Ergodic Theory, Hyperbolic Dynamics and Dimension Theory

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Author: Luis Barreira

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Universitext

Universitext Series Editors: Sheldon Axler San Francisco State University Vincenzo Capasso Università degli Studi di Milano Carles Casacuberta Universitat de Barcelona Angus J. MacIntyre Queen Mary, University of London Kenneth Ribet University of California, Berkeley Claude Sabbah ´ CNRS, Ecole Polytechnique Endre Süli University of Oxford Wojbor A. Woyczynski Case Western Reserve University

Universitext is a series of textbooks that presents material from a wide variety of mathematical disciplines at master’s level and beyond. The books, often well class-tested by their author, may have an informal, personal even experimental approach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teaching curricula, to very polished texts. Thus as research topics trickle down into graduate-level teaching, first textbooks written for new, cutting-edge courses may make their way into Universitext.

For further volumes: www.springer.com/series/223

Luis Barreira

Ergodic Theory, Hyperbolic Dynamics and Dimension Theory

123

Luis Barreira Departamento de Matemática Instituto Superior Técnico Lisboa Portugal

ISBN 978-3-642-28089-4 ISBN 978-3-642-28090-0 (eBook) DOI 10.1007/978-3-642-28090-0 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2012937287 Mathematics Subject Classification (2010): 37AXX, 37DXX, 37C45 c Springer-Verlag Berlin Heidelberg 2012 This work is subject to copyright. All rights are reserved 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. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. 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. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To Claudia

•

Preface

This book is an introduction to the interplay of three main areas of research: ergodic theory, hyperbolic dynamics, and dimension theory of dynamical systems. This includes an introduction to the thermodynamic formalism, which is an important tool in dimension theory. My main aim was to provide in a single volume a rigorous self-contained introduction to dimension theory of hyperbolic dynamics, including sufficiently high-level introductions to ergodic theory and the thermodynamic formalism. This caused that several topics of ergodic theory and hyperbolic dynamics had to be excluded, essentially to keep the size of the book under control. However, it should be emphasized that the same happened to several topics of dimension theory while making an effort to reach a good compromise between the various areas. On the other hand, it was necessary to include topics of ergodic theory and hyperbolic dynamics that are often absent in introductory texts, such as the construction of Markov partitions for repellers and an introduction to the thermodynamic formalism. The book is directed primarily to graduate students interested in dynamical systems, as well as researchers in other areas who wish to learn ergodic theory or dimension theory of hyperbolic dynamics at an intermediate level and in a sufficiently detailed manner. In particular, the text can be used as a basis for a graduate course on ergodic theory and the thermodynamic formalism (using Chaps. 2–5) and for a graduate course on dimension theory of hyperbolic dynamics (using Chaps. 6–9, eventually referring to Chaps. 2–5 for any prerequisites). The book can also be used for independent study: it is self-contained, and with the exception of some basic well-known statements and results from other areas, all statements are included with detailed proofs. Moreover, each chapter can essentially be read independently. I only assume some familiarity with basic material from measure theory and integration theory, which anyway is recalled in Appendix A. The material is divided into four parts: * Part I is dedicated to the foundations of ergodic theory and its interplay with symbolic dynamics and topological dynamics. In Chap. 2, we introduce the basic vii

viii

Preface

notions and results of ergodic theory, including Poincaré’s recurrence theorem and Birkhoff’s ergodic theorem. We also give a large number of examples. In Chap. 3, we discuss additional topics of ergodic theory, including the existence of invariant measures for a continuous transformation of a compact metric space. * Part II is an introduction to entropy theory and the thermodynamic formalism. In Chap. 4, we introduce the notions of metric entropy and topological entropy. We also establish the Shannon–McMillan–Breiman theorem and the variational principle for the topological entropy. Chapter 5 is an introduction to the thermodynamic formalism. In particular, we establish the variational principle for the topological pressure. * Part III is dedicated to hyperbolic dynamics. In Chap. 6, we start by discussing some basic properties of hyperbolic sets. In addition, we discuss in detail some properties of the Smale horseshoe and of the hyperbolic automorphisms of T2 . We also construct Markov partitions for any repeller. In Chap. 7, after establishing the existence of stable and unstable manifolds, we use the shadowing property to construct Markov partitions for any locally maximal hyperbolic set. * Finally, Part IV is an introduction to dimension theory of hyperbolic dynamics. In Chap. 8 we give an introduction to the basic notions of dimension theory. In particular, we introduce the notions of Hausdorff dimension and box dimension. We also show how pointwise dimension can be used to estimate the dimension of a measure. In Chap. 9, we study the dimension of repellers and hyperbolic sets for conformal transformations using Markov partitions. In addition, the book contains more than 150 exercises of variable level of difficulty. There are no words that can adequately express my gratitude to Claudia Valls for her help, patience, encouragement, and inspiration, without which it would be impossible for this book to exist. Lisbon, Portugal December 2011

Lu´ıs Barreira

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Dynamical Systems and Hyperbolicity .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Ergodic Theory and Nontrivial Recurrence . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Hyperbolicity and Dimension Theory .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Geometric Constructions and Limit Sets . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 Classical Thermodynamic Formalism .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.6 Dimension Theory in Hyperbolic Dynamics .. . . . .. . . . . . . . . . . . . . . . . . . . 1.7 Multifractal Analysis and Irregular Sets. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.8 Quantitative Recurrence and Dimension . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

Part I

1 1 2 3 5 6 8 12 15

Ergodic Theory

2 Basic Notions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Invariant Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.1 The Notion of Invariant Measure .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.2 Rotations and Translations of Rn . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.3 Circle Rotations and Interval Translations .. . . . . . . . . . . . . . . . . . . 2.2.4 Expanding Maps of the Circle . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.5 Toral Endomorphisms.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.6 Piecewise Linear Expanding Maps . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Poincaré’s Recurrence Theorem . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Invariant Sets and Invariant Functions.. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Birkhoff’s Ergodic Theorem .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5.1 Formulation of the Theorem .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5.2 Conditional Expectation . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5.3 Proof of Birkhoff’s Ergodic Theorem .. . . .. . . . . . . . . . . . . . . . . . . . 2.6 Ergodicity .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6.1 Basic Notions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6.2 Fourier Coefficients and Ergodicity . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6.3 The Case of Invariant Measures . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

19 19 21 21 23 24 27 28 31 33 36 37 37 39 40 43 43 45 48 ix

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2.7

Applications to Number Theory . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7.1 Fractional Parts of Polynomials . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7.2 Continued Fractions .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Exercises .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

48 48 51 58

3 Further Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Existence of Invariant Measures .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.1 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.2 Existence of Invariant Measures. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Unique Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.1 Basic Notions and Uniform Convergence.. . . . . . . . . . . . . . . . . . . . 3.3.2 Criteria for Unique Ergodicity and Examples . . . . . . . . . . . . . . . . 3.4 Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5 Symbolic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.1 Basic Notions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.2 Markov Measures and Bernoulli Measures . . . . . . . . . . . . . . . . . . . 3.5.3 Topological Markov Chains and Markov Measures .. . . . . . . . . 3.5.4 The Case of Two-Sided Sequences .. . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6 Topological Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.7 Exercises .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

65 65 66 67 71 72 72 74 80 84 85 87 91 92 94 96

2.8

Part II

Entropy and Pressure

4 Metric Entropy and Topological Entropy .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Metric Entropy.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.1 The Notion of Metric Entropy .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.2 Conditional Entropy.. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.3 Generators and Examples .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Shannon–McMillan–Breiman Theorem .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Topological Entropy .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.1 Basic Notions and Some Properties . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.2 Topological Nature of the Entropy . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 Variational Principle .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.6 Exercises .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

107 107 109 109 115 121 125 132 132 137 138 143

5 Thermodynamic Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Topological Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 Symbolic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4 Variational Principle .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.5 Equilibrium Measures .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.6 Exercises .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

147 147 148 149 153 158 164

Contents

Part III

xi

Hyperbolic Dynamics

6 Basic Notions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Hyperbolic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.1 The Notion of Hyperbolicity . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.2 Some Properties of Hyperbolic Sets. . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Smale Horseshoe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.1 Construction of the Horseshoe . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.2 Symbolic Dynamics .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 Hyperbolic Automorphisms of T2 . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4.1 Construction of a Coding Map . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4.2 Induction of a Markov Measure . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.5 The Case of Repellers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6 Exercises .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

171 171 173 173 176 178 178 182 185 185 190 193 197

7 Invariant Manifolds and Markov Partitions .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 Stable and Unstable Manifolds . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3 Ergodicity and Hopf’s Argument .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4 Product Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.5 The Shadowing Property .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.6 Construction of Markov Partitions.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.7 Exercises .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

201 201 202 211 215 217 219 228

Part IV

Dimension Theory

8 Basic Notions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Dimension of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2.1 Basic Notions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3 Dimension of Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3.1 Basic Notions and Examples . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3.2 Dimension Estimates via Pointwise Dimension . . . . . . . . . . . . . . 8.4 Exercises .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

235 235 236 236 237 242 242 244 249

9 Dimension Theory of Hyperbolic Dynamics . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 Repellers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2.1 Conformal Maps and Examples . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2.2 Dimension of Repellers .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3 Hyperbolic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3.1 Dimension Along the Invariant Manifolds .. . . . . . . . . . . . . . . . . . . 9.3.2 Dimension of Hyperbolic Sets. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4 Exercises .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

253 253 254 255 256 265 265 271 272

xii

Contents

A Notions from Measure Theory.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.1 Measure Spaces.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.2 Outer Measures and Measurable Sets . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.3 Measures in Topological Spaces . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.4 Measurable Functions, Integration, and Convergence . . . . . . . . . . . . . . . . A.5 Absolutely Continuous Measures. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.6 Product Spaces.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

277 277 278 278 279 281 282

References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 283 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 287

Chapter 1

Introduction

This chapter is a first introduction to the rich interplay of ergodic theory, hyperbolic dynamics, and dimension theory—the three main areas considered in the book. The main aim is to describe briefly the topics under consideration and illustrate their importance for the theory of dynamical systems. The exposition is nontechnical but also rigorous.

1.1 Dynamical Systems and Hyperbolicity One of the paradigms of the theory of dynamical systems is that the local instability of trajectories influences the global behavior of the system and opens the way to the existence of stochastic behavior. Mathematically, the instability of trajectories corresponds to some amount of hyperbolicity. Let f W M ! M be a diffeomorphism and let M be a compact f -invariant set. We say that is a hyperbolic set for f if for every point x 2 there exists a decomposition of the tangent space Tx M D E s .x/ ˚ E u .x/ satisfying dx f E s .x/ D E s .f .x//

and dx f E u .x/ D E u .f .x//;

and there exist constants 2 .0; 1/ and c > 0 such that kdx f n jE s .x/k cn

and kdx f n jE u .x/k cn

for every x 2 and n 2 N.

L. Barreira, Ergodic Theory, Hyperbolic Dynamics and Dimension Theory, Universitext, DOI 10.1007/978-3-642-28090-0 1, © Springer-Verlag Berlin Heidelberg 2012

1

2

1 Introduction

A hyperbolic set possesses a very rich structure and in particular families of stable and unstable manifolds. Given x 2 M and " > 0, we consider the sets ˚ V s .x/ D y 2 B.x; "/ W d.f n .y/; f n .x// < " for every n > 0 and

˚ V u .x/ D y 2 B.x; "/ W d.f n .y/; f n .x// < " for every n < 0 ;

where d is the distance on M and B.x; "/ M is an open ball of radius ". Theorem 1.1 (Hadamard–Perron). If is a hyperbolic set for a C 1 diffeomorphism, then there exists " > 0 such that for each x 2 , the sets V s .x/ and V u .x/ are manifolds containing x such that Tx V s .x/ D E s .x/

and Tx V u .x/ D E u .x/:

The manifolds V s .x/ and V u .x/ are called, respectively, stable and unstable manifolds. Under the assumptions of Theorem 1.1, one can also show that the sizes of V s .x/ and V u .x/ are uniformly bounded away from zero, that is, there exists D ."/ > 0 such that V s .x/ B s .x; /

and V u .x/ B u .x; /

for every x 2 , where B s .x; / and B u .x; / are the open balls of radius with respect to the distances induced by d , respectively, on V s .x/ and V u .x/. The continuous dependence of the spaces E s .x/ and E u .x/ in x 2 guarantees that there exists ı D ı."/ > 0 such that if d.x; y/ < ı for some points x; y 2 , then the intersection V s .x/ \ V u .y/ consists exactly of one point. When the whole manifold M is a hyperbolic set for f , we say that f is an Anosov diffeomorphism. This class of diffeomorphisms was introduced and studied by Anosov in [4]. The notion of hyperbolic set was introduced by Smale in his seminal paper [99]. In a certain sense, Anosov diffeomorphisms and more generally diffeomorphisms with a hyperbolic set possess the strongest possible hyperbolicity.

1.2 Ergodic Theory and Nontrivial Recurrence Now we introduce the notion of invariant measure, which is another fundamental departure point for the study of stochastic behavior. Namely, the existence of a finite invariant measure ensures that there is a nontrivial recurrence. Let T W X ! X be a measurable transformation. We say that a measure in X is T -invariant if .T 1 A/ D .A/

1.3 Hyperbolicity and Dimension Theory

3

for every measurable set A X . The study of measure-preserving transformations is the main theme of ergodic theory. In order to describe rigorously the concept of nontrivial recurrence, we recall one of the basic but also fundamental results of ergodic theory—Poincaré’s recurrence theorem. It states that any dynamics preserving a finite measure exhibits a nontrivial recurrence in any set A of positive measure, in the sense that the orbit of almost every point in A returns infinitely often to A. Theorem 1.2 (Poincaré’s recurrence theorem). Let T W X ! X be a measurable transformation and let be a T -invariant finite measure in X . If A X is a measurable set of positive measure, then cardfn 2 N W T n .x/ 2 Ag D 1 for -almost every x 2 A. A modified version of Theorem 1.2 was first established by Poincaré in his seminal memoir on the three-body problem [78]. The simultaneous existence of hyperbolicity and nontrivial recurrence causes a very rich orbit structure (see [11, 43]). Roughly speaking, the nontrivial recurrence implies that there exist orbits returning arbitrarily close to the initial point. On the other hand, the existence of stable and unstable manifolds at these points together with their transversality often causes the existence of transverse homoclinic points, thus leading to (topological) Smale horseshoes and an enormous complexity. In other words, the ergodic theory of hyperbolic dynamics, bringing together hyperbolicity and nontrivial recurrence, is a natural playground for the study of stochastic behavior. Unfortunately, real-life situations are often much more complicated. In particular, the notion of hyperbolicity in Sect. 1.1 can be too stringent for the dynamics, thus leading to the consideration of weaker notions such as nonuniform hyperbolicity (see [10, 11]).

1.3 Hyperbolicity and Dimension Theory We describe briefly in this section the relation between hyperbolicity and dimension. We first introduce some basic notions of dimension theory. Let X be a separable metric space. Given Z X and ˛ 2 R, we define m.Z; ˛/ D lim inf "!0 U

X

.diam U /˛ ;

U 2U

where the infimum is taken over all finite or countable covers of Z by open sets of diameter at most ". The Hausdorff dimension of Z is defined by dimH Z D inff˛ W m.Z; ˛/ D 0g:

4

1 Introduction

The lower and upper box dimensions of Z are defined, respectively, by dimB Z D lim inf "!0

log N.Z; "/ log "

and dimB Z D lim sup "!0

log N.Z; "/ ; log "

where N.Z; "/ denotes the number of balls of radius " needed to cover Z. It is easy to verify that dimH Z dimB Z dimB Z: (1.1) In general, these inequalities may be strict, and the coincidence of the three dimensions is a relatively rare phenomenon (see [8, 29, 73]). Now let be a finite measure in X . The Hausdorff dimension and the lower and upper box dimensions of are defined, respectively, by dimH D lim inffdimH Z W .Z/ .X / ıg; ı!0

dimB D lim inffdimB Z W .Z/ .X / ıg; ı!0

dimB D lim inffdimB Z W .Z/ .X / ıg: ı!0

In general, these quantities need not coincide, respectively, with the Hausdorff dimension and the lower and upper box dimensions of the support of the measure, and thus, they contain additional information about the way in which is distributed on its support. It follows from (1.1) that dimH dimB dimB : As in (1.1), in general, these inequalities may be strict. The following criterion for equality was established by Young in [105]: if is a finite measure in X and lim

r!0

log .B.x; r// Dd log r

(1.2)

for -almost every x 2 X , then dimH D dimB D dimB D d: The limit in (1.2), when it exists, is called the pointwise dimension of at x. There is a vast theory relating hyperbolicity and dimension. In particular, the following result is due to Barreira, Pesin and Schmeling [12]. Theorem 1.3. Let f W M ! M be a C 1C˛ diffeomorphism, for some ˛ > 0, and let be a finite f -invariant measure whose support is a hyperbolic set. Then the limit lim

r!0

exists for -almost every x 2 M .

log .B.x; r// log r

(1.3)

1.4 Geometric Constructions and Limit Sets

5

Theorem 1.3 also builds on seminal work of Ledrappier and Young in [54]. It plays a role in dimension theory that is similar to the central role played by the Shannon–McMillan–Breiman theorem (Theorem 4.4) in entropy theory. Under the assumptions of Theorem 1.3, when is ergodic, it follows from the above criterion of Young that dimH D dimB D dimB : In fact, the almost everywhere existence of the limit in (1.3) guarantees the coincidence not only of these three dimensions but also of many other characteristics of dimension type (see [12, 73, 105]).

1.4 Geometric Constructions and Limit Sets There are important differences between the dimension theory of invariant sets and the dimension theory of invariant measures. In particular, while virtually all dimensional characteristics of invariant measures on hyperbolic sets coincide, the study of the dimension of hyperbolic sets revealed that different dimensional characteristics frequently depend on properties of geometrical and number-theoretical nature. This justifies the interest in simpler models. We start with the description of a geometric construction in R. Consider constants 1 ; : : : ; p 2 .0; 1/ and disjoint closed intervals 1 ; : : : ; p R of lengths 1 ; : : : ; p . For each k D 1; : : : ; p, choose again p disjoint closed intervals k1 ; : : : ; kp k of lengths k 1 ; : : : ; k p . Iterating this Q procedure, for each n 2 N, we obtain p n disjoint closed intervals i1 in of lengths nkD1 ik . The limit set of the construction is defined by F D

1 [ \

i1 in :

(1.4)

nD1 i1 in

In [62], Moran showed that dimH F D s, where s is the unique root of the equation p X

sk D 1:

(1.5)

kD1

It is remarkable that the Hausdorff dimension of the set F does not depend on the location of the intervals i1 in but only on their lengths. Pesin and Weiss [75] extended the result of Moran to arbitrary symbolic dynamics in Rm , using the thermodynamic formalism (see Sect. 1.5). We also consider geometric constructions described in terms of a general symbolic dynamics. Given p 2 N, consider the family of sequences Xp D f1; : : : ; pgN and equip this space with the distance d.!; ! 0 / D

1 X kD1

e k j!k !k0 j:

(1.6)

6

1 Introduction

We also consider the shift map W Xp ! Xp such that .!/n D !nC1 for each n 2 N. A geometric construction in Rm is defined by: 1. A compact set Q Xp such that 1 Q Q for some p 2 N. 2. A decreasing sequence of compact sets !1 !n Rm for each ! 2 Q such that diam!1 !n ! 0 where n ! 1. We also assume that int i1 in \ int j1 jn ¤ ∅ whenever .i1 in / ¤ .j1 jn /. The limit set F of the geometric construction is defined by (1.4) with the union taken over all vectors .i1 ; : : : ; in / such that ik D !k for each k D 1; : : : ; n and some ! 2 Q. Now we consider the particular case when all sets i1 in are balls. Write ri1 in D diami1 in . The following result was established by Barreira [7]. Theorem 1.4 (Dimension of the limit set). For a geometric construction modeled by Q Xp such that all sets i1 in are balls, if there exists a constant ı > 0 such that ri1 inC1 ıri1 in and ri1 inCm ri1 in rinC1 im for every .i1 i2 / 2 Q and n; m 2 N, then dimH F D dimB F D dimB F D s; where s is the unique root of the equation X 1 log ris1 in D 0: n!1 n i i lim

1

n

This result contains as particular cases the results of Moran and of Pesin and Weiss (see also Sect. 1.5), for which ri1 in D

n Y

ik

kD1

for some numbers 1 ; : : : ; p . The value of the dimension is also independent of the location of the sets i1 in . The proof of Theorem 1.4 is based on a nonadditive version of the thermodynamic formalism (see [9]).

1.5 Classical Thermodynamic Formalism This section is a brief introduction to the classical thermodynamic formalism, considering only the case of symbolic dynamics. Let Q Xp be a compact set such that 1 Q Q. Given a continuous function 'W Q ! R, the topological pressure of ' (with respect to ) is defined by

1.5 Classical Thermodynamic Formalism

7

! n1 X X 1 P .'/ D lim log exp sup ' ı k ; n!1 n i i 1

(1.7)

kD0

n

where the supremum is taken over all sequences .j1 j2 /2Q such that .j1 jn /D .i1 in /. The topological entropy of jQ is defined by h.jQ/ D P .0/: Topological pressure is the most basic notion of the thermodynamic formalism. It was introduced by Ruelle in [83] for expansive transformations and by Walters in [102] in the general case. Now we present an equivalent description of the topological pressure. Let be a -invariant probability measure in Q and let be a finite or countable partition of Q into measurable sets. We write H ./ D

X

.C / log .C /;

C 2

with the convention that 0 log 0 D 0. The Kolmogorov–Sinai entropy of jQ with respect to is defined by ! n1 _ 1 k h .jQ/ D sup lim H ; n!1 n kD0

where

Wn1 kD0

k is the partition of Q formed by the sets Ci1 in D

n1 \

k CikC1

kD0

with Ci1 ; : : : ; Cin 2 . The topological pressure satisfies the variational principle Z P .'/ D sup h .jQ/ C ' d ;

(1.8)

Q

where the supremum is taken over all -invariant probability measures in Q. A -invariant probability measure in Q is called an equilibrium measure for ' (with respect to jQ) if the supremum in (1.8) is attained at this measure, that is, if Z P .'/ D h .jQ/ C

' d: Q

There is a very close relation between dimension theory and the thermodynamic formalism. To illustrate this relation, we consider numbers 1 ; : : : ; p and the function 'W Q ! R defined by

8

1 Introduction

'.i1 i2 / D log i1 :

(1.9)

We have n X X 1 exp s log ik P .s'/ D lim log n!1 n i i 1

X 1 log n!1 n i i 1

kD1

n

D lim

n

n Y

ik s

kD1

p X 1 D lim log i s n!1 n i D1

D log

p X

!

!n

si :

i D1

Therefore, (1.5) is equivalent to P .s'/ D 0:

(1.10)

Equation (1.10) was introduced by Bowen in [21]. It has a rather universal character: virtually all known equations used to compute or to estimate the dimension of an invariant set are particular cases of (1.10) or of an appropriate generalization (see [9]). For example, the result of Pesin and Weiss in [75] mentioned in Sect. 1.4 can be formulated as follows: Theorem 1.5 (Dimension of the limit set). For a geometric Q construction modeled by Q Xp such that the sets i1 in are balls of diameter nkD1 ik , we have dimH F D dimB F D dimB F D s; where s is the unique root of the equation P .s'/ D 0 with ' as in (1.9).

1.6 Dimension Theory in Hyperbolic Dynamics As observed above, one of the main motivations for the study of geometric constructions is the study of the dimension of hyperbolic sets. This approach can be effected using Markov partitions. We first consider expanding maps. These are a noninvertible version of diffeomorphisms with a hyperbolic set. Let gW M ! M be a differentiable map of a smooth manifold. We also consider a g-invariant compact set J M . We say that J

1.6 Dimension Theory in Hyperbolic Dynamics

9

is a repeller of g and that g is an expanding map on J if there exist constants c > 0 and ˇ > 1 such that kdx g n vk cˇ n kvk for every n 2 N, x 2 J and v 2 Tx M . Now let J be a repeller of the map g. A finite cover of J by nonempty closed sets R1 ; : : : ; Rp is called a Markov partition of J if: 1. int Ri D Ri for each i . 2. int Ri \ int Rj D ∅ whenever i ¤ j . 3. g.Ri / Rj whenever g.int Ri / \ int Rj ¤ ∅. The interior of each set Ri is computed with respect to the topology induced on J . Any repeller has Markov partitions of arbitrarily small diameter (see [85]). Now we use Markov partitions to model repellers by geometric constructions. Let J be a repeller of a map g and let R1 ; : : : ; Rp be the elements of a Markov partition of J . We consider the p p matrix A D .aij / with entries ( 1 if g.int Ri / \ int Rj ¤ ∅, aij D 0 if g.int Ri / \ int Rj D ∅. Consider the space of sequences Xp D f1; : : : ; pgN and the shift map W Xp ! Xp (see Sect. 1.4). We call topological Markov chain with transition matrix A to the restriction of the shift map to the set ˚ XA D .i1 i2 / 2 Xp W ain inC1 D 1 for every n 2 N : One can define a coding map W XA ! J by .i1 i2 / D

1 \

g k RikC1 :

kD0

The map is surjective, satisfies ı D g ı ;

(1.11)

and is Hölder continuous (with respect to the distance in (1.6)). Even though in general is not invertible, identity (1.11) allows one to see as a dictionary transferring the symbolic dynamics jXA (and often the results at the level of symbolic dynamics) to the dynamics of g on J . In particular, the repeller can be seen as the limit set of a geometric construction (see Sect. 1.4), defined by the sets i1 in D

n1 \ kD0

g k RikC1 :

10

1 Introduction

The map g is said to be conformal on J if dx g is a multiple of an isometry for every x 2 J . When J is a repeller of a conformal map of class C 1C˛ , one can show that there is a constant C > 0 such that C 1

n1 Y

exp '.g k .x// diam i1 in C

kD0

n1 Y

exp '.g k .x//

kD0

for every x 2 i1 in , where the function 'W J ! R is defined by '.x/ D logkdx gk: The topological pressure defined by (1.7) with Q D XA can be used to compute the dimension of the repeller. Theorem 1.6 (Dimension of conformal repellers). If J is a repeller of a C 1C˛ transformation g, for some ˛ > 0, such that g is conformal on J , then dimH J D dimB J D dimB J D s; where s is the unique root of the equation P .s'/ D 0. Ruelle showed in [85] that dimH J D s. The coincidence between the Hausdorff and box dimensions is due to Falconer [31]. It was also shown by Ruelle in [85] that if is the unique equilibrium measure of s', then dimH J D dimH :

(1.12)

In fact, he showed that is equivalent to the s-dimensional Hausdorff measure on J . Now we move to the study of the dimension of hyperbolic sets. Let be a hyperbolic set for a diffeomorphism f W M ! M . We consider the functions 's W ! R and 'u W ! R defined by 's .x/ D logkdx f jE s .x/k

and 'u .x/ D logkdx f jE u .x/k:

The set is said to be locally maximal if there exists an open neighborhood U of such that \ D f n .U /: n2Z

The following result is a version of Theorem 1.6 for hyperbolic sets: Theorem 1.7 (Dimension of hyperbolic sets on surfaces). If is a locally maximal hyperbolic set for a C 1 surface diffeomorphism, and dimE s .x/ D dimE u .x/ D 1 for every x 2 , then dimH D dimB D dimB D ts C tu ; where ts and tu are the unique real numbers such that

1.6 Dimension Theory in Hyperbolic Dynamics

11

P .ts 's / D P .tu 'u / D 0: It follows from work of McCluskey and Manning [59] that dimH D ts Ctu . The coincidence between the Hausdorff and box dimensions is due to Takens [101] for C 2 diffeomorphisms and to Palis and Viana [68] in the general case. The result in Theorem 1.7 can be readily extended to the more general case of conformal maps. We say that f W M ! M is conformal on a hyperbolic set if dx f jE s .x/ and dx f jE u .x/ are multiples of isometries for every x 2 (e.g., if M is a surface and dimE s .x/ D dimE u .x/ D 1 for every x 2 , then f is conformal on ). One can also ask whether there is an appropriate generalization of property (1.12) in the present context, that is, whether there exists an invariant measure supported on such that dimH D dimH . The answer to this question is almost always negative. More precisely, McCluskey and Manning [59] showed that such a measure exists if and only if there is a continuous function W ! R such that ts 's tu ' D

ıf f

on . By Livschitz’s theorem (see, e.g., [44]), this happens if and only if kdx f jE s .x/kts kdx f jE u .x/ktu D 1 for every x 2 and n 2 N such that f n .x/ D x. The study of the dimension of repellers and hyperbolic sets for nonconformal maps is much less developed than the corresponding study for conformal maps. The main difficulty is the possibility of existence of distinct Lyapunov exponents associated to directions that may change from point to point. There exist however some partial results, for certain classes of repellers and hyperbolic sets, starting essentially with the seminal work of Douady and Oesterlé [28]. Falconer [32] computed the Hausdorff dimension of a class of nonconformal repellers (see also [30]), while Hu [41] computed the box dimension of a class of nonconformal repellers leaving invariant a strong unstable foliation. Related ideas were applied by Simon and Solomyak in [91] to compute the Hausdorff dimension of a class of hyperbolic sets in R3 . Falconer also studied a class of limit sets of geometric constructions obtained from the composition of affine transformations that are not necessarily conformal [30]. In another direction, Bothe [17] and Simon [90] studied the dimension of solenoids. A solenoid is a hyperbolic set of the form D

1 \

f n .T /;

nD1

where T R3 is diffeomorphic to a solid torus S 1 D for some closed disk D R2 and f W T ! T is a diffeomorphism such that for each x 2 S 1 , the intersection f .T / \ .fxg D/ is a disjoint union of p sets homeomorphic to a closed disk.

12

1 Introduction

1.7 Multifractal Analysis and Irregular Sets By Theorem 1.3, if f W M ! M is a C 1C˛ diffeomorphism and is a finite f -invariant measure supported on a hyperbolic set, then the limit lim

r!0

log .B.x; r// log r

exists for -almost every x 2 M . Multifractal analysis studies the properties of the level sets log .B.x; r// x 2 M W lim D˛ r!0 log r for ˛ 2 R. We present in this section the main components of multifractal analysis. Birkhoff’s ergodic theorem—another basic but also fundamental result of ergodic theory—states that if S W X ! X is a measurable transformation preserving a finite measure in X , then for each integrable function ' 2 L1 .X; /, the limit 1X '.S k .x// n!1 n n1

'S .x/ D lim

kD0

exists for -almost every x 2 X . Furthermore, if is ergodic, then 'S .x/ D

1 .X /

Z ' d

(1.13)

X

for -almost every x 2 X . Of course, this does not mean that (1.13) holds for every x 2 X for which 'S .x/ is well defined. Given ˛ 2 R, we consider the level set (

) n1 1X k '.S x/ D ˛ : K˛ .'/ D x 2 X W lim n!1 n kD0

We also consider the set (

) n1 n1 1X 1X k k K.'/ D x 2 X W lim inf '.S x/ < lim sup '.S x/ : n!1 n n!1 n kD0

Clearly, X D K.'/ [

kD0

[

K˛ .'/:

(1.14)

˛2R

We call the decomposition of X in (1.14) a multifractal decomposition. One way to measure the complexity of the sets K˛ .'/ is to compute their Hausdorff dimension. We define a function

1.7 Multifractal Analysis and Irregular Sets

13

DW f˛ 2 R W K˛ .'/ ¤ ∅g ! R by D.˛/ D dimH K˛ .'/: We also consider the numbers Z Z ' d and ˛ D sup ' d; ˛ D inf

X

X

where the infimum and supremum are taken over all S -invariant probability measures in X . It is easy to verify that K˛ .'/ D ∅ whenever ˛ 62 Œ˛; ˛ . We also define a function T W R ! R by T .q/ D P .q'/ qP .'/; where P denotes the topological pressure. For topological Markov chains (see Sect. 1.6), the function T is analytic (see [84]). Under the assumptions in Theorem 1.8 below, there exists a unique equilibrium measure q of q' (see Sect. 1.5). The following result shows that in the case of topological Markov chains, the set K˛ .'/ is nonempty for every ˛ 2 .˛; ˛/ and that the function D is analytic and strictly convex. Theorem 1.8 (Multifractal analysis of Birkhoff averages). If the topological Markov chain jX is topologically mixing and 'W X ! R is Hölder continuous, then: 1. K˛ .'/ is dense in X for each ˛ 2 .˛; ˛/. 2. The function DW .˛; ˛/ ! R is analytic and strictly convex. 3. The function D is the Legendre transform of T , that is, D.T 0 .q// D T .q/ qT 0 .q/ for each q 2 R. 4. If q 2 R, then q .KT 0 .q/ .'// D 1 and log q .B.x; r// D T .q/ qT 0 .q/ r!0 log r lim

for q -almost every point x 2 KT 0 .q/ .'/. Theorem 1.8 reveals an enormous complexity of multifractal decompositions that is not foreseen by Birkhoff’s ergodic theorem. In particular, it shows that the multifractal decomposition in (1.14) is composed of an uncountable number of (pairwise disjoint) dense invariant sets, each of them of positive Hausdorff dimension. Statement 1 is an exercise. The remaining statements in Theorem 1.8

14

1 Introduction

follow from results of Pesin and Weiss in [76]. In [86], Schmeling showed that the domain of D coincides with Œ˛; ˛ , that is, that K˛ .'/ ¤ ∅ if and only if ˛ 2 Œ˛; ˛ . The concept of multifractal analysis was suggested in [38]. The first rigorous approach is due to Collet, Lebowitz and Porzio in [25] for a class of measures invariant under one-dimensional Markov maps. Lopes [56] considered the measure of maximal entropy for hyperbolic Julia sets, and Rand [80] studied Gibbs measures for a class of repellers. We refer the reader to the books [8, 73] for detailed discussions and further references. Now let M be a surface and let M be a locally maximal hyperbolic set for a C 1C˛ diffeomorphism f W M ! M . We assume that f is topologically mixing on . Consider an equilibrium measure of a Hölder continuous function 'W ! R. We define functions Ts W ! R and Tu W ! R by Ts .q/ D P .q logkdf jE s k C q'/ qP .'/ and Tu .q/ D P .q logkdf jE u k C q'/ qP .'/: In [92], Simpelaere showed that dimH

log .B.x; r// D ˛ D Ts .q/ qTs0 .q/ C Tu .q/ qTu0 .q/; x 2 M W lim r!0 log r

where q 2 R is the unique real number such that ˛ D Ts0 .q/ Tu0 .q/: Again we observe an enormous complexity that is not precluded by the -almost everywhere existence of the pointwise dimension in Theorem 1.3. Now we consider the irregular set K.'/ in (1.14). When 'W X ! R is a continuous function, it follows from Birkhoff’s ergodic theorem that the set K.'/ has zero measure with respect to any S -invariant finite measure in X . Therefore, at least from the point of view of measure theory, the set K.'/ is very small. Remarkably, from the point of view of dimension theory, this set is as large as the whole space. Let S W X ! X be a continuous transformation of a topological space X . Two continuous functions '1 W X ! R and '2 W X ! R are said to be cohomologous if there exist a continuous function W X ! R and a constant c 2 R such that '1 '2 D ı S C c on X . If the function ' is cohomologous to a constant, then K.'/ D ∅. The following result of Barreira and Schmeling in [14] shows that if ' is not cohomologous to a constant, then K.'/ is as large as the whole space from the points of view of topological entropy and Hausdorff dimension. We recall that h.f jX / denotes the topological entropy of f jX (see Sect. 1.5).

1.8 Quantitative Recurrence and Dimension

15

Theorem 1.9 (Irregular sets). If X is a repeller of a C 1C˛ transformation, for some ˛ > 0, such that f is conformal and topologically mixing on X , and 'W X ! R is a Hölder continuous function, then the following properties are equivalent: 1. ' is not cohomologous to a constant. 2. K.'/ is a nonempty dense set with h.f jK.'// D h.jX /

and dimH K.'/ D dimH X:

(1.15)

A priori, Property 1 in Theorem 1.9 could be rare. However, precisely the opposite happens. Let C .X / be the space of Hölder continuous functions in X with Hölder exponent 2 .0; 1 equipped with the norm j'.x/ '.y/j k'k D supfj'.x/j W x 2 Xg C sup C > 0 W C for all x; y 2 X : d.x; y/

It is shown in [14] that for each 2 .0; 1 , the family of functions in C .X / that are not cohomologous to a constant is an open dense set. Therefore, given 2 .0; 1

and a generic function ' in C .X /, the set K.'/ is dense and satisfies the identities in (1.15). S Now let K D older continuous ' K.'/, with the union taken over all H¨ functions 'W X ! R. Under the hypotheses of Theorem 1.9, we have h.jK/ D h.jX /

and dimH K D dimH X:

These identities were established by Pesin and Pitskel in [74] when is a Bernoulli shift with two symbols, that is, for the transition matrix A D 11 11 . A related result of Shereshevsky in [88] shows that for a generic C 2 surface diffeomorphism with a locally maximal hyperbolic set , and an equilibrium measure of a Hölder continuous function that is generic in the C 0 topology, the set log .B.x; r// log .B.x; r// < lim sup I D x 2 W lim inf r!0 log r log r r!0 has positive Hausdorff dimension. In fact, it follows from results in [14] that dimH I D dimH (under those generic assumptions).

1.8 Quantitative Recurrence and Dimension Poincaré’s recurrence theorem (Theorem 1.2) is one of the fundamental results of the theory of dynamical systems. Unfortunately, it only provides qualitative information. In particular, it does not consider:

16

1 Introduction

1. With which frequency the orbit of a point visits a given set of positive measure. 2. With which rate the orbit of a point returns to an arbitrarily small neighborhood of the initial point. Birkhoff’s ergodic theorem gives a fairly complete answer to the first problem. Now we briefly describe some results concerning the second problem. Given a transformation f W M ! M , the (first) return time of a point x 2 M to the ball B.x; r/ is given by

r .x/ D inffn 2 N W d.f n .x/; x/ < rg: The lower and upper recurrence rates of x are defined by R.x/ D lim inf r!0

log r .x/ log r

and R.x/ D lim sup r!0

log r .x/ : log r

(1.16)

When R.x/ D R.x/, we denote the common value by R.x/ and call it the recurrence rate of x. In the present context, the study of quantitative recurrence started with the work of Ornstein and Weiss [67], closely followed by the work of Boshernitzan [16]. In [67], the authors considered the case of symbolic dynamics (and thus the corresponding symbolic metric in (1.6)) and for an ergodic -invariant probability measure showed that R.x/ D h ./ for -almost every x. On the other hand, Boshernitzan considered an arbitrary metric space M and showed in [16] that R.x/ dimH (1.17) for -almost every x 2 M . In the particular case of hyperbolic sets, the following result of Barreira and Saussol in [13] shows that (1.17) is often an identity. Theorem 1.10 (Quantitative recurrence). For a C 1C˛ diffeomorphism with a hyperbolic set , for some ˛ > 0, if is an ergodic equilibrium measure of a Hölder continuous function, then R.x/ D lim

r!0

log .B.x; r// log r

(1.18)

for -almost every x 2 . We note that identity (1.18) relates two quantities of very different nature. In particular, R.x/ does not depend on the measure, and the pointwise dimension does not depend on the map. Putting together (1.16) and (1.18), we obtain lim

r!0

log r .x/ log .B.x; r// D lim r!0 log r log r

for -almost every point x 2 . Therefore, the return time r .x/ is approximately equal to 1=.B.x; r// when r is sufficiently small.

Part I

Ergodic Theory

•

Chapter 2

Basic Notions and Examples

We introduce in this chapter the basic notions and results of ergodic theory, starting with the notion of invariant measure with respect to a measurable transformation. In particular, we establish two basic but also fundamental results of ergodic theory: Poincaré’s recurrence theorem and Birkhoff’s ergodic theorem. We also discuss the notion of ergodicity as well as its consequences. All the notions and results are illustrated with a number of examples. These include rotations and translations of Rn , rotations and expanding maps of the circle, toral endomorphisms, etc. We conclude this chapter with some applications of ergodic theory to number theory, namely, to fractional parts of polynomials and continued fractions. All the necessary material from measure theory is recalled in Appendix A.

2.1 Introduction Ergodic theory can be described as the study of measurable maps and flows and more generally group actions, preserving a certain measure. Some emphasis is given to the study of the stochastic properties of the dynamics, such as ergodicity and mixing with respect to a given invariant measure. The origins of ergodic theory go back to statistical mechanics with an attempt to apply probability theory to conservative mechanical systems with many degrees of freedom. For conservative systems defined by a Hamiltonian, it follows from Liouville’s theorem that the volume in phase space is invariant under the dynamics. More precisely, let @H @H ; p0 D (2.1) q0 D @p @q be the Hamiltonian equations defined by an autonomous Hamiltonian H D H.p; q/ in Rn Rn , where q gives the positions of some particles and q gives their corresponding momenta. Since the divergence of the vector field .@H=@p; @H=@q/ is zero (provided that H is of class C 2 ), the Hamiltonian flow 't defined by (2.1) L. Barreira, Ergodic Theory, Hyperbolic Dynamics and Dimension Theory, Universitext, DOI 10.1007/978-3-642-28090-0 2, © Springer-Verlag Berlin Heidelberg 2012

19

20

2 Basic Notions and Examples

preserves volume, that is, v.'t .A// D v.A/ for any t 2 R and any measurable set A, where Z v.A/ D dp dq: A

Due to the ubiquity of Hamiltonian equations in physics, this yields a large class of examples of measure-preserving flows and in fact also of measure-preserving maps, simply by taking the corresponding time-t maps. Since the Hamiltonian H is invariant under the Hamiltonian flow, that is, H ı 't D H

for every t 2 R;

it is sufficient to study the restriction of the dynamics to each level set of constant energy Lc D f.p; q/ W H.p; q/ D cg; on which the volume v induces a corresponding invariant volume vc . Given a function F , Boltzmann’s ergodic hypothesis corresponds to assume that points x in a given level set Lc (assuming that the volume vc is finite) satisfy 1 t !C1 t

Z

t

lim

F .'s .x// ds D

0

1 vc .Lc /

Z F dvc

(2.2)

Lc

or some appropriate version of this requirement. Rigorous versions of Boltzmann’s ergodic hypothesis were established independently by Birkhoff and von Neumann. In particular, Birkhoff’s ergodic theorem states that for an integrable function F , the limit in the left-hand side of (2.2) exists for vc -almost every x 2 Lc . However, in general, the limit in (2.2) may depend on x (and may not exist). In order to obtain independence with respect to the point, we need to consider the notion of ergodicity, which means that from the point of view of ergodic theory, that is, from the point of view of an invariant measure, the space cannot be decomposed into invariant sets of positive measure. When the measure vc is ergodic, it follows from Birkhoff’s ergodic theorem that identity (2.2) holds for vc -almost every x 2 Lc . In another direction, the existence of a finite invariant measure naturally gives rise to the concept of nontrivial recurrence. Let T be the time-t map of a Hamiltonian flow. It also preserves the Liouville volume v. For a level set Lc of finite vc -volume, a simple yet striking property was established by Poincaré: the transformation T exhibits a nontrivial recurrence in any set A Lc of positive measure, in the sense that the trajectory of almost every point in A returns infinitely often to A. In other words, n vc lim sup T A D vc .A/: n!1

This is Poincaré’s recurrence theorem. The result follows from the fact that due to the invariance and finiteness of the measure, it is impossible that successive iterations of a set of positive measure only intersect on a set of zero measure.

2.2 Invariant Measures

21

2.2 Invariant Measures We introduce in this section the most basic notion of ergodic theory: the notion of invariant measure with respect to a dynamical system. Roughly speaking, a measure is invariant if the evolution of the dynamics does not change the value of the measure. We also illustrate the concept with several instructive examples. We recall that a triple .X; A; / is said to be a measure space if A is a -algebra of subsets of X and is a measure in A. We say that the measure is finite if .X / < 1.

2.2.1 The Notion of Invariant Measure Let .X; A; / be a measure space. We say that a transformation T W X ! X is Ameasurable if ˚ T 1 B WD x 2 X W T .x/ 2 B 2 A

for every B 2 A;

that is, if the preimage of any set in A is also in A. Whenever there is no danger of confusion, we simply say that T is measurable without any explicit reference to the -algebra. Definition 2.1. Let .X; A; / be a measure space and let T W X ! X be an Ameasurable transformation. We say that is T -invariant and that T preserves if .T 1 B/ D .B/

for every B 2 A:

(2.3)

1

When the transformation T is invertible and T is A-measurable, we note that T 1 B 2 A if and only if B 2 A, and thus, in this case, condition (2.3) is equivalent to .T .B// D .B/ for every B 2 A: In order to present a characterization of the invariance of a finite measure, we start by recalling that the characteristic function B W X ! f0; 1g of a set B X is defined by ( 1 if x 2 B; (2.4) B.x/ D 0 if x 62 B: One can easily verify that B ı T D T 1 B ;

(2.5)

and thus, property (2.3) is equivalent to Z

Z .B ı T / d D

X

B d for every B 2 A: X

(2.6)

22


Now let L1 .X; / be the set of all -integrable functions, that is, the set of all measurable functions 'W X ! R such that Z j'j d < 1: X

Proposition 2.1. Let T W X ! X be a measurable transformation and let be a finite measure in X . Then is T -invariant if and only if for every function ' 2 L1 .X; /, we have ' ı T 2 L1 .X; / and Z

Z .' ı T / d D X

' d:

(2.7)

X

Proof. We first assume that (2.7) R holds. Given a measurable set B X , we consider the function ' D B . Since X j'j d D .B/ < 1, we have B 2 L1 .X; /, and using (2.5), it follows from (2.7) that (2.3) holds. Now we assume that is T -invariant. Then (2.6) holds. If s is a simple function, that is, a finite linear combination of characteristic functions, then it follows from (2.6) that Z Z .s ı T / d D s d: (2.8) X

X

Now we consider an arbitrary function ' 2 L .X; /. We can write it in the form ' D ' C ' , where ' C and ' are the integrable functions 1

' C D maxf'; 0g and ' D maxf'; 0g: Since ' C ; ' 0, it is sufficient to prove the result for functions ' 2 L1 .X; / with ' 0. Thus, let sn be a sequence of simple functions such that 0 sn snC1 '

for each n 2 N;

with sn ! ' pointwise and Z

Z sn d ! X

' d X

when n ! 1. Since sn ı T % ' ı T when n ! 1, it follows from Fatou’s lemma (Theorem A.1) and (2.8) that Z Z .' ı T / d lim inf .sn ı T / d n!1

X

X

Z

D lim inf n!1

Z D

sn d X

' d < 1: X


23

Fig. 2.1 Rotations and translations of Rn

This shows that the function ' ı f is integrable. Moreover, it follows from the monotone convergence theorem (Theorem A.2) that Z

Z .' ı T / d D lim X

n!1 X

.sn ı T / d Z

Z

D lim

n!1 X

sn d D

This completes the proof of the proposition.

' d: X

t u

2.2.2 Rotations and Translations of Rn Here and in the following sections, we present several examples of transformations with invariant measures. We start by considering a class of linear transformations. Let S.n; R/ be the set of n n matrices A with entries in R such that jdet Aj D 1. Given A 2 S.n; R/ and b 2 Rn , we define a transformation T W Rn ! Rn by T .x/ D Ax C b

(2.9)

for each x 2 Rn . For example, all rotations and translations are of this form. Moreover, any composition of rotations and translations (see Fig. 2.1) can also be written as in (2.9) for some orthogonal matrix A. This means that A A D Id, where A is the transpose of A. Clearly, each transformation T in (2.9) is invertible and differentiable, with derivative dx T D A for each x 2 Rn . Now let m be the Lebesgue measure in Rn (see Appendix A for the definition). If B Rn is a measurable set, then

24


Fig. 2.2 The transformation .x; y/ 7! .2x; y=2/

y

x

Z m.T .B// D

jdet dx T j d m.x/ Z

B

(2.10) jdet Aj d m D m.B/:

D B

Since the transformation T is invertible, it follows from (2.10) that the Lebesgue measure m is T -invariant. In particular, any composition of rotations and translations preserves the Lebesgue measure. We note that the converse does not hold. For an example, consider the transformation .x; y/ 7! .2x; y=2/ (see Fig. 2.2).

2.2.3 Circle Rotations and Interval Translations Now we consider the rotations of the circle ˚ S 1 D z 2 C W jzj D 1 : We can identify S 1 with the quotient T D Œ0; 1=f0; 1g by the one-to-one transformation hW T ! S 1 given by h./ D e 2 i :

(2.11)

Definition 2.2. Given w 2 S 1 , we define the circle rotation Rw W S 1 ! S 1 by Rw .z/ D wz; and given 2 R, we define the interval translation T W T ! T (see Fig. 2.3) by T .x/ D x C mod 1 D x C bx C c;

(2.12)


25

Fig. 2.3 An interval translation 1

τ

1

where bxc denotes the integer part of x. We note that if w D e 2 i , then .Rw ı h/./ D h. C / D .h ı T /./

(2.13)

for every 2 R. Now let m be the probability measure in T induced by the Lebesgue measure in Œ0; 1 R. Since the latter is invariant under translations (see Sect. 2.2.2), we have m.B C / D m.B/

(2.14)

for every measurable set B Œ0; 1, where B C D f C W 2 Bg: Therefore, any interval translation preserves the Lebesgue measure. We also consider the Lebesgue measure in S 1 defined by .h.B// D m.B/ for each measurable set B Œ0; 1. Proposition 2.2. Any circle rotation preserves the Lebesgue measure. Proof. By (2.13), for any measurable set B Œ0; 1, we have Rw .h.B// D h.T .B//;

(2.15)

26


and hence, .Rw .h.B/// D .h.B C // D m.B C /: It follows from (2.14) and (2.15) that .Rw .h.B/// D .h.B//; t u

and since Rw is invertible, we obtain the desired statement.

Now we look at the periodic points, which sometimes may indicate the existence of some complexity for a given transformation (see also Sect. 2.2.4). Definition 2.3. Given m 2 N, we say that x 2 X is an m-periodic point of T if T m .x/ D x. We also say that x is a periodic point of T if it is m-periodic for some m. We note that an m-periodic point is km-periodic for every k 2 N. Given an mperiodic point x 2 X of a transformation T W X ! X , we can define a T -invariant probability measure in X by D

m1 1 X ıT k .x/ ; m

(2.16)

kD0

where ıx is the probability measure such that ıx .fxg/ D 1. The measure is concentrated on the periodic orbit ˚ OT .x/ D T k .x/ W k D 0; : : : ; m 1 ; in the sense that .X / D .OT .x//. Now we study the periodic points in the particular case of the circle rotations. Proposition 2.3. Given w D e 2 i 2 S 1 , the following properties hold: 1. If 62 Q, then Rw has no periodic points. 2. If 2 Q, then Rw has m-periodic points if and only if m 2 Z. 3. If there is a n-periodic point of Rw , then all points of S 1 are n-periodic. Proof. We note that a point z 2 S 1 is m-periodic for Rw if and only if e 2 i m z D z, that is, if and only if e 2 i m D 1. Property 3 follows immediately from this observation. Furthermore, e 2 i m D 1 if and only if m 2 Z. This yields Properties 1 and 2. t u Let w D e 2 i 2 S 1 and consider the interval translation T in (2.12). Definition 2.4. When 2 Q, we say that Rw is a rational circle rotation and that T is a rational interval translation. When 62 Q, we say that Rw is an irrational circle rotation and that T is an irrational interval translation.


27

It follows from Proposition 2.3 and the construction in (2.16) that each rational circle rotation and each rational interval translation have uncountably many invariant probability measures.

2.2.4 Expanding Maps of the Circle We consider in this section a class of transformations in the circle that expand distances. Definition 2.5. Given q 2 Z, we define a transformation Eq W S 1 ! S 1 by Eq .z/ D zq for each z 2 S 1 . When jqj > 1, we say that Eq is an expanding map of the circle. Proposition 2.4. Any expanding map of the circle preserves the Lebesgue measure. Proof. For each ˛ 2 .0; 1/, we have Eq1 .h.Œ0; ˛// D

q [ i ˛Ci ; h q q i D1

(2.17)

Eq1 .h.Œ0; ˛// D

jqj [ ˛Ci i ; h q q i D1

(2.18)

whenever q > 0, and

whenever q < 0. We note that both unions in (2.17) and (2.18) are disjoint. Hence, jqj

X ˛ Eq1 .h.Œ0; ˛// D D˛ jqj i D1 D .h.Œ0; ˛// for every ˛ 2 .0; 1/. Since the intervals Œ0; ˛ generate the Borel -algebra of Œ0; 1, we conclude that .Eq1 B/ D .B/ t for any measurable set B S 1 . Therefore, Eq preserves the Lebesgue measure. u Now we study the periodic points of the expanding maps of the circle. Since .Eq /m D Eq m , a point e 2 i 2 S 1 is m-periodic for Eq if and only if e 2 i.q

m 1/

D 1;

that is, if and only if D j=jq m 1j

for some

j D 1; : : : ; jq m 1j:

(2.19)

28


Fig. 2.4 The eight 2-periodic points of E3

Hence, the number of m-periodic points of Eq is equal to jq m 1j, and they are uniformly distributed in S 1 (see Fig. 2.4 for an example). In particular, each expanding map Eq has m-periodic points for every m 2 N. Proposition 2.5. The periodic points of an expanding map Eq are dense in S 1 , that is, the set ˚ z 2 S 1 W z is m-periodic for Eq for some m 2 N is dense in S 1 . Proof. The statement is an immediate consequence of (2.19).

t u

In order to measure the complexity of a transformation from the point of view of the periodic points, we now introduce the notion of periodic entropy. Definition 2.6. Given a transformation T W X ! X , we define the periodic entropy of T by ˚ 1 (2.20) p.T / D lim sup logC card x 2 X W T m .x/ D x ; m m!1 where logC a D maxf0; log ag (with the convention that log 0 D 1). For example, for the expanding maps of the circle, we have p.Eq / D lim sup m!1

1 logC jq m 1j D logjqj > 0: m

(2.21)

2.2.5 Toral Endomorphisms We consider in this section a higher-dimensional version of the expanding maps of the circle. Let G.n; Z/ be the set of n n matrices A with entries in Z such that det A ¤ 0. We note that A.Zn / Zn (2.22) for each A 2 G.n; Z/. Since A defines a linear transformation of Rn , this is equivalent to Ay Ax 2 Zn whenever y x 2 Zn : (2.23)


29 (5;4)

(4;3)

(2;2) (3;2)

(0;1)

(0;0)

(1;1)

(1;0)

Fig. 2.5 Image of the square Œ0; 12 under an endomorphism of T2 (the broken line divides the parallelogram into two identical parallelograms)

Now we consider the sets ˚ Œx D y 2 Rn W y x 2 Zn for each x 2 Rn and the n-torus Tn D Rn =Zn D fŒx W x 2 Rn g: It follows from (2.23) that if y 2 Œx, then Ay 2 ŒAx. Therefore, one can define a map TA of the n-torus by TA Œx D ŒAx. Definition 2.7. Given a matrix A 2 G.n; Z/, we define the toral endomorphism TA W Tn ! Tn by TA Œx D ŒAx for each Œx 2 Tn : We also say that TA is the toral endomorphism induced by A. We note that, in general, the map TA need not be invertible, even though the matrix A 2 G.n; Z/ is always invertible. As an example, we represent in Fig. 2.5 the noninvertible endomorphism of T2 induced by the matrix 32 22 . A necessary and sufficient condition for the invertibility of the toral endomorphism TA is that Ay Ax 2 Zn

if and only if

y x 2 Zn :

30

2 Basic Notions and Examples (3;2)

(0;1)

(1;1) (2;1)

(0;0)

(1;0)

Fig. 2.6 Image of the square Œ0; 12 under an automorphism of T2

Since A defines a linear transformation of Rn , this is equivalent to require that A.Zn / D Zn . It follows from (2.22) that TA is invertible if and only if A.Zn / Zn and thus if and only if A1 .Zn / Zn . We note that this last inclusion holds if and only if A1 has only integer entries. In this case, both det A and det.A1 / are integers, which implies that jdet Aj D 1. On the other, if jdet Aj D 1, then clearly A1 has only integer entries. Let S.n; Z/ G.n; Z/ be the set of n n matrices A with jdet Aj D 1. It follows from the former discussion that TA is invertible if and only if A 2 S.n; Z/:

(2.24)

Definition 2.8. For each matrix A 2 S.n; Z/, the endomorphism TA W Tn ! Tn is also called a toral automorphism. By (2.24), each toral automorphism is invertible. Moreover, the inverse of a toral automorphism TA is also a toral automorphism, with .TA /1 D TA1 . As an 2 we represent in Fig. 2.6 the automorphism of T induced by the matrix example, 2 1 2 S.2; Z/. We note that the parallelogram 2 1 Œ0; 12 can be decomposed 11 11 into four triangles, which can then be used to reassemble the square. Now let m be the Lebesgue measure in Rn . We define the Lebesgue measure in the torus Tn by .ŒB/ D m.B/ for each measurable set B Œ0; 1n , where ŒB D fŒx W x 2 Bg. Proposition 2.6. Any toral endomorphism preserves the Lebesgue measure. Proof. Let A 2 G.n; Z/. Each point x 2 Tn has a number k D jdet Aj of preimages under the toral endomorphism TA . Moreover, for each sufficiently small r > 0 and each x 2 Tn , the preimage TA1 B.x; r/ of the open ball B.x; r/ of radius r centered at x consists of a number k of connected components B1 ; : : : ; Bk . We consider the


31

corresponding local inverses Si W B.x; r/ ! Bi of TA for i D 1; : : : ; k. Then TA ı Si D Id in B.x; r/, and thus, dy Si D A1 for each y 2 B.x; r/ and i D 1; : : : ; k. Therefore, .TA1 B.x; r// D

k X

.Bi / D

k X

i D1

D

k Z X i D1

.Si B.x; r//

i D1

jdet dy Si j d .y/ B.x;r/

Z

jdet A1 j d

D jdet Aj B.x;r/

D .B.x; r//: Since the family of open balls B.x; r/ with x 2 Tn and r > 0 sufficiently small generates the Borel -algebra of Tn , we conclude that is TA -invariant. t u

2.2.6 Piecewise Linear Expanding Maps Given a 2 .0; 1/, we consider the piecewise linear transformation T W Œ0; 1 ! Œ0; 1 defined by ( x=a if 0 x a; T .x/ D (2.25) .x a/=.1 a/ if a < x 1 (see Fig. 2.7). For each ˛ 2 Œ0; 1 (see Fig. 2.8), we have T 1 .0; ˛/ D .0; a˛/ [ .a; a C ˛ a˛/; with a disjoint union. Therefore, m.T 1 .0; ˛// D m..0; ˛//

for every ˛ 2 Œ0; 1:

Since the intervals of the form .0; ˛/ generate the Borel -algebra of Œ0; 1, we conclude that T preserves the Lebesgue measure. We note that in the definition of the transformation T , the interval Œ0; 1 is divided into two disjoint subintervals: .0; a/ and .a; 1/. More generally, we can divide the interval Œ0; 1 into a (finite or infinite) countable number of disjoint subintervals .ak ; bk / Œ0; 1 such that

32


Fig. 2.7 A piecewise linear expanding map 1

a

Fig. 2.8 Preimage of the interval Œ0; ˛, where b D a C ˛ a˛

1

1

α

aa

X .bk ak / D 1:

a

b

1

(2.26)

k

Definition 2.9. Given a (finite or infinite) countable number of disjoint subintervals .ak ; bk / Œ0; 1 satisfying (2.26), any transformation T W Œ0; 1 ! Œ0; 1 such that T .x/ D

x ak bk ak

whenever x 2 .ak ; bk / for some k is called a piecewise linear expanding map of the interval. S We note that for each x 2 k fak ; bk g, the value T .x/ is arbitrary. Moreover, for each piecewise linear expanding map of the interval, there exists > 1 such that

2.3 Poincaré’s Recurrence Theorem

33

T 0 .x/ >

for every x 2

[

.ak ; bk /:

k

A similar argument to that for the particular linear expanding map of the interval in (2.25) establishes the following: Proposition 2.7. Any piecewise linear expanding map of the interval preserves the Lebesgue measure. Proof. For each ˛ 2 Œ0; 1, we have T 1 .0; ˛/ D

[

.ak ; ak C ˛.bk ak //;

k

with a disjoint union. Therefore, by (2.26), m.T 1 .0; ˛// D

X m .ak ; ak C ˛.bk ak // k

D

X

˛.bk ak / D ˛

k

D m .0; ˛/ : Since the intervals .0; ˛/ generate the Borel -algebra of Œ0; 1, we conclude that the map T preserves the Lebesgue measure. t u

2.3 Poincaré’s Recurrence Theorem We establish in this section a basic but also fundamental result of ergodic theory— Poincaré’s recurrence theorem. Essentially, it says that the existence of a finite invariant measure causes a nontrivial recurrence for the dynamics. Let T W X ! X be a measurable transformation and let be a finite T -invariant measure in X . Let also A X be a measurable set with .A/ > 0. If the sets A; T 1 A; T 2 A; : : : are pairwise disjoint (see Fig. 2.9), that is, if T m A \ T n A D ∅ for every m ¤ n; then because is T -invariant, we have

1 [ kD1

! T

k

A D

1 X kD1

.T k A/ D

1 X

.A/ D 1:

kD1

But this is impossible because is a finite measure. Therefore, there exist positive integers m < n such that

34


Fig. 2.9 Pairwise disjoint preimages X A

T −1A

T −2A

T m .A \ T mn A/ D T m A \ T n A ¤ ∅: Hence, A \ T .nm/ A ¤ ∅, and some preimage of the set A must intersect A. In other words, whenever there exists a finite invariant measure, there is a nontrivial recurrence in each set of positive measure. Now we formulate a much stronger statement. Theorem 2.1 (Poincaré’s recurrence theorem). Let T W X ! X be a measurable transformation and let be a finite T -invariant measure in X . If A X is measurable, then the set ˚ B D x 2 A W T n .x/ 2 A for infinitely many integers n 2 N has measure .B/ D .A/. Proof. We have B D A \ lim sup T n A n!1

1 1 [ \

DA\

T k A

(2.27)

nD1 kDn

DAn

1 [

An

nD1

1 [

! T

k

A :

kDn

Moreover, An

1 [

T k A

kDn

1 [

T k A n

kD0

D

1 [ kD0

1 [

T k

kDn

T

k

AnT

n

1 [ kD0

(2.28) T

k

A:

2.3 Poincaré’s Recurrence Theorem

35

Fig. 2.10 Intervals Ii1 and Ii1 i2 for the base-2 representation of numbers in Œ0; 1

I00

I01

I10

I0

0

I11 I1

1

On the other hand, since is T -invariant, we have 1 [

! T

k

A D T

n

kD0

!

1 [

T

k

A :

(2.29)

kD0

Moreover, since 1 [

T k A

kD0

1 [

T k A D T n

kDn

1 [

T k A;

kD0

and is finite, it follows from (2.28) and (2.29) that An

1 [

! T

k

A

kDn

1 [

! T

k

A T

kD0

n

1 [

! T

k

A D0

kD0

for every n 2 N. We conclude from (2.27) that .B/ D .A/.

t u

Exercises 2.7 and 2.8 show that the finiteness of an invariant measure is crucial for the existence of nontrivial recurrence. The following is an application of Poincaré’s recurrence theorem to base-2 representations: Example 2.1. For each n 2 N and .i1 ; : : : ; in / 2 f0; 1gn, we consider the interval Ii1 in D

n n X ij X ij 1 ; C n j j 2 j D1 2 2 j D1

! Œ0; 1:

We note that Ii1 in has length 1=2n and that for each fixed n, the intervals Ii1 in are pairwise disjoint (see Fig. 2.10). A point x 2 Œ0; 1 is in the closure of Ii1 in if and only if the first n digits of some base-2 representation of x are equal to i1 in . Incidentally, we note that the base-2 representation is unique for Lebesgue-almost every point. Now we consider the piecewise linear expanding map of the interval (see Sect. 2.2.6) defined by ( T .x/ D

2x

if 0 x 1=2;

2x 1 if 1=2 < x 1:

36


By Proposition 2.7, the transformation T preserves the Lebesgue measure. Hence, it follows from Theorem 2.1 that ˚ card m 2 N W T m .x/ 2 Ii1 in D 1

(2.30)

for Lebesgue-almost every x 2 Ii1 in . Now we observe that if 0:i1 i2 is a base-2 representation of x, then 0:imC1 imC2 is a base-2 representation of T m .x/. Thus, it follows from (2.30) that for Lebesgue-almost every x 2 Ii1 in , the digits i1 in appear infinitely often, in this order, in some base-2 representation of x.

2.4 Invariant Sets and Invariant Functions We already know that one of the basic notions of ergodic theory is the notion of invariant measure. We are also interested in other invariant objects. Here we introduce the notions of invariant set and invariant function. Definition 2.10. Given a transformation T W X ! X , we say that: 1. A set A X is T -invariant if T 1 A D A. 2. A function 'W X ! R is T -invariant if '.T .x// D '.x/ for every x 2 X . We note that if ' is a T -invariant function, then '.y/ D '.x/ for every y 2 T 1 x since T .y/ D x. We also note that when T is invertible, a set A X is T invariant if and only if T .A/ D A. For example, any periodic orbit of an invertible transformation is an invariant set. Example 2.2. Consider the piecewise linear expanding map of the interval (see Sect. 2.2.6) defined by

T .x/ D

8 ˆ ˆ3x
0 and .X n E/ > 0, we consider the -subalgebra F D f∅; E; X n E; X g A; and we find explicitly 'F . By Proposition 2.9, we have Z

Z 'F d D A

B d D .A \ B/ A

(2.37)

40


for every set A 2 F. We note that a function is F-measurable if and only if it is constant on E and X n E. Since 'F is F-measurable, letting A D E and A D X n E in (2.37), we thus obtain ( 'F .x/ D

.E \ B/=.E/

if x 2 E;

..X n E/ \ B/=.X n E/

if x 2 X n E:

(2.38)

Now we consider the -subalgebra of invariant sets of a given transformation. Example 2.4. Let .X; A; / be a finite measure space and let T W X ! X be an A-measurable transformation. We consider the -subalgebra of T -invariant sets ˚ F D A 2 A W T 1 A D A :

(2.39)

Let also 'W X ! R be an A-measurable function. We show that ' is F-measurable if and only if the sets ' 1 ˛ are T -invariant for any ˛ 2 R. Indeed, ' is F-measurable if and only if ' 1 B is T -invariant for every measurable set B R and thus if and only if .' ı T /1 B D T 1 .' 1 B/ D ' 1 B (2.40) for every measurable set B R. Since f˛g R is measurable for each ˛ 2 R, it follows from (2.40) that .' ı T /1 ˛ D ' 1 ˛ (2.41) for every ˛ 2 R. Now we assume that (2.41) holds for every ˛ 2 R. Then .' ı T /1 B D .' ı T /1 D

[

[

f˛g

˛2B

.' ı T /1 ˛

˛2B

D

[

˛2B

D ' 1

' 1 ˛ [

f˛g D ' 1 B

˛2B

for any set B R. This implies that ' is F-measurable. It follows from Proposition 2.8 that ' is F-measurable if and only if is T invariant. In particular, any conditional expectation 'F of an A-measurable function ' 2 L1 .X; / is a T -invariant function.

2.5.3 Proof of Birkhoff’s Ergodic Theorem We are now ready to prove Birkhoff’s ergodic theorem (Theorem 2.2).

2.5 Birkhoff’s Ergodic Theorem

41

2 L1 .X; /, for each n 2 N, we consider the function

Proof. Given

D max

n

X `1

ıT W1`n : k

kD0

Clearly,

nC1

n

for each n. Since j

nj

n1 X

j

ı T kj

kD0

and is T -invariant, it follows from Proposition 2.1 that Z j

n j d

X

n1 Z X

j

ı T k j d

kD0 X

Z

Dn

j j d < 1: X

Therefore, . Moreover,

n /n2N

is a nondecreasing sequence of -integrable functions. (

nC1

D max

)

` X

ıT W1`n k

kD0

(

D

C max 0;

` X

)

(2.42)

ıTk W 1 ` n

kD1

D

C maxf0;

n

ı T g:

This implies that lim

n!1

nC1 .x/

and hence,

D C1

if and only if

n A D x 2 X W lim

n!1

lim

n!1

n .x/

n .T .x//

D C1

D C1;

o

is a T -invariant set. It also follows from (2.42) that nC1

and hence,

n

ıT D

C maxf0;

D

minf0;

n n

ıTg

ı T g;

n

ıT

(2.43)

42


nC1

ıT &

n

in A when n ! 1:

Moreover, again by (2.43), we have 0

nC1

n

ıT

2

1

ıT

for every n 2 N. Since 2 1 ıT is -integrable, it follows from the dominated convergence theorem (Theorem A.3) and Proposition 2.1 that Z 0

.

nC1

n / d

A

Z D

Z .

nC1

n

ı T / d !

A

d A

when n ! 1, and thus,

Z d 0:

(2.44)

A

Now let F be the -subalgebra F of T -invariant sets in (2.39). Given " > 0, we consider the function D ' 'F ": By (2.36), since A is T -invariant, we have Z d D ".A/; A

and it follows from (2.44) that .A/ D 0. Hence, lim

n!1

n .x/

< C1

for -almost every x 2 X . Since 1X n n1

ıTk

kD0

we obtain

1X n

n

n

;

n1

lim sup n!1

ıTk 0

kD0

-almost everywhere in X , and since 'F is T -invariant (see Example 2.4), we conclude that n1 1X lim sup ' ı T k 'F C " (2.45) n!1 n kD0

2.6 Ergodicity

43

-almost everywhere in X . Now we observe that .'/F D 'F . Therefore, replacing ' by ' in (2.45), we obtain 1X ' ı T k 'F " n n1

lim inf n!1

(2.46)

kD0

-almost everywhere in X . By (2.45) and (2.46), there is a set X" X with measure .X" / D .X / on which 'F " lim inf n!1

1X 1X ' ı T k lim sup ' ı T k 'F C ": n n!1 n

Therefore,

n1

n1

kD0

kD0

1X ' ı T k D 'F n!1 n n1

lim

(2.47)

kD0

in the set of full -measure

1 \

X1=m X:

mD1

By (2.47), the number 'T .x/ in (2.34) is well defined for -almost every x 2 X , and 'T D 'F -almost everywhere. Since 'F 2 L1 .X; /, we conclude that 'T 2 L1 .X; /, and it follows from (2.36) that Z

Z

Z

'T d D X

'F d D X

This completes the proof of the theorem.

' d: X

t u

2.6 Ergodicity In the theory of dynamical systems, one is often interested in decomposing the phase space into invariant sets that are indecomposable in some appropriate sense. In ergodic theory, we are also interested in a corresponding notion.

2.6.1 Basic Notions We first introduce the notion of ergodic measure.

44


Definition 2.13. Let T W X ! X be a measurable transformation. A measure in X is said to be ergodic with respect to T if the measure of any T -invariant measurable subset A X satisfies .A/ D 0 or .X n A/ D 0. In this case, we also say that T is ergodic with respect to . We note that an ergodic measure is not necessarily invariant. We start with a simple example. Example 2.5. Given an invertible measurable transformation T W X ! X , let x 2 X be an m-periodic point which is not k-periodic for any other k j m. Then the T invariant measure in (2.16) is ergodic since the only nonempty T -invariant set contained in the orbit of x is the orbit itself. Now we present a criterion for ergodicity in terms of invariant functions. More precisely, we use the notion of T -invariance almost everywhere introduced in Definition 2.11. Proposition 2.10. If T W X ! X be a measurable transformation and is a measure in X , then the following properties are equivalent: 1. The measure is ergodic with respect to T . 2. If 'W X ! R is a measurable function that is T -invariant almost everywhere, then it is constant almost everywhere, that is, there exists ˛ 2 R such that '.x/ D ˛ for -almost every x 2 X . Proof. We first assume that is ergodic. Let 'W X ! R be a measurable function that is T -invariant almost everywhere. Proceeding in a similar manner to that in the proof of Proposition 2.8, we can show that ' 1 A is T -invariant almost everywhere for every measurable subset A R. Indeed, let B X be a T -invariant measurable set with .X n B/ D 0 such that 'jB is .T jB/-invariant. If A R is measurable, then by Proposition 2.8, we have B \ ' 1 A D B \ D

[

[

' 1 ˛

˛2A

.'jB/1 ˛

˛2A

D

[

.T jB/1 .'jB/1 ˛

˛2A

D

[

.'jB ı T jB/1 ˛

˛2A

D .' ı T /jB

1

A D B \ .' ı T /1 A:

Therefore, by ergodicity, .' 1 A/ D 0

or .X n ' 1 A/ D 0:

(2.48)

2.6 Ergodicity

45

For each n 2 Z and m 2 N, we consider the interval Amn D Œn=2m ; .n C 1/=2m/. For a fixed m, the sets ˚ ' 1 Amn D x 2 X W n=2m '.x/ < .n C 1/=2m S are pairwise disjoint and satisfy n2Z ' 1 Amn D X . It follows from (2.48) that there exists a unique n D T nm 2 Z such that .X n ' 1 Amnm / D 0. Now we observe that the intersection A D m2N Amnm contains at most one point since each interval Amnm has length 1=2m. Moreover, .X n '

1

A/ D X n

\

! '

1

m2N

D

[

Amnm !

.X n Amnm / D 0:

m2N

This shows that A is nonempty, and thus, it consists of a single point say ˛. In other words, there exists ˛ 2 R such that '.x/ D ˛ for -almost every x 2 X . Now we assume that all measurable functions that are T -invariant almost everywhere are constant almost everywhere. In particular, if a set A X is measurable and T -invariant, then its characteristic function A is also measurable and T -invariant, and thus, A D 0 -almost everywhere or A D 1 -almost everywhere in X . We conclude that .A/ D 0 or .X n A/ D 0. Therefore, is ergodic. t u The following is a simple consequence of the proof of Proposition 2.10 for finite measures: Proposition 2.11. If T W X ! X be a measurable transformation and is a finite measure in X , then the following properties are equivalent: 1. The measure is ergodic with respect to T . 2. If ' 2 L1 .X; / is T -invariant almost everywhere, then it is constant almost everywhere.

2.6.2 Fourier Coefficients and Ergodicity We illustrate in this section how the Fourier coefficients can be used to establish the ergodicity of some transformations. Given a function ' 2 L1 .T; m/, where m is the Lebesgue measure in Œ0; 1, we define the Fourier coefficients of ' by

46


Z

1

ak .'/ D

e 2 i kx '.x/ d m.x/

0

for each k 2 Z. We briefly recall some properties of these numbers. Two functions '; 2 L1 .T; m/ are equal Lebesgue-almost everywhere if and only if ak .'/ D ak . / for every k 2 Z. In other words, the Fourier coefficients of a function determine it completely on a set of full Lebesgue measure. In particular: 1. A function ' 2 L1 .T; m/ is zero Lebesgue-almost everywhere if and only if ak .'/ D 0 for every k 2 Z. 2. A function ' 2 L1 .T; m/ is constant almost everywhere if and only if ak .'/ D 0 for every k 2 Z n f0g, in which case '.x/ D a0 .'/ for Lebesgue-almost every x 2 T. Furthermore, ak .'/ ! 0 when jkj ! 1, by the Riemann–Lebesgue lemma. Example 2.6. Given ˛ 2 R n Q, let T˛ W T ! T be the irrational interval translation defined by T˛ .x/ D x C ˛ mod 1 (see Sect. 2.2.3). For each ' 2 L1 .T; m/ and k 2 Z, we have Z

1

ak .' ı T˛ / D Z

e 2 i kx '.T˛ .x// d m.x/

0 1

D

e 2 i k.x˛/ '.x/ d m.x/

(2.49)

0

De

2 i k˛

ak .'/:

Hence, the function ' is T˛ -invariant almost everywhere if and only if ak .' ı T˛ / D ak .'/

for every k 2 Z:

Since ˛ 2 R n Q, we have e 2 i k˛ ¤ 1 for every k ¤ 0, and thus, by (2.49), the function ' is T˛ -invariant almost everywhere if and only if ak .'/ D 0 for every k ¤ 0. This shows that all functions in L1 .T; m/ that are T˛ -invariant almost everywhere are constant almost everywhere. By Proposition 2.11, the Lebesgue measure is ergodic with respect to T˛ . One the other hand, the Lebesgue measure is not ergodic with respect to the rational translations of the interval. Indeed, consider the interval translation T˛ with ˛ D p=q 2 Q and p; q 2 Z. It is sufficient to observe that the function 'W T ! R defined by '.x/ D cos.2qx/ is T˛ -invariant, that is, ' ı T˛ D ', but it is clearly not constant almost everywhere. In conclusion, the Lebesgue measure is ergodic with respect to an interval translation T˛ if and only if ˛ 2 R n Q. We can also consider a multidimensional version of the interval translations. Given a function ' 2 L1 .Tn ; /, where is the Lebesgue measure in Tn , we define the Fourier coefficients of ' by

2.6 Ergodicity

47

Z ak .'/ D

e 2 i hk;xi '.x/ d .x/

Tn

for each k 2 Zn , where hk; xi D k1 x1 C C kn xn

(2.50)

is the standard inner product in Rn . We recall that two functions '; 2 L1 .Tn ; / are equal Lebesgue-almost everywhere if and only if ak .'/ D ak . / for every k 2 Zn . Moreover, ak .'/ ! 0 when kkk ! 1 by the Riemann–Lebesgue lemma. Example 2.7. Let TA be the automorphism of the torus Tn induced by a matrix A. We assume that det.Am Id/ ¤ 0

for every m 2 Z n f0g:

(2.51)

Now let ' 2 L1 .Tn ; /. Since jdet Aj D 1, for each k 2 Zn , we have Z ak .' ı TA / D

Tn

Z D

Tn

Z D

Tn

e 2 i hk;xi '.Ax/ d .x/ 1 xi

e 2 i hk;A

'.x/ d .x/

1 / k;xi

e 2 i h.A

'.x/ d .x/ D a.A1 / k .'/:

The function ' is TA -invariant almost everywhere (with respect to ) if and only if ak .'/ D a.A1 / k .'/

(2.52)

for every k 2 Zn . Now we observe that if k D .A /m k for some m; k 2 Z, with k ¤ 0, then .A /m and thus also Am would have 1 as eigenvalue, which contradicts (2.51). This implies that the function m 7! .A /m k is injective for each k ¤ 0, and hence, k.A /m kk ! 1 when m ! 1: It follows from the Riemann–Lebesgue lemma that a.A /m k .'/ ! 0 when m ! 1, and using (2.52), we conclude that ak .'/ D 0 for every k ¤ 0. Therefore, '.x/ D a0 .'/ for -almost every x 2 Tn . By Proposition 2.11, the measure is ergodic with respect to TA . 2.8. If a matrix A 2 S.n; Z/ has no eigenvalues in S 1 , such as A D Example 2 1 , then no power of A has 1 as eigenvalue, and (2.51) holds. It follows from 11 Example 2.7 that if A 2 S.n; Z/ has no eigenvalues in S 1 , then is ergodic with respect to the toral automorphism TA .

48


2.6.3 The Case of Invariant Measures Now we consider the particular case of invariant ergodic measures, and we obtain additional information to that given by Birkhoff’s ergodic theorem (Theorem 2.2). Proposition 2.12. Let T W X ! X be a measurable transformation and let be a T -invariant ergodic measure in X with .X / < 1. If ' 2 L1 .X; /, then 1X 1 '.T k .x// D n!1 n .X / n1

Z ' d

lim

kD0

X

for -almost every x 2 X . Proof. By Theorem 2.2, the function 'T defined -almost everywhere by (2.34) is T -invariant almost everywhere. Therefore, by Proposition 2.11, we conclude that 'T is constant almost everywhere. The desired result follows now readily from (2.35). t u Under the assumptions of Proposition 2.12, using (2.32) and (2.33), we find that if A X is a measurable set, then lim

n!1

.A/ n .A; x/ D n .X /

for -almost every x 2 X . In other words, with respect to an invariant ergodic measure, the frequency with which the orbit of a “typical” point visits a given set is proportional to the measure of the set.

2.7 Applications to Number Theory We present briefly in this section some applications of ergodic theory to problems of number theory. In particular, we consider fractional parts of polynomials and continued fractions.

2.7.1 Fractional Parts of Polynomials Consider the polynomial P .t/ D a0 t r C C ar ; where r 2 N and a0 ; : : : ; ar 2 R.

(2.53)

2.7 Applications to Number Theory

49

Definition 2.14. The numbers Pn D P .n/ mod 1 2 Œ0; 1/, for n 2 N, are called the fractional parts of the polynomial P . Now we introduce the notion of uniformly distributed sequence. Definition 2.15. We say that a sequence .yn /n2N Œ0; 1 is uniformly distributed if 1X lim '.yk / D n!1 n n

kD1

Z

1

'.x/ dx 0

for every continuous function 'W Œ0; 1 ! R. We want to illustrate how ergodic theory can be used to study the distribution of the fractional parts of a polynomial. Example 2.9. Given r D 1, a0 D ˛ 2 R n Q, and a1 D ˇ 2 R, we have Pn D T n .ˇ/, where T .x/ D x C ˛ mod 1. By Example 2.6, the irrational translation T is ergodic. Therefore, by Proposition 2.12, for any continuous function 'W Œ0; 1 ! R, we have 1X 1X lim '.Pk / D lim '.T k .ˇ// D n!1 n n!1 n n

n

kD1

kD1

Z

1

'.x/ dx 0

for Lebesgue-almost every ˇ 2 Œ0; 1. This shows that if ˛ 62 Q, then the sequence .Pn /n2N of fractional parts of the polynomial P .t/ D ˛t Cˇ is uniformly distributed for Lebesgue-almost every ˇ 2 R. Now we consider polynomials of degree 2. Proposition 2.13. Given ˛ 2 R n Q, the sequence .Pn /n2N of fractional parts of the polynomial P .t/ D ˛t 2 C ˇt C is uniformly distributed for Lebesgue-almost every .ˇ; / 2 R2 . Proof. Let T W T2 ! T2 be defined by T .x; y/ D .x C ˛; y C 2x C ˛/ mod 1: Since det d.x;y/ T D 1 for .x; y/ 2 T2 , the transformation T preserves the Lebesgue measure. Moreover, one can use induction to show that T n .x; y/ D .x C n˛; ˛n2 C 2nx C y/ mod 1 D .x C n˛; Pn / mod 1 for every n 2 N, with ˇ D 2x and D y. Now we show that the Lebesgue measure in T2 is ergodic with respect to T . Take ' 2 L1 .T2 ; /. The Fourier coefficients of the function ' ı T are given by

50


Z ak` .' ı T / D Z

T2

D T2

e 2 i.kxC`y/ '.x C ˛; y C 2x C ˛/ d .x; y/ e 2 i Œ.k2`/xC`y e 2 i.k`/˛ '.x; y/ d .x; y/

D e 2 i.k`/˛ ak2`;` .'/: If ' is T -invariant almost everywhere (with respect to ), then e 2 i.k`/˛ ak2`;` .'/ D ak` .'/

(2.54)

for every k; ` 2 Z. In particular, if ` D 0, then e 2 i k˛ ak0 .'/ D ak0 .'/

for every k 2 Z:

Since ˛ 2 R n Q, we have e 2 i k˛ ¤ 1, and hence, ak0 .'/ D 0 for every k 2 Z n f0g. On the other hand, if ak` .'/ ¤ 0 for some ` ¤ 0, then it follows from (2.54) that jak2`;` .'/j D jak` .'/j ¤ 0

for every k 2 Z;

and hence, jak2n`;` .'/j D jak` .'/j ¤ 0 for every k 2 Z and n 2 N. But this contradicts the Riemann–Lebesgue lemma, and we conclude that ak` .'/ D 0 for every .k; `/ 2 Z2 n f.0; 0/g: Therefore, ' D a00 .'/ Lebesgue-almost everywhere, and so ' is constant almost everywhere. This shows that the Lebesgue measure is ergodic with respect to T . For each continuous function W Œ0; 1 ! R, we define a continuous function in T2 by '.x; y/ D .y/. By Proposition 2.12, we have 1X 1X .Pk / D lim '.T k .x; y// n!1 n n!1 n kD1 kD1 Z D ' d n

n

lim

Z

T2

1

D

./ d 0

for -almost every .x; y/ 2 T2 and thus for Lebesgue-almost every .ˇ; / 2 R2 . This yields the desired result. t u


51

Fig. 2.11 Gauss transformation 1

1

2.7.2 Continued Fractions Now we consider the relation between ergodic theory and the theory of continued fractions. We first introduce a transformation that is closely related to the theory of continued fractions. Definition 2.16. We define the Gauss transformation T W Œ0; 1/ ! Œ0; 1/ by ( T .x/ D

1=x mod 1

if x ¤ 0;

0

if x D 0

(see Fig. 2.11). Given an irrational number x 2 .0; 1/, we can define positive integers nj .x/ D

1

(2.55)

T j 1 .x/

for each j 2 N. Since T j .x/ D T T j 1 .x/ D

1 T j 1 .x/

nj .x/;

we obtain T j 1 .x/ D for each j 2 N. Therefore,

1 nj .x/ C T j .x/

52


xD

1

1 D n1 .x/ C T .x/

n1 .x/ C

1 n2 .x/ C T 2 .x/

1

D

1

n1 .x/ C n2 .x/ C

1 n3 .x/ C T 3 .x/

and so on. Furthermore, one can show that the sequence 1 ; n1 .x/

1 n1 .x/ C

1 n2 .x/

1

;

;

1

n1 .x/ C

n2 .x/ C

:::

1 n3 .x/

converges to x. So we simply write xD

1

: 1 n1 .x/ C n2 .x/ C

(2.56)

The right-hand side of (2.56) is called the continued fraction of x. Now we consider the problem of computing the frequency with which an integer k 2 N occurs in the sequence n1 .x/; n2 .x/; : : :. We first show how to translate this problem into the language of ergodic theory. It follows from (2.55) that nj .x/ D k if and only if 1 1 j 1 ; : .x/ 2 T kC1 k Therefore, given an irrational number x 2 .0; 1/, the frequency with which k 2 N occurs in the sequence n1 .x/; n2 .x/; : : : is given by ˚ 1 card j 2 f1; : : : ; ng W nj .x/ D k n n 1X D lim 1 ; 1 i .T j 1 .x//; n!1 n kC1 k j D1

k .x/ D lim

n!1

whenever the limit exists. We will show that k .x/ is well defined for Lebesguealmost every x 2 .0; 1/. We start by considering an appropriate measure.


53

Definition 2.17. The Gauss measure is the probability measure in Œ0; 1/ defined by Z dx 1 .A/ D log 2 A 1 C x for each measurable set A Œ0; 1/. We note that

1 1 < .k C 1/2 4: x When n is even, we obtain ˇ ˇ j.T n /0 .x/j D ˇŒ.T 2 /n=2 0 .x/ˇ 4n=2 D 2n ; and when n is odd, ˇ ˇ j.T n /0 .x/j D ˇ.T n1 ı T /0 .x/ˇ ˇ ˇ D ˇ.T n1 /0 .T .x//ˇ jT 0 .x/j 4.n1/=2 D 2n1 : Therefore, j.T n /0 .x/j 2n1 for every x 2

S1

kD1 Ik .


55

Now take x; y 2 Œ0; 1, n 2 N, and i0 ; : : : ; in1 2 N. By the mean value theorem, we obtain ˇ jx yj D ˇT n .

i0 in1 .x//

D j.T n /0 .z/j j 2n1 j

T n.

i0 in1 .x/

i0 in1 .x/

i0 in1 .y//

ˇ ˇ

i0 in1 .y/j

i0 in1 .y/j

for some point z 2 Ii0 in1 , and thus, j

i0 in1 .x/

i0 in1 .y/j

21n jx yj:

(2.59)

By (2.58), it follows from (2.59) that the interval Ii0 in1 has length at most 21n . Therefore, given a measurable set B Œ0; 1/ and " > 0, there is a disjoint union J of intervals in the family ˚ F D Ii0 in1 W n 2 N and i0 ; : : : ; in1 2 N

(2.60)

such that .B n J / < " and .J n B/ < ": Now we observe that ˇ 0 ˇ ˇ .x/ˇ ˇ i00 in1 ˇ D ˇ .y/ˇ

ˇ ˇ n 0 ˇ.T / . i i .y//ˇ 0 n1 ˇ ˇ ˇ.T n /0 . i i .x//ˇ 0 n1 ˇ ˇ n1 Y ˇT 0 . ij in1 .y//ˇ ˇ ˇ D ˇT 0 . i i .x//ˇ

i0 in1

j D0

n1 Y

j

1C

ˇ 0 ˇT .

j D0

n1 Y j D0

n1

ij in1 .y//

ˇ ˇT 0 .

T 0.

ij in1 .x//

ij in1 .x//

ˇ ˇ

ˇ! ˇ

jT 00 .zj /j j ij in1 .y/ ij in1 .x/j ˇ ˇ 1C ˇT 0 . i i .x//ˇ j

!

n1

for some zj 2 Iij in1 . Moreover, j

ij C1 in1 .y/

ij C1 in1 .x/j

D jT 0 .wj /j j

ij in1 .y/

for some wj 2 Iij in1 , and since Iij in1 Iij , we obtain

ij in1 .x/j

56


ˇ ˇT 0 .

2 jT 00 .zj /j ˇ D 0 ˇ ij in1 .x// jT .wj /j

ij in1 .x/

2

w2j

z3j

2.ij C 1/3 ij 4 1 3 2 1C D 16 ij ij

Therefore, it follows from (2.59) that ˇ ˇ ˇ ˇ

ˇ

0 ˇ i0 in1 .x/ ˇ 0 ˇ i0 in1 .y/

n1 Y

.1 C 16 22nCj jx yj/

j D0

1 Y

.1 C 2j /

j D4

D exp

1 X

log.1 C 2j /

j D4

exp

1 X

2j D e 2 DW c 5

j D4

since log.1 C x/ x for every x 0. Let again m be the Lebesgue measure in Œ0; 1. For any measurable set A Œ0; 1, it follows from (2.57) that .Ii0 in1 \ T

n

1 A/ D log 2

Z Ii0 in1 \T n A

1 m.Ii0 in1 \ T n A/ 2 log 2

1 m. 2 log 2 Z 1 D j 2 log 2 A

D

i0 in1 .A// 0 i0 in1 j d m

˚ 1 m.A/ inf j 2 log 2 Z 1 1 m.A/ j 2c log 2 0

dx 1Cx

0 i0 in1 .x/j

W x 2 .0; 1/

0 i0 in1 j d m


D

57

1 m.A/ 2c log 2

Z 1 dm Ii0 in1

log 2 .A/.Ii0 in1 /: 2c

Now we assume that the set A is T -invariant. By former inequality, we obtain .Ii0 in1 \ A/

log 2 .A/.Ii0 in1 /: 2c

(2.61)

As observed above, given " > 0, there is a disjoint union J of intervals in the family F (see (2.60)) such that setting X D Œ0; 1/, .J n .X n A// < " and ..X n A/ n J / < ": By (2.61), we conclude that .J \ A/

log 2 .A/.J /; 2c

and since J n .X n A/ D J \ A, we obtain ">

log 2 .A/..X n A/ "/: 2c

Letting " ! 0 yields .A/.X n A/ D 0, and thus, the measure is ergodic.

t u

With slight changes, the method of proof of Theorem 2.3 can be applied to a large class of transformations with an invariant measure that is absolutely continuous with respect to the Lebesgue measure. See in particular Exercise 3.28. It follows from Theorem 2.3 and Proposition 2.12 that 1X 1 ; 1 i .T j 1 .x// n!1 n kC1 k j D1 Z 1 ; 1 i d D n

k .x/ D lim

Œ0;1/

D D

1 log 2

kC1 k

Z

1=k 1=.kC1/

dx 1Cx

.k C 1/2 1 log log 2 k.k C 2/

for -almost every x 2 Œ0; 1/. Moreover, since

58


0 .x C 1/2 2 0, all frequencies k .x/ are distinct, and 1 .x/ > 2 .x/ > 3 .x/ > for -almost every x 2 Œ0; 1/.

2.8 Exercises Exercise 2.1. Show that if Rw is an irrational circle rotation, then for each z 2 S 1 , the set fRwn .z/ W n 2 Ng is dense in S 1 . Exercise 2.2. Given an integer q > 1, show that if T is a piecewise linear expanding map of the interval obtained from the subintervals 1 1 2 q1 0; , , : : :, ; ;1 ; q q q q then h ı T ı h1 D Eq in the set ˚ S 1 n e 2 i k=q W k D 0; : : : ; q 1 ; where hW T ! S 1 is the one-to-one map in (2.11). Exercise 2.3. Show that if T is a piecewise linear expanding map of the interval, obtained from a (finite or infinite) number q of subintervals, then p.T / D log q. Exercise 2.4. For the toral endomorphism of T2 induced by the matrix 31 11 , show that: 1. The orbit of each point with rational coordinates contains at least one periodic point. 2. Not all points with rational coordinates are periodic. 3. .1=3; 1=3/ is a periodic point of period 8. Exercise 2.5. the periodic entropy of the automorphism of T2 induced by 2 1 Compute the matrix 1 1 . 2 by the Exercise Find all periodic points of the automorphism of T induced 2 2.6. 1 matrix 1 1 . Hint: consider the set of points with rational coordinates Q2 =Z2 T2 .

Exercise 2.7. Show that the statement in Theorem 2.1 in general does not hold for infinite measures. Hint: consider the map T W R ! R given by T .x/ D x C 1.

2.8 Exercises

59

Exercise 2.8. Find a continuous transformation T W X ! X of a compact metric space and an infinite measure in X for which the statement in Theorem 2.1 does not hold. Exercise 2.9. Let T W X ! X be a measurable transformation and let be a T invariant probability measure in X . Show that: 1. is ergodic if and only if [ 1 n T A D1 nD1

for every measurable set A X with .A/ > 0. 2. If is ergodic and 'W X ! RC 0 is a measurable function such that 1X '.T k .x// < 1 n n1

lim sup n!1

kD0

for -almost every x 2 X , then ' 2 L1 .X; /. Hint: use Theorem A.1. Exercise 2.10. Let x 2 X be an m-periodic point of a transformation T W X ! X . Show that the limit n1 1X lim '.T k .x// n!1 n kD0

exists for any function 'W X ! R. Also, compute theRlimit explicitly and express it in terms of the measure in (2.16). Hint: verify that X ' d ıx D '.x/. Exercise 2.11. Let T W X ! X be a measurable transformation preserving an R ergodic finite measure in X . Show that if ' 2 L1 .X; / has integral X ' d ¤ 0, then ˇ n1 ˇ ˇX ˇ 1 ˇ ˇ lim log ˇ '.T k .x//ˇ D 1 n!1 log n ˇ ˇ kD0

for -almost every x 2 X . Exercise 2.12. Show that for Lebesgue-almost every x 2 Œ0; 1, the frequency of the number 1 in the base-2 representation of x is equal to 1=2. Hint: show that the transformation x 7! 2x mod 1 is ergodic with respect to the Lebesgue measure. Exercise 2.13. Show that if T W X ! X is a measurable transformation preserving a finite measure in X and ' 2 L1 .X; /, then '.T n .x// D0 n!1 n lim

for -almost every x 2 X .

60


Exercise 2.14. Given ˛1 ; : : : ; ˛n 2 R, let T˛1 ;:::;˛n W Tn ! Tn be defined by T˛1 ;:::;˛n .x1 ; : : : ; xn / D .x1 C ˛1 ; : : : ; xn C ˛n / mod 1:

(2.62)

The transformation T˛1 ;:::;˛n is called a toral translation. Describe a necessary and sufficient condition in terms of ˛1 ; : : : ; ˛n so that the Lebesgue measure in Tn is ergodic with respect to T˛1 ;:::;˛n . Exercise 2.15. Let .X; A; / be a finite measure space and let F A be a finite -subalgebra. Given an A-measurable function ' 2 L1 .X; / and a set A 2 F with .A/ > 0 such that no proper subset of A in F has positive measure, show that R 'F .x/ D

' d .A/

A

for -almost every x 2 A. This yields a generalization of formula (2.38) for arbitrary functions. Exercise 2.16. Let T W X ! X be a measurable transformation preserving a finite measure in X with .X / D 1 (see Appendix A for the definition of -finite measure). Show that if ' 2 L1 .X; /, then: 1. The limit 'T .x/ in (2.34) exists for -almost every x 2 X . 2. The function 'T is measurable and T -invariant almost everywhere. Hint: first obtain a version of Proposition 2.9 for -finite measures (see Theorems A.5 and A.6). This yields a generalization of Birkhoff’s ergodic theorem to some infinite measure spaces. Exercise 2.17. Let T W X ! X be a measurable transformation preserving an ergodic measure in X which is infinite but -finite. Show that if ' 2 L1 .X; /, then n1 1X lim '.T k .x// D 0 n!1 n kD0

for -almost every x 2 X . Hint: use Exercise 2.16. Exercise 2.18. Let 'W S 1 ! R be the function defined by ' D h.Œ0;1=2/ (see Fig. 2.12), with h as in (2.11). For the Lebesgue measure in S 1 , find the conditional expectation 'F with respect to the -subalgebra F of E2 -invariant sets. Exercise 2.19. Show that for each integer n > 1, the Lebesgue measure m in T is ergodic with respect to the transformation T .x/ D nx mod 1. Hint: show that the Fourier coefficients of a function ' 2 L1 .T; m/ satisfy ( ak .' ı T / D

ak=n .'/ if n j k, 0

if n − k.

2.8 Exercises

61

Fig. 2.12 Conditional expectation in the circle

ϕ=1

ϕ=0

Exercise 2.20. Show that for each integer n > 1, the Lebesgue measure in S 1 is ergodic with respect to the transformation z 7! zn . Exercise 2.21. Let T W Rn ! Rn be a C 1 diffeomorphism. Show that: 1. T preserves the Lebesgue measure if and only if jdet dx T j D 1 for every x 2 Rn . 2. For a measure in Rn that is absolutely continuous with respect to the Lebesgue measure, with Radon–Nikodym derivative (see Appendix A for the definition), the diffeomorphism T preserves if and only if

D . ı T /jdet d T j

(2.63)

Lebesgue-almost everywhere in Rn . 3. If is defined everywhere and is continuous, then T preserves if and only if (2.63) holds everywhere. Exercise 2.22. Let T W Rn ! Rn be a C 1 diffeomorphism preserving a measure that is absolutely continuous with respect to the Lebesgue measure, with a continuous Radon–Nikodym derivative defined everywhere. We also assume that 0 < inff .x/ W x 2 Rn g supf .x/ W x 2 Rn g < 1:

1. Show that jdet dx T m j D 1 for each m-periodic point x 2 Rn . 2. Show that the set fdet dx T n W x 2 Rn and n 2 Ng is bounded. Hint: use Exercise 2.21.

62


Exercise 2.23. Given a transformation T W X ! X , we say that a point x 2 X has period m if x is m-periodic, but is not k-periodic for any other k j m. Let q.T / D lim sup m!1

˚ 1 logC card x 2 X W x has period m : m

Compute q.T / for: 1. A rational circle rotation. 2. An expanding map of the circle. 3. The toral automorphism of T2 induced by the matrix 21 11 . Exercise 2.24. Let ni D ni .x/ for i 2 N be the integers in the continued fraction p of a number x 2 .0; 1/. Show that the geometric average k n1 nk converges Lebesgue-almost everywhere to a constant when k ! 1, and compute this constant as explicitly as possible. Exercise 2.25. Let T W X ! X be a measurable transformation preserving a probability measure in X . We also assume that T is invertible and that its inverse is measurable. By Poincaré’s recurrence theorem (Theorem 2.1), given a measurable set A X with .A/ > 0, the integer ˚ nA .x/ D min n 2 N W T n .x/ 2 A is well defined for -almost every x 2 A. It is called the (first) return time of the orbit of x to A. Define a transformation SA W A ! A for -almost every x 2 A by SA .x/ D T nA .x/ .x/: This is called the induced transformation by T on A. Show that: 1. SA preserves the probability measure A D =.A/ in A, called the induced invariant measure. 2. The function x 7! nA .x/ defined -almost everywhere in the set A is A integrable. R 3. If is ergodic, then A nA d D 1. Hint: consider the decomposition of A, up to a set of zero measure, into the pairwise disjoint subsets fx 2 A W nA .x/ D ng for n 2 N. Exercise 2.26. Draw the graph of the transformation SŒ0;1=2/ induced by the interval translation T .x/ D x C ˛ mod 1 on the interval Œ0; 1=2/. Exercise 2.27. Show that any transformation induced by an ergodic transformation is ergodic with respect to the induced invariant measure in Exercise 2.25. Exercise 2.28. Consider the transformation T W Œ0; 1/ ! Œ0; 1/ given by

2.8 Exercises

63

Fig. 2.13 The transformation in Exercise 2.28 1

1

8 ˆ ˆ <x C 1=2

p T .x/ D x C 1=2 1= 2 ˆ ˆ :x 1=p2

if 0 x < 1=2;

p if 1=2 x < 1= 2; p if 1= 2 x < 1

(see Fig. 2.13). Show that: 1. T preserves the Lebesgue measure. 2. T is ergodic but T 2 is not ergodic. Hint: the interval Œ0; 1=2/ is T 2 -invariant. Exercise 2.29. Given a transformation T W X ! X with ˚ an WD card x 2 X W T n .x/ D x < 1 for every n 2 N, we define its zeta function by .z/ D exp

1 X an zn nD1

n

(2.64)

for each z 2 C with a convergent power series. Show that: 1. If an > 0 for every n 2 N, then the radius of convergence of the power series is equal to e p.T / with p.T / as in (2.20). 2. If jzj < e p.T / , then Y 1 .z/ D ; 1 z`. /

64


with the product taken over all periodic orbits , where `. / is the period of . Hint: note that setting ˚ bk D card x 2 X W x has period k ; we have an D

P kjn

bk , and hence, .z/ D exp

1 X 1 X bk zj k kD1 j D1

D

1 Y kD1

0 @exp

jk

1 jk X z j D1

j

1bk =k A

:

Exercise 2.30. Compute the zeta function of the toral automorphism of T2 induced 2 1 by the matrix 1 1 .

Notes The books [26, 44, 57, 103] are excellent sources for ergodic theory. A slightly modified version of Theorem 2.1 was established by Poincaré in his seminal work on the three-body problem [78]. Incidentally, Lebesgue’s theory of integration [53] only appeared afterwards. Theorem 2.2 is due to Birkhoff [15], and our proof is based on [44]. The statements in Example 2.9 and Proposition 2.13 are special cases of a result of Weyl [104] saying that if ak 2 R n Q for some k 2 f0; : : : ; rg, for the polynomial P in (2.53), then the sequence .Pn /n2N of fractional parts of P is uniformly distributed. See, for example, [26, Sect. 7.2] for a proof using ergodic theory. The zeta function in (2.64) was introduced by Artin and Mazur in [6]. We refer to [70] for a detailed description.

Chapter 3

Further Topics

We discuss in this chapter several additional topics of ergodic theory. In particular, we establish the existence of finite invariant measures for any continuous transformation of a compact metric space. We also discuss the notions of unique ergodicity and mixing. In particular, we show how unique ergodicity can be characterized in terms of uniform convergence in Birkhoff’s ergodic theorem. We conclude this chapter with brief introductions to symbolic dynamics and topological dynamics, with emphasis on their relations to ergodic theory. This includes the construction of Markov measures and Bernoulli measures, as well as the relation between Markov measures and topological Markov chains. We also show that each ergodic property implies a corresponding topological property.

3.1 Introduction We already described several classes of measure-preserving transformations, but we discussed no procedure to find invariant measures of a given transformation. There is however a simple yet powerful procedure due to Krylov and Bogolubov to find finite invariant measures. For a brief description, let T W X ! X be a continuous transformation of a compact metric space. Given a finite measure in X , we consider the sequence of measures n defined by 1X .T k A/ n n1

n .A/ D

kD0

obtained from averaging over the pullbacks of the measure under the dynamics. One can show that any sublimit of the sequence n (which always has sublimits) is a finite T -invariant measure. In another direction, the notion of ergodicity was already discussed in the former chapter. It turns out that it can be strengthen in several ways. In particular, L. Barreira, Ergodic Theory, Hyperbolic Dynamics and Dimension Theory, Universitext, DOI 10.1007/978-3-642-28090-0 3, © Springer-Verlag Berlin Heidelberg 2012

65

66

3 Further Topics

if a given transformation has a single invariant probability measure (such as an irrational rotation of the circle), then this measure is ergodic. This justifies why such transformation is called uniquely ergodic. The existence of a single invariant probability measure turns out to imply that the convergence in Birkhoff’s ergodic theorem is uniform. In particular, this property has applications to number theory, among many other applications of the theory of dynamical systems to number theory. Still another way to strengthen the notion of ergodicity is motivated by an equivalent description: a finite T -invariant measure is ergodic if and only if 1X .T k A \ B/ D .A/.B/ n!1 n n1

lim

kD0

for any measurable sets A and B. This can be described as an asymptotic independence in the average of the events T n A and B when n ! 1. The stronger notion of mixing is obtained by discarding the average: we say that a measure is mixing if lim .T n A \ B/ D .A/.B/

n!1

for any measurable sets A and B. This can be described as an asymptotic independence of the events T n A and B when n ! 1. There are several other possibilities of strengthening the notion of ergodicity, although a comprehensive description falls out of the scope of this book. We emphasize that ergodic theory is a confluence of many areas and not only from the theory of dynamical systems. Among them is symbolic dynamics. Our related primary interest is the possibility of coding repellers and hyperbolic sets, which in many situations allows to give simpler proofs. The symbolic coding is effected by dividing the phase space into appropriate pieces, forming what is called a Markov partition, and then coding each trajectory by enumerating successively the pieces of the partition containing the elements of the trajectory. It turns out that the symbolic coding of a measure-preserving dynamics induces shift-invariant measures on the symbolic dynamics, such as Markov measures and Bernoulli measures.

3.2 Existence of Invariant Measures In Chap. 2, we described several examples of measure-preserving transformations. However, we did not discuss the problem of the existence of (finite) invariant measures for a given transformation. For example, this is crucial for Poincaré’s recurrence theorem (Theorem 2.1) and Birkhoff’s ergodic theorem (Theorem 2.2). We address in this section the problem of the existence of invariant measures for continuous transformations of compact metric spaces.

3.2 Existence of Invariant Measures

67

3.2.1 Preliminary Results Given a compact metric space X , we denote by C.X / the vector space of continuous functions 'W X ! R. We define a norm in C.X / by ˚ k'k1 D max j'.x/j W x 2 X :

(3.1)

Proposition 3.1. If X is a compact metric space, then with the norm in (3.1), the space C.X / is separable, that is, it contains a countable dense set. Proof. Since X is compact, there exists a sequence .xn /n2N dense in X . Now for each m; n 2 N, we consider the continuous function ( nm .x/ D

1=m .x; xn /

if .x; xn / 1=m;

0

if .x; xn / > 1=m;

where is the distance in X . Let also A C.X / be the smallest algebra containing the constant function 1 and all the functions nm . This means that A is the smallest subset of C.X / which contains these functions such that ' C ; ' ; ' 2 A whenever '; 2 A and 2 R. Now we observe that given two distinct points x; y 2 X , there exist n; m 2 N such that nm .x/ ¤ nm .y/. Indeed, taking ı D .x; y/;

xn 2 B.x; ı=N /;

and m 2 .N=..N 1/ı/; N=ı/ \ N

(the interval is nonempty provided that N is sufficiently large), we have nm .x/ D 1=m .x; xn / > 0 and nm .y/ D 0: Hence, the functions nm and thus also those in A distinguish between the points of X . Now we can apply Stone–Weierstrass theorem which (for our purposes) says that if X is a compact metric space and A C.X / is an algebra containing the constant function 1 and distinguishing between the points of X , then A D C.X /. Finally, we consider the countable subset B A formed by all functions of the form N X c0 C ci ni mi i D1

with c0 ; c1 ; : : : ; cN 2 Q and ni ; mi 2 N for i D 1; : : : ; N , and all their finite products. One can verify that B is dense in A and thus also in C.X /. This shows that C.X / is separable. t u Now we denote by M.X / the set of all Borel probability measures in X . We want to show that M.X / is metrizable. Proposition 3.1 guarantees the existence of a sequence of continuous functions .'n /n2N with closure equal to the ball

68

3 Further Topics

˚ ' 2 C.X / W k'k1 1 : We define a function d W M.X / M.X / ! RC 0 by ˇZ ˇ Z 1 X ˇ 1 ˇˇ d.; / D 'n d 'n d ˇˇ : 2n ˇ X

nD1

Clearly, d.; /

1 X 2k'n k1 nD1

(3.2)

X

2n

2

for every ; 2 M.X /, and thus, d is well defined. Proposition 3.2. If X is a compact metric space, then d is a distance in M.X /. Moreover, given a sequence of measures .n /n2N in M.X / and 2 M.X /, we have d.n ; / ! 0 when n ! 1 if and only if Z

Z lim

n!1 X

' dn D

' d X

for every function ' 2 C.X /. Proof. For the first property, we note that d.; / 0 and d.; / D d.; / for any measures and . It is also immediate that d satisfies the triangle inequality. Thus, to show that d is a distance, it remains to verify that D whenever d.; / D 0. If d.; / D 0, then Z Z 'm d D 'm d X

X

for every m 2 N. By the choice of the sequence .'m /m2N , given a nonzero function ' 2 C.X / and " > 0, there exists m 2 N such that ' < ": ' (3.3) m k'k 1 1 Therefore,

ˇZ ˇ ˇZ ˇ Z Z ˇ ˇ ˇ 1 ˇˇ ˇ ˇ 'm d ˇ C 2" D 2": ' d ' d ' d m ˇ ˇ ˇ ˇ k'k1 X X X X

Since " is arbitrary, we conclude that Z Z ' d D ' d X

X

for every ' 2 C.X /. Hence, D , and d is a distance.


69

For the second property, we first assume that d.n ; / ! 0 when n ! 1. Then Z

Z lim

n!1 X

'm dn D

'm d X

for every m 2 N. Again, given a nonzero function ' 2 C.X / and " > 0, there exists m 2 N such that inequality (3.3) holds. Thus, for each n; m 2 N, we have ˇZ ˇ ˇZ ˇ Z Z ˇ ˇ ˇ 1 ˇˇ ˇ ˇ ' dn ' dˇ ˇ 'm dn 'm dˇˇ C 2": k'k1 ˇ X X X X Letting n ! 1, we obtain ˇZ ˇ Z ˇ ˇ 1 lim sup ˇˇ ' dn ' dˇˇ 2"; k'k1 n!1 X X and since " is arbitrary, we conclude that Z

Z lim

n!1 X

' dn D

' d:

(3.4)

X

Now we assume that (3.4) holds for any function ' 2 C.X /. We want to show that d.n ; / ! 0 when n ! 1. Given " > 0, for each m 2 N, there exists km 2 N such that if n > km , then ˇZ ˇ Z ˇ ˇ ˇ 'i dn ˇ km . Taking m D m."/ 2 N such that 1=2m1 < ", we obtain d.n ; / < 2" for every n > km."/ : Letting " ! 0, we conclude that d.n ; / ! 0 when n ! 1. This completes the proof of the proposition. t u We also show that with the distance d in (3.2), the space M.X / is compact.

70

3 Further Topics

Theorem 3.1. If X is a compact metric space, then M.X / is compact. Proof. Since M.X / is a metric space, it is sufficient to show that each sequence .n /n2N in M.X / has a convergent subsequence. Let .'n /n2N be again the sequence used to define the distance d in (3.2). For each n 2 N, we consider the bounded sequence ˇn W N ! Œ1; 1 defined by Z ˇn .k/ D

'k dn : X

By Tychonoff’s theorem, the space of sequences Œ1; 1N with the product topology is compact, and there exists a subsequence .ˇmn /n2N converging in this topology. R This implies that the sequence . X 'k dmn /n2N converges for each k 2 N. On the other hand, given a nonzero function ' 2 C.X / and " > 0, there exists k 2 N such that ' < ": ' k k'k 1 1 Then ˇZ ˇ ˇZ ˇ Z Z ˇ ˇ ˇ 1 ˇˇ ˇ ˇ ˇ ' d ' d ' d ' d mn m` ˇ k mn k m` ˇ C 2" ˇ ˇ k'k1 X X X X for every n; ` 2 N. We conclude that if n and ` are sufficiently large, then ˇZ ˇ Z ˇ ˇ ˇ ' dm ˇ ' d m` ˇ 3"k'k1 ; n ˇ X

X

R

and . X ' dmn /n2N is a Cauchy sequence. Therefore, one can define a linear functional in C.X / by Z ' dmn : (3.5) J.'/ D lim n!1 X

Clearly, jJ.'/j k'k1 , and in particular, J is continuous. Moreover, J.1/ D 1, and J is positive, that is, J.'/ 0 whenever ' 0. By Riesz’s representation theorem (Theorem A.4), there exists a unique measure 2 M.X / such that Z J.'/ D

' d X

for every function ' 2 C.X /. By Proposition 3.2 and (3.5), we conclude that d.mn ; / ! 0 when n ! 1. This shows that the space M.X / is compact. t u


71

3.2.2 Existence of Invariant Measures The following result shows that M.X / contains at least one T -invariant probability measure when T is continuous: Theorem 3.2 (Krylov–Bogolubov). Let X be a compact metric space. If the transformation T W X ! X is continuous, then there exists a T -invariant probability measure in X . Proof. Consider the transformation T W M.X / ! M.X / defined by .T /.A/ D .T 1 A/:

(3.6)

We note that a measure is T -invariant if and only if it is a fixed point of T . Moreover, one can easily verify that Z Z ' d.T / D .' ı T / d X

X

for every function ' 2 C.X /. By Proposition 3.2, if the sequence of measures .n /n2N M.X / converges to , then for each ' 2 C.X /, we have Z Z ' dn ! ' d X

X

when n ! 1. Since T is continuous, we conclude that if ' 2 C.X /, then Z Z ' d.T n / D .' ı T / dn X

X

Z !

Z .' ı T / d D

X

' d.T / X

when n ! 1, that is, the sequence .T n /n2N converges to T . This shows that the transformation T is continuous. Now given 2 M.X /, we consider the sequence of measures 1X k T 2 M.X /: n n1

n D

kD0

By Theorem 3.1, there exists a convergent subsequence .mn /n2N , say with limit 2 M.X /. Since T mn D

mn 1 X Tmn Tk D mn C ; mn mn mn

(3.7)

kD1

and T is continuous, letting n ! 1 in (3.7), we obtain T D .

t u

72

3 Further Topics

3.3 Unique Ergodicity We consider in this section the transformations having a unique invariant probability measure—the so-called uniquely ergodic transformations—and we study their ergodic properties. In particular, we show how unique ergodicity leads to uniform convergence in Birkhoff’s ergodic theorem.

3.3.1 Basic Notions and Uniform Convergence Clearly, if is an invariant measure, then c is also an invariant measure for any c > 0. In the following definition, we normalize the measure in order to avoid this duplication. Definition 3.1. We say that a transformation T W X ! X is uniquely ergodic if T has exactly one T -invariant probability measure in X . For example, rational circle rotations, rational interval translations, expanding maps of the circle, and linear expanding maps of the interval are not uniquely ergodic (see Exercise 3.3). The following statement justifies the expression “uniquely ergodic”: Proposition 3.3. The unique invariant probability measure of a uniquely ergodic transformation is ergodic. Proof. Let be the unique invariant probability measure of a uniquely ergodic transformation in X . If is not ergodic, then there is an invariant measurable set A X with 0 < .A/ < 1. We define new invariant probability measures 1 and 2 by 1 .B/ D

.B \ A/ .A/

and 2 .B/ D

.B \ .X n A// : .X n A/

Clearly, 1 ¤ 2 (e.g., 1 .A/ D 1 and 2 .A/ D 0). This contradicts the unique ergodicity of the transformation, and we obtain the desired statement. t u The following statement shows that Birkhoff’s ergodic theorem can be considerably strengthened for the uniquely ergodic continuous transformations of a compact metric space. We recall that C.X / denotes the space of all continuous functions 'W X ! R. Theorem 3.3. Let T W X ! X be a uniquely ergodic continuous transformation of a compact metric space. If ' 2 C.X /, then the sequence of functions 1X ' ıTk n n1

kD0

(3.8)

3.3 Unique Ergodicity

73

R

converges uniformly when n ! 1 to the constant T -invariant probability measure in X .

X

' d, where is the unique

Proof. Given x 2 X , we consider the sequence of probability measures 1X ıT k .x/ : n n1

n D

kD0

By Exercise 3.1, each accumulation point of this sequence is a T -invariant probability measure, and hence, it is equal to . This shows that .n /n2N converges to the measure . Given a function ' 2 C.X /, since Z

1X '.T k .x//; n n1

' dn D X

kD0

we thus obtain 1X '.T k .x// D n!1 n n1

Z ' d:

lim

kD0

(3.9)

X

If the convergence in (3.9) was not uniform, then for some sufficiently small " > 0, it would exist a sequence .mn /n2N N with mn % 1 such that Z n 1 1 mX k 'ıT ' d mn X

"

1

kD0

for every n 2 N. Therefore, since X is compact, there would exist a sequence .xn /n2N X such that ˇ ˇ Z n 1 ˇ 1 mX ˇ ˇ ˇ k '.T .xn // ' dˇ " ˇ ˇ mn ˇ X

(3.10)

kD0

for every n 2 N. On the other hand, it follows from Theorem 3.1 that the sequence of measures mn 1 1 X n D ıT k .xn / (3.11) mn kD0

has a subsequence .`n /n2N converging to a measure 2 M.X /. We show that is T -invariant. Given 2 C.X /, we have

74

3 Further Topics

Z

Z .

ı T / d D lim

n!1 X

X

.

ı T / d`n

m`n 1 1 X .T kC1 .x`n // n!1 m` n

D lim

kD0

1 n!1 m` n

D lim

mX `n 1

.T k .x`n // C

.T m`n .x`n //

.x`n /

kD0

m`n 1 1 X .T k .x`n //; n!1 m` n

D lim

kD0

and thus, by (3.11), we obtain Z

Z

Z . X

ı T / d D lim

n!1 X

d`n D

d: X

It follows from Proposition 2.1 that is T -invariant. But since T is uniquely ergodic, we have D , and in view of (3.10), we obtain the contradiction ˇ ˇ ˇ ˇ ˇZ ˇ m`n 1 Z Z ˇ 1 X ˇ ˇ ˇ k ˇ ˇ ˇ "ˇ '.T .x`n // ' dˇ D ˇ ' d`n ' dˇˇ ! 0 m X X X ˇ `n kD0 ˇ when n ! 1. This completes the proof of the theorem.

t u

3.3.2 Criteria for Unique Ergodicity and Examples In order to determine whether a given transformation is uniquely ergodic, it is convenient to have the following criterion: Theorem 3.4. Let T W X ! X be a continuous transformation of a compact metric space. If for each ' 2 C.X / the sequence of functions in (3.8) converges uniformly to a constant when n ! 1, then T is uniquely ergodic. Proof. We define a linear functional J W C.X / ! R by 1X ' ı T k: n!1 n n1

J.'/ D lim

(3.12)

kD0

One can easily verify that J is continuous and positive (i.e., J.'/ 0 whenever ' 0), and that J.1/ D 1. By Riesz’s representation theorem (Theorem A.4), there


75

is a unique probability measure in X such that Z ' d for every ' 2 C.X /:

J.'/ D X

On the other hand, it follows readily from (3.12) that J.' ı T / D J.'/

for every ' 2 C.X /:

Therefore, by Proposition 2.1, we conclude that is T -invariant. Now we consider the sequence of continuous functions 'n W X ! R in (3.8). Since by hypothesis Z 'n !

' d uniformly when n ! 1; X

if is a T -invariant probability measure in X , then Z

Z Z

Z

'n d ! X

' d d D X

X

' d X

when n ! 1. On the other hand, since is T -invariant, by Proposition 2.1, we have Z Z n1 Z 1X k 'n d D .' ı T / d D ' d: n X X X kD0

Therefore,

Z

Z ' d D

X

' d X

for every function ' 2 C.X /, and D . This shows that T is uniquely ergodic. t u It follows from Theorems 3.3 and 3.4 that if T W X ! X a continuous transformation of a compact metric space, then the following properties are equivalent: 1. T is uniquely ergodic. 2. For each ' 2 C.X /, the sequence of functions in (3.8) converges uniformly to a constant when n ! 1. In fact, in order to show that a given transformation is uniquely ergodic, it is sufficient to consider certain subfamilies of continuous functions for which the sequence in (3.8) converges uniformly to a constant. Given a family of functions ˚ C.X /, we denote by L.˚/ the set of all finite linear combinations of functions in ˚. Theorem 3.5. Let T W X ! X be a continuous transformation of a compact metric space, and let ˚ C.X / be a family of functions such that L.˚/ is dense in C.X /.

76

3 Further Topics

If for each ' 2 ˚ the sequence in (3.8) converges uniformly to a constant when n ! 1, then T is uniquely ergodic. Proof. Given ' 2 C.X /, there exist functions m k1

k'

!0

m

2 L.˚/ for m 2 N such that

when m ! 1:

Furthermore, for each m 2 N, there exists cm 2 R such that k

mn

cm k1 ! 0 when n ! 1;

where

1X n

(3.13)

n1

mn

D

m

ı T k:

kD0

Indeed, if m

D am1 1 C C amp p

for some 1 ; : : : ; p 2 ˚ and am1 ; : : : ; amp 2 R, then we can take cm D am1 d1 C C amp dp ; where the constants d1 ; : : : ; dp 2 R are such that 1X k ı T j ! dk n j D0 n1

uniformly when n ! 1. On the other hand, given m; ` 2 N, we have 1X k n n1

k

mn

`n k1

m

ıTk

`

ı T k k1

m

Dk

kD0

1X k n n1

` k1

m

` k1 ;

kD0

and hence, jcm c` j kcm

mn k1

Ck

mn

`n k1

kcm

mn k1

Ck

m

` k1

m

` k1 ;

Ck

Letting n ! 1, it follows from (3.13) that jcm c` j k

Ck `n

`n

c` k1

c` k1 :


77

and hence, .cm /m2N is a Cauchy sequence, say with limit c. Now let .'n /n2N be the sequence of functions in (3.8). We have k'n ck1 k'n k'

mn k1 m k1

Ck

Ck

mn

mn

cm k1 C jcm cj

cm k1 C jcm cj:

(3.14)

On the other hand, given " > 0, for any sufficiently large m, we have k'

m k1

< " and jcm cj < ":

Hence, for any such m, letting n ! 1 in (3.13), it follows from (3.14) that lim supk'n ck1 2": n!1

Since " is arbitrary, this implies that k'n ck1 ! 0 when n ! 1. Now we can apply Theorem 3.4 to obtain the desired statement. t u We describe two families of functions ˚ that can be used in Theorem 3.5. Example 3.1. For the space X D Œa; b, let ˚ D f1g [ f'k W k 2 Ng C.Œa; b/; where 'k .x/ D x k for each k 2 N. Then the set of polynomials L.˚/ in Œa; b is dense in C.Œa; b/. Example 3.2. For the space X D Œ0; 1, let ˚ D f1g [ f'k W k 2 Ng [ f

k

W k 2 Ng C.Œ0; 1/;

where 'k .x/ D cos.2 kx/

and

k .x/

D sin.2 kx/

(3.15)

for each k 2 N. Then L.˚/ is dense in C.Œ0; 1/. Now we apply the criterion for unique ergodicity in Theorem 3.5 to circle rotations. Example 3.3. Consider the circle rotation Rw (see Sect. 2.2.3) and the functions k .z/ D zk for each k 2 Z. We have 0 D 1, k .e 2 ix / C k .e 2 ix / D 2'k .x/; and k .e 2 ix / k .e 2 ix / D 2i

k .x/;

78

3 Further Topics

with 'k and k as in (3.15). By Example 3.2, the set of all finite linear combinations of the functions k is dense in C.S 1 /. Moreover, k .Rw .z// D wk k .z/: If Rw is an irrational rotation, that is, if w D e 2 i for some 62 Q, then wk ¤ 1 for every k 2 Z n f0g, and hence, ˇ ˇ ˇ ˇ ˇ ˇ X ˇ ˇ X n1 ˇ ˇ ˇ ˇ 1 n1 1 kj j ˇ ˇ ˇ k .Rw .z//ˇ D ˇ w k .z/ˇˇ ˇn n ˇ ˇ j D0 ˇ ˇ j D0 D

j1 wk n j nj1 wk j

2 !0 nj1 wk j

when n ! 1. On the other hand, 1X 0 .Rw j .z// D 1 n j D0 n1

for every n 2 N. Thus, by Theorem 3.5, the rotation Rw is uniquely ergodic. In other words, any irrational circle rotation is uniquely ergodic. (3.16) Moreover, since the circle rotations preserve the Lebesgue measure, the unique Rw invariant probability measure is the Lebesgue measure. In addition, one can easily verify that the one-to-one map hW T ! S 1 defined by h./ D e 2 i (see Sect. 2.2.3) is a homeomorphism when T has the topology induced from the standard topology in R. Hence, it follows from (2.13) and (3.16) that any irrational interval translation is uniquely ergodic, (3.17) and again the unique invariant probability measure of an irrational interval translation is the Lebesgue measure. The following is an application of property (3.17): Example 3.4. Consider the sequence `1 D 2, `2 D 4, `3 D 8, `4 D 1; : : : of the first digits in base 10 of the powers of 2. We want to find the frequency with which each k 2 f1; : : : ; 9g occurs in the sequence `n . More precisely, we want to show that the limit ˚ card j 2 f1; : : : ; ng W `j D k lim (3.18) n!1 n exists for each k, and to compute its value explicitly.


79

Consider the interval translation T W T ! T given by T .x/ D x C log10 2 mod 1: Since log10 2 is irrational, it follows from (3.17) that T is uniquely ergodic. On the other hand, 2n D 10n log10 2bn log10 2c 10bn log10 2c D 10T 0 10bn log10 2c : n

Therefore, `n D k if and only if T n .0/ 2 Œlog10 k; log10 .k C 1//: Now, let ' D Œlog10 k;log10 .kC1// : We consider continuous functions ap ; bp W T ! Œ0; 1 for p 2 N with ap ' bp ;

p 2 N;

such that Z

Z

1

0

Z

1

ap d m ! 0

Z

1

1

bp d m !

' d m and 0

' dm

(3.19)

0

when p ! 1, where m is the Lebesgue measure in Œ0; 1. We observe that n1 n1 n1 1X 1X 1X ap .T j .0// '.T j .0// bp .T j .0//: n j D0 n j D0 n j D0

(3.20)

Since T is uniquely ergodic, it follows from Theorem 3.3 that 1X ap .T j .0// ! n j D0

Z

1X bp .T j .0// ! n j D0

Z

n1

and

n1

1

ap d m 0

1

bp d m 0

(3.21)

80

3 Further Topics

when n ! 1. By (3.20) and (3.21), we obtain Z

1X '.T j .0// n j D0 n1

1

ap d m lim inf n!1

0

1X '.T j .0// n j D0 n1

lim sup n!1

Z

1

bp d m: 0

Letting p ! 1, it follows from (3.19) that 1X '.T j .0// D n!1 n j D0 n1

Z

1

' d m:

lim

0

Therefore, the frequency with which the integer k occurs in the sequence `n is given by Z

1

Z

1

' dm D

0

0

Œlog10 k;log10 .kC1// d m

1 : D log10 1 C k

3.4 Mixing We consider in this section an ergodic property which is stronger than the ergodicity of an invariant measure. We first give an alternative characterization of ergodicity. Proposition 3.4. Let T W X ! X be a measurable transformation preserving a probability measure in X . The measure is ergodic if and only if 1X .T k A \ B/ D .A/.B/ n!1 n n1

lim

(3.22)

kD0

for any measurable sets A; B X . Proof. We first assume that is ergodic. Let A; B X be measurable sets. By Proposition 2.12, we obtain 1X A .T k .x// D n!1 n n1

Z A d D .A/

lim

kD0

A

3.4 Mixing

81

for -almost every x 2 X . Therefore, by the dominated convergence theorem (Theorem A.3), we conclude that Z .A/.B/ D .A/ B d X

Z D

.A/B d X

Z D

1X A .T k .x//B .x/ d.x/ X n!1 n n1

lim

kD0

D lim

n!1

1 n

n1 Z X

A .T k .x//B .x/ d.x/:

kD0 X

Since A .T k .x//B .x/ D 1 if and only if T k .x/ 2 A and x 2 B, that is, if and only if x 2 T k A \ B, we have .A ı T k /B D T k A\B ;

(3.23)

and hence, 1X .T k A \ B/: n!1 n n1

.A/.B/ D lim

kD0

Now we assume that property (3.22) holds. Let A X be a T -invariant measurable set. Then 1X .A/.X n A/ D lim .T k A \ .X n A// n!1 n n1

kD0

1X .A \ .X n A// D 0; n!1 n n1

D lim

kD0

and thus, .A/ D 0 or .X n A/ D 0. This shows that is ergodic.

t u

Whenever we consider a convergence stronger than that in (3.22), we obtain a property stronger than ergodicity. In particular, we consider the following notion: Definition 3.2. Let T W X ! X be a measurable transformation and let be a T invariant probability measure in X . We say that is mixing with respect to T if

82

3 Further Topics

lim .T n A \ B/ D .A/.B/

n!1

for any measurable sets A; B X . One can easily verify that if is mixing, then is ergodic. In order to describe a criterion for mixing, given a probability measure in X we consider the space L2 .X; / of all measurable functions 'W X ! R such that Z

1=2 ' 2 d < 1;

k'k2 WD X

equipped with the norm ' 7! k'k2 . We note that L2 .X; / L1 .X; /. Indeed, by the Cauchy–Schwarz inequality, we have Z

Z j'j d D X

1 j'j d (3.24)

X

.X /

1=2

k'k2 D k'k2 < 1:

Given a family of functions ˚ L2 .X; / we denote by L.˚/ the set of all finite linear combinations of functions in ˚. Theorem 3.6. Let T W X ! X be a measurable transformation preserving a probability measure in X , and let ˚ L2 .X; / be a family of functions such that L.˚/ is dense in L2 .X; /. If Z

Z lim

n!1 X

Z

.' ı T n / d D

' d X

d

(3.25)

X

2 ˚, then the measure is mixing.

for any ';

Proof. We note that by (3.24), the integrals in the right-hand side of (3.25) are well defined. Moreover, by the Cauchy–Schwarz inequality and Proposition 2.1, we have 1=2 Z .' ı T n /2 d

Z

Z j.' ı T n / j d X

X

Z D

1=2 d

X

1=2 Z ' 2 d

X

2

2

1=2 d < 1;

X

and hence, the integral in the right-hand of (3.25) side is also well defined. Now we proceed with the proof of the theorem. We first observe that since (3.25) holds for any '; 2 ˚, this property also holds for any '; 2 L.˚/. Given '; N N 2 L2 .X; / and " > 0, we consider functions '; 2 L.˚/ such that k' 'k N 2 < " and k

N k2 < "

(3.26)

3.4 Mixing

83

(which is always possible because L.˚/ is dense in L2 .X; /). Using the Cauchy– Schwarz inequality and (3.24), we obtain ˇ ˇZ Z Z ˇ ˇ n N ˇ N Dn D ˇ .'N ı T / d dˇˇ 'N d X X X ˇZ ˇ ˇZ ˇ ˇ ˇ ˇ ˇ ˇˇ .'N ı T n /. N / dˇˇ C ˇˇ .'N ı T n ' ı T n / dˇˇ X X ˇ ˇZ Z Z ˇ ˇ n ˇ ' d dˇˇ C ˇ .' ı T / d X

X

ˇZ Z ˇ ˇ C ˇ .' '/ N d X

X

ˇ ˇZ Z ˇ ˇ ˇ ˇ dˇ C ˇ 'N d .

X

X

X

ˇ ˇ N / dˇˇ

k'N ı T k2 k N k2 C k.'N '/ ı T n k2 k k2 ˇZ ˇ Z Z ˇ ˇ C ˇˇ .' ı T n / d ' d dˇˇ n

X

X

C k'N 'k2 k k2 C k'k N 2k

X

N k2 :

Hence, it follows from Proposition 2.1 and (3.26) that ˇZ Z Z ˇ n ˇ Dn 2"k'k N 2 C 2"k k2 C ˇ .' ı T / d ' d X

X

X

ˇ ˇ dˇˇ :

Letting n ! 1, we conclude that N 2 C k N k2 C " ; lim sup Dn 2" k'k n!1

and it follows from the arbitrariness of " that (3.25) holds for '; 2 L2 .X; /. Now we consider measurable sets A; B X . Clearly, A ; B 2 L2 .X; /. By identity (3.23), setting ' D A and D B , it follows from (3.25) that is mixing. t u We use this criterion to show that some toral automorphisms have the mixing property. Proposition 3.5. Let TA be the automorphism of the torus Tn induced by a matrix A such that (2.51) holds. Then the Lebesgue measure is mixing with respect to TA . Proof. By Theorem 3.6, it is sufficient to consider the family of functions ˚ D fk W k 2 Zn g; where k .x/ D e 2 i hk;xi with hk; xi as in (2.50). Indeed, one can show that for this family, the set L.˚/ is dense in L2 .X; /, where is the Lebesgue measure in Tn .

84

3 Further Topics

Now we study the integrals Z Tn

.k ı T p /` d:

Z

We first note that

Tn

k ` d D ık1 `1 ıkn `n ; (

where ıab D

1 if

a D b;

0 if

a ¤ b:

If k D ` D 0, then Z

Z

Tn

.k ı T p /` d D 1 D

(3.27)

Z Tn

k d

Tn

` d:

Now we assume that k ¤ 0. We have .k ı T p /.x/ D e 2 i hk;A

p xi

D .A /p k .x/:

On the other hand, it follows readily from (2.51) that .A /p1 k ¤ .A /p2 k for any positive integers p1 > p2 since otherwise .A /p1 p2 .A /p2 k D .A /p2 k; and .A /p2 k would be a nonzero eigenvector of .A /p1 p2 with eigenvalue 1. In particular, .A /p k ¤ ` for any sufficiently large p, and it follows from (3.27) that Z

Z .k ı T /` d D p

Tn

Tn

.A /p k ` d D 0:

Since k ¤ 0, we also have Z

Z Tn

k d

Tn

` d D 0:


t u

3.5 Symbolic Dynamics We give in this section a brief introduction to symbolic dynamics and to its relations to ergodic theory. In particular, we construct Markov measures and Bernoulli measures, which are shift-invariant probability measures. We also describe the relation between Markov measures and topological Markov chains.

3.5 Symbolic Dynamics

85

Fig. 3.1 Graph associated to a topological Markov chain 0

1

3.5.1 Basic Notions We first develop the theory of one-sided sequences. See Sect. 3.5.4 for the case of two-sided sequences. For each k 2 N, we consider the set XkC D f1; : : : ; kgN of one-sided sequences of numbers in f1; : : : ; kg that we write in the form ! D .i1 .!/i2 .!/ /: Definition 3.3. We define the shift map W XkC ! XkC by .i1 i2 / D .i2 i3 /: Clearly, the map W XkC ! XkC is not invertible. We also consider a particular class of shift-invariant set in XkC . Namely, given a k k matrix A D .aij / with aij 2 f0; 1g for each i and j , we consider the set ˚ XAC D ! 2 XkC W ain .!/inC1 .!/ D 1 for every n 2 N : It is easy to verify that .XAC / XAC . Definition 3.4. Given a k k matrix A with entries in f0; 1g, the restriction of the shift map jXAC W XAC ! XAC is called the (one-sided) topological Markov chain or subshift of finite type with transition matrix A. Example 3.5. If k D 2 and A D 11 10 , then ˚ XAC D ! 2 X2C W .in .!/inC1 .!// ¤ .11/ for every n 2 N is the set of sequences in which the symbol 1 is always isolated (when it occurs). This set can be represented by the graph in Fig. 3.1, where an arrow indicates the possibility of a transition from i to j . In particular, we want to find the number of m-periodic points of a topological Markov chain. We first observe that ! 2 XAC is m-periodic if and only if m .!/ D !, that is, if and only if inCm.!/ D in .!/

for every n 2 N:

86

3 Further Topics

Thus, in order to find the number of m-periodic points, it is sufficient to find how many finite sequences .i1 im / of length m with aim i1 D 1 can occur in the elements of XAC . We start with an example. Example 3.6. For the matrix A D 11 10 considered in Example 3.5, we note that the possible transitions from i to the same symbol i trough j are 1 ! 1 ! 1, 1 ! 2 ! 1, and 2 ! 1 ! 2. Therefore, the shift map has exactly the 2-periodic points . 11111 /, . 21212 /, and . 12121 /. Now we translate this observation into a more algebraic approach. We note that there is a transition from i to the same symbol i trough j if and only if aij aj i D 1. Therefore, the number of 2-periodic points in XAC is equal to 2 2 X X i D1 j D1

aij aj i D

2 X .A2 /i i D tr.A2 / D 3: i D1

Now we find the number of m-periodic points of an arbitrary topological Markov chain. We recall that the spectral radius of a square matrix A is given by ˚ .A/ D max j j W 2 Sp.A/ ; where Sp.A/ denotes the set of eigenvalues of A. We also recall the notion of periodic entropy introduced in Definition 2.6. Proposition 3.6. For the one-sided topological Markov chain jXAC , we have: 1. For each m 2 N, ˚ card ! 2 XAC W m .!/ D ! D tr.Am /: 2. p. jXAC / D log .A/. Proof. In a similar manner to that in Example 3.6, the number of m-periodic points of the topological Markov chain jXAC is given by k X i1 ;:::;im D1

ai1 i2 ai2 i3 aim i1 D

k X

.Am /i i D tr.Am /:

i D1

This establishes the first statement. The second statement follows immediately from the first. t u


87

Example 3.7. Consider the one-sided topological Markov chain with k k transition matrix A. In view of Proposition 3.6, the zeta function of jXAC (see Exercise 2.29) is given by

.z/ D exp

1 X tr.An / nD1

n

zn :

Now let 1 ; : : : ; k be the eigenvalues of A, counted with their multiplicities. We have tr.An / D n1 C C nk ; and hence,

.z/ D exp

1 k X X nj j D1 nD1

D exp

k X j D1

D

log

n

zn

1 1 j z

k Y

1 : 1 j z j D1

Therefore,

.z/ D

1 : det.Id zA/

In particular, this shows that the zeta function of a topological Markov chain is always a rational function.

3.5.2 Markov Measures and Bernoulli Measures We discuss in this section some relations between symbolic dynamics and ergodic theory. We start by introducing the notion of stochastic matrix. Let P D .pij / be a nonnegative k k matrix. This means that pij 0 for every i; j D 1; : : : ; k. Definition 3.5. We say that P is a stochastic matrix if: P 1. kj D1 pij D 1 for i D 1; : : : ; k. 2. There exist numbers p1 ; : : : ; pk 2 .0; 1/ such that

88

3 Further Topics k X

k X

pi D 1 and

i D1

pi pij D pj

(3.28)

i D1

for j D 1; : : : ; k. We then say that p D .p1 ; : : : ; pk / is a probability vector associated to P , and that .P; p/ is a stochastic pair. If P is a stochastic matrix, then 1 is an eigenvalue of P : simply note that .1; : : : ; 1/ is an eigenvector with eigenvalue 1. We also note that the second condition in (3.28) can be written in matrix form as pP D p. Now to each stochastic pair, we associate a measure in XkC . We first need to introduce a -algebra in XkC . Given m 2 N and i1 ; : : : ; im 2 f1; : : : ; kg, we define the cylinder set ˚ Ci1 im D .j1 j2 / 2 XkC W j` D i` for ` D 1; : : : ; m ; and we consider the -algebra in XkC generated by these sets. Definition 3.6. Given a stochastic pair .P; p/, we define a measure in XkC by requiring that .Ci1 im / D pi1 pi1 i2 pim1 im for each m 2 N and i1 ; : : : ; im 2 f1; : : : ; kg. We then say that is the (one-sided) Markov measure associated to the pair .P; p/. One can easily verify that any Markov measure in XkC is well defined. Indeed, since the cylinder sets generate the -algebra of XkC , and since Ci1 im D

k [

Ci1 im j

(3.29)

j D1

is a disjoint union, it is sufficient to note that k X

pi1 pi1 i2 pim1 im pim j D pi1 pi1 i2 pim1 im

j D1

k X j D1

pim j

(3.30)

D pi1 pi1 i2 pim1 im : Proposition 3.7. Any one-sided Markov measure is -invariant. Proof. Let be the Markov measure associated to a stochastic pair .P; p/. Since

1

Ci1 im D

k [ j D1

Cj i1 im ;


89

we obtain . 1 Ci1 im / D

k X

.Cj i1 im / D

j D1

k X

pj pj i1 pi1 i2 pim1 im

j D1

D pi1 pi1 i2 pim1 im D .Ci1 im /: Since the cylinder sets generate the -algebra this yields the desired result.

t u

Example 3.8. Consider the matrix P D

1=2 1=2 : 1 0

Then p D .2=3; 1=3/ is an eigenvector of P with eigenvalue 1. Therefore, p D pP , and .P; p/ is a stochastic pair. If is the corresponding Markov measure in X2C , then, for example, .C121 / D p1 p12 p21 D

1 2 1 1D ; 3 2 3

and

1 1 1 1 D : 3 2 6 Now we consider a particular class of Markov measures. .C211 / D p2 p21 p11 D

Definition 3.7. We say that a measure in XkC is a (one-sided) Bernoulli measure if there exist numbers p1 ; : : : ; pk 2 .0; 1/ such that: P 1. kiD1 pi D 1. 2. .Ci1 im / D pi1 pim for each m 2 N and i1 ; : : : ; im 2 f1; : : : ; kg. To see that any Bernoulli measure is indeed a Markov measure, it is sufficient to take pij D pj for each i and j . In particular, by Proposition 3.7, all Bernoulli measures are invariant under the shift map. Moreover, the following property holds: Proposition 3.8. Any one-sided Bernoulli measure is mixing. Proof. Again it is sufficient to consider cylinder sets. We have [

. n Ci1 ik \ Cj1 j` / D

!

Cj1 j` m1 mn` i1 ik

m1 mn`

D

X

pj1 pj` pm1 pmn` pi1 pik

m1 mn`

D pj 1 pj ` pi1 pik D .Cj1 j` /.Ci1 ik / for any sufficiently large n 2 N. This yields the desired result.

t u

90

3 Further Topics

Example 3.9. Let be an ergodic Markov measure (e.g., by Proposition 3.8, all Bernoulli measures are ergodic). By Proposition 2.12, for each i D 1; : : : ; k, we have Z n1 1X j lim Ci . .!// D Ci d D .Ci / D pi n!1 n XkC j D0 for -almost every ! 2 XkC . We also give a criterion for a nonnegative matrix to be a stochastic matrix. Theorem 3.7 (Perron–Frobenius). Let P D .pij / be a nonnegative k k matrix such that: P 1. kj D1 pij D 1 for i D 1; : : : ; k. 2. For each i; j D 1; : : : ; k, there exists m D m.i; j / 2 N such that the entry .i; j / of P m is positive. Then there exists one and only one p 2 Rk such that .P; p/ is a stochastic pair. Proof. Let with the norm kvk D tion F W S ! S by

Pk

˚ k S D v 2 .RC 0 / W kvk D 1 ;

i D1 jvi j.

By the first condition, we can define a transforma-

F .v/ D P v=kP vk:

Since S is homeomorphic to the closed unit ball in Rk1 and F is continuous, it follows from Brouwer’s fixed point theorem that F has a fixed point p 2 S . We thus obtain P p D p, where D kP pk D

k X

.P p/j

j D1

D

k k X X

pi pij D

j D1 i D1

D

k X

k X i D1

pi

k X

pij

j D1

pi D 1:

i D1

Therefore, to show that .P; p/ is a stochastic pair, it remains to verify that p has only positive entries. Since p 2 S , there exists j 2 f1; : : : ; kg such that pj > 0. On the other hand, by the second condition, for each i D 1; : : : ; k, there exists m D m.i; j / such that the entry .i; j / of .P /m is positive. Since .P /m p D p, we thus obtain k X m pi D .P /m (3.31) i ` p` .P /ij pj > 0 `D1


91

for i D 1; : : : ; k. Now we consider another vector q 2 Rk with kqk D 1

and

minfqi W i D 1; : : : ; kg > 0

such that P q D q. We want to show that q D p. Otherwise, there would exist t 2 R for which p C tq would be a nonzero eigenvector with nonnegative entries, at least with one entry equal to zero. On the other hand, one can repeat the argument in (3.31) to show that all entries of pCtq must be positive. This contradiction shows that q D p. t u

3.5.3 Topological Markov Chains and Markov Measures Now we introduce a distance and thus also a topology in XkC . Namely, given ˇ > 1, for each !; ! 0 2 XkC , we set ( 0

dˇ .!; ! / D

ˇ n

if ! ¤ ! 0 ;

0

if ! D ! 0 ;

(3.32)

where n D n.!; ! 0 / 2 N is the smallest integer such that in .!/ ¤ in .! 0 /. One can easily verify that dˇ is indeed a distance in XkC . For additional properties, we refer to Exercises 3.14–3.16. Now we describe the relation between Markov measures and topological Markov chains. Given a stochastic matrix P D .pij /, we consider the topological Markov chain jXAC W XAC ! XAC where the k k transition matrix A D .aij / is defined by ( aij D

1 if pij > 0; 0 if pij D 0:

(3.33)

We recall that the support of a measure in XkC is the set supp XkC formed by all points x 2 XkC such that .U / > 0 for any open set U containing x (see also Proposition 3.12 below). Proposition 3.9. If is the one-sided Markov measure associated to a stochastic pair .P; p/, then supp D XAC , where A is the transition matrix defined by (3.33). Proof. For each ! D .i1 i2 / 2 XAC and m 2 N, we have .Ci1 im / D pi1 pi1 i2 pim1 im > 0: Since the cylinder sets generate the topology, this shows that ! 2 supp . Now we assume that ! D .i1 i2 / 62 XAC . Then there exists m 2 N such that aim imC1 D 0. Therefore,

92

3 Further Topics

pim imC1 D 0

and .Ci1 imC1 / D 0:

Since ! 2 Ci1 imC1 , this shows that ! 62 supp .

t u

By Proposition 3.9, the support of any Bernoulli measure is XkC .

3.5.4 The Case of Two-Sided Sequences We develop in this section a corresponding theory of two-sided sequences, highlighting the differences with respect to the theory of one-sided sequences. For each k 2 N, we consider the set Xk D f1; : : : ; kgZ of two-sided sequences of numbers in f1; : : : ; kg. Given a sequence ! 2 Xk , we write it in the form ! D . i1 .!/i0 .!/i1 .!/ /: Definition 3.8. We define the shift map W Xk ! Xk by .!/ D ! 0 , where in .! 0 / D inC1 .!/

for each n 2 Z:

Clearly, the map W Xk ! Xk is invertible. Given a k k matrix A D .aij / with aij 2 f0; 1g for each i and j , we consider the set ˚ XA D ! 2 Xk W ain .!/inC1 .!/ D 1 for every n 2 Z : It is easy to verify that .XA / D XA . Definition 3.9. The restriction jXA W XA ! XA is called the (two-sided) topological Markov chain or subshift of finite type with transition matrix A. We note that the statement in Proposition 3.6 also holds for two-sided topological Markov chains, that is, ˚ card ! 2 XA W m .!/ D ! D tr.Am /

(3.34)

for each m 2 N, and hence, p. jXA / D log .A/:

(3.35)

Moreover, to each stochastic pair we associate a measure in Xk . Given m 2 N and im ; : : : ; im 2 f1; : : : ; kg, we define the cylinder set ˚ Cim im D . j0 / 2 Xk W j` D i` for ` D m; : : : ; m ; and we consider the -algebra in Xk generated by these sets.

(3.36)


93

Definition 3.10. Given a stochastic pair .P; p/, we define a measure in Xk by requiring that .Cim im / D pim pim imC1 pim1 im for each m 2 N and im ; : : : ; im 2 f1; : : : ; kg. We then say that is the (two-sided) Markov measure associated to the pair .P; p/. Proceeding as in (3.29) and (3.30), one can easily verify that any Markov measure in Xk is well defined. Proposition 3.10. Any two-sided Markov measure is -invariant. Proof. Let be the Markov measure associated to a stochastic pair .P; p/. In view of the identity k k [ [ 1 Cim im D Cj ìm im ; j D1 `D1

we obtain . 1 Cim im / D

k k X X

.Cj ìm im /

j D1 `D1

D

k k X X

pj pj ` pìm pim1 im

j D1 `D1

D pim pim imC1 pim1 im D .Cim im /: Since the cylinder sets generate the -algebra, this yields the desired result.

t u

We also consider the class of Bernoulli measures. Definition 3.11. We say that a measure in Xk is a (two-sided) Bernoulli measure if there exist numbers p1 ; : : : ; pk 2 .0; 1/ such that: P 1. kiD1 pi D 1. 2. .Cim im / D pim pim for each m 2 N and im ; : : : ; im 2 f1; : : : ; kg. Now we introduce a distance and thus also a topology in Xk . Namely, given ˇ > 1, for each !; ! 0 2 Xk , we set ( 0

dˇ .!; ! / D

ˇ n

if ! ¤ ! 0

0

if ! D ! 0 ,

(3.37)

where n D n.!; ! 0 / 2 N [ f0g is the smallest integer such that in .!/ ¤ in .! 0 / or in .!/ ¤ in .! 0 /. Then dˇ is a distance in Xk .

94

3 Further Topics

Finally, we describe the relation between Markov measures and topological Markov chains. Proposition 3.11. If is the two-sided Markov measure associated to a stochastic pair .P; p/, then supp D XA , where A is the transition matrix defined by (3.33). Proof. For each ! D . i0 / 2 XA and m 2 N, we have .Cim im / D pim pim imC1 pim1 im > 0: Since the cylinder sets generate the topology, this shows that ! 2 supp . Now we assume that ! D . i0 / 62 XA . Then there exists m 2 Z such that aim imC1 D 0. Therefore, pim imC1 D 0 and .Cim1 imC1 / D 0 when m 0, and .Cim im / D 0 when m < 0. This shows that ! 62 supp .

t u

3.6 Topological Dynamics We give in this section a brief introduction to the relation between topological dynamics and ergodic theory. In particular, we show that each ergodic property implies a corresponding topological property. We start with an auxiliary statement concerning the support of a Borel probability measure in a metric space X . We recall that the support of is the set supp X formed by all points x 2 X such that .U / > 0 for any open set U containing x. Proposition 3.12. If X is a metric space and is a Borel probability measure in X , then the following properties hold: 1. 2. 3. 4.

supp is the complement of the largest open set of zero -measure. supp is a closed set. .X n supp / D 0. If A is a measurable set with .X n A/ D 0, then A supp .

Proof. Let V X be the largest open set of zero -measure. If x 2 X nsupp , then there exists an open set U X containing x with .U / D 0. Thus, x 2 U V , and we conclude that X n supp V . On the other hand, if x 2 V , then clearly x 62 supp . This yields Property 1. Properties 2 and 3 follow immediately from Property 1. For Property 4, we note that if supp n A ¤ ∅, then there would exist an open set U X disjoint from A with .U / > 0. On the other hand, .U / D .U \ supp / .supp n A/ .X n A/ D 0: This contradiction shows that A supp .

t u

3.6 Topological Dynamics

95

Now we recall some basic notions of topological dynamics. We continue to assume that X is a metric space. Definition 3.12. Let T W X ! X be a continuous transformation. 1. We define the !-limit set of a point x 2 X by !.x/ D

1 \

fT k .x/ W k ng:

nD1

2. We say that x is a recurrent point of T if x 2 !.x/. Clearly, !.x/ is a closed set. One can easily verify that y 2 !.x/ if and only if there exists an increasing sequence .nk /k2N N such that T nk .x/ ! y when k ! C1. Therefore, x is a recurrent point of T if and only if for each open set U containing x, there exists n 2 N such that T n .x/ 2 U . We also introduce some topological properties. Definition 3.13. Let T W X ! X be a continuous transformation. 1. We say that T is topologically transitive if for any nonempty open sets U; V X , there exists n 2 N such that T n U \ V ¤ ∅. 2. We say that T is topologically mixing if for any nonempty open sets U; V X , there exists m 2 N such that T n U \ V ¤ ∅ for every n > m. Clearly if T is topologically mixing, then T is topologically transitive. The following result makes explicit the relations between the ergodic properties of and the topological properties on the support of the measure: Theorem 3.8. Let T W X ! X be a homeomorphism of a separable metric space and let be a T -invariant Borel probability measure in X . Then the following properties hold: 1. 2. 3. 4.

supp is T -invariant. If is ergodic, then T j supp is topologically transitive. If is mixing, then T j supp is topologically mixing. If X is compact and T j supp is uniquely ergodic, then the two-sided orbit of any point in supp is dense in supp .

Proof. Let x 2 T 1 .supp /. Given an open set U X containing x, the open set T .U / contains the point T .x/ 2 supp . Therefore, .U / D .T .U // > 0, and x 2 supp . This shows that T 1 .supp / supp : Now let x 2 supp . Given an open set U X containing T .x/, the open set T 1 .U / contains x 2 supp . Therefore, .U / D .T 1 .U // > 0 and T .x/ 2 supp . This shows that supp T 1 .supp /:

96

3 Further Topics

Now we assume that is ergodic. Let U and V be nonempty open sets intersecting supp . By Proposition 3.4, we have 1X .T k U \ V / D .U /.V / > 0: n!1 n n1

lim

kD0

This implies that there exist infinitely many integers n with T n U \ V ¤ ∅, and in particular, T j supp is topologically transitive. Now we assume that is mixing. Let again U and V be nonempty open sets intersecting supp . Then .U /; .V / > 0, and by the mixing property, we obtain .T n U \ V / ! .U /.V / > 0 when n ! 1. Therefore, T n U \ V ¤ ∅ for all sufficiently large n. Finally, in order to proceed by contradiction, we assume that there exists x 2 supp such that its two-sided orbit ˚ A D T n .x/ W n 2 Z is not dense in supp . Since supp is T -invariant, A is a proper subset of supp . On the other hand, since A is compact and T jA is continuous, by the Krylov– Bogolubov theorem (Theorem 3.2), there exists a T -invariant probability measure in X with supp A. Since A is a proper subset of supp , we conclude that ¤ . But this contradicts the unique ergodicity of T j supp . t u Example 3.10. Consider a Bernoulli measure in XkC . By Proposition 3.8, the measure is mixing. Since supp D XkC , it follows from Theorem 3.8 that T is topologically mixing.

3.7 Exercises Exercise 3.1. Let T W X ! X be a continuous transformation of a compact metric space. Show that for each x 2 X , the sequence of probability measures 1X ıT k .x/ n n1

(3.38)

kD0

has at least one accumulation point in M.X / and that any of the accumulation points is a T -invariant probability measure. Hint: we have Tk ıx D ıT k .x/ with T as in (3.6) and thus replacing by ıx in (3.7) yields

3.7 Exercises

T

97 mn 1 1 X ıT k .x/ mn kD0

! D

mn 1 ıT mn .x/ 1 X ıx ıT k .x/ C : mn mn mn kD0

Exercise 3.2. Show that when x is an m-periodic point, the measure in (2.16) is the unique accumulation point of the sequence of measures in (3.38). Exercise 3.3. Show that a uniquely ergodic transformation has at most one periodic orbit. Hint: consider the measure in (2.16). Exercise 3.4. For each n 2 N, say whether there exists a uniquely ergodic toral automorphism of Tn . Exercise 3.5. Take ˛1 ; : : : ; ˛n 2 R such that the set f1; ˛1 ; : : : ; ˛n g is rationally independent. This means that if k0 C k1 ˛1 C C kn ˛n D 0 for some integers k0 ; : : : ; kn 2 Z, then k0 D D kn D 0. Show that the toral translation T˛1 ;:::;˛n W Tn ! Tn in (2.62) is uniquely ergodic and that the unique T˛1 ;:::;˛n -invariant probability measure is the Lebesgue measure. Exercise 3.6. Given p 2 N, consider the sequence of the first digits in base-10 of the powers of p. Determine whether the limit in (3.18) exists, and if so, compute its value explicitly. Exercise 3.7. Show that if ˛ 62 Q, then the sequence of fractional parts of the polynomial P .t/ D ˛t C ˇ is uniformly distributed for every ˇ 2 R. Exercise 3.8. Given a 2 R, consider the set Ka D fbnac W n 2 Ng; where bnac denotes the integer part of na. 1. Show that if a; b 2 .1; 1/ are such that f1; a1 ; b 1 g is rationally independent (see Exercise 3.5), then lim

n!1

1 card.Ka \ Kb \ Œ1; n/ D : n ab

2. Show that in general, the set Ka \ Kb may be empty. Hint: take a > 1 such that a1 C a2 D 1. Exercise 3.9. Given ˛ 2 R n Q, we define a transformation S W X ! X in the space X D T Œ0; 1 by S.x; t/ D .x C ˛; t/ mod 1: Show that for each ' 2 C.X /, the sequence of functions in (3.8) converges uniformly, but that S is not uniquely ergodic. Hint: fix t 2 Œ0; 1 and observe that

98

3 Further Topics

for the continuous function

.x/ D '.x; t/, the sequence of functions 1X n n1

ı T k;

kD0

R1 where T .x/ D x C ˛ mod 1, converges uniformly to 0 .x/ dx when n ! 1. This shows that the sequence of functions in (3.8) may converge uniformly even when T is not uniquely ergodic, although of course not to a constant. Exercise 3.10. Let ˚ D f't W X ! X gt 2R be a group of measurable transformations, that is, a family of measurable transformations such that '0 .x/ D x for every x 2 X and 't Cs D 't ı 's for every t; s 2 R. We say that a measure in X is ˚-invariant if .'t .A// D .A/ for every t 2 R and every measurable set A X . 1. Show that if is a ˚-invariant finite measure in X and g 2 L1 .X; /, then for -almost every x 2 X , the limit 1 t !C1 t

Z

t

g.'s .x// ds

lim

0

exists. Hint: consider the sequence 1 n

Z

n

g.'s .x// ds 0

and use Birkhoff’s ergodic theorem (Theorem 2.2). 2. Show that if is a ˚-invariant finite measure in X and g; h 2 L1 .X; / with inffh.x/ W x 2 X g > 0, then for -almost every x 2 X , the limit Rt lim R0t

t !C1

0

g.'s .x// ds h.'s .x// ds

exists. Exercise 3.11. Show that if ˚ D f't W T2 ! T2 gt 2R is the group of diffeomorphisms generated by the solutions of the differential equation .x; y/0 D .˛; ˇ/ in T2 for some ˛ ¤ 0, then there exists a unique ˚-invariant probability measure in T2 (see Exercise 3.10) if and only if ˇ=˛ 62 Q. Exercise 3.12. Given q 2 Z with jqj > 1, let Eq W S 1 ! S 1 be the expanding map of the circle given by Eq .z/ D zq . Show that the Lebesgue measure is mixing with respect to Eq .

3.7 Exercises

99

Exercise 3.13. Prove or disprove the following statement: if .T; / is mixing, then .T n ; / is mixing for each n 2. Exercise 3.14. Equipping XkC with the distance dˇ in (3.32), show that: 1. The shift map W XkC ! XkC is continuous. 2. Each open ball is closed. 3. The cylinder sets are simultaneously open and closed, and they generate the topology of XkC . Exercise 3.15. We recall that two distances d and d 0 are said to be equivalent if there exists a constant c > 0 such that c 1 d.!; ! 0 / d 0 .!; ! 0 / cd.!; ! 0 / for every !; ! 0 2 XkC . Moreover, two distances d and d 0 are said to be Hölderequivalent if there exist constants c > 0 and ˛ 2 .0; 1 such that c 1 d.!; ! 0 /1=˛ d 0 .!; ! 0 / cd.!; ! 0 /˛ for every !; ! 0 2 XkC . Given distinct numbers ˇ; ˇ 0 > 1: 1. Show that the distances dˇ and dˇ0 are not equivalent. 2. Show that the distances dˇ and dˇ0 are Hölder-equivalent and conclude that they generate the same topology. Exercise 3.16. Equipping f1; : : : ; kg with the discrete topology, that is, the topology generated by all subsets of f1; : : : ; kg, show that the corresponding product topology in XkC D f1; : : : ; kgN coincides with the topology induced by each distance dˇ . It follows from Tychonoff’s theorem that the space XkC is compact. Exercise 3.17. A k k matrix A is called irreducible if for each i; j 2 f1; : : : ; kg, there exists n 2 N (possibly depending on i and j ) such that the entry .i; j / of An is positive. Show that the graph associated to the topological Markov chain jXAC has an oriented path between any two vertices if and only if A is irreducible. Exercise 3.18. Show that any two-sided Bernoulli measure is mixing. Exercise 3.19. Let be a Markov measure with support XAC for some matrix A. Show that if is mixing, then there exists m 2 N such that all entries of the matrix Am are positive. Exercise 3.20. Equipping Xk with the distance dˇ in (3.37), show that Xk is compact, and that W Xk ! Xk is a homeomorphism. Exercise 3.21. Consider the set of sequences Y f1; 2gZ in which 1 appears only in pairs. Since .Y / Y , we can consider the restriction jY W Y ! Y . 1. Show that jY is not a topological Markov chain. 2. Verify that the periodic entropy of jY is positive.

100

3 Further Topics

3. Show that the zeta function of jY is given by

.z/ D

1Cz : 1 z z2

Exercise 3.22. For the shift map W XkC ! XkC , find a recurrent point ! 2 XkC which is not periodic. Exercise 3.23. We say that two transformations Ti W Xi ! Xi preserving a measure i in Xi , for i D 1; 2, are equivalent if there is a measurable transformation hW X1 ! X2 such that: 1. h is bijective almost everywhere. 2. h ı T1 D T2 ı h 1 -almost everywhere in X1 . 3. 1 .h1 A/ D 2 .A/ for every measurable set A X2 . Show that if T1 and T2 are equivalent, then: 1. 1 is ergodic if and only if 2 is ergodic. 2. 1 is mixing if and only if 2 is mixing. Exercise 3.24. Consider the expanding map of the circle Eq W S 1 ! S 1 together with the Lebesgue measure and the shift map W XkC ! XkC together with the Bernoulli measure with probability vector .1=q; : : : ; 1=q/. Show that: 1. The measurable transformation hW XkC ! S 1 defined by h.i1 i2 / D exp 2 i

1 X

! .ik 1/q

k

kD1

is bijective almost everywhere. 2. Eq ı h D h ı everywhere in XkC . 3. .h1 A/ D .A/ for any measurable set A S 1 . This shows that the two transformations are equivalent (see Exercise 3.23). Exercise 3.25. Consider the transformation in T W R ! R given by ( T .x/ D

.x 1=x/=2 if x ¤ 0; if x D 0:

0

1. Show that T preserves the Borel probability measure 1 .A/ D

Z A

dx : 1 C x2

3.7 Exercises

101

Fig. 3.2 The transformation in Exercise 3.28 1

a1

a2

1

2. Show that the transformation T together with the measure is equivalent to the shift map W X2C ! X2C together with the Bernoulli measure with probability vector .1=2; 1=2/. Exercise 3.26. For the expanding map of the circle E2 , find a point z 2 S 1 such that !.z/ D S 1 . Exercise 3.27. Under the assumptions of Theorem 3.8, show that if is ergodic, then for -almost every x 2 X , the orbit of x is dense in supp . Hint: consider a countable base of open sets Un for the induced topology in supp (with .Un / > 0 for each n). Exercise 3.28. For a transformation T W Œ0; 1 ! Œ0; 1, we assume that there exist p 2 N and a0 ; : : : ; apC1 2 R with 0 D a0 < a1 < < ap < apC1 D 1 such that:

Sp 1. T is of class C 2 in i D0 .ai ; ai C1 /. 2. T ..ai ; ai C1 // D .0; 1/ for i D 0; : : : ; p. Sp 3. There exists > 1 such that jT 0 .x/j > for every x 2 i D0 .ai ; ai C1 /. 4.

00 jT .x/j W x; y 2 .a d1 WD sup ; a / and i D 0; : : : ; p < 1: i i C1 jT 0 .y/j See Fig. 3.2 for an example with p D 2. Other examples are given by the piecewise linear expanding maps (see Sect. 2.2.6). For each n 2 N, the transformation T n is strictly monotone in each interval

102

3 Further Topics

Ii0 in1 D

n1 \

T k .aik ; aik C1 /;

kD0

for i0 ; : : : ; in1 2 f0; : : : ; pg, and T n .Ii0 in1 / D .0; 1/. We denote by i0 in1 W .0; 1/

! Ii0 in1

the corresponding local inverse of T n , that is, the unique transformation of class C 2 in .0; 1/ such that i0 in1 ..0; 1// D Ii0 in1 and T n . i0 in1 .x// D x for every x 2 .0; 1/. We also assume that T preserves a Borel probability measure that is absolutely continuous with respect to the Lebesgue measure m in Œ0; 1. In addition, we assume that the Radon–Nikodym derivative D d=d m is defined everywhere in Œ0; 1, and that it satisfies ˚ ˚ d2 WD inf .x/ W x 2 Œ0; 1 > 0 and d3 WD sup .x/ W x 2 Œ0; 1 < 1: Under these assumptions, show that: 1. For each x; y 2 .0; 1/, n 2 N, and i0 ; : : : ; in1 2 f0; : : : ; pg, we have j

i0 in1 .x/

i0 in1 .y/j

n jx yj:

2. There exists c1 > 0 such that if x; y 2 .0; 1/, n 2 N, and i0 ; : : : ; in1 2 f0; : : : ; pg, then ˇ 0 ˇ ˇ ˇ i0 in1 .x/ ˇ c1 : ˇ 0 ˇ .y/ˇ i0 in1

Hint: note that n1 Y jT 0 .T k .z//j j.T n /0 .z/j D j.T n /0 .w/j jT 0 .T k .w//j kD0

n1 Y kD0

jT 0 .T k .z// T 0 .T k .w//j : 1C jT 0 .T k .w//j

3. There exists c2 > 0 such that if n 2 N, i0 ; : : : ; in1 2 f0; : : : ; pg and x 2 Ii0 in1 , then 1 .Ii0 in1 / c2 : c2 j.T n /0 .x/j1 4. There exists c3 > 0 such that if A Œ0; 1 is measurable and i0 ; : : : ; in1 2 f0; : : : ; pg, then

3.7 Exercises

103

.Ii0 in1 \ T n A/ c3 .Ii0 in1 /.A/: Z

Hint: note that m.Ii0 in1 \ T

n

A/ D

j A

5. The measure is ergodic. 6. For -almost every xD

1 \

0 i0 in1 j d m:

Ii0 in1 2 Œ0; 1;

nD0

we have lim

n!1

log .Ii0 in1 / D n

Z

1

logjT 0 j d:

0

Hint: show that the function logjT 0 j is -integrable. 7. The set of periodic points of T is countable and is dense in Œ0; 1. 8. The periodic entropy of SpT is positive. 9. If T 0 .x/ > 0 for x 2 i D0 .ai ; ai C1 / and T .ai / D 0 for i D 0; : : : ; p C 1, then the zeta function of T (see Exercise 2.29) has a single pole.

Notes For additional topics of ergodic theory, in addition to [26, 44, 57, 103], we recommend [23, 51, 66, 69, 77, 79, 82, 96, 97, 100]. The books [48, 55] are excellent sources for (abstract) symbolic dynamics. For the relation between symbolic dynamics and the theory of dynamical systems, see, for example, [20, 44]. Hadamard [37] laid the foundations of symbolic dynamics, later developed by Morse and Hedlund [63, 64]. Theorem 3.2 on the existence of invariant measures for a continuous transformation of a compact metric space is due to Krylov and Bogolubov [52]. Our proof of Theorem 3.7 is based on [24].

•

Part II

Entropy and Pressure

•

Chapter 4

Metric Entropy and Topological Entropy

This chapter is dedicated to the study of metric entropy, including its relation to topological entropy. After establishing some basic properties of metric entropy, we consider the notion of conditional entropy, and we show how generators can be used to compute metric entropy. We then establish the Shannon–McMillan– Breiman theorem, which can be seen as the fundamental theorem of entropy theory. In particular, it shows that metric entropy can be computed in terms of an invariant local quantity. We also introduce the notion of topological entropy for a continuous transformation of a compact metric space, and we establish the variational principle showing that the topological entropy is the supremum of the metric entropies over all invariant probability measures.

4.1 Introduction The mathematical notion of entropy has its roots in thermodynamics with the work of Clausius, Maxwell, Boltzmann, and others. It appeared in the work of Shannon as the main concept of information theory. Let p D .p1 ; : : : ; pn / be a P vector with nonnegative entries such that niD 1 pi D 1. The Shannon entropy of p is defined by n X H.p/ D pi log pi ; i D1

with the convention that 0 log 0 D 0 (in fact with log2 instead of log, but for simplicity, we shall take this form). Clearly, H.p/ D

n X

f .pi /;

where f .x/ D x log x:

i D1


107

108

4 Metric Entropy and Topological Entropy

One can show that up to a multiplicative constant, f is the unique function satisfying m n X X

f .pi qj / D

i D1 j D1

for all pi ; qj 0 such that in the form

n X

f .pi / C

i D1

Pn

i D1

pi D

Pm

j D1

m X

f .qj /

(4.1)

j D1

qj D 1. Identity (4.1) can be written

H.p q/ D H.p/ C H.q/; where p D .p1 ; : : : ; pn /, q D .q1 ; : : : ; qm / and p q 2 RnCm is any vector with components pi qj . Now let be a probability measure in a space X and let D fA1 ; : : : ; An g be a measurable partition of X , that is, a collection of pairwise disjoint measurable sets whose union is X . The information function I associated to is defined by I.x/ D log .A/ for x 2 A: The entropy of the partition is the mean value of the information, that is, Z H ./ D

I d D H..A1 /; : : : ; .An //:

(4.2)

X

The notion of metric entropy is due to Kolmogorov and was extended to all dynamical systems by Sinai. It examines how a partition and its entropy vary under a dynamics. Namely, given a measurable transformation T W X ! X preserving a probability measure in X , the entropy of T with respect to measurable partition is defined by 1 h .T; / D inf H .n /; n2N n where n is the partition of X formed by the sets C1 \ T 1 C2 \ \ T .n1/ Cn

(4.3)

with C1 ; : : : ; Cn 2 . Roughly speaking, the speed with which the transformation T cuts smaller and smaller pieces under iteration is measured by the entropy. In a similar manner to that in (4.2), the entropy h .T; / can be expressed as the integral of a local quantity. This is due to the Shannon–McMillan–Breiman theorem, which can be seen as the fundamental theorem of entropy theory. For ergodic measures, the theorem implies that most sets in (4.3) have approximately equal measure for large n (in a rigorous sense to be made precise later on). There is a corresponding notion of entropy in topological dynamics, called topological entropy. Roughly speaking, it measures the exponential growth rate of the number of orbits that can be distinguished as time increases. For continuous transformations of a compact metric space, the so-called variational principle for the topological entropy says that the supremum of the metric entropies over all invariant probability measures is equal to the topological entropy. In fact, this identity could be used as a (working) definition for the notion of topological entropy.

4.2 Metric Entropy

109

4.2 Metric Entropy We introduce in this section the notion of metric entropy, starting with the notion of entropy with respect to a measurable partition. We also establish some of its basic properties.

4.2.1 The Notion of Metric Entropy We first introduce the notion of entropy with respect to a measurable partition. Let .X; A; / be a measure space with .X / D 1. Definition 4.1. We say that a finite family A is a pre-partition of .X; A; / or simply of X (with respect to ) if: S 1. . C 2 C / D 1. 2. .C \ D/ D 0 for any distinct C; D 2 . We emphasize that we only consider finite pre-partitions. Now we introduce an equivalence relation. Given pre-partitions and of X , we write if: 1. For every C 2 , there exists D 2 such that .C n D/ D 0. 2. For every D 2 , there exists C 2 such that .D n C / D 0. In other words, two pre-partitions are equivalent if and only if their elements differ at most by sets of zero measure. It is easy to verify that this is indeed an equivalence relation. Definition 4.2. Each equivalence class of this equivalence relation is called a measurable partition of X . Since the elements of any pre-partition representing a given equivalence class are well defined up to sets of zero measure, in what follows, we shall never distinguish between any two pre-partitions representing the same measurable partition. In particular, we always consider pre-partitions without elements of zero measure. Definition 4.3. The entropy of a measurable partition of X (with respect to ) is given by X H ./ D .C / log .C /: C 2

The number H ./ can be interpreted as the amount of information obtained by cutting X into the pieces in . We shall see that the smaller the pieces (in terms of measure), the largest amount of information we obtain. Now we introduce the notion of refinement. Definition 4.4. Given two measurable partitions and of X , we say that is a refinement of if for every D 2 there exists C 2 such that .D n C / D 0.

110


We note that if is a refinement of , then each element of (or more precisely of any pre-partition representing ) is a union of elements of (or more precisely of any pre-partition representing ) up to sets of zero measure. Example 4.1. Consider the space X D Œ0; 1 with the Lebesgue measure m. For each k 2 N, we consider the measurable partition ˚ k D .j=k; .j C 1/=k/ W j D 0; : : : ; k 1 of X . We have Hm .k / D

k1 X 1 1 log D log k: k k j D0

In particular, if k j `, then ` is a refinement of k , and we have Hm .k / Hm .` /, with equality if and only if k D `. Now let

W Œ0; 1 ! R be the continuous function defined by ( .x/ D

x log x

if 0 < x 1;

0

if x D 0:

We note that H ./ D

X

(4.4)

..C //:

C 2

Since hence,

00

.x/ D 1=x > 0 for every x > 0, the function ! p p X X ai xi ai .xi / i D1

is strictly convex, and (4.5)

i D1

Pp for any numbers x1 ; : : : ; xp , a1 ; : : : ; ap 2 Œ0; 1 such that i D 1 ai D 1. Moreover, (4.5) is an equality if and only if all numbers xi corresponding to a nonzero ai are equal. In particular, setting k D card , we obtain H ./ D k

X1 ..C // k C 2

k

X .C / C 2

D k

1 k

!

k D log k;

that is, H ./ log card : We also have the following property:

(4.6)

4.2 Metric Entropy

111

Proposition 4.1. Let and be measurable partitions of X . If is a refinement of , then H ./ H ./. Proof. We have H ./ D

X

..D//

D2

D

XX

(4.7) ..D//:

C 2 DC

For each C 2 , since

P

.D/ D .C /, we obtain

DC

X

..C // D X

D

! .D/

DC

.D/ log

DC

X

X

.D/

DC

.D/ log .D/ D

DC

X

..D//;

DC

and it follows from (4.7) that H ./

X

..C // D H ./:

C 2

t u

This yields the desired inequality.

Now we start discussing what happens to the entropy of a partition in the presence of a dynamics. Given two measurable partitions and of X , we define a new measurable partition ˚ _ D C \ D W C 2 ; D 2 : As explained above, and without loss of generality, by discarding sets of zero measure, one can always assume that all elements of the partition _ have positive measure. Proposition 4.2. If T W X ! X is a measurable transformation preserving a probability measure in X and is a measurable partition of X , then the limit lim

n!1

1 1 H .n / D inf H .n / n2N n n

(4.8)

exists, where n D

n1 _ kD0

T k

for each n 2 N:

(4.9)

112


Proof. We first prove some auxiliary lemmas. Lemma 4.1. If and are measurable partitions of X , then H . _ / H ./ C H ./: Proof of the lemma. We have X .C \ D/ log .C \ D/ H . _ / D C 2;D2

X

D

C 2;D2

D

XX

.C \ D/ C log .C / .C \ D/ log .C / .C /

D2 C 2

X

.C \ D/ .C /

.C \ D/ log .C /:

C 2;D2

By (4.5), we obtain H . _ / D

X

X

D2

C 2

X

.C \ D/ .C / .C /

!

X

.C / log .C /

C 2

..D// C H ./

D2

D H ./ C H ./: t u

This completes the proof of the lemma. Now we consider the measurable partition ˚ T 1 D T 1 C W C 2 : Lemma 4.2. If is a measurable partition of X , then H .T 1 / D H ./. Proof of the lemma. Since is T -invariant, we obtain X H .T 1 / D .T 1 C / log .T 1 C / C 2

D

X

.C / log .C /

C 2

D H ./; which yields the desired identity.

t u

4.2 Metric Entropy

113

Using these two lemmas, given n; m 2 N, we obtain H .nCm / D H .n _ T n m / H .n / C H .T n m /

(4.10)

D H .n / C H .m /: We also need the following result: Lemma 4.3. If .an /n2N is a sequence of positive numbers such that anCm an Cam for every n; m 2 N, then the limit na o an k D inf Wk2N 0 n!1 n k lim

(4.11)

exists. Proof of the lemma. Take k 2 N. Then n D qk C r for some integers q 2 N and r 2 f0; : : : ; k 1g. We have aqk C ar an qak C ar : n qk C r qk C r Thus, letting n ! 1 (and hence q ! 1), we obtain lim sup n!1

Therefore, lim sup n!1

ak an : n k

na o an an k inf W k 2 N lim inf : n!1 n n k

This establishes the existence of the limit in (4.11).

t u

By (4.10), it follows from Lemma 4.3 that the sequence H .n /=n converges and that identity (4.8) holds. t u We use the limit in Proposition 4.2 to introduce the notion of entropy of a measure-preserving transformation. Definition 4.5. Let T W X ! X be a measurable transformation preserving a probability measure in X . We define the (measure-theoretical or metric) entropy of T with respect to and a measurable partition of X by h .T; / D lim

n!1

1 H .n /; n

(4.12)

with n as in (4.9). We also define the (measure-theoretical or metric) entropy of T with respect to by

114


˚ h .T / D sup h .T; / W is a measurable partition of X :

(4.13)

We give two examples of the computation of the entropy. Example 4.2. Let T W T ! T be the interval translation T .x/ D x C ˛ mod 1 and let be the Lebesgue measure in T. Given a measurable partition of T by intervals, we have card n n card for each n 2 N: S i Indeed, the number of boundary points of the intervals in in1 , and thus the D0 T number of cuts made by these intervals in T, is at most n card . Therefore, by (4.6), we have 1 H .n / n 1 lim log.n card / D 0: n!1 n

h .T; / D lim

n!1

It follows from Theorem 4.2 below that h .T / D 0. Example 4.3. Let Eq .z/ D zq be an expanding map of the circle for some q > 1. For each m 2 N, we consider the measurable partition of S 1 given by

j j C1 m W j D 0; : : : ; q ; 1 ; m D h qm qm

(4.14)

where h is the map in (2.11), with respect to the Lebesgue measure in S 1 . It is easy to verify that m1 _ Eqi 1 D m : i D0

Therefore, n1 _ 1 h .Eq ; / D lim H Eqi m n!1 n i D0

!

m

1 H . mCn / n 1 1 mCn 1 mCn log mCn D lim q n!1 n q q D lim

n!1

D log q: It follows from Theorem 4.2 below that h .Eq / D log q:

4.2 Metric Entropy

115

Now we consider the powers of a transformation. Proposition 4.3. We have h .T k / D kh .T / for each k 2 N. Proof. We note that n1 _ 1 H T k` k n

!

`D0

1 D H n

nk1 _

! T

i

:

i D0

Letting n ! 1, we obtain h .T k ; k / D kh .T; /; and hence, h .T k / sup h .T k ; k / D kh .T /:

On the other hand, H

n1 _

! T

ki

H

i D0

nk1 _

! T

i

:

i D0

Therefore, 1 h .T k ; / lim H n!1 n

nk1 _

! T i

D kh .T; /:

i D0

Taking the supremum in , this implies that h .T k / kh .T /.

t u

4.2.2 Conditional Entropy We consider in this section the notion of conditional entropy. In particular, it allows us to give alternative characterizations of the notion of metric entropy. Definition 4.6. Given measurable partitions and of X , we define the conditional entropy of with respect to by H .j/ D

X C 2;D2

.C \ D/ log

.C \ D/ : .D/

Given a measurable partition of X and a measurable set D X , we denote by jD the measurable partition of D defined by ˚ jD D C \ D W C 2 :

116


We also consider the conditional measure .C jD/ D

.C \ D/ : .D/

In particular, for the probability measure D D =.D/ in D, we have HD .jD/ D

X

D .C / log D .C /

C 2

D

X

.C jD/ log .C jD/:

C 2

Therefore, given measurable partitions and of X , we can write H .j/ D

X

.D/

D2

D

X

X

..C jD//

C 2

.D/HD .jD/:

D2

Theorem 4.1. If T W X ! X is a measurable transformation preserving a probability measure in X and is a measurable partition of X , then h .T; / D lim H j n!1

n _

! T

i

:

(4.15)

i D1

Proof. We first show that the limit in (4.15) exists. For that, we establish the following auxiliary result: Lemma 4.4. If is a refinement of , then H .j/ H .j/: Proof of the lemma. We have H .j/ D

X

.C \ D/ log

C 2;D2

D

X

.D \ E/

C 2;D2;E2

D

X C 2;E2

.E/

.C \ D/ .D/ .C \ D/ .C \ D/ log .D/ .D/

X .D \ E/ .C \ D/ : .E/ .D/

D2

4.2 Metric Entropy

117

Thus, it follows from (4.5) that X

H .j/

! X .D \ E/ .C \ D/ : .E/ .D/

.E/

C 2;E2

D2

But since is a refinement of , we have X

X .D \ E/ .C \ D/ D .E/ .D/

D2

D2;DE

.C \ E/ .C \ D/ D ; .E/ .E/

and hence, X

H .j/

.E/

C 2;E2

X

D

.C \ E/ .E/

.C \ E/ log

C 2;E2

.C \ E/ D H .j/; .E/

which yields the desired inequality. t u WnC1 i Wn Since i D 1 T is a refinement of i D 1 T i , it follows from Lemma 4.4 that the sequence ! n _ i H j T i D1

is nonincreasing in n, and thus, there exists the limit in (4.15). To complete the proof, we also need the following result: Lemma 4.5. If , , and are measurable partitions of X , then H . _ j/ D H .j _ / C H .j/: Proof of the lemma. We have H . _ j/ D

X

.C \ D \ E/ log

.C \ D \ E/ .D/

.C \ D \ E/ log

.C \ D \ E/ .D \ E/

C 2;D2;E2

D

X C 2;D2;E2

X C 2;D2;E2

.C \ D \ E/ log

.D \ E/ ; .D/

118


with the convention that 0 log 0 D 0. Therefore, X

H . _ j/ D H .j _ /

.D \ E/ log

D2;E2

.D \ E/ .D/

D H .j _ / C H .j/; t u

which yields the desired inequality. In particular, for the trivial partition D fX g, it follows from Lemma 4.5 that H . _ / D H .j/ C H ./:

(4.16)

Now we use induction to show that n1 X

H .n / D H ./ C

H j

j D1

j _

! T

i

(4.17)

i D1

for every n 2 N. The identity is clear when n D 1. Now we assume that it holds for some n. Then, by Lemma 4.2 and (4.16), we have ! n _ i H .nC1 / D H T _ i D1

D H

n _

! T i C H j

i D1

n _

! T i

i D1

D H .n / C H j

!

n _

T

i

i D1

D H ./ C

n X

H j

j D1

j _

!

T i ;

i D1

using the induction hypothesisWin the last line. Since the sequence H .j niD 1 T i / is convergent, it follows from (4.17) that h .T; / D lim

n!1

1 H .n / n

_ 1X H j T i D lim n!1 n j D1 i D1 ! n _ T i : D lim H j j

n1

n!1


!

i D1

t u

4.2 Metric Entropy

119

We also use the notion of conditional entropy to show that it is sufficient to consider increasing sequences of partitions generating the -algebra of X . Given a measurable partition of X , we denote by A./ the -algebra generated by , that is, the smallest -algebra containing all the elements of . Moreover, we denote by W n2I A.n / the -algebra generated by the measurable partitions n for n 2 I . In particular, when I is a finite set, we have A

_

! D

n

n2I

_

A.n /:

n2I

Theorem 4.2. Let T W X ! X be a measurable transformation preserving a probability W1 measure in X . If .˛n /n2N is a sequence of measurable partitions of X with n D 1 A.˛n / D A such that ˛nC1 is a refinement of ˛n for each n 2 N, then h .T / D lim h .T; ˛n / D sup h .T; ˛n /: n!1

(4.18)

n2N

Proof. We first observe that the limit in (4.18) exists. Indeed, it follows from Proposition 4.1 that H

n1 _

! T

i

˛m

H

i D0

n1 _

! T

i

˛mC1

i D0

for any m; n 2 N, and hence, h .T; ˛m / h .T; ˛mC1 /

for each m 2 N:

Now we establish an auxiliary result. Lemma 4.6. If and are measurable partitions of X , then h .T; / h .T; / C H .j/:

(4.19)

Proof of the lemma. By Proposition 4.1 and (4.16), we have H .n / H .n _ n / D H .n / C H .n jn /; where n D

n1 _ i D0

T i and n D

n1 _

T i :

i D0

On the other hand, it follows from Lemmas 4.4 and 4.5 that H .˛ _ ˇj / H .˛j / C H .ˇj /

(4.20)

(4.21)

120


for any measurable partitions ˛, ˇ, and of X . Therefore, by (4.20), (4.21), and Lemma 4.4, we obtain H .n / H .n / C

n1 X

H .T i jn /

i D0

H .n / C

n1 X

H .T i jT i /

i D0

D H .n / C nH .j/; using the invariance of the measure in the last inequality. Thus, dividing by n and taking limits when n ! 1 yields inequality (4.19). t u To complete the proof, we use the following lemma: Lemma 4.7. For any measurable partition of X , we have H .j˛n / ! 0

when n ! 1:

Proof of the lemma. Write D fC1 ; : : : ; Ck g. For each n 2 N and i D 1; : : : ; k, let Din Ci be a set in A.˛n / such that .Ci n Din / ! 0 when n ! 1. We set ˇn D fD0n ; D1n ; : : : ; Dkn g; where D0n D X n

Sk

i D1

Din . Clearly, .Ci \ D0n / ! 0 when n ! 1. Since

Ci \ Din D Din

and Ci \ Djn D ∅

for i D 1; : : : ; k and j ¤ i , we obtain H .jˇn / D

k X

.Ci \ Din / log

i D1

k X

.Ci \ D0n / log

i D1

k X X

.Ci \ D0n / .D0n /

.Ci \ Djn / log

j D1 i ¤j

D

k X i D1

.Ci \ Din / .Din /

.Ci \ D0n / log

.Ci \ Djn / .Djn /

.Ci \ D0n / !0 .D0n /

(4.22)

4.2 Metric Entropy

121

when n ! 1. Since ˛n is a refinement of ˇn , it follows from Lemma 4.4 that H .j˛n / H .jˇn / ! 0

when n ! 1; t u

which yields the desired statement. Setting D ˛n in (4.19), it follows from Lemma 4.7 that h .T; / lim h .T; ˛n /: n!1

t u


4.2.3 Generators and Examples Sometimes it is possible to compute the entropy of a measure-preserving transformation using a single partition, as we describe in this section. Definition 4.7. Given a measure space .X; A; /, let T W X ! X be a measurable transformation and let be a measurable partition of X . 1. We say that is a one-sided generator (with respect to T ) if C1 _

A.T k / D A:

kD0

2. We say that is a two-sided generator (with respect to T ) if C1 _

A.T k / D A:

kD1

Generators may be hard to find, but when they exist, the computation of the entropy reduces to the computation of the entropy with respect to the generator. Theorem 4.3 (Kolmogorov–Sinai). Let T W X ! X be a measurable transformation preserving a probability measure in X . Then the following properties hold: 1. If is a one-sided generator, then h .T / D h .T; /. 2. If is a two-sided generator and T is invertible almost everywhere, then h .T / D h .T; /. Proof. We first assume that is a one-sided generator. It is sufficient to show that h .T; / h .T; / for any measurable partition of X . Indeed, this yields h .T; / h .T / D sup h .T; / h .T; /:

122


We proceed with the proof. Setting D nC1 in (4.19), we obtain h .T; / h .T; nC1 / C H .jnC1 /: On the other hand, we have 1 H h .T; nC1 / D lim m!1 m

m1 _

! T

k

nC1

kD0

(4.23)

1 H .mCn / m!1 m

D lim

D h .T; /; and hence, h .T; / h .T; / C H .jnC1 /:

(4.24)

Since is a one-sided generator and A

n _

! T

i

D

i D0

n _

A.T i /;

i D0

it follows from Lemma 4.7 that H .jnC1 / ! 0 when n ! 1: Thus, we conclude from (4.24) that h .T; / h .T; /. Now we assume that is a two-sided generator and that T is invertible almost everywhere. As in the case of one-sided generators, it is sufficient W to show that h .T; / h .T; / for any measurable partition of X . Setting D niD n T i in (4.19), we obtain h .T; / h T;

n _

! T

i

C H j

i Dn

n _

! T

i

:

i Dn

We also have h T;

n _

! T i

D h .T; 2nC1 / D h .T; /

i Dn

by the invariance of the measure and (4.23). Therefore, h .T; / h .T; / C H j

n _ i Dn

! T

i

:

(4.25)

4.2 Metric Entropy

123

On the other hand, by Lemma 4.7, we obtain ! n _ i T ! 0 when n ! 1; H j i Dn

and it follows from (4.25) that h .T; / h .T; /.

t u

Now we give several examples of the computation of the entropy using generators. Example 4.4. Let Eq .z/ D zq be an expanding map of the circle, for some integer q > 1, and let be the Lebesgue measure in S 1 . For each n 2 N, we consider again the measurable partitions m of S 1 in (4.14). Since n1 D

n1 _

Eq1 1 D n ;

i D0

the partition 1 is a one-sided generator (with respect to Eq ). By Theorem 4.3 and Example 4.3, we thus obtain h .Eq / D h .Eq ; 1 / D log q: Example 4.5. Let W Xk ! Xk be the shift map and let be the (two-sided) Bernoulli measure in Xk generated by the numbers p1 ; : : : ; pk . The measurable partition into cylinder sets D fC1 ; : : : ; Ck g is a two-sided generator. Indeed, n _

˚ j D Cin in W in ; : : : ; in 2 f1; : : : ; kg ;

j Dn

and the cylinder sets generate the -algebra of Xk . By Theorem 4.3, we obtain ! n1 _ 1 ` h ./ D lim H n!1 n `D0

1 X .Ci0 in1 / log .Ci0 in1 / D lim n!1 n i i 0

n1

1 X D lim pi0 pin1 log.pi0 pin1 / n!1 n i i 0

n1

1 X D lim pi0 pin1 .log pi0 C C log pin1 / n!1 n i i 0

n1

k k X 1 X D lim n pi log pi D pi log pi ; n!1 n i D1 i D1

124


that is, h ./ D

k X

pi log pi :

(4.26)

i D1

Since the function

in (4.4) is strictly convex, we obtain k X 1 .pi / k i D1 ! k X pi k k i D1 1 D log k D k k

h ./ D k

for any Bernoulli measure , with equality if and only if p1 D D pk D 1=k. In particular, ˚ sup h ./ W is a Bernoulli measure D log k: (4.27) Example 4.6. We consider the shift map 1 W f1; 2; 3; 4gZ ! f1; 2; 3; 4gZ with the Bernoulli measure 1 with the probabilities 14 ; 14 ; 14 ; 14 . We also consider the shift map 2 W f1; 2; 3; 4; 5gZ ! f1; 2; 3; 4; 5gZ with the Bernoulli measure 2 with probabilities 12 ; 18 ; 18 ; 18 ; 18 . It follows from (4.26) that h1 .1 / D h2 .2 / D 2 log 2: Now we compute the entropy of a Markov measure. Example 4.7. Let W Xk ! Xk be the shift map and let be the (two-sided) Markov measure associated to a stochastic pair .P; p/. As in Example 4.5, we use the twosided generator D fC1 ; : : : ; Ck g. We have H

n1 _

!

`

D

X

.Ci0 in1 / log .Ci0 in1 /

i0 in1

`D0

D

X

pi0 pi0 i1 pin2 in1 log.pi0 pi0 i1 pin2 in1 /

i0 in1

D

X

pi0 pi0 i1 pin2 in1 log pi0

i0 in1

X

pi0 pi0 i1 pin2 in1

i0 in1

D

k X i D1

pi log pi .n 1/

n2 X

log pij ij C1

j D0 k k X X i D1 j D1

pi pij log pij ;

4.3 Shannon–McMillan–Breiman Theorem

125

and by Theorem 4.3, we obtain n1 _ 1 ` h ./ D lim H n!1 n

! D

k k X X

pi pij log pij :

i D1 j D1

`D0

Furthermore, h ./ D

k k X X

pi .pij /

i D1 j D1

D

k X

pi

i D1

k X

k X

! .pij /

j D1

pi k

i D1

D

k X

k X pij j D1

pi k

i D1

!

k

1 D log k; k

with equality if and only if pij D 1=k for every i and j , that is, if and only if is a Bernoulli measure with p1 D D pk D 1=k. Therefore (compare with (4.27)), ˚ sup h ./ W is a Markov measure D log k:

4.3 Shannon–McMillan–Breiman Theorem We describe in this section a different approach to the definition of metric entropy, based on the existence of a certain local quantity. The metric entropy is the integral of this local quantity. We first note that if is a measurable partition of X with respect to a measure , then for -almost every x 2 X , there exists a single element n .x/ 2 n D

n1 _

T k

kD0

such that x 2 n .x/. Theorem 4.4 (Shannon–McMillan–Breiman). If T W X ! X is a measurable transformation preserving a probability measure in X and is a measurable partition of X , then the limit

126


1 h .T; ; x/ WD lim log .n .x// n!1 n exists for -almost every x 2 X . Moreover, the function x 7! h .T; ; x/ is T invariant almost everywhere, is -integrable, and Z h .T; / D

h .T; ; x/ d.x/: X

Proof. Given a measurable set A X , we denote by x 7! .Aj/.x/ the conditional expectation of A with respect to the -algebra B./ generated by (see Sect. 2.5.2). This is a B./-measurable function such that Z .Aj/ d D .A \ B/ B

for any set B 2 B./. Lemma 4.8. We have log .n .x// D log ..T n1 .x/// C

n1 X

log .T j 1 .x//j

j D1

_

nj

! T k

(4.28)

kD1

for every n 2 N and -almost every x 2 X . Proof of the lemma. We first observe that if A; B X are measurable sets, then .Aj/ D

X .A \ B/ .B/

B2

B

(4.29)

-almost everywhere. Indeed, the right-hand side of (4.29) is B./-measurable. Moreover, given C 2 , we have Z X .A \ B/ C B2

.B/

Z

B d D .A \ C / D

.Aj/ d: C

This establishes (4.29). Now we use induction to establish (4.28). When n D 1, identity is clear. W (4.28)k and Now we assume that it holds for some n 2 N. Setting ˛n D n1 k D1 T using (4.29), we obtain .Aj˛n / D

X .A \ B/

B ; .B/ B2˛ n


and thus, log .Aj˛n / D

127

X

log

B2˛n

.A \ B/

B .B/

(4.30)

-almost everywhere for every A 2 . Therefore, log .nC1 .x// D

X

log .A \ B/ A\B .x/

A2;B2˛n

D

X

log .B/ B .x/ A .x/ C

A2;B2˛n

D

X

log .B/ B .x/ C

B2˛n

X

X

log .Aj˛n / A .x/

A2

log .Aj˛n / A .x/

A2

D log .˛n .x// C log ..x/j˛n /; and since ˛n .x/ D n .T .x//, we obtain log .nC1 .x// D log .n .T .x/// C log ..x/j˛n /: By the induction hypothesis, we conclude that log .nC1 .x// D log ..T n1 .T .x//// C

n1 X

log .T j 1 .T .x//j˛nC1j

j D1

C log ..x/j˛n / D log ..T n .x/// C

n1 X

log .T j 1 .x/j˛nC1j

j D2

C log ..x/j˛n / D log ..T n .x/// C

n1 X

log .T j 1 .x/j˛nC1j :

j D1

t u

This establishes identity (4.28). We have thus shown that log .n .x// D

n1 X

Fnj .T j .x//;

j D0

where Fk .x/ D log ..x/j˛k /;

(4.31)

128


with the convention that ˛1 D fX g. Now we observe that the function 'W X ! R 0 defined -almost everywhere by '.x/ D log ..x// is in L1 .X; /, since Z X j'j d D .C / log .C / D H ./ < 1: X

C 2

Therefore, it follows from the increasing martingale theorem (Theorem A.7) that the sequence .Fk /k2N converges -almost everywhere and in L1 .X; / to the function F .x/ D log ..x/j˛1 /; W where ˛1 D 1 n D 1 A.˛n / is the -algebra generated by the partitions ˛n . On the other hand, it follows from (4.31) that n1 1 1X log .n .x// D Fnj .T j .x// n n j D0 n1 n1 1X 1X D F .T j .x// C .Fnj F /.T j .x//: n j D0 n j D0

(4.32)

By Birkhoff’s ergodic theorem (Theorem 2.2), the limit 1X F .T j .x// n!1 n j D0 n1

.x/ D lim

exists for -almost every x 2 x. Moreover, the function everywhere, is in L1 .X; /, and Z Z d D F d: X

(4.33) is T -invariant -almost

X

We want to show that the second term in the right-hand sice of (4.32) converges to zero -almost everywhere. For this, we consider the function F D supk1 jFk j. Lemma 4.9. The function F is in L1 .X; /. Proof of the lemma. Given a measurable set A X , c > 0, and n 2 N, we consider the set BnA X composed of the points x 2 X such that log .Aj˛k / c

for k D 1; : : : ; n 1

and log .Aj˛n / > c. Clearly, 1 ˚ X x 2 A W F .x/ > c D .BnA \ A/: nD1


129

Moreover, since BnA 2 B.˛n /, we obtain Z .BnA \ A/ D Z

Z BnA

BnA

XA d D

BnA

.Aj˛n / d

e c d D e c .BnA /;

and thus, 1 ˚ X x 2 A W F .x/ > c D .BnA \ A/ nD1

e c

1 X

.BnA / e c

nD1

since BnA \ BmA D ∅ for n ¤ m. Therefore, Z

F d D X

XZ A2

D

XZ A2

F d A 1

˚ x 2 A W F .x/ > c dc

0

XZ

log .A/

Z .A/ dc C

0

A2

1 log .A/

e c dc

X .A/ log .A/ C .A/ D A2

D H ./ C 1 < 1; t u

which yields the desired result.

Now we return to the last term in the right-hand side of (4.32). Given k 2 N, we set Gk D jFk F j and Gk D supnk Gn . We note that n1 nk 1X 1 1X Gnj .T j .x// D Gnj .T j .x// C n j D0 n j D0 n

1X j 1 Gk .T .x// C n j D0 n nk

n1 X

Gnj .T j .x//

j DnkC1 n1 X

.F C F /.T j .x//:

j DnkC1

Letting n ! 1, it follows from Lemma 4.9 and Birkhoff’s ergodic theorem (see also Exercise 2.13) that

130


lim sup n!1

n1 n1 1X 1X j Gnj .T j .x// lim Gk .T .x// DW n!1 n n j D0 j D0

k .x/

for -almost every x 2 X , for some T -invariant almost everywhere function L1 .X; / with Z Z k

d D

X

X

k

2

Gk d:

Since Gk ! 0 -almost everywhere when k ! 1, and Gk F C F for k 2 N, with F C F 2 L1 .X; /, it follows from the dominated convergence theorem (Theorem A.3) that Z

1X .Gnj ı T j / d n j D0 n1

lim sup X

n!1

Z X

Gk d ! 0

when k ! 1. Therefore, 1X Gnj .T j .x// D 0 n!1 n j D0 n1

lim

for -almost every x 2 X , and by (4.32) and (4.33), we obtain 1 lim log .n .x// D n!1 n

.x/

for -almost every x 2 X . This establishes the first statement in the theorem as well as the T -invariance -almost everywhere and the -integrability of the function x 7! h .T; ; x/. Moreover, Z

1 lim log .n .x// d.x/ D X n!1 n

Z

Z d D X

F d: X

Since Fn ! F in L1 .X; / when n ! 1, it follows from (4.30) and Theorem 4.1 that Z Z Z F d D lim Fn d D lim log ..x/j˛n / d.x/ X

n!1 X

n!1 X

D lim

n!1

D lim

n!1

XZ A2

log .Aj˛n / d A

XZ X A2

A B2˛ n

log

.A \ B/

B d .B/


131

D lim n!1

X

.A \ B/ log

A2;B2˛n

.A \ B/ .B/

D lim H .j˛n / D h .T; /: n!1

t u

This completes the proof of the theorem. Now we give some examples.

Example 4.8. Let Eq .z/ D zq be an expanding map of the circle, for some q > 1, and let be the Lebesgue measure in S 1 . We consider again the measurable partitions n of S 1 in (4.14). For -almost every x 2 S 1 , we have n1 .x/ D n .x/, and hence, 1 1 1 log .n .x// D log n D log q: n n q It follows from Theorem 4.4 that h .Eq ; 1 / D log q. Example 4.9. Let W XkC ! XkC be the shift map and let be the (one-sided) Bernoulli measure generated by the numbers p1 ; : : : ; pk . We also consider the measurable partition into cylinder sets D fC1 ; : : : ; Ck g. Given ! D .i1 i2 / 2 XkC , we have n .!/ D Ci1 in , and hence, log .n .!// D log .Ci1 in / D

n X

log pij D

j D1

n1 X

'. j .!//;

j D0

where 'W XkC ! R is the continuous function given by '.i1 i2 / D log pi1 . Since is ergodic (see Proposition 3.8), for -almost every ! 2 XkC , we have 1 log .n .!// ! n D

Z XkC

k X

' d D

k X

.Cj /'jCj

j D1

.Cj / log .Cj / D H ./:

j D1

By Theorem 4.4, we obtain Z h .T; / D

XkC

1 lim log .n .!// d.!/ D H ./: n!1 n

Moreover, since n1 _ j D0

˚ j D Ci1 in W i1 ; : : : ; in 2 f1; : : : ; kg ;

132


the partition is a one-sided generator, and it follows from Theorem 4.3 that h .T / D h .T; / D H ./ D

k X

pi log pi :

i D1

Example 4.10. For a transformation T W Œ0; 1 ! Œ0; 1 and a T -invariant probability measure as in Exercise 3.28, we consider the measurable partition ˚ D .ai ; ai C1 / W i D 0; : : : ; p : Using the notation in that exercise, we know that 1 lim log .Ii0 in1 / D n!1 n

Z

1

log jT 0 j d

0

T for -almost every x D 1 n D 0 Ii0 in1 . Since is a one-sided generator, it follows from Theorems 4.3 and 4.4 that Z h .T / D h .T; / D 0

1

1 lim log .Ii0 in1 / d.x/ D n!1 n

Z

1

log jT 0 j d:

0

4.4 Topological Entropy We introduce in this section the notion of topological entropy, and we establish some of its basic properties. Since the emphasis of this chapter is on the notion of metric entropy, we develop the material in a somewhat pragmatic manner. For completeness, some additional results correspond to exercises at the end of this chapter. For simplicity of the exposition, we only consider continuous transformations of a compact metric space, and not of an arbitrary compact topological space.

4.4.1 Basic Notions and Some Properties Let T W X ! X be a continuous transformation of a compact metric space .X; d /. For each n 2 N, we introduce a new distance in X by ˚ dn .x; y/ D dn;T .x; y/ D max d.T k .x/; T k .y// W 0 k n 1 :

(4.34)

Moreover, given " > 0, we denote by N.d; "/ the maximum number of points in X at a d -distance at least ".

4.4 Topological Entropy

133

Definition 4.8. We define the topological entropy of T by h.T / D hd .T / D lim lim sup "!0 n!1

1 log N.dn ; "/: n

(4.35)

We note that since the function " 7! lim sup n!1

1 log N.dn; "/ n

is nondecreasing, the limit in (4.35) when " ! 0 is well defined. Now we give several examples. Example 4.11. If T is an isometry, then d.T n .x/; T n .y// D d.x; y/, and hence, dn .x; y/ D d.x; y/

for every n 2 N:

Therefore, N.dn; "/ D N.d; "/, and h.T / D lim lim sup "!0 n!1

1 log N.d; "/ D 0: n

Example 4.12. Let T W Tn ! Tn be the toral translation T .x1 ; : : : ; xn / D .x1 C ˛1 ; : : : ; xn C ˛n / mod 1: Since T is an isometry, we have h.T / D 0. Example 4.13. Now we consider the expanding map of the circle Eq W S 1 ! S 1 , for some q > 1, with the distance d in S 1 obtained from normalizing the length. We note that if d.x; y/ < q n , then dn .x; y/ D d.Eqn1 .x/; Eqn1 .y// D q n1 d.x; y/:

(4.36)

Now given k 2 N, we consider the points xi D i=q nCk for i D 0; : : : ; q nCk 1. By (4.36), we have dn .xi ; xi C1 / D q .kC1/ for each i . Therefore, dn .xi ; xj / q .kC1/ whenever i ¤ j , and we conclude that N.dn ; q .kC1/ / q nCk : On the other hand, given a set S in S 1 with at least q nCk C 1 points, there exist x; y 2 S with x ¤ y such that d.x; y/ < q .nCk/ , and hence, also dn .x; y/ < q .kC1/ . This shows that N.dn ; q .kC1/ / D q nCk

for n; k 2 N;

134


and h.Eq / D lim lim sup k!1 n!1

D lim lim sup k!1 n!1

1 log N.dn ; q .kC1/ / n nCk log q n

D log q: By (2.21) and Example 4.4, we have h.Eq / D h .Eq / D p.Eq /; where is the Lebesgue measure and where p is the periodic entropy. Now we establish several properties of the topological entropy. Proposition 4.4. If T W X ! X is a continuous transformation of a compact metric space, then the following properties hold: 1. If X S is a closed T -invariant set, then h.T j/ h.T /. 2. If X D m k D 1 k where k is a closed T -invariant set for k D 1; : : : ; m, then ˚ h.T / D max h.T jk / W 1 k m : Proof. Let X be a closed T -invariant set. Then T jW ! is also a continuous transformation of a compact metric space. Since N.dn;T j ; "/ N.dn;T ; "/, the first property is immediate. For the second property, we first note that N.dn;T ; "/

m X

N.dn;T jk ; "/:

kD1

Therefore, for each n and ", there exists k D k.n; "/ 2 N such that N.dn;T jk ; "/

1 N.dn;T ; "/: m

(4.37)

Now we take a sequence .ni /i 2N N such that 1 1 log N.dni ;T ; "/ D lim sup log N.dn;T ; "/: i !1 ni n!1 n lim

Since f1; : : : ; mg is a finite set, we can also assume that k.ni ; "/ takes the same value, say j , for all i 2 N. By (4.37), we thus obtain


lim sup n!1

135

1 1 log N.dn;T jj ; "/ lim sup log N.dni ;T jj ; "/ n n i !1 i lim sup

1 log N.dni ;T ; "/ ni

D lim sup

1 log N.dn;T ; "/: n

i !1

n!1

Therefore, h.T jj / h.T /, and it follows from Property 1 that ˚ h.T / max h.T jk / W 1 k m h.T jj / h.T /: This completes the proof of the proposition.

t u

We also give an equivalent description of the topological entropy. Given " > 0, let D.d; "/ be the minimum number of sets of d -diameter less than " that are needed to cover X . Theorem 4.5. If T W X ! X is a continuous transformation of a compact metric space, then 1 h.T / D lim lim log D.dn ; "/: (4.38) "!0 n!1 n Proof. It is easy to verify that N.dn ; 2"/ D.dn ; "/ N.dn; "=2/

(4.39)

for every n 2 N and " > 0. Lemma 4.10. If m; n 2 N and " > 0, then D.dmCn ; "/ D.dm ; "/D.dn ; "/: Proof of the lemma. Let fA1 ; : : : ; Ak g be a cover of X by sets of dn -diameter less than ", where k D D.dn ; "/. Let also fB1 ; : : : ; B` g be a cover of X by sets of dm diameter less than ", where ` D D.dm ; "/. Then each set Ai \ T n Bj has dmCn diameter less than ", and ˚ U D Ai \ T n Bj W i D 1; : : : ; k and j D 1; : : : ; ` is a cover of X . Therefore, D.dmCn ; "/ card U `k D D.dm ; "/D.dn ; "/; which yields the desired inequality.

t u

136


It follows from Lemmas 4.3 and 4.10 that the limit lim

n!1

1 log D.dn ; "/ n

exists. Hence, by (4.39), we conclude that identity (4.38) holds.

t u

Finally, we compute the topological entropy of the powers of a given transformation. Theorem 4.6. If T W X ! X is a continuous transformation of a compact metric space, then the following properties hold: 1. h.T k / D kh.T / for each k 2 N. 2. If T is a homeomorphism, then h.T 1 / D h.T /. Proof. We first note ˚ dn;T k .x; y/ D max d.T i k .x/; T i k .y// W 0 i n 1 ˚ max d.T i .x/; T i .y// W 0 i nk 1 D dnk;T .x; y/: Therefore, N.dn;T k ; "/ N.dnk;T ; "/, and h.T k / lim lim sup "!0 n!1

1 log N.dnk;T ; "/ kh.T /: n

For the converse inequality, we note that by the uniform continuity of T , given " > 0 there exists ı."/ 2 .0; "/ such that dk .x; y/ < " whenever d.x; y/ < ı."/. Equivalently, d.x; y/ ı."/ whenever dk .x; y/ ", and hence dn;T k .x; y/ ı."/ whenever dk n .x; y/ ": Therefore, N.dk n ; "/ N.dn;T k ; ı."//, and lim lim sup

"!0 n!1

1 1 log N.dk n ; "/ lim lim sup log N.dn;T k ; ı."// D h.T k /; (4.40) "!0 n!1 n n

where the last equality is due to the fact that ı."/ ! 0 when " ! 0. Finally, it follows from (4.39) and (4.40) that h.T k / lim lim sup "!0 n!1

lim lim

"!0 n!1

1 log N.dk n ; "/ n

1 log D.dk n ; 2"/ n


137

1 log D.dk n ; 2"/ kn 1 D k lim lim log D.dn ; 2"/ D kh.T /; "!0 n!1 n

D k lim lim

"!0 n!1

which yields the first property. For the second property, we observe that dn;T .x; y/ D dn;T 1 .T n1 .x/; T n1 .y//: This shows that if x1 ; : : : ; xk are at a dn;T -distance at least ", then the points T n1 .x1 /; : : : ; T n1 .xk / are at a dn;T 1 -distance at least ", and vice versa. Therefore, N.dn;T 1 ; "/ D N.dn;T ; "/; and hence, h.T 1 / D h.T /.

t u

4.4.2 Topological Nature of the Entropy The results described in this section highlight the topological nature of the topological entropy. We first show that h.T / only depends on the topology induced by the distance d . Proposition 4.5. If the distances d and d 0 generate the same topology on X , then 0 hd .T / D hd .T / for every continuous function T W X ! X . Proof. Given " > 0, we consider the set ˚ D" D .x; y/ 2 X X W d.x; y/ " : Since the function d W X X ! R is continuous, the set D" is compact. Moreover, since d 0 W X X ! R is also continuous, we have ˚ ı."/ WD min d 0 .x; y/ W .x; y/ 2 D" > 0; and ı."/ ! 0 when " ! 0. We note that dn0 .x; y/ ı."/ whenever dn .x; y/ ": Therefore, N.dn0 ; ı."// N.dn ; "/, and 0

hd .T / D lim lim sup "!0 n!1

lim lim sup "!0 n!1

1 log N.dn0 ; ı."// n 1 log N.dn ; "/ D hd .T /: n 0

Reversing the roles of d and d 0 , we also obtain hd .T / hd .T /.

t u

138


We also show that the topological entropy is a topological invariant, in the sense that topologically equivalent transformations have the same topological entropy. We recall that two continuous transformations T W X ! X and S W Y ! Y are topologically equivalent if there exists a homeomorphism W X ! Y such that

ı T D S ı in X . In other words, the diagram T

X ! ? ?

y

X ? ?

y

(4.41)

S

Y ! Y is commutative. Proposition 4.6. If two continuous transformations T W X ! X and S W Y ! Y of compact metric spaces are topologically equivalent, then h.T / D h.S /. Proof. Let d be the distance in X . We introduce a distance in Y by d 0 .x; y/ D d. 1 .x/; 1 .y//; where is the homeomorphism in (4.41). We note that W .X; d / ! .Y; d 0 / is an isometry, and hence, 0 N.dn;T ; "/ D N.dn;S ; "/: (4.42) Moreover, since is a homeomorphism, the distance d 0 generates the original topology of Y . Therefore, by Proposition 4.5 and (4.42), we obtain 0

h.S / D hd .S / D hd .T / D h.T /; which yields the desired result.

t u

4.5 Variational Principle We end this chapter by establishing an important relation between the metric entropy and the topological entropy—the so-called variational principle. It says that the supremum of the metric entropies over all invariant probability measures is equal to the topological entropy. Theorem 4.7 (Variational principle for the topological entropy). If T W X ! X is a continuous transformation of a compact metric space, then ˚ h.T / D sup h .T / W is a T -invariant probability measure in X :

(4.43)

Proof. Let D fC1 ; : : : ; Ck g be a measurable partition of X . Given ı > 0, for each i D 1; : : : ; k, let Di Ci be a compact set such that .Ci n Di / < ı. Now

4.5 Variational Principle

139

S let ˇ D fD0 ; D1 ; : : : ; Dk g, where D0 D X n kiD 1 Di . Clearly, ˇ is a measurable partition of X . Proceeding as in (4.22) with Din replaced by Di , we conclude that H .jˇ/ < 1 for any sufficiently small ı. Thus, it follows from (4.19) that h .T m ; / h .T m ; ˇ/ C H .jˇ/ < h .T m ; ˇ/ C 1

(4.44)

for each m 2 N. Now we consider the open cover of X given by U D fD0 [ D1 ; : : : ; D0 [ Dk g; as well as the open covers Umn D

n1 \

T

i m

Ui W U0 ; : : : ; Un1

2U

i D0

for each m; n 2 N. Since each element D0 [ Di of U intersects at most the elements D0 and Di of ˇ, we have card ˇmn 2n card Umn ; where ˇmn D

Wn1 i D0

T i m ˇ. Moreover, by (4.6), we obtain

H .ˇmn / log card ˇmn n log 2 C log card Umn :

(4.45)

We observe that if " is the Lebesgue number of U (i.e., if every ball of radius r < " is contained in some element of U), then " is the Lebesgue number of the cover Umn with respect to the distance dn;T m . Moreover, by construction, each open set U 2 Umn contains at least a point xU that is not in any other element of Umn . Therefore, Bdn;T m .xU ; "/ U for each U 2 Umn ; and thus, dn;T m .xU ; xV / " for any V 2 Umn distinct from U . This shows that card Umn N.dn;T m ; "/; and it follows from (4.45) that 1 H .ˇmn / n 1 log 2 C lim sup log N.dn;T m ; "/: n n!1

h .T m ; ˇ/ D lim

n!1

By (4.44), letting " ! 0 yields h .T m ; / h.T m / C log 2 C 1;

140


and hence, h .T m / h.T m / C log 2 C 1: Therefore, by Proposition 4.3 and Theorem 4.6, 1 h .T m / m 1 h.T m / C log 2 C 1 m 1 D h.T / C .log 2 C 1/ m

h .T / D

for each m 2 N. Letting m ! 1, we conclude that h .T / h.T /. To complete the proof, given " > 0, for each n 2 N, we consider a set En of points at a d -distance at least " such that card En D N.dn ; "/. We then define measures n1 X 1 1X i n D ıx and n D T n ; card En x2E n i D0 n

with T as in (3.6). Given a sequence .kn /n2N N such that lim

n!1

1 1 log card Ekn D lim sup log card En ; kn n!1 n

(4.46)

let be any accumulation point of the sequence of measures .kn /n2N , which exists by Theorem 3.1. Moreover, if .mn /n2N is a subsequence of .kn /n2N such that .mn /n2N converges to , then it follows from (3.7) that is T -invariant. We note that in general, the measure may depend on ". Now we consider some particular measurable partitions. Let fB1 ; : : : ; Bk g be an open cover of X by balls of radius less than "=2 such that .@Bi / D 0 for i D 1; : : : ; k. This is always possible since for each x 2 X , there are at most countably many values of r > 0 such that [email protected]; r// > 0. We define a partition D fC1 ; : : : ; Ck g of X by C1 D B1

and Ci D Bi n

i 1 [

Bj

for i D 2; : : : ; k:

j D1

S Then diam Ci < " and .@Ci / D 0 for each i since @Ci kj D 1 @Bj . Now we W i contains at most observe that since diam Ci < ", each element of n D n1 i D0 T one point in En . Therefore, there is exactly a number card En of elements of n with n -measure equal to 1= card En , and thus, H n .n / D log card En :

(4.47)


141

Given m; n 2 N, we write n D q m C r, where q 0 and 0 r < m. We have _

q1

n D qmCr D

T j m m _

_

qmCr1

T j ;

j Dqm

j D0

and thus, for each i D 0; : : : ; m 1, the measurable partition _

q1

D

T

j mi

_

!

qmCr1

m _

T

j

_ i

j Dqm

j D0

is a refinement of n . Since _

qmCr1

card

! T j _ i

.card /2m ;

j Dqm

it follows from Lemma 4.1 and (4.6) that H n .n / H n ./

q1 X

H n .T j mi m / C 2m log card :

(4.48)

j D0

Now we observe that since the function

in (4.4) is convex, we have

m1 q1 n1 1X 1 XX j mi H .T m / H n .T ` m / n i D0 j D0 n n `D0

D

n1 XX 1 n .T ` A/ n

A2m `D0

D

X

n1 X

A2m

`D0

X

1 n .T ` A/ n

!

.n .A// D Hn .m /;

A2m

and thus, by (4.48), 2m2 m H n .n / Hn .m / C log card : n n

(4.49)

142


It follows from (4.47) with n replaced by mn that 1 1 2m log card Emn Hmn .m / C log card : mn m mn On the other hand, by (4.46), letting n ! 1 yields lim sup n!1

1 1 log N.dn ; "/ D lim sup log card En n n!1 n 1 1 lim Hmn .m / D H .m /: n!1 m m

(4.50)

Indeed, let A X be a measurable set with .@A/ D 0. Since .mn /n2N converges to , if 'k W X ! RC 0 is a sequence of continuous functions decreasing to A when k ! 1, then Z lim sup mn .A/ lim sup 'k dmn n!1

n!1

Z D

X

'k d ! .A/ X

when k ! 1. Therefore, since .@A/ D 0, we have lim sup mn .A/ lim sup mn .A/ .A/ D .A/: n!1

n!1

Similarly, since @.X n A/ D @A, we also have lim sup mn .X n A/ .X n A/; n!1

which yields lim inf mn .A/ .A/: n!1

Together with (4.51), this implies that lim mn .A/ D .A/;

n!1

and hence, lim Hmn .m / D H .m /:

n!1

Letting m ! 1 in (4.50), we obtain lim sup n!1

1 log N.dn ; "/ h .T; / h .T / sup h .T /; n

(4.51)

4.6 Exercises

143

where the supremum is taken over all T -invariant probability measures in X . Letting " ! 0 yields h.T / sup h .T /:

t u


We note that identity (4.43) could be used as an alternative definition of topological entropy.

4.6 Exercises Exercise 4.1. Let T W X ! X be a measurable transformation preserving a probability measure in X . Show that if T is invertible almost everywhere and T 1 is measurable, then h .T 1 / D h .T /: Exercise 4.2. Let T W Rm ! Rm be a diffeomorphism preserving a probability measure in Rm such that Z h .T / D n

Rm

logkdx T n k d

for each n 2 N. Show that h .T / lim

n!1

1 log sup kdx T n k: n x2Rm

Exercise 4.3. Consider the transformation T W f0; 1gN ! f0; 1gN defined by ( .T .x//n D

1 xn

if n D 1 or if xm D 1 for every m < n;

xn

otherwise:

It is called an infinite adding machine since it can be written in the form T .x/ D x C .10 /, where each element is in base 2 with the addition effected from the left to the right. Show that: 1. T preserves the Bernoulli measure with probability vector .1=2; 1=2/. 2. h .T / D 0. Exercise 4.4. For i D 1; 2, let Ti W Xi ! Xi be a measurable transformation preserving a probability measure i in Xi . Show that if .T1 ; 1 / and .T2 ; 2 / are equivalent (see Exercise 3.23), then h1 .T1 / D h2 .T2 /. Exercise 4.5. Find a function hW f1; 2; 3; 4gZ ! f1; 2; 3; 4; 5gZ as in Exercise 3.23 for the shift maps 1 and 2 in Example 4.6.

144


Exercise 4.6. Say if there exist transformations such that: 1. .T; / and .T 2 ; / are ergodic, but .T 3 ; / is not ergodic. 2. .T; / and .T 3 ; / are ergodic, but .T 2 ; / is not ergodic. 3. .T; / is ergodic, but .T 2 ; / and .T 3 ; / are not ergodic. Exercise 4.7. Show that H .j/ D 0 if and only if is a refinement of . Hint: if H .j/ D 0, then .C \ D/ log

.C \ D/ D 0 for C 2 ; D 2 ; .D/

and hence, either .C \ D/ D 0 or .C \ D/ D .D/ for each C 2 and D 2 . Exercise 4.8. Show that d.; / D H .j/ C H .j/

(4.52)

is a distance (Rohklin’s distance) in the space of measurable partitions of X . Hint: by Proposition 4.1 and Lemmas 4.4 and 4.5, we have H .j/ H . _ j/ D H .j _ / C H .j/ H .j/ C H .j/: Exercise 4.9. Show that if T W X ! X is a measurable transformation and and are measurable partitions of X , then jh .T; / h .T; /j d.; /; where d is the distance in (4.52). Hint: use inequality (4.19). Exercise 4.10. Compute h .T / for the map T and the measure in Exercise 3.28. Exercise 4.11. Compute the entropy of a one-sided Markov measure. Exercise 4.12. Let T W X ! X be a measurable transformation preserving a probability measure in X . Show that if is a one-sided generator and T is invertible almost everywhere, then h .T / D 0. Exercise 4.13. Let T W X ! X be a measurable transformation preserving a probability measure in X and let be a finite (one-sided or two-sided) generator. 1. Show that h .T / log card . 2. When h .T / D log card , show that for each n 2 N, the measurable partition W k n D n1 has exactly .card /n elements all with the same measure. k D0 T Hint: the function defined by (4.4) is strictly convex.

4.6 Exercises

145

3. Show that if h .T n / log n C an for every n 2 N, then h .T / a. Exercise 4.14. For the shift map W XkC ! XkC , show that h./ D log k. Exercise 4.15. For the one-sided topological Markov chain jXAC W XAC ! XAC , show that h./ D log .A/, where .A/ is the spectral radius of the matrix A. Exercise 4.16. For a continuous transformation T W X ! X of a compact metric space, show that 1 h.T / D lim lim inf log N.dn ; "/: "!0 n!1 n Hint: see the proof of Theorem 4.5. Exercise 4.17. Let T W X ! X be a continuous transformation of a compact metric space. Given " > 0, we denote by M.d; "/ the least number of points p1 ; : : : ; pm 2 X such that any x 2 X satisfies d.x; pi / < " for some i . Show that: 1. For each " > 0, D.d; 2"/ M.d; "/ N.d; "/ M.d; "=2/ D.d; "=2/: 2. h.T / D lim lim sup "!0 n!1

D lim lim inf "!0 n!1

1 log M.dn ; "/ n

1 log M.dn ; "/: n

Exercise 4.18. Given a matrix A 2 S.2; Z/ without eigenvalues in S 1 , compute the topological entropy of the toral automorphism TA W T2 ! T2 . Hint: use Exercise 4.17. Exercise 4.19. Let T W X ! X be an invertible measurable transformation with measurable inverse, preserving a probability measure in X . Given a function 'W X ! R, we define a new function UT .'/W X ! R by UT .'/ D 'ıT . Show that: 1. UT W L2 .X; / ! L2 .X; / is a linear transformation with norm kUT k D 1. 2. T is ergodic if and only if 1 is an eigenvalue of UT . 3. For a function ' 2 L2 .X; / such that Z

Z lim

n!1 X

'UTn .'/ d D

2 ' d ;

(4.53)

X

if S is the smallest closed subspace of L2 .X; / containing the functions 1 and UTn .'/ for every n 2 N, then both S and S ? (the orthogonal complement of S ) are contained in

146


Z

Z 2 L .X; / W lim 2

n!1 X

UTn .'/ d

Z

D

d X

' d :

X

4. is mixing if and only if (4.53) holds for every function ' 2 L2 .X; /. Exercise 4.20. Let X be a compact topological space and let T W X ! X be a continuous transformation. Show that for each finite open cover U of X , the limit _ 1 log card T k U n!1 n n1

lim

kD0

exists, where n1 _ kD0

( T k U D

n1 \

) T k Uk W U0 ; : : : ; Un1 2 U :

kD0

Exercise 4.21. Let X be a compact metric space and let T W X ! X be a continuous transformation. Show that _ 1 log card T k U; n!1 n n1

h.T / D sup lim U

kD0

where the supremum is taken over all finite open covers U of X .

Notes The notion of metric entropy is due to Kolmogorov [49, 50]. It was extended to all dynamical systems by Sinai [93] in the form (4.12)–(4.13). Theorem 4.4 was proven successively in more general forms by several authors. Shannon [87] considered Markov measures, although the statement was only derived rigorously by Khinchin [46] (see also [47]). McMillan [60] obtained the L1 convergence, and Breiman [22] obtained the convergence almost everywhere. Our proof of Theorem 4.4 is based on [77]. The original definition of topological entropy is due to Adler, Konheim and McAndrew [1] (in the form described in Exercises 4.20 and 4.21). The definition in (4.35) was introduced independently by Bowen [19] and Dinaburg [27]. The variational principle for the topological entropy in Theorem 4.7 is a combination of work of Goodwyn [36] (showing that the topological entropy bounds the metric entropy), Dinaburg [27] (for a finite-dimensional space X ), and Goodman [34] (for an arbitrary space). Our proof of Theorem 4.7 is based on [77], which follows the simpler proof of Misiurewicz [61].

Chapter 5

Thermodynamic Formalism

This chapter is an introduction to the thermodynamic formalism. We first introduce the notion of topological pressure, which includes topological entropy as a special case. In particular, we establish a somewhat explicit formula for the topological pressure in the case of symbolic dynamics. This formula is particularly useful in dimension theory of hyperbolic dynamics. We also establish the variational principle for the topological pressure. Finally, we show that there exist equilibrium measures for any expansive transformation. These are invariant probability measures attaining the supremum in the variational principle.

5.1 Introduction The notion of topological pressure, which is the most basic notion of the thermodynamic formalism, was introduced by Ruelle for expansive transformations and by Walters in the general case. The thermodynamic formalism (following Ruelle’s original expression) can be described as a rigorous study of certain mathematical structures inspired in thermodynamics. For a continuous transformation T W X ! X of a compact metric space, the topological pressure of a continuous function 'W X ! R is defined by P .'/ D lim lim sup "!0 n!1

n1 X X 1 log sup exp '.T k .x//; n E x2E kD0

where the supremum is taken over all .n; "/-separated sets E X (see Sect. 5.2 for the definition). For example, taking ' D 0, we recover the notion of topological entropy 1 h.T / D lim lim sup log Nn;" ; "!0 n!1 n where Nn;" is the maximal number of elements of an .n; "/-separated set E X . L. Barreira, Ergodic Theory, Hyperbolic Dynamics and Dimension Theory, Universitext, DOI 10.1007/978-3-642-28090-0 5, © Springer-Verlag Berlin Heidelberg 2012

147

148

5 Thermodynamic Formalism

The variational principle relating topological pressure to Kolmogorov–Sinai entropy was established by Ruelle for expansive transformations and by Walters in the general case. It says that Z ' d ; P .'/ D sup h .T / C

X

where the supremum is taken over all T -invariant probability measures in X . The theory also includes a discussion of the existence and uniqueness of equilibrium and Gibbs measures. In particular, a T -invariant probability measure is called an equilibrium measure for ' if Z P .'/ D h .T / C

' d: X

As we already mentioned earlier, the possibility of coding repellers and hyperbolic sets via symbolic dynamics often allows one to give simpler proofs. Thus, it is of interest to have explicit formulas for the topological pressure with respect to the shift map. One can show that the topological pressure of a continuous function 'W XkC ! R is given by n1 X X 1 log exp sup ' ı k; n!1 n C i i n 1 i i

P .'/ D lim

1

kD0

n

where Ci1 in are the cylinder sets. For example, for the function '.i1 i2 / D log i1 ; which occurs later on in dimension theory of hyperbolic dynamics, we have P .'/ D log

k X

j :

j D1

5.2 Topological Pressure We first introduce the notion of topological pressure. Let T W X ! X be a continuous transformation of a compact metric space .X; d /. For each n 2 N, we consider the distance dn in X defined by (4.34). Definition 5.1. Given " > 0, a set E X is called .n; "/-separated if dn .x; y/ > " for every x; y 2 E with x ¤ y.


149

We note that since X is compact, each .n; "/-separated set E is finite. The notion of topological pressure can now be introduced as follows: Definition 5.2. The topological pressure of a continuous function 'W X ! R (with respect to T ) is defined by PT .'/ D lim lim sup "!0 n!1

n1 X X 1 exp '.T k .x//; log sup n E x2E

(5.1)

kD0

where the supremum is taken over all .n; "/-separated sets E X . Since the function " 7! lim sup n!1

n1 X X 1 log sup exp '.T k .x// n E x2E kD0

is nondecreasing, the limit in (5.1) when " ! 0 is well defined. The following example shows that the topological entropy is a particular case of the topological pressure: Example 5.1. For the constant function ' D c, and any .n; "/-separated set E, we have n1 X X X exp '.T k .x// D e nc D e nc card E: x2E

x2E

kD0

Therefore, sup

X

exp

E x2E

n1 X

'.T k .x// D e nc N.dn ; "/;

kD0

with N.dn ; "/ as in Sect. 4.4.1. By (4.35), we obtain PT .c/ D lim lim sup "!0 n!1

1 log e nc N.dn ; "/ D c C h.T /; n

and in particular, PT .0/ D lim lim sup "!0 n!1

1 log N.dn; "/ D h.T /: n

(5.2)

5.3 Symbolic Dynamics We consider in this section the particular case of symbolic dynamics. Let W XkC ! XkC be the (one-sided) shift map. We recall that XkC is a compact metric space with the distance d D dˇ in (3.32) for each ˇ > 1. Given " > 0, let m D m."/ 2 N be the largest integer such that

150


m < log "= log ˇ;

ˇ m > ":

that is,

Then the .n; "/-separated sets are exactly the collections of sequences !i1 imCn1 2 Ci1 imCn1

for i1 ; : : : ; imCn1 2 f1; : : : ; kg:

Now we give a formula for the topological pressure with respect to the shift map. Theorem 5.1. For each continuous function 'W XkC ! R, we have n1 X X 1 log exp sup ' ı k: n!1 n C i i 1 n i i

P .'/ D lim

1

(5.3)

kD0

n

Proof. Since XkC is compact, the function ' is uniformly continuous, and thus, for each ı > 0, there exists n 2 N such that sup ' inf ' < ı Ci1 in

Ci1 in

for every i1 ; : : : ; in 2 f1; : : : ; kg. Writing for simplicity Dn D Ci1 in , we thus have ın WD max sup ' inf ' ! 0 i1 in

Dn

Dn

when n ! 1. This implies that X

0 log

exp sup 'n log

i1 imCn1

DmCn1

X

exp inf

DmCn1

i1 imCn1

'n nım

for each m; n 2 N, where 'n D

n1 X

' ı k:

kD0

On the other hand, given m; n 2 N and a .n; "/-separated set E XkC , we have X i1 imCn1

exp inf 'n DmCn1

X x2E

exp 'n .x/

X i1 imCn1

exp sup 'n ; DmCn1


151

and thus, ım C lim sup n!1

lim sup n!1

lim sup n!1

X 1 log n i i 1

exp sup 'n DmCn1

mCn1

X 1 log exp 'n .x/ n x2E X 1 log n i i 1

exp sup 'n : DmCn1

mCn1

Letting m ! 1, we obtain P .'/ D lim lim sup m!1 n!1

m!1 n!1

Sn D

exp sup 'n

X 1 log n i i

exp inf 'n :

1

D lim lim sup

Moreover, writing

X 1 log n i i

1

X

X

Ci1 in

exp sup 'n k m1 Sn ; DmCn1

i1 imCn1

and

X

exp sup 'n e .m1/k'k1 SmCn1 ; DmCn1

i1 imCn1

where

DmCn1

mCn1

exp sup 'n ;

i1 in

we have

DmCn1

mCn1

˚ k'k1 D sup j'.!/j W ! 2 ˙kC :

This implies that lim lim sup

m!1 n!1

X 1 log n i i 1

exp sup 'n lim sup

mCn1

DmCn1

n!1

1 log Sn ; n

and lim lim sup

m!1 n!1

X 1 log n i i 1

mCn1

exp sup 'n lim lim sup DmCn1

m!1 n!1

D lim sup m!1

1 log SmCn1 n

1 log Sn ; n

152


that is, X 1 log n i i

lim lim sup

m!1 n!1

1

exp sup 'n D lim sup n!1

DmCn1

mCn1

1 log Sn : n

(5.4)

Finally, since sup Ci1 imCn

'mCn

sup Ci1 imCn

'm C

sup 'm C

sup

Ci1 im

we have SmCn

X i1 imCn

D

X

CimC1 imCn

exp

'n ı T m 'n ;

sup 'm C

Ci1 im

Ci1 im

'n

sup CimC1 imCn

X

exp sup 'm

i1 im

sup Ci1 imCn

exp

sup CimC1 imCn

imC1 imCn

'n D Sm Sn :

The formula for the topological pressure P .'/ in (5.3) follows now readily from Lemma 4.3 together with identity (5.4). t u The following are applications of Theorem 5.1: Example 5.2. Given numbers 1 ; : : : ; k > 0, we consider the continuous function 'W XkC ! R defined by '.i1 i2 / D log i1 : (5.5) It follows from (5.3) that n X X 1 log exp log ik n!1 n i i

P .'/ D lim

1

kD1

n

n XY 1 log ik n!1 n i i

D lim

1

1 D lim log n!1 n D log

k X

n

kD1

k X

!n

(5.6)

j

j D1

j :

j D1

We also consider functions depending on the first two symbols of each sequence. Example 5.3. Given numbers ij > 0 for i; j D 1; : : : ; k, we consider the continuous function 'W XkC ! R defined by '.i1 i2 / D log i1 i2 :


153

It follows again from (5.3) that n X X 1 log exp max log ik ikC1 n!1 n inC1 i i

P .'/ D lim

1

kD1

n

X 1 D lim log max n!1 n inC1 i i 1

n

n Y

(5.7)

ik ikC1 :

kD1

Since n1 Y kD1

! ik ikC1 min ij max i;j

inC1

n Y

n1 Y

ik ikC1

kD1

! ik ikC1 max ij ;

kD1

i;j

considering the matrix B D .ij /, it follows from (5.7) that X n1 Y 1 ik ikC1 P .'/ D lim log n!1 n i i 1

n

kD1

1 D lim log tr.B n1 / n!1 n p D log lim n tr.B n / n!1

D log .B/; where .B/ is the spectral radius of B.

5.4 Variational Principle We establish in this section the variational principle for the topological pressure that includes as a particular case the variational principle for the topological entropy in Theorem 4.7. Theorem 5.2 (Variational principle for the topological pressure). Given a continuous transformation T W X ! X of a compact metric space, if 'W X ! R is a continuous function, then

Z

PT .'/ D sup h .T / C

' d ;

(5.8)

X

where the supremum is taken over all T -invariant probability measures in X .

154


Proof. The argument is an elaboration of the proof of Theorem 4.7. We consider again a measurable partition D fC1 ; : : : ; Ck g of X . Given ı > 0, for each i D 1; : : : ; k, let Di Ci be a compact set such that .Ci n Di / < ı. We S also consider the measurable partition ˇ D fD0 ; D1 ; : : : ; Dk g, where D0 D X n kiD1 Di . As in the proof of Theorem 4.7, for any sufficiently small ı, we have h .T; / < h .T; ˇ/ C 1: Now let

˚ D inf d.x; y/ W x 2 Di ; y 2 Dj ; i ¤ j :

Clearly, > 0. Moreover, take " 2 .0; =2/ such that j'.x/ '.y/j < 1 For each n 2 N and C 2 ˇn WD

whenever d.x; y/ < ":

Wn1

j D0 T

j

(5.9)

ˇ, there exists xC 2 C such that

˚ 'n .xC / D sup 'n .x/ W x 2 C ; where 'n D

n1 X

' ı Tj:

j D0

If E is a .n; "=2/-separated set with card E D N.dn ; "=2/, then for each C , there exists pC 2 E such that dn .xC ; pC / < ". By (5.9), we thus obtain 'n .xC / 'n .pC / C n:

(5.10)

On the other hand, since " < =2, for each x 2 E and j 2 f0; : : : ; n 1g, the point T j .x/ can be at most in two elements of ˇ. Therefore, ˚ card C 2 ˇn W pC D x 2n :

(5.11)

To proceed with the proof, we need the following auxiliary result: P Lemma 5.1. For any pi 0 with kiD1 pi D 1, and any ci 2 R, we have k X

pi . log pi C ci / log

i D1

with equality if and only if pi D e ci =

k X

e ci ;

i D1

Pk

i D1

e ci for i D 1; : : : ; k.

(5.12)


155

Proof of the lemma. Setting k pi X ci and xi D c e e i i D1

e ci

ai D Pk

i D1 e

for each i , we obtain

k X

ci

ai xi D

i D1

k X

p1 D 1:

i D1

Since the function

in (4.4) is convex, we obtain ! k k X X 0D ai xi ai .xi / i D1 k X

i D1

k k pi X ci pi X ci D e log e Pk ci e ci i D1 e ci i D1 i D1 e i D1 ! k k X X ci D pi log pi ci C log e

e ci

i D1

D log

!

i D1 k X i D1

e ci

k X

pi . log pi C ci /:

i D1

This establishes inequality (5.12). Moreover, by the strict convexity of the function , this is an identity if and only if x1 D D xk D d for some d 0, that is, if and only if de ci for i D 1; : : : ; k: pi D P k ci i D1 e Summing over Pi yields d D 1, and hence, inequality (5.12) is an identity if and only t u if pi D e ci = kiD1 e ci for i D 1; : : : ; k. By Lemma 5.1 together with (5.10) and (5.11), we obtain Z H .ˇn / C

'n d X

X

.C / log .C / C 'n .xC /

C 2ˇn

log

X

e 'n .xC /

C 2ˇn

log

X

e 'n .pC /Cn

C 2ˇn

n C log 2n

X x2E

! e 'n .x/ ;

156


and hence, 1 H .ˇn / C n

Z

1 1 ' d D H .ˇn / C n n X 1 C log 2 C

Z 'n d X

X 1 log sup e 'n .x/ : n E x2E

This implies that Z

Z

h .T; / C

' d < h .T; ˇ/ C 1 C X

' d X

2 C log 2 C lim sup n!1

X 1 log sup e 'n .x/ ; n E x2E

and letting " ! 0 yields Z ' d D sup h .T; / C ' d

Z h .T / C

X

X

(5.13)

2 C log 2 C PT .'/: Since PT m .'m / D mPT .'/, replacing T by T m and ' by 'm in (5.13), we obtain Z h .T / C

Z 1 h .T m / C 'm d m X 1 2 C log 2 C PT m .'m / m 2 C log 2 C PT .'/: D m

' d D X

Finally, letting m ! 1 yields Z h .T / C

' d PT .'/: X

To complete the proof, given " > 0, for each n 2 N, we consider a set En of points at a dn -distance at least " such that log

X x2En

e 'n .x/ > log sup

X

e 'n .x/ 1;

(5.14)

E x2E

where the supremum is taken over all .n; "/-separated sets. We then define probability measures


157

P x2E n D P n

e 'n .x/ ıx

x2En

and

e 'n .x/

1X i T n ; n i D0 n1

n D

with T as in (3.6). Given a sequence .kn /n2N N such that X X 1 1 log e 'kn .x/ D lim sup log e 'n .x/ ; n!1 kn n!1 n x2E x2E lim

(5.15)

n

kn

let be any accumulation point of the sequence of measure .kn /n2N . Then is a T -invariant measure. As in the proof of Theorem 4.7, we consider a partition of X such that diam C < " and .@C / D 0 for each C 2 . Write En D fx1 ; : : : ; xk g, pi D n .fxi g/, and ci D 'n .xi / for i D 1; : : : ; k. We note that pi D P

e 'n .xi / e ci D : P N 'n .x/ ci x2En e i D1 e

By Lemma 5.1, this ensures that (5.12) is an identity, and hence, Z Hn . n / C n

Z X

' dn D Hn . n / C D

X

'n dn X

n .fxg/ log n .fxg/ C 'n .x/

x2En

D log

X

e 'n .x/ :

x2En

Now given m; n 2 N, we write n D q m C r, where q 0 and 0 r < m. Then by (4.48) and (4.49), we have Z X m m 'n .x/ log e D Hn . n / C m ' dn n n X x2E n

Z m1 1X D H . n / C m ' dn n i D0 n X

Z m1 q1 1 XX 2m2 log card C m Hn .T j mi m / C ' dn n i D0 j D0 n X

2m2 log card C m Hn . m / C n

Z

' dn : X

158


Therefore, by (5.15), in a similar manner to that in the proof of Theorem 4.7, we obtain Z X 1 1 'kn .x/ lim log e H . m / C ' d: n!1 kn m X x2E kn

Letting m ! 1 yields Z X 1 log e 'kn .x/ h .T; / C ' d n!1 kn X x2E lim

kn

Z

h .T / C

' d; X

and hence, lim sup n!1

Z X 1 log e 'n .x/ sup h .T / C ' d ; n X x2E n

with the supremum taken over all T -invariant probability measures in X . Finally, by (5.14), letting " ! 0 yields Z PT .'/ sup h .T / C ' d :

X

t u


We note that identity (5.8) can be used as an alternative definition of the topological pressure. Setting ' D 0, one recovers the variational principle for the topological entropy in Theorem 4.7.

5.5 Equilibrium Measures We consider in this section the class of measures at which the supremum in (5.8) is attained. Let T W X ! X be a continuous transformation of a compact metric space. Definition 5.3. Given a continuous function 'W X ! R, a T -invariant probability measure in X is called an equilibrium measure for ' (with respect to T ) if Z PT .'/ D h .T / C

' d: X

We give an example in the particular case of symbolic dynamics. Example 5.4. Given numbers 1 ; : : : ; k > 0, we consider the continuous function 'W XkC ! R defined by (5.5). By (5.6), its topological pressure is given by

5.5 Equilibrium Measures

159

P .'/ D log

k X

j :

j D1

Thus, it follows from the variational principle in Theorem 5.2 that log

k X

Z j D sup h ./ C

XkC

j D1

D sup h ./ C

k X

' d

(5.16)

.Ci / log i ;

i D1

where the supremum is taken over all -invariant probability measures in XkC and where C1 ; : : : ; Ck are cylinder sets. Now let be the (one-sided) Bernoulli measure generated by the numbers p1 ; : : : ; pk . By (4.26), we have h ./ D

k X

pi log pi :

i D1

Hence, by Lemma 5.1 (or again by Theorem 5.2), we obtain Z h ./ C

XkC

' d D

k X

pi . log pi C log i /

i D1

log

k X

(5.17) i D P .'/:

i D1

It also Pfollows from Lemma 5.1 that we have an equality in (5.17) if and only if pi D i = kiD1 i for i D 1; : : : ; k. In particular, this shows that the suprema in (5.16) are attained at the Bernoulli measure generated by these numbers. In other words, this is an equilibrium measure for the function ', and no other Bernoulli measure is an equilibrium measure. Now we consider a particular class of transformations. Definition 5.4. A transformation T W X ! X is said to be one-sided expansive if there exists " > 0 such that if d.T n .x/; T n .y// < "

for every n 2 N [ f0g;

then x D y. Example 5.5. No isometry T is one-sided expansive, since it satisfies d.T n .x/; T n .y// D d.x; y/

for every n 2 N:

(5.18)

160


Example 5.6. For the shift map W XkC ! XkC , we note that if d. n .!/; n .! 0 // < 1

for every n 2 N [ f0g;

then it follows readily from (3.32) that ! D ! 0 . Therefore, the shift map in XkC is one-sided expansive. Example 5.7. Let us consider the expanding map Eq W S 1 ! S 1 . If d.z; w/ < 1=q 2 with z ¤ w (where d is the distance in S 1 ), then there exists n 2 N such that d.Eqn .z/; Eqn .w// D q n d.z; w/

1 : q2

This implies that if d.Eqn .z/; Eqn .w//
0 such that if d.T n .x/; T n .y// < "

for every n 2 Z;

then x D y. Example 5.8. The shift map W Xk ! Xk is invertible. We note that if d. n .!/; n .! 0 // < 1

for every n 2 Z;

then it follows readily from (3.37) that ! D ! 0 . Therefore, the shift map in Xk is two-sided expansive. The following statement establishes the existence of equilibrium measures for any two-sided expansive homeomorphism: Theorem 5.5. If T W X ! X is a two-sided expansive homeomorphism of a compact metric space, then any continuous function 'W X ! R has at least one equilibrium measure.

164


Proof. For " as in Definition 5.5, let be a finite measurable partition of X with diam < ". We can showWin a similar manner to that in the proof of Theorem 5.3 that the partitions 0n D nkDn T k satisfy diam 0n ! 0 when n ! 1. This implies that is a two-sided generator (see Definition 4.7), and thus, it follows from Theorem 4.3 that (5.20) holds. This allows us to repeat arguments in the proof of Theorem 5.3 to show that in this new situation, the transformation 7! h .T / is also upper semicontinuous. Therefore, for each continuous function 'W X ! R, the R transformation 7! h .T / C X ' d is upper semicontinuous, and hence, ' has at least one equilibrium measure. t u Similarly, the following is an immediate consequence of Theorem 5.5: Theorem 5.6. If T W X ! X is a two-sided expansive homeomorphism of a compact metric space, then there exists a T -invariant probability measure in X with h .T / D h.T /.

5.6 Exercises Exercise 5.1. Let T W X ! X be a continuous transformation of a compact metric space, and let '; W X ! R be continuous functions. Show that: 1. If ' , then PT .'/ PT . /. 2. If PT .'/ and PT . / are finite, then jPT .'/ PT . /j k' 3. PT .' C

k1 :

ı T / D PT .'/.

Exercise 5.2. Let T W X ! X be a continuous transformation of a compact metric space and let 'W X ! R be aP continuous function. Show that PT n .'n / D nPT .'/ k for every n 2 N, where 'n D n1 kD0 ' ı T . Exercise 5.3. Let T W X ! X be a homeomorphism of a compact metric space. Show that PT .'/ D PT 1 .'/ for every continuous function 'W X ! R. Exercise 5.4. Given a continuous transformation T W X ! X of a compact metric space and a continuous function 'W X ! R, for each n 2 N and " > 0, let Rn .'; "/ D inf V

and Sn .'; "/ D inf V

X V 2V

X V 2V

exp inf V

exp sup V

n1 X

' ıTk

kD0

n1 X kD0

' ı T k;

5.6 Exercises

165

where the infimum in V is taken over all finite open covers V of X by dn -balls of radius ". Show that PT .'/ D lim lim sup

1 log Rn .'; "/ n

D lim lim sup

1 log Sn .'; "/: n

"!0 n!1

"!0 n!1

Exercise 5.5. Show that if 'W XkC ! R is a continuous function, then X 1 log exp inf 'n ; n!1 n Ci1 in i i

P .'/ D lim

1

where 'n D

Pn1 kD0

n

' ı k . Hint: note that X X inf 'n exp inf 'n ; i1 imCn1

Ci1 imCn1

Ci1 in

i1 in

and X i1 imCn1

X

'n e .m1/k'k1

inf

Ci1 imCn1

exp

i1 imCn1

inf

Ci1 imCn1

'mCn1 :

Exercise 5.6. Show that if 'W XkC ! R is a continuous function, then n1 X X 1 log exp '. k .!//; n!1 n !2P

P .'/ D lim

kD0

n

where

˚ Pn D ! 2 ˙AC W n .!/ D !

is the set of n-periodic points. Exercise 5.7. Show that if 'W Xk ! R is a continuous function, then n1 X X 1 log exp sup ' ı k: n!1 n C i1 in i i

P .'/ D lim

1

kD0

n

Exercise 5.8. Given a k k matrix A with entries in f0; 1g, show that: 1. If 'W XAC ! R is a continuous function, then n1 X X 1 log exp ' ı k; n!1 n i i

P jX C .'/ D lim A

1

n

kD0

166


where the supremum is taken over all finite sequences i1 in that are the first n elements of some sequence in XAC . 2. If 'W XA ! R is a continuous function, then n1 X X 1 log exp ' ı k; n!1 n i i

P jXA .'/ D lim

1

kD0

n

where the supremum is taken over all finite sequences i1 in that are the elements i1 .!/ in .!/ of some sequence ! 2 XA . Exercise 5.9. Show that if 'n W XkC ! R are continuous functions such that 'nCm 'n C 'm ı n for every m; n 2 N, then there exists the limit X 1 log exp sup 'n : n!1 n Ci1 in i i

p D lim

1

n

Exercise 5.10. Given numbers 1 ; : : : ; k > 0, show that the continuous function 'W XAC ! R defined by (5.5) has topological pressure P jX C .'/ D log .AB/; A

where B is the k k diagonal matrix with entries 1 ; : : : ; k in the diagonal. Exercise 5.11. Show that if a continuous transformation T W X ! X is uniquely ergodic and is the unique T -invariant probability measure in X , then Z PT .'/ D h .T / C

' d X

for every continuous function 'W X ! R. Exercise 5.12. Let T W X ! X be a continuous transformation of a compact metric space. Show that if h.T / D 1, then there is a T -invariant probability measure in X with h .T / D 1. Hint: take T -invariant probability measures n with hn .T / > 2n and consider the measure D

1 X 1 n : 2n nD1

Exercise 5.13. Show that any toral automorphism induced by a matrix without eigenvalues in S 1 is two-sided expansive.

5.6 Exercises

167

Exercise 5.14. Prove or disprove the following statement: any toral automorphism as in Exercise 5.13 is one-sided expansive. Exercise 5.15. Let T W X ! X be a continuous transformation of a compact metric space. Show that if T is one-sided expansive, then its topological entropy is given by 1 log N.dn ; ı/ n 1 D lim log M.dn ; ı/ n!1 n 1 D lim log D.dn ; ı/ n!1 n

h.T / D lim

n!1

for any sufficiently small ı > 0, with D.dn ; ı/ as in (4.39) and M.dn ; ı/ as in Exercise 4.17. Hint: show that given positive numbers ı < ˛ < ", with " as in Definition 5.4, there exists m D m.ı; ˛/ 2 N such that if d.x; y/ ı, then d.f i .x/; f i .y// > ˛

i 2 f0; : : : ; mg

for some

and conclude that N.dn ; ı/ N.dnC2m; ˛/. Exercise 5.16. Show that if T W X ! X is a two-sided expansive homeomorphism of a compact metric space, with ˚ an WD card x 2 X W T n .x/ D x < 1 for every n 2 N, then h.T / lim sup n!1

1 log an : n

Exercise 5.17. Given a continuous transformation T W X ! X of a compact metric space and continuous functions '; W X ! R, show that lim inf t !0

PT .' C t / PT .'/ t

Z d; X

where is any equilibrium measure for '. Exercise 5.18. Let T W X ! X be a continuous transformation of a compact metric space with h.T / < 1. Show that if is a T -invariant probability measure in X with Z h .T / D inf PT .'/ '

' d ; X

where the infimum is taken over all continuous functions 'W X ! R, then the map 7! h .T / is upper semicontinuous at D .

168


Notes The notion of topological pressure was introduced by Ruelle [83] for expansive transformations and by Walters [102] in the general case. They also established corresponding versions of the variational principle for the topological pressure (Theorem 5.2). Our proof of Theorem 5.2 is based on [103], which follows the simpler proof of Misiurewicz [61]. Theorem 5.3 is due to Ruelle [83] (for ' D 0, the statement was first established by Goodman [35]). The argument for the upper semicontinuity of the entropy in the proof of Theorem 5.3 is based on [103]. Ruelle’s book [84] contains a detailed discussion of the relation between the thermodynamic formalism and the theory of dynamical systems. For further developments, we refer to [44, 45, 70, 103].

Part III

Hyperbolic Dynamics

•

Chapter 6


We introduce in this chapter the basic notions of hyperbolic dynamics, starting with the concept of hyperbolicity. We also establish several basic properties of hyperbolic sets, including the continuous dependence of the stable and unstable subspaces on the base point. In addition, we discuss several examples of hyperbolic sets. These include hyperbolic fixed points, the Smale horseshoe, and hyperbolic automorphisms of the 2-torus. We also construct coding maps via symbolic dynamics. Finally, we consider noninvertible transformations and their repellers, and we construct corresponding Markov partitions. We refer to the following chapter for the more elaborate construction of Markov partitions for hyperbolic sets.

6.1 Introduction The study of hyperbolicity goes back to seminal work of Hadamard on the geodesic flow in the unit tangent bundle of a surface with negative curvature, in particular revealing its instability with respect to initial conditions. In the case of constant negative curvature, the geodesic flow can be described as follows. Consider the upper-half plane H D fz 2 C W =z > 0g; with the inner product in the tangent space Tz H D C given by hv; wiz D

hv; wi ; .=z/2

where hv; wi is the standard inner product in R2 . Consider also the group G of matrices A D ac db with real entries and determinant 1 or 1 and define Möbius transformations TA in H by


171

172


TA .z/ D

az C b cz C d

or TA .z/ D

aNz C b ; cNz C d

respectively, when detA is 1 or 1. Then G=fId; Idg is the group of isometries of H . Now take ˚ .z; v/ 2 SH D .z; v/ 2 H C W jvjz D 1 : One can show that there exists a Möbius transformation T such that T .z/ D i and T 0 .z/v D i , thus taking the geodesic passing through z with tangent v onto the geodesic iet traversing the positive part of the imaginary axis. The geodesic flow 't W SH ! SH is given by 't .z; v/ D ..t/; 0 .t//; where .t/ D T 1 .i e t /. One can show that the geodesic flow preserves volume and thus it also exhibits a nontrivial recurrence. The case of nonconstant negative curvature is more elaborate, but the geodesic flow still preserves volume (in fact, any geodesic flow is a Hamiltonian flow). A considerable activity took place during the 1920s and 1930s in particular with the important contributions of Hedlund and Hopf who established several topological and ergodic properties of geodesic flows. Moreover, the geodesic flow on a surface with negative curvature is hyperbolic. Hyperbolicity corresponds to the existence of complementary transverse subspaces, called stable and unstable, exhibiting, respectively, expansion and contraction. More precisely, the expansion and contraction is required for the linear maps approximating the dynamics. Hyperbolicity gives rise to a very rich structure and in particular to the existence of stable and unstable manifolds: it follows from a substantial generalization of the Hadamard–Perron theorem that for every point x in a hyperbolic set of a C 1 diffeomorphism f and any sufficiently small " > 0, the sets ˚ V s .x/ D y 2 B.x; "/ W d.f n .x/; f n .y// < " for every n 0 and

˚ V u .x/ D y 2 B.x; "/ W d.f n .x/; f n .y// < " for every n 0

are invariant manifolds tangent, respectively, to the stable and unstable subspaces. There is a corresponding theory for noninvertible transformations, in which case the notion of hyperbolic set is replaced by the notion of repeller, with the dynamics exhibiting only expansion. While there are some important differences between the two theories, it is sometimes simpler to first consider some notions for repellers and then consider appropriate elaborations for hyperbolic sets. This is the case for example with the construction of Markov partitions for repellers. The corresponding construction for hyperbolic sets is an elaboration of this approach.

6.2 Hyperbolic Sets

173

6.2 Hyperbolic Sets We introduce in this section the notion of hyperbolicity. We also discuss some of the basic properties of hyperbolic sets.

6.2.1 The Notion of Hyperbolicity Let f W M ! M be a C 1 diffeomorphism of a smooth manifold M . For each point x 2 M , we consider the inner product h; ix and the corresponding norm kkx in the tangent space Tx M . Whenever there is no danger of confusion, we simply write h; i and kk, without making explicit the dependence on x. Definition 6.1. A compact f -invariant set M is said to be a hyperbolic set for f if there exist 2 .0; 1/, c > 0 and a decomposition Tx M D E s .x/ ˚ E u .x/

(6.1)

for each x 2 such that dx f E s .x/ D E s .f .x//; kdx f n vk c n kvk and

dx f E u .x/ D E u .f .x//;

(6.2)

whenever v 2 E s .x/;

kdx f n vk c n kvk whenever v 2 E u .x/

for every x 2 and n 2 N. We then call E s .x/ and E u .x/, respectively, the stable and unstable subspaces at x. As first examples, we consider fixed points and periodic points. Definition 6.2. Let f be a diffeomorphism. 1. A fixed point x D f .x/ of f is said to be hyperbolic if fxg is a hyperbolic set. 2. An m-periodic point x D f m .x/ of f is said to be hyperbolic if its orbit ˚ Of .x/ D f k .x/ W k D 0; : : : ; m 1 is a hyperbolic set. We denote by Sp.A/ the spectrum of a square matrix A, that is, the set of its eigenvalues. Note that when x is a fixed point of f , we have dx f .Tx M / Tx M , and thus, the symbol Sp.dx f / is well defined. The following is a characterization of the hyperbolicity of a fixed point x in terms of the spectrum of dx f :

174


Proposition 6.1. A fixed point x of a diffeomorphism f is hyperbolic if and only if Sp.dx f / \ S 1 D ∅. Proof. We consider the complexification ˚ Tx M C D u C i v W u; v 2 Tx M of Tx M , equipped with the norm ku C i vk D

p kuk2 C kvk2 ;

u; v 2 Tx M;

and the linear operator AW Tx M C ! Tx M C defined by A.u C i v/ D dx f u C idx f v

for each u; v 2 Tx M:

We first assume that fxg is a hyperbolic set. If v 2 Tx M C and 2 S 1 are such that Av D v, then kAn vk D jjn kvk D kvk: (6.3) Writing v D vs C vu , where vs 2 E s .x/C and vu 2 E u .x/C , we obtain kvk kAn vu k kAn vs k c 1 n kvu k c n kvs k:

(6.4)

Letting n ! 1 in (6.4) yields vu D 0, and thus, v 2 E s .x/C . Then it follows from (6.3) that v D 0. This shows that is not an eigenvalue of dx f , and thus, Sp.dx f / \ S 1 D ∅. Now we assume that dx f has no eigenvalues in S 1 . Let F s Tx M C be the linear subspace generated by all vectors v 2 Tx M C satisfying .A Id/k v D 0

(6.5)

for some k 2 N and some 2 Sp.dx f / with jj < 1. Similarly, let F u Tx M C be the linear subspace generated by all vectors v 2 Tx M C satisfying (6.5) for some k 2 N and some 2 Sp.dx f / with jj > 1. We can easily verify that AF s F s

and AF u F u :

(6.6)

Indeed, it is sufficient to note that .A Id/k Av D A.A Id/k v: Moreover, since dx f has no eigenvalues in S 1 , we have Tx M C D F s ˚ F u . Therefore, setting E s .x/ D F s \ Tx M

and E u .x/ D F u \ Tx M;

6.2 Hyperbolic Sets

we obtain

175

Tx M D Tx M C \ Tx M D E s .x/ ˚ E u .x/;

and it follows from (6.6) that dx f E s .x/ D E s .x/

and dx f E u .x/ D E u .x/:

Hence, property (6.2) holds. Now we take 2 .0; 1/ such that ˚ .; 1=/ \ jj W 2 Sp.dx f / D ∅: Using Jordan’s canonical form, it is easy to verify that there is a constant c > 0 such that kdx f n vk cn kvk for v 2 E s .x/; n 2 N; and

kdx f n vk cn kvk

for v 2 E u .x/; n 2 N:

Therefore, fxg is a hyperbolic set.

t u

Now we describe an example where the hyperbolic set is the whole manifold. Example Let T W T2 ! T2 be the toral automorphism induced by the matrix 2 16.1. B D 1 1 , which has eigenvalues D .3 C

p p 5/=2 > 1 and 1 D .3 5/=2 < 1;

both outside S 1 . We denote respectively by F u and F s the eigenspaces of B corresponding to the eigenvalues and 1 . Since dx T D B for every x 2 T2 , we obtain kdx T n vk D n kvk for v 2 F u ; n 2 N; and

kdx T n vk D n kvk

for v 2 F s ; n 2 N:

Furthermore, F u ˚ F s D R2 D Tx T2 , dx TF s D F s ;

and dx TF u D F u :

Therefore, T2 is a hyperbolic set for T . This example can be readily generalized to the class of toral automorphisms induced by matrices without eigenvalues in S 1 . Definition 6.3. We say that a square matrix B is hyperbolic if Sp.B/ \ S 1 D ∅, and we say that a toral automorphism of Tn is hyperbolic if Tn is a hyperbolic set for the automorphism.

176


6.2.2 Some Properties of Hyperbolic Sets We study in this section some elementary properties of hyperbolic sets. We start with the study of the dependence of the stable and unstable subspaces on the base point. We denote by †.E; F / the angle between two subspaces E and F . Proposition 6.2. If is a hyperbolic set, then the following properties hold: 1. The spaces E s .x/ and E u .x/ vary continuously with x 2 . 2. inff†.E s .x/; E u .x// W x 2 g > 0. Proof. Let .xp /p2N be a sequence converging to x 2 when p ! 1. We consider a subsequence .yp /p2N such that the numbers dimE s .yp / and dimE u .yp / are independent of p (this is always possible because the dimension of the subspaces can only take finitely many values). Now let v1p ; : : : ; vkp 2 E s .yp / be an orthonormal basis of E s .yp /, where k D dimE s .yp /. By the compactness of the unit tangle bundle S M , we can also assume that for each i D 1; : : : ; k, there exists vi 2 Sx M such that vip ! vi when p ! 1. On the other hand, for i D 1; : : : ; k and n 2 N, we have kdyp f n vip k c n kvip k: Letting p ! 1, we conclude that kdx f n vi k c n kvi k; which implies that v1 ; : : : ; vk 2 E s .x/, and hence, dimE s .x/ k. One can show in a similar manner that dimE u .x/ dimM k. It follows from (6.1) that dimE s .x/ D k

and

dimE u .x/ D dimM k:

Therefore, for any sufficiently large p, the numbers dimE s .xp / and dimE u .xp / are independent of p. This completes the proof of Property 1. For the second property, we note that the function x 7! †.E s .x/; E u .x// 2 .0; =2 is the composition of the continuous functions .E; F / 7! †.E; F /

and x 7! E s .x/ ˚ E u .x/:

Therefore, Property 2 follows readily from the compactness of .

t u

Now we show that it is always possible to redefine the inner product so that one can take c D 1 in Definition 6.1.

6.2 Hyperbolic Sets

177

Proposition 6.3. If is a hyperbolic set for a diffeomorphism f , then there exists a inner product h; i0 in TM with respect to which the following properties hold: 1. †0 .E s .x/; E u .x// D =2 for every x 2 . 2. There exists 2 .; 1/ such that for each x 2 , we have kdx f vk0 kvk0 and

whenever v 2 E s .x/;

kdx f 1 vk0 kvk0

whenever v 2 E u .x/:

Proof. Fix " > 0. Given v; w 2 E s .x/, we define hv; wi0x D

1 X

hdx f n v; dx f n wi 2n e 2"n :

(6.7)

nD0

To show that the series converges, it is sufficient to observe that 1 X

jhdx f v; dx f wij n

n

2n 2"n

e

nD0

1 X

c 2 e 2"n kvk kwk < 1:

nD0

For each v 2 E .x/, we have s

1 2 X kdx f vk0f .x/ D kdx f nC1 vk2 2n e 2"n nD0

1 X kdx f n vk2 2.n1/ e 2".n1/ nD0

D 2 e 2" .kvk0x /2 ; and hence,

kdx f vk0f .x/ e " kvk0x :

(6.8)

Similarly, given v; w 2 E .x/, we define u

hv; wi0x D

1 X hdx f n v; dx f n wi 2n e 2"n ;

(6.9)

nD0

and for each v 2 E u .x/, we have kdx f 1 vk0f 1 .x/ e " kvk0x :

(6.10)

Now we extend h; i0x to the whole tangent space Tx M . Namely, given v D vs C vu and w D ws C wu in Tx M with vs ; ws 2 E s .x/ and vu ; wu 2 E u .x/, we define hv; wi0x D hvs ; ws i0x C hvu ; wu i0x ;

(6.11)

178


where hvs ; ws i0x and hvu ; wu i0x are given, respectively, by (6.7) and (6.9). If v 2 E s .x/ and w 2 E u .x/, then vu D ws D 0, and it follows from (6.11) that hv; wi0x D 0. This establishes the first property. Setting D e " for some " > 0 such that < 1, the second property follows readily from (6.8) and (6.10). t u In fact, to prove Proposition 6.3, one can also use finite sums, instead of the series in (6.7) and (6.9) (see Exercise 6.4).

6.3 Smale Horseshoe We consider in this section a more elaborate example of hyperbolic set. It will also help us motivating the use of symbolic dynamics.

6.3.1 Construction of the Horseshoe We consider a diffeomorphism f in an open neighborhood of the square Q D Œ0; 12 with the behavior indicated in Fig. 6.1. To define f more explicitly, we consider the horizontal strips (see Fig. 6.2)

1 4 ; H1 D Œ0; 1 9 9

5 8 and H2 D Œ0; 1 ; ; 9 9

and the vertical strips (see Fig. 6.2) V1 D

1 4 Œ0; 1 ; 9 9

and V2 D

5 8 Œ0; 1: ; 9 9

Now we define a transformation gW H1 [ H2 ! R2 by ( g.x; y/ D

.x=3; 3y/ C .1=9; 1=3/

if .x; y/ 2 H1 ;

.x=3; 3y/ C .8=9; 8=3/ if .x; y/ 2 H2 ;

(6.12)

and we consider any diffeomorphism f with the behavior indicated in Fig. 6.1 such that f j.H1 [ H2 / D g: We note that f transforms horizontal strips into vertical strips, that is, f .H1 / D V1

and f .H2 / D V2 :

6.3 Smale Horseshoe

179

f (Q)

Q

f

V1

V2

Fig. 6.1 The diffeomorphism f

Q

f (H1) = V1

f (H2) = V2

H2 f

H1

Fig. 6.2 Horizontal and vertical strips

Applying f a second time, we obtain the set f 2 .Q/ in Fig. 6.3. We note that the intersection Q \ f 2 .Q/ is composed of four vertical strips of width 1=9. Similarly, applying successively f 1 , we obtain the sets in Fig. 6.4. Definition 6.4. The set Q defined by D

\ n2Z

is called a Smale horseshoe.

f n .Q/

(6.13)

180


f 2(Q)

Q

f2

Fig. 6.3 Second iterate of the diffeomorphism

f −1(Q)

f −2 (Q)

Fig. 6.4 First iterates of the inverse f 1

6.3 Smale Horseshoe

181

Fig. 6.5 The sets Q, Q1 , and Q2

We note that is the intersection of the decreasing sequence of closed sets Qn D

n \

f k .Q/;

n 2 N;

kDn

that is, D

\

Qn :

n2N

This implies that is compact and nonempty. Each set Qn is the union of 4n disjoint squares with sides of length 3n . In Fig. 6.5, we represent Q1 and Q2 , respectively, with white and gray backgrounds. It follows from (6.13) that is f -invariant, that is, f 1 ./ D . Since f is a diffeomorphism, this is equivalent to f ./ D . In particular, one can consider the restriction f jW ! . Proposition 6.4. The Smale horseshoe is a hyperbolic set for the diffeomorphism f . Proof. By (6.12), we have ( dx f D

B

if x 2 H1 ;

B

if x 2 H2 ;

where BD

(6.14)

1 3 0 : 0 3

Now we consider the decomposition Tx R2 D E s .x/ ˚ E u .x/; where E s .x/ and E u .x/ are, respectively, the horizontal and vertical axes. Since the matrix B is diagonal, this decomposition satisfies (6.2). Furthermore, by (6.14),

182


( kdx f vk D

31 kvk

if v 2 E s .x/;

3kvk

if v 2 E u .x/;

and we can take D 31 and c D 1 in Definition 6.1. This shows that is a hyperbolic set for f . u t

6.3.2 Symbolic Dynamics We consider again the Smale horseshoe and the diffeomorphism f introduced in Sect. 6.3.1. We show that f j has a certain symbolic representation. This is one of the important properties of the Smale horseshoe. In particular, it can be used to obtain information that otherwise would be difficult to obtain. We first recall the relevant material from Sect. 3.5.4. Let X D X2 D f1; 2gZ be the space of two-sided sequences ! D . i1 .!/i0 .!/i1 .!/ / of numbers in f1; 2g. Given ˇ > 1, we equip X with the distance dˇ in (3.37). Let also W X ! X be the shift map defined by .!/ D ! 0 , where in .! 0 / D inC1 .!/ for each n 2 Z. Since ˇ 1 dˇ .!; ! 0 / dˇ ..!/; .! 0 // ˇdˇ .!; ! 0 / for every !; ! 0 2 X , the shift map is a homeomorphism. For each ! 2 X and n 2 N, we define the set Qn .!/ D

n1 \

f k .Vik .!/ /:

(6.15)

kDn

We note that Qn .!/ is a square with sides of length 3n (see Fig. 6.5). Furthermore, let \ \ H.!/ D f n .Vin .!/ / D Qn .!/: n2Z

n2N

It follows from (6.13) that H.!/ . Moreover: 1. card H.!/ 1 since Qn .!/ is a decreasing sequence of closed sets. 2. card H.!/ 1 since diam Qn .!/ ! 0 when n ! 1. Therefore, card H.!/ D 1 for every ! 2 X . Denoting the single element of H.!/ by h.!/, we can define a transformation hW X ! by

6.3 Smale Horseshoe

183

\

h.!/ D

f n .Vin .!/ /:

(6.16)

n2Z

Proposition 6.5. The following properties hold: 1. The transformation h is a homeomorphism. 2. The inverse of h is given by h1 .x/ D . i1 i0 i1 /, where ( in D

1

if f n .x/ 2 V1 ;

2

if f n .x/ 2 V2

(6.17)

for each n 2 Z. 3. h ı D f ı h, that is, the diagram

X ! ? ? hy

X ? ? yh

f

! is commutative. Proof. It follows from the definition of the Smale horseshoe that D

\

f n .V1 [ V2 /

n2Z

D

[\

f n .Vin .!/ / D h.X /;

!2X n2Z

and hence, h is onto. To show that h is injective, we take sequences !; ! 0 2 X with ! ¤ ! 0 . Then im .!/ ¤ im .! 0 / or equivalently Vim .!/ \ Vim .! 0 / D ∅, for some m 2 Z. Therefore, ! ! \ \ 0 n n H.!/ \ H.! / D f .Vin .!/ / \ f .Vin .! 0 / / D ∅; n2Z

n2Z

and h.!/ ¤ h.! 0 /, which shows that h is injective. To show that h is a homeomorphism, let n \ Qn0 .!/ D f k .Vik .!/ /: kDn

Then h1 .Qn0 .!// is a cylinder set (see (3.36)). Since it is open and since the sets Qn0 .!/ generate the topology of , we conclude that h is continuous. Moreover, h.Cin .!/in .!/ / D \ int Qn0 .!/

184


is open in , and since the cylinder sets generate the topology of X , the transformation h1 is continuous. Identity (6.17) follows readily from (6.16). Finally, for the third property, we note that \ h..!// D f n .VinC1 .!/ / n2Z

D

\

f 1n .Vin .!/ / D f .h.!//:

n2Z


t u

The transformation h is called a coding map (of the Smale horseshoe) and acts as a dictionary between the dynamics f j on the Smale horseshoe and the symbolic dynamics jX given by the shift map. In many situations, a coding map allows one to study in a somewhat easier manner some aspects of the dynamics. For example, by Proposition 6.5, we have h ı m D f m ı h for each m 2 N, and hence, m .!/ D !

if and only if f m .h.!// D h.!/:

(6.18)

In other words, the study of the periodic points of f j can be reduced to the study of the periodic points of . We first consider the shift map. Proposition 6.6. The following properties hold: 1. For each m 2 N,

˚ card ! 2 X W m .!/ D ! D 2m :

2. The set of periodic points of is dense in X . Proof. The transition matrix of jX is A D 11 11 . Therefore, tr.Am / D 2m for each m 2 N, and the first property follows readily from (3.34). For the second property, since the cylinder sets generate the topology of X , it is sufficient to show that each set Cin in contains a periodic point. One can easily verify that the sequence ! 2 Cin in obtained from repeating the finite sequence .in in / satisfies 2nC1 .!/ D !. Therefore, ! is a periodic point. t u Now we study the periodic points of f j. Proposition 6.7. The following properties hold: 1. For each m 2 N,

˚ card x 2 W f m .x/ D x D 2m :

2. The set of periodic points of f j is dense in .

6.4 Hyperbolic Automorphisms of T2

185

Proof. It follows from (6.18) and Property 1 in Proposition 6.6 that ˚ ˚ card x 2 W f m .x/ D x D card ! 2 X W m .!/ D ! D 2m for each m 2 N. Since h is a homeomorphism, the second property follows readily from Proposition 6.6. t u By Propositions 6.6 and 6.7, we have p.f j/ D p./ D log 2 > 0; where p is the periodic entropy introduced in Definition 2.6.

6.4 Hyperbolic Automorphisms of T2 We show in this section how to introduce a coding map for a hyperbolic toral automorphism of T2 . In order to make the construction more explicit, we consider a particular automorphism.

6.4.1 Construction of a Coding Map Let TW T2! T2 be the hyperbolic automorphism of T2 induced by the matrix B D 21 11 . This matrix has eigenvalues D

p 3C 5 2

and 1 D

p 3 5 2

with orthogonal eigendirections defined by the vectors

p p 1 5 1C 5 ;1 and ;1 : 2 2

We consider a partition of T2 into closed rectangles S1 and S2 . This means that T2 D S1 [ S2

and S1 \ S2 D @S1 [ @S2 :

Namely, we consider the partition indicated in Fig. 6.6, with the sides of the rectangles S1 and S2 parallel to the stable and unstable directions or more precisely to the eigendirections of the matrix B. Applying the automorphism T to the

186


S1

S2

Fig. 6.6 Partition of the torus into rectangles

R5 R4

R3

R5

R2

R1 R4

R2

R3

Fig. 6.7 Another partition of the torus into rectangles

rectangles S1 and S2 , we obtain a new partition of T2 into five closed rectangles R1 , R2 , R3 , R4 , and R5 with disjoint interiors such that (see Fig. 6.7) T .S1 / D R3 [ R5

and T .S2 / D R1 [ R2 [ R4 :


187

T(R i )

Rj

Rj T(R i )

Fig. 6.8 Two types of intersections of rectangles

One can easily verify that: 1. If int T .Ri / \ int Rj ¤ ∅, then T .Ri / intersects Rj along the whole unstable direction. 2. If int T 1 .Ri / \ int Rj ¤ ∅, then T 1 .Ri / intersects Rj along the whole stable direction. This is called the Markov property of the partition (with respect to T ). See Fig. 6.8 for two possible types of intersections. The first type satisfies the Markov property, while the second one does not. We shall see that the Markov property is crucial in order to obtain a coding map for the automorphism T . The following is a particular case of a more general notion introduced in Sect. 7.6: Definition 6.5. Given a hyperbolic automorphism T of T2 , let R1 ; : : : ; Rk T2 be closed rectangles with sides parallel to the stable and unstable directions such that T2 D

k [

Ri

and Ri \ Rj D @Ri \ @Rj

i D1

˚ whenever i ¤ j . If the Markov property holds, then Ri W i D 1; : : : ; k is called a Markov partition of T2 (with respect to T ). Now we return to the hyperbolic automorphism T D TB . We define a 5 5 transition matrix A with entries ( 1 if int T .Ri / \ int Rj ¤ ∅; aij D (6.19) 0 if int T .Ri / \ int Rj D ∅:

188


One can easily verify that

0

1 010 0 1 0C C C 0 1 0C : C 1 0 1A 101

11 B1 1 B B A D B1 1 B @0 0 00

(6.20)

For each ! 2 XA X5 (see Sect. 3.5.4), we define a set H.!/ T2 by H.!/ D

\

T n Rin .!/ :

n2Z

Since the sides of the rectangles are parallel to the stable and unstable directions, there is a constant c > 0 such that for each n 2 N, the closed set Qn .!/ D

n \

T k Rik .!/

(6.21)

kDn

has diameter at most c n . This ensures that card H.!/ 1. We use the Markov property to show that in fact card H.!/ D 1 for every ! 2 XA . Let Ri , Rj , and Rk be rectangles such that int T .Ri / \ int Rj ¤ ∅

and

int T .Rj / \ int Rk ¤ ∅:

By the Markov property, int T 2 .Ri / intersects int T .Rj / along the whole unstable direction of Rk . This implies that int T 2 .Ri / \ int T .Rj / \ int Rk ¤ ∅: One can use induction to show that if int T .Rik / \ int RikC1 ¤ ∅

for k D n; : : : ; n 1;

then int T 2n .Rin / \ int T 2n1 .RinC1 / \ \ int Rin ¤ ∅: Therefore, the closed set Qn .!/ in (6.21) is nonempty for each n 2 N. This implies that card H.!/ D 1 for every ! 2 XA , and hence, we can define a coding map hW XA ! T2 by \ T n Rin : (6.22) h.!/ D n2Z

Incidentally, one can use a similar construction to obtain a Markov partition and thus also a coding map for any hyperbolic automorphism of T2 . Repeating arguments in the proof of Proposition 6.5, we obtain the following statement:


189

Proposition 6.8. The transformation hW XA ! T2 is continuous and onto, and it satisfies T ı h D h ı , that is, the diagram

XA ! ? ? hy

XA ? ? yh

T

T2 ! T2 is commutative. Contrarily to what happens in the case of the Smale horseshoe (see Sect. 6.3.2), the coding map h in (6.22) is not injective. This can be seen as follows. The fixed points of jXA are the constant sequences. By direct inspection of the matrix A in (6.20), the fixed points are thus . 000 /, . 111 /, and . 444 /. However, one can show that 0 is the unique fixed point of T (see also (6.26) below), and thus, h. 000 / D h. 111 / D h. 444 / D 0: (6.23) We also consider the periodic points of the hyperbolic automorphism T D TB . We first observe that the matrix A in (6.20) has eigenvalues (counted with their multiplicities) p p 3C 5 3 5 ; ; 0; 0; 0: 2 2 By (3.34), we have ˚ card ! 2 XA W m .!/ D ! D tr.Am / p p 3 5 m 3C 5 m C D 2 2

(6.24)

for each m 2 N. In particular, the periodic entropy of jXA is equal to p 3C 5 > 0: p.jXA / D log 2 To find the number of m-periodic points of T , we first note that T m .x; y/ D .x; y/ if and only if .B m Id/.x; y/ D .0; 0/ mod 1; (6.25) 2 1 m where B D 1 1 . Since det.B Id/ ¤ 0, the number of solutions .x; y/ of (6.25) is equal to the number of representatives of the equivalence class of .0; 0/ 2 T2 in .B m Id/Œ0; 1/2 , which is equal to jdet.B m Id/j. Therefore,

190


˚ card x 2 T2 W T m .x/ D x D jdet.B m Id/j ˇ m ˇ ˇ ˇ 0 1 ˇ D ˇˇdet m 0 1 ˇ D j. 1/. m

D C m

m

m

(6.26)

1/j

2 D tr.B m / 2:

It follows from (6.26) that p 1 3C 5 m log.tr.B / 2/ D log > 0: p.T / D lim m!1 m 2 Note that p.T / D p.jXA /. We emphasize that this is not a coincidence. Indeed, we shall see in Sect. 7.6 how to establish this identity for a much larger class of dynamics without computing either p.T / or p.jXA /. Incidentally, by (6.24), we have ˚ ˚ card x 2 T2 W T m .x/ D x D card ! 2 XA W m .!/ D ! 2: We note that the discrepancy is due to the fixed points of jXA (see (6.23)). More precisely, one can easily verify that ˚ ˚ card x 2 T2 W T m .x/ D x; T .x/ ¤ x D card ! 2 XA W m .!/ D !; .!/ ¤ ! for any m 2 N.

6.4.2 Induction of a Markov Measure We show in this section that any Markov partition of a hyperbolic automorphism of T2 gives rise to a Markov measure (see Definition 3.10). Let T W T2 ! T2 be a hyperbolic automorphism of T2 . We consider a Markov partition fRi W i D 1; : : : ; kg of T2 with respect to T and the associated topological Markov chain jXA W XA ! XA , where A D .aij / is the k k transition matrix with the entries in (6.19). We also define a coding map hW XA ! T2 by h.!/ D

\

T n Rin .!/ :

n2Z

It is easy to verify that the statement in Proposition 6.8 also holds in this setting.


191

Now we define a measure in Xk by .h1 C / D .C /

(6.27)

for each measurable set C T2 , where is the Lebesgue measure in T2 . Proposition 6.9. The measure is a -invariant Markov measure in Xk . Proof. To show that is -invariant, we note that since T ı h D h ı we have 1 .h1 C / D .h ı /1 C D .T ı h/1 C D h1 .T 1 C / for any measurable set C T2 , and thus, . 1 .h1 C // D .h1 .T 1 C // D .T 1 C / D .C / D .h1 C /; using the T -invariance of the measure . Now we show that is a Markov measure. We first observe that .Cin in / D

!

n \

T

`

Ri`

`Dn

D

2n \

T

! `

RinC` ;

`D0

using the T -invariance of the measure . Set pi D .Ri /

and pij D

.Ri \ T 1 Rj /

.Ri /

for i; j D 1; : : : ; k. Clearly, pi > 0 for each i , and k X i D1

pi pij D

k X

Pk

i D1

pi D 1. Moreover,

.Ri \ T 1 Rj /

i D1

D .T 1 Rj / D .Rj / D pj :

(6.28)

192

6 Basic Notions and Examples Ri

Ri

0

–1

T −1R i

1

T −1

T −1R i T −2R i

0

1

Fig. 6.9 Intersection of rectangles along the unstable direction

Therefore, .P; p/ is a stochastic pair. It remains to show that the associated Markov measure is indeed . We note that .Ci1 i0 i1 / D .Ri1 \ T 1 Ri0 \ T 2 Ri1 / D .Ri1 /

.Ri1 \ T 1 Ri0 / .Ri1 \ T 1 Ri0 \ T 2 Ri1 /

.Ri1 /

.Ri1 \ T 1 Ri0 /

D pi1 pi1 i0

.Ri1 \ T 1 Ri0 \ T 2 Ri1 / :

.Ri1 \ T 1 Ri0 /

We also have pi0 i1 D

.T 1 Ri0 \ T 2 Ri1 /

.Ri0 \ T 1 Ri1 / D :

.Ri0 /

.T 1 Ri0 /

By the Markov property of the Markov partition, if int Ri1 \ int T 1 Ri0 \ int T 2 Ri1 ¤ ∅; then the rectangle Ri1 intersects T 1 Ri0 and T 2 Ri1 along the whole unstable direction (see Fig. 6.9). Therefore, pi0 i1 D

.Ri1 \ T 1 Ri0 \ T 2 Ri1 / ;

.Ri1 \ T 1 Ri0 /

and .Ci1 i0 i1 / D pi1 pi1 i0 pi0 i1 : Similarly, one can show that .Cin in / D pin pin inC1 pin1 in : This completes the proof of the proposition.

t u

6.5 The Case of Repellers

193

6.5 The Case of Repellers In the former sections, we have illustrated with the Smale horseshoe and with the hyperbolic automorphisms of T2 how certain symbolic representations of a hyperbolic set can be used to obtain information that otherwise would be difficult to obtain. Also as a preparation for the full generalization of this approach to an arbitrary hyperbolic set, we consider in this section noninvertible transformations and the notion of repeller, which can be considered a somewhat simpler version of hyperbolic sets in which only expansion is present. In particular, we detail the construction of Markov partitions for repellers. The corresponding construction for hyperbolic sets in Sect. 7.6 is an elaboration of this approach. Let f W U ! M be a C 1 map, where U is an open subset of a smooth manifold M . Definition 6.6. Given a compact f -invariant set J U , we say that f is expanding on J and that J is a repeller for f if there exist c > 0 and > 1 such that kdx f n vk c n kvk

(6.29)

for every x 2 J , v 2 Tx M , and n 2 N. Example 6.2. The expanding map of the circle Eq .z/ D zq , for some integer q > 1, satisfies kdz Eq k D q > 1. Therefore, Eq is expanding on S 1 , and S 1 is a repeller for Eq . Example 6.3. Let T be a toral endomorphism of Tn induced by a matrix A with all eigenvalues with absolute value greater than 1. For example, A can be a diagonal matrix with the entries in the diagonal with values in N n f1g. Then T is expanding on Tn , and Tn is a repeller for T . We also consider a version of Markov partition for repellers. Definition 6.7. Let J be a repeller for a C 1 map f . A collection of closed sets R1 ; : : : ; Rk J is called a Markov partition of J (with respect to f ) if: S 1. J D kiD1 Ri , and Ri D int Ri for each i . 2. int Ri \ int Rj D ∅ whenever i ¤ j . 3. If int f .Ri / \ int Rj ¤ ∅, then f .Ri / Rj . We note that the interiors are taken with respect to the induced topology on J . We describe several examples of Markov partitions. Example 6.4. For the expanding map of the circle Eq , a Markov partition of S 1 is composed of the sets h.Œ2k=q; 2.k C 1/=q/ for k D 0; : : : ; q 1, where h is the transformation in (2.11).

194


Fig. 6.10 Expanding quadratic map 1

1

Example 6.5. Let f W Œ0; 1 ! R be the quadratic map f .x/ D ax.1 x/ for a > 4 (see Fig. 6.10). We consider the f -invariant closed set J D

1 \

f k Œ0; 1:

kD0

It is straightforward to verify that f .x/ D 1 for x D .1 ˙

p

1 4=a/=2, and hence,

p jf 0 .x/j D aj1 2xj a 1 4=a for every x 2 J . Therefore, if a is sufficiently large, then J is a repeller for some extension of f . A Markov partition of J is composed of the sets

0; .1

p

1 4=a/=2 \ J

and

p

.1 C 1 4=a/=2; 1 \ J:

The following result establishes the existence of Markov partitions for any repeller: Theorem 6.1. Any repeller has Markov partitions of arbitrarily small diameter. Proof. Since f is a local diffeomorphism (by the inverse function theorem) and J is compact, there exists ı > 0 such that the preimage of any connected open set U with diameter diam U < ı is composed of a finite number of connected components, each of them diffeomorphic to U . Now we construct a finite open cover U of J (with respect to the induced topology on J ) with the property that if f .U / \ V ¤ ∅ for some U; V 2 U, then f .U / V . Let U0 be a finite open cover of J by balls of radius r such that

6.5 The Case of Repellers

195

2r 1 C

1 c. 1/

< ı:

(6.30)

Let also V1 be the open cover of J composed of the connected components of f 1 U for U 2 U0 . We consider the finite open cover U1 of J composed of the sets [

U .1/ D

V

V 2V1 WV \U ¤∅

for U 2 U0 . Now we proceed inductively to construct a sequence of finite open covers Um of J . Namely, for each m 2 N, let VmC1 be the family of connected components of f 1 V for V 2 Vm such that V \ @U ¤ ∅ for some U 2 Um1 . We consider the finite open cover UmC1 of J composed of the sets [

U .mC1/ D U .m/ [

V

V 2VmC1 WV \U .m/ ¤∅

for U .m/ 2 Um . We have diam U .mC1/ diam U .m/ C 2rc 1 .mC1/ ; and hence, by (6.30), diam U

.mC1/

2r C 2rc

1

mC1 X

i

2r 1 C

i D1

1 c. 1/

0. Exercise 6.7. Show that the transformation h in (6.22) in a dense Gı S is invertible S set of full Lebesgue measure. Hint: consider the set n2Z T n . 5iD1 @Ri /. Exercise 6.8. Construct a Markov partition and a coding map for: 1. The hyperbolic automorphism of T2 induced by the matrix 52 21 . 2. The hyperbolic automorphism of T2 induced by the matrix 51 41 . Exercise 6.9. Find explicitly the numbers pi and pij in (6.28) for the Markov measure in (6.27) obtained from the Markov partition constructed in Sect. 6.4.1. Exercise 6.10. Prove or disprove the following statements: 1. In the Smale horseshoe, the periodic points of period odd are dense. 2. In the Smale horseshoe, the periodic points of period prime are dense. 3. In the Smale horseshoe, the periodic points of period at least 100 are dense. Exercise 6.11. Show that the transformation h in (6.34) is continuous and onto and that it satisfies h ı D f ı h in XAC . Exercise 6.12. Let ˚ U D T .x; y/ 2 R2 W x 2 C y 2 1 :

6.6 Exercises

199

Given 2 .0; 1=2/, we define a transformation f W U ! U by 1 1 f .; x; y/ D 2; x C cos.2/; y C sin.2/ : 2 2 Show that: 1. f is injective. T 2. The solenoid D n2N f n .U / is a closed f -invariant set. 3. is a hyperbolic set for f . Exercise 6.13. Let T W Tn ! Tn be a hyperbolic automorphism induced by a matrix B. Show that X logj j; h.T / D h .T / D p.T / D

2Sp.B/Wj j>1

where is the Lebesgue measure in Tn and where p is the periodic entropy (see Definition 2.6). Hint: to compute the topological entropy, use Exercise 4.17. Exercise 6.14. Prove or disprove the following statement: if T is a hyperbolic automorphism of Tn , then the topological entropy h.T / can take any value in RC . Exercise 6.15. Let i be a hyperbolic set for a diffeomorphism fi W Mi ! Mi for i D 1; 2. Show that the product 1 2 is a hyperbolic set for the diffeomorphism f W M1 M2 ! M1 M2 defined by f .x; y/ D .f1 .x/; f2 .y//: Exercise 6.16. Let J be a repeller for a C 1 map f and let 'W J ! R be a continuous function. Show that for any Markov partition R1 ; : : : ; Rk of J , we have n1 X X 1 log exp sup ' ı f k; n!1 n R i i n 1 i i j D0

Pf jJ .'/ D lim

1

where Ri1 in D

n1 \

n

f j Rij C1

(6.35)

j D0

and where the sum is taken over all i1 ; : : : ; in 2 f1; : : : ; kg such that Ri1 in ¤ ∅.

Notes The books [42, 44, 65, 81, 89] are excellent sources for hyperbolic dynamics. We refer to [10,11] for the theory of nonuniform hyperbolicity, also called Pesin theory.

200


The study of hyperbolicity goes back to Hadamard [37], in connection with the geodesic flow on a surface with negative curvature. Perron [71] studied the existence of bounded solutions under bounded perturbations of linear equations, thus also preparing the ground for the stable manifold theorem. The notion of hyperbolic set was introduced by Smale in his seminal paper [99], where he also laid the foundations of the theory. Anosov [3] introduced the class of systems that now bears his name and in particular made several important developments in [4]. The Smale horseshoe was introduced in [98]. The construction of Markov partitions for a hyperbolic automorphism of T2 in Sect. 6.4.1 is due to Adler and Weiss [2].

Chapter 7

Invariant Manifolds and Markov Partitions

We continue in this chapter our study of hyperbolic dynamics, starting with the construction of stable and unstable manifolds for any point of a hyperbolic set. The optimal regularity of the invariant manifolds is obtained using invariant cone families. In particular, we use the stable and unstable manifolds to define a product structure on any locally maximal hyperbolic set. In addition, we construct Markov partitions with a substantial elaboration of the corresponding construction for repellers. The shadowing property is also presented and is used as a tool in the construction of the Markov partitions. Moreover, we describe Hopf’s argument to establish the ergodicity of Lebesgue measure for a hyperbolic toral automorphism.

7.1 Introduction We already mentioned earlier that hyperbolicity gives rise to a very rich structure and in particular to the existence of stable and unstable manifolds at each point of a hyperbolic set. This is a substantial generalization of the Hadamard–Perron theorem for a hyperbolic fixed point and has several nontrivial consequences. As a first application, we describe Hopf’s argument to establish the ergodicity of Lebesgue measure for a hyperbolic toral automorphism. This approach should be compared to the earlier one in terms of Fourier analysis. For arbitrary transformations, in general, one is not able to apply this other method to establish the ergodicity of an invariant measure. On the other hand, Hopf’s argument can be applied with success in many other situations, although the details fall out of the scope of this book. In particular, the method depends crucially on the so-called absolute continuity property. In another direction, the shadowing property of a locally maximal hyperbolic set tell us that there is a real orbit close to any sequence of points such that the image of each of them is sufficiently close to the next. A consequence is that if two points x and f m .x/ of a given orbit in a hyperbolic set are sufficiently close, then there exists an m-periodic point close to them. The shadowing property, together L. Barreira, Ergodic Theory, Hyperbolic Dynamics and Dimension Theory, Universitext, DOI 10.1007/978-3-642-28090-0 7, © Springer-Verlag Berlin Heidelberg 2012

201

202

7 Invariant Manifolds and Markov Partitions

with the product structure defined by the stable and unstable manifolds, can be used to construct Markov partitions for any locally maximal hyperbolic set, with an elaboration of the corresponding construction for repellers.

7.2 Stable and Unstable Manifolds We construct in this section local stable and unstable manifolds at any point of a hyperbolic set. We first consider the case of linear diffeomorphisms. Namely, let AW Rn ! Rn be the invertible linear transformation A.x; y/ D .x; y/;

(7.1)

where .x; y/ 2 Rp Rq D Rn and ; 1 2 .0; 1/. We note that the horizontal and vertical axes E s D Rp f0g and E u D f0g Rq satisfy ˚ E s D .x; y/ 2 Rn W kAm .x; y/k ! 0 when m ! C1 ; ˚ E u D .x; y/ 2 Rn W kAm .x; y/k ! 0 when m ! 1 : Clearly, AE s D E s

and AE u D E u ;

that is, the linear subspaces E s and E u are A-invariant sets (see Sect. 2.4 for the definition). Now we consider arbitrary diffeomorphisms and their hyperbolic sets. We start with the particular case of a hyperbolic fixed point. We recall that the stable and unstable subspaces at a point x are denoted, respectively, by E s .x/ and E u .x/. Theorem 7.1 (Hadamard–Perron). If x is a hyperbolic fixed point of a C 1 diffeomorphism f , then there exist C 1 manifolds V s .x/ and V u .x/ containing x such that Tx V s .x/ D E s .x/; Tx V u .x/ D E u .x/ and f .V s .x// V s .x/;

f 1 .V u .x// V u .x/:

The sets V s .x/ and V u .x/ are called, respectively, (local) stable manifold and (local) unstable manifold at x. The local behavior is indicated in Fig. 7.1. Theorem 7.1 is a particular case of Theorem 7.2, which establishes the existence of stable and unstable manifolds for arbitrary hyperbolic sets. Let be a hyperbolic set for a C 1 diffeomorphism f . Given " > 0, for each x 2 , we consider the sets ˚ V s .x/ D V"s .x/ D y 2 B.x; "/ W d.f n .x/; f n .y// < " for every n 0 (7.2) and

7.2 Stable and Unstable Manifolds

203

E s(x)

x V s(x)

E u(x)

V u(x)

Fig. 7.1 Stable and unstable manifolds of a hyperbolic fixed point

˚ V u .x/ D V"u .x/ D y 2 B.x; "/ W d.f n .x/; f n .y// < " for every n 0 ; (7.3) where d is the distance in M and B.x; "/ is the ball of radius " centered at x. Theorem 7.2. If is a hyperbolic set for a C 1 diffeomorphism f , then for any sufficiently small " > 0, the following properties hold: 1. For each x 2 , the sets V s .x/ and V u .x/ are C 1 manifolds containing x such that Tx V s .x/ D E s .x/; Tx V u .x/ D E u .x/ (7.4) and f .V s .x// V s .f .x//;

f 1 .V u .x// V u .f 1 .x//:

(7.5)

2. There exists D ."/ > 0 such that V s .x/ B s .x; /

and V u .x/ B u .x; /

(7.6)

for every x 2 , where B s .x; / and B u .x; / are balls with respect to the induced distances on V s .x/ and V u .x/. 3. For each > , there exists C > 0 such that

204


d.f n .x/; f n .y// C n d.x; y/; and

y 2 V s .x/

d.f n .x/; f n .y// C n d.x; y/;

y 2 V u .x/

(7.7)

(7.8)

for every x 2 and n 2 N (with as in Definition 6.1). Proof. We only establish the result for the sets V s .x/. The argument for V u .x/ is entirely analogous. We want to show that ˚ V s .x/ D expx .v;

s x .v//

W v 2 B"s ;

where xs W B"s ! E u .x/ is a C 1 map in a ball B"s E s .x/ of radius " centered at the origin such that xs .0/ D 0 and d0 xs D 0. Here expx is the exponential map, given by expx v D .1/, where is the unique geodesic such that .0/ D x and 0 .0/ D v. Since is a hyperbolic set, for each x 2 , the transformation s u fx D exp1 f .x/ ıf ı expx W B" B" ! Tf .x/ M

can be written in the form fx .v; w/ D ANx v C gN x .v; w/; BN x w C hN x .v; w/ ; where .v; w/ 2 B"s B"u , for some invertible linear transformations ANx W E s .x/ ! E s .f .x//;

BNx W E u .x/ ! E u .f .x//;

and some maps gN x W B"s B"u ! E s .x/;

hN x W B"s B"u ! E u .x/

such that gN x .0; 0/ D 0;

hN x .0; 0/ D 0;

d.0;0/ gN x D 0;

and d.0;0/ hN x D 0:

(7.9)

By eventually considering an inner product as in Proposition 6.3, we can assume that there exists 2 .; 1/ such that kANx k <

and kBN x1 k <

for every x 2 :

Given x 2 and n 2 Z, we consider the transformation Fn D ff n .x/ , which we write in the form Fn .v; w/ D An v C gn .v; w/; Bn w C hn .v; w/ ;


205

where An D ANf n .x/ ;

Bn D BN f n .x/ ;

gn D gN f n .x/ ;

and hn D hNf n .x/ :

Setting Ens D E s .f n .x//

and Enu D E u .f n .x//;

the linear transformations s An W Ens ! EnC1

satisfy kAn k <

u and Bn W Enu ! EnC1

and kBn1 k <

for every n 2 Z. Moreover, since f is a C 1 diffeomorphism and is compact, u s the family .dgn ; dhn /n2Z is equicontinuous in B2" B2" (by eventually making " smaller), and we have kd.v;w/ gn k K

and kd.v;w/ hn k K

u s B2" , where K D K."/ ! 0 when " ! 0, in for every n 2 Z and .v; w/ 2 B2" view of (7.9). Now we consider an appropriate space of Lipschitz functions. Namely, given 2 .0; 1, let X be the space of two-sided sequences ' D .'n /n2Z such that each function 'n W B"s ! Enu satisfies 'n .0/ D 0 and

k'n .v/ 'n .w/k kv wk;

v; w 2 B"s :

One can easily verify that X is a complete metric space with the distance ˚ d.'; / D sup k'n .v/

n .v/k

W v 2 B"s ; n 2 Z :

Our aim is to find a sequence ' 2 X such that ˚ V s .f n .x// D expx .v; 'n .v// W v 2 B"s for each n 2 Z. The sequence will be obtained as the unique fixed point of a contraction operator. We first proceed formally in order to motivate the introduction of the operator. Since Fn .v; 'n .v// D An v C gn .v; 'n .v//; Bn 'n .v/ C hn .v; 'n .v// ; in view of (7.5), we must have 'nC1 An v C gn .v; 'n .v// D Bn 'n .v/ C hn .v; 'n .v//;

206

that is,


'n .v/ D Bn1 'nC1 An v C gn .v; 'n .v// Bn1 hn .v; 'n .v//

for each n 2 Z and v 2 B"s . Reciprocally, if the functions 'n satisfy this identity, then (7.5) holds. This leads us to define an operator T by T .'/n .v/ D Bn1 'nC1 An v C gn .v; 'n .v// Bn1 hn .v; 'n .v// for each ' 2 X , n 2 Z, and v 2 B"s . We want to show that T has a unique fixed point in X . We first show that T is well defined. Take n 2 Z and v 2 B"s . Since 'n .v/ D 'n .v/ 'n .0/; we have k'n .v/k kvk ", and hence, kAn v C gn .v; 'n .v//k " C Kk.v; 'n .v//k < " C .1 C /K" < "; provided that " is sufficiently small. This allows one to compute the first term in T .'/n .v/. Furthermore, T .'/n.0/ D 0, and kT .'/n .v/ T .'/n .w/k 'nC1 An v C gn .v; 'n .v// 'nC1 An w C gn .w; 'n .w// C khn .v; 'n .v// hn .w; 'n .w//k kAn .v w/ C gn .v; 'n .v// gn .w; 'n .w//k C Kk.v; 'n .v// .w; 'n .w//k 2 C .1 C /2 K kv wk for each n 2 Z and v; w 2 B"s . Provided that " is sufficiently small, we can make K so small that kT .'/n .v/ T .'/n .w/k kv wk for each n 2 Z and v; w 2 B"s . This shows that T .X / X . Now we show that T is a contraction. Given '; 2 X , n 2 Z, and v 2 B"s , we have kT .'/n .v/ T . /n .v/k 'nC1 An v C gn .v; 'n .v//

An v C gn .v;

n .v//

C khn .v; 'n .v// hn .v; n .v//k 'nC1 An v C gn .v; 'n .v// 'nC1 An v C gn .v;

n .v//

nC1


207

C 'nC1 An v C gn .v; C Kk'n .v/

n .v//

nC1

An v C gn .v;

n .v//

n .v/k

kgn .v; 'n .v// gn .v; n .v//k C d.'; / C Kd.'; / 1 C .1 C /K d.'; /: Therefore, provided that " is so small that 0 D 1 C .1 C /K < 1, we obtain d.T .'/; T . // 0 d.'; /; and T is a contraction in X . Hence, T has a unique fixed point ' 2 X . It remains to show that each function 'n is of class C 1 . Given v; w 2 B"s , set v;w 'n D

.w; 'n .w// .v; 'n .v// k.w; 'n .w// .v; 'n .v//k

and let tv 'n be the set of vectors w 2 Tf n .x/ M such that v;vm 'n ! w when m ! 1 for some sequence .vm /m2N converging to v. We also consider the set ˚ .v;'n .v// Vns D w W w 2 tv 'n and 2 R ; where

˚ Vns D .v; 'n .v// W v 2 B"s :

One can easily verify that 'n is differentiable if and only if .v;'n .v// Vns is a subspace of dimension dim Ens . Now we observe that v;vm 'n ! w when m ! 1 if and only if d.v;'n .v// Fn w Fn .vm ; 'n .vm // Fn .v; 'n .v// D : lim m!1 kFn .vm ; 'n .vm // Fn .v; 'n .v//k kd.v;'n .v// Fn wk This implies that .d.v;'n .v// Fn /.v;'n .v// Vns D Fn .v;'n .v// Fn .Vns /

(7.10)

for every n 2 Z. To proceed with the proof of the theorem, we need some auxiliary results. Given 2 .0; 1/, we consider the cones ˚ Cns D .v; w/ 2 Ens Enu W kwk < kvk [ f.0; 0/g; and

˚ Cnu D .v; w/ 2 Ens Enu W kvk < kwk [ f.0; 0/g:

Lemma 7.1. For any sufficiently small " > 0, we have

208

7 Invariant Manifolds and Markov Partitions s d.v;w/ .Fn1 /CnC1 Cns

u and d.v;w/ Fn Cnu CnC1

(7.11)

for every n 2 Z and .v; w/ 2 B"s B"u . Proof of the lemma. Given .p; q/ 2 Ens Enu , let .p 0 ; q 0 / D d.v;w/ Fn .p; q/ D An p C d.v;w/ gn .p; q/; Bn q C d.v;w/ hn .p; q/ : We obtain and For .p; q/ 2

Cnu

kp 0 k kpk C Kk.p; q/k;

(7.12)

kq 0 k 1 kqk Kk.p; q/k:

(7.13)

n f.0; 0/g, we have kpk < kqk, and thus, kp 0 k kqk C K.1 C /kqk;

and

kq 0 k 1 kqk K.1 C /kqk:

Thus, provided that " is sufficiently small, we obtain kp 0 k < kq 0 k for q 0 ¤ 0. This establishes the second inclusion in (7.11). A similar argument yields the first inclusion. t u For each n 2 Z and .v; w/ 2 B"s B"u , we consider the intersections 1 \

Ens .v; w/ D

1 s d.v;w/ .Fn1 ı ı FnCj 1 /CnCj

j D0

and Enu .v; w/ D

1 \

u d.v;w/ .Fn1 ı ı Fnj /Cnj :

j D0

It follows from Lemma 7.1 that Ens .v; w/ Cns

and Enu .v; w/ Cnu :

Lemma 7.2. Provided that " and are sufficiently small, the sets Ens .v; w/ and Enu .v; w/ are subspaces of dimensions respectively dim Ens and dim Enu , vary continuously with .v; w/, and satisfy Ens .v; w/ ˚ Enu .v; w/ D Tf n .x/ M; and s d.v;w/ Fn Ens .v; w/ D EnC1 .Fn .v; w//; u d.v;w/ Fn Enu .v; w/ D EnC1 .Fn .v; w//:

(7.14)


209

Proof of the lemma. We observe that the set u Hj D d.v;w/ .Fn1 ı ı Fnj /Cnj

(7.15)

contains a subspace, say Fju , of dimension k D dim Enu . Let vj1 ; : : : ; vj k be an orthonormal basis of Fju for each j . Since the closed unit ball in Enu is compact, there exists a sequence .kj /j 2N N such that v kj i ! v i

when j ! 1;

for i D 1; : : : ; k;

where v1 ; : : : ; vk is an orthonormal set in Enu .v; w/ (we note that the sequence of sets Hj in (7.15) is nondecreasing in j ). This shows that Enu .v; w/ contains a subspace Gnu of dimension k. A similar argument shows that Ens .v; w/ contains a subspace Gns of dimension dim M k. Moreover, since Cns \ Cnu D f.0; 0/g, we have Gns ˚ Gnu D Tf n .x/ M:

(7.16)

Now we show that Gns D Ens .v; w/

and Gnu D Enu .v; w/:

For .p; q/ 2 Cnu n f.0; 0/g, we have kpk < kqk, and hence, by (7.12) and (7.13), k.p 0 ; q 0 /k kq 0 k kp 0 k 1 kqk Kk.p; q/k kpk Kk.p; q/k .1 /kqk 2Kk.p; q/k 1 2K k.p; q/k: 1C

(7.17)

Similarly, for .p; q/ 2 Cns n f.0; 0/g, we have k.p 0 ; q 0 /k kp 0 k C kq 0 k kpk C Kk.p; q/k C 1 kqk C Kk.p; q/k . C 1 /kpk C 2Kk.p; q/k C 1 C 2K k.p; q/k: 1

(7.18)

If there exists p 2 Enu .v; w/ n Gnu , then it can be written in the form p D ps C pu where ps 2 Gns n f0g and pu 2 Gnu . By (7.17) and (7.18), we obtain

210


kps k D

C 1 C 2K 1

m

C 1 C 2K 1

d.v;w/ .F 1 ı ı F 1 /ps nm n1

m

d.v;w/ .F 1 ı ı F 1 /.p pu / nm n1

. C 1 /=.1 / C 2K .1 /=.1 C / 2K

m kp pu k ! 0

when m ! 1, provided that " and are sufficiently small. This contradiction shows that ps D 0, and hence, Gnu D Enu .v; w/. A similar argument shows that Gns D Ens .v; w/. By (7.16), this yields property (7.14). It remains to establish the continuity of the spaces in .v; w/. We first note that by (7.17) and (7.18), d.v;w/ .FmCn1 ı ı Fn /.p; q/

C 1 C 2K 1

m k.p; q/k

for every .p; q/ 2 Ens .v; w/ and m n, and d.v;w/ .F 1 ı ı F 1 /.p; q/ nm n1

1 2K 1C

m

k.p; q/k

for every .p; q/ 2 Enu .v; w/ and m n. Therefore, we can repeat the argument in the proof of Proposition 6.2 to show that the subspaces Ens .v; w/ and Enu .v; w/ vary continuously with .v; w/. t u To complete the proof of the theorem, we note that taking D yields the inclusion .v;'n .v// Vns Cns , and thus, it follows from (7.10) and Lemma 7.2 that .v;'n .v// Vns Ens .v; 'n .v// for every n 2 Z and v 2 B"s . On the other hand, since the set .v;'n .v// Vns projects onto Ens .v; 'n .v// (because Vns is a graph over B"s ), it is a subspace with the dimension of Ens .v; 'n .v//, and hence, .v;'n .v// Vns D Ens .v; 'n .v//: This shows that 'n is differentiable. Since v 7! Ens .v; 'n .v// is continuous, the set Vns is a C 1 manifold. Setting v D 0, we obtain T0 Vns D Ens . This establishes the first two statements in Theorem 7.2. For the last statement, we observe that kFn .v; 'n .v//k An v C gn .v; 'n .v//; 'nC1 .An v C gn .v; 'n .v/// .1 C /kAn v C gn .v; 'n .v//k .1 C /.kvk C Kk.v; 'n .v//k/ k.v; 'n .v//k;

7.3 Ergodicity and Hopf’s Argument

211

where D

1C C .1 C /K: 1

Since can be made arbitrarily close to by taking " and sufficiently small, this completes the proof of the theorem. t u Properties (7.7) and (7.8) motivate the following definition: Definition 7.1. The sets V s .x/ and V u .x/ are called, respectively, (local) stable manifold and (local) unstable manifold at the point x.

7.3 Ergodicity and Hopf’s Argument Here and in the following sections, we describe several applications of the existence of stable and unstable manifolds. We first describe Hopf’s argument to establish the ergodicity of the Lebesgue measure for a hyperbolic toral automorphism. Theorem 7.3. The Lebesgue measure is ergodic with respect to any hyperbolic toral automorphism. Proof. Let be the Lebesgue measure in the torus Tn . Given a continuous function 'W Tn ! R, by Birkhoff’s ergodic theorem (Theorem 2.2), the functions m1 1 X '.T k .x// m!1 m

' C .x/ D lim

kD0

and

m1 1 X '.T k .x// m!1 m

' .x/ D lim

kD0

are well-defined Lebesgue-almost everywhere. Lemma 7.3. We have ' C D ' Lebesgue-almost everywhere. Proof of the lemma. We consider the set ˚ Y D x 2 Tn W ' C .x/ > ' .x/ : Since ' C and ' are invariant almost everywhere, Y is also invariant almost everywhere. Again by Theorem 2.2, we have .' Y /C D ' C Y

and .' Y / D ' Y

Lebesgue-almost everywhere. Furthermore, by (2.35),

212


Z

' C d D Y

Z Z

Tn

D Z

Tn

D Tn

Z

that is,

' C Y d D

Z Tn

Z ' Y d D ' Y d D

Tn

.' Y /C d

.' Y / d

Z

' d; Y

.' C ' / d D 0: Y

Since ' C ' > 0 in Y , we conclude that .Y / D 0. One can show in a similar manner that ˚ x 2 Tn W ' C .x/ < ' .x/ D 0: Therefore, ' C D ' Lebesgue-almost everywhere.

t u

Now let X T be the set of full Lebesgue measure defined by n

˚ X D x 2 Tn W ' C .x/ and ' .x/ are well defined and ' C .x/ D ' .x/ : Lemma 7.4. The following properties hold: 1. If ' C .x/ is well defined, then ' C .y/ is well defined for every y 2 V s .x/ and ' C .y/ D ' C .x/. 2. If ' .x/ is well defined, then ' .y/ is well defined for every y 2 V u .x/ and ' .y/ D ' .x/. Proof of the lemma. Since Tn is compact, given " > 0, there exists ı > 0 such that j'.x/ '.y/j < "

whenever d.x; y/ < ı;

where d is the distance in Tn . On the other hand, if y 2 V s .x/, then d.T k .y/; T k .x// ! 0 when k ! 1 by (7.7). Therefore, taking ` 2 N such that d.T k .y/; T k .x// < ı for k `, we obtain 0 lim sup m!1

D lim sup m!1

m1 1 X j'.T k .y// '.T k .x//j m kD0

m1 1 X j'.T k .y// '.T k .x//j ": m kD`

7.3 Ergodicity and Hopf’s Argument

213

Fig. 7.2 Relation between two stable manifolds

V u(z) z

π (z)

x V s(x) y V s(y)

Since " is arbitrary, we conclude that ' C .y/ is well defined if and only if ' C .x/ is well defined, in which case ' C .y/ D ' C .x/. The desired result follows from covering Tn with balls of diameter ı. The proof of the second statement is analogous. t u The next step is to relate two stable manifolds V s .x/ and V s .y/, as in Fig. 7.2. Lemma 7.5. For . /-almost every .x; y/ 2 X X , there exist points z 2 V s .x/ \ X and w 2 V s .y/ \ X such that w 2 V u .z/. Proof of the lemma. We consider a set R D Rs Ru Tn , where Rs and Ru are closed rectangles, respectively, along the stable and unstable directions. Let also ms and mu be the Lebesgue measures in these directions. We note that Z ms .V s .x/ \ R/ d mu .x/;

.R/ D

(7.19)

Ru

and since .X / D 1, it follows from Lemma 7.4 that Z .R/ D .R \ X / D

ms .V s .x/ \ R \ X / d mu .x/:

(7.20)

Ru

We conclude from (7.19) and (7.20) that ms .Rs / D ms .V s .x/ \ R/ D ms .V s .x/ \ R \ X /

(7.21)

for mu -almost every x 2 Ru . Now for each x; y 2 R, we define a function W V s .x/ \ R ! V s .y/ \ R by .z/ D V u .z/ \ V s .y/

214


(see Fig. 7.2). It follows from (7.21) that ms .V s .x/ \ R \ X / \ .V s .y/ \ R \ X / D ms .Rs / for . /-almost every .x; y/ 2 R R. In particular, there exist z 2 V s .x/ \ R \ X

and w 2 V s .y/ \ R \ X

such that .z/ D w.

t u

Taking points x, y, z, and w as in Lemma 7.5, it follows from Lemma 7.4 that ' C .x/ D ' C .z/ D ' .z/ D ' .w/ D ' C .w/ D ' C .y/;

(7.22)

and hence, ' C .x/ D ' C .y/ for . /-almost every .x; y/ 2 Tn Tn . In other words, ' C is constant -almost everywhere. Now let W Tn ! R be a T -invariant function in L1 .Tn ; /. Given ` > 0, there exists a continuous function '` W Tn ! R such that Z j Tn

Since

'` j d < 1=`:

is T -invariant, we have .x/

C ` .x/

m1 1 X .T k .x// '` .T k .x// m!1 m

D lim

kD0

for -almost every x 2 X , and by the dominated convergence theorem (Theorem A.3), Z j Tn

1 '`C j d D lim m!1 m lim sup m!1

Z D

j Tn

Z

1 m

ˇ m1 ˇX ˇ ˇ n T ˇ kD0

m1 XZ kD0

j. Tn

ˇ ˇ ˇ '` / ı T k ˇ d ˇ '` / ı T k j d

'` j d < 1=`:

By (7.22), the functions '`C are constant -almost everywhere, and hence, letting ` ! 1, we conclude that '`C ! -almost everywhere when ` ! 1 and that is constant -almost everywhere. It follows from Proposition 2.11 that the Lebesgue measure is ergodic. t u

7.4 Product Structure

215

The alternative proof of Theorem 7.3 given by Example 2.7 (in fact in a more general setting) uses Fourier analysis. However, for arbitrary transformations, in general, one is not able to apply with success this other method to establish the ergodicity of an invariant measure. On the other hand, Hopf’s argument can be applied in many other situations (see in particular Exercise 7.11), although the technical details fall out of the scope of this book. Certainly, the simpler proof of Theorem 7.3 in the special case of hyperbolic toral automorphisms substantially hides some complications.

7.4 Product Structure Given a hyperbolic set, the families of stable and unstable manifolds V s .x/ and V u .x/ induce a product structure, in the following sense: Definition 7.2. We say that a hyperbolic set has a product structure if there exist " > 0 and ı > 0 such that (see Fig. 7.3) card.V"s .x/ \ V"u .y// D 1 whenever x; y 2 and d.x; y/ ı. When the hyperbolic set has a product structure, we write Œx; y D V"s .x/ \ V"u .y/

[x;y]

x

y V u( y)

V s(x)

Fig. 7.3 Product structure induced by the stable and unstable manifolds

216


for each x; y 2 with d.x; y/ ı. We thus obtain a transformation ˚ Œ; W .x; y/ 2 W d.x; y/ ı ! M:

(7.23)

A consequence of Theorem 7.2 is the following: Proposition 7.1. Any hyperbolic set has a product structure, and the transformation in (7.23) is continuous. Proof. By Proposition 6.2, the stable and unstable subspaces E s .x/ and E u .x/ vary continuously with x 2 and are uniformly transverse. By their continuity, the same happens between any two spaces E s .x/ and E u .y/ with x; y 2 sufficiently close. It follows from (7.4) that for " sufficiently small, the stable and unstable manifolds V s .x/ and V u .y/ are transverse, and thus, the set Œx; y consists of at most one point for each sufficiently close x; y 2 . Furthermore, by (7.6), there exists > 0 such that V s .x/ B s .x; /

and V u .y/ B u .y; /

for x; y 2 . Therefore, there exists ı > 0 such that if x; y 2; and d.x; y/ ı, then Œx; y ¤ ∅. This shows that the hyperbolic set has a product structure. The continuity of the product structure follows from the continuity of the map .x; y/ 7! E s .x/ E u .y/ and from the fact that the sets V s .x/ and V u .y/ are smooth manifolds tangent, respectively, to E s .x/ and E u .y/. t u Now we consider a particular class of hyperbolic sets. Definition 7.3. A hyperbolic set for a diffeomorphism f is said to be locally maximal if there is an open set U such that D

\

f n .U /:

n2Z

In other words, a hyperbolic set for a diffeomorphism f is locally maximal if all orbits of f in a sufficiently small open neighborhood of are in fact in the hyperbolic set. Example 7.1. It is easy to verify that any hyperbolic set formed by a single fixed point is locally maximal. For a locally maximal hyperbolic set, one can always replace M by in (7.23), that is, we obtain a transformation ˚ Œ; W .x; y/ 2 W d.x; y/ ı ! :

(7.24)

7.5 The Shadowing Property

217

Proposition 7.2. For a locally maximal hyperbolic set , if two points x; y 2 are sufficiently close, then Œx; y 2 . Proof. By Proposition 7.1, the hyperbolic set has a product structure. Take x; y 2 with d.x; y/ ı such that Œx; y consists of a single point in M . We must show that Œx; y 2 . By the continuity of the transformation .x; y/ 7! Œx; y, provided that ı is sufficiently small, the image of Œ; is contained in some open set V with V U , where U is any open set as in Definition 7.3. Moreover, by Property 3 in Theorem 7.2, provided that ı is sufficiently small, the open set V is so small that f n .Œx; y/ 2 U

for every n 2 Z

since Œx; y 2 V s .x/; V u .y/. This shows that Œx; y 2 .

t u

7.5 The Shadowing Property We describe in this section the so-called shadowing property of a locally maximal hyperbolic set. Roughly speaking, it tell us that there is a real orbit close to any sequence of points such that the image of each is sufficiently close to the next. Let f W M ! M be a C 1 diffeomorphism of a manifold M . Let also a 2 Z [ f1g and b 2 Z [ fC1g. Definition 7.4. Given ˛ > 0, a sequence .xn /anb M is called an ˛-orbit of f if d.f .xn /; xnC1 / < ˛ for every n 2 Œa; b/: Definition 7.5. Given ˇ > 0, a point x 2 M is said to ˇ-shadow .xn /anb M if d.f n .x/; xn / ˇ

for every n 2 Œa; b:

We also say that the sequence .xn /anb is ˇ-shadowed by the point x. Now we establish the shadowing property. Theorem 7.4. Let be a locally maximal hyperbolic set for a diffeomorphism f . For each ˇ > 0, there exists ˛ > 0 such that each ˛-orbit .xn /anb of f is ˇ-shadowed by some point x 2 . Proof. Let " > 0 and ı 2 .0; "/ be as in Definition 7.2 so that Œx; y 2 whenever x; y 2 (see Proposition 7.2). Let also C and be as in Theorem 7.2 and take m 2 N such that C m " < ı=2. Moreover, take ˛ > 0 with the property that if .yn /0nm is an ˛-orbit, then d.f n .y0 /; yn / < ı=2 for every n 2 Œ0; m:

218


Now we consider an ˛-orbit .xn /0npm for some p 2 N. By the choice of m and ˛, we can define recursively the points x00 D x0 , and 0 0 x.nC1/m D Œx.nC1/m ; f m .xnm /;

n 2 Œ0; p/:

0 Indeed, since xnm 2 V s .xnm /, we have

0 /; f m .xnm / C m " < ı=2; d f m .xnm and by the choice of ˛, we have d f m .xnm /; x.nC1/m < ı=2: Therefore,

0 /; x.nC1/m/ < ı; d.f m .xnm

0 0 is well defined. By the choice of ı, we also have xnm 2 for n 2 Œ0; p. and x.nC1/m pm 0 Now let x D f .xpm / 2 . For each n 2 Œ0; pm, taking q 2 N such that n 2 Œq m; .q C 1/m/, we obtain p X n nqm 0 0 0 d f .x/; f .xqm / d f nkm .xkm /; f n.k1/m .x.k1/m / kDqC1

p X

C kmn "

kDqC1

C " 1

0 0 0 2 V u .f m .x.k1/m // for each k. On the other hand, since xqm 2 V s .xqm /, since xkm and hence, 0 f nqm .xqm / 2 V s .f nqm .xqm //;

we have

0 d f nqm .xqm /; f nqm .xqm / < ";

and by the choice of ˛, we have d f nqm .xqm /; xn < ı=2 < "=2: Therefore, 0 0 d.f n .x/; xn / d f n .x/; f nqm .xqm / C d f nqm .xqm /; f nqm .xqm / C d f nqm .xqm /; xn
0, if x 2 and d.f m .x/; x/ < ˛, then there exists y 2 such that f m .y/ D y and d.f n .x/; f n .y// < "

for every n 2 Œ0; m:

Proof. For some ˇ < "=2, take ˛ 2 .0; "=2/ as in Theorem 7.4. For each n 2 N, we set xn D f k .x/ if n D k mod m with k 2 Œ0; m/. Then .xn /n2N is an ˛-orbit of f , and by Theorem 7.4, it is ˇ-shadowed by some point y 2 . We note that since the sequence xn is m-periodic, it is also ˇ-shadowed by the point f m .y/. Therefore, d f n .y/; f n .f m .y// d.f n .y/; xn / C d xn ; f n .f m .y// 2ˇ < " for every n 2 Z. It follows from (7.2) and (7.3) that f m .y/ 2 V s .y/ \ V u .y/; and hence, f m .y/ D y. Moreover, d.f n .x/; f n .y// d.f n .x/; xn / C d.xn ; f n .y// d.f m .x/; x/ C d.xn ; f n .y// < ˛ C ˇ < " for every n 2 Œ0; m. This completes the proof of the theorem.

t u

7.6 Construction of Markov Partitions In this section, we use the product structure determined by the stable and unstable manifolds to construct Markov partitions for any locally maximal hyperbolic set.

220


Fig. 7.4 Intersection of a stable manifold with a rectangle

V s(x)

R V s(x; R)

Let be a locally maximal hyperbolic set for a C 1 diffeomorphism f and take ı > 0 (and " > 0) such that the transformation Œ; in (7.24) is well defined. Definition 7.6. A closed set R is called a rectangle if: 1. diam R < ı and R D int R (with the interior taken with respect to the induced topology on ). 2. Œx; y 2 R whenever x; y 2 R. This notion of rectangle includes the rectangles used in the construction of a coding map for a hyperbolic automorphism of T2 (see Sect. 6.4.1). Now we introduce the notion of Markov partition. Given a rectangle R and a point x 2 R, we consider the sets (see Fig. 7.4) V s .x; R/ D V s .x/ \ R

and V u .x; R/ D V u .x/ \ R:

Definition 7.7. A collection of rectangles R1 ; : : : ; Rk is called a Markov partition of (with respect to f ) if: 1. int Ri \ int Rj D ∅ whenever i ¤ j . 2. If x 2 int Ri and f .x/ 2 int Rj , then f V u .x; Ri / V u .f .x/; Rj /

and f 1 V s .f .x/; Rj / V s .x; Ri /:

As for the hyperbolic automorphisms of T2 , the second property is called the Markov property of the partition. Example 7.2. The Markov partitions in Definition 6.5, for a hyperbolic automorphism of T2 , are Markov partitions in the sense of Definition 7.7. The following result establishes the existence of Markov partitions for any locally maximal hyperbolic set:

7.6 Construction of Markov Partitions

221

Theorem 7.6. Any locally maximal hyperbolic set has Markov partitions of arbitrarily small diameter. Proof. Let be a locally maximal hyperbolic set for a diffeomorphism f . Take ˛; ˇ > 0 as in Theorem 7.4. Take also r 2 .0; ˛=2/ such that d.f .x/; f .y// < ˛=2 whenever d.x; y/ < r: We consider a set ˙ D fx1 ; : : : ; xN g such that each point in is at a distance at most r of some element of ˙, and we let ˚ ˙ 0 D .yn /n2Z 2 ˙ Z W d.f .yn /; ynC1 / < ˛ for n 2 Z : For each sequence y D .yn /n2Z 2 ˙ 0 , there exists a unique point py 2 that ˇ-shadows y. The existence follows from Theorem 7.4. For the uniqueness, we observe that if x 2 is another point that ˇ-shadows y, then d f n .x 0 /; f n .py / d f n .x 0 /; yn C d yn ; f n .py / 2ˇ

(7.25)

for every n 2 Z. Provided that 2ˇ < ", it follows from (7.2) and (7.3) that x 0 2 V s .py / \ V u .py /, and thus, x 0 D py . Moreover, since r < ˛=2, for each x 2 , there exists y 2 ˙ 0 such that x D py . Lemma 7.6. The transformation hW ˙ 0 ! defined by h.y/ D py is continuous. Proof of the lemma. Otherwise, there would exist ı > 0 and for each m 2 N points zm ; z0m 2 ˙ 0 such that zmi D z0mi for every i 2 Œm; m, with d.pzm ; pz0m / ı:

(7.26)

On the other hand, d f n .pzm /; f n .pz0m / 2ˇ

for n 2 Œm; m:

(7.27)

Eventually taking subsequences, we may assume that pzm ! x and pz0m ! x 0 when m ! 1. It thus follows from (7.27) that d f n .x/; f n .x 0 / 2ˇ

for n 2 Z:

As in (7.25), provided that 2ˇ < ", this implies that x D x 0 , while (7.26) yields that d.x; x 0 / ı. This contradiction shows that the transformation h is continuous. u t For each i D 1; : : : ; N , we consider the set ˚ Ti D py W y 2 ˙ 0 and y0 D xi D h fy 2 ˙ 0 W y0 D xi g :

222


Since the cylinder set fy 2 ˙ 0 W y0 D xi g is closed, it follows from Lemma 7.6 that Ti is closed for i D 1; : : : ; N . Moreover, for each y; y 0 2 ˙ 0 with y0 D y00 , we define a point y y 0 2 ˙ 0 by ( 0

.y y /n D We note that and

yn

if n 0;

yn0

if n 0:

d f n .pyy 0 /; f n .py / 2ˇ

for n 0

d f n .pyy 0 /; f n .py 0 / 2ˇ

for n 0:

Provided that 2ˇ < ", this implies that pyy 0 2 V s .py / \ V u .py 0 /;

that is,

pyy 0 D Œpy ; py 0 :

Let x D py ; x 0 D py 0 2 Ti , where y0 D y00 D xi . Then Œx; x 0 D Œpy ; py 0 D pyy 0 2 Ti :

(7.28)

To show that the sets Ti satisfy a certain Markov property, we assume that y1 D xj and that x 0 2 V s .x; Ti / WD V s .x/ \ Ti . Then x 0 D Œx; x 0 D pyy 0 ; and since .y y 0 /0 D xi , we obtain f .x 0 / D p .yy 0 / 2 Tj ; where is the shift map. Therefore, f .x 0 / 2 V s .f .x/; Tj /, and hence, f V s .x; Ti / V s .f .x/; Tj //:

(7.29)

One can show in a similar manner that f V u .x; Ti / V u .f .x/; Tj /: Now for each i D 1; : : : ; N , we set ˚ @s Ti D x 2 Ti W x 62 int V u .x; Ti / and

˚ @u Ti D x 2 Ti W x 62 int V s .x; Ti / :

(7.30)


223

Lemma 7.7. For i D 1; : : : ; N , we have @Ti D @s Ti [ @u Ti :

(7.31)

Proof of the lemma. If x 2 int Tj , then V u .x; Ti / D Ti \ .V u .x/ \ / is an open neighborhood of x in V u \ , and hence, x 2 int V u .x; Ti /. We show in a similar manner that x 2 int V s .x; Ti /. Now we assume that x 2 int V u .x; Ti / and x 2 int V s .x; Ti /. Given y 2 sufficiently close to x, the points Œx; y 2 V s \ and Œy; x 2 V u .x/ \ are well defined and vary continuously with y. This implies that Œx; y; Œy; x 2 Ti for any y 2 sufficiently close to x. It follows from (7.28) that y D Œy; x; Œx; y 2 Ti ; and hence, x 2 int Ti . This establishes (7.31).

t u

The sets Ti shall be used to construct the Markov partition. For this, given x 2 , we consider the families ˚ ˚ and S.x/ D Tj W Tj \ Ti ¤ ∅ for some Ti 2 T .x/ : T .x/ D Ti W x 2 Ti We can proceed in a similar manner to that in the proof of Lemma 7.7 to show that the set ˚ A D x 2 W V s .x/ \ @s T D ∅ and V u .x/ \ @u T D ∅ for every T 2 S.x/ is open and dense in . Given i; j D 1; : : : ; N with Ti \ Tj ¤ ∅, let ˚ Tij1 D x ˚ Tij2 D x ˚ Tij3 D x ˚ Tij4 D x

2 Tj W V u .x; Ti / \ Tj ¤ ∅; V s .x; Ti / \ Tj ¤ ∅ ; 2 Tj W V u .x; Ti / \ Tj ¤ ∅; V s .x; Ti / \ Tj D ∅ ; 2 Tj W V u .x; Ti / \ Tj D ∅; V s .x; Ti / \ Tj ¤ ∅ ; 2 Tj W V u .x; Ti / \ Tj D ∅; V s .x; Ti / \ Tj D ∅ :

We note that all these sets are open in . Moreover, if x; y 2 Ti , then V s Œx; y; Ti D V s .x; Ti / and V u Œx; y; Ti D V u .y; Ti /:

224


Therefore, each set B D int Tijk has the property that Œx; y 2 B whenever x; y 2 B. Furthermore, each point x 2 Ti \ A is in int Tijk for some j and k. For each x 2 A, we consider the open set R.x/ D

\˚

int Tijk W Ti \ Tj ¤ ∅ and x 2 Tijk :

One can easily verify that Œy; z 2 R.x/

whenever y; z 2 R.x/:

T Now let y 2 R.x/ \ A. Since R.x/ D x2Ti Ti and R.x/ \ Ti D ∅ for Ti 62 T .x/, we have T .y/ D T .x/. Moreover, given Ti 2 T .x/ D T .y/ with Ti \ Tj ¤ ∅, the points x and y are in the same set Tijk since Tijk R.x/. This shows that R.y/ D R.x/. Furthermore, if R.x/ \ R.y/ ¤ ∅, for some x; y 2 A, then there exists z 2 R.x/ \ R.y/ \ A, and we obtain R.x/ D R.y/ D R.z/. Now we consider the family ˚ R D R.x/ W x 2 A : This will be our Markov partition. For each x; y 2 A, we have R.x/ D R.y/ This implies that

or R.x/ \ R.y/ D ∅:

R.x/ n R.x/ \ A D ∅:

Since A is dense in , the set R.x/ n R.x/ has empty interior, and hence, R.x/ D int R.x/. This shows that each closure R.x/ is a rectangle (see Definition 7.6). We show that R is a Markov partition. Given R.x/ ¤ R.y/ with x; y 2 A, we have int R.x/ \ int R.y/ D R.x/ \ R.y/ D ∅: It remains to verify the Markov property. We start with an auxiliary statement. Lemma 7.8. For each x; y 2 A \ f 1 .A/ with R.x/ D R.y/ and y 2 V s .x/, we have R.f .x// D R.f .y//. Proof of the lemma. We first show that T .f .x// D T .f .y//. Otherwise, if f .x/ 2 Ti and f .y/ 62 Ti , let f .x/ D p .y/ , with y0 D x` and y1 D xi . Then x D py 2 T` , and it follows from (7.29) that f .y/ 2 f V s .x; T` / V s .f .x/; Ti /; which contradicts the assumption that f .y/ 62 Ti . Now we show that R.f .x// D R.f .y//. Let f .x/; f .y/ 2 Ti and assume that Ti \ Tj ¤ ∅. Since f .y/ 2 V s .f .x//, we have


225

V s .f .y/; Ti / D V s .f .x/; Ti /: In order to proceed by contradiction, we assume that V u .f .y/; Ti / \ Tj D ∅

and V u .f .x/; Ti / \ Tj ¤ ∅;

(7.32)

and we take a point f .z/ in the second intersection. Let again f .x/ D py with y0 D x` and y1 D xi . It follows from (7.30) that f .z/ 2 V u .f .x/; Ti / f .V u .x; T` //; and hence, z 2 V u .x; T` /. Let also f .z/ D p .y 0 / with y00 D xm and y10 D xj . Then z 2 Tm and f .V s .z; Tm // V s .f .z/; Tj /: We thus have T` 2 T .x/ D T .y/ and z 2 Tm \ T` ¤ ∅. Since z 2 V u .x; T` / \ Tm , k there exists w 2 V u .y; T` / \ Tm (note that x and y are in the same set T`m ). We thus obtain Œz; y D Œz; w 2 V s .z; Tm / \ V u .y; T` /; and f .Œz; y/ D Œf .z/; f .y/ 2 V s .f .z/; Tj / \ V u .f .y/; Ti /; which contradicts (7.32). This shows that R.f .x// D R.f .y//.

t u

Now we consider the sets C D

(

[

Vıs .z/

Wz2

N [

) s

@ Ti

i D1

and DD

[

( Vıu .z/

Wz2

N [

) u

@ Ti ;

i D1

for some ı > 0 sufficiently small. Repeating former arguments, one can show that C and D are closed and have empty interior. Then the set n .C [ D/ A is open and dense. If x 62 .C [ D/ \ f 1 .C [ D/; then x 2 A \ f 1 .A/, and the set ˚ y 2 V s .x; R.x// W y 2 A \ f 1 .A/ is open and dense in V s .x; R.x// (with respect to the induced topology on the set V s .x/ \ ). By Lemma 7.8, for each such y, we have R.f .y// D R.f .x//, and thus, by continuity, f .V s .x; R.x/// R.f .x//:

226


Since f .V s .x; R.x/// V s .f .x//; we thus obtain f .V s .x; R.x/// V s .f .x/; R.f .x///:

(7.33)

If int Ri \ f 1 .int Rj / ¤ ∅ for some Ri ; Rj 2 R, then there exist x and y as above such that Ri D R.x/ and Rj D R.f .x//. For each y 2 Ri \ f 1 .Rj /, we have ˚ V s .y; Ri / D Œy; z W V s .x; Ri / ; and hence, by (7.33), ˚ f .V s .y; Ri // D Œf .y/; f .z/ W z 2 V s .x; Ri / ˚ Œf .y/; w W w 2 V s .f .x/; Rj / V s .f .y/; Rj /: The second inclusion in the Markov property can be obtained in a similar manner. t u In a similar manner to that for a hyperbolic automorphism of T2 , to each Markov partition of a locally maximal hyperbolic set, we can associate a coding map. Namely, given a Markov partition R1 ; : : : ; Rk of a locally maximal hyperbolic set for a diffeomorphism f , we define a k k matrix A D .aij / by ( aij D

1

if int f .Ri / \ int Rj ¤ ∅;

0

if int f .Ri / \ int Rj D ∅:

Using the Markov property and the expansion and contraction along the stable and unstable manifolds, one can define a coding map hW XA ! by h.!/ D

\

f n .Rin .!/ /:

n2Z

Theorem 7.7. For any Markov partition of a locally maximal hyperbolic set and its associated coding map h, the following properties hold: 1. h is continuous and onto. 2. h ı D f ı h in XA . 3. h is injective in the set n

k [[ n2Z i D1

where

f n .@Ri / D n

k [[ n2Z i D1

f n .@s Ri [ @u Ri /;


227

˚ @s Ri D x 2 Ri W x 62 int V u .x; Ri / and

˚ @u Ri D x 2 Ri W x 62 int V s .x; Ri / :

4. card h1 x k 2 for every x 2 . Proof. Properties 1 and 2 can be obtained with similar arguments to those in the proof of Proposition 6.5. Property 3 is a simple consequence of the definitions (see also the proof of Lemma 7.7). It remains to establish Property 4. We assume that card h1 x > k 2 for some x 2 , and we proceed by contradiction. Take n 2 N such that 0 .in in / ¤ .in in0 /

(7.34)

for some sequences 0 ! D . i1 i0 i1 /; ! 0 D . i1 i00 i10 / 2 card h1 x: 0 Since the points .in ; in / take at most k 2 values, we can also assume that in D in 0 0 and in D in . Since h.!/ D h.! /, we have

Rik \ Rik0 ¤ ∅

for k D n; : : : ; n:

(7.35)

Now we take points x; y 2 such that f k .x/ 2 int Rik

and f k .y/ 2 int Rik0

for k D n; : : : ; n, which exist since n \

f k .int Rik /

and

kDn

n \

f k .int Rik0 /

kDn

are nonempty open sets (with respect to the induced topology on ). Provided that ı is sufficiently small, it follows from (7.35) that Œf k .x/; f k .y/ is well defined, and thus, f k .Œx; y/ D Œf k .x/; f k .y/ for k D n; : : : ; n: 0 and in D in0 , we have On the other hand, since in D in 0 Œf n .x/; f n .y/ 2 int Rin D int Rin

and Œf n .x/; f n .y/ 2 int Rin D int Rin0 : Therefore, since Œf n .x/; f n .y/ 2 V s .f n .x/; Rin /

and Œf n .x/; f n .y/ 2 V u .f n .y/; Rin0 /;

228


it follows from the Markov property that Œf k .x/; f k .y/ D f kCn .Œf n .x/; f n .y// 2 int V s .f k .x/; Rik / int Rik and Œf k .x/; f k .y/ D f kn .Œf n .x/; f n .y// 2 int V u .f k .x/; Rik0 / int Rik0 for k D n; : : : ; n. This shows that Rik D Rik0 for each k. On the other hand, by (7.34), there is k such that int Rik \ int Rik0 D ∅. This contradiction shows that card h1 x k 2 . t u Example 7.3. Let T W Tn ! Tn be a hyperbolic automorphism induced by a matrix B. By Property 4 in Theorem 7.7, if A is the transition matrix associated to a Markov partition, then ˚ ˚ card x 2 Tn W T m .x/ D x card ! 2 XA W m .!/ D ! ˚ k 2 card x 2 Tn W T m .x/ D x for each m 2 N. Therefore, by (3.35), p.T / D p. jXA / D log .A/: It follows from Exercise 6.13 that h.T / D h .T / D p.T / D log .A/ D

X

logjj;

2Sp.B/Wjj>1

where is the Lebesgue measure in Tn .

7.7 Exercises Exercise 7.1. Show that any local stable manifold and any local unstable manifold contain at most one periodic point. Exercise 7.2. Let T be a hyperbolic automorphism of T2 induced by a matrix A. 1. Show that the eigenvalues of A are real if and only if jtr Aj 2. In particular, if all entries of A are positive, then both eigenvalues are real. 2. Show that the stable and unstable manifolds are dense in T2 . Exercise 7.3. Find all invariant curves of the linear transformation A in (7.1) with p D q D 1. Hint: show that the graph

7.7 Exercises

229

Fig. 7.5 Invariant curves of a linear transformation

C D f.x; '.x// W x 2 Rg of a function 'W R ! R is A-invariant if and only if '.x/ D '.x/

for every x 2 R

and consider the functions '.x/ D cx log = log for c 2 R (see Fig. 7.5). Exercise 7.4. Let be a hyperbolic set for a diffeomorphism f W M ! M . Show that there exists " > 0 such that if x 2 and y 2 M are distinct, then d.f n .x/; f n .y// > " for some n 2 Z: Exercise 7.5. Show that the Smale horseshoe is a locally maximal hyperbolic set. Exercise 7.6. Construct a Markov partition for the Smale horseshoe. Exercise 7.7. Let be a hyperbolic set with product structure Œ; such that if two points x; y 2 are sufficiently close, then Œx; y 2 . Show that is locally maximal. Exercise 7.8. Show that the solenoid in Exercise 6.12 is locally maximal. Exercise 7.9. Construct a Markov partition for the solenoid. Exercise 7.10. Let be a locally maximal hyperbolic set for a diffeomorphism f . Show that for each number ˇ > 0, there exists ˛ > 0 such that if x 2 satisfies d.f n .x/; x/ < ˛ for some n 2 N, then there exists y 2 with f n .y/ D y and d f k .y/; f k .x/ ˇ

for every k 2 Œ0; n:

230


T

Fig. 7.6 The transformation T in Example 7.11

Exercise 7.11. Consider the transformation T W Œ0; 1/2 ! Œ0; 1/2 defined by ( .2x; y=2/ if 0 x < 1=2; T .x; y/ D .2x 1; y=2 C 1=2/ if 1=2 x < 1 (see Fig. 7.6). 1. Show that T preserves the Lebesgue measure in Œ0; 1/2 . 2. Use Hopf’s argument to show that is ergodic. 3. Say if the pair .T; / is equivalent to a shift map with a one-sided or a two-sided Bernoulli measure. 4. Compute the entropies h .T / and h.T /. Exercise 7.12. Let f W Tm ! Tm be a C 1 diffeomorphism such that Tm is a hyperbolic set for f . Let also be an f -invariant probability measure in Tm absolutely continuous with respect to the Lebesgue measure, with continuous and positive Radon–Nikodym derivative . 1. Show that

.f n .y// D1 n!1 .f n .x// lim

for every x 2 Tm and y 2 V s .x/. Hint: note that .f n .y// .f n .x// C j.f n .y// .f n .x//j: 2. Show that if is Hölder continuous, then there exists > 0 such that ˇ ˇ ˇ .f n .y// ˇ 1 ˇ 1ˇˇ < lim sup log ˇ n .f .x// n!1 n for every x 2 Tm and y 2 V s .x/. 3. Show that 1 Y det df k .y/ f det dy f n .y/ D D lim n!1 det dx f n .x/ det df k .x/ f kD0

for Lebesgue-almost every .x; y/ 2 Tm Tm such that y 2 V s .x/. Hint: use Exercise 2.21.

7.7 Exercises

231

Notes We refer to Anosov [4] for a historical account of Theorem 7.1. Theorem 7.2 is due to Hirsch and Pugh [40]. For further developments of Hopf’s argument, we refer to Anosov and Sinai [5] for Anosov systems (see also [44]) and to Pesin [72] for nonuniformly hyperbolic systems (see also [10, 11]). Our proof of the shadowing property (Theorem 7.4) is based on [20]. Theorem 7.5 is due to Anosov [4] in the case of Anosov diffeomorphisms and to Bowen [19] in the general case. The construction of Markov partitions is due to Sinai [94, 95] in the case of Anosov diffeomorphisms and to Bowen [18] in the general case (using the shadowing property). Our proof of the existence of Markov partitions for a locally maximal hyperbolic set (Theorem 7.6) is based on [20]. The argument for the proof of the last statement in Theorem 7.7 is taken from [89].

•

Part IV

Dimension Theory

•

Chapter 8


This chapter is a self-contained introduction to the basic notions and results of dimension theory. The presentation is oriented primarily towards the applications to dimension theory of hyperbolic dynamics in the following chapter. We first introduce the notions of Hausdorff dimension and of lower and upper box dimensions, both for sets and measures. We then consider the notions of lower and upper pointwise dimensions, and show how they can be used to estimate and sometimes compute the dimension of a measure. All the notions and results are illustrated with various examples, including model examples of repellers and hyperbolic sets.

8.1 Introduction The notion of Hausdorff dimension, introduced by Hausdorff in 1918, attributes a possibly noninteger dimension to a set that may not be an Euclidean space or even a manifold. In particular, it plays an important role in geometric measure theory and in the theory of dynamical systems. Certainly, being a single number, it fails to give information about any nonuniform mass distribution on the set. On the other hand, it is still able to distinguish many sets with a complicated structure. A practical drawback of the Hausdorff dimension, which is defined in terms of Hausdorff measures, is that it is often difficult to compute. This partly caused the introduction of several other characteristics of dimension type that sometimes are easier to compute. This is the case, for example, of the lower and upper box dimensions of a set. On the other hand, these various notions of dimension need not coincide, and a considerable part of the theory is dedicated to describe criteria either for their coincidence or for their noncoincidence. In this chapter, mostly we shall be concentrated in obtaining general relations between the Hausdorff dimension and the lower and upper box dimensions, in view of the applications to dimension theory of dynamical systems. In particular, we shall avoid discussing specific classes of sets that have no dynamical nature.


235

236


8.2 Dimension of Sets We introduce in this section the Hausdorff dimension and the lower and upper box dimensions of a given set. We also give several examples that illustrate the computation of the dimensions. Sometimes this is a daunting task, which may also depend on the particular class of sets under consideration.

8.2.1 Basic Notions Let X be a subset of a smooth manifold, and let d be the distance in X . We define the diameter of a collection U of subsets of X by ˚ diam U D sup diam U W U 2 U ; where

˚ diam U D sup d.x; y/ W x; y 2 U

is the diameter of the set U . Given Z X and ˛ 2 R, we define the ˛-dimensional Hausdorff measure of Z by m.Z; ˛/ D lim inf "!0 U

X

.diam U /˛ ;

(8.1)

U 2U

where the infimum is taken over all finite or countable covers U of the set Z with diam U ". One can easily verify that the function ˛ 7! m.Z; ˛/ jumps from C1 to 0 at a single point, and thus, the notion of Hausdorff dimension can be introduced as follows: Definition 8.1. The Hausdorff dimension of a set Z X is defined by ˚ dimH Z D inf ˛ 2 R W m.Z; ˛/ D 0 ˚ D sup ˛ 2 R W m.Z; ˛/ D C1 : The lower and upper box dimensions of Z X are defined, respectively, by dimB Z D lim inf "!0

log N.Z; "/ log "

and dimB Z D lim sup "!0

log N.Z; "/ ; log "

where N.Z; "/ denotes the least number of balls of radius " that are needed to cover the set Z. Proposition 8.1. We have dimH Z dimB Z dimB Z:

(8.2)

8.2 Dimension of Sets

237

Proof. The second inequality is clear. For the first, we note that by considering only the covers V of the set Z by balls of radius " we obtain m.Z; ˛/ lim inf "!0 V

.diam U /˛

V 2V

D lim inf "!0 V

X X

.2"/˛

V 2V

D lim .2"/˛ N.Z; "/ : "!0

Now take ˛ < dimH Z. Then m.Z; ˛/ D C1, and in particular, .2"/˛ N.Z; "/ < 1 for any sufficiently small " > 0. Taking logarithms, we obtain dimB Z D lim inf "!0

lim inf "!0

log N.Z; "/ log " ˛ log.2"/ D ˛: log "

Letting ˛ ! dimH Z thus yields dimB Z dimH Z.

t u

8.2.2 Examples The following are examples of the computation of the Hausdorff dimension and the lower and upper box dimensions. Sometimes this may be an overwhelming task, particularly since it is difficult to use the same method for different classes of sets. Example 8.1. We consider the set J Œ0; 1 composed of the points with a base-3 representation without the digit 1. One can easily verify that J is a closed set with empty interior and without isolated points. We shall compute the box dimensions of J . Take n 2 N. Given " 2 .3.nC1/ ; 3n , we have 2nC1 > N.J; "/ 2n : Therefore,

that is,

log N.J; "/ log.2nC1 / log.2n / < < ; log.3.nC1/ / log " log.3n / n log 2 log N.J; "/ n C 1 log 2 < < ; n C 1 log 3 log " n log 3

238


for each n 2 N. Letting " ! 0 and thus n ! 1, we obtain dimB J D dimB J D

log 2 : log 3

(8.3)

We show in Example 8.3 that dimH J D log 2= log 3. Example 8.2. Let R2 be the Smale horseshoe (see Sect. 6.3.1 for the definition). Proceeding in a similar manner to that in Example 8.1, for each n 2 N and " 2 .3.nC1/ ; 3n , we have 4nC1 > N.; "/ 4n : Therefore,

log 4 log N.; "/ n C 1 log 4 n < < n C 1 log 3 log " n log 3

for each n 2 N, and hence, dimB D dimB D

log 4 : log 3

We show in Example 8.8 that dimH D log 4= log 3. Now we consider a more elaborate example. We construct a subset of Œ0; 1 as follows. Given numbers a1 ; a2 2 Œ0; 1/ and 1 ; 2 2 .0; 1/, we consider the functions fi .x/ D i x C ai for i D 1; 2: (8.4) We always assume that fi .Œ0; 1/ Œ0; 1 for i D 1; 2 and f1 .Œ0; 1/ \ f2 .Œ0; 1/ D ∅:

(8.5)

Up to reordering the functions f1 and f2 , this is equivalent to assume that 1 C a1 < a2

and 2 C a2 1:

We also consider the compact set J D

1 [ \

i1 in ;

nD1 i1 in

where the union is taken over all i1 ; : : : ; in 2 f1; 2g and where

(8.6)

8.2 Dimension of Sets

239

i1 in D .fi1 ı ı fin /.Œ0; 1/: We notice that i1 in is a closed interval of length i1 in and that i1 in \ j1 jn D ∅

whenever .i1 in / ¤ .j1 jn /:

One can also show that

[

J D

(8.7)

h.!/;

!2˙2C

where h.!/ D lim .fi1 ı ı fin /.p/ n!1

(8.8)

for any p 2 Œ0; 1 (i.e., the limit is independent of p). Proposition 8.2. We have dimH J D dimB J D dimB J D s;

(8.9)

where s 2 .0; 1/ is the unique root of the equation s1 C s2 D 1:

(8.10)

Proof. To verify that s is well defined, it is sufficient to observe that the function '.s/ D s1 C s2 is strictly decreasing and that '.0/ D 2 and '.1/ D 1 C 2 < 1, in view of (8.5). In order to show that s is a lower bound for dimH J , we define a probability measure in J by requiring that .i1 in / D .i1 in /s for every n 2 N and i1 ; : : : ; in 2 f1; 2g. Since 2 [

i1 in D i1 in1

in D1

and [ 2 2 X i1 in D .i1 in / in D1

in D1

(8.11)

240


D

2 X

.i1 in /s

in D1

D .i1 in1 /

s

2 X in D1

sin

D .i1 in1 /; the measure is well defined. Moreover, by (8.7) and (8.10), we have .J / D

X

.i1 in /

i1 in

D

X

.i1 in /s

i1 in

D .s1 C s2 /n D 1: We also construct a special cover of the set J . Given a sequence ! D .i1 i2 / 2 ˙2C and r 2 .0; 1/, let n D n.!; r/ be the unique integer such that i1 in < r i1 in1 :

(8.12)

Q We denote by .!; r/ the set i1 in with n D n.!; r/. Let also .!; r/ be the largest interval containing h.!/ (see (8.8)) such that: Q Q r/. 1. .!; r/ D .! 0 ; r/ for some ! 0 2 ˙2C with h.! 0 / 2 .!; 0 0 Q Q 2. .! ; r/ .!; r/ whenever h.! / 2 .!; r/. Q By construction, for a fixed r, the intervals .!; r/ are pairwise disjoint. Moreover, they form a cover of the set J . On the other hand, it follows from (8.12) that Q r=c diam .!; r/ < r;

(8.13)

where c D 1= minf1 ; 2 g. This implies that for any interval I of length r, there is Q at most a number c of sets .!; r/ that intersect I , say D1 ; : : : ; Dk with k c. By (8.11) and (8.12), we have .Di / < r s

for i D 1; : : : ; k:

Therefore, .I /

k X i D1

.Di /
0, where cn is a constant depending only on n. Then d .x/ D d .x/ D n for every x 2 Rn : Example 8.5. Let J Œ0; 1 be the set in Example 8.1. We construct a probability measure in J as follows. For each interval Ii1 in D 0:i1 in ; 0:i1 in 22 \ J; with i1 ; : : : ; in 2 f0; 2g, we set .Ii1 in / D 2n . Since Ii1 in D Ii1 in 0 [ Ii1 in 2

and Ii1 in 0 \ Ii1 in 2 D ∅;

the measure is well defined. Now take x 2 J , n 2 N and r 2 .3.nC1/ ; 3n . Then B.x; r/ B.x; 3n /, and since the ball B.x; 3n / intersects at most one set Ii1 in , we have .B.x; r// 2n :

(8.18)

One the other hand, we have B.x; r/ B.x; 3.nC1/ /, and one can easily verify that B.x; 3.nC1/ / contains some set Ij1 jnC2 . Therefore, .B.x; r// 2.nC2/: It follows from (8.18) and (8.19) that log 2 log .B.x; r// n C 2 log 2 n < < : n C 1 log 3 log r n log 3

(8.19)

244


Finally, letting r ! 0 and thus n ! 1, we obtain d .x/ D d .x/ D

log 2 log 3

for every x 2 J:

(8.20)

8.3.2 Dimension Estimates via Pointwise Dimension The following result relates the Hausdorff dimension with the lower pointwise dimension: Theorem 8.1. The following properties hold: 1. If d .x/ ˛ for -almost every x 2 X , then dimH ˛. 2. If d .x/ ˛ for every x 2 Z X , then dimH Z ˛. 3. We have ˚ dimH D ess sup d .x/ W x 2 X : Proof. Set

˚ Y D x 2 X W d .x/ ˛ :

Given " > 0, for each x 2 Y , there exists r.x/ > 0 such that .B.x; r// .2r/˛"

(8.21)

for every r 2 .0; r.x//. Given > 0, set ˚ Y D x 2 Y W r.x/ : Clearly,

Y1 Y2

for 1 2 ;

and Y D

[

Y :

>0

Since .X nY / D 0, there exists > 0 such that .Y / .X /=2. Now let Z Y be a set of full -measure and let U be a cover of Z \ Y by open balls. Without loss of generality, we assume that U \ Y ¤ ∅ for every U 2 U. Then for each U 2 U, there exists xU 2 U \ Y , and we consider the new cover ˚ V D B.xU ; diam U / W U 2 U of the set Z \ Y . It follows from (8.21) that X X .diam U /˛" D 2"˛ .diam V /˛" U 2U

V 2V

2"˛

X

.V /

V 2U

2"˛ .Y / 2"˛ .X /=2:

8.3 Dimension of Measures

245

Since U is arbitrary, we obtain m.Z \ Y ; ˛ "/ 2"˛ .X /=2: This implies that dimH .Z \ Y / ˛ ", and by the arbitrariness of ", we conclude that dimH Z dimH .Z \ Y / ˛: Now we establish the second property. For each x 2 Z and " > 0, there exists a sequence rn D rn .x; "/ & 0 when n ! 1 such that .B.x; rn // .2rn /˛C"

(8.22)

for every n 2 N. Consider a cover ˚ U B.x; rn .x; "// W x 2 Z and n 2 N of the set Z. We note that the diameter of U can be made arbitrarily small. To proceed with the proof, we need the following result: Lemma 8.1 (Besicovitch’s covering lemma). Given a set Z Rm and a bounded function rW Z ! RC , the cover fB.x; r.x// W x 2 Zg of Z contains a countable subcover V of finite multiplicity, that is, there exists a constant K > 0 such that cardfV 2 V W x 2 V g K

for every

x 2 Z:

By Lemma 8.1 and Whitney’s embedding theorem, there exists a subcover V U of Z and a constant K > 0 such that cardfV 2 V W x 2 V g K for every x 2 Z. Therefore, by (8.22), X

.diam V /˛C"

V 2V

X

.B.x; r// K.X /;

V 2V

and since the diameter of the cover V can be made arbitrarily small, we conclude that m.Z; ˛ C "/ K.X /: This implies that dimH Z ˛ C ", and since " is arbitrary, we obtain dimH Z ˛. Finally, for the third property, let ˚ ˚ ˛ D ess sup d .x/ W x 2 X and Z D x 2 X W d .x/ ˛ :

246


We have .Z/ D .X /, and by the second property, dimH dimH Z ˛: Now given " > 0, let ˚ Z" D x 2 X W d .x/ ˛ " : We have .Z" / > 0, and it follows from the first property that dimH dimH .jZ" / ˛ ": Since " is arbitrary, we obtain dimH ˛.

t u

We give several applications of Theorem 8.1. Example 8.6. Let J be the set in (8.6). For the measure in J defined by (8.11), we have the estimate in (8.14) for any set U intersecting J . In particular, .B.x; r// c.2r/s ; and hence, d .x/ lim inf r!0

logŒc.2r/s Ds log r

for every x 2 J . Therefore, it follows from Theorem 8.1 that dimH J dimH s: This shows that the argument used in the proof of Proposition 8.2 to obtain the lower bound for the Hausdorff dimension can be reformulated in terms of the lower pointwise dimension. Example 8.7. Let J and be the set and the measure in Example 8.5. By (8.20), we have d .x/ D log 2= log 3 for every x 2 J: Thus, it follows from Theorem 8.1 that dimH J D dimH D

log 2 : log 3

(8.23)

This yields another proof that the set J in Example 8.1 has Hausdorff dimension log 2= log 3. Example 8.8. Let R2 be the Smale horseshoe. We have already shown in Example 8.2 that the lower and upper box dimensions of are equal to log 4= log 3. Now we compute dimH . In a similar manner to that in Example 8.5, we first

8.3 Dimension of Measures

247

construct a measure in . Set .Qn .!// D 4n

for each ! 2 X2 ; n 2 N;

with Qn .!/ as in (6.15). Proceeding also in a similar manner to that in Example 8.5, one can show that d .x/ D d .x/ D

log 4 log 3

for every x 2 :

Thus, it follows from Theorems 8.1 and 8.2 together with Example 8.2 that dimH D dimB D dimB D

log 4 log 3

dimH D dimB D dimB D

log 4 : log 3

and

We also give a criterion for the coincidence of the Hausdorff dimension and the lower and upper box dimensions of a measure. Theorem 8.2 (Young). If is a finite measure in X and there exists d 0 such that log .B.x; r// lim Dd (8.24) r!0 log r for -almost every x 2 X , then dimH D dimB D dimB D d: Proof. By Theorem 8.1, we have dimH d . Now we show that dimB d and thus the result follows from (8.17). Set ˚ Z D x 2 X W d .x/ d : Given " > 0, for each x 2 Z, there exists r.x/ > 0 such that if r 2 .0; r.x//, then .B.x; r// .2r/d C" : Given > 0, we consider the set ˚ Y D x 2 Z W r.x/ : Clearly, Y1 Y2

for 1 2 ;

and Z D

[ >0

Y :

248


Therefore, since .X n Z/ D 0, we have .Y / % .X / when ! 0. For each r < , the balls B.x; r/ form a cover U of the set Y . By Lemma 8.1, there exists a subcover V U of Y of finite multiplicity K. Therefore, X

.diam V /d C"

V 2V

Since

X

X

.B.x; r// K.X /:

V 2V

.diam V /d C" D .2r/d C" card V .2r/d C" N.Y ; r/;

V 2V

we obtain N.Y ; r/

K.X / ; .2r/d C"

and hence, dimB Y D lim sup r!0

log N.Y ; r/ d C ": log r

Since .Y / % .X / when ! 0, we conclude that dimB lim sup dimB Y d C ": !0

Finally, since " is arbitrary, we obtain dimB d . This completes the proof of the theorem. t u Definition 8.4. The limit in (8.24), when it exists, is called the pointwise dimension of the measure at x, and we denote it by d .x/. The following is an application of Theorem 8.2: Example 8.9. Let again J and be as in Example 8.5. By (8.20), we have d .x/ D log 2= log 3 for every x 2 J: Thus, it follows from Theorem 8.2 together with (8.3) and (8.23) that dimH J D dimB J D dimB J D

log 2 ; log 3

dimH D dimB D dimB D

log 2 : log 3

and

8.4 Exercises

249

8.4 Exercises Exercise 8.1. Show that dimH Z1 dimH Z2 and dimH

[

whenever Z1 Z2

Zi D sup dimH Zi : i 2N

i 2N

Exercise 8.2. Show that: 1. If Z1 Z2 , then dimB Z1 dimB Z2 2. dimB

[

and dimB Z1 dimB Z2 :

Zi sup dimB Zi

and dimB

i 2N

i 2N

[

Zi sup dimB Zi :

i 2N

i 2N

Exercise 8.3. Show that for any set Z, dimB Z D dimB Z

and dimB Z D dimB Z:

Exercise 8.4. Show that dimB

n [

Zi D max dimB Zi : i D1;:::;n

i D1

Exercise 8.5. Verify that if dimB Z D dimB Z D s, then s D lim

n!1

1 log N.Z; e n /: n

Exercise 8.6. Compute the Hausdorff dimension and the lower and upper box dimensions of the set Q \ Œ0; 1. Exercise 8.7. Show that the inequalities dimH X C dimH Y dimH .X Y / and dimB .X Y / dimB X C dimB Y hold for any subsets X and Y of Rm .

250


Exercise 8.8. Let f W X ! Y be a Lipschitz map. Show that dimH f .A/ dimH A for any set A X . Exercise 8.9. Let f W X ! Y be a diffeomorphism. Show that dimH f .A/ D dimH A for any set A X . Exercise 8.10. Show that d .x/ D lim inf

log .B.x; ae n // n

d .x/ D lim sup

log .B.x; ae n // n

n!1

and n!1

for each a > 0 and x 2 X . Hint: for each r > 0, take n D n.r/ 2 N such that ae .nC1/ r < ae n < 1: Exercise 8.11. Show that if T W X ! X is a diffeomorphism and is a T -invariant probability measure in X , then d .f .x// D d .x/

and d .f .x// D d .x/

for every x 2 X . Exercise 8.12. Show that if there exist a finite Borel measure in a set Z and constants c; d > 0 such that .B.x; r// cr d for every x 2 Z and all sufficiently small r > 0, then dimH Z d . Exercise 8.13. Given numbers 1 ; 2 2 .0; 1/ with 1 C 2 < 1, for each n 2 N and i1 ; : : : ; in 2 f1; 2g, let i1 in Œ0; 1 be a closed interval of diameter diam i1 in D i1 in

(8.25)

satisfying condition (8.7). Show that (8.9) holds for the set J in (8.6), with the same s. This shows that the location of the sets i1 in plays no role in the statement of Proposition 8.2. P Exercise 8.14. Given numbers 1 ; : : : ; k 2 .0; 1/ with kiD1 i < 1, for each n 2 N and i1 ; : : : ; in 2 f1; : : : ; kg, let i1 in Œ0; 1 be a closed interval of diameter as in (8.25) satisfying condition (8.7). Show P that (8.9) holds for the set J in (8.6), where s is the unique root of the equation kiD1 si D 1. Exercise 8.15. Let f be a C 1 diffeomorphism in some open neighborhood of the square Œ0; 12 and let Hi and Vi for i D 1; 2 be, respectively, horizontal and vertical strips such that:

8.4 Exercises

251

1. f .Hi / D Vi for i D 1; 2. 2. There exist i ; i 2 .0; 1/ and .ai ; bi / 2 Œ0; 1/ with .f jHi /.x; y/ D .i x C ai ; i y C bi /

for i D 1; 2:

Show that: T 1. D n2Z f n .H1 [ H2 / is a hyperbolic set for f . 2. dimH D dimB D dimB D ˛ C ˇ; where ˛ and ˇ are, respectively, the unique roots of the equations ˛1 C ˛2 D 1

ˇ

ˇ

and 1 C 2 D 1:

Hint: use Exercise 8.7. Exercise 8.16. Use Exercise 8.15 to show that the Hausdorff dimension of the Smale horseshoe is log 4= log 3. Exercise 8.17. Compute the Hausdorff dimension and the lower and upper box dimensions of the solenoid in Exercise 6.12.

Notes We refer to the books [29, 33, 58, 73] for further topics of dimension theory. See, for example, [58] for Besicovitch’s covering lemma (Lemma 8.1). Theorem 8.2 is due to Young [105].

•

Chapter 9

Dimension Theory of Hyperbolic Dynamics

We study in this chapter the dimension of hyperbolic invariant sets of conformal transformations, both invertible and noninvertible. This means that the derivative of the map along the stable and unstable directions is a multiple of an isometry at every point. More precisely, we compute the Hausdorff dimension and the lower and upper box dimensions of repellers and hyperbolic sets for a conformal dynamics. The dimension of the invariant sets is expressed, as explicitly as possible, in terms of the topological pressure. It turns out that Markov partitions are a principal element of the proofs. In particular, they allow us to reduce effectively some of the arguments and computations to the special case of symbolic dynamics.

9.1 Introduction The dimension theory of dynamical systems progressively developed during the last 2 decades into an independent and quite active field of research. The theory concerns the study of dynamical systems from the point of view of dimension. In other words, one does not consider topics of dimension theory that are not of dynamical nature, of course, independently of their importance. Roughly speaking, the main objective is to measure the complexity from the dimensional point of view of the objects that remain invariant under the dynamics, such as invariant sets and invariant measures. The thermodynamic formalism has a privileged relation to the dimension theory of dynamical systems. This is due to the fact that the unique solution s of the equation P .s'/ D 0; (9.1) where ' is a certain function associated to a given invariant set, is often related to the Hausdorff dimension of the set. This equation was introduced by Bowen and is usually called Bowen’s equation. It is also appropriate to call it the Bowen–Ruelle equation, taking into account not only the fundamental role of the thermodynamic formalism developed by Ruelle but also his study of the Hausdorff dimension of L. Barreira, Ergodic Theory, Hyperbolic Dynamics and Dimension Theory, Universitext, DOI 10.1007/978-3-642-28090-0 9, © Springer-Verlag Berlin Heidelberg 2012

253

254

9 Dimension Theory of Hyperbolic Dynamics

repellers of a conformal dynamics (this means that the derivative of the map is a multiple of an isometry at every point). For a repeller J of a conformal map f , he showed that dimH J D s; where s is the solution of (9.1) for the function 'W J ! R defined by '.x/ D log kdx f k: To a certain extent, the study of the dimension of hyperbolic sets is analogous. Indeed, assuming that the derivatives of the diffeomorphism along the stable and unstable directions are multiples of isometries, starting with the work of McCluskey and Manning, it was possible to develop a corresponding theory. However, there are nontrivial differences between the theory for repellers and for hyperbolic sets. For example, each conformal repeller J has a unique invariant measure of full Hausdorff dimension, that is, an invariant probability measure such that ˚ dimH D inf dimH Z W .J n Z/ D 0 D dimH J: On the other hand, unless some cohomology relations hold, there is no invariant measure of full dimension concentrated on a conformal hyperbolic set. Virtually all known equations used to compute or estimate the dimension of an invariant set, both for an invertible and an noninvertible dynamics, are particular cases of (9.1) or of appropriate generalizations. Nevertheless, despite many significant developments, only the case of conformal dynamics is completely understood. Some motivations for the study of dimension in the context of the theory of dynamical systems come from the study of attractors in infinite-dimensional spaces. The longtime behavior of many dynamical systems, such as those coming from partial differential equations and delay differential equations, can essentially be described in terms of a global attractor. An important question is how many degrees of freedom are necessary to specify the dynamics on the attractor. It turns out that a large class of attractors have finite Hausdorff dimension or even finite box dimension. Hence, the dynamics on the attractor is essentially finite-dimensional.

9.2 Repellers We start our study with the somewhat simpler case of repellers. After introducing the notion of conformal map, we give several examples of repellers of conformal maps. We then compute the Hausdorff dimension and the lower and upper box dimensions of any such repeller in terms of the topological pressure. More precisely, the dimension is the root of an equation involving the topological pressure.

9.2 Repellers

255

9.2.1 Conformal Maps and Examples Let f W U ! M be a C 1 map in an open subset of a smooth manifold M and let J U be a repeller for f (see Definition 6.6). Definition 9.1. We say that f is conformal on J if dx f is a multiple of an isometry for every x 2 J . We give several examples of repellers of conformal maps. Example 9.1. Consider: 1. k disjoint closed intervals 1 ; : : : ; k R. S 2. A C 1 map f W U ! R, where U is an open neighborhood of D kiD1 i . We assume that f is expanding on U and that f .@/ @ and i f .j / whenever @i \ @f .j / ¤ ∅. Then the set J D

1 \

f j C1

j D1

is a repeller for f . Moreover, since dx f is a single number for each x, the map f is conformal on J . Example 9.2 (Hyperbolic Julia sets). Let S be the Riemann sphere and let RW S ! S be a rational map of degree greater than one. Since R is holomorphic, it is conformal. We say that an n-periodic point z 2 S of R is repelling if j.Rn /0 .z/j > 1. The Julia set J of R is the closure of the set of the repelling periodic points of R. If R is expanding on J , then the Julia set is a repeller for a conformal map. Example 9.3. We construct a repeller in R as follows. Let 1 ; : : : ; k Œ0; 1 be disjoint closed intervals of lengths, respectively, 1 ; : : : ; k > 0. Let also gi W i ! Œ0; 1 be affine maps such that gi .i / D Œ0; 1 for i D 1; : : : ; k. We define a map fW

k [

i ! Œ0; 1

i D1

for i D 1; : : : ; k. We note that f is the restriction of an expanding by f ji D gi S map to the set kiD1 i , and that J D

1 \

f

n

[ k

nD0

i

(9.2)

i D1

is a repeller for that map. Indeed, jf 0 ji j D 1 i for each i . We also note that when k D 2, we recover the set J in (8.6) by taking fi D gi1 D .f ji /1

for i D 1; 2:

256


9.2.2 Dimension of Repellers For conformal maps, the following result gives a formula for the Hausdorff dimension and the lower and upper box dimensions of any repeller in terms of the topological pressure. We recall that f is said to be of class C 1C˛ , for some ˛ 2 .0; 1, if f is of class C 1 and that there exists c > 0 such that kdx f dy f k cd.x; y/˛

(9.3)

for every x; y 2 M . Theorem 9.1 (Dimension of repellers). If J is a repeller for a C 1C˛ transformation f , for some ˛ 2 .0; 1, such that f is conformal on J , then dimH J D dimB J D dimB J D s;

(9.4)

where s is the unique real number such that Pf jJ .s'/ D 0

(9.5)

for the function 'W J ! R defined by '.x/ D log kdx f k:

(9.6)

Proof. The argument is an elaboration of the proof of Proposition 8.2, although the required modifications are substantial. The uniqueness of the number s follows from the strict monotonicity of the function s 7! Pf jJ .s'/ that can be established as follows. Take m 2 N such that c m > 1, with c and as in (6.29). For each n 2 N, we write n D q m C r, where q D bn=mc and r 2 f0; : : : ; m 1g. Then n1 X kD0

'.f k .x// D

n1 X

log kdf k .x/ f k

kD0

Y

qmCr1

D log

kdf k .x/ f k

kD0

D log kdx f qmCr k log kdx f r k.c m /q log min min kdx f r k r

x2J

n m

1 log.c m /

9.2 Repellers

257

for every x 2 J and n 2 N. Therefore, given t s and n 2 N, we have s

n1 X

' ıfk t

kD0

n1 X kD0

n ' ı f k C .s t/ a log.c m / ; m

where a D log min min kdx f r k C log.c m /: r

x2J

Writing P D Pf jJ , this implies that P .s'/ P .t'/ .s t/

1 log.c m /; m

(9.7)

and thus, the function s 7! P .s'/ is strictly decreasing. It also follows from (9.7) that lim P .s'/ D 1 and lim P .t'/ D C1: t !1

s!C1

Therefore, there exists a unique real number s such that P .s'/ D 0. Now let R1 ; : : : ; Rk be the elements of a Markov partition of J (by Theorem 6.1, any repeller has Markov partitions). We also consider the associated topological Markov chain jXAC W XAC ! XAC , with the entries of the k k transition matrix A given by (6.33), and the coding map hW XAC ! J defined by (6.34). For each ! D .i1 i2 / 2 ˙AC and r 2 .0; 1/, let n D n.!; r/ 2 N be the unique integer such that kdh.!/ f n k1 < r kdh.!/ f n1 k1 (9.8) and set .!; r/ D

n1 \

f j Rij C1 :

j D0

By (9.3), for each x; y 2 .!; r/, it follows from (6.29) that n1 Y kdf j .x/ f k kdx f n k D kdy f n k kdf j .y/ f k j D0

kdf j .x/ f df j .y/ f k 1C kdf j .y/ f k j D0 n1 Y

n1 Y

˛

1 C Cd f j .x/; f j .y/

j D0

n1 Y j D0

˛ 1 C Cc˛ d f j .x/; f j .y/ ˛.j n/ ;

258


where C D 1= infz2J kdz f k. Since

d f j .x/; f j .y/ max diam Ri DW ; i D1;:::;k

there exists a constant C 0 > 0 such that n1 Y

kdx f n k 1 C C 0 ˛.j n/ n kdy f k j D0

1 Y

1 C C 0

˛l

(9.9)

DW D < 1:

lD1

Now let g be the local inverse of f n j.!; r/, where n D n.!; r/. Since .dx f n /1 D df n .x/ g and f is conformal on J , we have kdf n .x/ gk D kdx f n k1 :

(9.10)

By (9.9), we thus obtain diam .!; r/ D

d.x; y/

sup x;y2.!;r/

sup

d f n .x/; f n .y/

x;y2.!;r/

sup

n 1

kdw f k

sup

kdz gk

z2f n ..!;r//

(9.11)

w2.!;r/

Dkdx f n k1 < r for every x 2 .!; r/, provided that is sufficiently small, that is, provided that the diameter of the Markov partition is sufficiently small. The next step is the construction of a special cover of J . This is an elaboration of a corresponding construction in the proof of Proposition 8.2. We first note that h.!/ 2 .!; r/ and that if h.! 0 / 2 .!; r/ and n.! 0 ; r/ n.!; r/, then Q .! 0 ; r/ .!; r/. For each ! 2 ˙AC and r > 0, let .!; r/ be the largest set containing h.!/ such that: Q Q r/. 1. .!; r/ D .! 0 ; r/ for some ! 0 2 ˙AC with h.! 0 / 2 .!; 0 0 Q Q 2. .! ; r/ .!; r/ whenever h.! / 2 .!; r/. Q By construction, for a fixed r, the sets .!; r/ intersect at most along their Q boundaries. The cover formed by the sets .!; r/ is called a Moran cover of J . Since each set Ri is the closure of its interior, there exists > 0 such that Ri contains a ball of radius for i D 1; : : : ; k. Using again (9.10), this implies that each Q Q set .!; r/ contains a ball of radius D 1 kdx f n k1 for some point x 2 .!; r/.

9.2 Repellers

259

Moreover, by (9.8), we conclude that there exists a constant 2 .0; 1/ (independent Q of ! and r) such that each set .!; r/ contains a ball of radius r. On the other hand, it also follows from (9.8) that there exists a constant 0 > 1 such that each Q Q set .!; r/ has diameter at most 0 r. Since the sets .!; r/ intersect at most along their boundaries, there exists a constant C > 0 (independent of r) such that each ball Q B.x; r/ intersects at most a number C of the sets .!; r/. Now we establish the identities in (9.4) involving the Hausdorff dimension and the lower and upper box dimensions. Given ı; r; > 0, there exists a countable cover U of J with diam U < r such that X .diam U /dimH J Cı < : U 2U

For each U 2 U we consider the family of sets ˚ UU D .!; diam U / W .!; diam U / \ U ¤ ∅ and ! 2 ˙AC ; which is a cover of U . Then ˚ V D V W V 2 UU and U 2 U is a cover of J , and X

.diam V /dimH J C"

V 2V

X X

.diam V /dimH J C"

U 2U V 2UU

X X

.diam U /dimH J C"

(9.12)

U 2U V 2UU

X

C.diam U /dimH J C" < C :

U 2U

Now given ˛ 2 R and N 2 N, we set M.˛; N / D inf C

X

0

X

n.!;r/1

exp @˛ n.!; r/ C sup .!;r/

.!;r/2C

1 ' ı f jA;

j D0

where the infimum is taken over all finite covers C of J by sets in the family ˚ HN D .!; r/ W n.!; r/ > N : Lemma 9.1. We have o n P .'/ D inf ˛ W lim M.˛; N / D 0 : N !1

260


Proof of the lemma. Clearly, M.˛; N /

X

0 exp @˛N C sup

N 1 X

1 ' ı f jA;

(9.13)

i1 iN j D0

i1 iN

where

N 1 \

i1 iN D

f j Rij C1 :

j D0

On the other hand, given " > 0, there exists C > 0 such that X

exp sup

N 1 X

' ı f j C e .P .'/C"/N

i1 iN j D0

i1 iN

for every N 2 N. Therefore, provided that ˛ > P .'/ and " is sufficiently small, it follows from (9.13) that M.˛; N / e ˛N C e .P .'/C"/N ! 0 when N ! 1. Hence, o n p WD inf ˛ W lim M.˛; N / D 0 P .'/: N !1

Now let ˛ > p. There exist N 2 N and a finite cover C HN of J , say with elements Q 1 ; : : : ; Q q , such that N.˛; C/ WD

X

0 exp @˛ jj C sup

jj1 X

2C

1 ' ı f j A < 1;

(9.14)

j D0

using the notation ji1 iN j D N . Let I1 ; : : : ; Iq be the finite sequences such that Q i D Ii

for i D 1; : : : ; q:

(9.15)

For each n 2 N, we consider the cover Cn of J formed by the sets Ii1 Iin with i1 ; : : : ; in 2 f1; : : : ; qg, and we write i1 in D Ii1 Iin . Since j i1 in j1

sup i1 in

X

'ıfj

j D0

n X `D1

n X `D1

j i` j1

sup i` in

X

'ıfj

j D0

j i` j1

sup i`

X

j D0

' ı f j;

9.2 Repellers

261

it follows from (9.15) that N.˛; Cn /

n X Y

0 exp @˛ jj C sup

`D1 2C

jj1 X

1 ' ı f j A D N.˛; C/n :

j D0

Therefore, by (9.14), we obtain N.˛; C1 /

X

N.˛; C/n < 1;

n2N

S

where C1 D n2N Cn . Now let m be the maximal length of the finite sequences Ii . For each n 2 N and ! 2 ˙AC , there exists I 2 C1 such that h.!/ 2 I

and n jI j < n C m:

Let also C be the family of all sets K such that K is a finite sequence consisting of the first n elements of some finite sequence I such that I 2 C1 . We have N.˛; C / N.˛; C1 /e msupj'j maxf1; e ˛m g < 1: Since N.˛; C / D

X

e ˛ n exp sup

i1 in

n1 X

' ı f j;

i1 in j D0

we obtain P .'/ ˛ 0, and thus, p P .'/.

t u

The lemma allows us to show that s is a lower bound for the Hausdorff dimension of J . Indeed, it follows from (9.12) that lim M.0; N / D 0;

N !1

and by Lemma 9.1, we obtain

P .dimH J C ı/' 0: Since the function t 7! P .t'/ is strictly decreasing, we conclude that dimH J Cı s, and it follows from the arbitrariness of ı that dimH J s. Now we show that s is an upper bound for the upper box dimension of J . It follows from (9.8) that n.!; r/

log r C a; log

262


with as in (6.29), and where a D 1 log c= log (one can always assume that c 1, and hence that a 1). Let .!j ; r/ for j D 1; : : : ; N be the elements of the Moran cover. It follows from (9.11) that diam .!j ; r/ < r for each j . Therefore, X

˚ card j W n.!j ; r/ D m N.F; r/:

m2N

This implies that there exists m D m.r/ such that ˚ card j W n.!j ; r/ D m

N.J; r/ : log r= log C a

On the other hand, for each ı > 0, there exists a sequence rn & 0 such that N.J; rn / > rnıdimB J : Setting m D m.rn / and using again (9.8), we obtain k

X i1 im1

kdx f m1 k˛

sup x2i1 im1

X

kdx f m1 k˛

sup

i1 im x2i1 im1

rn˛

rnıdimB J log rn = log C a

log 2ıC˛dimB J r 2a n

for all sufficiently large n (since 1= log r r ˛ for all sufficiently small r > 0). Therefore, provided that ˛ < dimB J 2ı, we obtain X i1 im1

exp

sup x2i1 im1

˛

m2 X

! 'ıf

l

1

(9.16)

lD0

for all sufficiently large n, where again m D m.rn/. Furthermore, it also follows from (9.8) that log r ; (9.17) n.!; r/ log 0 where

0 D supfkdx f k W x 2 J g:

We note that 0 > 1. Otherwise, we would have kdx f n vk kvk for every n 2 N, x 2 J and v 2 Tx M , which contradicts (6.29). It follows from (9.17) that min n.!j ; r/ ! 1 when r ! 0; j

9.2 Repellers

263

and thus, m.rn / ! 1 when n ! 1. Since n can be taken arbitrarily large, it follows from (9.16) that P .˛'/ 0, and thus, ˛ s whenever ˛ < dimB J 2ı. This implies that dimB J 2ı s, and it follows from the arbitrariness of ı that dimB J s. t u Now we give several examples. Example 9.4. Let J be a repeller for a C 1C˛ transformation and let ' be the function in (9.6). We note that n1 X

'.f k .x// D

kD0

n1 X

log kdf k .x/ f k

kD0

D log

n1 Y

kdf k .x/ f k

kD0

D log kdx f n k: Now let R1 ; : : : ; Rk be a Markov partition of J . It follows from Exercise 6.16 that ! n1 X X 1 k exp sup s 'ıf ; Pf jJ .s'/ D lim log n!1 n Ri1 in i i 1

kD0

n

where the sum is taken over all finite sequences i1 in such that Ri1 in ¤ ∅, with Ri1 in as in (6.35). Therefore, X

1 log sup kdx f n ks : n!1 n i i Ri1 in

Pf jJ .s'/ D lim

1

(9.18)

n

Example 9.5. Let J Œ0; 1 be the set constructed in Example 8.1. We first note that J is a repeller. Indeed, let f W Œ0; 1=3 [ Œ2=3; 1 ! R be defined by ( f .x/ D

x 2 Œ0; 1=3;

3x;

3x 2; x 2 Œ2=3; 1:

This is the restriction of a C 1C˛ transformation to the set K D Œ0; 1=3 [ Œ2=3; 1: Moreover, f 0 .x/ D 3 for every x, and J D

1 \ nD0

f n K:

264


This shows that J is a repeller for some extension of f . The function 'W J ! R in (9.6) is given by '.x/ D log jf 0 .x/j D log 3: Now we observe that the sets R1 D J \ Œ0; 1=3 and R2 D J \ Œ2=3; 1 form a Markov partition of J . Thus, it follows from (9.18) that X 1 log exp.ns log 3/; n!1 n i i

Pf jJ .s'/ D lim

1

n

where i1 ; : : : ; in 2 f1; 2g. Therefore,

1 log 2n e ns log 3 D log.2 3s /: n!1 n

Pf jJ .s'/ D lim

Solving the equation Pf jJ .s'/ D 0 yields s D log 2= log 3, and thus, it follows from Theorem 9.1 that dimH J D dimB J D dimB J D

log 2 : log 3

We note that this is also a consequence of Examples 8.1 and 8.3. Example 9.6. We consider again the repeller J in (9.2). With the notation in Example 9.3, one can easily verify that the sets Ri D J \ i , for i D 1; : : : ; k, form a Markov partition of J . For the function ' in (9.6), we have 'jRi D log jf 0 ji j D log i ; and it follows from (9.18) that X n1 X 1 log exp sup s 'ıfj n!1 n Ri1 in i i j D0

Pf jJ .s'/ D lim

1

n

X n1 X 1 log exp s 'jRij C1 in n!1 n i i j D0

D lim

1

n

X n X 1 D lim log exp s log ij n!1 n i i j D1 1

n

n XY 1 log sij n!1 n i i j D1

D lim

1

n

9.3 Hyperbolic Sets

265

1 D lim log n!1 n D log

k X

X k

n sj

j D1

sj :

j D1

Thus, in this case, the equation Pf jJ .s'/ D 0 in (9.5) reduces to When k D 2, this formula was already obtained in Proposition 8.2.

Pk

s j D1 j

D 1.

9.3 Hyperbolic Sets We study in this section the dimension of the hyperbolic sets of a conformal diffeomorphism. Here the conformality means that the diffeomorphism is conformal along the stable and unstable directions. More precisely, for a locally maximal hyperbolic set, we compute the Hausdorff dimension and the lower and upper box dimensions along the stable and unstable manifolds. We then use this result to compute the dimension of the hyperbolic set.

9.3.1 Dimension Along the Invariant Manifolds Let f W M ! M be a diffeomorphism and let M be a hyperbolic set for f . Definition 9.2. We say that f is conformal on if the linear transformations dx f jE s .x/ and dx f jE u .x/ are multiples of isometries for every x 2 . We note that if M is a surface and dim E s .x/ D dim E u .x/ D 1

for every x 2 ;

then f is conformal. More explicit examples are: 1. The Smale horseshoe in Sect. 6.3.1. 2. The solenoid in Example 6.12. 3. The general horseshoes in Exercise 8.15. The following result is a version of Theorem 9.1 along the stable and unstable manifolds of a locally maximal hyperbolic set. Let 's W ! R and 'u W ! R be the functions defined by 's .x/ D log kdx f jE s .x/k

and 'u .x/ D log kdx f jE u .x/k:

By Proposition 6.2, the spaces E s .x/ and E u .x/ vary continuously with x, and thus, the functions 's and 'u are continuous (because they are compositions of

266


continuous functions). We also recall the notion of topologically mixing transformation in Definition 3.13. Theorem 9.2. Let be a locally maximal hyperbolic set for a C 1C˛ diffeomorphism, for some ˛ 2 .0; 1, such that f is conformal and topologically mixing on . Then dimH .V s .x/ \ / D dimB .V s .x/ \ / D dimB .V s .x/ \ / D ts ;

(9.19)

dimH .V u .x/ \ / D dimB .V u .x/ \ / D dimB .V u .x/ \ / D tu ;

(9.20)

and

where ts and tu are the unique real numbers such that Pf j .ts 's / D Pf j .tu 'u / D 0:

(9.21)

Proof. Let R1 ; : : : ; Rk be the elements of a Markov partition of with diameter smaller than the size of the local stable and unstable manifolds (by Theorem 7.6 the set has Markov partitions of arbitrarily small diameter). Given x; y 2 in the s same rectangle Ri , let Hx;y W V u .x/ ! V u .y/ be the holonomy map defined by s .z/ D Œz; y: Hx;y s By Proposition 7.1, the map Hx;y depends continuously on x and y. Furthermore, by results of Hasselblatt in [39], since f is conformal on , the product structure is a Lipschitz homeomorphism with Lipschitz inverse. This implies that the holonomy s map Hx;y is Lipschitz. Now we establish identities (9.19) and (9.20) for the dimensions along the stable and unstable manifolds. We only consider V u .x/ since the arguments for V s .x/ are entirely analogous. For i D 1; : : : ; k, let Vi be segments of local unstable manifolds such that Vi \ D V u .xi / \ \ Ri (9.22)

for some point xi 2 Ri . We note that Vi \ int Rj D ∅ whenever j ¤ i: Since the holonomy maps are Lipschitz, the numbers dimH .Vi \ /, dimB .Vi \ /, and dimB .Vi \ / are independent of xi . Furthermore, since f is topologically mixing on , we have dimH .Vi \ / D dimH .Vj \ /; dimB .Vi \ / D dimB .Vj \ /; dimB .Vi \ / D dimB .Vj \ /

(9.23)

9.3 Hyperbolic Sets

267

for every i and j . Set V D Ri0 in D

Sk

i D1

n \

Vi . We define

f j Rij

and Vi0 in D V \ Ri0 in

j D0

for every . i1 i0 i1 / 2 ˙A and n 2 N, where A is the transition matrix obtained from the Markov partition. We first show that tu is an upper bound for the upper box dimension of . Note that f n .Vi0 in / V u .f n .xi0 // \ Rin ; and Hfs n .xi

0 /;xin

f n .Vi0 in \ / D Vin \ :

Furthermore, each point in Vi \ has exactly one preimage under the holonomy map Hfs n .xi /;xi . Hence, if U is a cover of Vin \ , then 0

n

[ U 2U

f n Hxsin ;f n .xi / U Vi0 in \ : 0

This implies that N.Vi0 in \ ; r/ N.Vin \ ; K 1 ri0 in / for every r > 0, where K 1 is a Lipschitz constant for the holonomy map and where ˚ i0 in D max kdx f n jE u .x/k W x 2 Ri0 in : Therefore, N.V \ ; r/

X

N.Vi0 in \ ; r/

i0 in

X

N.V \ ; K 1 ri0 in /:

i0 in

Now take s > dimB .V \ /. Then there exists r0 > 0 such that N.V \ ; r/ < r s Setting cn .s/ D

for every r 2 .0; r0 /: X

s

i0 in ;

i1 in

we thus obtain

N.V \ ; r/ r s K s cn .s/

268


for every r < n Kr0 , where n D mini0 in i0 in . By induction, we conclude that N.V \ ; r/ r s K s cn .s/m for every r < .n K/m r0 . Now we note that n K < 1 for all sufficiently large n. Therefore, m log cn .s/ log N.V \ ; r/ sC log r log r sC

m log cn .s/ ; logŒ.n K/m r0

and dimB .V \ / s C lim sup m!C1

Ds

m log cn .s/ logŒ.n K/m r0

log cn .s/ : log.n K/

Letting s & dimB .V \ /, we conclude that cn .dimB .V \ // 1: On the other hand, for each s > dimB .V \ /, we have cn .s/ D

X

s

i0 in D

i0 in

X i0 in

X n k exp max s 'u .f .x// ; x2Ri0 in

kD0

and hence,

1 log cn .s/ 0 D P .tu 'u /; n where P D Pf j . Since the function s 7! P .s'u / is strictly decreasing, it follows that s tu for every s > dimB .V \ /, and thus, P .s'u / D lim

n!1

dimB .V \ / tu :

(9.24)

Now we show that tu is a lower bound for the Hausdorff dimension of . Assume on the contrary that dimH .V \ / < tu and let s be a positive number such that dimH .V \ / < s < tu :

(9.25)

9.3 Hyperbolic Sets

269

Then m.V \ ; s/ D 0, and since V \ is compact, for each ı > 0, there is a finite cover U of V \ by open balls such that X

.diam U /s < ı s :

(9.26)

U 2U

For each n 2 N, let ın be a positive number such that ˚ pn .U / WD card .i0 in / W U \ Ri0 in ¤ ∅ < k whenever diam U < ın (the existence of ın follows from the properties of the Markov partition). We note that ın ! 0 when n ! 1. It follows from (9.26) with ı D ın that diam U < ın and hence that pn .U / < k

for every U 2 U:

Let N D nCm1, for some m 2 N such that Am > 0 (recall that f is topologically mixing on ). For each . i1 i0 i1 / 2 ˙A and n 2 N, we consider the cover Ui0 iN of V composed of the projections along the stable leaves into V of the sets f N .U / with U 2 U such that U \ Ri0 in ¤ ∅. We have X U 2Ui0 iN

where

X

.diam U /s i0 iN s

.diam U /s ;

U 2U; U \Ri0 in ¤∅

˚ i0 in D min kdx f n jE u .x/k W x 2 Ri0 in :

Now let us assume that

X

.diam U /s ın s

U 2Ui0 iN

for every . i1 i0 i1 / 2 ˙A and n 2 N. Then kın s > k

X U 2U

D

X

.diam U /s

X

X

pn .U /.diam U /s

U 2U

.diam U /s

i0 in U 2U; U \Ri0 in ¤∅

k mC1

X

X

.diam U /s

(9.27)

i0 iN U 2U; U \Ri0 in ¤∅

k

mC1

X i0 iN

i0 iN

k mC1 ın s

X

i0 iN

X U 2Ui0 iN

i0 iN s :

.diam U /

s

270


On the other hand, by a property analogous to that in (9.9), there is a constant C > 0 (independent of n 2 N and .i0 in /) such that C 1

i0 in C: i0 in

Therefore, by (9.27), 1 X s i0 in N !1 N i i

P .s'u / D lim

0

N

1 X lim i0 in s 0: N !1 N i i 0

(9.28)

N

Since the function s 7! P .s'u / is strictly decreasing and P .tu 'u / D 0, we conclude that (9.28) contradicts (9.25). Hence, we must have X

.diam U /s < ın s

(9.29)

U 2Ui0 iN

for some sequence .i0 iN / and all sufficiently large n (recall that N D nCm1). Now we restart the process using the cover V1 D Ui0 iN to find inductively finite covers V` of V \ for each ` 2 N. By (9.29), we have diam V` < ın , and hence, pn .U / < k for every U 2 V` . This implies that card V`C1 < card V` , and hence, card V` D 1 for some ` D `.n/. Writing V`.n/ D fUng, we thus obtain diam.V \ / diam Un < ın ! 0 when n ! 1; which is impossible. This contradiction shows that dimH .V \ / tu :

(9.30)

By (9.24) and (9.30), we obtain dimH .V \ / D dimB .V \ / D dimB .V \ / D tu : By (9.22) and (9.23), this establishes the identities in (9.19).

t u

We emphasize that all dimensions in identities (9.19) and (9.20) are independent of the point x. Example 9.7. Let be the general horseshoe constructed in Exercise 8.15. At each point x 2 , the stable and unstable manifolds of sufficiently small size (possibly depending on x) are respectively horizontal and vertical segments. Moreover, the

9.3 Hyperbolic Sets

271

projection of the intersection V s .x/ \ to the horizontal axis coincides with the set J constructed in Example 9.3 by taking k D 2 and gi .x/ D

x ai i

for i D 1; 2:

By Example 9.6, we have dimH .V s .x/ \ / D dimB .V s .x/ \ / D dimB .V s .x/ \ / D ˛;

(9.31)

where ˛ is the unique root of the equation ˛1 C˛2 D 1. Similarly, one can show that dimH .V u .x/ \ / D dimB .V u .x/ \ / D dimB .V u .x/ \ / D ˇ; ˇ

(9.32)

ˇ

where ˇ is the unique root of the equation 1 C 2 D 1.

9.3.2 Dimension of Hyperbolic Sets We show in this section that the Hausdorff dimension and the lower and upper box dimensions of a locally maximal hyperbolic set are equal to the sum of the dimensions along the stable and unstable manifolds. Theorem 9.3 (Dimension of hyperbolic sets). Let be a locally maximal hyperbolic set for a C 1C˛ diffeomorphism, for some ˛ 2 .0; 1, such that f is conformal and topologically mixing on . Then dimH D dimB D dimB D ts C tu ;

(9.33)

with ts and tu as in (9.21). Proof. Since f is conformal on , the product structure is a Lipschitz homeomorphism with Lipschitz inverse, and thus, dimH ŒV s .x/ \ ; V u .x/ \ D dimH ..V s .x/ \ / .V u .x/ \ //; dimB ŒV s .x/ \ ; V u .x/ \ D dimB ..V s .x/ \ / .V u .x/ \ //;

(9.34)

dimB ŒV .x/ \ ; V .x/ \ D dimB ..V .x/ \ / .V .x/ \ //: s

u

s

u

By Exercise 8.7, it follows from (9.19), (9.20), and (9.34) that dimH ŒV s .x/ \ ; V u .x/ \ D dimB ŒV s .x/ \ ; V u .x/ \ D dimB ŒV s .x/ \ ; V u .x/ \ D ts C tu :

(9.35)

272


On the other hand, since is locally maximal, we have ŒV s .x/ \ ; V u .x/ \ for every x 2 , and thus, there exist points x1 ; : : : ; xN 2 such that

D

N [

ŒV s .xn / \ ; V u .xn / \ :

nD1

The identities in (9.33) follow now immediately from (9.35).

t u

Example 9.8. Let be the general horseshoe constructed in Exercise 8.15. It follows from Theorem 9.3 together with (9.31) and (9.32) that the Hausdorff dimension and the lower and upper box dimensions of are equal to ˛ C ˇ. This value was already obtained in Exercise 8.15.

9.4 Exercises Exercise 9.1. For a repeller J of a C 1C˛ transformation f such that f is conformal on J , show that if a kdx f k b for every x 2 J; (9.36) then

h.f jJ / h.f jZ/ dimH J : (9.37) b a Exercise 9.2. For a repeller J as in Exercise 9.1, show that the inequalities in (9.37) remain true when condition (9.36) is replaced by c 1 an kdx f n k cb n

for every x 2 J; n 2 N:

Exercise 9.3. Show that a repeller for a map f is also a repeller for any power f n , with n 2 N. Exercise 9.4. Find a nonconformal map f such that f 2 is conformal. Exercise 9.5. Let J be a repeller for a C 1C˛ transformation f . Show that if some power f n of the map f is conformal on J , then (9.4) holds, where s is now the unique root of the equation

Pf n jJ s logkdf n k D 0: Hint: use Exercise 9.3.

9.4 Exercises

273

Exercise 9.6. For a repeller J of a C 1C˛ transformation such that f is conformal on J , show that if is an equilibrium measure for the function 'W J ! R defined by '.x/ D log kdx f k, then dimH J D dimB J D dimB J D R J

h .f / : log kdf k d

Exercise 9.7. Use Theorem 9.3 to show that the Hausdorff dimension of the Smale horseshoe is log 4= log 3. Exercise 9.8. Show that for each d 2 .0; 2/, there is a horseshoe such that dimH D dimB D dimB D d: Hint: change the construction of the Smale horseshoe. Exercise 9.9. Show that for each ds ; du 2 .0; 1/, there is a horseshoe such that dimH .V s .x/ \ / D dimB .V s .x/ \ / D dimB .V s .x/ \ / D ds and dimH .V u .x/ \ / D dimB .V u .x/ \ / D dimB .V u .x/ \ / D du for every x 2 . Exercise 9.10. For a repeller J of a C 1C˛ transformation f such that f is conformal on J , let R1 ; : : : ; Rk be the elements of a Markov partition and let i1 in D

n1 \

f j Rij C1

j D0

for each n 2 N and i1 ; : : : ; in 2 f1; : : : ; kg. We assume that there exist a probability measure in J and constants D1 ; D2 > 0 such that D1

.i1 in / D2 kdx f n k1

for every n 2 N and x 2 i1 in . Show that: 1. is equivalent to the Hausdorff measure m.; s/, where s D dimH J , with Radon–Nikodym derivative bounded and bounded away from zero. 2. 0 < m.J; s/ < 1. 3. d .x/ D d .x/ D s for every x 2 J . 4. dimH D dimB D dimB D s.

274


Exercise 9.11. For a repeller J of a C 1C˛ transformation f such that f is conformal on J , let R1 ; : : : ; Rk be the elements of a Markov partition with transition matrix A. We assume that there exists q 2 N such that all entries of the matrix Aq are positive. Given a Hölder continuous function 'W J ! R, we consider the numbers ˚ ai1 in D max exp 'n .x/ W x 2 i1 in ; where 'n D

Pn1 lD0

' ı f l , and also, ˛n D

X

ai1 in :

i1 in

Show that: 1. ˛nCl ˛n ˛l for every n; l 2 N. 2. There exists D > 0 such that j'n .x/ 'n .y/j < D for every n 2 N and x; y 2 i1 in . 3. There exists D 0 > 0 such that ˛nCl D 0 ˛n ˛l for every n; l 2 N. 4. 1 1 Pf jJ .'/ D lim log ˛n D inf log ˛n : n!1 n n2N n 5. For each l 2 N and any probability measure l satisfying l .i1 il / D ai1 il =˛l for each .i1 i2 / 2 ˙AC , there exist C1 ; C2 > 0 such that C1

l i1 in

C2 exp nPf jJ .'/ C 'n .x/

for every n; l 2 N and x 2 i1 in . 6. Any sublimit of the sequence .l /l satisfies C1

i1 in


for every n 2 N and x 2 i1 in . 7. For any sublimit of the sequence of measures 1X ı f j ; n j D0 n1

n D

9.4 Exercises

275

where is any sublimit of the sequence .l /l , there exist C10 ; C20 > 0 such that C10

i1 in


for every n 2 N and x 2 i1 in . 8. is an equilibrium measure for ', that is, is an f -invariant probability measure in J and Z Pf jJ .'/ D h .f / C ' d: J

Notes We refer to the books [8, 9, 73] for further topics of dimension theory of dynamical systems. Equation (9.5) was introduced by Bowen [21] in his study of quasicircles. For a repeller J of a conformal map f , Ruelle [85] showed that dimH J D s (under the assumption that f is topologically mixing on J ). The equality between the Hausdorff and box dimensions is due to Falconer [31]. Our proof of Theorem 9.1 is based on [8]. McCluskey and Manning [59] obtained (9.19) and (9.20) for the Hausdorff dimension. The equality between the Hausdorff and box dimensions of a hyperbolic set is due to Takens [101] for C 2 diffeomorphisms and to Palis and Viana [68] in the general case. Our proof of Theorem 9.2 is based on [7], which is inspired in arguments in [101].

•

Appendix A

Notions from Measure Theory

We recall in this appendix all the necessary material from measure theory.

A.1 Measure Spaces Let A be a family of subsets of a set X . We say that A is a -algebra of X if: 1. ∅, X 2 A. 2. B S 2 A whenever X n B 2 A. 3. 1 kD1 Bk 2 A whenever Bk 2 A for every k 2 N. Each element of a -algebra A is called an A-measurable set. Given a family A of subsets of X , the smallest -algebra containing A, that is, the intersection of all -algebras containing A, is called the -algebra of X generated by A. Now let A be a -algebra of X . We say that a function W A ! Œ0; C1 is a measure in X with respect to A provided that: 1. .∅/ D 0. 2. If Bk 2 A for every k 2 N and Bk \ B` D ∅ whenever k ¤ `, then ! 1 1 [ X Bk D .Bk /: kD1

kD1

The triple .X; A; / is then called a measure space. We note that a measure is completely determined by its values in any family of sets generating the -algebra. We say that the measure space .X; A; / is: 1. Finite if .X / < 1. 2. Infinite if .X / D 1. S 3. -finite if there exist sets Bk 2 A for k 2 N such that X D 1 kD1 Bk and .Bk / < 1 for every k 2 N. 4. A probability space and in this case that is a probability measure if .X / D 1. L. Barreira, Ergodic Theory, Hyperbolic Dynamics and Dimension Theory, Universitext, DOI 10.1007/978-3-642-28090-0, © Springer-Verlag Berlin Heidelberg 2012

277

278

A Notions from Measure Theory

We say that a given property holds -almost everywhere if the set of points where it does not hold has zero -measure.

A.2 Outer Measures and Measurable Sets Let A be the family of all subsets of a set X . A function W A ! Œ0; C1 is called an outer measure in X provided that: 1. .∅/ D 0. 2. .B/ .C / whenever B C . 3. If Bk 2 A for every k 2 N, then

1 [

! Bk

kD1

1 X

.Bk /:

kD1

In particular, when .X; A; / is a measure space, one can define an outer measure in X by (

.B/ D inf

1 X kD1

.Bk / W B

1 [

) Bk and Bk 2 A for every k 2 N :

(A.1)

kD1

We call the outer measure associated to .X; A; /. Given an outer measure in X we say that a set A X is -measurable if .B/ D .B \ A/ C .B n A/ for any B X . If is an outer measure in X , then the following properties hold: 1. The family A of -measurable sets is a -algebra, and the restriction of to A is a measure in X . 2. If is an outer measure associated to a measure space .X; A0 ; 0 /, then A0 A and 0 .B/ D .B/ for every B 2 A0 .

A.3 Measures in Topological Spaces Now let X be a topological space, and let A be the family of all open subsets of X . The -algebra of X generated by A is called the Borel -algebra of X , and its elements are called Borel-measurable sets. Given a measure in X its support is the complement of the largest open set U with .U / D 0, and we denote it by supp .

A.4 Measurable Functions, Integration, and Convergence

279

We also consider the particular case of the set X D Rn with its usual topology. Let A0 A be the family of open rectangles, that is, the family of sets of the form .a1 ; b1 / .an ; bn /: Since each open subset of Rn is a countable union of element of A0 , the -algebra of Rn generated by A0 is the Borel -algebra of Rn . In particular, the Borel -algebra of R is generated by the open intervals. If B is the Borel -algebra of Rn , then there exists a unique measure W B ! Œ0; C1 such that n Y .a1 ; b1 / .an ; bn / D .bi ai / i D1

for every ak < bk and k D 1; : : : ; n. Using (A.1), one can construct an outer measure associated to . The -algebra of -measurable sets is called the Lebesgue -algebra of Rn , and its elements are called Lebesgue-measurable sets. The restriction of to this -algebra is called the Lebesgue measure in Rn . For any set B Rn with .B/ < 1, the following properties are equivalent: 1. The set B is Lebesgue-measurable. 2. Given " > 0, there exist a compact set K B and an open set U B such that .U n K/ < ". 3. Given " > 0, there exist rectangles R1 ; : : : ; Rm such that [ m m [ Ri < " and Ri n B < ": Bn

i D1

i D1

In particular, if B Rn is a Lebesgue-measurable set with .B/ < 1, then .B/ D inff.U / W U B is openg and

.B/ D supf.K/ W K B is compactg:

A.4 Measurable Functions, Integration, and Convergence Let X be a set and let A be a -algebra of X . A transformation T W X ! X is said to be A-measurable if T 1 B 2 A whenever B 2 A. A function 'W X ! R is said to be A-measurable if ' 1 B 2 A whenever B is a Borel-measurable set. Given A-measurable sets B1 ; : : : ; Bn X and numbers a1 ; : : : ; an 2 R, we define a simple function by n X 'D ak Bk ; kD1

280


where B is the characteristic function B W X ! f0; 1g of the set B X , given by ( B .x/ D

1

if x 2 B;

0

if x 62 B:

Given a measure in X with respect to A and an A-measurable function ' 0, we define the Lebesgue integral of ' in X by (

Z ' d D sup X

n X

n X

ak .Bk / W

kD1

) ak Bk ' :

kD1

For an arbitrary A-measurable function ', we say that ' is -integrable if Z

' C d < 1 and

Z

X

where

' d < 1; X

' C D maxf'; 0g and ' D maxf'; 0g:

In this case, we define the Lebesgue integral of ' in X by Z

Z ' d D X

Z

' C d

X

' d: X

One can show that ' is -integrable if and only if Z j'j d < 1: X

We denote by L1 .X; / the set of -integrable functions 'W X ! R. We note that Z L1 .X; / 3 ' 7! ' d X

is a linear transformation. Theorem A.1 (Fatou’s lemma). measurable functions, then

If 'n W X ! R is a sequence of nonnegative

Z

Z lim inf 'n d lim inf

X n!1

n!1

'n d: X

Theorem A.2 (Monotone convergence theorem). If 'n W X ! RC 0 is a nondecreasing sequence of measurable functions, then Z

Z lim 'n d D lim

X n!1

n!1 X

'n d:

A.5 Absolutely Continuous Measures

281

Theorem A.3 (Dominated convergence theorem). If 'n W X ! R is a sequence of measurable functions such that the limit ' D lim 'n n!1

W X ! RC 0 such that j'n j

exists, and there is a -integrable function every n 2 N, then ' is -integrable and

for

Z

Z ' d D lim

n!1 X

X

'n d:

When X is a topological space, we denote by C.X / the space of continuous functions 'W X ! R. We recall that an operator J W C.X / ! R is said to be positive if J.'/ 0 whenever ' 0. Theorem A.4 (Riesz’s representation theorem). If X is a compact metric space and J W C.X / ! R is a positive continuous linear operator, then there exists exactly one measure in the Borel -algebra of X such that Z J.'/ D

' d X

for every ' 2 C.X /. If, in addition, J.1/ D 1, then is a probability measure.

A.5 Absolutely Continuous Measures Let and be measures in X with respect to the same -algebra. We say that is absolutely continuous with respect to , and we write , if .B/ D 0 for every B 2 A with .B/ D 0. For example, if ' 2 L1 .X; /, then the measure defined by Z .B/ D

' d B

for each B 2 A is absolutely continuous with respect to . Theorem A.5 (Radon–Nikodym). If .X; A; / and .X; A; / are -finite measure spaces and is absolutely continuous with respect to , then there exists an A-measurable function 'W X ! R, uniquely determined in a set of full -measure, such that Z ' d .B/ D B

for every B 2 A with .B/ < 1.

282


Any function ' as in Theorem A.5 is called a Radon–Nikodym derivative of with respect to , and we represent it by d=d. We note that if .X; A; / and .X; A; / are finite measure spaces and , then d=d 2 L1 .X; /. Theorem A.6. Let .X; A; / be a -finite measure space and let F A be a subalgebra. For each -integrable A-measurable function 'W X ! R there exists an F-measurable function 'F W X ! R such that Z

Z 'F d D B

' d for every

B 2 F:

B

The function 'F is called a conditional expectation of ' with respect to F. See Proposition 2.9 for a proof of Theorem A.6 in the case of finite measure spaces. Theorem A.7 (Increasing martingale theorem [69]). If 'W X ! R is a -integrableS function, and Fn is a nondecreasing sequence of -algebras such that the union 1 nD1 Fn generates the -algebra F, then 'Fn ! 'F -almost everywhere and Z j'Fn 'F j d ! 0 when n ! 1: X

A.6 Product Spaces Now let .X; A; / and .Y; B; / be measure spaces. We consider the product space .X Y; C; /, where C is the -algebra generated by the family of sets ˚ A B D A B W A 2 A and B 2 B and where is the measure in C such that . /.A B/ D .A/.B/ for every A B 2 A B. The measure is called a product measure. Theorem A.8 (Fubini). If ' 2 L1 .X Y; /, then: 1. The function x 7! '.x; y/ is -integrable for -almost every y 2 Y . 2. The function y 7! '.x; y/ is -integrable for -almost every x 2 X . 3. We have Z Z Z ' d. / D '.x; y/ d.x/ d.y/ X Y

Y

X

X

Y

Z Z D

'.x; y/ d.y/ d.x/:

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Index

Absolutely continuous measure, 281 Adding machine, 143 Almost everywhere, 278 -algebra, 277 Borel, 278 Lebesgue, 279 ˛-orbit, 217 A-measurable function, 279 transformation, 279 Angle, 176 Automorphism, 30 hyperbolic, 175

Delta measure, 26 Diameter, 236 Dimension box, 236 Hausdorff, 236, 242 lower box, 236, 242 lower pointwise, 243 pointwise, 248 theory, 235 upper box, 236, 242 upper pointwise, 243 Dominated convergence theorem, 281

Bernoulli measure, 89, 93 Birkhoff’s ergodic theorem, 37, 38, 40 Borel -algebra, 278 measurable set, 278 Box dimension, 236 lower, 242 upper, 242

Endomorphism, 29 Entropy conditional, 115 measure-theoretical, 113 metric, 109, 113 periodic, 28 topological, 133 Equilibrium measure, 158 Equivalence between distances, 99 topological, 138 Ergodic measure, 44 theorem, 38 Ergodicity, 43 criterion, 44 Expanding map, 27, 193 piecewise linear, 32 Expansive one-sided, 159 two-sided, 163

Characteristic function, 21, 280 Circle, 24 expanding map, 27 rotation, 24, 26, 78 Coding map, 184, 188, 197, 226 Conditional entropy, 115 expectation, 39, 282 Conformal map, 253, 255, 265 Continued fraction, 51, 52 Cylinder set, 88, 92

L. Barreira, Ergodic Theory, Hyperbolic Dynamics and Dimension Theory, Universitext, DOI 10.1007/978-3-642-28090-0, © Springer-Verlag Berlin Heidelberg 2012

287

288 Fatou’s lemma, 280 Fourier coefficient, 45, 46 Fractional part, 49 Fubini’s theorem, 282 Function A-measurable, 279 characteristic, 21, 280 invariant, 36 invariant almost everywhere, 37 -integrable, 280 simple, 279 G.n; Z/, 28 Gauss measure, 53 transformation, 51 Generator, 121 one-sided, 121 two-sided, 121 Hadamard–Perron theorem, 202 Hausdorff dimension, 236, 242 measure, 236 Holonomy map, 266 Hopf’s argument, 211 Horseshoe, 179 Hyperbolic fixed point, 173 matrix, 175 periodic point, 173 set, 173 toral automorphism, 175 Increasing martingale theorem, 282 Induced invariant measure, 62 transformation, 62 Infinite adding machine, 143 Integrable function, 280 Integral, 280 Interval translation, 24, 26 Invariant function, 36 function almost everywhere, 37 measure, 21, 48, 66, 71 set, 36 set almost everywhere, 37 Irrational circle rotation, 26, 78 interval translation, 26 Irreducible matrix, 99

Index Julia set, 255

Krylov–Bogolubov theorem, 71

Lebesgue -algebra, 279 integral, 280 measurable set, 279 measure, 279 Locally maximal, 216 Lower box dimension, 236, 242 pointwise dimension, 243

Manifold local stable, 202, 211 local unstable, 202, 211 Markov chain, 85, 91, 92 measure, 88, 91, 93 partition, 187, 193, 220 property, 187, 220 Matrix hyperbolic, 175 irreducible, 99 stochastic, 87 transition, 85, 92, 197 Measurable partition, 109 set, 277 transformation, 21 Measure, 277 absolutely continuous, 281 Bernoulli, 89, 93 delta, 26 equilibrium, 158 ergodic, 44 Gauss, 53 Hausdorff, 236 invariant, 21, 48, 66, 71 Lebesgue, 279 Markov, 88, 93 outer, 278 probability, 277 Measure space, 277 -finite, 277 finite, 277 infinite, 277 probability, 277 Measure-theoretical entropy, 113 Metric entropy, 109, 113

Index Mixing, 80, 81 topologically, 95 Monotone convergence theorem, 280 Moran cover, 258

Number theory, 48

!-limit set, 95 One-sided expansive, 159 generator, 121 Outer measure, 278

Partition Markov, 187, 193, 220 measurable, 109 Period, 62 Periodic entropy, 28 hyperbolic point, 173 orbit, 26 point, 26 repelling point, 255 Perron–Frobenius theorem, 90 Poincaré’s recurrence theorem, 34 Point m-periodic, 26 periodic, 26 recurrent, 95 Pointwise dimension, 248 Pre-partition, 109 Pressure, 149 Probability measure, 277 space, 277 vector, 88 Product measure, 282 space, 282 structure, 215

Radon–Nikodym derivative, 282 theorem, 281 Rational circle rotation, 26 interval translation, 26 Rectangle, 220 Recurrence theorem, 34

289 Recurrent point, 95 Refinement, 109 Repeller, 193 Repelling periodic point, 255 Return time, 62 Riemann–Lebesgue lemma, 47 Riesz’s representation theorem, 281 Rohklin’s distance, 144 Rotation, 23, 24 irrational, 26, 78 rational, 26 S.n; R/, 23 Separated set, 148 Sequence one-sided, 85 two-sided, 92 uniformly distributed, 49 Set A-measurable, 277 Borel-measurable, 278 hyperbolic, 173 invariant, 36 invariant almost everywhere, 37 Julia, 255 Lebesgue-measurable, 279 measurable, 278 separated, 148 Shadowing property, 217 Shannon–McMillan–Breiman theorem, 125 Shift map, 85, 92, 182 Simple function, 279 Smale horseshoe, 179 Solenoid, 199 Spectral radius, 86 Spectrum, 173 Stable manifold, 202, 211 subspace, 173 Stochastic matrix, 87 pair, 88 Subshift of finite type, 85, 92 Subspace stable, 173 unstable, 173 Support, 91, 94, 278 Symbolic dynamics, 84 T, 24 Tn , 29

290 Thermodynamic formalism, 148 Topological dynamics, 94 entropy, 133 equivalence, 138 Markov chain, 85, 91, 92 mixing, 95 pressure, 149 transitivity, 95 Toral automorphism, 30 endomorphism, 28, 29 translation, 60 Torus, 29 Transformation A-measurable, 279 conformal, 255, 265 Gauss, 51 induced, 62 infinite adding machine, 143 measurable, 21 measure-preserving, 21 mixing, 81 topologically mixing, 95 topologically transitive, 95 uniquely ergodic, 72

Index Transition matrix, 85, 92, 197 Translation, 23 irrational, 26 rational, 26 Two-sided expansive, 163 generator, 121 sequence, 92

Uniform convergence, 72 Uniformly distributed sequence, 49 Unique ergodicity, 72 criterion, 74, 75 Unstable manifold, 202, 211 subspace, 173 Upper box dimension, 236, 242 pointwise dimension, 243

Variational principle, 138

Zeta function, 63

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