0 and a > 1, given ρ > 0, there is c > 0 such that, for all ϕ ∈ Hρ and ψ ∈ L ∞ (m), there is C  > 0 such that a

 a

Cor μ (ϕ, ψ ◦ f n ) ≤ C  e−c n . Proof By Corollary 6.24, there is an expanding Gibbs-Markov map f R defined on an open ball 0 ⊂  ⊂ with the corresponding decays for m{R > n} in terms of m{h H > n}, for the rates under consideration. The conclusions then follow from  Corollary 3.57 applied to f | : → . Finally, we obtain the classical result on exponential decay of correlations with respect to the unique SRB measure for an expanding map, given by Corollary 6.22. Corollary 6.28 Let f : M → M be a C 1+η expanding map of a connected manifold and μ its unique SRB measure. Given ρ > 0, there is c > 0 such that, for all ϕ ∈ Hρ and ψ ∈ L ∞ (m), there is C > 0 such that Cor μ (ϕ, ψ ◦ f n ) ≤ Ce−cn . Proof Take H = = M and h H (x) = 1, for all x ∈ M. Using Lemma 2.13, Lemma 6.21 and Corollary 6.27, we get the conclusion; recall also (2.15). 

212

6 Nonuniformly Expanding Attractors

Note that, although the Lebesgue measure of {h H > n} decays much faster than exponentially to zero (it is actually equal to zero for n ≥ 2), we cannot get better than an exponential rate in Corollary 6.28. In some specific cases, one can obtain superexponential decay of correlations for analytic observables using spectral methods; see, for example [8, Sect. 2.5].

6.6 Applications In this section, we present two open classes of maps where some results obtained in this chapter can be applied. The first one is a class of local diffeomorphisms on the torus introduced in [3] displaying strong nonuniform expansion Lebesgue almost everywhere, with exponential decay on the tail of expansion. The second one is a class of multidimensional nonuniformly expanding maps with critical sets introduced by Viana in [24]. Subsequent works [1, 7] gave that such maps are topologically mixing and have a unique ergodic SRB measure. In [7], it is also proved the statistical stability of these maps, meaning that the SRB measures depend continuously on the map.

6.6.1 Derived from Expanding Here, we describe an open set in the C 1 topology of C 2 nonuniformly expanding local diffeomorphisms introduced in [3]. Consider a Riemannian manifold M and a cover of M by small Borel sets B0 , B1 , . . . , B p . Given constants 0 < σ0 < 1 < σ1 and a small δ > 0 (to be specified in (6.36), depending on σ0 , σ1 and p), let D be a set of C 2 local diffeomorphisms f : M → M such that, for all f ∈ D, (D1 ) (D2 ) (D3 ) (D4 ) (D5 )

f | Bi is injective, for all 0 ≤ i ≤ p; | det D f (x)| ≥ σ1 , for all x ∈ M; D f (x)−1  ≤ 1 + δ, for all x ∈ B0 ; D f (x)−1  ≤ σ0 , for all x ∈ M \ B0 ; f is locally eventually onto.

At the end of this subsection, we will explain how to obtain open sets in the C 1 topology of C 2 maps which satisfy (D1 )–(D5 ) and are not uniformly expanding. In Proposition 6.33 below, we show that, for a convenient choice of the constants, the maps in D are strongly nonuniformly expanding on a set with full Lebesgue measure on M. Using Proposition 6.33, we will be able to prove the next theorem, which is actually the main result of this section. Theorem 6.29 Each f ∈ D has a unique SRB measure μ. Moreover, μ is ergodic, the support of μ coincides with M, its basin covers m almost all of M and, given η > 0, there is c > 0 such that, for all ϕ ∈ Hη and ψ ∈ L ∞ (m), there is C > 0 for which Cor μ (ϕ, ψ ◦ f n ) ≤ Ce−cn .

6.6 Applications

213

Our first goal is to show that the maps in D are strongly nonuniformly expanding. We start with a useful lemma that provides an upper bound for the binomial coefficients. We will use Stirling’s formula, which states that n! ∼

√

2π n (n/e)n ,

(6.28)

where the sign ∼ means that the ratio of the two quantities tends to 1 as n → ∞. Lemma 6.30 There is C > 0 such that, for all 0 < θ < 1/e and n, k ∈ N with k ≤ θ n,   n ≤ Ceτ n , k with τ > 0 depending only on θ and, moreover, τ → 0 when θ → 0. Proof Using Stirling’s formula, we easily get a uniform constant C0 > 0 such that   n n! = k k!(n − k)!

√  n n n ≤ C0 √  k √ e  n−k k k e n − k n−k e  n−k    n n k n = C0 . k(n − k) n − k k

(6.29)

Now, assume 1 ≤ k ≤ θ n. This implies n − k ≥ (1 − θ )n, and so n n 1 e ≤ = ≤ . k(n − k) (1 − θ )n 1−θ e−1

(6.30)

Also, 

n n−k

n−k

 ≤

1 1−θ

n

= e τ1 n ,

(6.31)

with τ1 = − log(1 − θ ). Finally, considering f (x) = (n/x)x , we have f an increasing function for 1 ≤ x ≤ n/e. This gives for k ≤ θ n ≤ n/e  n k k

 θn 1 ≤ = e τ2 n , θ

(6.32)

log θ . The conclusion follows from (6.29), (6.30), (6.31) and (6.32), with τ2 = −θ √ with C = C0 e/(e − 1) and τ = τ1 + τ2 . Note that τ > 0 and τ → 0 when θ → 0. 

214

6 Nonuniformly Expanding Attractors

Given n ≥ 1 and i = (i 0 , . . . , i n−1 ) ∈ {0, . . . , p}n , set G n (i) = #{0 ≤ j < n : i j ≥ 1}.

(6.33)

  In (θ ) = i ∈ {0, . . . , p}n : G n (i) < θ n .

(6.34)

Set also for 0 < θ < 1

Corollary 6.31 Given σ > 1, there are C > 0 and θ > 0 such that, for all n ≥ 1, In (θ ) ≤ Cσ n . Proof We have #In (θ ) =

 n  k 0 sufficiently small such that the conclusion of Corollary 6.31 holds for some σ > 1 with σ σ1−(1−θ) < 1.

(6.35)

Note that the factor σ1−(1−θ) converges to σ1−1 < 1 when θ goes to 0, and so it is possible to take σ > 0 and θ > 0 in the conditions above. Then, choose δ > 0 small enough so that (6.36) σ0θ (1 + δ)1−θ < 1. Corollary 6.32 There are constants C, c > 0 and, for all n ≥ 1, a set E n ⊂ M with m(E n ) ≤ Ce−cn such that, for all x ∈ M \ E n ,  1  # 0 ≤ j < n : f j (x) ∈ B1 ∪ · · · ∪ B1 ≥ θ. n Proof Given n ≥ 1 and i = (i 0 , . . . , i n−1 ) ∈ {0, . . . , p}n , set [i] = Bi0 ∩ f −1 (Bi1 ) ∩ · · · ∩ f −n+1 (Bin−1 ) and En =



[i].

i∈In (θ)

6.6 Applications

215

By (D1 ) and (D1 ), we have for all i ∈ In (θ ) m([i]) ≤ σ1−(1−θ)n m(M), which together with Corollary 6.31 yields n  m(E n ) ≤ #In (θ )σ1−(1−θ)n m(M) ≤ C σ σ1−(1−θ) m(M). Recalling the choice of σ > 1 in (6.35) we finish the proof.



In the next result, we show that each local diffeomorphism in D is strongly nonuniformly expanding on a set H with full Lebesgue measure with an exponential estimate for the tail of the function h H defined in (6.26). Proposition 6.33 For each f ∈ D, there is a set H ⊂ M with full m measure on M such that f is strongly nonuniformly expanding on H . Moreover, m{h H > n} decays exponentially fast to 0 with n. Proof By Corollary 6.32, there are C, c > 0 and, for all n ≥ 1, a set E n ⊂ M with m(E n ) ≤ Ce−cn such that, for all x ∈ M \ E n ,  1  # 0 ≤ j < n : f j (x) ∈ B1 ∪ · · · ∪ B p ≥ θ. n

(6.37)

Take λ > 0 such that σ θ (1 + δ)1−θ < e−λ ; recall (6.36). Conditions (D3 ) and (D4 ) together with (6.37) imply that, for all n ≥ 1 and x ∈ M \ E n , n−1

D f ( f j (x))−1  ≤ σ0θn (1 + δ)(1−θ)n < e−λn .

(6.38)

j=0

Set E=



E n and H = M \ E.

k≥1 n≥k

It follows from (6.38) that, for all k ≥ 1 and all x ∈ M \

 n≥k

E n , we have

n−1 1 −1 log D f ( f i (x))  < −λ, for all n ≥ k. n i=0

This obviously implies that f is strongly nonuniformly expanding and {h H > k} ⊂



E n , for all k ≥ 1.

n≥k

Recalling the m measure of the proof.

 n≥k

E n decays exponentially fast to 0 with k, we finish 

216

6 Nonuniformly Expanding Attractors

Let us now complete the proof of Theorem 6.29. Recall that f is locally eventually onto, by (D5 ), and therefore topologically mixing and transitive on the whole M, by Lemma 2.13. It follows from Proposition 6.33 and Corollary 6.20 that f has a unique SRB measure μ. Moreover, μ is ergodic, the support of μ coincides with M and its basin covers m almost all of M. In addition, Corollary 6.27 gives the conclusion on the decay of correlations. This concludes the proof of Theorem 6.29. Finally, we explain how to obtain open sets in the C 1 topology of C 2 maps which satisfy (D1 )–(D5 ) and are not uniformly expanding. Let M be the d-dimensional torus Rd /Zd , for some d ≥ 2, and f 0 the quotient of a linear map on Rd with a diagonal matrix having integer entries, all greater than one in the diagonal. Let B0 be a small neighbourhood of 0 such that f 0 | B0 is injective. We modify f 0 inside B0 in only one direction, making its derivative have an eigenvalue equal to one, thus creating a repelling neutral fixed point in that direction, as in Sect. 3.5.1. In this way, we obtain a map f 1 : M → M which is topologically conjugate to f 0 . In particular, f 1 is locally eventually onto. For this map f 1 , the conditions (D1 )–(D4 ) are clearly satisfied, possibly with δ = 0 in (D3 ). Now, observe that λ in the proof of Proposition 6.33 can be taken uniform, as long as we take δ > 0 and θ > 0 sufficiently small. Therefore, by Remark 6.6, the radius δ1 > 0 for the images of hyperbolic pre-balls in the respective hyperbolic time can also be uniform. Since f 1 is locally eventually onto, we can find an integer n 1 > 0 such that f n 1 (B) = M, for any ball B of radius δ1 and any f in a sufficiently small neighbourhood N of f 1 in the C 1 topology. By the choice of δ1 , given any f close to f 1 and any open set U in M, by Proposition 6.33 there are points in U with arbitrarily large hyperbolic times. In particular, we can have the preballs contained in U and growing to a ball of radius δ1 in the respective hyperbolic time. This implies that (D5 ) holds for f ∈ N, by the choices of δ1 and N. Next, we consider a partition of M \ B0 into Borel sets B1 , . . . , B p such that f | B j is injective for all 1 ≤ j ≤ p and f ∈ N. Finally, observe that arbitrarily close to f , we have maps for which the point zero is a saddle point. The existence of saddle points is an obvious topological obstruction for the uniform expansion.

6.6.2 Viana Maps In this subsection, we describe an open class of strongly nonuniformly expanding maps with slow recurrence to a critical set defined in the two-dimensional cylinder S 1 × R. These maps have been introduced by Viana in [24] and have natural extensions to higher dimensions. Our main goal here is to obtain Theorem 6.34 and use it to deduce Corollary 6.35 as an application of some of our previous results. Since it would be impossible to provide all the details in a reasonable number of pages, we only sketch the main steps of the proof of Theorem 6.34; see [1, 7, 24] for details. We start by the describing the family of Viana maps. Let a0 ∈ (1, 2) be a parameter such that the critical point x = 0 is pre-periodic for the quadratic map Q(x) = a0 − x 2 . Consider the circle S 1 = R/Z and b : S 1 → R a Morse function, for instance, b(s) = sin(2π s). For small α > 0, define

6.6 Applications

217

fˆ : S 1 × R, −→  S 1 × R (s, x) −→ g(s), ˆ q(s, ˆ x) , where gˆ : S 1 → S 1 is the uniformly expanding map defined, for some d ≥ 16, by g(s) ˆ = ds (mod 1) and q(s, ˆ x) = a(s) − x 2 , with a(s) = a0 + αb(s). It is easily verified that, for any small α > 0, there exists an interval I ⊂ (−2, 2) for which fˆ(S 1 × I ) is contained in the interior of S 1 × I . This implies that any f sufficiently close to fˆ in the C 0 topology still has S 1 × I as a forward invariant region. Thus, f has an attractor

f =



f n (S 1 × I ).

(6.39)

n≥0

We will see in (6.49) that f actually coincides with f 2 (S 1 × I ) for I conveniently chosen and f sufficiently close to fˆ. Recalling the expression of fˆ, it is not difficult to check that C = {x = 0} is a critical set for fˆ and C is a nondegenerate set. By an implicit function argument, we can see that any f sufficiently close to fˆ in the C 2 topology still has a non degenerate critical set C f close to C. We define the family V of Viana maps as the set of C 3 maps in a sufficiently small neighbourhood of fˆ : S 1 × I → S 1 × I in the C 2 topology. As usual, let m be the Lebesgue measure on S 1 × I and consider h H a function as in (6.26). Theorem 6.34 For every f ∈ V, there is a set H with full m measure on S 1 × I such that f is strongly nonuniformly expanding and has slow √recurrence to C f on H . Moreover, there are C, c > 0 such that m{h H > n} ≤ Ce−c n and f is locally eventually onto on f . The strong nonuniform expansion and the measure estimate for {h H > n} have essentially been obtained in [24]; see also [2, Sect. 6]. The fact that Viana maps are locally eventually onto on the attractor was proved in [7] using previous estimates in [1]. It is worth mentioning that the imposition d ≥ 16 on the degree of the base map g has been weakened to d ≥ 2 in [10], but only for open sets of maps in the C ∞ topology. It remains an interesting open question to know whether the measure estimate for {h H > n} is optimal or just a limitation of the method used in [24]. From Corollaries 6.20 and 6.27, we easily get the next consequence of Theorem 6.34, on the existence of a unique SRB measure and stretched exponential decay of correlations for Viana maps. Corollary 6.35 Each f ∈ V has a unique ergodic SRB measure μ whose basin covers m almost all of S 1 × I . Moreover, given η > 0, there exists c > 0 such that, √for all ϕ ∈ Hη and ψ ∈ L ∞ (m), there is C > 0 for which Cor μ (ϕ, ψ ◦ f n ) ≤ Ce−c n .

218

6 Nonuniformly Expanding Attractors

The existence of an ergodic SRB measure and the stretched exponential decay of correlations with respect to this SRB measure have been obtained in [1] and [16], respectively. See also [7] for the uniqueness of the SRB measure and the continuity of its density in the L 1 -norm with the map f ∈ V (statistical stability). In the next three subsections, we present the main ideas behind the proof of Theorem 6.34. Nonuniform Expansion and Slow Recurrence First, we present the main steps of the proof of Theorem 6.34 assuming that f has a skew-product form f (s, x) = (g(s), q(s, x)), (6.40) with

and

∂x q(s, x) = 0 ⇐⇒ x = 0

(6.41)

 f − fˆC 2 ≤ α, on S 1 × I,

(6.42)

for α sufficiently small, where  C 2 stands for the norm in the C 2 topology. In Sect. 6.6.2, we explain how to extend the conclusions to the general case. The estimates on the derivative √ rely √ on a statistical analysis of the returns of orbits to the neighbourhood S 1 × (− α, α ) of the critical set C = {(s, x) : x = 0}. From here on, we only consider points (s, x) ∈ S 1 × I whose orbit does not hit C. This is not an obstacle to our conclusions, since the set of those points has full Lebesgue measure. For each r ≥ 0, set √ √ J (0) = I \ (− α, α) and J (r ) = {x ∈ I : |x| < e−r }. For each integer j ≥ 0, set (s j , x j ) = f j (s, x) and   r j (s, x) = min r ≥ 0 : x j ∈ J (r ) . Consider, for some small constant 0 < η < 1/4,     1 1 − 2η log . G = 0 ≤ j < n : r j (s, x) ≥ 2 α Set

 √  B0 (n) = (s, x) : there is 1 ≤ j < n with x j ∈ J ([ n]) ,

√ √ where [ n] stands for the integer part of n. The results in [24, Sect. 2.4] show that, for n ≥ 1 sufficiently large (only depending on α > 0), we have √

m(B0 (n)) ≤ Ce−

n/4

(6.43)

6.6 Applications

219

for some uniform constant C > 0 and, for some small c > 0 (only depending on the quadratic map Q), log

n−1

|∂x q(s j , x j )| ≥ 2cn −



j=0

r j (s, x),

(6.44)

j∈G

for all (s, x) ∈ / B0 (n). Moreover, setting for γ > 0    / B2 (n) : r j (s, x) ≥ γ n , B1 (n) = (s, x) ∈ j∈G

for γ > 0 sufficiently small there is ξ > 0 such that  m B1 (n) ≤ e−ξ n .

(6.45)

Recalling the definitions of J (r ) and r j , this shows that for δ = (1/2 − 2η) log(1/α), we have n−1  − log dδ ( f j (x), C) ≤ γ n, j=0

for all (s, x) ∈ / B0 (n) ∪ B1 (n). This gives that the orbits of almost all points in S 1 × I have slow recurrence to C. On the other hand, we have for all (s, x) ∈ S 1 × I −1

D f (s, x)

1 = ∂x q(s, x)∂s g(s)



 0 ∂x q(s, x) . −∂s q(s, x) ∂s g(s)

(6.46)

Take the norm of D f (s, x)−1 as the maximum of the absolute values of its entries. From (6.40)–(6.42) we deduce that, for small α > 0, |∂s g| ≥ d − α, |∂s q| ≤ α|b | + α ≤ 8α and |∂x q| ≤ |2x| + α ≤ 4, which together with (6.46) gives



D f (s, x) −1 = |∂x q(s, x)|−1 , as long as we take α > 0 sufficiently small. This implies that n−1  j=0

log D f (s j , x j ))−1  = −

n−1 

log |∂x q(s j , x j )|

j=0

for every (s, x) ∈ S 1 × I . If we choose γ < c, then we have

(6.47)

220

6 Nonuniformly Expanding Attractors n−1 

log |∂x q(s j , x j )| = log

j=0

n−1

|∂x q(s j , x j )| ≥ cn

(6.48)

j=0

for every (s, x) ∈ / B0 (n) ∪ B1 (n); recall (6.44) and the definition of B1 (n). From (6.47) and (6.48), we conclude that n−1 

log D f (s j , x j ))−1  ≤ −cn,

j=0

for all (s, x) ∈ / B0 (n) ∪ B1 (n). In view of the estimates on the Lebesgue measure of B0 (n) and B1 (n) in (6.43) and (6.45), this proves that f is strongly nonuniformly expanding on a set H ⊂ S 1 × I as in Theorem 6.34. General Case Here, we explain how the previous conclusions are drawn in [24] without assuming (6.41). Since fˆ is uniformly expanding in the horizontal direction, using classical ideas on normal hyperbolicity from [13], it can be shown that any map f sufficiently close to fˆ admits a unique invariant central foliation F c of S 1 × I by smooth curves uniformly close to vertical segments; see [24, Sect. 2.5]. Actually, the foliation F c is obtained as the set of integral curves of a vector field (ξ c , 1) in S 1 × I with ξ c uniformly close to zero. Now, the analysis in the previous subsection can be carried out in terms of the expansion of f along this central foliation F c . More precisely, |∂x q(s, x)| is replaced by |∂c q(s, x)| = |D f (s, x)vc (s, x)|, where vc (s, x) is a unit vector tangent to the leaf in foliation F c at (s, x). The previous observations imply that vc is uniformly close to (0, 1) if f is close to fˆ. Moreover, it is no restriction to suppose |∂c q(s, 0)| ≡ 0, so that ∂c q(s, x) ≈ |x|, as in the unperturbed case. Indeed, considering the critical set of f C = {(s, x) ∈ S 1 × I : ∂c q(s, x) = 0}, by an easy implicit function argument it is shown in [24, Sect. 2.5] that C is the graph of some C 2 map η : S 1 → I arbitrarily C 2 -close to zero for α sufficiently small. This means that, up to a change of coordinates C 2 -close to the identity map, we may suppose that η ≡ 0 and, hence, write for α > 0 small ∂c q(s, x) = xψ(s, x), with |ψ + 2| close to zero. This provides an analog to assumption (6.41). At this point, the arguments apply with ∂x q(s, x) replaced by ∂c q(s, x), to show that orbits have slow approximation to the n−1 |∂c q(si , xi )| grows exponentially fast for Lebesgue almost critical set C and i=0 1 every (s, x) ∈ S × I . A matrix formula for D f n (s, x)−1 similar to that in (6.46) can

6.6 Applications

221

be obtained replacing the vector (0, 1) in the canonical basis of the tangent space to S 1 × I at (s, x) by vc (s, x), and consider the matrix of D f n (s, x)−1 with respect to the new basis. Locally Eventually Onto Here, we give the main steps of the proof in [7, Sect. 6] that each f ∈ V is locally eventually onto on the respective attractor f . The first step is to introduce a new system of coordinates (t, y) in S 1 × I related to the original coordinates (s, x) in the following way. As we have seen in the previous subsection, each f ∈ V admits an invariant central foliation whose leaves are smooth submanifolds close to line segments {s = const}. It was also mentioned in the previous subsection that the critical set of f may be supposed to coincide with S 1 × {0}. By definition, (t, y) represents the point where the central leaf through the point (t, 0) intersects the circle S 1 × {y}. Thus, central leaves correspond to vertical line segments corresponding to t equal to a constant in these new coordinates. Moreover, the map f has the form f (t, y) = (g(t), ˜ q˜t (y)). Since the central foliation is usually not transversely smooth, the map g˜ is only continuous, and q˜t depends only continuously on the variable t. On the other hand, the leaves themselves are at least C 2 (initially they are C 3 , but the change of coordinates that brings the critical set to {x = 0} is only C 2 ). This ensures that every q˜t is a C 2 map. Moreover, this map is C 2 close to the one-dimensional quadratic map Q. Our arguments in the sequel always refer to the coordinates (t, y). The next step is to prove that, for all f ∈ V,

f = f 2 (S 1 × I ),

(6.49)

where f is the attractor introduced in (6.39). To see this, observe first that the interval J = [Q 2 (0), Q(0)] is forward invariant for the map Q, and Q 3 (0) is in the interior of J . Then we may take I ⊂ (−2, 2) slightly larger than J , so that Q(I ) is contained in the interior of I and Q 2 (I ) = J . Using that every q˜t is C 2 close to Q, and that its critical point is located at y = 0, we conclude that the first image q˜t (I ) is contained in I and the second one q˜t2 (I ) coincides with the vertical segment J (t) = {g˜ 2 (t)} × [q˜t2 (0), q˜ g(t) ˜ (0)]. By induction, it follows that q˜tn (I ) coincides with J (g˜ n−2 (t)) for every n ≥ 2. Thus, for any τ ∈ S 1 and n ≥ 2, f n (S 1 × I ) ∩ ({τ } × I ) =



J (τ  ) = f 2 (S 1 × I ) ∩ ({τ } × I )

τ

where the union is taken over all the τ  in S 1 such that g˜ 2 (τ  ) = τ . This proves (6.49).

222

6 Nonuniformly Expanding Attractors

The next step is [7, Proposition 6.2], which establishes that there exist an integer M = M(α) ≥ 1, a sequence (Rn )n of m mod 0 partitions of S 1 × I with diam(Rn ) → 0 when n → ∞ and, for each ωn ∈ Rn , a positive integer Rn (ωn ) such that f Rn (ωn )+M (ωn ) = f .

(6.50)

The partitions in the horizontal direction are obtained considering the refinements of the natural partition in S 1 associated with the expanding map in the base of the cylinder. In the vertical direction, the proof of this is divided into four parts. Firstly, it is proved that the height of f R(ω) (ω) (the length of its intersection with vertical lines) is larger than const α 1−2η , where const is used to denote some uniform positive 1−2η becomes a constant. Then, it is shown that any √ vertical segment of length const α vertical segment of length const α, after a finite number of iterates. In the third part it is shown that, again in a finite number of iterates, the length of any such segment becomes larger than some constant independent of α. In the last part is used the fact that f˜ is close to Q and the map Q is locally eventually onto. Finally, using (6.50), we are able to prove that each f ∈ V is locally eventually onto on f . In fact, consider A an arbitrary open subset of S 1 × I . Since the diameter of Rn converges to zero when n goes to infinity, we may find n ≥ 1 and ωn ∈ Rn such that ωn ⊂ A. It follows from (6.50) that there is k ≤ Rn (ωn ) + M such that f k (A) = f . This gives that f is locally eventually onto on f .

References 1. J.F. Alves, SRB measures for non-hyperbolic systems with multidimensional expansio Ann. Sci. École Norm. Sup. (4) 33(1), 1–32 (2000) 2. J.F. Alves, V. Araújo, Random perturbations of nonuniformly expanding maps. Astérisque (286)xvii, 25–62 (2003) 3. J.F. Alves, C. Bonatti, M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140(2), 351–398 (2000) 4. J.F. Alves, C.L. Dias, S. Luzzatto, V. Pinheiro, SRB measures for partially hyperbolic systems whose central direction is weakly expanding. J. Eur. Math. Soc. (JEMS) 19(10), 2911–2946 (2017) 5. J.F. Alves, X. Li, Gibbs-Markov-Young structures with (stretched) exponential tail for partially hyperbolic attractors. Adv. Math. 279, 405–437 (2015) 6. J.F. Alves, S. Luzzatto, V. Pinheiro, Markov structures and decay of correlations for nonuniformly expanding dynamical systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 22(6), 817– 839 (2005) 7. J.F. Alves, M. Viana, Statistical stability for robust classes of maps with non-uniform expansion. Ergodic Theory Dynam. Syst. 22(1), 1–32 (2002) 8. V. Baladi, Positive Transfer Operators and Decay of Correlations, vol. 16, Advanced Series in Nonlinear Dynamics (World Scientific Publishing Co., Inc., River Edge, NJ, 2000) 9. H. Bruin, J. Hawkins, Examples of expanding C 1 maps having no σ -finite invariant measure equivalent to Lebesgue. Israel J. Math. 108, 83–107 (1998) 10. J. Buzzi, O. Sester, M. Tsujii, Weakly expanding skew-products of quadratic maps. Ergodic Theory Dynam. Syst. 23(5), 1401–1414 (2003)

References

223

11. J.T. Campbell, A.N. Quas, A generic C 1 expanding map has a singular S-R-B measure. Comm. Math. Phys. 221(2), 335–349 (2001) 12. P. Góra, B. Schmitt, Un exemple de transformation dilatante et C 1 par morceaux de l’intervalle, sans probabilité absolument continue invariante. Ergodic Theory Dynam. Syst. 9(1), 101–113 (1989) 13. M.W. Hirsch, C.C. Pugh, M. Shub, Invariant Manifolds, vol. 583, Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1977) 14. K. Krzy˙zewski, W. Szlenk, On invariant measures for expanding differentiable mappings. Studia Math. 33, 83–92 (1969) 15. M. Shub, Endomorphisms of compact differentiable manifolds. Amer. J. Math. 91, 175–199 (1969) 16. S. Gouëzel, Decay of correlations for nonuniformly expanding systems. Bull. Soc. Math. France 134(1), 1–31 (2006) 17. V. Pinheiro, Sinai-Ruelle-Bowen measures for weakly expanding maps. Nonlinearity 19(5), 1185–1200 (2006) 18. V. Pinheiro, Expanding measures. Ann. Inst. H. Poincarè Anal. Non Linèaire 28(6), 889–993 (2011) 19. V.A. Pliss, On a conjecture of Smale. Differencial’nye Uravnenija 8, 268–282 (1972) 20. A.N. Quas, A C 1 expanding map of the circle which is not weak-mixing. Israel J. Math. 93, 359–372 (1996) 21. A.N. Quas, Non-ergodicity for C 1 expanding maps and g-measures. Ergodic Theory Dynam. Syst. 16(3), 531–543 (1996) 22. A.N. Quas, Most expanding maps have no absolutely continuous invariant measure. Studia Math. 134(1), 69–78 (1999) 23. D. Ruelle, The thermodynamic formalism for expanding maps. Comm. Math. Phys. 125(2), 239–262 (1989) 24. M. Viana, Multidimensional nonhyperbolic attractors. Inst. Hautes Études Sci. Publ. Math. 85, 63–96 (1997) 25. M. Viana, Dynamics: a probabilistic and geometric perspective, in Proceedings of the International Congress of Mathematicians, vol. 597. I (Berlin, 1998), number Extra Vol. I, pp. 557–578 (electronic) (1998)

Chapter 7

Partially Hyperbolic Attractors

In this chapter, we obtain some results for a class of partially hyperbolic attractors introduced in [1], with a uniformly contracting direction and non-uniform expansion in the centre-unstable direction. We obtain in particular the existence of SRB measures and rates for the decay of correlations with respect to these SRB measures, presenting results that contribute to an already quite general theory on this type of attractors. The dual case of partially hyperbolic attractor with a uniformly expanding direction and nonuniform contraction in the center-stable direction will not be considered here. In such case, the existence of SRB measures has already been obtained under very general conditions in [7], based on previous [12, 19]. However, the results on the decay of correlations with respect to the SRB measures are still far from the most general case. In this direction, we mention [15] in dimension three and [13, 14] for higher dimensional systems with some geometric constraints or a one-dimensional unstable direction. In Sect. 7.2, we obtain a finite number of transitive attractors for most points in a forward invariant partially hyperbolic set with a positive Lebesgue measure subset of points displaying nonuniform expansion in the centre-unstable direction. In Sect. 7.4, we deduce a measure theoretical version of this result with the transitive attractors replaced by SRB measures. For this, we previously obtain in Sect. 7.3 a set with a Young structure for each transitive attractor. Finally, in Sect. 7.5, we use this Young structure to derive estimates on the decay of correlations for the SRB measures. For proving the existence of the attractors, the SRB measures and the Young structures we use ideas from [1, 2, 20]. For the decay of correlations, we follow the strategy in [3], which uses ideas from [16] and extends to the (stretched) exponential case previous results obtained in [5] for the polynomial case.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. F. Alves, Nonuniformly Hyperbolic Attractors, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-62814-7_7

225

226

7 Partially Hyperbolic Attractors

7.1 Dominated Splitting Let f : M → M be a C 1+η diffeomorphism of a finite dimensional Riemannian manifold M. We say that a forward invariant compact set K ⊂ M has a dominated splitting if there are a D f -invariant decomposition TK M = E cs ⊕ E cu and 0 < λ < 1 such that, for some choice of a Riemannian metric on M, D f |E xcs  · D f −1 |E cu f (x)  ≤ λ, for all x ∈ K . We call E cs the centre-stable direction and E cu the centre-unstable direction. We say that E cs is uniformly contracting if D f |E xcs  ≤ λ, for all x ∈ K , and E cu is uniformly expanding if D f −1 |E cu f (x)  ≤ λ, for all x ∈ K . If E cs is uniformly contracting, we call it the strong-stable direction and denote it by E s . Similarly, if E cu is uniformly expanding, we call it the strong-unstable direction and denote it by E u . We say that a forward invariant compact set K ⊂ M is partially hyperbolic if it has dominated splitting TK M = E cs ⊕ E cu for which E cs is uniformly contracting or E cu is uniformly expanding. If both directions have uniform behaviour, K is called a hyperbolic set.

7.1.1 Hölder Control of Centre-Unstable Disks Given K ⊂ M with a dominated splitting TK M = E cs ⊕ E cu , we fix continuous extensions of the bundles E cs and E cu to some neighbourhood U of K , that we denote by E˜ cs and E˜ cu . We do not require these extensions to be D f -invariant. We take the neighbourhood U of K and ζ > 0, both sufficiently small in such a way that, for some λ ∈ (λ, 1), 1+ζ ≤ λ , for all x ∈ U ∩ f −1 (U ). D f | E˜ xcs  · D f −1 | E˜ cu f (x) 

(7.1)

  Given 0 < a < 1, define the centre-unstable cone field Cacu = Cacu (x) x∈U by   Cacu (x) = v1 + v2 ∈ E˜ xcs ⊕ E˜ xcu such that v1  ≤ av2  .

(7.2)

  The centre-stable cone field Cacs = Cacs (x) x∈U is defined in a similar way, just reversing the roles of the bundles E˜ cs and E˜ cu in (7.2). Note that the centre-unstable

7.1 Dominated Splitting

227

cone field is forward invariant: D f (x)Cacu (x) ⊂ Cacu ( f (x)), for all x ∈ U ∩ f −1 (U ).

(7.3)

Actually, the domination property together with the D f -invariance of E cu imply that cu ( f (x)) ⊂ Cacu ( f (x)), D f (x)Cacu (x) ⊂ Cλa

for every x ∈ K , and this extends to any x ∈ U ∩ f −1 (U ), by continuity. We say that an embedded C 1 submanifold N ⊂ U is tangent to the centre-unstable cone field if the tangent subspace to N at each point x ∈ N is contained in the corresponding cone Cacu (x). It follows from (7.3) that f (N ) is also tangent to the centre-unstable cone field, if it is contained in U . A disk D ⊂ U tangent to the centre-unstable cone field will be simply called a cu-disk. Let N ⊂ U be a submanifold of M tangent to the centre-unstable cone field. We choose δ > 0 sufficiently small so that the inverse of the exponential map expx is defined on the δ neighbourhood of every x ∈ U . From now on, we identify this neighbourhood of x with the corresponding neighbourhood Ux of the origin in Tx N , through the local chart defined by exp−1 x . Accordingly, we identify x with the zero vector in Tx N . Reducing δ > 0, if necessary, we may suppose that E˜ xcs is contained in Cacs (y) for every y ∈ Ux . In particular, the intersection of Cacu (y) with E˜ xcs is reduced to the zero vector. Then, Ty N is parallel to the graph of a unique linear map A x (y) : Tx N → E˜ xcs . Now, fix ζ > 0 small enough so that f is C 1+ζ and (7.1) holds. Given C > 0, we say that the tangent bundle of N is (C, ζ )-Hölder if A x (y) ≤ Cdx (y)ζ , for all y ∈ N ∩ Ux and x ∈ U,

(7.4)

where dx (y) denotes the distance from x to y measured along N ∩ Ux . We define κ(N ) = inf{C > 0 : the tangent bundle of N is (C, ζ )-H¨older}.

(7.5)

Proposition 7.1 There exist 0 < α0 < 1 and C0 > 0 such that for every C 1 submanifold N ⊂ U ∩ f −1 (U ) tangent to the centre-unstable cone field we have κ( f (N )) ≤ α0 κ(N ) + C0 . Proof Fixing x ∈ N , we use (u, s) ∈ Tx N ⊕ E˜ xcs and (u 1 , s1 ) ∈ T f (x) f (N ) ⊕ E˜ csf(x) as local coordinates in Ux and U f (x) , respectively. In these local coordinates, we have f (u, s) = (u 1 (u, s), s1 (u, s)). Note that if x ∈ K , then the partial derivatives of u 1 and s1 at 0 ∈ Tx N are ∂u u 1 (0) = D f |Tx N , ∂s u 1 (0) = 0, ∂u s1 (0) = 0, ∂s s1 (0) = D f | E˜ xcs .

228

7 Partially Hyperbolic Attractors

This is because E xcs = E˜ xcs is mapped to E csf(x) = E˜ csf(x) under D f (x) and, similarly, Tx N is mapped to T f (x) N . Then, given any ε0 > 0 small, we have ∂u u 1 (y) − D f |Tx N  < ε0 , ∂s u 1 (y) < ε0 , ∂s s1 (y) − D f | E˜ xcs  < ε0 , (7.6) for every x ∈ U and y ∈ Ux , as long as δ and U are small enough. Taking the cone width a also small, we get D f |Ty N − D f | E˜ xcu  ≤ ε0 and D f −1 |T f (y) f (N ) − D f −1 | E˜ cu f (x)  ≤ ε0 , (7.7) for all x ∈ U and y ∈ Ux . Since f is C 1+ζ , there is also some C1 > 0 such that ∂u s1 (y) ≤ C1 dx (y)ζ .

(7.8)

Now, given y1 ∈ U f (x) , let A f (x) (y1 ) be the linear map from T f (x) f (N ) to E˜ csf(x) whose graph is parallel to Ty1 f (N ). We are going to show that for ε0 sufficiently small, A f (x) (y1 ) satisfies (7.4) for any C > α0 κ(N ) + C0 , with convenient α0 and C0 . Firstly, note that A f (x) (y1 ) is bounded by some uniform constant C2 > 0, since f (N ) is tangent to the centre-unstable cone field. Let D f −1  = sup{D f −1 (z) : z ∈ Uw , w ∈ U }, where the norms are taken with respect to the Riemannian metrics in the local charts. We choose C0 ≥ C2 /(δ/D f −1 )ζ , so that (7.4) is immediate when d f (x) (y1 ) ≥ δ/D f −1 . In fact, A f (x) (y1 ) ≤ C2 ≤ C0 (δ/D f −1 )ζ ≤ C0 d f (x) (y1 )ζ . So we may assume d f (x) (y1 ) < δ/D f −1 . Let γ1 be a curve in f (N ) ∩ U f (x) joining f (x) to y1 and whose length is d f (x) (y1 ). Then γ = f −1 (γ1 ) is a curve in N ∩ Ux joining x to y = f −1 (y1 ), with length less than δ. Using (7.7), we get dx (y) ≤ length(γ ) ≤ (D f −1 | E˜ cu f (x)  + ε0 ) length(γ1 ) = (D f −1 | E˜ cu f (x)  + ε0 )d f (x) (y1 ). Now, observe that    −1 A f (x) (y1 ) = ∂u s1 (y) + ∂s s1 (y) · A x (y) · ∂u u 1 (y) + ∂s u 1 (y) · A x (y) . On the one hand, by (7.6) and (7.8),   ∂u s1 (y) + ∂s s1 (y) · A x (y) ≤ C1 dx (y)ζ + D f | E˜ xcs  + ε0 κ(N )dx (y)ζ     ≤ C1 + D f | E˜ xcs  + ε0 κ(N ) dx (y)ζ .

7.1 Dominated Splitting

229

On the other hand, ∂s u 1 (y) · A x (y) ≤ ε0 C2 . It follows from (7.6) and (7.7) that  −1  ∂u u 1 (y) + ∂s u 1 (y) · A x (y)  ≤ D f −1 | E˜ cu f (x)  + ε1 , where ε1 can be made arbitrarily small by reducing ε0 . Putting these estimates together, we conclude that A f (x) (y1 ) d f (x) (y1 )−ζ is less than (D f | E˜ xcs  + ε0 )(D f −1 | E˜ cu C1 (D f −1 | E˜ cu f (x)  + ε1 ) f (x)  + ε1 ) κ(N ) + . cu cu −1 −ζ −1 ˜ ˜ (D f | E f (x)  + ε0 ) (D f | E f (x)  + ε0 )−ζ So, choosing δ, U , a sufficiently small, we can make ε0 , ε1 , sufficiently close to zero so that the factor multiplying κ(N ) is less than some α0 ∈ (λ , 1); recall (7.1). Moreover, the second term in the expression above is bounded by some constant that  depends only on f . Take C0 larger than this constant. Corollary 7.2 There is C1 > 0 such that for any C 1 submanifold N ⊂ U tangent to the centre-unstable cone field, there is n 0 ≥ 1 such that 1. if n ≥ n 0 and f k (N ) ⊂ U for all 0 ≤ k ≤ n, then κ( f n (N )) ≤ C1 ; 2. if κ(N ) ≤ C1 and f k (N ) ⊂ U for all 0 ≤ k ≤ n, then κ( f n (N )) ≤ C1 ; moreover, each function   Jk : f k (N )  x → log | det D f |Tx f k (N ) | is (Z , ζ )-Hölder continuous, with Z > 0 depending only on C1 and f . Proof Take C0 > 0 given by Proposition 7.1 and C1 ≥ C0 /(1 − α0 ).



7.1.2 Hyperbolic Times and Predisks Let f : M → M be a C 1+η diffeomorphism of a finite dimensional manifold M and K ⊂ M a forward invariant compact set with a dominated splitting TK M = E cs ⊕ E cu . We say that f is nonuniformly expanding along the E cu direction on a set H ⊂ K if there exists c > 0 such that lim inf n→+∞

n 1 log D f −1 |E cu f j (x)  < −c, for all x ∈ H. n j=1

(7.9)

We fix an open neighbourhood U of K and a, δ > 0 sufficiently small so that the results of the previous section hold. In particular, there exists C1 > 0 for which the conclusions of Corollary 7.2 hold. Given σ < 1, we say that n is a σ -hyperbolic time for x ∈ K if n k D f −1 |E cu (7.10) f j (x)  ≤ σ , for all 1 ≤ k ≤ n. j=n−k+1

230

7 Partially Hyperbolic Attractors

By the continuity of D f , we may fix δ1 > 0 sufficiently small such that, for all x ∈ K , D f −1 ( f (y))v ≤ σ −1/4 D f −1 |E cu f (x)  v,

(7.11)

whenever d(x, y) ≤ δ1 and v ∈ Cacu (y). Proposition 7.3 Given a cu-disk D with κ(D) ≤ C1 , there exist C, ζ > 0 such that for all x ∈ K ∩ D with a σ 3/4 -hyperbolic time n there exists a neighbourhood Vn (x) of x in D such that 1. f n maps Vn (x) to a cu-disk of radius δ1 around f n (x); 2. n is a σ 1/2 -hyperbolic time for all y ∈ Vn (x); 3. for all y, z ∈ Vn (x) we have a. d f n−k (D) ( f n−k (y), f n−k (z)) ≤ σ k/2 d f n (D) ( f n (y), f n (z)), for all 1 ≤ k ≤ n; det D f n |Ty D ≤ Cd f n (D) ( f n (y), f n (z))ζ . b. log det D f n |Tz D Proof Let η0 be a curve of minimal length in f n (D) connecting f n (y) to f n (x). For 1 ≤ k ≤ n write ηk = f n−k (η0 ). We prove the first item by induction on k. Let 1 ≤ k ≤ n and assume that length(η j ) ≤ δ1 , for all 0 ≤ j ≤ k − 1. Denote by η˙ 0 (t) the tangent vector to the curve η0 at a point t. Then, in view of the choice of δ1 in (7.11) and the definition of σ -hyperbolic times, we have n

D f −k (η0 (t))η˙ 0 (t) ≤ σ −k/4 η˙ 0 (t)

k/2 D f −1 |E cu η˙ 0 (t). f j (x)  ≤ σ

j=n−k+1

Hence, length(ηk ) ≤ σ k/2 length(η0 ) = σ k/2 d f n−k (D) ( f n−k (y), f n−k (x)) ≤ δ1 . This gives the inductive step, thus proving the first item. This also gives the second item and the first part of the third item. We are left to prove the bounded distortion property. Set Jk (y) = | det D f |T f k (y) f k (D)|, for 0 ≤ k < n and y ∈ D. By Corollary 7.2, each log Jk is (Z , ζ )-Hölder continuous, with Z , ζ > 0 not depending on k. Then, log

n−1  det D f n |Ty D = (log Jk (y) − log Jk (z)) det D f n |Tz D k=0

≤

n−1 

Z d f k (D) ( f k (y), f k (z))ζ .

k=0

Use item 3.a and take C = Z /(1 − σ ζ /2 ).



7.1 Dominated Splitting

231

Remark 7.4 From the proof of Proposition 7.3, we can easily see that the σ 3/4 hyperbolic time n for x was used only to deduce that n is a σ 1/2 -hyperbolic time for all points on Vn (x). This is the only information needed to prove the first part of the third item. The same calculations give more generally that, if n is a σ0 -hyperbolic time for all points on a cu-disk γ , then d f n−k (D) ( f n−k (y), f n−k (z)) ≤ σ0k d f n (D) ( f n (y), f n (z)), for all 1 ≤ k ≤ n and all y, z ∈ γ . We will refer to the sets Vn (x) given by Proposition 7.3 as hyperbolic predisks. Notice that f n (Vn (x)) is a cu-disk of radius δ1 . As in Sect. 5.1, we will also consider sets

n (x) ⊂ Vn (x) (7.12) Wn (x) ⊂ W

n (x) to the disk of radius such that f n maps Wn (x) to the disk of radius δ1 /9 and W n δ1 /3, both centred at f (x). The next result is a simple consequence of Lemma 6.2. Lemma 7.5 Given c > 0, there exists θ > 0 such that if for x ∈ K we have n 1 log D f −1 |E cu f j (x)  < −c, n j=1

then x has e−c/2 -hyperbolic times 1 ≤ n 1 < · · · < n ≤ n with ≥ θ n. Proof Define a j = − log D f −1 |E cu  − c/2, for each 1 ≤ j ≤ n. Taking A = f j (x)  −1 cu maxx∈K | log D f |E x |, we have a j ≤ A for all 1 ≤ j ≤ n. Moreover, nj=1 a j ≥ cn/2. Choosing θ = c/(2 A), it follows  i from Lemma 6.2 that there are 1 ≤ n 1 < a j ≥ 0, for all 1 ≤ k ≤ n i and 1 ≤ i ≤ . · · · < n ≤ n, with ≥ θ n, such that nj=k This implies that ni 

c log D f −1 |E cu f j (x)  ≤ − (n i − k + 1), 2 j=k

(7.13)

for all 1 ≤ k ≤ n i and 1 ≤ i ≤ . Taking exponential on both sides of (7.13), we  easily see that this is equivalent to each n i a e−c/2 -hyperbolic time for x. Proposition 7.6 Let f : M → M be a C 1+η diffeomorphism, K ⊂ M have a dominated splitting TK M = E cs ⊕ E cu and f be nonuniformly expanding along the E cu direction on a set H ⊂ K . Then, for every A ⊂ H with m(A) > 0 and f (A) ⊂ A there are a cu-disk D ⊂ U with κ(D) ≤ C1 , integers 1 ≤ n 1 < n 2 < · · · and Wn k (xk ) ⊂ D for each k ≥ 1 such that for k = f n k (Wn k (xk )) we have lim

k→∞

m k (A ∩ k ) = 1. m k (k )

232

7 Partially Hyperbolic Attractors

Proof We claim that there exists a cu-disk D ⊂ U with κ(D) ≤ C1 such that m D (A ∩ D) > 0. To see this, consider an m density point p of A. Notice that cu T p M = E cs p ⊕ E p . Thus, we may consider a neighbourhood of the origin in T p M foliated by disks parallel to the E cu direction, whose images under the exponential map exp p are cu-disks in the manifold. Since exp p is a local diffeomorphism, exp−1 p (A) has positive volume in T p M (and full density at the origin). By Fubini Theorem, at least one of the cu-disks parallel to the E cu direction must intersect exp−1 p (A) in a set with positive relative volume. Thus, its image D under the exponential map must intersect A in positive m D measure and satisfies κ(D) ≤ C1 . Now, by Lemma 7.5, there exists σ ∈ (0, 1) such that every point in H has infinitely many σ 3/4 -hyperbolic times. Consider Hn the set of points x ∈ H ∩ D such that n is a σ 3/4 -hyperbolic time for x. Since A ⊂ H , it follows that every point in A ∩ D belongs in infinitely many Hn ’s. Moreover, Proposition 7.3 gives that (I2 ) holds for the sequence of sets (Hn )n in D. Using Proposition 5.3 we get the conclusions. 

7.1.3 Partial Hyperbolicity Here, we obtain some results which are specific for partially hyperbolic sets. Assume that f : M → M is a C 1+η diffeomorphism and  ⊂ M is a compact invariant set with a partially hyperbolic splitting T M = E s ⊕ E cu . It is a standard fact for partially hyperbolic sets that, for all x ∈ , there are embedded disks Wεs (x) and Wεcu (x) through x, with Tx Wεcu (x) = E xcu and Tx Wεs (x) = E xs , such that • f (Wεs (x)) ⊂ Wεs (x) and f contracts distances in Wεs (x) by a uniform factor; • f (Wεcu (x)) ∩ Bε ( f (x)) ⊂ Wεcu ( f (x)), where Bε ( f (x)) is the ball of radius ε > 0 around f (x) ∈ ; • {Wεs (x)}x∈ and {Wεcu (x)}x∈ are continuous families of C 1 disks; see for example [19, Theorem IV.1]. In this case, the disk Wεs (x) is uniquely determined and coincides with the Pesin stable disk γεs (x). On the contrary, the disk Wεcu (x) is not unique in general. We will refer to Wεs (x) as a stable disk and to Wεcu (x) as a centre-unstable disk. In the next result, we show that Wεcu (x) contains the Pesin unstable disk γδu (x), for points x ∈  where γδu (x) exists and has the same dimension of Wεcu (x). Therefore, Wεcu (x) is unique for these points. Proposition 7.7 Let f : M → M be a C 1+η diffeomorphism and  ⊂ M an invariant compact set with a partially hyperbolic splitting T M = E s ⊕ E cu . If γδu (x) ⊂  and dim γδu (x) = dim Wεcu (x), for some x ∈ , then γδu (x) ⊂ Wεcu (x), for small δ > 0. Proof Take ε > 0 small enough so that Ty Wεcu (x) is contained in the centre-unstable cone at y, for all y ∈ Wεcu (x) and x ∈ ; recall (7.2). Taking δ > 0 also small, we have that the stable disk Wεs (yn ) through yn = f −n (y) intersects Wεcu ( f −n (x)) in a

7.1 Dominated Splitting

233

unique point wn , for all y ∈ γδu (x) and n ≥ 0. By (1.2), there is C > 0 and 0 < λ < 1 such that, for all n ≥ 1, d( f −n (y), f −n (x)) ≤ Cλn d(y, x).

(7.14)

Let W0 be the neigbourhood of x in Wεcu (x) defined by the intersections of Wεs (y) with Wεcu (x), when y ranges over γδu (x). Noting that Wεcu ( f −n (x)) and f −n (γεs (yn )) are both tangent to the centre-unstable cone (the latter is actually tangent to E cu ), there is some uniform constant C0 > 0 such that, for all w ∈ W0 and n ≥ 1, d( f −n (w), f −n (x)) ≤ C0 d( f −n (y), f −n (x)) ≤ C0 Cλn d(y, x) ≤ C02 Cλn d(w, x). This shows that the points in W0 belong in an unstable disk. Since W0 is a neighbourhood of x in Wεcu (x) and we assume that dim γδu (x) = dim Wεcu (x), we have shown  that Wεcu (x) contains an unstable disk. The requirement that the unstable disk be contained in  is generally satisfied when  is an attractor. This will be further explored in Sect. 7.2. Remark 7.8 The constant δ > 0 given by Proposition 7.7 depends on the point x. However, this can be chosen uniformly over a set of points x ∈  with the same constant C > 0 in (7.14), by the uniqueness of Pesin unstable disks. Since we assume f a C 1+η diffeomorphism, it is a standard fact that the fiber bundles E s and E cu are Hölder continuous on a compact invariant set  with a partially hyperbolic splitting T M = E s ⊕ E cu ; see [10]. In the next subsection we extend this Hölder control to disks in a forward invariant set with a dominated splitting whose tangent space is contained in a centre-unstable cone.

7.2 Attractors Our objective in this section is to prove Theorem 7.9 below, on the existence of finitely many attractors for almost all points in a partially hyperbolic set whose orbits have nonuniform expansion along the centre-unstable direction. We say that  ⊂ M is an elementary set for a diffeomorphism f : M → M if •  is compact set for which f () ⊂ ; • there is some point in  whose forward orbit is dense in . Note that this definition coincides with that given in Sect. 6.2 in the case C = ∅. Moreover, in the present setting, the existence of some point in  whose forward orbit is dense in  is equivalent to the transitivity of f | ; recall Lemma 2.13. Theorem 7.9 Let f : M → M be a C 1+η diffeomorphism and K ⊂ M a set with f (K ) ⊂ K and a partially hyperbolic splitting TK M = E s ⊕ E cu such that f is

234

7 Partially Hyperbolic Attractors

nonuniformly expanding along the E cu direction on a set H ⊂ K with m(H ) > 0. Then, there are elementary sets 1 , . . . ,  ⊂ K such that, for m almost every x ∈ H , there is 1 ≤ j ≤ for which ω(x) =  j . Moreover, each  j contains an unstable disk  j such that f is nonuniformly expanding along the E cu direction, for m  j almost every point in  j . The rest of this section is devoted to the proof of Theorem 7.9. First, in the next subsection, we introduce some preliminary notions and general results. The proof of Theorem 7.9 will be completed in Sect. 7.2.2.

7.2.1 Ergodic Components In (1.3), we have introduced the stable set of a point with negative Lyapunov exponents. Here, we consider a more general version. Given a map f : M → M, we define the stable set of a point x ∈ M as   W s (x) = y ∈ M : lim d( f n (x), f n (y)) = 0 , n→∞

and the stable set of a set S ⊂ M as W s (S) =



W s (x).

x∈S

Lemma 7.10 If S, S  ⊂ M, then W s (S) \ W s (S  ) = W s (S \ W s (S  )). Proof We first prove that W s (S) \ W s (S  ) ⊂ W s (S \ W s (S  )). Take x ∈ W s (S) such that x ∈ / W s (S  ). This means that there exists y ∈ S such that x ∈ W s (y), and s  also x ∈ / W (y ), for any y  ∈ S  . To obtain the first inclusion, we only have to see that y ∈ / W s (S  ). Arguing by contradiction, assume y ∈ W s (S  ). Then, y ∈ W s (z), for some z ∈ S  . It is easily verified that x ∼ y ⇐⇒ x ∈ W s (y) gives an equivalence relation in M. It follows from the transitivity of the equivalence relation ∼ that x ∈ W s (z), and so a contradiction. Now, we prove that W s (S) \ W s (S  ) ⊃ W s (S \ W s (S  )). Take x ∈ W s (S \ W s (S  )). This gives x ∈ W s (y), for some y ∈ S \ W s (S  ). It just remains to show that x ∈ / W s (S  ). Sups  s pose by contradiction that x ∈ W (S ). Then, we have x ∈ W (z), for some z ∈ S  . It follows once more from the transitivity of ∼ that y ∈ W s (z), which contradicts the fact that y ∈ S \ W s (S  ).  We say that a set S ⊂ M is s-saturated if W s (S) = S. We say that a Borel set E ⊂ M is an s-saturated ergodic component of f if

7.2 Attractors

235

• E is an s-saturated invariant set; • for any s-saturated invariant set E  ⊂ E we have m(E  ) = 0 or m(E  ) = m(E). Recall that an invariant set has been defined in Subsection 6.2.1 as a set E ⊂ M for which f −1 (E) = E. Lemma 7.11 If E is an s-saturated ergodic component of f , then there is a compact set  ⊂ M with f () =  such that ω(x) = , for m almost every x ∈ E. Proof Let E ⊂ M be an s-saturated ergodic component of f . Given an open set A ⊂ M, set A∗ = {x ∈ E : ω(x) ∩ A = ∅}. It is easily verified that A∗ is an s-saturated invariant set contained in E. Therefore, it must be m(A∗ ) = 0 or m(A∗ ) = m(E). Now, consider X 1 = M and A1 a finite cover of M by open balls of radius 1. By the previous considerations, for every A ∈ A1 we have m(A∗ ) = 0 or m(A∗ ) = m(E). Since A1 has only a finite number of elements, there exists at least one A∗ ∈ A1 such that m(A∗ ) = m(E). Let A1 = {A ∈ A1 : m(A∗ ) = 0} and X 2 = X 1 \

A∈A1

A.

Note that X 2 is a non-empty compact set and ω(x) ⊂ X 2 , for m-almost every x ∈ E. We can therefore repeat the procedure above with a finite cover A2 of X 2 by open balls of radius 1/2. By induction, we construct sequences (An )n , (An )n and (X n )n such that M = X 1 ⊃ X 2 ⊃ ... is a sequence of non-empty compact sets and ω(x) ⊂ X j , for m almost every x ∈ E. Set

= Zn. n≥1

We have ω(x) ⊂ , for m almost every x ∈ E. We claim that  ⊂ ω(x), for m almost every x ∈ E. Indeed, given y ∈ , we have that y ∈ X n for every n, and therefore  there exists some  An ∈ An \ An such that y ∈ An . Since diam(An ) → 0 as n → ∞, it follows that n An = {y}. Moreover, as An ∈ An \ An , we have m(A∗n ) = m(E), for every n. Therefore, ω(x) ∩ An = ∅, for m almost all x ∈ E and all n. This implies that y ∈ ω(x), for m almost all x ∈ E. Therefore,  ⊂ ω(x), for m almost all x ∈ E. Since each ω(x) is a compact set such that f (ω(x)) = ω(x), we have completed the proof. 

7.2.2 Unshrinkable Sets We say that a Borel set X ⊂ M is cu-unshrinkable if X is an invariant set with m(X ) > 0 for which there exists δ > 0 such that any invariant set S ⊂ X with m(S) > 0 necessarily has m(W s (S)) > δ.

236

7 Partially Hyperbolic Attractors

Lemma 7.12 If X ⊂ M is cu-unshrinkable, then X is (m mod 0) contained in the union of a finite number of s-saturated ergodic components. Proof Let X 1 = X and F1 = {W s (S) : S ⊂ X 1 , f −1 (S) = S and m(W s (S)) > 0 }. Note that F1 is non-empty because W s (X 1 ) ∈ F1 . We claim that W, W  ∈ F1 and m(W \ W  ) > 0 =⇒ W \ W  ∈ F1 .

(7.15)

Actually, given invariant sets S, S  ⊂ X 1 , take W = W s (S) and W  = W s (S  ). Notice that S \ W s (S  ) ⊂ X 1 and S \ W s (S  ) is invariant because both S, W s (S  ) are invariant. Therefore, Lemma 7.10 gives (7.15). Now, consider the partial  order on F1 defined by strict inclusion in terms of measure, meaning that W  W  if W ⊃ W  and m(W \ W  ) > 0. We claim that every totally ordered family of F1 is finite, and in particular it has a lower bound. Arguing by contradiction, suppose that there is an infinite sequence W1  W2  · · · in F1 . This implies W1 ⊃ W2 ⊃ · · · with m(Wk \ Wk+1 ) > 0, for all k ≥ 1. Hence, 

m(Wk \ Wk+1 ) = m(W1 ) < ∞,

k≥1

and therefore m(Wk \ Wk+1 ) → 0 as k → ∞. It follows from (7.15) that Wk \ Wk+1 ∈ F1 for all k. This contradicts the fact that X 1 = X is cu-unshrinkable. Hence, every totally ordered subset of F1 has a lower bound. By Zorn’s Lemma, there exists at least one minimal element W s (S1 ) ∈ F1 , which necessarily is an s-saturated ergodic component of f . Now let X 2 = X 1 \ W s (S1 ). This is again an invariant set. If m(X 2 ) = 0, then X = X 1 is essentially contained in W s (S1 ), which is an s-saturated ergodic component, and we are done. If on the other hand m(X 2 ) > 0, we repeat the argument above to produce a set S2 ⊂ X 2 and an s-saturated ergodic component W s (S2 ). Inductively, we construct a collection of disjoint ergodic components W s (S1 ), . . . , W s (Sk ) and continue as long as m(X \ (W s (S1 ) ∪ · · · ∪ W s (Sk ))) > 0. However, since X is cuunshrinkable, there exists δ > 0 such that m(W s (S j )) ≥ δ, for all 1 ≤ j ≤ k. Thus, this process must stop in a finite number of steps.  Now, we complete the proof of Theorem 7.9. Consider a compact set K ⊂ M with f (K ) ⊂ K and a partially hyperbolic splitting TK M = E s ⊕ E cu such that f nonuniformly expanding along the E cu direction on a set H ⊂ K with m(H ) > 0. Set

= f n (H ). H n∈Z

is cu-unshrinkable. Lemma 7.13 H

7.2 Attractors

237

is clearly invariant and m( H

) > 0. We need to show that there exists δ > 0 Proof H

with m(S) > 0 we have m(W s (S)) > δ. We such that for every f -invariant set S ⊂ H remark that in the proof of this assertion we will only use that S is forward invariant.

This allows us to assume without loss of generality that S ⊂ H . Indeed, if S ⊂ H

, we must have m(S ∩ is an invariant set with m(S) > 0, then by definition of H H ) > 0. Since H is forward invariant, also S ∩ H is forward invariant. Clearly, if m(W s (S ∩ H )) > δ, then we also have m(W s (S)) > δ. Now let S ⊂ H ⊂ K be a forward invariant set with m(S) > 0. By Proposition 7.6, there are δ > 0, a cu-disk D ⊂ U be such that m D (S) > 0, a sequence of sets (Wk )k in D and a sequence of positive integers (n k )k such that k = f n k (Wk ) is a cu-disk of radius δ and the relative measure of S in k converges to 1 when k → ∞. Since S ⊂ K , all points of S have local stable manifolds of uniform size. Moreover, as each k is a disk of radius δ, it easily follows that there exists δ0 > 0 such that

is cu-unshrinkable.  m(W s (S)) ≥ δ0 . This shows that H It follows from Lemmas 7.11, 7.12 and 7.13 that there exist compact sets 1 , ...,  with f ( j ) =  j , for all 1 ≤ j ≤ , such that, for m almost every x ∈ H , we have ω(x) =  j , for some 1 ≤ j ≤ . Let us see that each  j as all the properties stated in Theorem 7.9. Fix 1 ≤ j ≤ and set  =  j . For all n ≥ 1, set 

 1 An = x ∈ H : d( f (x), ) ≤ , for all k ≥ 0 . n k

Notice that each An is a subset of H such that f (An ) ⊂ An . Without loss of generality, we may assume that the set of points x ∈ H with ω(x) =  has positive m measure. This implies that m(An ) > 0, for all n ≥ 1. It follows from Proposition 7.6 that, for all n ≥ 1, there is a cu-disk Dn ⊂ U with m Dn (An ∩ Dn ) > 0, integers 1 ≤ i n,1 < i n,2 · · · and, for each k ≥ 1, a point xn,k ∈ Hin,k such that n,k = f in,k (Win,k (xn,k )) is a cu-disk of radius δ1 /9 and the relative m measure of An in n,k converges to one when k → ∞. Since each Win,k (xn,k ) is contained in hyperbolic predisk Vin,k (xn,k ), Proposition 7.3 gives that f − |n,k is a σ /2 -contraction for all 1 ≤ ≤ i n,k . Let pn,k denote the centre of each disk n,k . Up to taking a subsequence, we may assume that the sequence ( pn,k )k converges to a point pn ∈ K . Up to taking a further subsequence, using Arzelá-Ascoli Theorem and the fact that each n,k is a cu-disk, we may assume that the sequence (n,k )k converges uniformly to some cu-disk n of radius δ1 /9 when k → ∞. Note that each n is necessarily contained in K and in a 1/n-neighbourhood of . Moreover, f − |n is a σ /2 -contraction for all ≥ 1. We claim that f is nonuniformly expanding along the E cu direction for m n almost every point in n . To see this, recall first of all that the nonuniform expansion is an asymptotic property. Therefore, if it is satisfied by a point x, then it is satisfied by every y ∈ W s (x). Moreover, every point in n ⊂ K has a local stable manifold of uniform size, and the foliation by those local stable manifolds is absolutely continuous; see [11, Sect. 3]. Since some subsequence of (n,k )k converges uniformly to n when k → ∞, the disks in that subsequence must necessarily intersect the stable foliation through the points of n in a subset with relative m n measure converging

238

7 Partially Hyperbolic Attractors

to one as k → ∞. We have f nonuniformly expanding along the E cu direction, for m n almost every point in n , inasmuch as An ⊂ H and the relative measure of An in n,k converges to one when k → ∞. Now, arguing as above, we can consider a subsequence of (n )n converging uniformly to some cu-disk  of radius δ1 /9, and obtain that f is nonuniformly expanding along the E cu direction, for m  almost all points in  in the same way as before. Moreover, f − j | is a σ j/2 -contraction for all j ≥ 1, thus obtaining that  is an unstable disk. As each n is contained in a 1/n-neighbourhood of  and  is closed, it follows that  ⊂ . It remains to prove the transitivity of f | . By construction, there exists some point (in fact, a positive m measure subset) in H whose ω-limit coincides with . The orbit of any such point must eventually hit the stable manifold of some point in  ⊂ . As points in the same stable manifold have the same ω-limit sets, we easily conclude that there exists some point in  whose orbit is dense in .

7.3 Young Structure In this section, we show that each elementary set given by Theorem 7.9 contains a set with a Young structure. Despite its intrinsic interest, Theorem 7.14 below will be particularly helpful in Sect. 7.4, where it will be used to obtain SRB measures supported on the attractors given by Theorem 7.9. Theorem 7.14 Let f : M → M be a C 1+η diffeomorphism and  ⊂ M an elementary set with a partially hyperbolic splitting T M = E s ⊕ E cu . If there exists an unstable disk  ⊂  such that f is nonuniformly expanding along the E cu direction, for m  almost every point in , then f has a set  ⊂  with a full Young structure and integrable recurrence times. We prove Theorem 7.14 in the remainder of this section. The idea is to show that there exists a disk 0 ⊂  for which conditions (I1 )–(I3 ) in Sect. 5.1 hold and apply Theorem 5.1. Such a disk will be obtained in the next subsection. In Sect. 7.3.2, we use 0 to build a set  with a product structure. In the subsequent subsections we show that  has a full Young structure with integrable recurrence times.

7.3.1 Inducing Domain Let  ⊂  ⊂ M be as in Theorem 7.14. Choose δ1 > 0 as in (7.11) and 0 < ε0 < δ1 such that the stable disk γεs0 (x) of size ε0 is defined for all x ∈ . Note that these local stable disks depend continuously on the point x ∈  in the C 1 topology; see for example [21, Theorem VI.6]. Let H ⊂  be a set with full m  measure such that f is nonuniformly expanding along the E cu direction on H . By Lemma 7.5, there are 0 < σ < 1 and θ > 0 such that, for every x ∈ H ,

7.3 Young Structure

239

1 lim sup # {1 ≤ j ≤ n : j is a σ -hyperbolic time for x} ≥ θ. n→∞ n

(7.16)

Set for each n ≥ 1   Hn = x ∈ H : n is a σ 3/4 -hyperbolic time for x .

(7.17)

The choice of σ 3/4 in the definition of Hn will become clear in Sect. 7.3.3. It easily follows from the definition of hyperbolic time in (7.10) that x ∈ Hn =⇒ f j (x) ∈ Hn− j ,

for all 0 ≤ j < n.

(7.18)

Moreover, since every σ -hyperbolic is a σ 3/4 -hyperbolic, it follows from (7.16) that m  almost every point in  belongs in infinitely many Hn ’s. By Proposition 7.3, each x ∈ Hn has a neighbourhood Vn (x) in  that grows to a disk of radius δ1 in n iterates. A priori, we do not know the localization of this image disk. We will use the transitivity in  to bring it close to  in a finite number of iterates. Given a disk D ⊂ , define the cylinder C(D) =



γεs0 (x).

(7.19)

x∈D

Let  : C() →  be the projection along stable disks. We say that an unstable disk γ crosses the cylinder C(D) if  maps some connected component of γ ∩ C(D) bijectively to D. In the next result, we consider Wn (x) as in (7.12). Lemma 7.15 There are p ∈  and L ≥ 1 such that, for small enough δ0 > 0, given any Wn (x) ⊂ , there is 0 ≤ ≤ L for which f n+ (Wn (x)) intersects γεs0 /2 ( p). Moreover, denoting D2δ0 ( p) ⊂  the disk of radius 2δ0 centred at p, a connected component of f n+ (Wn (x)) ∩ C(D2δ0 ( p)) crosses C(D2δ0 ( p)) and intersects γεs0 /2 ( p). Proof First of all, note that the angles of the two sub-bundles in the dominated splitting T M = E s ⊕ E cu are uniformly bounded away from zero. Therefore, given any c > 0, there is a > 0, with a → 0 as c → 0, for which the following property holds: if x, y are points in  are such that d(x, y) < c and d D (y, ∂ D) > δ1 /9, for some disk D ⊂  with y ∈ D, then γεs0 (x) intersects D in a point z with dγεs0 (x) (z, x) < a and d D (z, y) < δ1 /18. Choose c > 0 sufficiently small so that 4a < ε0 . Since f | is transitive, we may take q ∈  and L ∈ N such that • γεs0 /4 (q) intersects  in a point p with d ( p, ∂) > 0; • { f −L (q), . . . , f −1 (q), q} is c-dense in . Given any Wn (x) ⊂ , we have that f n (Wn (x)) is an unstable disk of radius δ1 /9 centred at y = f n (x) contained in . Take 0 ≤ ≤ L such that d( f − (q), y) < c. It follows from the choice of c that γεs0 ( f − (q)) intersects f n (Wn (x)) in a point z such that

240

7 Partially Hyperbolic Attractors

dγεs ( f − (q)) (z, f − (q)) < a < 0

ε0 δ1 and d f n (Wn (x)) (z, y) < . 4 18

In particular, f n (Wn (x)) contains an unstable disk D of radius δ1 /18 centred at z. We claim that there is δ2 = δ2 (L , δ1 ) > 0 such that f (D) contains an unstable disk of radius δ2 centred at f (z), for all 1 ≤ ≤ L. Assume = 1. Let z be the centre of the unstable disk D and f (y) a point in ∂ f (D) minimising the distance from f (z) to ∂ f (D). Let η1 be a curve of minimal length in f (D) from f (z) to f (y). Consider η0 = f −1 (η1 ) and η˙ 1 (x) the tangent vector to the curve η1 at the point x. We have D f −1 (w)η˙ 1 (x) ≤ C η˙ 1 (x),   where C = maxx∈M D f −1 (x) ≥ 1. It follows that length(η0 ) ≤ C length(η1 ). Since η0 is a curve connecting z to y ∈ ∂ D, we have length(η0 ) ≥ δ1 /18. Hence, length(η1 ) ≥

1 δ1 length(η0 ) ≥ . C 18C

This shows that f (D) contains the unstable disk D1 of radius C −1 δ1 /18 centred at f (z). This gives the claim for = 1. Now, putting D1 in place of D and f 2 (z) in place of f (z), the argument above shows that f (D1 ) contains an unstable disk of radius C −2 δ1 /182 centred at f 2 (z). Inductively, we show that f (D) contains an unstable disk of radius C − δ1 /18 ≥ C −L δ1 /18 L centred at f (z), for each 1 ≤ ≤ L. Taking δ2 = C −L δ1 /18 L , we have proved the claim. By the claim, f (D) contains an unstable disk of radius δ2 > 0 centred at f (z) ∈ γεs0 ( p). As the distance between any two points in a stable disk is contracted under forward iterates, we get dγεs0 ( p) ( f (z), p) ≤ dγεs0 ( p) ( f (z), q) + dγεs0 ( p) (q, p) ≤

ε0 . 2

Therefore, for sufficiently small δ0 > 0 (depending only on δ2 ), a connected component of f n+ (Wn (x)) ∩ C(D2δ0 ( p)) intersects γεs0 /2 ( p) and crosses  C(D2δ0 ( p)). Take p ∈ , L ∈ N and δ0 > 0 for which the conclusion of Lemma 7.15 holds. Set (7.20) 0 = Dδ0 ( p) and 1 = D2δ0 ( p). Notice that 0 is an unstable disk. Consider the corresponding cylinders C0 = C(0 ) and C1 = C(1 ).

(7.21)

Proposition 7.16 Given N0 ≥ 1, there are an m 0 mod 0 partition P of 0 and a function R : 0 → {N0 , N0 + 1, . . . } constant in the elements of P such that 1. for every n ≥ 1, there are finitely many ω ∈ P with R(ω) = n; 2. f R maps each ω ∈ P to a cu-disk that crosses C0 and f R (C(ω)) ⊂ C0 ;

7.3 Young Structure

241

3. there are S1 , S2 , · · · ⊂ 0 with

 n≥1

m 0 (Sn ) < ∞ such that, for all n ≥ 1,

Hn ∩ {R > L + n} ⊂ Sn 4. if h θ is defined for some θ as in (5.2), then there are E 1 , E 2 , · · · ⊂ 0 with m 0 (E n ) converging to zero exponentially fast with n such that, for all n ≥ 1, {R > n + L} ⊂ {h θ > n} ∪ E n . Proof Our goal is to verify conditions (I1 )-(I3 ) in Sect. 5.1 on the disk 0 ⊂  and use Theorem 5.1. For each n ≥ 1, take Hn as in (7.17). It follows immediately from (7.16) that (I1 ) is satisfied. Moreover, taking for each x ∈ Hn the hyperbolic preballs Vn (x) given by Proposition 7.3, property (I2 ) is also satisfied. Now, we ωn, that satisfy property (I3 ). Using Lemma 7.15, we describe the domains ωn, and

easily deduce that, for each Wn (x) ⊂ Vn (x), there exists an integer 0 ≤ ≤ L such that a connected component of f n+ (Wn (x)) ∩ C1 intersects γεs0 /2 ( p) and crosses C1 . Let

ωn, be the part of Wn (x) that f n+ maps diffeomorphically to that connected ωn, that f n+ (ωn, ) ⊂ C0 and f n+ (ωn, ) component. Let also ωn, be the part of

crosses C0 . We have (7.22) f n+ (ωn, ) ∩ γεs0 /2 (x) = ∅. Notice that f n+ (ωn, ) and f n+ (

ωn, ) are not exactly disks of radii δ0 and 2δ0 as required in (I3 ). However, according to Remark 5.4, this is not an obstruction for the validity of Theorem 5.1, since the stable disks depend continuously on the base point x ∈ 1 in the C 1 topology. The two final requirements in (I3 ) are clearly satisfied by the centre-unstable disks

ωn, , since f is a diffeomorphism and we have at most

≤ L iterates. Using Theorem 5.1, we obtain an m 0 mod 0 partition P of 0 into domains ωn, . Moreover, taking R(x) = n + for each x ∈ ωn, ∈ P, we have that f R maps ωn, to an unstable disk that crosses C0 . By Remark 5.10, given any k ≥ 1, there is a finite number of elements ω ∈ P for which R(ω) = k. Thus, we have the first item. Moreover, according to (5.35), we have R ≥ N0 , for some positive integer N0 that can be taken arbitrarily large. Since (7.22) holds, taking N0 large enough, we guarantee f R (C(ωn, )) ⊂ C0 . This proves the second item. The other items follow directly from Theorem 5.1.  Remark 7.17 The proof of Proposition 7.16 provides additional useful information: every ω ∈ P is contained in a hyperbolic pre-disk Vn (x) with n ≥ 1 and x ∈ Hn . In particular, n is a σ 3/4 -hyperbolic time for x. It follows from Proposition 7.3 that n is a σ 1/2 -hyperbolic time for all points in ω. We could have included in Proposition 7.16 conclusions on expansion and bounded distortion for the elements ω in the partition P of the unstable disk 0 ⊂ C0 , similar to those in item 1 of Theorem 5.1. This will be achieved in the next subsection more generally for any unstable disk inside the cylinder C0 .

242

7 Partially Hyperbolic Attractors

7.3.2 Product Structure In this subsection, we introduce the families  s and  u which define the set  with a Young structure as in Theorem 7.14 and check properties (Y1 )-(Y5 ). The idea for building  u is similar to that used in Sect. 4.6 for the solenoid with intermittency. Let 0 be an unstable disk as in (7.20). Consider an m 0 mod 0 partition P of 0 and R : 0 → N as in Proposition 7.16. Assume the elements P listed as ω1 , ω2 , . . . Set for each i ≥ 1 (7.23) Ci = C(ωi ) and Ri = R(ωi ). Remark 7.18 Some previous results will be applied here to unstable disks γ ⊂ C0 , giving conclusions with respect to the distance measured along f j (γ ∩ Ci ), for 0 ≤ j ≤ Ri . Since the diameter of these disks is uniformly bounded and they are tangent to the centre-unstable cone field, using Corollary 7.2, we guarantee that the distance measured along f j (γ ∩ Ci ) is bounded from above and below by the distance in M, up to a uniform factor. Lemma 7.19 There are C > 0 and 0 < α < 1 such that, for every i ≥ 1 and every unstable disk γ ⊂ C0 that crosses C0 , d( f Ri −k (x), f Ri −k (y)) ≤ Cα k d( f Ri (x), f Ri (y)), for all 1 ≤ k ≤ Ri and x, y ∈ Ci ∩ γ . Proof Given i ≥ 1, we have for ωi ∈ P Ci = C(ωi ) and Ri = n + , for some 0 ≤ ≤ L and n a σ 1/2 -hyperbolic time for all points on ωi ; recall Remark 7.17. Since we assume 0 < ε0 < δ1 and local stable disks are uniformly contracted in forward time, it follows from the choice of δ1 in (7.11) that n is a σ 1/4 -hyperbolic time for all points on γ . According to Remark 7.4, we have d f n−k (γ ) ( f n−k (x), f n−k (y)) ≤ σ k/4 d f n (γ ) ( f n (x), f n (y)), for all 1 ≤ s ≤ n and x, y ∈ γ . Since 0 ≤ ≤ L, then there exists some uniform constant C0 > 0 such that d f Ri −k (γ ) ( f Ri −k (x), f Ri −k (y)) ≤ C0 σ k/4 d f Ri (γ ) ( f Ri (x), f Ri (y)),

(7.24)

for all 1 ≤ k ≤ Ri and x, y ∈ γ . Using (7.24) and Remark 7.18, we obtain some uniform constant C > 0 such that d( f Ri −k (x), f Ri −k (y)) ≤ Cσ k/4 ( f Ri (x), f Ri (y)), for all 1 ≤ k ≤ Ri and x, y ∈ γ . Take α = σ 1/4 .



7.3 Young Structure

243

Now, consider the sequence (n )n≥0 , defined inductively in the following way. We start with 0 = {0 } . Assuming that n−1 is defined for some n ≥ 1, set n =



 f Ri (Ci ∩ γ ) : γ ∈ n−1 .

i≥1

Note that, by the first item of Proposition 7.16, each γ ∈ n is an unstable disk inside C0 that crosses C0 . Set n and ∞ = γ. ∞ = n≥0 γ ∈n

n≥0

Since ∞ is not necessarily a compact set, we still need to take accumulation points. We will see that the accumulation points are contained in centre-unstable disks as introduced in Sect. 7.1. Acting as in Sect. 4.6.3, this could be used to prove that the accumulation points ∞ also belong in C 1 disks (not guaranteed unstable) accumulated by disks in ∞ . However, with a little more effort, we can prove that these accumulation points also belong to unstable disks. Lemma 7.20 For every accumulation point z of ∞ , there is an unstable disk contained in the closure of ∞ that contains z and crosses C0 . Proof Given an accumulation point z of ∞ , there are a sequence of positive integers ( jk )k , disks γ jk ∈  jk and points z jk ∈ γ jk converging to z, when k → ∞. Taking an appropriate local chart and using the domination property, we may think of these unstable disks as graphs of C 1 functions with uniformly bounded derivative. By Arzelá-Ascoli Theorem, some subsequence of (γ jk )k converges uniformly to a disk γ∞ containing z. To simplify notation, we still denote this subsequence by (γ jk )k . In principle, γ∞ is just a topological disk. Now we show that γ∞ is an unstable disk (and therefore a smooth disk). We may assume, without loss of generality, that the terms of (γ jk )k are all distinct. Thus, possibly with the only exception of one term which is equal to 0 , by definition of  jk , for each γ jk ∈  jk , there are i jk ≥ 1 and an unstable disk γˆ jk such that γ jk = f

Ri j

k

(Ci jk ∩ γˆk ),

for some i k ≥ 1 and an unstable disk γˆk . By the first item of Proposition 7.16, it is no restriction to assume that Ri j1 < Ri j2 < · · · It follows from Lemma 7.19 that d( f

Ri j − j k

(x), f

Ri j − j k

(y)) ≤ Cα j d( f

Ri j

k

(x), f

Ri j

k

(y)),

for all 1 ≤ j ≤ Ri jk and x, y ∈ Ci jk ∩ γˆ jk . This is equivalent to say that d( f − j (x), f − j (y)) ≤ Cα j d(x, y),

(7.25)

244

7 Partially Hyperbolic Attractors

for all 1 ≤ j ≤ Ri jk and x, y ∈ γ jk . Since the disks γ jk accumulate on γ∞ , we easily see that (7.25) still holds for all x, y ∈ γ∞ and 1 ≤ j ≤ Ri jk . Noting that Ri jk → +∞  when k → +∞, we conclude that γ∞ is an unstable disk. Let  be the family of all accumulation disks given by Lemma 7.20. Set  u =  ∪ ∞ . Since the elements in  u are all unstable disks, they are necessarily pairwise disjoint. Now we show that  u is a continuous family of C 1 disks. Recall that 0 is contained in the compact set  ⊂ M with a partially hyperbolic splitting T M = E s ⊕ E cu . Therefore, the disks in  u are all contained . Moreover, for each x ∈ , there is an embedded centre-unstable C 1 disk Wεcu (x) such that Tx Wεcu (x) = E xcu and f (Wεcu (x)) ∩ Bε ( f (x)) ⊂ Wεcu ( f (x)), where Bε ( f (x)) is the ball of radius ε > 0 around f (x) ∈ . In addition, the disks Wεcu (x) depend continuously on x ∈  in the C 1 topology; see Sect. 7.1. By construction, each γ ∈  u is an unstable disk of radius δ0 and intersects in a single point the stable disk γεs0 ( p) used to define the cylinder C0 . Consider K ⊂ γεs0 ( p) the compact set formed by all these intersection points. Choosing δ0 suffciently small, it follows from Proposition 7.7 that γ ⊂ Wεcu (x), for all x ∈ K . Since the disks Wεcu (x) depend continuously on x ∈ K in the C 1 topology, we conclude that  u is a continuous family of C 1 disks. The family of stable disks is naturally defined as    s = γεs0 (x) : x ∈ 0 .

(7.26)

The contraction property (Y2 ) is immediate with this choice, since the stable disks are uniformly contracted under forward iterations. Moreover,  s is a continuous family of C 1 disks, by standard hyperbolic theory; see for example [19, Theorem IV.1]. The product product structure is also immediate, as long as ε0 > 0 and δ0 > 0 are small enough. Finally, take γ. = γ ∈ u

Considering Ci and Ri as in (7.23), for each i ≥ 1, set i = Ci ∩  and R|i = Ri . Proposition 7.21 The set  has a full Young structure with recurrence time R. Proof Clearly,  has a full product structure. The properties (Y1 )–(Y2 ) are easily checked, by the construction of  u and  s . Recall that, by Proposition 7.16, we may choose Ri ≥ N0 , for any positive integer N0 . Taking N0 sufficiently large, it follows from Lemma 7.19 that there is 0 < β < 1 such that, for all i ≥ 1 and γ ∈  u ,

7.3 Young Structure

245

d(x, y) ≤ βd( f R (x), f R (y)).

(7.27)

This gives the expansion property (Y3 ); recall Remark 4.1. Now, we check the Gibbs property (Y4 ). Take γ ∈  u and x, y ∈ γ ∩ i , for some i ≥ 1. Given 0 ≤ k < Ri , the function log | det D f |T f k (x) f k (γ )| is (Z , ζ )-Hölder continuous, with uniform constants Z , ζ > 0, by Corollary 7.2. Using Lemma 7.19, we get R i −1  det D f Ri |Tx γ log Z d f k (γ ) ( f k (x), f k (y))ζ ≤ det D f Ri |Ty γ k=0

≤

CZ d( f Ri (x), f Ri (y))ζ . 1 − αζ

(7.28)

This actually gives (Y4 ), which together with (Y3 ) implies (Y4 ). Finally, we obtain the regularity of the stable holonomy (Y5 ), through Theorem 4.20 and Corollary 4.21. We will use Lemmas 4.22 and 4.23 to check conditions (Y0 ) and (Y4 ) of Sect. 4.5. Note that  is contained in the compact invariant set  ⊂ M with a partially hyperbolic splitting T M = E s ⊕ E cu . Since f is a C 1+α diffeomorphism, the fiber bundles E s and E cu are Hölder continuous on ; see [10]. Recalling that the unstable disks are contained in , the Hölder continuity of the E cu direction and the uniform contraction on stable disks enable us to apply Lemma 4.22, thus obtaining (Y0 ). Property (Y4 ) is a consequence of (7.27) and (7.28), together with Lemma 4.23. Applying Theorem 4.20 and Corollary 4.21, we get (Y5 ). Hence, the set  has a full Young structure with recurrence time R. 

7.3.3 Recurrence Times In this subsection, we obtain the integrability of the recurrence time R associated with the Young structure obtained in Sect. 7.3.2, given by the families of disks  s and  u . We will use Proposition 5.12 for this purpose. Consider 0 ∈  u as in (7.20). By Proposition 7.16, there are sets S1 , S2 , · · · ⊂ 0 with 

m 0 (Sn ) < ∞

n≥1

such that, for all n ≥ 1,

Hn ∩ {R > L + n} ⊂ Sn .

Note that, in this case, we have 0 ∩  = 0 . Consider the quotient map F : 0 → 0 as in (4.5). By Proposition 4.2, we have that F is a Gibbs-Markov map. In order to apply Proposition 5.12, we just need to obtain a frequent sequence (Hn∗ )n of sets in 0 that is F-concatenated in (Hn )n ; recall (5.37) and (5.38). Set for each n ≥ 1

246

7 Partially Hyperbolic Attractors

Hn∗ = {x ∈ 0 : n is a σ -hyperbolic time for x} .

(7.29)

It follows from (7.16) that, for m 0 almost every x ∈ 0 , we have  1  lim sup # 1 ≤ j ≤ n : x ∈ H j∗ ≥ θ. n→∞ n Thus, (Hn∗ )n is a frequent sequence of sets in 0 . On the other hand, as long as we take ε0 < δ1 , it follows from (7.11) and (7.18) that x ∈ Hn∗ , y ∈ γεs0 ( f j (x)) =⇒ y ∈ Hn− j , for all 0 ≤ j < n;

(7.30)

recall that Hn has beed defined as the set of points x such that n is a σ 3/4 -hyperbolic time for x. Now, take R0 = 0 and, for each i ≥ 1, Ri =

i−1 

R ◦ Fk.

k=0

Given x ∈ Hn∗ , let i ≥ 0 be such that Ri (x) ≤ n < Ri+1 (x). We have F i (x) = 0 ◦ f Ri (x) (x), where 0 is the projection from C0 to 0 along stable disks. Therefore, F i (x) ∈ γεs0 ( f Ri (x) (x)). It follows from (7.30) that F i (x) ∈ Hn−Ri . This shows that (Hn∗ )n is F-concatenated in (Hn )n . The integrability of R follows from Proposition 5.12.

7.4 SRB Measures In Theorem 7.22 below, we prove that each elementary set obtained in Theorem 7.9 supports a unique SRB measure. Corollary 7.23 can be interpreted as a measure theoretical version of the topological decomposition given by Theorem 7.9, with the attractors replaced by SRB measures. Theorem 7.22 Let f : M → M be a C 1+η diffeomorphism and  ⊂ M an elementary set with a partially hyperbolic splitting T M = E s ⊕ E cu . If  contains some unstable disk  such that f is nonuniformly expanding along the E cu direction on a subset of  with positive m  measure, then f has some ergodic SRB measure μ whose support coincides with . Moreover, the basin of μ contains m γ almost all points in some unstable disk γ ⊂ .

7.4 SRB Measures

247

Proof Let 0 ⊂  be a set of points with m  (0 ) > 0 such that f is nonuniformly expanding along E cu on the set 0 . Considering H the union of the stable disks through the points in 0 , we have m(H ) > 0. Here, we need to use the absolute continuity of the stable holonomy, which holds in general for partially hyperbolic sets; see [11]. Applying Theorem 7.9 to H , we get elementary sets 1 , . . . ,  ⊂ M such that, for m almost every x ∈ H , there is some 1 ≤ j ≤ for which ω(x) =  j . Since  is a compact set with f | transitive and H is a union of stables disks through points in , then the sets 1 , . . . ,  are necessarily contained in . Fix some 1 ≤ i ≤ . Theorem 7.9 also gives that f is nonuniformly expanding along the E cu direction, for m i almost all points in some disk i ⊂ i . It follows from Theorem 7.14 that f has a set  with a full Young structure and integrable recurrence times. Consider the return map f R :  →  associated with this structure. Let ν0 be the unique SRB measure for f R and ν=

∞ 

f ∗j (ν0 |{R > j}).

(7.31)

j=0

Since the Young structure has integrable recurrence times, then ν is finite, by Theorem 4.9. It also follows from Theorem 4.9 that μ = ν/ν(M) is an ergodic SRB measure for f . We are going to see that μ satisfies the desired conclusions. By Proposition 4.10, the support of μ coincides with . We are left to check that the basin of μ contains m γ almost all points in some unstable disk γ ⊂ . By Proposition 2.12, μ almost all of  is contained in the basin of μ. In particular, μ almost all of  is contained in the basin of μ. Furthermore, by Theorem 4.7, the conditional measures of ν0 on the unstable disks of  u are almost all equivalent to the conditional measures of m. Since μ = ν/ν(M), it follows from (7.31) that the conditional measures of μ on the unstable disks of  u are almost all equivalent to the conditional measures of m. This implies that there exists some disk γ ∈  u such that m γ almost all of γ is contained in the basin of μ. In fact, this happens for all disks in  u , by the absolute continuity of the stable holonomy and the fact that all points in the stable disk of a point belonging in the basin of a measure still belong in that basin.  From Theorem 7.22, we deduce the following measure theoretical version of the topological decomposition given by Theorem 7.9, with the attractors replaced by SRB measures. Corollary 7.23 Let f : M → M be a C 1+η diffeomorphism and K ⊂ M be a compact set with f (K ) ⊂ K and a partially hyperbolic splitting TK M = E s ⊕ E cu . If f is nonuniformly expanding along the E cu direction on a set H ⊂ K with m(H ) > 0, then f has ergodic SRB measures μ1 , . . . , μ such that, for m almost every x ∈ H , there is 1 ≤ j ≤ for which x belongs in the basin of μ j and ω(x) coincides with the support of μ j . Moreover, the basin of μ j contains m γ j almost all points of some unstable disk γ j contained in the support of μ j . Proof By Theorem 7.9, there are elementary sets 1 , . . . ,  ⊂ M such that, for m almost every x ∈ H , there is 1 ≤ j ≤ such that ω(x) =  j . Setting

248

7 Partially Hyperbolic Attractors

H j = {x ∈ H : ω(x) =  j }, for all 1 ≤ j ≤ , we have H=



H j , m mod 0.

(7.32)

j=1

It follows from Theorem 7.22 that, for each 1 ≤ j ≤ , there is an ergodic SRB measure μ j whose support coincides with  j . Moreover,  j contains an unstable disk γ j such that m γ j almost all of γ j is contained in the basin of μ j . Let B j be the union of the stable disks through the points in γ j , for each 1 ≤ j ≤ . Note that B j intersects  j in a set with nonempty interior. Since ω(x) =  j for every x ∈ H j , it follows that ∞ Hj = f −n (B j ), n=0

for all 1 ≤ j ≤ . Together with (7.32), this gives that, for m almost every x ∈ H , there is 1 ≤ j ≤ for which x belongs in the basin of μ j and ω(x) coincides with  j .  Since the support of μ j is equal to  j , the proof is completed. Corollary 7.24 Let f : M → M be a C 1+η diffeomorphism for which M has a partially hyperbolic splitting T M = E s ⊕ E cu for which f is nonuniformly expanding along the E cu direction on m almost all of M. If f is transitive, then f has a unique ergodic SRB measure whose support coincides with M and its basin covers m almost all of M. Proof Using Corollary 7.23 with K = M, we obtain ergodic SRB measures μ1 , . . . , μ whose basins cover m almost all of M. Moreover, the basin of each μ j contains m γ j almost all points of some unstable disk γ j . Considering the union of the stable disks through the points in each γ j we obtain a set with nonempty interior such that almost all of its points belong in the basin of μ j . Using the fact that f is a transitive diffeomorphism, we easily see that the basins of any two of these ergodic SRB measures necessarily have nonempty intersection, and therefore the measures all coincide.  From Theorem 7.22, we can also derive the classical result on the existence and uniqueness of SRB measures supported on transitive hyperbolic attractors. In particular, Axiom A attractors. Let f : M → M be a C 1+η diffeomorphism of a Riemannian manifold M. It is known that, for a transitive compact hyperbolic set  ⊂ M, the following conditions are equivalent: • there is a positive Lebesgue measure set of points whose ω-limit is equal to ; • the ω-limit of all points in a neighbourhood of  coincides with ; • there is some local unstable disk contained in .

7.4 SRB Measures

249

See, for example, [4] for the not so well-known fact that the first condition implies the second one. A set  with the properties above will be called a hyperbolic attractor. Note that if f is only C 1 , then the conditions above are not necessarily equivalent; see [9] for a C 1 diffeomorphism having a horseshoe with positive Lebesgue measure. Corollary 7.25 Let f : M → M be a C 1+η diffeomorphism and  ⊂ M a transitive compact hyperbolic attractor. Then, f has an ergodic SRB measure whose support coincides with  and its basin covers m almost all of a neighbourhood of . Proof Since  contains some unstable disk where f is necessarily uniformly expanding along the unstable direction, it follows from Theorem 7.22 that f has some ergodic SRB measure μ whose support coincides with . Moreover, the basin of μ contains m γ almost all points in some unstable disk γ ⊂ . Considering the union of the stable disks through the points in γ we obtain a set with nonempty interior such that m almost all of its points belongs in the basin of μ. From the transitivity of f , the invariance of the basin of μ and the fact that f is a diffeomorphism we easily see that any point in  has a neighborhood such that m almost all of its points are contained in the basin of μ. Considering the union of all these neighbourhoods we finish the proof. 

7.5 Decay of Correlations Here, we obtain rates for the decay of correlations with respect to the SRB measure obtained in Theorem 7.22. Similarly to Sect. 6.5, we need a condition of nonuniform expansion along the E cu direction stronger than that in (7.9). Let f : M → M be a C 1+η diffeomorphism and K ⊂ M with a partially hyperbolic splitting TK M = E s ⊕ E cu . We say that f is strongly nonuniformly expanding along the E cu direction direction on a set H ⊂ K if there is c > 0 such that lim sup n→+∞

n 1 log D f −1 |E cu f j (x)  < −c, for all x ∈ H. n j=1

(7.33)

Clearly, (7.33) implies (7.9). Fixing c > 0 as in (7.33), we may define for each x ∈ H 

n−1 1 h H (x) = min N ≥ 1 : log D f −1 |E cu f j (x)  < −c, ∀n ≥ N n i=0

 .

(7.34)

Note h H depends on the choice of c > 0, but we will not explicit this dependency in the notation, for the sake of simplicity. In the proof of the next result we show that h H is related to a function h θ as in (5.2), for 0 < θ ≤ 1 given by Lemma 7.5. In fact, the strong nonuniform expansion introduced above can be interpreted as a practical way of finding a function h θ as in (5.2).

250

7 Partially Hyperbolic Attractors

Proposition 7.26 Let f : M → M be a C 1+η diffeomorphism and  ⊂ M an elementary set with a partially hyperbolic splitting T M = E s ⊕ E cu . If there is an unstable disk  ⊂  such that f is nonuniformly expanding along the E cu direction, for m  almost all points in , then there exist L > 0 and a set with a full Young structure containing an unstable disk 0 ⊂  such that {R > n + L} ⊂ {h H > n} ∪ E n , for some sequence (E n )n of sets in 0 with m 0 (E n ) → 0, exponentially fast with n. Proof The Young structure has already been obtained in Theorem 7.14, under the weaker assumption of nonuniform expansion along E cu in (7.9). Note that, for defining h H in (7.34), we have fixed c > 0 as in (7.33). Consider 0 < θ ≤ 1 associated with c given by Lemma 7.5. It easily follows from the definition of h H that, for all x ∈ H and n ≥ h H (x), there are σ -hyperbolic times 1 ≤ n 1 < · · · < n ≤ n for x, with σ = e−c/2 and ≥ θ n. Set Hn = {x ∈ H : n is a σ -hyperbolic time for x} , for each n ≥ 1. Clearly, n ≥ h H (x) =⇒

 1  # 1 ≤ j ≤ n : x ∈ H j ≥ θ. n

This shows that a function h θ as in (5.2) can be defined and, moreover, h θ ≤ h H . By the third item of Theorem 5.1, we have {R > n + L} ⊂ {h θ > n} ∪ E n , for a sequence (E n )n of sets in 0 such that m 0 (E n ) decays exponentially fast with n.  Since {h θ > n} ⊂ {h H > n}, we finish the proof. Under the assumptions of the next result, Theorem 7.22 provides a unique SRB measure whose support coincides with . Actually, Theorem 7.22 is proved under the weaker condition (7.9) of nonuniform expansion along the E cu direction. Theorem 7.27 Let f : M → M be a C 1+η diffeomorphism and  ⊂ M an elementary set with a partially hyperbolic splitting T M = E s ⊕ E cu for which there is an unstable disk  ⊂  such that f is strongly nonuniformly expanding along the E cu direction on a set H ⊂  with total m  measure. If μ is the unique ergodic SRB measure for f whose support coincides with , then there are 1 ≤ p ≤ q and exact SRB measures μ1 , . . . , μ p for f q with f ∗ μ1 = μ2 ,…, f ∗ μ p = μ1 and μ = (μ1 + · · · + μ p )/ p such that, for all 1 ≤ i ≤ p, 1. if m  {h H > n} ≤ Cn −a for some C > 0 and a > 1, then for all ϕ, ψ ∈ Hη there is C  > 0 such that Cor μi (ϕ, ψ ◦ f qn ) ≤ C  n −a+1 ; 2. if m  {h H > n} ≤ Ce−bn for some C, c > 0 and 0 < a ≤ 1, then given η > 0, there is c > 0 such that, for all ϕ, ψ ∈ Hη , there is C  > 0 for which a

7.5 Decay of Correlations

251  a

Cor μi (ϕ, ψ ◦ f qn ) ≤ C  e−c n . Proof By Proposition 7.26, there exist L > 0 and a full Young structure containing an unstable disk 0 ⊂  whose recurrence times satisfy {R > n + L} ⊂ {h H > n} ∪ E n ,

(7.35)

where (E n )n is a sequence of sets in 0 such that m 0 (E n ) converges to 0 exponentially fast with n. Since the decay rates of m 0 {h H > n} under consideration are not faster than exponential, it follows from (7.35) that m 0 {R > n} decays at least at the same speed of m 0 {h H > n}. Taking q = gcd(R) and applying Theorem 4.15, we get the conclusions.  Recall that Corollary 7.24 gives a unique ergodic SRB measure for a diffeomorphism in the conditions of the corollary below. In fact, under a weaker assumption of transitivity. Assuming that the system is topologically mixing, we get the following consequence of Theorem 7.27 on the decay of correlations. Corollary 7.28 Let f : M → M be a topologically mixing C 1+η diffeomorphism with a partially hyperbolic splitting T M = E s ⊕ E cu for which there is an unstable disk  such that f is strongly nonuniformly expanding along the E cu direction on a set H ⊂  with full m  measure. If μ is the unique ergodic SRB measure for f , then 1. if m  {h H > n} ≤ Cn −a for some C > 0 and a > 1, then for all ϕ, ψ ∈ Hη there is C  > 0 such that Cor μ (ϕ, ψ ◦ f n ) ≤ C  n −a+1 ; 2. if m  {h H > n} ≤ Ce−bn for some C, c > 0 and 0 < a ≤ 1, then given η > 0, there is c > 0 such that, for all ϕ, ψ ∈ Hη , there is C  > 0 for which a

 a

Cor μ (ϕ, ψ ◦ f n ) ≤ C  e−c n . Proof By Theorem 7.27, there are 1 ≤ p ≤ q and exact SRB measures μ1 , . . . , μ p for f q with f ∗ μ1 = μ2 ,…, f ∗ μ p = μ1 and μ = (μ1 + · · · + μ p )/ p. Note that f topologically mixing implies f q topologically mixing, and therefore transitive, by (2.15). It follows from Corollary 7.24 that the measures μ1 , . . . , μ p are all equal, and therefore all equal to μ. The expected conclusions for Cor μ (ϕ, ψ ◦ f qn ) then follow from Theorem 7.27. Applying those conclusions to the observables  ψ ◦ f, · · · , ψ ◦ f q−1 in place of ψ, we finish the proof. In the next corollary, we obtain the classical result on exponential decay of correlations with respect to the unique SRB measure supported on a topologically mixing hyperbolic attractor. Recall that such an SRB measure was obtained in Corollary 7.24 under the weaker assumption of transitivity.

252

7 Partially Hyperbolic Attractors

Corollary 7.29 Let f : M → M be a C 1+η diffeomorphism and  ⊂ M a topologically mixing compact hyperbolic attractor. If μ is the unique ergodic SRB measure for f whose support coincides with , then there is c > 0 such that, for all ϕ, ψ ∈ Hη , there is C  > 0 such that Cor μ (ϕ, ψ ◦ f n ) ≤ Ce−cn . Proof Take any unstable disk  ⊂ M and H = . Set h H (x) = 1, for all x ∈ H . Using Corollary 7.28, we get the conclusion.

7.6 Application: Derived from Anosov Using ideas similar to those in Sect. 6.6.1, we can also construct robust classes of partially hyperbolic diffeomorphisms whose centre-unstable direction is nonuniformly expanding. Consider a decomposition T M = E 1 ⊕ E 2 and, for some a > 0, families of cone fields (Cacu (x))x∈M and (Cas (x))x∈M   Cacu (x) = v1 + v2 ∈ E x1 ⊕ E x2 : v1  ≤ av2  and

  Cas (x) = v1 + v2 ∈ E x1 ⊕ E x2 : v2  ≤ av1  .

Though there is still no dynamics involved, the notation indicates that these cones will be associated with centre-unstable and stable directions. As in Sect. 7.1.1, a disk in M whose tangent space at all its points is contained in the centre-unstable cone is called a cu-disk and a disk whose tangent space at all its points is contained in the stable cone is called a stable disk. Consider a cover of M by small Borel sets B0 , B1 , . . . , B p for which there is some constant C0 > 0 such that, for any cu-disk γ ⊂ M, (7.36) m γ (γ ∩ Bi ) ≤ C0 , for all 0 ≤ i ≤ p. Given constants 0 < σ0 < 1 < σ1 and δ > 0 (to be specified in (7.38) below, depending on σ0 , σ1 and p), let A be a set of C 2 diffeomorphisms f : M → M such that, for all f ∈ A, (A0 ) D f (x)Cacu (x) ⊂ Cσcu0 a ( f (x)) and D f −1 (x)Cas (x) ⊂ Cσs 0 a ( f −1 (x)), for all x ∈ M. (A1 ) D f |Tx γ s  ≤ σ0 , for any stable disk γ s and x ∈ γ s ; (A2 ) | det(D f |Tx γ cu )| ≥ σ1 , for any cu-disk γ cu and x ∈ γ cu ; (A3 ) (D f |Tx γ cu )−1  ≤ 1 + δ, for any cu-disk γ cu and x ∈ γ cu ∩ B0 ; (A4 ) (D f |Tx γ cu )−1  ≤ σ0 , for any cu-disk γ cu and x ∈ γ cu \ B0 ; (A5 ) f is topologically mixing. At the end of this subsection, we explain how to obtain open sets in the C 1 topology of C 2 maps which satisfy (A1 )–(A5 ) and are not uniformly hyperbolic. In Proposition 6.33 below, we show that, for a convenient choice of the constants, for

7.6 Application: Derived from Anosov

253

each diffeomorphism in A, there is a partially hyperbolic splitting the manifold M with strong nonuniform expansion along the centre-unstable direction. Using Proposition 6.33, we will be able to prove the next theorem, which is actually the main result of this section. Theorem 7.30 Every diffeomorphism f ∈ A has a unique ergodic SRB measure μ whose support coincides with M and its basin covers m almost all of M. Moreover, given any η > 0, there exists c > 0 such that, for all ϕ, ψ ∈ Hη , there exists C > 0 for which Cor μ (ϕ, ψ ◦ f n ) ≤ Ce−cn . First, we explain how to choose the constant δ > 0 in (A3 ). As in (6.33) and (6.34), given n ≥ 1 and i = (i 0 , . . . , i n−1 ) ∈ {0, . . . , p}n , set G n (i) = #{0 ≤ j < n : i j ≥ 1}. and, for 0 < θ < 1,   In (θ ) = i = (i 0 , . . . , i n−1 ) ∈ {0, . . . , p}n : G n (i) < θ n . Then, take θ > 0 sufficiently small such that the conclusion of Corollary 6.31 holds for some σ > 1 with (7.37) σ σ1−(1−θ) < 1. Finally, choose δ > 0 small enough so that σ0θ (1 + δ) < 1.

(7.38)

Note that the factor σ1−(1−θ) in (7.37) converges to σ1−1 < 1 when θ goes to 0. Therefore, it is possible to choose σ > 0 and θ > 0 in those conditions. Let us now prove Theorem 7.30. Lemma 7.31 There are C, c > 0 such that, for every cu-disk γ ⊂ M and n ≥ 1, there is a set E n ⊂ γ such that m γ (E n ) ≤ Ce−cn such that, for all x ∈ γ \ E n ,  1  # 0 ≤ j < n : f j (x) ∈ B1 ∪ · · · ∪ B p ≥ θ. n Proof Set for each i = (i 0 , . . . , i n−1 ) ∈ {0, . . . , p}n   [i] = x ∈ γ : f j (x) ∈ Bi j , for all 0 ≤ j < n and En =



[i].

i∈In (θ)

254

7 Partially Hyperbolic Attractors

It follows from (A1 ), (A2 ) and (7.36) that, for all i ∈ In (θ ), m γ ([i]) ≤ σ1−(1−θ)n m f n−1 (γ ) ( f n−1 (γ ) ∩ Bin−1 ) ≤ C0 σ1−(1−θ)n .

(7.39)

Recalling that σ1 > 1, choose σ > 1 and θ > 0 small enough so that σ σ1−(1−θ) < 1. From Corollary 6.31 and (7.39), we get n  m γ (E n ) ≤ #In (θ )C0 σ1−(1−θ)n ≤ C0 C σ σ1−(1−θ) . 

This finishes the proof.

In the next result, we use the function h H associated with a set H on which a map is strongly nonuniformly expanding along the centre-unstable direction; see (7.34) for its definition. Proposition 7.32 For every f ∈ A, there exists a partially hyperbolic splitting T M = E s ⊕ E cu . Moreover, for every cu-disk γ ⊂ M there is a set H ⊂ γ with full m γ measure on γ such that f is strongly nonuniformly expanding along the E cu direction on H and m γ {h H > n} decays exponentially fast to 0 with n. Proof The existence of a partially hyperbolic splitting T M = E s ⊕ E cu with respect to any f ∈ A is a consequence of the existence of the invariant cone fields, together with (A1 ). This is standard for hyperbolic sets and still holds for partially hyperbolic sets; see for example [6, Appendix B]. Consider now acu-disk γ ⊂ M tangent to the E cu direction. By Lemma 7.31, there are C, c > 0 and, for all n ≥ 1, a set E n ⊂ γ with m γ (E n ) ≤ Ce−cn such that, for all x ∈ γ \ E n ,  1  # 0 ≤ j < n : f j (x) ∈ B1 ∪ · · · ∪ B p ≥ θ. n Take any λ > 0 such that σ0θ (1 + δ)1−θ < e−λ ; recall (7.38). It follows from (A3 ) and (A4 ) that, for all n ≥ 1 and x ∈ γ \ E n , n−1

θn (1−θ)n D f −1 |E cu < e−λn . f j (x)  ≤ σ0 (1 + δ)

j=0

Set E=



E n and H = γ \ E.

k≥1 n≥k

By (7.40) that, for all k ≥ 1 and x ∈ γ \

 n≥k

E n , we have

n−1 1 log D f −1 |E cu f j (x)  < −λ, for all n ≥ k. n i=0

(7.40)

7.6 Application: Derived from Anosov

255

This obviously implies that f is strongly nonuniformly expanding and {h H > k} ⊂



En ,

n≥k

for all k ≥ 1. Since m γ (E n ) ≤ Ce−cn , the proof is completed.



Let us now finish the proof of Theorem 7.30. Consider any diffeomorphism f ∈ A. Since the conclusion of Proposition 7.32 is valid for any cu-disk, we easily deduce that f is nonuniformly expanding along the E cu direction on m almost all of M. Moreover, f is topologically mixing, by (A5 ), and therefore transitive, by (2.15). It follows from Corollary 7.24 that f has a unique ergodic SRB measure whose support coincides with M and its basin covers m almost all of M. We are left to obtain the estimate on the decay of correlations. Since f is nonuniformly expanding along the E cu direction on m almost all of M, Theorem 7.9 provides in particular some unstable disk γ ⊂ M. Applying Proposition 7.32 to this unstable disk, we get a set H ⊂ γ with full m γ measure such that m γ {h H > n} decays exponentially fast to 0 with n. Then, Corollary 7.28 gives the desired conclusion on the decay of correlations. Finally, we explain how to obtain open sets in the C 1 topology of C 2 diffeomorphisms which satisfy (A1 )–(A5 ) and are not hyperbolic. Let M be the d-dimensional torus Rd /Zd , for some d ≥ 3, and f 0 be a topologically mixing linear automorphism whose unstable fiber bundle has at least dimension two. Assume that the matrix A associated with f 0 has at least two real eigenvalues bigger than one. Consider families of D f 0 invariant cone fields (Cacu (x))x∈M and (Cas (x))x∈M associated with the hyperbolic structure of M with respect to f 0 . Let B0 be a small neighbourhood of zero. We modify f 0 inside B0 in only a one-dimensional subspace of the unstable direction, keeping it invariant, and making its derivative have an eigenvalue equal to one, thus creating a repelling neutral fixed point in that direction as in Subsection 3.5.1. In this way, we obtain a diffeomorphism f 1 : M → M topologically conjugate to f 0 . In particular, f 1 is topologically mixing. For this map f 1 we have properties (A1 )– (A4 ). In fact, we may even take δ = 0 in (A3 ). For the (A5 ) property, we only sketch the main ideas. First, we start with the Anosov f 0 diffeomorphism having a topologically mixing Markov partition. Then, the construction of the diffeomorphism f above can be performed in such a way that the big perturbation allowing a small contraction in some direction of E cu occurs in the neighbourhood of a saddle within some domain R0 of the Markov partition of f 0 . In addition, f coincides with some Anosov diffeomorphism close to f 0 out of R0 which keeps the property of being topologically mixing. See Example 2 in [13, Sect. 1.2] for details. We should mention that in [13] is considered the dual case of partially hyperbolic diffeomorphism with strong unstable direction. However, as the roles of the unstable and unstable directions are symmetric, we can apply the results in [13] to f −1 .

256

7 Partially Hyperbolic Attractors

References 1. J.F. Alves, C. Bonatti, M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140(2), 351–398 (2000) 2. J.F. Alves, C.L. Dias, S. Luzzatto, V. Pinheiro, SRB measures for partially hyperbolic systems whose central direction is weakly expanding. J. Eur. Math. Soc. (JEMS) 19(10), 2911–2946 (2017) 3. J.F. Alves, X. Li, Gibbs-Markov-Young structures with (stretched) exponential tail for partially hyperbolic attractors. Adv. Math. 279, 405–437 (2015) 4. J.F. Alves, V. Pinheiro, Topological structure of (partially) hyperbolic sets with positive volume. Trans. Amer. Math. Soc. 360(10), 5551–5569 (2008) 5. J.F. Alves, V. Pinheiro, Gibbs-Markov structures and limit laws for partially hyperbolic attractors with mostly expanding central direction. Adv. Math. 223(5), 1706–1730 (2010) 6. C. Bonatti, L.J. Díaz, M. Viana, Dynamics beyond uniform hyperbolicity, volume 102 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, A global geometric and probabilistic perspective (Mathematical Physics, III, 2005) 7. C. Bonatti, M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115, 157–193 (2000) 8. R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, vol. 470, Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1975) 9. R. Bowen, A horseshoe with positive measure. Invent. Math. 29(3), 203–204 (1975) 10. M. Brin, Hölder continuity of invariant distributions, in A. Katok, R. de la Llave, Y. Pesin, H. Weiss (eds.), Smooth Ergodic Theory and Its Applications, volume 69 of Proceedings of Symposia in Pure Mathematics (American Mathematical Society, 2001) 11. M.I. Brin, J.B. Pesin, Partially hyperbolic dynamical systems. Izv. Akad. Nauk SSSR Ser. Mat. 38, 170–212 (1974) 12. M. Carvalho, Sinai-Ruelle-Bowen measures for N -dimensional derived from Anosov diffeomorphisms. Ergodic Theory Dynam. Syst. 13(1), 21–44 (1993) 13. A. Castro, Backward inducing and exponential decay of correlations for partially hyperbolic attractors. Israel J. Math. 130, 29–75 (2002) 14. A. Castro, Fast mixing for attractors with a mostly contracting central direction. Ergodic Theory Dynam. Syst. 24(1), 17–44 (2004) 15. D. Dolgopyat, On dynamics of mostly contracting diffeomorphisms. Comm. Math. Phys. 213(1), 181–201 (2000) 16. S. Gouëzel, Decay of correlations for nonuniformly expanding systems. Bull. Soc. Math. France 134(1), 1–31 (2006) 17. R. Mañé, Ergodic theory and differentiable dynamics, vol. 8, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] (Springer-Verlag, Berlin, 1987) 18. Y. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory. Uspehi Mat. Nauk, 32(4 (196)):55–112, 287 (1977) 19. Y.B. Pesin, Y.G. Sinai, Gibbs measures for partially hyperbolic attractors. Ergodic Theory Dynam. Systems, 2(3-4), 417–438 (1983), 1982 20. V. Pinheiro, Expanding measures. Ann. Inst. H. Poincaré Anal. Non Linéaire 28(6), 889–939 (2011) 21. M. Shub, Global stability of dynamical systems (Springer-Verlag, New York, 1987). With the collaboration of Albert Fathi and Rémi Langevin, Translated from the French by Joseph Christy

Index

A Absolutely continuous foliation, 4, 105 ≈, 84 Attractor, 1

B Basin, 1, 18 Basis, 13 Bounded dependence, 48 lower, 48

C Centre-stable cone field, 226 direction, 226 Centre-unstable cone field, 226 direction, 226 disk, 232 Characteristic function, 30 Core, 182 Correlation function, 5 Crosses, 239 cu-disk, 227, 252 Cylinder, 239

D Discretisation, 126 Disintegration, 4 Dominated splitting, 226 Doubling map, 25

E Elementary set, 199, 233 Ergodic component, 199, 234 Extension, 48

F F-concatenated, 175 f ∗ m  m, 19 Frequent sequence, 175 Full branches, 24

G Gibbs, 24, 105 Gibbs-Markov map, 24 weak, 23

H Hölder continuous, 80 tangent bundle, 227 Holonomy map, 104 Hyperbolic attractor, 249 preball, 196 predisk, 231 time, 192, 229

I Indicator function, 30 Induced map, 38 expanding, 80 open, 83 Invariant set, 199 Invertibility domain, 17

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. F. Alves, Nonuniformly Hyperbolic Attractors, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-62814-7_1

257

258 J Jacobian, 17

L , 84 Locally eventually onto, 20 Long branches, 24 Lyapunov exponent, 3

M Markov, 23, 104 1 A , 30 Measure conditional, 4 ergodic, 2, 32 exact, 32 invariant, 1 mixing, 5, 32 physical, 1, 21 push-forward, 13 quotient, 4 Sinai-Ruelle-Bowen (SRB), 4, 84, 189 space, 9 support, 21

N Nondegenerate set, 190 Nonsingular, 24 Nonuniformly expanding, 190 along E cu , 229 strongly, 208 strongly along E cu , 249

O ω-limit, 198 P Palis Conjecture, 1 Partially hyperbolic set, 151, 226 Partition, 9 diameter, 14 generating, 10, 13 increasing sequence, 9 measurable, 15 mod 0, 9 refinement, 9 subordinate, 4 ≺, 9 Product structure, 104

Index full, 104

Q Quotient, 48 map, 106

R Rank, 184 Recurrence time, 38, 105 integrable, 105 Regular set, 3 Relatively compact, 30 Return map, 105 Return time simultaneous, 52

S Semiconjugacy, 46 Separability, 23 Separable measure space, 14 Separation time, 24, 56, 105 Slow recurrence, 191 s-saturated, 234 s-subset, 104 Stable disk, 3, 104, 232, 252 manifold, 4 set, 234 Stable holonomy regularity, 105 Stirling’s formula, 213 Strong-stable direction, 226 Strong-unstable direction, 226

T Topologically mixing, 20 Tower, 41, 117 base, 41 level, 41 map, 41, 117 roof, 41 Transitive map, 19 Trapping region, 147 , 10 Truncated distance, 191

Index U Unshrinkable, 200 cu-unshrinkable, 235 Unstable disk, 3, 104 manifold, 4 u-subset, 104

V Viana

259 Conjecture, 191 map, 217

X ξi -measurable, 53 Y Young structure, 104 full, 104