Smooth Ergodic Theory for Endomorphisms (Lecture Notes in Mathematics)

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1978

Min Qian · Jian-Sheng Xie · Shu Zhu

Smooth Ergodic Theory for Endomorphisms

123

Editors Shu ZHU c/o Jian-Sheng XIE School of Mathematical Sciences Peking University Beijing 100871 P. R. China

Min QIAN School of Mathematical Sciences Peking University Beijing 100871 P. R. China Jian-Sheng XIE School of Mathematical Sciences Fudan University Shanghai 200433 P. R. China [email protected]

ISBN: 978-3-642-01953-1 DOI: 10.1007/978-3-642-01954-8

e-ISBN: 978-3-642-01954-8

Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2009928105 Mathematics Subject Classification (2000): 37C40, 37C45, 37D20, 37H15 c Springer-Verlag Berlin Heidelberg 2009  This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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. Cover design: SPi Publisher Services Printed on acid-free paper springer.com

Comme un souvenir de mon amit´e avec le Professeur Y. -S. Sun, je voudrais mentioner que c’est lui qui m’a fait connaitre la formule de Pesin pour la premi`ere fois dans ma vie. Min Qian

Preface

Smooth ergodic theory of deterministic dynamical systems deals with the study of dynamical behaviors relevant to certain invariant measures under differentiable mappings or flows. The relevance of invariant measures is that they describe the frequencies of visits for an orbit and hence they give a probabilistic description of the evolution of a dynamical system. The fact that the system is differentiable allows one to use techniques from analysis and geometry. The study of transformations and their long-term behavior is ubiquitous in mathematics and the sciences. They arise not only in applications to the real world but also to diverse mathematical disciplines, including number theory, Lie groups, algorithms, Riemannian geometry, etc. Hence smooth ergodic theory is the meeting ground of many different ideas in pure and applied mathematics. It has witnessed a great progress since the pioneering works of Sinai, Ruelle and Bowen on Axiom A diffeomorphisms and of Pesin on non-uniformly hyperbolic systems, and now it becomes a well-developed field. In this theory, among the major concepts are the notions of Lyapunov exponents and metric entropy. Lyapunov exponent describes the exponential rate of expansion or contraction in certain direction along an orbit. Obviously, positive Lyapunov exponents corresponds to the local instability of trajectories. One of the paradigms of dynamical systems is that the local instability of trajectories may lead to the stochastic behavior of the system. Metric entropy, introduced by Kolmogorov and Sinai, is a purely measure-theoretic invariant, which measures the complexity of the dynamical system generated by iterations of the transformation. It has been studied a good deal in abstract ergodic theory, see [75]. The relationship between these two concepts has always been an important problem. A fundamental result concerning this problem is Margulis-Ruelle inequality, which states that the metric entropy can be bounded from above by the sum of positive Lyapunov exponents (See Chapter II). More deep results can be obtained when the system exhibits certain hyperbolicity. The strongest hyperbolicity occurs in the important class of Axiom A systems. In the ergodic theory of Axiom A diffeomorphisms developed by Sinai [88], Ruelle [76] and Bowen [10, 11], it was shown that

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for an Axiom A attractor there is a unique invariant measure which is characterized by each of the following properties: (1) The metric entropy is equal to the sum of positive Lypunov exponents. (2) The conditional measures of the invariant measure on unstable manifolds are absolutely continuous with respect to the Lebesgue measures on these manifolds. (3) Lebesgue almost every point in an open neighborhood of the attractor is generic to this measure. Property (1) is now known as Pesin’s entropy formula and property (2) is known as SRB property of the invariant measure. Each of these properties has been shown to be significant in its own right, but it is also remarkable that they are equivalent to each other in the case of an Axiom A attractor. In mid-seventies, in a series of papers Pesin developed a machinery to study non-uniformly hyperbolic systems [62, 63]. He obtained a general theorem on the existence and the absolute continuity of invariant families of stable and unstable manifolds of a smooth dynamical system, corresponding to its non-zero Lyapunov exponents. Meanwhile, he also studied the ergodic properties of smooth dynamical systems possessing an absolutely continuous invariant measure. The most striking result is that Pesin’s entropy formula also holds in this case. Then it was conjectured by Ruelle and later on proved by Ledrappier, Strelcyn and Young that for an invariant measure of a C2 diffeomorphism, Pesin’s entropy formula holds if and only if it satisfies the SRB property [41, 42]. In other words, the equivalence of properties (1) and (2) can hold in a more general circumstance. The above results have been successfully generalized to several frameworks. Among them are random iterations of diffeomorphisms and deterministic endomorphisms. For random diffeomorphisms, first initiated by Ledrappier and Young [44], Liu and Qian provided a systematic treatment on the subject [51]. However the results for deterministic system are still scattered in the literature. The main purpose of this monograph is to summarize these results and to provide a systematic treatment on this aspect for deterministic systems. The novelty of our treatment lies in the fact that we directly consider endomorphisms throughout the monograph. The results for diffeomorphisms can be obtained as a special case. It is interesting to point out that the method developed to attack Random Dynamical Systems [38, 44] can be adapted to treat the endomorphism case. It turns out to be the inverse sequence approach known in the dynamical system theory but it has never been detailed into a systematic treatment as one can see in [44]. Therefore, this monograph gives convincing evidence how deterministic theory can be benefited by probabilistic consideration. The monograph is organized as follows. We will review some fundamental concepts in Chapter I. Since the whole monograph mainly deals with endomorphisms with the help of inverse limit space, we also provide the simple relations of entropies and Lyapunov exponents between the base dynamical system and the induced dynamical system on the inverse limit space. Chapter II is devoted to the Margulis-Ruelle inequality. This inequality was first given by Margulis in the case of diffeomorphisms preserving a smooth measure.

Preface

ix

The general statement is due to Ruelle [77]. Rigorous proofs are available in several books for the case of diffeomorphisms only, see [32] and [57]. We present a short and rigorous proof for the general C1 maps in this chapter. In Chapter III, we study the simplest case—expanding maps. Although Pesin’s entropy formula is actually a consequence of the main theorem in Chapter VI, there are still some other nice results under weak conditions. This chapter follows from the work of H.-Y. Hu [27]. In Chapter IV, we study the strongest hyperbolic case, the ergodic theory for Axiom A endomorphisms. This chapter is from the work of Qian and Zhang [72]. Chapter V consists of the study of the structure of unstable manifolds. Since in general the unstable manifold at each point depends on the whole backward orbit, for different orbit there might be different unstable manifold at the same point. Therefore, there is no foliation structure of unstable manifolds in these case. We consider the structure of unstable manifolds in the inverse limit space. The source of this chapter is the work of S. Zhu [100] with slight modification. In Chapter VI we extend Pesin’s entropy formula to the general C2 endomorphisms. This is done by Liu [46] recently in a different approach. In Chapter VII we present a formulation of the SRB property for invariant measures of C2 endomorphisms of a compact manifold via their inverse limit spaces, and then prove that this property is sufficient and necessary for the entropy formula. This is a non-invertible version of the main theorem of [42]. As a nontrivial corollary of this result, an invariant measure of a C2 endomorphism has this SRB property if it is absolutely continuous with respect to the Lebesgue measure of the manifold. Invariant measures having this SRB property also exist on Axiom A attractors of C2 endomorphisms. Comparing with the case of diffeomorphisms, the major difficulty arises from non-invertibility. To overcome this deficiency, the inverse limit space has to be introduced. Notice that, when the inverse limit space is introduced, one can compare a full orbit with a sample orbit from random iteration of diffeomorphisms. Keep this in mind, with some necessary modifications, many ideas and techniques developed for the random diffeomorphisms, for which a systematic treatment is now available in [51], can be applied to our present study. The result was given by Qian and Zhu in [73], and we provide a detailed presentation in this chapter. In Chapter VIII, we study the ergodic hyperbolic attractors. This chapter follows from the work of Jiang and Qian [28]. Chapters IX and X may be viewed as the climax of this book. In Chapter IX, we present here a generalized entropy formula for any Borel probability measure invariant under a C2 endomorphism. It is a non-invertible endomorphisms version of a formula obtained by Ledrappier and Young [43], hence covers theirs as a consequence. The generalized entropy formula relates closely to Eckmann-Ruelle conjecture for the endomorphism version; In Chapter X, we apply this entropy formula to hyperbolic measures preserved respectively by expanding maps and diffeomorphisms, proving Eckmann-Ruelle conjecture in these two situations. These two chapters are rewritten from the work of Qian and Xie [71] and Liu and Xie [54]. The proof of Eckmann-Ruelle conjecture (for diffeomorphisms) was first presented

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by Barreira et al [7] (see also [64, pp. 279–292]); our proof is slightly different from theirs and seems more accessible. In Appendix A, we show that Pesin entropy formula still holds true for C2 random endomorphisms if the sample measures of the invariant measure are smooth. This result covers those obtained by Pesin [63] for C2 diffeomorphisms, Liu [46] for C2 endomorphisms, Ledrappier and Young [44] for i.i.d. random diffeomorphisms and Liu [48] for two-sided stationary random endomorphisms. In Appendix B, we present a large deviation theorem, it was included as one chapter in the first draft of the manuscript. Since the presentation is not selfcontained, we prefer to shift it to the end as Appendix B. This part of the manuscript was prepared by Y. Zhao, see [53]. We would like to thank Prof. P.-D. Liu for very useful discussions. We are deeply indebted to Prof. Ledrappier for his helpful comments on Qian and Xie’s work [71]. S. Zhu also wishes to thank Profs. Anatole Katok, Yakov Pesin, Luis Barreira and Wenxian Shen for many helpful conversations during his visit at Pennsylvania State University and Aub ur n University. During the long editing period of this monograph, M. Qian was partially support by the 973 Fund of China for Nonlinear Science and the NSFC of China; and J.-S. Xie is partially support by NSFC Grant No. 10701026. Beijing, Shanghai, February 2009

Min Qian Jian-Sheng Xie Shu Zhu

Contents

I

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . I.1 Metric Entropy.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.2 Multiplicative Ergodic Theorem.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.3 Inverse Limit Space.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4

II

Margulis-Ruelle Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9 II.1 Statement of the Theorem.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 II.2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 II.3 Proof of the Theorem.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

III

Expanding Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . III.1 Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.2 Proof of Theorem III.1.1.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.3 Basic Facts About Expanding Maps.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.4 Proofs of Theorems III.1.2 and III.1.3.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 17 19 22

IV

Axiom A Endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . IV.1 Introduction and Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.3 Volume Lemma and the H¨older Continuity of φ u . . . . . . . . . . . . . . . . . . IV.4 Equilibrium States of φ u on Λ f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.5 Pesin’s Entropy Formula.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.6 Large Ergodic Theorem and Proof of Main Theorems.. . . . . . . . . . . .

27 27 29 31 38 40 42

V

Unstable and Stable Manifolds for Endomorphisms . . . . . . . . . . . . . . . . . . . V.1 Preliminary Facts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.2 Fundamental Lemmas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.3 Some Technical Facts About Contracting Maps. . . . . . . . . . . . . . . . . . . V.4 Local Unstable Manifolds.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.5 Global Unstable Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . V.6 Local and Global Stable Manifolds.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 46 49 56 58 73 78

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V.7 V.8 V.9 VI

H¨older Continuity of Sub-bundles.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Absolute Continuity of Families of Submanifolds.. . . . . . . . . . . . . . . . 83 Absolute Continuity of Conditional Measures. . . . . . . . . . . . . . . . . . . . . 85

Pesin’s Entropy Formula for Endomorphisms . . . . . .. . . . . . . . . . . . . . . . . . . . VI.1 Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI.2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI.3 Proof of Theorem VI.1.1.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 87 88 91

VII SRB Measures and Pesin’s Entropy Formula for Endomorphisms . . . 97 VII.1 Formulation of the SRB Property and Main Results. . . . . . . . . . . . . . . 98 VII.2 Technical Preparations for the Proof of the Main Result. . . . . . . . . . 100 VII.3 Proof of the Sufficiency for the Entropy Formula.. . . . . . . . . . . . . . . . . 110 VII.4 Lyapunov Charts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 VII.5 Local Unstable Manifolds and Center Unstable Sets. . . . . . . . . . . . . . 116 VII.5.1 Local Unstable Manifolds and Center Unstable Sets. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 116 VII.5.2 Some Estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 VII.5.3 Lipschitz Property of Unstable Subspaces within Center Unstable Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 VII.6 Related Measurable Partitions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 VII.6.1 Partitions Adapted to Lyapunov Charts. . . . . . . . . . . . . . . . . . 125 VII.6.2 More on Increasing Partitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 VII.6.3 Two Useful Partitions.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 130 VII.6.4 Quotient Structure.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 VII.6.5 Transverse Metrics. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 132 VII.7 Some Consequences of Besicovitch’s Covering Theorem.. . . . . . . . 135 VII.8 The Main Proposition.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 VII.9 Proof of the Necessity for the Entropy Formula.. . . . . . . . . . . . . . . . . . . 144 VII.9.1 The Ergodic Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 VII.9.2 The General Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 VIII Ergodic Property of Lyapunov Exponents . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 151 VIII.1 Introduction and Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 VIII.2 Lyapunov Exponents of Axiom A Attractors of Endomorphisms . 153 VIII.3 Nonuniformly Completely Hyperbolic Attractors.. . . . . . . . . . . . . . . . . 163 IX

Generalized Entropy Formula . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . IX.1 Related Notions and Statements of the Main Results. . . . . . . . . . . . . . IX.1.1 Pointwise Dimensions and Transverse Dimensions.. . . . IX.1.2 Statements of the Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . IX.2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX.2.1 Some Estimations on Unstable Manifolds.. . . . . . . . . . . . . . IX.2.2 Related Partitions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˜ ηi−1 with 2 ≤ i ≤ u. . . . . . . IX.2.3 Transverse Metrics on ηi (x)/ IX.2.4 Entropies of the Related Partitions.. . . . . . . . . . . . . . . . . . . . . .

173 173 174 176 176 177 180 182 186

Contents

IX.3 IX.4

IX.5 X

xiii

Definitions of Local Entropies along Unstable Manifolds.. . . . . . . . Estimates of Local Entropies along Unstable Manifolds. . . . . . . . . . IX.4.1 Estimate of Local Entropy h1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX.4.2 Estimate of Local Entropy hi from Below with 2 ≤ i ≤ u. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX.4.3 Estimate of Local Entropy hi from Above with 2 ≤ i ≤ u. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The General Case: without Ergodic Assumption.. . . . . . . . . . . . . . . . . .

188 193 193 194 197 203

Exact Dimensionality of Hyperbolic Measures . . . . . .. . . . . . . . . . . . . . . . . . . . 205 X.1 Expanding Maps’ Case–Proof of Theorem X.0.1.. . . . . . . . . . . . . . . . . 206 X.2 Diffeomorphisms’ Case–Proof of Theorem X.0.2. . . . . . . . . . . . . . . . . 211 X.2.1 Reconstruction of a Special Partition.. . . . . . . . . . . . . . . . . . . 214 X.2.2 Preparatory Lemmas.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 X.2.3 Proof of Theorem X.2.2.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 X.2.4 The Case of Nonergodic Measures. . . . . . . . . . . . . . . . . . . . . . 233 X.2.5 Proofs of Requirements (X.60) and (X.61).. . . . . . . . . . . . . 235 X.3 Comments on Further Researches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

Appendix A Entropy Formula of Pesin Type for One-sided Stationary Random Maps . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 245 A.1 Basic Notions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 A.1.1 Set-up.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 A.1.2 Invariant Measures and Sample Measures . . . . . . . . . . . . . . 246 A.1.3 Entropy.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 A.1.4 Lyapunov Exponents.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 A.2 Statement of the Main Result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 A.2.1 Ruelle Inequality.. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 A.2.2 Pesin (Entropy) Formula.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 A.2.3 Pesin Formula for Some Particular RDS’s. . . . . . . . . . . . . . 251 A.3 Proof of Theorem A.2.2.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 252 Appendix B Large Deviations in Axiom A Endomorphisms . . . . . . . . . . . . . . . . . 261 B.1 Introduction and Statement of Main Results. . . . . . . . . . . . . . . . . . . . . . . 261 B.2 Proof of Theorem B.1.1.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 B.2.1 A Large Deviation Theorem from Kifer [37]. . . . . . . . . . . 264 B.2.2 Smale Spaces.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 B.2.3 Smale Space Property of Locally Maximal Hyperbolic Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 B.2.4 Proof of Theorem B.1.1.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 References .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 271 Index . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 275

Chapter I

Preliminaries

In this part we review some necessary concepts and results from ergodic theory, which will be frequently used in this monograph. Throughout this book, M is an m0 -dimensional, smooth, compact and connected Riemannian manifold without boundary. We use f ∈ Cr (O, M) to denote a Cr map from O to M, where O is an open subset of M, and we call f a Cr endomorphism on M if f ∈ Cr (M, M). We use T f to denote the tangent map induced by f when r ≥ 1. For any compact metrizable space X and continuous map T : X → X, We use MT (X) to denote the set of T -invariant Borel probability measures on X.

I.1 Metric Entropy Let X be a compact metrizable space, T : X → X a continuous map on X, and μ a T -invariant Borel probability measure   on X. For any finite partition η = Ci of X, define the entropy of η by Hμ (η ) = − ∑ μ (Ci ) log μ (Ci ). i

Let

1 Hμ (η ∧ T −1 η ∧ · · · ∧ T −n+1 η ). n Then define the metric entropy of T with respect to μ as h μ (T, η ) = lim

n→∞

hμ (T ) = sup{ h μ (T, η ) : η is a finite partition of X}. For properties of the metric entropy, we refer the reader to [92].

M. Qian et al., Smooth Ergodic Theory for Endomorphisms, Lecture Notes in Mathematics 1978, DOI 10.1007/978-3-642-01954-8 I, c Springer-Verlag Berlin Heidelberg 2009 

1

2

I Preliminaries

I.2 Multiplicative Ergodic Theorem From Oseledec’s theorem we have the following version of Multiplicative Ergodic Theorem for differentiable maps [92]. Theorem I.2.1 Let f be a C1 endomorphism on M. Then there exists a Borel subset Γ ⊂ M with f Γ ⊂ Γ and μ (Γ ) = 1 for any μ ∈ M f (M). Moreover, the following properties hold. (1) There is a measurable integer function r : Γ → Z+ with r ◦ f = r. (2) For any x ∈ Γ , there are real numbers +∞ > λ1 (x) > λ2 (x) > · · · > λr(x) (x) ≥ −∞, where λr(x) (x) could be −∞. (3) If x ∈ Γ , there are linear subspaces V (0) (x) = Tx M ⊃ V (1) (x) ⊃ · · · ⊃ V (r(x)) (x) = {0} of Tx M. (4) If x ∈ Γ and 1 ≤ i ≤ r(x), then lim

n→∞

  1 logTx f n ξ  = λi (x) n

for all ξ ∈ V (i−1) (x)\V (i) (x). Moreover,  r(x) 1  log det(Tx f n ) = ∑ λi (x)mi (x), n→∞ n i=1 lim

dimV (i)(x) for all 1 ≤ i ≤ r(x). where mi (x) = dimV (i−1) (x) −  (5) λi (x) is measurably defined on x ∈ Γ  r(x) ≥ i and f -invariant, i.e. λi ( f x) = λi (x).   (6) Tx f V (i) (x) ⊂ V (i) ( f x) if i ≥ 0.  r(x) The numbers λi (x) i=1 , given by Theorem I.2.1 are called the Lyapunov exponents of f at x, and mi (x) is called the multiplicity of λi (x). In many cases, we require that system (M, f , μ ) satisfies the following integrability condition   (I.1) logdet(Tx f ) ∈ L1 (M, μ ). By Multiplicative Ergodic Theorem, under condition (I.1) we have  M

   logdet(Tx f ) d μ (x) =

r(x)

∑ λi (x)mi (x) d μ (x).

Γ i=1

(I.2)

I.2

Multiplicative Ergodic Theorem

3

   Γ∞ = x ∈ Γ  Tx f is degenerate or λr(x) (x) = −∞ .

Define

The integrability condition (I.1) and identity (I.2) imply that

μ (Γ∞ ) = 0. Let

Γ =Γ\

∞

f −n (Γ∞ ).

(I.3)

(I.4)

n=0

It is easy to see that f (Γ ) ⊂ Γ and for any x ∈ Γ , Tx f is an isomorphism and λr(x) (x) > −∞. From (I.3) we have μ (Γ ) = 1. For x ∈ M and 1 ≤ k ≤ m0 , let (Tx M)∧k be the kth -exterior power space ofTx M, namely, (Tx M)∧k is the linear space of all linear combinations of elements in ξ1 ∧ . . . ∧ ξk : ξi ∈ Tx M, 1 ≤ i ≤ k in which the following relations hold: (1) for all α , β ∈ R and 1 ≤ i ≤ k,

ξ1 ∧ · · · ∧ (αξi + β ξi ) ∧ · · · ∧ ξk = αξ1 ∧ · · · ∧ ξi ∧ · · · ∧ ξk + β ξ1 ∧ · · · ∧ ξi ∧ · · · ∧ ξk (2) for all 1 ≤ i, j ≤ k,

ξ1 ∧ · · · ∧ ξi ∧ · · · ∧ ξ j ∧ · · · ∧ ξk = − ξ1 ∧ · · · ∧ ξ j ∧ · · · ∧ ξi ∧ · · · ∧ ξk    Obviously,  if ξi : 1 ≤ i ≤ m0∧ is a basisof Tx M, then ξi1 ∧ · · · ∧ ξik : 1 ≤ i1 ≤ · · · ≤ ik ≤ m0 is a basis of (Tx M) k . Now, if ei : 1 ≤ i ≤ m0 is an orthonormal basis of Tx M, then by letting def



< ei1 ∧ · · · ∧ eik , e j1 ∧ · · · ∧ e jk > =

1 if (i1 , · · · , ik ) = ( j1 , · · · , jk ) 0 otherwise

we can define an inner product < ·, · > on (Tx M)∧k , and  it is clearly independent of the choice of the orthonormal basis ei : 1 ≤ i ≤ m0 . We shall denote also by | · | the norm on (Tx M)∧k induced by this inner product. If f : M → M is a C1 map, we define for x ∈ M and 1 ≤ k ≤ m0 (Tx f )∧k : (Tx M)∧k → (T f x M)∧k ξ1 ∧ · · · ∧ ξk → (Tx f ξ1 ) ∧ · · · ∧ (Tx f ξk ) and define

m0

|(Tx f )∧ | = 1 + ∑ |(Tx f )∧k |. def

k=1

4

I Preliminaries

Then an important conclusion from [77] gives Proposition I.2.2 Let (M, f , μ ) be given. Then we have lim

n→+∞

and

1 log |(Tx f n )∧ | = ∑ λi (x)+ mi (x), n i

1 lim n→+∞ n



n ∧

log |(Tx f ) | d μ =



μ − a.e.

∑ λi(x)+ mi (x) d μ . i

I.3 Inverse Limit Space Let X be a compact metric space. For any continuous map T on X, let X T denote the subset of X Z consisting of all full orbits, i.e.,    X T = x˜ = {xi }i∈Z  xi ∈ X, T xi = xi+1 , ∀i ∈ Z . Obviously, X T is a closed subset of X Z (endowed with the product topology and the −|i| d(x , y ) for x˜ = {x } Z T metric d(x, ˜ y) ˜ = ∑+∞ i i i i∈Z , y˜ = {yi }i∈Z ∈ X ). X is called i=−∞ 2 the inverse limit space of system (X, T ). Let p denote the natural projection from X T to X, i.e., p (x) ˜ = x0 , ∀x˜ ∈ X T , and θ : X T → X T as the shift homeomorphism. Clearly the following diagram commutes, θ X T −−−−→ X T ⏐ ⏐ ⏐p p⏐

X −−−−→ X T

i.e. p ◦ θ = T ◦ p. It is a basic knowledge that p induces a continuous map from Mθ (X T ) to MT (X), usually still denoted by p, i.e. for any θ -invariant Borel probability measure μ˜ on X T , p maps it to a T -invariant Borel probability measure pμ˜ on X defined by pμ˜ (ϕ ) = μ˜ (ϕ ◦ p), ∀ϕ ∈ C(X). The following proposition guarantees that p is a bijection between Mθ (X T ) and MT (X). Proposition I.3.1 Let T be a continuous map on X. For any T -invariant Borel probability measure μ on X, there exists a unique θ -invariant Borel probability measure μ˜ on X T such that pμ˜ = μ . Before providing the proof of the above proposition, we first introduce two elementary lemmas.

I.3

Inverse Limit Space

5

Lemma I.3.2 Let X and Y be two compact metrizable spaces, and h : X → Y a continuous surjective map. Then for any Borel probability measure μ on Y , there exists a Borel probability measure ν on X such that hν = μ . Proof. Let    W = ψ ∈ C(X)  ∃ϕ ∈ C(Y ) such that ψ = ϕ ◦ h . Obviously W is a linear subspace of C(X). Define a bounded linear functional L on W as follows, Lψ = μ (ϕ ),

where ϕ ∈ C(Y ) such that ψ = ϕ ◦ h.

It is easy to see that L is a positive bounded linear functional with L1 = 1. By a modification of the Hahn-Banach Theorem L can be extended to a positive bounded linear functional on C(X) preserving the property L1 = 1. Then Rieze Representation Theorem implies that there is a Borel probability measure ν on X such that Lψ = ν (ψ ) for all ψ ∈ C(X). It is easy to verify that hν = μ .   Lemma I.3.3 Let X and Y be two compact metrizable spaces, and T : X → X and S : Y → Y measurable mappings on corresponding spaces. Suppose there is a continuous surjective map h : X → Y such that S ◦ h = h ◦ T . Then for any S-invariant Borel probability measure μ on Y , there is a T -invariant Borel probability measure ν on X such that hν = μ . Proof. From Lemma I.3.2, there is a Borel probability measure ν0 on X such that hν0 = μ . Let 1 n−1 νn = ∑ T i ν0 , n i=0 and suppose that νnk → ν as nk → +∞. It is then easy to see that ν ∈ MT (X) and hν = μ .   We are now ready to prove Proposition I.3.1.  n Proof of Proposition I.3.1. Let X0 = ∞ n=0 T (X). Obviously X0 is a compact subset of X, and T (X0 ) = X0 , μ (X0 ) = 1 for any μ ∈ MT (X). Therefore X T = X0T and p : X0T → X0 is continuous and surjective. As a consequence of Lemma I.3.3, there is μ˜ ∈ Mθ (X T ) such that pμ˜ = μ . Since X T is a compact subset of X Z , μ˜ can be uniquely determined by its values on all cylinder sets. For any Borel subsets A0 , A1 , . . . , An ⊂ M, we have      μ˜ [A0 , A1 , . . . , An ] = μ (A0 T −1 A1 · · · T −n An ),

where

   [A0 , A1 , . . . , An ] = x˜ ∈ X T  xi ∈ Ai , i = 0, 1, . . . , n

is a cylinder set in X T . This ensures that μ˜ is uniquely determined by μ . The proof is completed.  

6

I Preliminaries

Remark I.1. In the circumstances of Proposition I.3.1, it is not hard to see that (X T , θ , μ˜ ) is ergodic if and only if (X, T, μ ) is ergodic. The following proposition provides the relationship between the entropies of these two systems. Proposition I.3.4 Let T : X → X be a continuous map on the compact metric space X with an invariant Borel probability measure μ . Let X T be the inverse limit space of (X, T ), θ the shift homeomorphism and μ˜ the θ -invariant Borel probability measure on X T such that pμ˜ = μ . Then h μ (T ) = h μ˜ (θ ).

(I.5)

Proof. For each n ∈ N, take a maximal 1/n-separated set En of X. (Recall that a subset E of a metric space (X, d) is an ε -separated set of X iff d(x, y) ≥ ε for any distinct points x, y ∈ E. It is called a maximal ε -separated set of X if in addition E is maximal, i.e., for any point x ∈ E and y ∈ E, d(x, y) < ε . Given a transform T : X ← and a positive integer n, one can define a new metric dn as  dn (x, y) := max d(T k x, T k y) : 0 ≤ k ≤ n . Then an ε -separated set of (X, dn ) is called  an (n,ε )-separating set of X.) We define   x ∈ En of X such that ξn (x) ⊂ Int(ξn (x)) a measurable finite partition ξ = ξ (x) n n     and Int(ξn (x)) = y ∈ X d(y, x) < d(y, xi ) if x = xi ∈ En for every x ∈ En . Clearly Diamξn ≤ 1/n. By Theorem 8.3 of [92], h μ (T ) = lim hμ (T, ξn ). n→∞

(I.6)

Using ξn , we may construct a measurable finite partition ηn of X T by

ηn =

n 

θ i (p−1 ξn ).

i=−n

It is easy to see Diamηn → 0 as n → ∞, thus hμ˜ (θ ) = lim hμ˜ (θ , ηn ). n→∞

(I.7)

Notice that θ is invertible, by Theorem 4.12 (vii) of [92] we have hμ˜ (θ , ηn ) = hμ˜ (θ , p−1 ξn ) = h μ (T, ξn ). This together with (I.6) and (I.7) yields that identity (I.5) holds.   In the previous proposition, we see that the entropies of these two systems are in fact identical. Now we consider the relationship between the Lyapunov exponents of these two systems.

I.3

Inverse Limit Space

7

For any continuous endomorphism f on the manifold M, let M f denote the inverse limit space of system (M, f ). We still use p to denote the natural projection from M f to M, and θ to denote the shift homeomorphism. For any f -invariant Borel probability measure μ on M, we still use μ˜ to denote the θ -invariant Borel probability measure on M f such that pμ˜ = μ . Let E = p∗ T M for the pull back bundle of the tangent bundle T M by the projection p : M f → M, and p∗ Ex˜ = p∗x˜ T M  Tx0 M p∗x˜ for the natural isomorphisms between fibers Ex˜ and Tx0 M: p∗ ξ = (x, ˜ v)  v, ∀v ∈ Tx0 M, x˜ ∈ M f . p∗x˜ When f is C1 , a fiber preserving map on E, with respect to θ , can be defined as p∗θ x˜ ◦ T f ◦ p∗ : Ex˜ → Eθ x˜ ,

for each x˜ ∈ M f .

Since it is equivalent to T f on the fibers, we still denote it as T f , p T M ←−−∗−− E −−−−→ ⏐ ⏐ ⏐ ⏐ T f T f

Mf ⏐ ⏐

θ

T M −−−−→ E −−−−→ M f p∗ i.e. T f is a continuous bundle endomorphism covering the homeomorphism θ of the compact base M f , so T f is a linear map on each fiber and there is a constant K > 0 such that T f  ≤ K for any x˜ ∈ M f . Let +∞

  Δ = Mf\ θ n p−1 (Γ∞ ) . n=−∞

Obviously θ (Δ ) = Δ and for any x˜ = {xn }n∈Z ∈ Δ we have xn ∈ M\Γ∞ . When the integrability condition (I.1) is satisfied, we have μ˜ (Δ ) = 1. By Theorem 5.2 of [69], (1)-(3) of [83] and the Oseledec’s theorem in the Appendix of [33], we have the following fundamental result. Proposition I.3.5 There exists a Borel set Δ˜ ⊂ Δ , such that θ Δ˜ = Δ˜ and μ˜ (Δ˜ ) = 1. Furthermore, for every x˜ = {xn }n∈Z ∈ Δ˜ , there is a splitting of the tangent space Tx0 M Tx0 M = E1 (x) ˜ ⊕ E2 (x) ˜ ⊕ · · · ⊕ Er(x) ˜ ˜ (x) and numbers +∞ > λ1 (x) ˜ > λ2 (x) ˜ > · · · > λr(x) ˜ > −∞ and mi (x) ˜ (i = 1, 2, . . . , ˜ (x) r(x)), ˜ such that

8

I Preliminaries

1) Txn f : Txn M → Txn+1 M is an isomorphism, ∀n ∈ Z. 2) r(·), λi (·) and mi (·) are θ -invariant, i.e., r(θ x) ˜ = r(x), ˜

λi (θ x) ˜ = λi (x) ˜ and mi (θ x) ˜ = mi (x) ˜

for each i = 1, . . . , r(x). ˜ ˜ = mi (x) ˜ for all n ∈ Z and 1 ≤ i ≤ r(x). ˜ 3) dim Ei (x) 4) The splitting is invariant under T f , i.e., ˜ = Ei (θ n+1 x) ˜ Txn f Ei (θ n x) for all n ∈ Z and 1 ≤ i ≤ r(x). ˜ 5) For any n, m ∈ Z, let ⎧ if m > 0, ⎨ Txn f m , m if m = 0, Tn (x) ˜ = id, ⎩ −m −1 (Tn+m ) , if m < 0. Then,

  1 logTnm (x) ˜ ξ  = λi (x), ˜ m→±∞ m lim

for all 0 = ξ ∈ Ei (θ n x), ˜ 1 ≤ i ≤ r(x). ˜ 6) We introduce ρ (1) (x) ˜ ≥ ρ (2) (x) ˜ ≥ · · · ≥ ρ (m0 ) (x) ˜ ˜ . . . , λ1 (x), ˜ . . . , λi (x), ˜ . . . , λi (x), ˜ . . . , λr(x) ˜ . . . , λr(x) ˜ with to denote λ1 (x), ˜ (x), ˜ (x) λi (x) ˜ being repeated mi (x) ˜ times. Now if {ξ1 , . . . , ξm0 } is a basis of Tx0 M which satisfies   1 logT0m (x) ˜ ξi  = ρ (i) (x) ˜ lim m→±∞ m for every 1 ≤ i ≤ m0 , then for any two non-empty disjoint subsets P, Q ⊂  1, · · · , m0 we have   1 log γ T0m (x)E ˜ P , T0m (x)E ˜ Q =0 m→±∞ m lim

  where EP and EQ denote the subspaces of Tx0 M spanned by the vectors ξi i∈P   and ξ j j∈Q respectively and γ (·, ·) denotes the angle between the two associated subspaces. 7) x0 ∈ Γ and r(x) ˜ = r(x0 ), λi (x) ˜ = λi (x0 ) and mi (x) ˜ = mi (x0 ) for all i = 1, 2, . . . , r(x), ˜ where r(x0 ), λi (x0 ) and mi (x0 ) are as in Theorem I.2.1.  r(x) ˜ Definition I.3.1 The numbers λi (x) ˜ i=1 , given by Proposition I.3.5 are called the ˜ and mi (x) ˜ is called the multiplicity of λi (x). ˜ Lyapunov exponents of (M f , θ , μ˜ ) at x,

Chapter II

Margulis-Ruelle Inequality

Margulis-Ruelle inequality is a basic and important fact in the ergodic theory of smooth dynamical systems, which relates the two key concepts: metric entropy and Lyapunov exponents. It asserts that the entropy can be bounded above by the sum of the positive Lyapunov exponents. Margulis first proved the result for diffeomorphisms preserving a smooth measure. The general statement due to Ruelle [77]. Rigorous proofs are available in several books for the case of diffeomorphisms, see [32] and [57]. A generalization of the result to random diffeomorphisms is given by [51]. In this chapter we present a rigorous proof for the general case, which is valid for non-invertible C1 maps.

II.1 Statement of the Theorem Theorem II.1.1 Let f be a C1 map of a compact, smooth Riemannian manifold M. If μ is an f -invariant Borel probability measure on M, then hμ ( f ) ≤ where



∑ λi (x)+ mi (x) d μ (x),

(II.1)

M i

+∞ > λ1 (x) > λ2 (x) > · · · > λr(x) (x) ≥ −∞

are Lyapunov exponents of f at x and mi (x) is the multiplicity of λi (x) for each i = 1, 2, . . . , r(x).

II.2 Preliminaries Due to the compactness of the manifold M, there is a universal number ρ0 > 0 such that the exponential map M. Qian et al., Smooth Ergodic Theory for Endomorphisms, Lecture Notes in Mathematics 1978, DOI 10.1007/978-3-642-01954-8 II, c Springer-Verlag Berlin Heidelberg 2009 

9

10

II

Margulis-Ruelle Inequality

  expx : Tx M(ρ0 ) = ξ ∈ Tx M : |ξ | < ρ0 → B(x, ρ0 ) ⊂ M is a C∞ diffeomorphism for every x ∈ M, where B(x, ρ0 ) is the open ball in M centered at x. The following lemma is a basic fact in differential geometry. Lemma II.2.1 For any 0 < r ≤ ρ0 there is a number b = b(r) ≥ 1 such that for any x ∈ M, if y and z ∈ B(x, r), then −1 b−1 d(y, z) ≤ | exp−1 x y − expx z| ≤ bd(y, z).

Let X and Y be two m0 -dimensional Euclidean spaces and A : X → Y a linear map. Let A∗ denote the adjoint map of A. The combination of A and A∗ is a nonnegative definite symmetric operator of X. We use χi (A) to denote the non-negative square root of the i-th eigenvalue of A∗ A, in other words, 0 ≤ χ1 (A)2 ≤ χ2 (A)2 ≤ · · · ≤ χm0 (A)2 are eigenvalues of A∗ A. By standard knowledge of linear algebra, we have the following lemma. Lemma II.2.2 Suppose that X and Y are two m0 -dimensional Euclidean spaces and A : X → Y a linear map. Let b be a positive constant. There exists a constant C = C(m0 , b) which depends only on m0 and b, such that for any r > 0, m0   Vol B(AB(0, r), br) ≤ Crm0 ∏ max{χi (A), 1}. i=1

II.3 Proof of the Theorem Let n be a fixed positive Since M is compact and f n is C1 , there is ε ∈   integer. n (0, ρ0 /2) such that f B(x, ε ) ⊂ B( f n x, ρ0 /2) for all x ∈ M, and moreover, for any x, y ∈ M with d(x, y) ≤ ε we have   d f n y, exp f n x ◦Tx f n ◦ exp−1 x y ≤ d(x, y),

(II.2)

and

1 |χi (Tx f n ) − χi (Ty f n )| ≤ , 1 ≤ i ≤ m0 , (II.3) 2 where m0 is the dimension of the manifold. From (II.3), it is easy to verify that   1 max 1, χi (Tx f n )   ≤ 2, ≤ 2 max 1, χi (Ty f n )

1 ≤ i ≤ m0 .

(II.4)

II.3

Proof of the Theorem

11

For each k ∈  N, let Ek be a maximal ε /k-separated set of M. We define a finite partition ξk = ξk (x) : x ∈ Ek of M such that ξk (x) ⊂ Int(ξk (x)) and     Int ξk (x) = y ∈ M : d(y, x) < d(y, x ) if x = x ∈ Ek for every x ∈ Ek . Obviously, ξk (x) ⊂ B(x, ε /k) for all x ∈ Ek and the diameter of ξk satisfies Diam ξk ≤ 2ε /k. Consequently, nhμ ( f ) = hμ ( f n ) = lim hμ ( f n , ξk ). k→∞

(II.5)

For each finite partition ξ , we have 1 Hμ ( f −n+1 ξ ∨ · · · ∨ f −1 ξ ∨ ξ ) n  1 lim Hμ ( f −n+1 ξ ∨ · · · ∨ f −1 ξ |ξ ) + Hμ (ξ ) n→∞ n  1 lim Hμ ( f −n+1 ξ ∨ · · · ∨ f −2 ξ | f −1 ξ ∨ ξ ) + Hμ ( f −1 ξ |ξ ) n→∞ n  1 lim Hμ ( f −n+1 ξ ∨ · · · ∨ f −2 ξ | f −1 ξ ) + Hμ ( f −1 ξ |ξ ) n→∞ n ···  1 n−1 lim Hμ ( f −i ξ | f −i+1 ξ ) ∑ n→∞ n i=1

hμ ( f , ξ ) = lim

n→∞

= = ≤ ≤ ≤

= lim

n→∞

i. e.,

n−1 Hμ ( f −1 ξ |ξ ), n hμ ( f , ξ ) ≤ Hμ ( f −1 ξ |ξ ).

Applying (II.6) to f n and ξk yields hμ ( f n , ξk ) ≤ Hμ ( f −n ξk |ξk ) =

∑ μ (ξk (x))Hμ ( f −n ξk |ξk (x))

x∈Ek

≤

∑ μ (ξk (x)) log Kn (x)

x∈Ek

where Kn (x) = #{x ∈ Ek : f −n ξk (x ) = #{x ∈ Ek : ξk (x )





ξk (x) = 0} /

f n ξk (x) = 0}. /

(II.6)

12

II

Margulis-Ruelle Inequality

Now we estimate the number Kn (x). Let b = b(ρ0 ) given by Lemma II.2.1. By (II.2),   f n ξk (x) ⊂ f n expx B(0, ε /k) ⊂ exp f n x B(Tx f n B 0, ε /k), bε /k , where B(A, δ ) is the δ -neighborhood of A ⊂ T f n x M. If

ξk (x )



f n ξk (x) = 0/

for some x ∈ Ek , we then have  ε   ε ε / B x , exp f n x B Tx f n B(0, ), b = 0. 2k k k Therefore,   ε    ε ε

−1 ε B exp−1 ⊂ exp−1 ⊂ B Tx f n B(0, , 2b . f nx x , b f nx B x , 2k 2k k k

−1 ε ), x ∈ E are also disjoint. Since B(x , ε /2k), x ∈ Ek , are disjoint, B(exp−1 k f nx x , b 2k Hence



−1 ε ) : x ∈ Ek and ξk (x ) f n ξk (x) = 0} / Kn (x) ≤ #{B(exp−1 f nx x , b 2k   ε ε

ε ≤ Vol B(Tx f n B(0, ), 2b ) / min{Vol(B(exp−1 )) : x ∈ Ek } f nx x , k k 2bk

As a consequence of Lemma II.2.2, there is a constant C = C(m0 , b) such that m0

Kn (x) ≤ C ∏ max{χi (Tx f n ), 1}. i=1

By (II.3), we have for y ∈ ξk (x) log+ χi (Tx f n ) ≤ log 2 + log+ χi (Ty f n ) and hence

m0

log Kn (x) ≤ logC + m0 log 2 + ∑ log+ χi (Ty f n ). i=1

It then follows that Hμ ( f −n ξk |ξk ) ≤

∑



x∈Ek ξk (x)

log Kn (x) d μ (y)

≤ logC + m0 log 2 +



m0

∑ log+ χi (Ty f n ) d μ (y). M i=1

II.3

Proof of the Theorem

13

This together with (II.5) yields nh μ ( f ) ≤ logC + m0 log 2 +



m0

∑ log+ χi(Tx f n ) d μ (x). M i=1

Since for every x ∈ M |(Tx f n )∧k | = one has

m0

∏

i=m0 −k+1

χi (Tx f n ), 1 ≤ k ≤ m0 ,

m0

|(Tx f n )∧ | ≥ ∏ max{χi (Tx f n ), 1}. i=1

Hence nhμ ( f ) ≤ logC + m0 log 2 +

 M

  log(Tx f n )∧  d μ (x).

(II.7)

Dividing (II.7) by n and letting n → +∞, we obtain (II.1) from Proposition I.I.2.2. This completes the proof of the theorem.  

Chapter III

Expanding Maps

In the previous chapter, we showed the Margulis-Ruelle inequality, which says that the measure-theoretic entropy is bounded above by the sum of positive Lyapunov exponents. In 1977, Pesin [63] showed that for a C2 diffeomorphism f if its invariant Borel probability measure μ is absolutely continuous with respect to the Lebesgue measure on the manifold, then the equality in (II.II.1) holds, i.e. hμ ( f ) =



∑ λi (x)+ mi (x) d μ (x).

M i

(PEF)

Now identity (PEF) is known as Pesin’s entropy formula. In this chapter, we consider a simple case that this identity holds, namely expanding maps. It is simpler than the case of diffeomorphisms as the local unstable manifold of an expanding map at a point is a neighborhood of the point. The results of this chapter are from [27].

III.1 Main Results Throughout this chapter, we assume that f is a C2 endomorphism of M, Λ is an f invariant closed subset of M, i.e.f Λ = Λ ,and μ is an f -invariant Borel probability measure on Λ . Let φ (x) = − logdet(Tx f ) and λ (x) denote the sum of the positive Lyapunov exponents of f at x, i.e.

λ (x) = ∑ λi (x)+ mi (x). i

For ε > 0, let

  B(x, ε ) = y ∈ M : d(x, y) < ε ,

M. Qian et al., Smooth Ergodic Theory for Endomorphisms, Lecture Notes in Mathematics 1978, DOI 10.1007/978-3-642-01954-8 III, c Springer-Verlag Berlin Heidelberg 2009 

15

16

III

and

Expanding Maps

    Bn (x, ε ) = y ∈ M : d f i x, f i y < ε , i = 0, 1, · · · , n .

Bn (x, ε ) is called the (n, ε )-ball centered at x. And for Γ ⊂ M, let B(Γ , ε ) =



B(x, ε ),

Bn (Γ , ε ) =

x∈Γ



Bn (x, ε ).

x∈Γ

Definition III.1.1 A map f is called an expanding map on an invariant set Λ , if there exist κ0 > 1 and c > 0 such that for all x ∈ Λ and n ≥ 0  n  Tx f (v) ≥ cκ n |v|, 0

for all v ∈ Tx M.

(III.1)

We also say that Λ is an expanding set of f . Remark III.1. From the definition, it is easy to see that if f is an expanding map on an invariant set Λ , then for any x ∈ Λ   1 lim inf logTx f n (v) > 0, n→∞ n

for all v ∈ Tx M\{0}.

(III.2)

Therefore, any Lyapunov exponent of an expanding map is positive. More generally we have the following concept. Definition III.1.2 A map f is called a quasi-expanding map on an invariant set Λ , if there is no negative Lyapunov exponent on Λ . Remark III.2. If f is an expanding or quasi-expanding map, then

λ (x) = lim

n→∞

  1 logdet(Tx f n ) n

μ -a.e.

(III.3)

The main results of this chapter are the following theorems. Theorem III.1.1 If f is a quasi-expanding map on an invariant set Λ and μ is absolutely continuous with respect to the Lebesgue measure m of the manifold M, then Pesin’s entropy formula holds. Theorem III.1.2 If f is an expanding map on an invariant set Λ , and Pesin’s entropy formula holds for the Borel probability measure μ , then IntΛ = 0/ and m(Λ ) > 0. Theorem III.1.3 Under the circumstances of Theorem III.1.2, the connectness of M implies Λ = M, so f is an expanding map on M. Moreover, μ is absolutely continuous with respect to the Lebesgue measure m. Remark III.3. It is known that for an expanding map f on M there exists an f invariant Borel probability measure μ which is absolutely continuous with respect to the Lebesgue measure m, see [91]. Therefore, from Theorems III.1.1 and III.1.3 we can see that if Λ ⊂ M is an expanding set of f , then the following statements are equivalent.

III.2

Proof of Theorem III.1.1

17

(1) There exists an invariant Borel probability measure on Λ such that Pesin’s entropy formula (PEF) holds. (2) There exists an invariant Borel probability measure on Λ which is absolutely continuous with respect to the Lebesgue measure on M. (3) Λ = M.

III.2 Proof of Theorem III.1.1 For ε , δ > 0, let N(n, ε , δ ) denote the minimal number of (n, ε )-balls covering a set with μ -measure not less than 1 − δ . The following result is from Katok’s work [31]. It is generally true for any continuous map. Lemma III.2.1 For any ε > 0, δ > 0, 1 lim sup log N(n, ε , δ ) ≤ h μ ( f ). n→∞ n Proof. Let ξ be a finite measurable partition of Λ such that the maximum diameter −n of the element of ξ is less  than ε /2, and η = ξ ∨ · · · ∨ f ξ . Then for every η (x) k and 0 ≤ k ≤ n, f η (x) lies inside some member of ξ , and its diameter is less than ε /2. This means that η (x) is contained in some (n, ε )-ball. Let       An,ε ,γ = x ∈ Λ : μ η (x) > exp −n h μ ( f , ξ ) + γ . According to Shannon-McMillan-Breiman Theorem (see [58, Theorem 2.53]), for   every γ > 0, μ An,ε ,γ → 1 as n → ∞.   Take γ > 0. There exists N0 > μ An,ε ,γ > 1 − δ for all n > N0 . The  0 such that  set An,ε ,γ contains at most exp n hμ ( f , ξ )+ γ elements of partition η . Since every element lies in an (n,ε )-ball, the minimal  number of (n, ε )-balls that cover set An,ε ,γ  does not exceed exp n hμ ( f , ξ ) + γ . It follows that    N(n, ε , δ ) ≤ exp n hμ ( f , ξ ) + γ . Thus,

1 lim sup log N(n, ε , δ ) ≤ hμ ( f , ξ ) + γ . n→∞ n

Since γ can be taken arbitrarily small and hμ ( f , ξ ) ≤ hμ ( f ), we obtain the conclusion of the lemma.   Lemma III.2.2 Under the conditions of Theorem III.1.1, if

λ (x) ≥ λ then hμ ( f ) ≥ λ .

μ -a.e. x ∈ Λ ,

(III.4)

18

III

Expanding Maps

Proof. Take β > 0, δ ∈ (0, 1/2) and ε > 0 such that the following two conditions are satisfied. (i) The distance between Λ and the set of critical points of f is larger than 2ε . (ii) For every x ∈ Λ , if a measurable set B ⊂ B(x, 2ε ), then     (III.5) m f (B) ≥ (1 − β )det(Tx f )m(B). Let

  An = x ∈ Λ : | det(Tx f k )| ≥ exp[k(λ − β )], ∀k ≥ n .

then there exists N0 > 0 such that μ (An ) > 1 − δ as n > N0 . ˜ ε , 2δ ) denote the minimal number of (n, ε )-balls which cover An Let N˜ = N(n, whose μ -measure is equal to or greater than 1 − 2δ and suppose that the set of (n, ε )-balls   Bn (x, ε ) : x ∈ S ⊂ M, #S = N˜ has such property. Since μ is absolutely continuous with respect to m, there exists κ > 0 such that for any Γ ⊂ Λ and μ (Γ ) ≥ 1 − 2δ , one has m(Γ ) ≥ κ . Thus we have 

 m Bn (x, ε ) ≥ κ (III.6) by the assumption μ



x∈S



x∈S Bn (x, ε )

≥ 1 − 2δ .



For every x ∈ S there exists y = y(x) ∈ Bn (x, ε ) An . Then Bn (y, 2ε ) ⊃ Bn (x, ε ), and by the definition of Bn (y, 2ε ), one has   f n Bn (y, 2ε ) ⊂ B( f n y, 2ε ).

(III.7)

(III.8)

From (III.5), it yields n−1     m f n (Bn (y, 2ε )) ≥ (1 − β )n ∏ | det(T f k y f )| · m Bn (y, 2ε ) k=0

  = (1 − β )n| det(Ty f n )| · m Bn (y, 2ε )   ≥ (1 − β )nen(λ −β )m Bn (y, 2ε ) . Then by (III.7) and (III.8),     m B( f n y, 2ε ) ≥ (1 − β )nen(λ −β )m Bn (x, ε ) .   Let K = supz∈M m B(z, 2ε ) . Then for every x ∈ S the m-measure of Bn (x, ε ) is less than K(1 − β )−ne−n(λ −β ). Comparing it with (III.6), we obtain ˜ ε , 2δ ) ≥ κ K −1 (1 − β )nen(λ −β ). N(n,

III.3

Basic Facts About Expanding Maps

19

˜ ε , 2δ ), it follows that Since N(n, ε , δ ) ≥ N(n, 1 lim sup log N(n, ε , δ ) ≥ log(1 − β ) + λ − β ≥ λ − 2β . n→∞ n Thus hμ ( f ) ≥ λ − 2β by Lemma III.2.1. The fact that β can be taken arbitrarily small implies that hμ ( f ) ≥ λ . This completes the proof of the lemma.   Proof of Theorem III.1.1. By Margulis-Ruelle inequality, it suffices to show that  (III.9) hμ ( f ) ≥ λ (x) d μ (x). Take α > 0. Since f is a C2 endomorphism, there exists L > 0 such that | det(Tx f )| < L for all x ∈ M. Then by (III.3), λ (x) ≤ L. Let   Γk = x ∈ Λ : (k − 1)α ≤ λ (x) ≤ kα .  [L/α ]+1 is a finite measurable Clearly Γk is an f -invariant set and the family Γk k=1 partition of Λ . If μ (Γk ) > 0 for some k,  we may define the conditional probability measure of μ on Γk by putting μk (·) = μ (· Γk )/ μ (Γk ). Obviously, μk is f -invariant and absolutely continuous with respect to m. Then Lemma III.2.2 can be applied and yields that h μk ( f ) ≥ (k − 1)α .

(III.10)

We extend μk to a Borel probability measure by putting μk (Λ \Γk ) = 0. Then μ = ∑Nk=1 μ (Γk )μk , where N = [L/α ] + 1. Since the entropy map μ → hμ ( f ) is affine (see [92, §8.1]) and by (III.10) we have hμ ( f ) =

N

N

N

k=1

k=1

k=1

∑ μ (Γk ) · hμk ( f ) ≥ ∑ μ (Γk ) · (k − 1)α = ∑ kα · μ (Γk ) − α .

Therefore, hμ ( f ) ≥



λ (x) d μ (x) − α .

Since α can be taken arbitrarily small, we obtain (III.9). This completes the proof of the theorem.

III.3 Basic Facts About Expanding Maps In this section we consider some useful properties of expanding maps. Using Mather’s method (see Hirsch and Pugh [25]) with a little modification, we can choose an adapted metric of f on M such that the inequality (III.1) holds with c = 1. More precisely, we have the following proposition.

20

III

Expanding Maps

Proposition III.3.1 A Riemannian structure on M can be chosen such that there exists κ1 with κ0 ≥ κ1 > 1 satisfying   Tx f (v) ≥ κ1 |v|,

∀v ∈ Tx M

for any x ∈ Λ . Due to the continuity of the tangent map Tx f in x and the compactness of Λ , we have the following proposition. Proposition III.3.2 There exist κ ∈ (1, κ1 ), r > 0, and ε0 > 0, such that for any x ∈ B(Λ , r) we have   d f y, f z ≥ κ d(y, z),

for all y, z ∈ B(x, ε0 ).

(III.11)

Therefore, when ε ≤ ε0 , f i Bn (x, ε ) = Bn−i ( f i x, ε ), i = 0, 1, . . . , n, f n+1 Bn (x, ε ) ⊃ B( f n+1 x, ε ). Furthermore, for any 0 < ε ≤ ε0 , we can find a K such that f K (B(x, ε )) ⊃ B( f K x, ε0 ),

∀x ∈ Λ ,

where K = [log εε0 / log κ ] + 1. Since f expands on B(Λ , r) and B(Λ , r) is compact, for every x ∈ B(Λ , r) there  are only finite number of points in the set f −1 {x} B(Λ , r). So we have the following result. 

Proposition III.3.3 For any ε ≤ ε0 and x ∈ B(Λ , r), if x ∈ f −1 (x) B(Λ , r), then f |B(x ,ε ) : f −1 B(x, ε ) → B(x, ε ) is a diffeomorphism. n−1       Let us recall that φ (x) = − logdet(Tx f ), so ∑ φ f i x = − logdet(Tx f n ). i=0

Proposition III.3.4 There exists b > 0, such that for any x ∈ B(Λ , r), y ∈ B(x, ε ) (ε ≤ ε0 ), we have n−1   n−1      ∑ φ f i x − ∑ φ f i y  ≤ b, i=0

∀n ≥ 1.

i=0

Proof. Since | det(Tx f )| > 1 on B(Λ , r) and f is C2 , φ is C1 . Let   φ 1 = sup |Dφ (x)| : x ∈ B(Λ , r) .

III.3

Basic Facts About Expanding Maps

Then if d(x, y) ≤ ε ,

21

|φ (x) − φ (y)| ≤ φ 1 d(x, y).

When y ∈ Bn (x, ε0 ) for 0 ≤ i ≤ n, we have d( f i x, f i y) ≤ κ −(n−i) d( f n x, f n y) ≤ κ −(n−i)ε0 , and  n   n     ∑ φ f ix − ∑ φ f iy  ≤ i=0

i=0

n





∑ φ ( f i x) − φ ( f i y)

i=0

n

≤ φ 1 ∑ d( f i x, f i y) i=0

n

≤ φ 1 ε0 ∑ κ −(n−i) i=0

κ ≤ ε0 φ 1 . κ −1 Setting b = κε0 φ 1 /(κ − 1) completes the proof.

 

Proposition III.3.5 For every ε , δ < ε0 , there exists a = a(ε , δ ) > 0 such that for x ∈ Λ , y ∈ Bn (x, ε ), we have     m Bn (y, δ ) ≥ a m Bn (x, ε ) ,

∀n ≥ 0.

  Proof. Since f n Bn (x, ε ) = B( f n x, ε ),    m B( f n x, ε ) =

Bn (x,ε )

  det(Ty f n ) dm(y).

When y ∈ Bn (x, ε ), by Proposition III.3.4, e

−b

  det(Tx f n )  ≤ eb . ≤  det(Ty f n )

Consequently,

      m B( f n x, ε ) ≥ det(Tx f n )e−b m Bn (x, ε ) .

Similarly,

      m B( f n y, δ ) ≤ det(Ty f n )eb m Bn (y, δ ) .

From these three inequalities it follows that    m B( f n y, δ )   m Bn (x, ε ) . m(Bn (y, δ )) ≥ e−3b  m B( f n x, ε )

22

III

Expanding Maps

Therefore we can put   e−3b infy∈M m B(y, δ )   a = a(ε , δ ) = supx∈M m B(x, ε )  

to obtain the conclusion of the proposition.

III.4 Proofs of Theorems III.1.2 and III.1.3 Lemma III.4.1 For any α > 0 and β > 0, there exist  = (α , β ) ≥ 1 and a measurable set Λ ⊂ Λ , such that μ (Λ ) > 1 − β and for each x ∈ Λ , we have   −1 en(λ (x)−α ) ≤ det(Tx f n ) ≤ en(λ (x)+α ),

∀n ≥ 0.

Proof. By (III.3), for μ -a.e. x ∈ Λ there exists N = N(x) > 0 such that for all n > N,   e−nα < det(Tx f n )e−nλ (x) < enα . Let

  −1    Lα (x) = sup 1, det(Tx f n )e−n(λ (x)+α ), det(Tx f n ) en(λ (x)−α ) . n≥1

Then Lα (x) is measurable and Lα (x) < +∞ μ -a.e., and we have   n(λ (x)−α ) L−1 ≤ det(Tx f n ) ≤ Lα (x)en(λ (x)+α ) . α (x)e Since Λ = and Λ .



≥1

  x : Lα (x) ≤  (mod 0), it is clear that we can find the required   

Lemma III.4.2 If Λ is a nowhere dense set, then there exists 0 < γ < ε0 such that for all ε ∈ (0, γ ),   1 lim inf log m Bn (Λ , ε ) < 0. (III.12) n→∞ n Proof. Since Λ is a nowhere dense set, we may take γ > 0 such that for every x ∈ Λ there exists y ∈ B(x, ε0 ) with d(y,   Λ ) > 2γ .  Choose k > 0 so that f k B x, γ /2 ⊃ B f k x, ε0 for all x ∈ Λ . Since for every x∈Λ       f n+k Bn x, γ /2 = f k B f n x, γ /2 ⊃ B f n+k x, ε0 ,   there exists y = yn,x ∈ Bn x, γ /2 such that   d f n+k yn,x , Λ > 2γ . Take δ ∈ (0, γ /2) so that d( f k y, f k z) < γ whenever d(y, z) ≤ δ .

III.4

Proofs of Theorems III.1.2 and III.1.3

23

  If z ∈ Bn yn,x , δ , then for i = 0, 1, . . . , n, d( f i z, f i x) ≤ d( f i z, f i y) + d( f i y, f i x) ≤ i.e.

γ γ + = γ, 2 2

 Bn yn,x , δ ) ⊂ Bn (x, γ ).

(III.13)

On the other hand, d( f n+k z, Λ ) ≥ d( f n+k y, Λ ) − d( f n+k y, f n+k z) > γ > 0, i.e.

  Bn+k (Λ , γ ) = 0. / Bn yn,x , γ

(III.14)

set of Λwith maximal cardinality. Then the (n, γ )Let S be an (n, 2γ )-separating   balls Bn (x, γ ) : x ∈ S are disjoint, and Bn (x, 2γ ) : x ∈ S covers Λ , therefore

Bn (x, 3γ ) ⊃ Bn (Λ , γ ).

x∈S

By (III.13) and (III.14), Bn (Λ , γ ) ⊃ Bn+k (Λ , γ )



  Bn yn,x , δ

x∈S

and the sets on the right-hand side are disjoint. Using Proposition III.3.5 we have     m Bn (Λ , γ ) − m Bn+k (Λ , γ )   ≥ ∑ m Bn (yn,x , δ ) x∈S

  ≥ a(3γ , δ ) ∑ m Bn (x, 3γ ) x∈S

  ≥ a(3γ , δ )m Bn (Λ , γ ) . So

      m Bn+k (Λ , γ ) ≤ 1 − a(3γ , δ ) m Bn (Λ , γ ) .

Therefore, by induction we have    j   m Bn+ jk (Λ , γ ) ≤ 1 − a(3γ , δ ) m Bn (Λ , γ ) . Thus,

   1  1 lim inf log m Bn (Λ , γ ) ≤ log 1 − a(3γ , δ ) < 0. n→∞ n k     The result follows from m Bn (Λ , ε ) ≤ m Bn (Λ , γ ) when ε < γ .

 

24

III

Expanding Maps

Let us remark that if hμ ( f , x) is the local entropy of f at x, then we have the following properties (see [12]).   1 (a) h μ ( f , x) = lim lim inf − log μ Bn (x, ε ) ε →0 n→∞ n   1 = lim lim sup − log μ Bn (x, ε ) . ε →0 n→∞ n (b)  hμ ( f , x) is f -invariant. hμ ( f , x) d μ (x) = hμ ( f ).

(c)

If f is a C1 map, then the relationship between hμ ( f , x) and λ (x) is as follows. Lemma III.4.3

and

hμ ( f , x) ≤ λ (x),

μ -a.e.

hμ ( f , x) = λ (x),

μ -a.e.

if and only if hμ ( f ) =



λ (x) d μ (x).

Proof. Let

 1 . Γk = x : hμ ( f , x) − λ (x) ≥ k Suppose μ (Γk ) > 0 for some k. We define the conditional measure on Γk by putting  μ (· Γk ) . μk (·) = μ (Γk )     Since μk Bn (x, ε ) ≤ μ Bn (x, ε ) /μ (Γk ), by (a) we have hμk ( f , x) ≥ hμ ( f , x). Then for all x ∈ Γk , 1 h μk ( f , x) ≥ λ (x) + . k Integrating both sides over Γk and applying (c) yields hμk ( f ) ≥



1 λ (x) d μk (x) + . k

This contradicts the Margulis-Ruelle inequality. So we must have μ (Γk ) = 0 for all k.    The first assertion follows from x : hμ ( f , x) ≥ λ (x) = k≥1 Γk . And the second assertion is clear from the first one.   Proof of Theorem III.1.2. Take α > 0 and β ∈ (0, 1/2). By property (a) of local entropy, there exist B1 ⊂ Λ , δ > 0 and N > 0 such that μ (B1 ) > 1 − β and if ε < δ , n > N, then for all x ∈ B1

III.4

Proofs of Theorems III.1.2 and III.1.3

25

  1 − log μ Bn (x, ε ) > h μ ( f , x) − α . n

(III.15)

By Lemma III.4.1, there exist  > 0 and B2 = Λ ⊂ Λ such that μ (B2 ) > 1 − β , and for all n > 0, x ∈ B2   det(Tx f n ) ≤  exp{n(λ (x) + α )}.

(III.16)

By Lemma III.4.3, the validity of Pesin’s entropy formula (PEF) implies hμ ( f , x) = λ (x)

μ -a.e.

Let

  Γk = x ∈ Λ : (k − 1)α ≤ hμ ( f , x) = λ (x) < kα .   Then Γk : k = 1, . . . , [J/α] + 1 isa finite measurable partition of Λ , where J ≥  supx∈M | det(T f )|. Since μ B1 B2 > 1 − 2β there exists at least an integer k such  x that μ (Γk B1  B2 )> 0. Let Γ = Γk B1 B2 . Take ε < min{ε0 , γ , δ }, where δ is as above and γ is the same as in Lemma III.4.2. If x ∈ Γ , then by Propositions III.3.2, III.3.4 and (III.16),    m B( f n x, ε ) =

| det(Tz f n )| dm(z)   ≤ | det(Tx f n )|eb m Bn (x, ε )   ≤ en(λ (x)+α )eb m Bn (x, ε )   < eb en(k+1)α m Bn (x, ε ) ,

i.e.

Bn (x,ε )

    m Bn (x, ε ) > −1 e−b e−n(k+1)α inf m B(z, ε ) . z∈M

On the other hand, by (III.15) we have   μ Bn (x, ε ) < e−n(hμ ( f ,x)−α ) ≤ e−n(k−2)α . Thus there exist μ (Γ ) · en(k−2)α disjoint (n, ε )-balls with the centers at points in Γ (ε can be reduced smaller if necessary),  and every (n, ε )-ball has m-measure larger that −1 e−b · e−n(k+1)α · infz∈M m B(z, ε ) . Since Bn (Λ , ε ) contains all of the (n, ε )-balls, it follows that for n > N,   m Bn (Λ , ε ) > Ke−3nα ,   where K = −1 e−b · μ (Γ ) · infz∈M m B(z, ε ) independent of n. Therefore,   1 lim inf log m Bn (Λ , ε ) ≥ −3α . n→∞ n

26

III

Expanding Maps

As α can be taken arbitrarily small, we have   1 lim inf log m Bn (Λ , ε ) ≥ 0. n→∞ n If Λ is nowhere dense, then it contradicts Lemma III.4.2. This completes the proof of the theorem.   Proof of Theorem III.1.3. Let

Ω1 =

∞ 

  f i Ω ( f |Λ ) .

i=1

Then Ω1 ⊂ Λ , f Ω1 = Ω1 and μ (Ω1 ) = 1. using Theorem III.1.2 with set Ω1 replacing Λ we obtain IntΩ1 = 0. / Take ε < ε0 /3. Suppose x ∈ B(IntΩ1 , ε ). Then there exists y = y(x) ∈ IntΩ1 , such that d(x, y) < 2ε . Take η ∈ (0, ε ) such that B(y, 2η ) ⊂ IntΩ1 and take N > 0 such that κ N > 3ε /η . Since y ∈ Ω1 , there exist n > N and z ∈ B(y, η ) such that f n z ∈ B(y, η ), and d(x, f n z) ≤ d(x, y) + d(y, f n z) ≤ 2ε + η < 3ε ,   i.e. x ∈ B f n z, 3ε . By (III.11) and 3ε /κ n < 3ε /κ N < η , we have Bn−1 (z, 3ε ) ⊂ B(z, η ) ⊂ IntΩ1 . Hence, Therefore,

f n Bn−1 (z, 3ε ) ⊂ f n IntΩ1 ⊂ IntΩ1 . x ∈ B( f n z, 3ε ) ⊂ f n Bn−1 (z, 3ε ) ⊂ IntΩ1 .

Now we have obtained the relation B(IntΩ1 , ε ) ⊂ IntΩ1 . That is, IntΩ1 is not only closed but also open in M. Since M is connected, IntΩ1 = M. Then result Λ = M follows from Λ ⊃ IntΩ1 . The last statement of Theorem III.1.3 follows from the uniqueness of the equilibrium state of an expanding map [78]. The whole proof is completed.   Remark III.4. Expanding maps can be viewed as a special case of the next chapter, but the theorem about Markov partition has to be proved seperately, we refer this to [72]. The main result of this chapter is also a special case of Chapter V, since the existence of an invariant density is valid [39].

Chapter IV

Axiom A Endomorphisms

Axiom A system is an important class in smooth dynamical systems. In the ergodic theory of Anosov diffeomorphisms or of Axiom A attractors of diffeomorphisms, it was shown that there is an invariant Borel probability measure that is characterized by each of the following properties: (1) Pesin’s entropy formula (PEF) holds. (2) SRB property: its conditional measures on unstable manifolds are absolutely continuous with respect to the Lebesgue measures on the corresponding submanifolds. (3) Lebesgue a.e. point in an open set is generic with respect to this measure. (4) This measure is approximable by measures that are invariant under suitable stochastic perturbations. Each one of these properties has been shown to be significant in its own right, but more striking is the fact that they are all equivalent to each one another. Many of these ideas are due to Sinai, Ruelle and Bowen. We refer the reader to [88], [10], [76] and [79]. In this chapter, we study the Axiom A endomorphisms along the line of [72].

IV.1 Introduction and Main Results Let M be an m0 -dimensional, smooth, compact and connected Riemannian manifold without boundary, m the Lebesgue measure on M given by the Riemannian structure. Let O be an open subset of M, f ∈ C1 (O, M), and Λ = f Λ ⊂ O be a compact invariant set of f . As in Section I.2, we may define the inverse limit space Λ f , which is a closed subspace of M Z (endowed with the product topology and the −|i| d(x , y ) for x˜ = {x } Z metric d(x, ˜ y) ˜ = ∑+∞ i i i i∈Z , y˜ = {yi }i∈Z ∈ M ), and the i=−∞ 2 ∗ pull back bundle E = p TΛ M, where p is the natural projection map from Λ f to Λ . Recall that the tangent map T f on TΛ M induces a fiber preserving map on E, with respect to the left shift map θ : Λ f → Λ f , defined by p∗ ◦ T f ◦ p∗ , and still denoted by T f . M. Qian et al., Smooth Ergodic Theory for Endomorphisms, Lecture Notes in Mathematics 1978, DOI 10.1007/978-3-642-01954-8 IV, c Springer-Verlag Berlin Heidelberg 2009 

27

28

IV

Axiom A Endomorphisms

Definition IV.1.1 Λ is called a hyperbolic set of f if there is a continuous splitting E = E s ⊕ E u such that (1) T f (E s ) ⊂ E s , T f (E u ) ⊂ E u ; (2) |T f n (ξ s )| ≤ Aan |ξ s |, ∀ξ s ∈ E s , n ∈ Z+ , |T f n (ξ u )| ≥ A−1 a−n |ξ u |, ∀ξ u ∈ E u , n ∈ Z+ , where 0 < a < 1 and A are constants. Remark IV.1. In Definition IV.1.1, for each x˜ ∈ Λ f , the decomposition of the fiber Ex˜ = Exs˜ ⊕ Exu˜ may depend on the past. More precisely, it may happen that p∗ Exu˜ = ˜ However, from (2) of Definition IV.1.1 it is easy to p∗ Eyu˜ though x0 = y0 (but x˜ = y). see that such a phenomenon is impossible for p∗ Exs˜ , it depends only on x0 . We will denote p∗ Exs˜ as (p∗ E s )x0 , which defines a bundle p∗ E s on Λ . Remark IV.2. By a general argument, we may suppose that the constant A in Definition IV.1.1 is equal to 1. Definition IV.1.2 Λ is called an Axiom A basic set of f if (i) Λ is hyperbolic; (ii) P( f ) ⊃ Λ , where P( f ) is the set of periodic points of f ; (iii) f is topologically + transitive on Λ , i.e. there is an x ∈ Λ such that Orb+ (x) =  i +∞ f x i=0 is dense on Λ ;  i (iv) there exists an open set V ⊃ Λ satisfying +∞ i=−∞ f (V ) = Λ . An Axiom A basic set is called an Axiom A attractor if there exist arbitrarily small open neighborhoods U of Λ such that f (U) ⊂ U. Definition IV.1.3 The map f ∈ C1 (M, M) is called an Axiom A endomorphism if (i) (ii) (iii)

Ω ( f ) (the nonwandering set of f ) is hyperbolic; P( f ) = Ω ( f );   m(C f ) = 0, where C f = x ∈ M : | det(Tx f )| = 0 .

Remark IV.3. It is easy to see that m(C f ) = 0 if C f consists of a countable collection of M s submanifolds of which the dimension ≤ m0 − 1. The main results of this chapter are the following theorems. Theorem IV.1.1 Let f ∈ C2 (O, M) and Λ ⊂ O be an Axiom A basic set. Then Λ is an Axiom A attractor if and only if there is an f -invariant Borel probability measure μ on Λ satisfying Pesin’s entropy formula: hμ ( f ) =



∑ λi (x)+ mi (x) d μ (x).

Λ i

In addition, we have also the following results:

(IV.1)

IV.2

Preliminaries

29

(i) μ is ergodic; (ii) if ε > 0 is small enough and m(C f ) = 0, then for m-almost all x ∈ Bε (Λ ), 1 n−1 ∑ ψ ( f i x) = n→∞ n i=0 lim

 Λ

ψ dμ

(IV.2)

   for every ψ ∈ C(B ε (Λ )), where Bε (Λ ) = y ∈ M : d(y, Λ ) < ε and C f = x ∈ O : | det(Tx f )| = 0 . Theorem IV.1.2 Let f be a C2 Axiom A endomorphism. Then there is M ⊂ M such that m(M ) = m(M), and for any ψ ∈ C(M), 1 n−1 ∑ ψ ( f i x) n→∞ n i=0

ψ ∗ (x) = lim

(IV.3)

are well defined for all x ∈ M , and so defined ψ ∗ : M → R is a measurable function with finite values.

IV.2 Preliminaries In this section we present some basic dynamical properties for hyperbolic invariant sets. Let Λ be a hyperbolic invariant set of f ∈ Cr (O, M) (r ≥ 1) and E = E s ⊕ E u the hyperbolic splitting. For ρ > 0, denote  def  E s (ρ ) = ζ s ∈ E s : |ζ s | < ρ ,  def  E u (ρ ) = ζ s ∈ E u : |ζ u | < ρ , def

E(ρ ) = E s (ρ ) ⊕ E u(ρ ), and

 def  p∗ E ⊥ = ζ ∈ TΛ M : ζ is perpendicular to p∗ E s .

For small ε > 0 and x˜ ∈ Λ f , define the local stable and unstable manifolds respectively as  def  Wεs (x0 ) = y0 ∈ M : d( f n x0 , f n y0 ) < ε , ∀n ∈ Z+  Wεu (x) ˜ = z0 ∈ M : ∃˜z ∈ M f such that z0 = p (˜z),  d(x−n , z−n ) < ε , ∀n ∈ Z+ . We have the following stable and unstable manifold theorems (see [67, 81, 94]).

30

IV

Axiom A Endomorphisms

Theorem IV.2.1 (Stable manifold theorem) For ρ > 0 small enough, there is a fiber-preserving map H s : E s (ρ ) → E u (ρ ) satisfying ˜ s , D˜ 2 H s , . . . , D˜ r H s (i) H s is continuous on E s (ρ ), Cr along each fiber, and DH s i th ˜ are continuous on E (ρ ), where D (·) denotes the i order derivative along the fiber. ˜ s (0x˜ ) = 0 for each x˜ ∈ Λ f , where 0x˜ is the zero vector in Ex˜ ; (ii) H s (0x˜ ) = 0x˜ , DH   s (iii) Denoting Dx˜ = expx0 ◦p∗ ◦ (id, H s )Exs˜ (ρ ) for each x˜ ∈ Λ f , then Dsx˜ x∈ is a ˜ Λf continuous family of Cr -embedded disks and there exist constants ε > 0 and 0 < λ < 1 such that (1) ∀x˜ ∈ Λ f , Tx0 Dsx˜ = p∗ Exs˜ ;  (2) ∀x˜ ∈ Λ f and ∀y0 ∈ Dsx˜ B(x0 , ε ), d( f n x0 , f n y0 ) ≤ λ n d(x0 , y0 ) for any n ∈   def Z+ , where B(x0 , ε ) = z0 ∈ M : d(x0 , z0 ) < ε ;  (3) ∀x˜ ∈ Λ f , Wεs (x0 ) = Dsx˜ B(x0 , ε ). Remark IV.4. It was actually proved in [94] that there is a fiber-preserving map H0s : p∗ E s (ρ ) → p∗ E ⊥ (ρ ) such that the statement of Theorem IV.2.1 is still valid if the H s , x˜ and Λ f are replaced by H0s , x0 and Λ , respectively. Here ρ > 0 is small and p∗ E s (ρ ), p∗ E ⊥ (ρ ) denote the ρ -ball bundles similarly as E s (ρ ) and E u (ρ ). Therefore the family of local stable manifolds is continuous not only in x˜ ∈ Λ f but also in x0 ∈ Λ , which will be useful later. Theorem IV.2.2 (Unstable manifold theorem) For ρ > 0 small enough, there is a fiber-preserving map H u : E u (ρ ) → E s (ρ ) satisfying ˜ u , D˜ 2 H u , . . . , D˜ r H u (i) H u is continuous on E u (ρ ), Cr along each fiber, and DH are continuous on E u (ρ ); ˜ u (0x˜ ) = 0 for each x˜ ∈ Λ f ; (ii) H u (0x˜ ) = 0x˜ , DH   u (iii) Denoting Dx˜ = expx0 ◦p∗ ◦ (H u , id)Exu˜ (ρ ) for each x˜ ∈ Λ f , then Dux˜ x∈ is ˜ Λf a continuous family of Cr -embedded disks and there exist constants ε > 0 and 0 < λ < 1 such that (1) ∀x˜ ∈ Λ f , Tx0 Dux˜ = p∗ Exu˜  ; (2) ∀x˜ ∈ Λ f and ∀z0 ∈ Dux˜ B(x0 , ε ), there exists a unique z˜ ∈ M f satisfying n + p(˜z) = z0 and d(x−n , z −n ) ≤ λ d(x0 , z0 ) for each n ∈ Z ; f u u ˜ = Dx˜ B(x0 , ε ). (3) ∀x˜ ∈ Λ , Wε (x) Remark IV.5. From the proof of Theorems IV.2.1 and IV.2.2, we know that in fact  Wεs (x0 ) = y0 ∈ M : d( f n x0 , f n y0 ) < ε , ∀n ∈ Z+  and lim d( f n x0 , f n y0 ) = 0 . n→+∞

and  Wεu (x) ˜ = z0 ∈ M : ∃˜z ∈ M f such that z0 = p (˜z),

 d(x−n , z−n ) < ε , ∀n ∈ Z+ and lim d(x−n , z−n ) = 0 , n→+∞

where x˜ ∈ Λ f .

IV.3

Volume Lemma and the H¨older Continuity of φ u

31

Theorem IV.2.3 Assuming ε > 0 small enough, we have 0 < δ < ε satisfying ˜ will transversally in(1) Given y˜ ∈ Λ f and x0 ∈ Λ with d(y0 , x0 ) < δ , then Wεu (y) tersects with Wεs (x0 ) at a unique point zx0 ,y˜ ; (2) zx0 ,y˜ is continuous with respect to (x0 , y) ˜ ∈ Λ ×Λ f; (3) if we assume furthermore that Λ is an Axiom A basic set of f , then for arbitrary y˜ ∈ Λ f and x0 ∈ Λ with d(x0 , y0 ) < δ , there is a unique w˜ ∈ Λ f satisfying p (w) ˜ = zx0 ,y˜ and d(w−n , y−n ) < ε for all n ∈ Z+ . Remark IV.6. This theorem is called the local product structure theorem, from which the pseudo-orbit-shadowing lemma can be derived [81]. Theorem IV.2.4 If f is an Axiom A endomorphism, then Ω ( f ) can be spectrally decomposed into the union of a finite family of pairwise disjoint Axiom A basic sets of f . This theorem can be found in [81]. Using Theorem IV.2.1, Remark IV.4 and Theorem IV.2.3, we can prove the following proposition. For the basic line of its proof, one is referred to [10, Lemma 4.9]. Proposition IV.2.5 Suppose Λ is an Axiom A basic set of f . Then the following properties are equivalent. (1) Λ is an Axiom A attractor of f . (2) There is some x˜ ∈ Λ f and small ε > 0 such that Wεu (x) ˜ ⊂ Λ. (3) There is a small ε > 0 such that Wεu (x) ˜ ⊂ Λ for all x˜ ∈ Λ f . (4) There is a sufficiently small ε > 0 such that the set  Wεs (Λ ) = z0 ∈ Λ : d( f n z0 , Λ ) < ε for all n ∈ Z+  and lim d( f n z0 , Λ ) = 0 n→∞

is an open neighborhood of Λ . (5) There is a sufficiently small ε > 0 such that B(ε , ∞) =

x0 ∈Λ

Wεs (x0 )

is a neighborhood of Λ . If Λ is not an Axiom A attractor, for ε > 0 small enough there is 0 < rε < ε such that for each x˜ ∈ Λ f we have some y0 ∈ Wεu (x) ˜ with d(y0 , Λ ) > rε .

IV.3 Volume Lemma and the H¨older Continuity of φ u Let Λ be a hyperbolic invariant set of f ∈ C2 (O, M). φ u : Λ f → R is defined as   u   φ (x) ˜ = − log det(Tx0 f |Exu˜ ) for each x˜ ∈ Λ f . Before showing the main results of this section we present some lemmas.

32

IV

Axiom A Endomorphisms

Lemma IV.3.1 Assume ε > 0 small enough. We have small ρ > 0 and fiberpreserving map Φ¯ : E(ρ ) → E such that Φ¯ , D˜ Φ¯ and D˜ 2 Φ¯ are continuous on E(ρ ), and the map Φ : E(ρ ) → M defined as Φ |Ex˜(ρ ) = Φx˜ = expx0 ◦p∗ ◦ Φ¯ |Ex˜(ρ ) for each x˜ ∈ Λ f satisfies the following properties. (1) ∀x˜ ∈ Λ f , Φx˜ is a C2 -diffeomorphism to its image (so is a C2 -chart map); (2) ∀x˜ ∈ Λ f , Φx˜ (0x˜ ) = x0 , DΦx˜ (0x˜ ) = p∗ |Ex˜ ; (3) ∀x˜ ∈ Λ f , we have

Φx˜ (Exs˜ (ρ )) = Wεs (x0 ) Φx˜ (Exu˜ (ρ )) = Wεu (x) ˜





Φx˜ (Ex˜ (ρ )),

Φx˜ (Ex˜ (ρ ));

(4) there exists A1 > 1 such that for arbitrary x˜ ∈ Λ f , v ∈ Ex˜ (ρ ), ζ ∈ Ex˜ and orthonormal family of vectors ζ1 , . . . , ζm0 in Ex˜ , we have (i) A−1 1 |ζ | ≤ |DΦx˜ (v)(ζ )| ≤ A1 |ζ |; (ii) A−1 1 ≤ |DΦx˜ (v)(ζ1 ) ∧ · · · ∧ DΦx˜ (v)(ζm0 )| ≤ A1 ; so the distance and volume on Ex˜ (ρ ) are equivalent to those on Φx˜ (Ex˜ (ρ )) ⊂ M, i.e. they are equal respectively up to bounded factors. Remark IV.7. It is easy to see from the following proof that the estimate in (4) can in fact be more precise: one can find a B1 > 0 satisfying (i ) (1 − B1|v|)|ζ | ≤ |DΦx˜ (v)(ζ )| ≤ (1 + B1|v|)|ζ |, (ii ) (1 − B1|v|) ≤ |DΦx˜ (v)(ζ1 ) ∧ · · · ∧ DΦx˜ (v)(ζm0 )| ≤ (1 + B1|v|). Proof of Lemma IV.3.1. For small ρ > 0, define Φ¯ : E(ρ ) → E as

Φ¯ (v) = v + H s(vs ) + H u (vu ), for each v = vs ⊕vu ∈ E(ρ ). Here H s and H u have the same meanings as in Theorems IV.2.1 and IV.2.2. We see immediately that Φ¯ , D˜ Φ¯ and D˜ 2 Φ¯ are continuous on E(ρ ) and Φ¯ satisfies (1)–(3). To prove (4)(ii), writing X=

  Ex˜ (ρ ) × m0 -orthonormal frames in Ex˜ ,

x∈ ˜ Λf

we consider

|DΦx˜ (v)(ζ1 ) ∧ · · · ∧ DΦx˜ (v)(ζm0 )|

as a function of (v, ζ1 , . . . , ζm0 ) ∈ X. It is C1 along each fiber Xx˜ and its derivative along fibers is continuous on X. Since  |DΦx˜ (v)(ζ1 ) ∧ · · · ∧ DΦx˜ (v)(ζm0 )|v=0 = |ζ1 ∧ · · · ∧ ζm0 | = 1, x˜

investigating in local chart neighborhoods first and then noticing the compactness   of set (v, ζ1 , . . . , ζm0 ) ∈ X : v = 0 , we can find a B1 satisfying (ii) . Finally, (i)

follows from (ii) , which completes the proof.  

IV.3

Volume Lemma and the H¨older Continuity of φ u

33

For sufficiently small ρ > 0, define F : E(ρ ) → E, a fiber-preserving map with f respect to θ : Λ f → Λ f , as F(v) = Φθ−1 x˜ ◦ f ◦ Φx˜ (v) for x˜ ∈ Λ and v ∈ Ex˜ (ρ ). Then 2 s ˜ and D˜ F are continuous on E(ρ ), and F(E (ρ )) ⊂ E s , F(E u (ρ )) ⊂ E u . F, DF, Given x˜ ∈ Λ f and writing Fx˜ = F|Ex˜(ρ ) , we have ˜ x˜ (0x˜ ) = p∗θ x˜ ◦ Tx0 f ◦ p∗ = Tx0 f . DF

(IV.4)

Noticing the continuity of D˜ 2 F and the compactness of Λ f it is seen that there is an A2 > 1 satisfying ˜ x˜ (v1 ) − DF ˜ x˜ (v2 )| < A2 |v1 − v2 |, |DF

∀x˜ ∈ Λ f , v1 , v2 ∈ Ex˜ (ρ ).

(IV.5)

Since Λ f is compact, all of the Finsler constructions on E are equivalent. So there is a K > 1 satisfying K −1 |v| ≤ max{|vs |, |vu |} ≤ K|v| for all v = vs ⊕ vu ∈ E. Letting dim(Exu˜ ) = u for x˜ ∈ Λ f , next we will suppose ρ > 0 small enough to sat2 2 isfy ρ < min{K −1 , A−1 3 } and λ2 = max{a + ρ A2 K , a/(1 − aρ A2 K )} < 1, where def

u A3 = 2u max(A2 , 1.5)K u−1 (A2 + maxx∈ ˜ Λ |Tx0 f |) and the number a is given in f Definition IV.1.1. Then for all x˜ ∈ Λ , v ∈ Ex˜ (ρ ), wu ∈ Exu˜ (ρ ) and ws ∈ Exs˜ (ρ ) with v + wu and v + ws ∈ Ex˜ (ρ ), we have

|Fx˜u (v + wu ) − Fx˜u (v) − (Tx0 f )u (wu )| ≤ K|Fx˜ (v + wu ) − Fx˜(v) − Tx0 f (wu )| ≤ KA2 |v + ξ wu||wu | for some ξ ∈ (0, 1) ≤ ρ K 2 A2 |wu |. Consequently, |Fx˜u (v + wu ) − Fx˜u (v)| ≥ |(Tx0 f )u (wu )| − |Fx˜u (v + wu ) − Fx˜u (v) − (Tx0 f )u (wu )| ≥ a−1 |wu | − ρ K 2A2 |wu |, i.e. and similarly

|Fx˜u (v + wu ) − Fx˜u (v)| ≥ λ2−1 |wu |,

(IV.6)

|Fx˜s (v + ws ) − Fx˜s (v)| ≤ λ2 |ws |,

(IV.7)

where (Fx˜s , Fx˜u ) = Fx˜ . Taking wu = −vu and ws = −vs , one obtains |Fx˜u (v)| ≥ λ2−1 |vu | and |Fx˜s (v)| ≤ λ2 |vs |. Defining   ˜ n) = w ∈ Ex˜ : |(F k w)u | < ρ , |(F k w)s | < ρ , ∀k = 0, . . . , n , Dρ (x,

34

IV

we then have

|(F k v)u | ≤ λ2n−k ρ

Axiom A Endomorphisms

and |(F k v)s | < λ2k ρ

(IV.8)

˜ n) and k = 0, 1, . . . , n. Therefore we obtain the following lemma. for all v ∈ Dρ (x, Lemma IV.3.2 For given n ∈ Z+ , x˜ ∈ Λ f and v ∈ Dρ (x, ˜ n), we have n n  [n]  F 2 (v) ≤ K λ [ 2 ] ρ ≤ λ [ 2 ] . 2 2

Here [ n2 ] is the biggest integer not over n2 . Let x˜ ∈ Λ f , v ∈ Ex˜ (ρ ) and V be a linear subspace of Ex˜ with V ⊕ Exs˜ = Ex˜ . Then there is a unique linear operator LV : Exu˜ → Exs˜ such that V = Graph(LV ). Writu def ˜ x˜i (v)V ) for ing θ0 (V ) = LV  = sup{ |LV ζu | : 0 = ζ u ∈ E u } and θi (v,V ) = θ0 (DF |ζ |

i = 0, 1, . . . , n, then we have the following lemma.

Lemma IV.3.3 If θ0 (V ) ≤ 1, then θ1 (v,V ) ≤ aθ0 (V ) + K 2 A2 |v|. Proof. For any ζ ∈ V , from (IV.4), (IV.5) and Definition IV.1.1, we have |ζ | ≤ K max(|ζ s |, |ζ u |) ≤ K|ζ u | and            DF ˜ x˜ (0x˜ )ζ s  ˜ x˜ (0x˜ )ζ s  +  DF ˜ x˜ (v)ζ s − DF ˜ x˜ (v)ζ s  ≤  DF ≤ |Tx0 f (ζ s )| + KA2 |v||ζ | ≤ aθ0 (V )|ζ u | + K 2A2 |v||ζ u |. Therefore,

     DF ˜ x˜ (v)ζ s  ≤ aθ0 (V ) + K 2A2 |v| |ζ u |.

(IV.9)

On the other hand,             DF ˜ x˜ (0x˜ )ζ u  −  DF ˜ x˜ (v)ζ u − DF ˜ x˜ (v)ζ u  ≥  DF ˜ x˜ (0x˜ )ζ u  ≥ a−1 |ζ u | − K 2 A2 |v||ζ u |, i.e.

      ˜ x˜ (v)ζ u . ˜ x˜ (v)ζ u  ≤  DF |ζ u | ≤ λ2  DF

(IV.10)

This together with (IV.9) yields        DF ˜ x˜ (v)ζ s  ≤ aθ0 (V ) + K 2 A2 |v|  DF ˜ x˜ (v)ζ u . This completes the proof.   Through induction, it is easy to show the following lemma by Lemma IV.3.3 and (IV.8).

IV.3

Volume Lemma and the H¨older Continuity of φ u

35

Lemma IV.3.4 If θ0 (V ) ≤ 1 and v ∈ Dρ (x, ˜ n), then  θ0 (V )λ2i + iρ A

2 λ2i−1 , 1 ≤ i ≤ [ n2 ]; θi (v,V ) ≤ i−1 n−i+1 n n i





θ0 (V )λ2 + [ 2 ]ρ A2 λ2 + A2 ρλ2 , [ 2 ] ≤ i ≤ n;

+∞ 2i where A

2 = A2 K 3 and A

2 = A2 ∑i=0 λ2 .

Lemma IV.3.5 Assume θ0 (V ) ≤ A−1 3 . Then    | det(DF  ˜ x˜ (v)|V )|   − 1 ≤ A3 (θ0 (V ) + |v|).  ˜ x˜ (0x˜ )|E u )|  | det(DF  x˜

˜ x˜ (0x˜ ) = Tx0 f . Let e1 , . . . , eu be an orthonor˜ x˜ (v) and T0 = DF Proof. Write T = DF mal frame of Exu˜ . It can be seen from (IV.4) and (IV.5) that |T (e1 + LV e1 ) ∧ · · · ∧ T (eu + LV eu ) − T0 (e1 ) ∧ · · · ∧ T0 (eu )| ≤ |T (e1 + LV e1 ) ∧ · · · ∧ T (eu + LV eu ) − T (e1 ) ∧ · · · ∧ T (eu )| + |T (e1 ) ∧ · · · ∧ T (eu ) − T0(e1 ) ∧ · · · ∧ T0 (eu )| ≤ uK u−1 LV (|T0 | + ρ A2)u + uA2|v|(|T0 | + ρ A2)u−1 θ0 (V ) |v| + ), ≤ A3 ( 3 2 and |(e1 + LV e1 ) ∧ · · · ∧ (eu + LV eu ) − e1 ∧ · · · ∧ eu | ≤ uLV K u−1 A3 ≤ θ0 (V ). 3 Furthermore, since |T0 (e1 ) ∧ · · · ∧ T0 (eu )| = | det(Tx0 f |Exu˜ )| > 1, we have ˜ x˜ (v)|V )| |T (e1 + LV e1 ) ∧ · · · ∧ T (eu + LV eu )| | det(DF = ˜ u |(e1 + LV e1 ) ∧ · · · ∧ (eu + LV eu )| | det(DFx˜ (0x˜ )|Ex˜ )| 1 · |T0 (e1 ) ∧ · · · ∧ T0 (eu )| 1 + A3(θ0 (V )/3 + |v|/2) ≤ 1 − θ0 (V )A3 /3 ≤ 1 + (θ0 (V ) + |v|)A3 . It can be similarly shown that ˜ x˜ (v)|V )| | det(DF ≥ 1 − (θ0(V ) + |v|)A3 . ˜ | det(DFx˜ (0x˜ )|Exu˜ )| Using Lemmas IV.3.2, IV.3.4 and IV.3.5, we can prove the following lemma.

2

36

IV

Lemma IV.3.6 A−1 4 <

Axiom A Endomorphisms

˜ x˜n (v)|V )| | det(DF < A4 ˜ x˜n (0x˜ )|E u )| | det(DF x˜

if ρ > 0 is small enough, v ∈ Dρ (x, ˜ n) and θ0 (V ) ≤ A−1 3 , where A4 is a constant, and can be taken arbitrarily close to 1 when ρ and θ0 (V ) are small enough. We now come to the volume lemmas. For small ε > 0, n ∈ Z+ and x˜ ∈ Λ f , define   Bn (x0 , ε ) = y0 ∈ M : d( f k x0 , f k y0 ) < ε for k = 0, · · · , n . Proposition IV.3.7 (Volume lemma) For ε > 0 small enough, there is Aε > 1 such that   n   A−1 ε < m(Bn (x0 , ε )) det(Tx0 f |Exu˜ ) < Aε for all x˜ ∈ Λ f and n ∈ Z+ . Proof. Taking ρ1 > 0 and ρ2 > 0 such that

Φx˜ Dρ1 (x, ˜ n) ⊂ Bn (x0 , ε ) ⊂ Φx˜ Dρ2 (x, ˜ n) for each x˜ ∈ Λ f and n ∈ Z+ , we need only to prove that for sufficiently small ρ > 0 there exists Aρ > 1 such that   ˜ x˜n (0x˜ )|E u  < Aρ A−1 ˜ n))det(DF ρ < mx˜ (Dρ (x, x˜ for x˜ ∈ Λ f and x ∈ Z+ , where mx˜ is the Lebesgue measure of inner product space Ex˜ . Given x˜ ∈ Λ f , vs ∈ Exs˜ (ρ ) and n ∈ Z+ , define   ˜ vs , n) = vu ∈ Exu˜ : |(Fx˜i (vs + vu ))u | < ρ for i = 0, 1, . . . , n , Duρ (x,   ˜ vs , n) = v = vs + vu : vu ∈ Duρ (x, ˜ vs , n) . Cρu (x, 

˜ n) = vs ∈Exs˜ (ρ ) Cρu (x, ˜ vs , n). Let mx,u It is easy to see that Dρ (x, ˜ , mx,s ˜ and mx,v ˜ s ,n denote, respectively, the Lebesgue measures of the inner product spaces Exu˜ , Exs˜ and the submanifold Fx˜n (Cρu (x, ˜ vs , n)) of inner product space Eθ n x˜ . Next we first prove a

fact that there exist Aρ > 1 such that −1

n u s

A ρ < mx,v ˜ s ,n (Fx˜ (Cρ (x, v , n))) < A ρ

where x˜ ∈ Λ f , vs ∈ Exs˜ (ρ ) and n ∈ Z+ . Using (IV.4) and (IV.5) one can show that |Fx˜s (v) − Fx˜s (w)| < |Fx˜u (v) − Fx˜u (w)|

IV.3

Volume Lemma and the H¨older Continuity of φ u

37

if |vs − ws | < |vu − wu |, where x˜ ∈ Λ f , w and v ∈ Ex˜ (ρ ). Hence one has hs : ˜ x˜n (v)(Exu˜ ) = ˜ vs , n)) = Graph(hs ) and DF Eθun x˜ (ρ ) → Eθs n x˜ (ρ ) with properties Fx˜n (Cρu (x,  s  Graph Dh ((Fx˜n (v))u ) . Since from Lemma IV.3.4 there follows |Dhs ((Fx˜n (v))u )| = θn (v, Exu˜ ) < (nλ2n−2 A

2 + A

2 )ρ , defining h = (hs , id) : Eθun x˜ (ρ ) → Eθ n x˜ we have n u ˜ vs , n))) = mx,v ˜ s ,n (Fx˜ (Cρ (x,

 Eθu n x˜

u | det(Dh(wu ))| dmθ n x,u ˜ (w ).

So the ‘fact’ is valid. Noticing that n u ˜ vs , n))) = mx,v ˜ s ,n (Fx˜ (Cρ (x,

 Duρ (x,v ˜ s ,n)

u ˜ x˜n (vs + vu)|E u )| dmθ n x,u | det(DF ˜ (v ), x˜

it is seen from Lemma IV.3.6 that −1

u A ρ < mx,u ˜ vs , n))| det(Tx0 f n |Exu˜ )|A4 , ˜ (Dρ (x,

and

u ˜ vs , n))| det(Tx0 f n |Exu˜ )|A−1 A ρ > mx,u ˜ (Dρ (x, 4 .

On the other hand, since Λ f is compact, there is a constant K1 such that K1−1 <  Exs˜ (ρ )

˜ n)) m(Dρ (x, u s mx,u ˜ vs , n)) dmx,s ˜ (Dρ (x, ˜ (v )

Which completes the proof of Proposition IV.3.7. Similarly it can be shown that

< K1 .

 

Proposition IV.3.8 Given ε > 0 and δ > 0 small enough, we have Aε ,δ > 1 such that   n u   A−1 ε ,δ < m(Bε (y0 , n)) det(Tx0 f |Ex˜ ) < Aε ,δ , where x˜ ∈ Λ f , y0 ∈ Bδ (x0 , n), and n ∈ Z+ . Finally we will show the H¨older continuity of φ u . Proposition IV.3.9 φ u : Λ f → R is H¨older continuous. ˜ y˜ ∈ Proof. It follows from Lemma IV.3.1 that there is T1 > 0 such that for any x, Λf, y0 ∈ Φx˜ Ex˜ (ρ ) if d(x, ˜ y) ˜ < 2−T1 . −1 u f Let v0 = Φx−1 ˜ y˜ = (DΦx˜ (v0 )) ◦ p∗ Ex˜ . Since Λ is com˜ (y0 ) ∈ Ex˜ (ρ ), and define Vx, pact, we have T > T1 such that

38

IV

θ0 (Vx,˜ y˜ ) < 1 if

Axiom A Endomorphisms

d(x, ˜ y) ˜ < 2−T .

Moreover, we can pick up N > 0 such that d(x, ˜ y) ˜ < 2−T as d(xi , yi ) < 2−2T for f i = −N, . . . , −1, 0, 1, . . . , N. Given x, ˜ y˜ ∈ Λ , if ˜ y) ˜ ≤ 2−k−2T −N , 2−(k+1)−2T−N ≤ d(x, then

d(xi , yi ) ≤ 2−(k+N)+|i|−2T

for i = −(k + N), . . . , 0, . . . , k + N.

(θ −k x, ˜ 2k)

Therefore there is v−k ∈ Dρ satisfying Φθ −k x˜ (v−k ) = y−k . Evidently F k (v−k ) = v0 . Since θ0 (Vθ −k x,˜ θ −k y˜ ) < 1, we know by Lemma IV.3.4 that θ0 (Vx,˜ y˜ ) ≤ λ2k + kρ A

2 λ2k−1 . Hence there is A5 > 1 and α > 0 independent of x˜ and y˜ satisfying θ0 (Vx,˜ y˜ ) ≤ d(x, ˜ y) ˜ α . Then from Lemma IV.3.5 it is easy to complete the proof.  

IV.4 Equilibrium States of φ u on Λ f Let Λ be an Axiom A basic set of f ∈ C1 (O, M). In order to construct the f -invariant measure on M satisfying Pesin’s entropy formula, we take this section to define the equilibrium state of φ u on Λ f . Definition IV.4.1 Let (X, d) be a compact metric space and g : X → X a homeomorphism. (X, d, g) is calleda Smale space if there are ε¯ > 0, 0 < δ¯ < ε¯ , 0 < λ¯ < 1 and a continuous map [·, ·] : (x, y) ∈ X × X : d(x, y) < δ¯ → X satisfying (i) [x, x] = x and [[x, y], z] = [x, z] = [x, [y, z]] when both sides of these relations are defined; (ii) g([x, y]) = [g(x), g(y)] when both sides make sense; (iii) d(gn y, gn z) ≤ λ¯ n d(y, z) d(g−n y, g−n z) ≤ λ¯ n d(y, z)

for y, z ∈ Vε¯s (x, g) and n ∈ Z+ , for y, z ∈ Vε¯u (x, g) and n ∈ Z+ ,

where

  Vε¯s (x, g) = v : d(x, v) < ε¯ and v = [v, x]

and

  Vε¯u (x, g) = v : d(x, v) < ε¯ and v = [x, v] .

Let ε > 0, 0 < λ < 1 and δ > 0 as defined in Theorems IV.2.1, IV.2.2 and IV.2.3 for Λ and take N ∈ Z+ such that λ N < 1/2, and define  d1 (x, ˜ y) ˜ =

+∞

N 1  ∑ 2|i| d(xi , yi ) −∞

1/N for x, ˜ y˜ ∈ Λ f .

IV.4

Equilibrium States of φ u on Λ f

39

Then (Λ f , d1 ) is a metric space with the same topology as (Λ f , d). For each x, ˜ ˜ y) ˜ < δ , there is a unique z˜ ∈ Λ f satisfying p(˜z) = [y0 , x] ˜ and y˜ ∈ Λ f with d1 (x, d(z−n , x−n ) < ε for all n ∈ Z+ . Define [x, ˜ y] ˜ = z˜ and take 0 < δ¯ < ε¯ small enough and ¯λ = (1/2 + λ N )1/N , then it is easy to check that (Λ f , d1 , θ ) is a Smale space. On the other hand, since f is topologically +transitive on Λ , θ is topologically +transitive on Λ f . Therefore, by Theorems 7.6, 7.9 and Corollary 5.6 of [78], using the Markov partition we can obtain a topologically +transitive symbolic system (Ω , σ ) and a continuous onto map π : Ω → Λ f satisfying the following properties. (1) π ◦ σ = θ ◦ π . (2) φ u ◦ π admits a unique equilibrium state ρ with respect to σ , i.e., ρ is a σ invariant Borel probability measure with Pσ (φ u ◦ π ) = hρ (σ ) +

 Ω

φ u ◦ π dρ ,

where Pσ (φ u ◦ π ) is the topological pressure of φ u ◦ π according to σ . (3) μ˜ φ u = πρ is the unique equilibrium state of φ u with respect to θ and Pθ (φ u ) = Pσ (φ u ◦ π ). (4) π : (Ω , ρ , σ ) → (Λ f , μ˜ φ u , θ ) is an isomorphism. (5) ρ is ergodic and is the unique Gibbs state of φ u ◦ π with respect to σ .  +∞ Furthermore, for fixed small ε > 0, and given s∗i i=−∞ ∈ Ω , n ∈ Z+ , let x˜∗ =  +∞ π s∗i i=−∞ , then Bn (x˜∗ , ε ) ⊃ π ([s∗0 , · · · , s∗n ]), where

  ∗ [s∗0 , · · · , s∗n ] = {si }+∞ i=−∞ ∈ Ω : si = si for i = 0, 1, . . . , n

is the cylinder set in Ω and Bn (x˜∗ , ε ) is defined as Bn (x0 , ε ). By property (5) and Pθ (φ u ) = Pσ (φ u ◦ π ), there exists a bε > 1 such that     μ˜ φ u Bn (x˜∗ , ε ) ≥ ρ [s∗0 , · · · , s∗n ]   n−1 u u i ∗ ≥ b−1 exp − nP ( φ ) + φ ( θ x ˜ ) . θ ∑ ε i=0

Define μ = pμ˜ φ u . Then μ is ergodic and   n−1   u u i μ Bn (x0 , ε ) ≥ b−1 exp − nP ( φ ) + φ ( θ x) ˜ θ ∑ ε

(IV.11)

i=0

for all x˜ ∈ Λ f and n ∈ Z+ . This will be proved in the next section to satisfy Pesin’s entropy formula.

40

IV

Axiom A Endomorphisms

IV.5 Pesin’s Entropy Formula Let Λ be an Axiom A basic set of f ∈ C2 (O, M). It will be proved in this section that f admits an invariant Borel probability measure on Λ satisfying Pesin’s entropy formula if and only if Λ is an Axiom A attractor, and in this case the measure μ defined at the end of last section is the unique one to satisfy Pesin’s entropy formula. Several lemma are required first. Lemma IV.5.1 Given ε > 0 small enough, writing B(ε , n) = n ∈ Z+ , then   1 Pθ (φ u ) = lim log m B(ε , n) ≤ 0. n→+∞ n



x0 ∈Λ

Bn (x0 , ε ) for

Proof. For each ε > 0 and n ∈ Z+ , let Λ (n, ε ) be a (n, ε )-separating subset of Λ f

with maximum cardinality, and take  an x˜ = q(x0 ) ∈ Λ for each  x0 ∈ Λ (n, ε ) such f

that p (x) ˜ = x0 . Write Λ (n, ε ) = x˜ = q(x0 ) : x0 ∈ Λ (n, ε ) . It is easily seen that Λ f (n, ε ) is a (n, ε )-separated subset of Λ f . Since   n−1  def Pn (θ , φ u , ε ) = sup ∑ exp ∑ φ u (θ i x) ˜ : E is a x∈E ˜

i=0

(n, ε ) − separating set of Λ f , from Proposition IV.3.7 it follows that Pn (θ , φ u , ε ) ≥

∑

  det(Tx f n |E u )−1 0 x˜

∑

  A−1 ε +ε m Bε +ε (x0 , n)

x∈ ˜ Λ f (n,ε )

≥



exp

x∈ ˜ Λ f (n,ε )

=

 n−1

∑

x∈ ˜ Λ f (n,ε )

∑ φ u (θ i x)˜

i=0

  ≥ A−1 ε +ε m B(ε , n) . Therefore, it can be known from [92, Theorem 9.4] that   1 1 lim sup log Pn (θ , φ u , ε ) ≥ lim log m B(ε , n) . Pθ (φ u ) = lim

n→∞ n ε →0 n→∞ n the other hand, for each ε > 0 we have 0 < ε

< ε and k ∈ Z+ such that   On k k

˜ θ y˜ < ε if x, ˜ y˜ ∈ Λ f satisfying d(x0 , y0 ) < ε

. Consequently, if G f (n, ε ) d θ x,  

is a (n, ε ) separated subset of Λ f , then G(n, ε

) = p θ −k G f (n, ε ) is a (n, ε

)separating subset of Λ . Since Λ f is compact and φ u is continuous on Λ f , there ˜ < T for all x˜ ∈ Λ f . Therefore from Proposition exists a T > 1 such that |φ u (x)| IV.3.7 it follows that

IV.5

Pesin’s Entropy Formula

41

 n−1

∑

exp

x∈G ˜ f (n,ε )



∑ φ u (θ i x)˜

 n−1−k

∑

≤

exp

x∈G ˜ f (n,ε )

i=0

= ekT

φ u (θ i x) ˜ + kT



i=−k

∑

x∈G ˜ f (n,ε )

≤ ekT Aε

/2

∑

  det(D f n−k |E u )−1 θ x˜ −k

∑

y0 ∈G(n,ε

)

θ

x˜

  m Bn (y0 , ε

/2)

Since G(n, ε

) is a (n, ε

)-separating subset of Λ , Bn (y0 , ε

/2) y0 ∈ G(n, ε

) are disjoint subsets of B(ε , n). Hence

∑

 n−1 exp

x∈G ˜ f (n,ε )



∑ φ u (θ i x)˜

  ≤ Aε

/2 ekT m B(ε , n) .

i=0

It follows that

  1 log m B(ε , n) . 2 n→∞ n Proposition IV.5.2 Let ν be an f -invariant Borel probability measure on Λ . Then ν satisfies Pesin’s entropy formula if and only if ν = μ and Pθ (φ u ) = 0, where μ = pμ˜ φ u given in the above section. Pθ (φ u ) ≤ lim

Proof. Let ν˜ be a θ -invariant Borel probability measure on Λ f with pν˜ = ν . By Birkhoff’s ergodic theorem and Oseledec’s theory we have  Λf



1 n−1 u i  ∑ φ (θ x)˜ d ν˜ (x)˜ Λ f n→∞ n i=0     1 lim logdet(Tx0 f n |Exu˜ ) d ν˜ (x) ˜ =− Λ f n→∞ n

φ u d ν˜ =

=− =−



lim



r(x) ˜

∑ λi (x)˜ + mi (x)˜ d ν˜ (x)˜

Λ f i=1

 r(x0 )

∑ λi (x0 )+ mi (x0 ) d ν (x0 ),

Λ i=1

where +∞ > λ1 (x) ˜ > · · · > λr(x) ˜ ≥ −∞ are the Lyapunov exponents of (Λ f , θ ) at ˜ (x) ˜ . . . , mr(x) ˜ are multiplicities of λ1 (x), ˜ . . . , λr(x) ˜ respectively. Therex, ˜ m1 (x), ˜ (x) ˜ (x), fore, from Lemma IV.5.1 and the variational principle [92] it is easy to see that if ν satisfies Pesin’s entropy formula then 0 ≥ Pθ (φ u ) ≥

 Λf

φ u d ν˜ + hν˜ (θ ) = 0.

Hence ν = μ because of the uniqueness of the equilibrium state. On the other hand, it is easy to check that μ satisfies Pesin’s entropy formula if Pθ (φ u ) = 0.  

42

IV

Axiom A Endomorphisms

Proposition IV.5.3 Assume ε > 0 is small enough. Then the following properties are equivalent. (1) Λ is an Axiom A attractor of f .   1 (2) Pθ (φ u ) = lim log m B(ε , n) = 0. n→∞ n Proof. It is easy to see from Proposition IV.2.5 and Lemma IV.5.1 that (1) ⇒ (2). For (2) ⇒ (1) one is referred to [10, Theorem 4.11].  

IV.6 Large Ergodic Theorem and Proof of Main Theorems We will finish this chapter in this section by showing the large ergodic theorem and completing the proof of the main theorems. Proposition IV.6.1 Let Λ be an Axiom A attractor of f ∈ C2 (O, M), μ the unique f -invariant Borel probability measure satisfying Pesin’s entropy formula. assume that ε > 0 small enough and m(C f ) = 0. Then for m-almost all x0 ∈ B(Λ , ε ), 1 n−1  i  ∑ ψ f x0 = n→∞ n i=0 lim

 Λ

ψ dμ,

  ∀ψ ∈ C B(Λ , ε ) ,

  where C f = x0 ∈ O : | det(Tx0 f )| = 0 .



Proof. By Proposition   IV.2.5, we have ε > ε > 0 such that B(Λ , ε ) ⊂ B(ε , ∞). given ψ ∈ C B(Λ , ε ) and α > 0, define

 n−1      Mα = x ∈ B(Λ , ε ) : ∀N > 0, ∃n > N s.t. n−1 ∑ ψ f i x − ψ d μ  ≥ α . i=0

Λ

  Similarly as in the proof of [10, theorem 4.12], there is a β > 0 such that m Mα    B(β , ∞) = 0. Since B(Λ , ε ) is compact, there is T > 1 such that f T B(Λ , ε ) ⊂ B(β , ∞). For each k ≥ 1, denoting   Uk = x ∈ B(Λ , ε ) : | det(Tx f T )| ≥ k−1 , then Uk is compact and we have a δ (k) > 0 such that the map   f T : B(x, δ (k)) → f T B(x, δ (k)) are diffeomorphisms for all x ∈ Uk . Given x ∈ Uk ,    0 = m Mα B(β , ∞)      ≥ m f T Mα B(x, δ (k)) Uk

IV.6

Large Ergodic Theorem and Proof of Main Theorems

=

 Mα





B(x,δ (k)) Uk

43

  det(D fyT ) dm(y)

  1   ≥ m Mα B(x, δ (k)) Uk . k 

Since Uk can be covered by finite δ (k)-balls, we have m(Uk Mα ) = 0. On the other hand, by m(C f ) = 0 one can inductively prove that m( f −i (C f )) = 0 and m(C f i+1 ) = 0 for i = 1, 2, . . ., where C f i+1 is defined similarly as C f . Therefore, m(Mα ) = lim m(Uk k→∞



Mα ) = 0.

Define   1 n−1   M(ψ ) = x ∈ B(Λ , ε ) : lim ∑ ψ f i x = ψ d μ . n→∞ n Λ i=0

  Then m M(ψ ) = lim m(Mα ) = 0. α →0  ∞     Let ψi i=1 be a dense subset of C B(Λ , ε ) , then m ∞ i=1 M(ψi ) = 0 and for  each x ∈ B(Λ , ε )\ ∞ i=1 M(ψi ) we have 1 n−1   lim ∑ ψ f i x = n→∞ n i=0

 Λ

ψ dμ

  for all ψ ∈ C B(Λ , ε ) .

2

Now we come to the proof of the main theorems. Proof of Theorem IV.1.1. It is evident that Theorem IV.1.1 follows immediately from Propositions IV.5.2, IV.5.3 and IV.6.1.   Proof of Theorem IV.1.2. By Theorem IV.2.4 the nonwandering set Ω ( f ) of Axiom A endomorphism f ∈ C2 (M, M) can be spectrally decomposed into some    pairwise disjoint Axiom A basic sets, Ω ( f ) = Ω1 Ω2 · · · Ωk . It is known from  the theory of dynamical systems that M = ki=1 W s (Ωi ), where   W s (Ωi ) = x ∈ M : lim d( f n x, Ωi ) = 0 . n→∞

Proposition IV.2.5 implies that there exists If some Ωi is not an Axiom A attractor,   an ε > 0 such that m x∈Ωi Wεs (x) = 0. Using the pseudo-orbit-shadowing method (see Remark IV.6) and noticing Remark IV.4, we can find 0 < εi < εi satisfying  W2sεi (ωi ) ⊂ x∈Ωi Wεs (x), where i

 W2sεi (ωi ) = x ∈ M : d( f n x, Ωi ) < 2εi for all n ∈ Z+  and lim d( f n x, Ωi ) = 0 . n→∞

44

IV

Axiom A Endomorphisms

  Since W s (Ωi ) = i=1 f −n Wεsi (Ωi ) , it can be shown that m(W s (Ωi )) = 0 as in Proposition IV.6.1. For given ψ ∈ C(M), if some Ω j is an Axiom A attractor, let μ j be the unique f -invariant Borel probability measure on Ω j satisfying Pesin’s entropy formula. Let ∞

  1 n−1   ψ dμ j . M j (ψ ) = x ∈ W s (Ω j ) : lim ∑ ψ f i x = n→∞ n Ωj i=0

It follows from Proposition IV.6.1 that there exists ε j > 0 such that the measure   m M j (ψ ) B2ε j (ω j ) = 0. Therefore, we can prove that m(M j (ψ )) = 0. It is now easy to complete the proof.  

Chapter V

Unstable and Stable Manifolds for Endomorphisms

Stable and unstable manifolds for hyperbolic fixed points and sets have been well studied, see for instance [26]. Pesin developed a general theory of stable and unstable manifolds for diffeomorphisms corresponding to its non-vanishing Lyapunov exponents [62]. His work translated the linear theory of Lyapunov exponents (multiplicative ergodic theorem) into the non-linear theory of stable and unstable invariant manifolds. It was demonstrated that this theory provides a fundamental tool for the development of the smooth ergodic theory. Some extensions of Pesin’s result had been given [33, 79], and even in infinite dimensional spaces [56, 80]. More recently, Liu and Qian developed a rigorous theory of stable and unstable manifolds for random diffeomorphisms [51]. There have been some partial results concerning invariant manifolds for endomorphisms [69, 83, 80]. However, those results are not enough for the study of smooth ergodic theory of endomorphisms. In order to study the relationship between entropy and Lyapunov exponents, we will need more subtle results. The purpose of this chapter is to an unstable manifold theorem of this kind for non-invertible differentiable maps of finite dimensional manifolds by borrowing the techniques given in [51]. Readers familiar with [51] can easily see that when the inverse limit space is introduced, some situations are very close to those in [51]. Nevertheless, for the convenience of a larger group of readers who may not wish to involve any randomness, we still give a detailed presentation. The source of this chapter is mainly [100] except Section 6-9 which follow from [51, Chapter III]. This chapter is organized as follows. In Section 1, we introduce some preliminary facts which will be used in the subsequent sections. Fundamental lemmas concerning tangent mappings are given in Section 2. The local unstable manifold theorem and its detailed proof are presented in Section 4. Section 5 is devoted to the global unstable sets. Theorems for stable manifolds are in Section 6. In this chapter, we assume that the system (M, f , μ ) always satisfies the following integrability condition   (V.1) logdet(Tx f ) ∈ L1 (M, μ ).

M. Qian et al., Smooth Ergodic Theory for Endomorphisms, Lecture Notes in Mathematics 1978, DOI 10.1007/978-3-642-01954-8 V, c Springer-Verlag Berlin Heidelberg 2009 

45

46

V Unstable and Stable Manifolds for Endomorphisms

V.1 Preliminary Facts In what follows, we always use Lip( · ) to denote the Lipschitz constant of a Lipschitz map. Lemma V.1.1 Let (E,  · ) be a Banach space. If A : E → E is an invertible linear  −1 map, ψ : E → E is a Lipschitz map with Lip(ψ ) < A−1  , then g = A + ψ is invertible and g−1 = (A + ψ )−1 is also a Lipschitz map with Lip(g−1 ) ≤

1 . A−1 −1 − Lip(ψ )

Proof. See page 90 of [99].   Using a simple method of algebraic topology, one can prove the following lemma. Lemma V.1.2 Let D ⊂ Rm be an open domain, h : D → h(D) ⊂ Rm a homeomorphism with h(D) ⊃ B(y, ρ ) for some y ∈ Rm , ρ > 0. Let g : D → Rm be continuous and   g(x) ≤ ρ , ∀x ∈ D. 2 Then (h + g)(D) ⊃ B(y, ρ /2). of f and the compactness of M, there is ρ1 > 0 such that  By the continuity  f B(x, ρ1 ) ⊂ B f x, ρ0 for  all x ∈ M, where ρ0 is as defined in Section II.2. Thus for any 0 < ρ ≤ ρ2 = min 1, ρ0 , ρ1 , map x = exp−1 ◦ f ◦ expx : Tx M(ρ ) → T f x M(ρ0 ) H fx is well defined. Lemma V.1.3 There is a measurable function κ : M → [1, +∞) such that 1) log κ ∈ L1 (M, μ ). 2) For any x ∈ Γ , the maps   x : Tx M κ (x)−1 → T f x M, H x and the inverse map of H   x−1 : T f x M κ (x)−1 → Tx M H are well defined and

x (·)) ≤ κ (x), Lip(DH x−1 (·)) ≤ κ (x), Lip(DH

x−1 denote the derivative of H x and H x−1 respectively. x and DH where DH

V.1 Preliminary Facts

47

Proof. We know that for any 0 < ρ ≤ ρ2 , x = exp−1 ◦ f ◦ expx : Tx M(ρ ) → T f x M(ρ0 ) H fx is well defined. Since f is C2 , there is C0 ≥ 1 such that x (·)) ≤ C0 Lip(DH

on Tx M(ρ ),

(V.2)

for any 0 < ρ ≤ ρ2 and x ∈ M. Consequently, x − Tx f ) ≤ C0 ρ Lip(H

on Tx M(ρ ).

(V.3)

For x ∈ Γ , Tx f is nondegenerate. Because of (V.3), by Lemma V.1.1, when C0 ρ <   (Tx f )−1 −1 , the map x : Tx M(ρ ) → T f x M H is injective. In this case  (Tx f )(Tx M(ρ )) ⊃ T f x M

ρ . (Tx f )−1 

 −1 If ρ ≤ 2C0 (Tx f )−1  , for |ξ | < ρ , by (V.3)   (H x − Tx f )(ξ ) ≤ C0 ρ |ξ | ≤

ρ . 2(Tx f )−1 

Hence by Lemma V.1.2 x (Tx M(ρ )) ⊃ T f x M H



ρ . 2(Tx f )−1 

  Therefore, for γ , ζ ∈ T f x M (2(Tx f )−1 )−1 ρ there are ξ , η ∈ Tx M(ρ ) such that x (ξ ) and ζ = H x (η ). Thus, from (V.3) we have γ =H     γ − ζ  = H x (η ) x (ξ ) − H     x (ξ ) − Tx f (ξ )] − [H x (η ) − Tx f (η )] ≥ Tx f (ξ ) − Tx f (η ) − [H  −1 −1 1 ≥ (Tx f )−1  |ξ − η | − (Tx f )−1  |ξ − η | 2   1 −1 = (Tx f )−1  |ξ − η | 2  −1  −1 1 x (γ ) − H x−1 (ζ ). = (Tx f )−1  H 2

48

V Unstable and Stable Manifolds for Endomorphisms

It follows that       DH x (ξ ))−1  ≤ 2(Tx f )−1  x−1 (γ ) = (DH

(V.4)

  for all γ ∈ T f x M (2(Tx f )−1 )−1 ρ . Therefore,     DH x (ξ ))−1 − (DH x−1 (γ ) − DH x−1 (ζ ) = (DH x (η ))−1    x (ξ ))−1 (DH x (η ) − DH x (ξ ))(DH x (η ))−1  = (DH    x (ξ ))−1  (DH x (η ))−1 C0 |ξ − η | ≤ (DH  2 −1 x (γ ) − H x−1(ζ )| ≤ 4C0 (Tx f )−1  |H  3 ≤ 8C0 (Tx f )−1  |γ − ζ |. As a result,

    x−1 (·) ≤ 8C0 (Tx f )−1 3 . Lip DH

Now we choose    ρ (x) = min ρ2 , 2(Tx f )−1 ρ2 ,

1 1  , ,   2C0 (Tx f )−1  4C0 (Tx f )−1 2

  so that (2(Tx f )−1 )−1 ρ (x) ≤ ρ2 for all x ∈ Γ . Define   2 (Tx f )−1  ,C0 κ (x) = max ρ (x) for x ∈ Γ and κ (x) = 1 for x ∈ Γ . Then it is easy to see that the lemma holds for this function κ .   Remark V.1. There exist a Borel set Γ0 ⊂ M (independent of ε ) and a measurable function κˆ : Γ0 → [1, ∞) such that μ (Γ0 ) = 1, f Γ0 ⊂ Γ0 and the following hold true: 1) For each x ∈ Γ0 and n ∈ Z+ , the map Fx,n = exp−1 ◦ f ◦ expx : T f n x M(κˆ ( f n x)−1 ) → T f n+1 x M f n+1 x def

is well defined and

Lip(DFx,n (·)) ≤ κˆ ( f n x);

2) κˆ ( f n x) ≤ κˆ (x)eε n for n ∈ Z+ and x ∈ Γ0 . Lemma V.1.4 Let E be an m0 -dimensional vector space with two two inner products < ·, · > and < ·, · > , and let | · | and | · | be the induced norms respectively.

V.2 Fundamental Lemmas

49

Suppose there is a basis {ξi : 1 ≤ i ≤ m0 } of E such that |ξ1 | = |ξ2 | = · · · = |ξm0 | = 1 and < ξi , ξ j > = δi j for all i, j. Then for each v ∈ E one has |ξ1 ∧ · · · ∧ ξm0 |

1 |v| √ |v| ≤ |v| ≤ m0 (m0 −1)/2 m0 |ξ1 ∧ · · · ∧ ξm0 | Proof. 0 (i) Let v = ∑i=1 xi ξi . Then

m

m0

m0

m0

i=1

i=1 m0

i=1

m0 |v| = m0 ∑ x2i ≥ ( ∑ |xi |)2 = ( ∑ |xi ||ξi |)2 2

≥ | ∑ xi ξi |2 = |v|2 i=1

Hence |v| ≥ √1 |v| for each v ∈ E. m0 def

(ii) Now let 0 = v ∈ E. Put v1 = v/|v| and choose v2 , · · · , vm0 ∈ E such that {v j : 1 ≤ j ≤ m0 } is an orthonormal basis of (E, < ·, · > ). Using (i) one has |v| |v1 ∧ v2 ∧ · · · ∧ vm0 | = |v ∧ v2 ∧ · · · ∧ vm0 | ≤ |v||v2 | · · · |vm0 | ≤ m0 (m0 −1)/2 |v||v2 | |v3 | · · · |vm0 |

= m0 (m0 −1)/2 |v|. Since {v1 , · · · , vm0 } and {ξ1 , · · · , ξm0 } are both orthonormal basis of (E, < ·, · > ), it follows that |ξ1 ∧ · · · ∧ ξm0 | = 1 and |ξ1 ∧ · · · ∧ ξm0 | = |v1 ∧ · · · ∧ vm0 |. Hence |v| ≤ m0 (m0 −1)/2

|ξ1 ∧ · · · ∧ ξm0 |

|v|. |ξ1 ∧ · · · ∧ ξm0 |

 

V.2 Fundamental Lemmas Let Δ˜ be defined by Proposition I.I.3.5. We have the following lemma. Lemma V.2.1 Given ε > 0. There exists a measurable function Cε : Δ˜ → [1, +∞) such that for any x˜ ∈ Δ˜ and n ∈ Z we have 1) Cε (x) ˜ −1 exp(nλi (x)− ˜ ε2 |n|)|v| ≤ |T0n (x)v| ˜ ≤ Cε (x) ˜ exp(nλi (x)+ ˜ ε2 |n|)|v|, for each ˜ and 1 ≤ i ≤ r(x); ˜ v ∈ Ei (x)

50

V Unstable and Stable Manifolds for Endomorphisms

2) if v = v1 + · · · + vr(x) ˜ i = 1, · · · , r(x), ˜ then ˜ with vi ∈ Ei (x), ˜ |v1 | + · · · + |vr(x) ˜ | ≤ Cε (x)|v| and 2 |v1 |2 + · · · + |vr(x) ˜ 2 |v|2 ˜ | ≤ Cε (x)

3) Cε (θ n x) ˜ ≤ Cε (x)e ˜ |n|ε . ˜ by T0n for simplicity of notations. We choose a Proof. Let x˜ ∈ Δ˜ . Denote T0n (x) ˜ 1 ≤ j ≤ mi (x)} ˜ of Tx0 M from E1 (x), ˜ ..., basis {ξ1 , · · · , ξm0 } = {ξi, j : 1 ≤ i ≤ r(x), Er(x) ˜ such that ξi, j ∈ Ei (x) ˜ and|ξi, j | = 1 for each i, j. For each n ∈ Z, define an ˜ (x) ˜ 1≤ j≤ inner product < ·, · >n on Txn M such that {T0n ξi, j /|T0n ξi, j | : 1 ≤ i ≤ r(x), mi (x)} ˜ is an orthonormal basis of (Txn M, < ·, · >n ). The induce norm of (Txn M, < ·, · >n ) is denoted by | · |n . Then by Lemma V.1.4 one has for each v ∈ Txn M |T n ξ1 ∧ T0n ξ2 ∧ · · · ∧ T0n ξm0 |n 1 √ |v| ≤ |v|n ≤ m0 (m0 −1)/2 0 n |v|. m0 |T0 ξ1 ∧ T0n ξ2 ∧ · · · ∧ T0n ξm0 |

(V.5)

Since |T0n ξ1 ∧ T0n ξ2 ∧ · · · ∧ T0n ξm0 |n = |T0n ξ1 ||T0n ξ2 | · · · |T0n ξm0 | and ˜ mi (x) ˜ m0 1 1 r(x) log ∏ |T0n ξi | = lim ∑ ∑ log |T0n ξi, j | n→±∞ n n→±∞ n i=1 i=1 j=1

lim

r(x) ˜

=

˜ ∑ mi (x)˜ λi (x),

i=1

lim

1

n→±∞ n

log |T0n ξ1 ∧ T0n ξ2 ∧ · · · ∧ T0n ξm0 | = lim

n→±∞

1 log | det(T0n )| n

r(x) ˜

=

˜ ∑ mi (x)˜ λi (x),

i=1

we have

|T n ξ1 ∧ T0n ξ2 ∧ · · · ∧ T0n ξm0 |n 1 log 0 n = 0. n→±∞ n |T0 ξ1 ∧ T0n ξ2 ∧ · · · ∧ T0n ξm0 | lim

Hence for each ε > 0 there is C1,ε (x) ˜ such that for each n ∈ Z and v ∈ Txn M 1 √ |v| ≤ |v|n ≤ C1,ε (x)e ˜ ε |n| |v|. m0 n ˜ i = 1, · · · , r(x), ˜ then If v = v1 + · · · + vr(x) ˜ with vi ∈ Ei (θ x),

√ m0 [|v1 |n + · · · + |vr(x) ˜ |n ] 1/2  ! 2 ˜ |v1 |2n + · · · + |vr(x) ≤ m0 r(x) ˜ |n

|v1 | + · · · + |vr(x) ˜ |≤

≤ m0C1,ε (x)e ˜ ε |n| |v|

(V.6)

V.2 Fundamental Lemmas

51

and 2 2 2 2 |v1 |2 + · · · + |vr(x) ˜ | ≤ m0 [|v1 |0 + · · · + |vr(x) ˜ |0 ] = m0 |v|0

≤ m0C1,ε (x) ˜ 2 e2ε |n| |v|2 . Clearly the right sides of the above two inequality depends on the choice of the basis m0 . Now we need to rectify this situation to give basis-independent estimations {ξk }k=1 by letting  |v | + · · · + |v | 1 r(x) ˜ e−ε |n| : n ∈ Z, |v| r(x) ˜ n˜ . 0 = v = v1 + v2 + · · · + vr(x) ˜ ∈ Txn M = ⊕i=1 Ei (θ x)

def

˜ = sup Cε (x)

(V.7)

From the definition, one has

˜ |v1 | + · · · + |vr(x) ˜ | ≤ Cε (x)|v|

and

2

˜ 2 |v|2 , |v1 |2 + · · · + |vr(x) ˜ | ≤ Cε (x)

where v = v1 + · · · + vr(x) ˜ Obviously Cε (·) is measurably defined on ˜ ∈ ⊕i=1 Ei (x). ˜ Δ and ˜ ≤ Cε (x)e ˜ ε |n| , ∀n ∈ Z. Cε (θ n x) r(x) ˜

˜ and 1 ≤ j ≤ mi (x) ˜ Since for each x˜ ∈ Δ˜ , 1 ≤ i ≤ r(x) lim

1

n→±∞ n

log |T0n ξi, j | = λi (x), ˜

there exists C2,ε (x) ˜ ≥ 1 such that C2,ε (x) ˜ −1 exp(nλi (x) ˜ −

|n| |n| ε ) ≤ |T0n ξi, j | ≤ C2,ε (x) ˜ exp(nλi (x) ˜ + ε) 4 4

holds true for each i, j and n ∈ Z. Therefore for each n, k ∈ Z C2,ε (x) ˜ −2 e

nλi (x)− ˜ ε (|n+k|+|k|)

4

≤

|T0n+k ξi, j | |T0k ξi, j |

≤ C2,ε (x) ˜ 2e

nλi (x)+ ˜ ε (|n+k|+|k|)

4

˜ let v = ∑ j α j ξi, j , then one has Hence for each n, k ∈ Z and v ∈ Ei (x), |T0n+k v| ≤ ∑ |α j ||T0n+k ξi, j | j

≤

√ nλi (x)+ ˜ ε (|n+k|+|k|) 4 m0C2,ε (x) ˜ 2e ∑ |α j ||T0k ξi, j |k j

.

52

V Unstable and Stable Manifolds for Endomorphisms

≤

 1/2 ! nλi (x)+ ˜ ε (|n+k|+|k|) 2 k 2 4 m0 mi (x)C ˜ 2,ε (x) ˜ 2e | α | |T ξ | j i, j ∑ 0 k j

! nλi (x)+ ˜ ε (|n+k|+|k|) k 4 = m0 mi (x)C ˜ 2,ε (x) ˜ 2e |T0 v|k ε nλi (x)+ ˜ (|n+k|+2|k|) k 4 ≤ m0C1,ε /4 (x)C ˜ 2,ε (x) ˜ 2e |T0 v|. Define

 n+k −nλi (x)− ˜ ε (|n+k|+|k|) |T (x)v| ˜ def 2 ˜ = sup |T0 k (x)v| e : n, k ∈ Z, Cε

(x) ˜ 0 ˜ , 1 ≤ i ≤ r(x), ˜ and 0 = v ∈ Ei (x)

(V.8)

˜ = max{Cε (x),C ˜ ε

(x)} ˜ Cε (x)

(V.9)

and put def

Then it is easy to verify that Cε (x) ˜ ≥ 1 and Cε (·) is measurably defined on Δ˜ satisfying 1)– 3).   Now let [a, b], 0 ≤ a < b, be a closed interval of R. Let  Δa,b = x˜ ∈ Δ˜ : λi (x) ˜ ∈ [a, b], i = 1, · · · , r(x) ˜ . Clearly θ Δa,b = Δa,b . For x˜ ∈ Δa,b and n, l ∈ Z we sometimes use the following notations: E j (x), ˜ E(x) ˜ = ⊕λi (x)b ˜ ˜ Obviously, ˜ θ n x) ˜ = H(θ n−l x), ˜ Tn−l (x)H(

Tn−l (x)E( ˜ θ n x) ˜ = E(θ n−l x)( ˜ x). ˜

 1  (b − a) and assume that the set We now fix k ∈ {1, · · · , m0 } and 0 < ε ≤ min 1, 200  def  Δa,b,k = x˜ ∈ Δa,b : dim H(x) ˜ = k = 0. / Remark V.2. For any x˜ ∈ Δa,b,k and n, l ∈ Z+ we have   −l 1) T−n (x) ˜ ξ  ≤ Cε /4 (θ −n x) ˜ 2 e(−b+ε /8)l |ξ |, ξ ∈ H(θ −n x); ˜  l  ˜ η  ≤ Cε /4 (θ −n x) ˜ 2 e(a+ε /8)l |η |, η ∈ E(θ −n x); ˜ 2) T−n (x)  −l  3) T−n (x) ˜ η  ≥ Cε /4 (θ −n x) ˜ −2 e−(a+5ε /8)l |η |, η ∈ E(θ −n x); ˜ Let C ≥ 1 be a number such that the set  def  C

Δa,b,k, ˜ ≤ C = 0. / ε = x˜ ∈ Δ a,b,k : Cε /4 (x)

C Lemma V.2.2 H(x) ˜ and E(x) ˜ depend continuously on x˜ ∈ Δa,b,k, ε.

C Proof. Suppose {x˜n }+∞ n=1 is a sequence of points in Δ a,b,k,ε such that x˜n converges C

to x˜ ∈ Δa,b,k,ε and H(x˜n ) converges to a subspace of Tx0 M as n → +∞. Because of

V.2 Fundamental Lemmas

53

Remark V.2, one can easily verify that this subspace coincides with H(x). ˜ From this C

there follows the continuity of H(x), ˜ and also of E(x), ˜ with respect to x˜ ∈ Δa,b,k, ε.   Let x˜ ∈ Δ˜ and n ∈ Z. Lemma V.2.1 allows us to introduce an inner product ·, ·x,n ˜ on Txn M such that ξ , ξ x,n ˜ =

def

+∞

∑

" l # ˜ ε |l| Tn (x) e−2λi (x)l−2 ˜ ξ , Tnl (x) ˜ ξ ,

ξ , ξ ∈ Ei (θ n x) ˜

(V.10)

l=−∞ n ˜ are orthogonal with respect to ·, · . We denote the and E1 (θ n x), ˜ . . . , Er(x) x,n ˜ ˜ (θ x) induced norm of this inner product ·, ·x,n ˜ by  · x,n ˜ , i.e. 1

def

ξ x,n ˜ = [ξ , ξ x,n ˜ ]2 ,

ξ ∈ Ei (θ n x), ˜ 1 ≤ i ≤ r(x) ˜

def

1 2

2 + · · · + ξ 2 ] , ξ x,n ˜ = [ξ1 x,n r(x) ˜ x,n ˜ ˜ n ˜ ⊕ ···⊕ E n˜ ξ = ξ1 + · · · + ξr(x) ˜ ∈ E1 (θ x) r(x) ˜ (θ x).

(V.11) (V.12)

+∞ ˜ We call the sequence of norms { · x,n ˜ }n=−∞ a Lyapunov metric at the point x. +∞ Lemma V.2.3 Let x˜ ∈ Δ˜ . The sequence of norms { · x,n ˜ }n=−∞ satisfies for each n, l ∈ Z

˜ 1) for 1 ≤ i ≤ r(x) ˜ and all ξi ∈ Ei (θ n x),  l  ˜ ε ˜ ε  ˜ ξi  el λi (x)−|l| ξi x,n ≤ el λi (x)+|l| ξi x,n ˜ ≤ Tn (x) ˜ ; x,n+l ˜ 2) √1 |ξ | ≤ ξ x,n ˜ ε )eε |n|/2 |ξ |, ξ ∈ Txn M, where ˜ ≤ A(x; m0 4 A(x; ˜ ε ) = Cε /4 (x) ˜2 ε and A(θ n x; ˜ ε ) ≤ eε |n|/2 A(x; ˜ ε ) for all n ∈ Z and x˜ ∈ Δ˜ . Proof. For any ξ = ξ1 + · · · + ξr ∈ Txn M = E1 (θ n x) ˜ ⊕ · · · ⊕ Er(x) ˜ from (V.10) ˜ (x), and (V.11), we have 2  k Tn (x) ˜ ξi x,n+k = ˜ =

+∞

∑

 2 ˜ ε |l|  l Tn+k (x)T e−2λi (x)l−2 ˜ nk (x) ˜ ξi 

l=−∞ +∞

∑

 2 ˜ ε |l|  k+l Tn (x) e−2λi (x)l−2 ˜ ξi 

l=−∞

=

+∞

∑

l=−∞

 2 ˜ ε |l|  l ˜ ε Tn (x) e−2λi (x)l−2 ˜ ξi  e2kλi (x)+2(|l|−|l−k|) .

54

V Unstable and Stable Manifolds for Endomorphisms

Hence 1) follows. From (V.10)–(V.12) it can be easily seen that |ξ | ≤ ∑ |ξi | ≤ ∑ ξi x,n ˜ ≤ i

√

m0 ξ x,n ˜

i

which implies the left inequality in 2). We now prove the right one. By Lemma V.2.1 and (V.11) ξi 2x,n ˜ ≤

+∞

∑

|ξi |2Cε /4 (θ n x) ˜ 2 e−7|l|ε /4

l=−∞

≤

2 1−e

C (θ −7ε /4 ε /4

n

x) ˜ 2 |ξi |2 ,

Using inequality 7 x e−x ≤ 1 − , ∀x ∈ [0, ] 8 4 one obtain

64 16 C (θ n x) ˜ 2 |ξi |2 ≤ 2 Cε /4 (θ n x) ˜ 2 |ξi |2 . 7ε ε /4 ε Hence from 2) of Lemma V.2.1 one has ξi 2x,n ˜ ≤

1/2  4 2 Cε /4 (θ n x) ˜ |ξ1 |2 + · · · + |ξr(x) ˜ | ε 4 ≤ Cε /4 (θ n x) ˜ 2 |ξ | ε |n| 4 ≤ Cε /4 (x) ˜ 2 e 2 ε |ξ | ε

ξ x,n ˜ ≤

= A(x; ˜ ε )e

|n| 2 ε

|ξ |.

The proof is completed.   on T M for Remark V.2 allows us to introduce another inner product ·, · x,n x−n ˜ C

+ such that each x˜ ∈ Δa,b,k, and n ∈ Z ε ξ , ξ  x,n ˜ =

def

η , η  x,n ˜ =

def

+∞

∑ e2l(b−ε )

l=0 n

" −l # −l T−n (x) ˜ ξ , T−n (x) ˜ ξ ,

∑ e−2l(a+ε )

" l # l T−n (x) ˜ η , T−n (x) ˜ η ,

ξ , ξ ∈ H(θ −n x) ˜ η , η ∈ E(θ −n x) ˜

l=0

˜ and E(θ −n x) ˜ are orthogonal with respect to ·, · x,n and H(θ −n x) ˜ . The norm induced

. It follows from Lemma V.2.2 and (V.10) that for each by ·, ·x,n is denoted by  · ˜ x,n ˜ C

fixed n ∈ Z+ the inner product ·, · x,n ˜ depends continuously on x˜ ∈ Δ a,b,k,ε .

V.2 Fundamental Lemmas

55

C

+∞ Remark V.3. Let x˜ ∈ Δa,b,k,ε . The sequence of norms { ·  x,n ˜ }n=0 satisfies for each + n∈Z  −1 

−n x); 1) T−n (x) ˜ ξ x,n+1 ≤ e−b+ε ξ  x,n ˜ ˜ , ξ ∈ H(θ  ˜  −1 −a−ε

−n   ˜η ≥e η  , η ∈ E(θ x); ˜ 2) T−n (x) x,n ˜

x,n+1 ˜

ε n/2 |ζ |, ζ ∈ T M, where A = 4C 2 /ε . 3) √1 |ζ | ≤ ζ  x,n x−n ˜ ≤ Ae 2

Finally we prove the following important lemma. The norm we use is | · | except mentioned otherwise. Lemma V.2.4 There exist a Borel set Δ0 ⊂ Δ˜ and a measurable function κ˜ : Δ0 → [1, +∞) such that μ˜ (Δ0 ) = 1, θ (Δ0 ) = Δ0 with the following properties. 1) For any x˜ ∈ Δ0 , the map def  −1 Gx,0 ˜ −1 ) → Tx−1 M ˜ = Hx−1 : Tx0 M(κ˜ (x)

are well defined and

˜ (x), ˜ Lip(DGx,0 ˜ (·)) ≤ κ

where DGx,0 ˜ −1 ). ˜ (·) : ξ → DGx,0 ˜ (ξ ), ξ ∈ Tx0 M(κ˜ (x) 2) The map −1 fx˜−1 = expx−1 ◦Gx,0 ˜ −1 ) → M ˜ ◦ expx0 : B(x0 , κ˜ (x) is well defined and

f ◦ fx˜−1 = id|B(x0 ,κ˜ (x) ˜ −1 ) .

˜ ≤ κ˜ (x)e ˜ ε n , n ∈ Z+ , x˜ ∈ Δ0 . 3) κ˜ (θ −n x) Proof. For any x˜ ∈ M f , let κˆ (x) ˜ = κ (x−1 ). By Lemma V.1.3, for any x˜ ∈ Δ˜  −1 Gx,0 ˜ −1 ) → Tx−1 M ˜ = Hx−1 : Tx0 M(κˆ (x) is well defined and

˜ Lip(DGx,0 ˜ (·)) ≤ κˆ (x).

L1 (M f , B(M f ), μ˜ ),

Since log κˆ (x) ˜ ∈ by Birkhoff ergodic theorem, there is Δ0 ⊂ Δ˜ satisfying θ Δ0 = Δ0 and μ˜ (Δ0 ) = 1 such that lim

n→+∞

1 log κˆ (θ −n x) ˜ =0 n

(V.13)

for any x˜ ∈ Δ0 . From (V.13) it follows that def κ˜ (x) ˜ = sup{κˆ (θ −n x)e ˜ −nε | n ≥ 0}

is finite at each point x˜ ∈ Δ0 . Then it is easy to verify that the Borel set Δ0 and the function κ˜ : Δ0 → [1, +∞) satisfy the requirements of this lemma.  

56

V Unstable and Stable Manifolds for Endomorphisms

The above lemma tells us that the map f is locally invertible along a full orbit x˜ = {xn : n ∈ Z} ∈ Δ˜ . On M, the map fx˜−1 can be defined along a trajectory x˜ to be the ’inverse’ map of f which maps x0 to x−1 , wherever it makes sense. We define for each n ≥ 1 def −1 fx˜−n = fθ−1 −n+1 x˜ ◦ · · · ◦ f x˜ def

with fx˜0 = idM wherever it makes sense.

V.3 Some Technical Facts About Contracting Maps In this section we introduce some additional technical facts concerning contracting maps which will be used later on. First we have the following simple result. Lemma V.3.1 Let X and Y be two complete metric spaces and let X × Y endowed with the product metric, i.e. for (x, y), (x , y ) ∈ X × Y ,     d (x, y), (x , y ) = max d(x, x ), d(y, y ) . Let ϑ : X ×Y → Y be a continuous map. Suppose that ϑ is a uniform λ -contraction for some 0 < λ < 1 on the second factor, i.e.   d ϑ (x, y), ϑ (x, y ) ≤ λ d(y, y ),

∀x ∈ X, ∀y, y ∈ Y.

For any x ∈ X, let ϑx denote the map ϑx : Y → Y , y → ϑ (x, y) and let ϕ (x) be the unique fixed point of ϑx . Then we have the following results. 1) The map ϕ : X → Y , x → ϕ (x) is continuous; 2) When ϑ is Lipschitz, ϕ is also Lipschitz with Lip(ϕ ) ≤ Lip(ϑ )/(1 − λ ). Moreover, if Lip(ϑ ) ≤ λ , then Lip(ϕ ) ≤ λ . Proof. For any two points x and x ∈ X, we have     d ϕ (x), ϕ (x ) = d ϑ (x, ϕ (x)), ϑ (x , ϕ (x ))     ≤ d ϑ (x, ϕ (x)), ϑ (x , ϕ (x)) + d ϑ (x , ϕ (x)), ϑ (x , ϕ (x ))     ≤ d ϑ (x, ϕ (x)), ϑ (x , ϕ (x)) + λ d ϕ (x), ϕ (x ) . Therefore

  d ϕ (x), ϕ (x ) ≤

  1 d ϑ (x, ϕ (x)), ϑ (x , ϕ (x)) , 1−λ from which there follows 1). If ϑ is Lipschitz, from (V.14) we obtain   d ϕ (x), ϕ (x ) ≤

1 Lip(ϑ )d(x, x ). 1−λ

(V.14)

V.3 Some Technical Facts About Contracting Maps

57

Hence ϕ is Lipschitz with Lip(ϕ ) ≤ 1−1 λ Lip(ϑ ). Furthermore, if Lip(ϑ ) ≤ λ , we have   d(ϕ (x), ϕ (x )) = d ϑ (x,ϕ (x)), ϑ (x , ϕ (x ))   ≤ λ max d(x, x ), d ϕ (x), ϕ (x ) . Since λ < 1,

  d ϕ (x), ϕ (x ) ≤ λ d(x, x ).  

Thus 2) is proved.

Lemma V.3.2 Under the circumstances of Lemma V.3.1, suppose moreover that X and Y are closed subsets of two Banach spaces (E,  · ) and (F,  · ) respectively and that ϑ is Lipschitz. We restrict below ϑ and ϕ respectively to Int(X × Y ) and Int(X) ∩ ϕ −1 Int(Y ), where Int(·) denotes the interior of a set. If ϑ is of class C1 , then ϕ is also of class C1 . In addition, if Dϑ is Lipschitz, then Dϕ is also Lipschitz with Lip(Dϕ ) ≤ CLip(Dϑ ), where C = (1 − λ )−3[1 + Lip(ϑ )]2 . Proof. Let (x, y) ∈ Int(X ×Y ). Let D1 ϑ (x, y) denote the partial derivative of ϑ with respect to x at the point (x, y). D2 ϑ (x, y) is defined analogously. We first remark that D2 ϑ (x, y) ≤ λ since ϑ is a uniform λ -contraction on the second factor. Consequently id − D2 ϑ (x, y) is invertible and its inversion can be given by the formula  −1 id − D2ϑ (x, y) =

+∞ 

∑

n D2 ϑ (x, y) .

(V.15)

n=0

We now prove that ϕ is of class C1 with   −1  Dϕx = id − D2 ϑ (x, ϕ (x)) D1 ϑ x, ϕ (x) .

(V.16)

  In fact, x, ϕ (x) ∈ Int(X × Y ) since x ∈ Int(X) ∩ ϕ −1 Int(Y ). Let x ∈ E such that x + x ∈ Int(X), then we have       ϕ (x + x) − ϕ (x) − id − D2 ϑ (x, ϕ (x)) −1 D1 ϑ x, ϕ (x) x  −1   ϕ (x + x) − ϕ (x) ≤  id − D2 ϑ x, ϕ (x)       −D2 ϑ x, ϕ (x) ϕ (x + x) − ϕ (x) − D1 ϑ x, ϕ (x) x   −1    ϑ x + x, ϕ (x + x) − ϑ (x, ϕ (x)) ≤  id − D2 ϑ x, ϕ (x)   −Dϑ(x,ϕ (x)) x, ϕ (x + x) − ϕ (x) .   This last expression is o( x, ϕ (x + x) − ϕ (x) ) and is hence o(x) since ϕ is Lipschitz. Therefore, there exists the derivative of ϕ at x and it can be given by formula (V.16). From (V.15) and (V.16) it follows that Dϕx depends continuously on x ∈ Int(X) ∩ ϕ −1 Int(Y ). Thus ϕ is of class C1 . Now suppose that Dϑ is Lipschitz. Let x, x ∈ Int(X) ∩ ϕ −1 Int(Y ), then by (V.15) and (V.16) we have

58

V Unstable and Stable Manifolds for Endomorphisms

  Dϕx − Dϕx   +∞   n +∞  n    ≤  ∑ D2 ϑ (x, ϕ (x)) − ∑ D2 ϑ (x , ϕ (x )) D1 ϑ (x, ϕ (x)) n=0  +∞ 

n=0

 n    + ∑ D2 ϑ (x , ϕ (x )) D1 ϑ (x, ϕ (x)) − D1 ϑ (x , ϕ (x )) n=0

≤

 +∞

∑ (n + 1)λ n

n=0  +∞

+

∑ λn



  Lip(Dϑ )Lip(ϑ )d (x, ϕ (x)), (x , ϕ (x ))

   Lip(Dϑ )d (x, ϕ (x)), (x , ϕ (x ))

n=0

≤ (1 − λ )−3[1 + Lip(ϑ )]2 Lip(Dϑ )d(x, x ). Thus Dϕ is Lipschitz with Lip(Dϕ ) ≤ (1 − λ )−3[1 + Lip(ϑ )]2 Lip(Dϑ ).

 

V.4 Local Unstable Manifolds First let us present a generalized stable manifold theorem for a sequence of maps. We will employ the theorem frequently in this book. Let {(En ,  · n )}∞ n=0 be a sequence of Banach spaces and A = {An : En → En+1 }∞ be a sequence of linear hyperbolic maps, i.e., there is a splitting n=0 En = Ens ⊕ Enu , ∀n ≥ 0 with s u = An Ens , En+1 = An Enu , ∀n ≥ 0 En+1

and a number τ ∈ (0, 1) such that def

xn = max{xs n , xu n } ∀x = xs + xu , xs ∈ Ens , xu ∈ Enu ; Asn  = (An |Ens ) ≤ τ , (Aun )−1  = (An |Enu )−1  ≤ τ . In the above case, we also say that the sequence {An }∞ n=0 is a hyperbolic sequence with a slope≤ τ . For each ρ > 0 define def

En (ρ ) = {x ∈ En : xn ≤ ρ }, def



def



Ens (ρ ) = {x ∈ Ens : xn ≤ ρ } = En (ρ ) Enu (ρ ) = {x ∈ Enu : xn ≤ ρ } = En (ρ ) and we will view En (ρ ) as Ens (ρ ) × Enu(b).

Ens , Enu ,

V.4 Local Unstable Manifolds

59

Lemma V.4.1 Let {An }∞ n=0 be a hyperbolic sequence with a slope≤ τ < 1. Let f n = An + ϕn : En (ρ ) → En+1 satisfies fn (0) = 0 and 1 Lip( fn − An) = Lip(ϕn ) < ε < (τ −1 − 1), ∀n ≥ 0. 2 If x , x ∈ En (ρ ) satisfy then

x u − xu n ≥ x s − xs n ,

 fnu (x ) − fnu (x)n+1 ≥  fns (x ) − fns (x)n+1

and

 fn (x ) − fn (x)n+1 ≥ (τ −1 − ε )x − xn,

where ( fns (x ), fnu (x )) and ( fns (x), fnu (x)) denote respectively the coordinates of s u . ⊕ En+1 fn (x ) and fn (x) with respect to the splitting En+1 = En+1 Proof. Notice that under the circumstance of the above lemma, one has x − xn = max{x s − xs n , x u − xu n } = x u − xu n ≥ x s − xs n .

Hence  fnu (x ) − fnu (x)n+1 = Aun x u − Aunxu + ϕnu(x ) − ϕnu(x)n+1

≥ Aun (x u − xu )n+1 − ϕnu (x ) − ϕnu (x)n+1

≥ (Aun )−1 −1 x u − xu n − Lip(ϕ )x − xn ≥ τ −1 x u − xu n − ε x − xn  fns (x ) − fns (x)n+1

= (τ −1 − ε )x − xn, = Asn x s − Asn xs + ϕns (x ) − ϕns (x)n+1 ≤ Asn x s − xs n + Lip(ϕ )x − xn ≤ τ x s − xs n + ε x − xn ≤ (τ + ε )x − xn .

Therefore  fnu (x ) − fnu (x)n+1 ≥ (τ −1 − ε )x − xn

≥ (τ + ε )x − xn ≥  fns (x ) − fns (x)n+1

and  fn (x ) − fn (x)n+1 =  fnu (x ) − fnu (x)n+1 ≥ (τ −1 − ε )x − xn .

 

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V Unstable and Stable Manifolds for Endomorphisms

Corollary V.4.1.1 Under the circumstance of the above lemma, if we assume furthermore fnl (x ), fnl (x) ∈ En+k (ρ ), k = 0, 1, 2, · · · , def

where fn0 = id and fnl = fn+l−1 ◦ · · · ◦ fn (defined wherever they make sense), then x = x. Corollary V.4.1.2 Let f = { fn = An + ϕn : En (ρ ) → En+1 }∞ n=0 satisfies 1 Lip( fn − An) = Lip(ϕn ) < ε < (τ −1 − 1), ∀n ≥ 0, 2

where {An }∞ n=0 is a hyperbolic sequence with a slope≤ τ < 1. If x , x ∈ En (ρ ) satisfy

1) fnl (x ), fnl (x) ∈ En+l (ρ ), l = 0, 1, 2, · · ·, 2) x s = xs , then x = x. Remark V.4. From Corollary V.4.1.2, for each xs ∈ Ens (ρ ) there exists at most one point xu = gn (xs ) ∈ Enu (ρ ) such that fnl (x) ∈ En+l (ρ ), l = 0, 1, 2, · · · , where x = (xs , xu ). Hence, if we define the local stable manifold of f = { fn = An + ϕn }∞ n=0 corresponding to En by def

WEsn (ρ ) (0, f ) = {x ∈ En (ρ ) : fnl (x) ∈ En+l (ρ ), l = 0, 1, 2, · · · and lim  fnl (x)n+l = 0}, l→+∞

then the local stable manifold of f corresponding to En is on the graph of some map gn : Ens (ρ ) → Enu (ρ ), i.e. WEsn (ρ ) (0, f ) ⊂ Graph(gn ). It follows clearly from Corollary V.4.1.1 that Lip(gn ) ≤ 1. Theorem V.4.2 (Generalized Stable Manifold Theorem) Let f = { fn = An + ϕn : ∞ En (ρ ) → En+1 }∞ n=0 be a sequence of maps, where A = {An }n=0 is a hyperbolic ∞ sequence with a slope≤ τ < 1 and {ϕn : En (ρ ) → En+1 }n=0 satisfies for each n ≥ 0 Lip(ϕn ) < ε < min{1 − τ , 12 (τ −1 − 1)}, ϕn (0) = 0. Then for each n ≥ 0 there exists a Lipschitz map gn : Ens (ρ ) → Enu (ρ )

V.4 Local Unstable Manifolds

61

such that (1) (2) (3) (4)

gn (0) = 0; Lip(gn ) ≤ τ + ε ;  l −1 Graph(gn ) = WEsn (ρ ) (0, f ) = +∞ l=0 ( f n ) En+l (ρ ); fnWEsn (ρ ) (0, f ) ⊂ WEs (b) (0, f ), and for each x, y ∈ WEsn (ρ ) (0, f ) n+1

 fn (x) − fn (y)n+1 ≤ (τ + ε )x − yn; (5) If ϕn is of class C2 on En (ρ ) for each n ≥ 0 with supn≥0 Lip(Dϕn (·)) < +∞, then gn is of class C1,1 and Lip(Dgn (·)) ≤ (1 − τ − ε )−3 (1 + τ + ε )2 sup Lip(Dϕl (·)); l≥n

(6) If ϕn is of class C1 and Dϕn (0) = 0 for each n ≥ 0, then Dgn (0) = 0 for each n ≥ 0. Hence the tangent space of WEsn (ρ ) (0, f ) at 0 is just Ens . Proof. We complete the proof by the following five steps. Step 1. Define s Γ s = { ξ = { ξl } ∞ l=1 : ξl ∈ En+l , and ξ  = sup ξl n+l }, def

def

l≥1

u def

Γ = {η

= {ηl }∞ l=0

:

u ηl ∈ En+l ,

def

and η  = sup ηl n+l }. l≥0

It is not difficult to see that (Γ s ,  · ) and (Γ u ,  · ) are both Banach spaces with respect to the natural operations of addition and scalar multiplication. Define def

Γ s (ρ ) = {ξ ∈ Γ s : ξ  ≤ ρ }, def

Γ u (ρ ) = {η ∈ Γ u : η  ≤ ρ }. They are obviously closed subsets of Γ s and Γ u respectively. Step 2. We now define

ϑ : Ens (ρ ) × Γ s (ρ ) × Γ u (ρ ) → Γ s (ρ ) × Γ u (ρ ), where

(xs , ξ , η ) −→ (ξ , η )



ξ1 = Asn xs + ϕns (xs , η0 ), s ξ = Asn+l−1 ξl−1 + ϕn+l−1 (ξl−1 , ηl−1 ), l ≥ 2,    l

u −1 u η0 = (An ) η1 − ϕn (xs , η0 ) ,  u (ξ , η ) , l ≥ 1, ηl = (Aun+l )−1 ηl+1 − ϕn+l l l

It is easy to verify that ϑ is well defined and is a Lipschitz map satisfying Lip(ϑ ) ≤ τ + ε < 1

62

V Unstable and Stable Manifolds for Endomorphisms

with respect to the product metrics on Ens (ρ ) × Γ s (ρ ) × Γ u (ρ ) and Γ s (ρ ) × Γ u (ρ ). We now show that if fn is of class C2 for each n ≥ 0 with supn≥0 Lip(Dϕn (·)) < +∞, then ϑ is C1 in Int(Ens (ρ ) × Γ s (ρ ) × Γ u (ρ )) and Dϑ is Lipschitz. Let (x∗s , ξ ∗ , η ∗ ) be a point in Int(En (ρ ) × Γ s (ρ ) × Γ u (ρ )). Define a map A(x∗s , ξ ∗ , η ∗ ) : Ens × Γ s × Γ u → Γ s × Γ u , (xs , ξ , η ) −→ (ξ

, η

) by letting 

and

ξ1

= Asn xs + Dϕns (x∗s , η0∗ )(xs , η0 ), s ∗ , η ∗ )(ξ ξl

= Asn+l−1 ξl−1 + Dϕn+l−1 (ξl−1 l−1 , ηl−1 ), l ≥ 2, l−1 $

  η0

= (Aun )−1 η1 − Dϕnu (x∗s , η0∗ )(xs , η0 ) ,   u (ξ ∗ , η ∗ )(ξ , η ) , l ≥ 1. ηl

= (Aun+l )−1 ηl+1 − Dϕn+l l l l l

It follows that A(x∗s , ξ ∗ , η ∗ ) is a well-defined bounded linear operator and is the derivative of ϑ at point (x∗s , ξ ∗ , η ∗ ). By a simple computation we obtain that Dϑ , which is defined on Int(Ens (ρ ) × Γ s (ρ ) × Γ u (ρ )), is a Lipschitz map such that Lip(Dϑ ) ≤ sup Lip(Dϕl (·)) < +∞. l≥n

And if ϕl is of class C1 with Dϕl (0) = 0 for each l ≥ 0, then D1 ϑ u (0, 0, 0) = 0, where (ϑ s (xs , ξ , η ), ϑ u (xs , ξ , η )) is the coordinate of ϑ (xs , ξ , η ) with respect to Γ s × Γ u and D1 ϑ u (xs , ξ , η ) denotes the partial derivative of ϑ u with respect to xs at the point (xs , ξ , η ). Step 3. Since Lip(ϑ ) ≤ τ + ε < 1, by Lemma V.3.1 there exists a Lipschitz map

ϕ : Ens (ρ ) → Γ s (ρ ) × Γ u (ρ ) with

Lip(ϕ ) ≤ τ + ε

such that for any xs ∈ Ens (ρ ), ϕ (xs ) is the unique fixed point of the map ϑxs : Γ s (ρ )× Γ u (ρ ) → Γ s (ρ ) × Γ u (ρ ), (ξ , η ) → ϑ (xs , ξ , η ). Since ϕ (0) = (0, 0) and Lip(ϕ ) ≤ τ + ε , we have ϕ (Int(Ens (ρ ))) ⊂ Int(Γ s (ρ )× Γ u (ρ )). Thus by Lemma V.3.2 the map Dϕ is well defined on Int(Ens (ρ )) and is a Lipschitz map such that Lip(Dϕ (·)) ≤ (1 − τ − ε )−3 (1 + τ + ε )2 Lip(Dϑ ).

V.4 Local Unstable Manifolds

63

And if ϕl is of class C1 with Dϕl (0) = 0 for each l ≥ 0, then Dϕ u (0) = 0, where ϕ u is defined analogous of ϑ u .   s Step 4. Let ϕ (xs ) = ϕ (xs ), ϕ u (xs ) for xs ∈ Ens (ρ ). Define gn : Ens (ρ ) → Enu (ρ ),

xs → ϕ u (xs )(0).

From Step 3 it follows that gn is a C1,1 map satisfying gn (0) = 0 and Lip(gn ) ≤ τ + ε , Lip(Dgn (·)) ≤ (1 − τ − ε )−3 (1 + τ + ε )2 Lip(Dϑ ).

(V.17) (V.18)

And if ϕl is of class C1 with Dϕl (0) = 0 for each l ≥ 0, then Dgn (0) = 0 for all n ≥ 0. Step 5. It is easy to see that Graph(gn ) ⊂

+∞ 

( fnl )−1 En+l (ρ ).

l=0



l −1

If x = (xs , xu ) ∈ +∞ l=0 ( f n ) En+l (ρ ), then x = (xs , gn (xs )) is also a point in l −1

k=0 ( f n ) En+l (ρ ). From Remark V.4 one has x = x . Hence def

+∞

Graph(gn ) =

+∞ 

( fnl )−1 En+l (ρ )

l=0

and fn Graph(gn ) ⊂ Graph(gn+1 ). For each x, y ∈ Graph(gn ), it is clear that x − yn = max(xs − ys n , gn (xs ) − gn(ys )n ) = xs − ys n and fn (x), fn (y) ∈ Graph(gn+1 ) for all n ≥ 0. Therefore  fn (x) − fn (y)n+1 =  fns (x) − fns (y)n+1 = Asn (xs − ys ) + ϕns (x) − ϕns (y) ≤ (τ + ε )x − yn. Hence

lim  fnl (x)n+l ≤ lim (τ + ε )l xn = 0.

l→+∞

l→+∞

64

V Unstable and Stable Manifolds for Endomorphisms

This implies Graph(gn ) ⊂ WEsn (ρ ) (0, f ). Then from Remark V.4 one has Graph(gn ) = WEsn (ρ ) (0, f ). Let r ≥ 2ρ2−1 be a number such that the Borel set

 

 C ,r def C

˜ ˜ < r = 0. / Δa,b,k, ε = x˜ ∈ Δ a,b,k,ε : x˜ ∈ Δ 0 , κ (x)



C ,r In this section we fix such a set Δa,b,k, ε and confine ourselves to it. For simplicity, we write C ,r

Δ = Δa,b,k, ε.

Remark V.5. For any x˜ ∈ Δ and n ∈ Z+ , it is not difficult to see from Lemma V.2.4 that  −1 −ε n  def  −1 → Tx−n−1 M e Gx,n ˜ = H x−n−1 : Tx−n M r  

−1 e−ε n as is well defined. Consequently, the map fθ−1 −n x˜ is well defined on B x−n , r   def −1

−1 −ε n fθ−1 →M e ˜ ◦ expx−n : B x−n , r −n x˜ = expx−n−1 ◦Gx,n and

f ◦ fθ−1 −n x˜ |B(x ,r −1 e−ε n ) = id|B(x ,r −1 e−ε n ) . −n −n   Definition V.4.1 Let X be a metric space and let Dx x∈X be a collection of subsets   of M. We call Dx x∈X a continuous family of C1 embedded k-dimensional discs in M if there is a finite open cover {Ui }li=1 of X such that for each Ui there exists a continuous map ϑi : Ui → Emb1 (Dk , M) such that ϑi (x)Dk = Dx , x ∈ Ui , where  k k D = ξ ∈ R : ξ 0 < 1 . About local unstable manifolds of f we have the following result. + 1,1 Theorem V.4.3 For each n ∈  Z , there exists a continuous family of C embedded ˜ x∈ in M and there exist numbers αn , βn and γn which k-dimensional discs Wn (x) ˜ Δ

depend only on a, b, k, ε ,C and r such that the followings hold for every x˜ ∈ Δ :

1) There exists a C1,1 map ˜ → E(θ −n x), ˜ hx,n ˜ : O−n (x)   where O−n (x) ˜ is an open subset of H(θ −n x) ˜ which contains ξ : |ξ | < αn , such that i) hx,n ˜ (0) = 0, Dhx,n ˜ (0) = 0;

V.4 Local Unstable Manifolds

65

ii) Lip(hx,n ˜ ) ≤ βn , Lip(Dh x,n ˜ ) ≤βn ;  −n x) ˜ = expx−n Graph(hx,n ˜ at the point x−n ; iii) Wn (x) ˜ ) which is tangent to H(θ     2) Wn (x) ˜ ⊂ B x−n , r −1 e−ε n and fθ−1 ˜ ⊂ Wn+1 (x); ˜ −n x˜ Wn (x) u (−b+2 ε )l u dθ −n x˜ (y−n , z−n ), y−n , z−n ∈ Wn (x), ˜ l ∈ Z+ , 3) dθ −n−l x˜ (y−n−l , z−n−l ) ≤ γn e −l −l u where y−n−l = fθ −n x˜ (y−n ), z−n−l = fθ −n x˜ (z−n ), dθ −m x˜ ( , ) is the distance along Wm (x) ˜ for m ∈ Z+ ; 4) αn+1 = αn e−3ε , βn+1 = βn e3ε , γn+1 = γn eε . Proof. We complete the proof by the following six steps. We fix x˜ ∈ Δ from Step 1 to Step 5. Step 1. Define for each n ≥ 0 def

def

En = Tx−n M, def

˜ Enu = E(θ −n x). ˜ Ens = H(θ −n x), Let 0 < ε ≤ 12 min{1, b−a 200 } and t = e−b+3ε . Now we define a new norm  · n on En by u

s u s u vn = t −n max(vs  x,n ˜ , v x,n ˜ ), ∀v = v + v ∈ En ⊕ En . def

Then it is easy to see t −n |v| ≤ vn ≤ Aeε n/2t −n |v|, ∀n ≥ 0, v ∈ En . 2

(V.19)

Setp 2. Let def

: En (ρ ) → En+1 , fn = Gx,−n ˜ −1 An = T−n (x) ˜ : En → En+1 , def def

ϕn = fn − An : En (ρ ) → En+1 , where ρ > 0 will be determined later. It is clear that ϕn (0) = 0, An = D fn (0), Dϕn (0) = 0 and Asn  u −1 (An ) 

≤ e−2ε = τ < 1, ≤ τ < 1.

By Lemma V.2.4, −n ˜ Lip(Dϕn (·)) = Lip(DGx,n ˜ (·)) ≤ κ˜ (θ x),

66

V Unstable and Stable Manifolds for Endomorphisms

where Lip(·) is defined with respect to | · |. Hence if ϕn is well defined on En (ρ ), then n+1 ˜ Lip· (Dϕn (·)) ≤ 4At n−1e 2 ε κ˜ (θ −n x), where Lip· (·) is defined with respect to  · n and  · n+1. Since x˜ ∈ Δ , one has Lip· (Dϕn (·)) ≤ 4Ar eb−2ε e−n(b−5ε ). Step 3. Fix n ≥ 0. We now determine ρ such that for each l ≥ 0, defined on En+l (ρ ) and

ϕn+l is well

Lip· (ϕn+l ) < ε ,

sup Lip· (Dϕn+l (·)) < +∞, l≥0

where Lip· (·) is defined with respect to  · n+l and  · n+l+1. First if v ∈ En+l (ρ ), then |v| ≤ 2t n+l ρ . Hence if

en(b−4ε ) , (V.20) 2r

then ϕn+l is well defined on En+l (ρ ) for each l ≥ 0. Suppose ρ satisfies (V.20). Let’s evaluate Lip(ϕn+l ) for each l ≥ 0 as bellow

ρ≤

ϕn+l (v) − ϕn+l (v )n+l+1 =  ≤

 1 0

 1 0

[Dϕn+l (λ v)v − ϕn+l (λ v )v ]d λ n+k+1 [Dϕn+l (λ v) − Dϕn+l (λ v )]v d λ n+l+1

+

 1 0

Dϕn+l (λ v)(v − v )d λ n+l+1

≤ 2ρ Lip· (Dϕn+l (·))v − v n+l , where v, v ∈ En+l (ρ ). Hence if let r0 =

ε e−b+2ε 8Ar

and put

ρn = r0 en(b−5ε ) , def

then ϕn+l : En+l (ρ ) → En+l+1 is well defined for all l ≥ 0 with ρ = ρn and Lip· (ϕn+l ) ≤ ε , ∀l ≥ 0, ε . sup Lip· (Dϕn+l (·)) ≤ 2ρn l≥0

V.4 Local Unstable Manifolds

67

Step 4. For l ∈ Z+ , let ˜ = id, G0n (x)

Gln (x) ˜ = Gx,n+l−1 ◦ · · · ◦ Gx,n ˜ , ˜

l > 0,

defined wherever they make sense. Notice that ρn+1 > ρn for each n ≥ 0. From Step 3, we can apply the Generalized Stable Manifold Theorem to { fn+k = An+k + ϕn+k }∞ k=0 . Hence for each n ≥ 0 and s u x˜ ∈ Δ , there exists a C1,1 map hx,n ˜ : En (ρn ) → En (ρn ) such that (i) hx,n ˜ (0) = 0, Dhx,n ˜ (0) = 0; ) ≤ τ + ε < e− ε ; (ii) Lip· (hx,n ˜ l ˜ ∈ En+l (ρn ), l = 0, 1, · · ·} and (iii) Graph(hx,n ˜ ) = {v ∈ En (ρn ) : Gn (x)(v) Gx,n ); ˜ Graph(hx,n ˜ ) ⊂ Graph(hx,n+1 ˜ (iv) for each v, v ∈ Graph(hx,n ˜ ),



Gx,n ˜ (v) − Gx,n ˜ (v )n+1 ≤ (τ + ε )v − v n ;

  −3 2 (v) Lip· Dhx,n ˜ (·) ≤ ε (1 − τ − ε ) (1 + τ + ε ) /(2ρn ).

Graph(hx,n }∞ ˜ ) can also express via the norms { · x,n+k ˜ k=0 : s u Graph(hx,n ˜ ) = {v = (v , v ) ∈ Tx−n M : l s 

˜ ≤ r0 e−2ε n e−(b−3ε )l , (Gn (x)(v)) x,n+l ˜ l u

˜ x,n+l ≤ r0 e−2ε n e−(b−3ε )l , (Gn (x)(v)) ˜ l = 0, 1, · · ·}.

(V.21)

From (ii) and (v), one has −ε (hx,n Lip· x,n ˜ ) ≤ τ +ε < e , ˜   Dh Lip· x,n (·) ≤ De2nε , x,n ˜ ˜

(V.22) (V.23)

where D = 4ε (1 − τ − ε )−3 r0−1 . Step 5. In this step we present the counterpart of the above results in terms of the norm | · |. Let x˜ ∈ Δ and n ∈ Z+ . Define  −2ε n On (x) ˜ = ξ ∈ H(θ −n x) ˜ : ξ  x,n . ˜ < r0 e ˜ = Ens (ρn ). Define Clearly On (x) ˜ = expx−n Graph(hx,n Wn (x) ˜ )

68

V Unstable and Stable Manifolds for Endomorphisms

and let

αn = A−1 r0 e−3ε n .

Then by 3) of Remark V.3 we have  ξ ∈ H(θ −n x) ˜ : |ξ | < αn ⊂ O−n (x). ˜ ˜ is tangent to H(θ −n x) ˜ at the point x−n . Moreover, And since Dhx,n ˜ (0) = 0, Wn (x) from Step 4, we obtain immediately   −1 Wn (x) ˜ ⊂ B x−n , r e−ε n Let

βn =

and

fθ−1 ˜ ⊂ Wn+1 (x). ˜ −n x˜Wn (x)

4 ε A2 e3nε . r0 (1 − τ − ε )3

Then from Step 4 and (V.19) it follows that Lip(hx,n ˜ ) ≤ βn ,

Lip(Dhx,n ˜ ) ≤ βn .

+ From (V.19) and Step 4 we obtain that for each v, v ∈ Graph(hx,n ˜ ) and l ∈ Z

|Gln (x)v ˜ − Gln(x)v ˜ | ≤ 2Aeε n e(−b+2ε )l |v − v | which implies that dθu −n−l x˜ (y−n−l , z−n−l ) ≤ γn e(−b+2ε )l dθu −n x˜ (y−n , z−n ), l ∈ Z+ for all y−n , z−n ∈ Wn (x), ˜ where y−n−l = fθ−l−n x˜ (y−n ), z−n−l = fθ−l−n x˜ (z−n ),

γn = 4A[b(ρ0/2)]2 eε n and b(ρ0 /2) is as introduced in Lemma II.II.2.1. Hence, for every x˜ ∈ Δ and n ∈ Z+ , Wn (x) ˜ and the numbers αn , βn and γn satisfy 1) – 4) of the   theorem. ˜ x∈ is a continuous Step 6. Now we complete the proof by showing that Wn (x) ˜ Δ

family of C1 embedded k-dimensional discs in M for each n ∈ Z+ . −n ˜ depend con˜ and Let n ∈ Z+ . By Lemma V.2.2 we know that H(θ −n x) θr x)  E(



tinuously on x˜ ∈ Δ . Then there exists a finite open cover Δl l=1 of Δ such that ˜ and a basis θ −n x), ˜ x˜ ∈ Δl which are for each Δl we can find a basis of H(θ −n x)  of E( r



continuous with respect to x˜ ∈ Δl . Let Δ p be a set in Δl l=1 . Since ·, · x,n ˜ depends continuously on x˜ ∈ Δ , then for each x˜ ∈ Δ p there exists, with respect to ·, · x,n ˜ , an  k  m0 −n orthonormal basis ξi (x) ˜ i=1 of H(θ x) ˜ and an orthonormal basis ξ j (x) ˜ j=k+1 −n

of E(θ x) ˜ such that they are continuous with respect to x˜ ∈ Δ p . For each x˜ ∈ Δ p , let

V.4 Local Unstable Manifolds

69

A(x) ˜ : Rk ⊕ Rm0 −k → H(θ −n x) ˜ ⊕ E(θ −nx) ˜ 0 be a linear map satisfying A(x)e ˜ s = ξr (x), ˜ 1 ≤ s ≤ m0 , where {es }m s=1 is the natural basis of Rk ⊕ Rm0 −k . Define a map

θ p : Δ p → Emb1 (Dk , M) by the formula

θ p (x) ˜ = expx−n ◦(id, hx,n ˜ Dk . ˜ ) ◦ A(x)|

˜ is a C1,1 embedding with Then it is clear that for each x˜ ∈ Δ p , θ p (x)

θ p (x)D ˜ k = Wn (x). ˜ Now we show that θ p is continuous. For x˜ ∈ Δ p , let h x,n ˜ −1 ◦ hx,n ˜ Dk . ˜ ◦ A(x)| ˜ = A(x)



Suppose that {x˜m }+∞ m=1 is a sequence of points in Δ p such that x˜m → x˜0 ∈ Δ p as m → +∞. According to Arzela-Ascoli Theorem, by (V.22)–(V.23) and (iii) of Step 4 we know that h x˜m ,n and Dh x˜m ,n converge uniformly to h x˜0 ,n and Dh x˜0 ,n respectively. This together with the continuity of A(x) ˜ with respect to x˜ ∈ Δ p implies that θ p is continuous.  



C ,r



Remark V.6. The set Δa,b,k, ε defined above depends only on a, b, k, ε , C and r . Denote Δ%a,b,k = Δa,b,k ∩ Δ0 (V.24)

+∞

and let {Cm }+∞ m=1 and {rm }m=1 be sequences of positive numbers such that Cm ↑ +∞

and rm ↑ +∞ as m → +∞. Then we have C

C ,r

,r

m+1 m+1 m m Δa,b,k, , ε ⊂ Δ a,b,k,ε

and

Δ%a,b,k =

+∞

m∈N

C ,r

m m Δa,b,k, ε.

m=1

If we write 

+∞  (an , bn ) = (a, b) : 0 ≤ a < b, a and b are rational n=1

and let

εn =

 b −a 1 n n min 1, , 2 200

(V.25)

70

V Unstable and Stable Manifolds for Endomorphisms

then

Δ0 =

m0 +∞  +∞





C ,r

Δanm,bnm,k,εn

 x˜ ∈ Δ0 : λi (x) ˜ ≤ 0, 1 ≤ i ≤ r(x) ˜ .

(V.26)

n=1 k=1 m=1 C ,r

Obviously, Theorem V.4.3 holds for any Δanm,bnm,k,εn . Remark V.7. For x˜ ∈ Δ , let  loc (x) ˜ = y˜ ∈ M f : y−n ∈ Wn (x), ˜ ∀n ≥ 0 . W

(V.27)

Due to Theorem V.4.3, it is clear that loc (x) ˜ → W0 (x) ˜ p:W

(V.28)

 (i) Δ0 = x˜ ∈ Δ0 : λi (x) ˜ >0 .

(V.29)

is bijective. Remark V.8. Let

(i)

From Theorem V.4.3 we know that for any x˜ ∈ Δ0 there exists a ki (x)-dimensional ˜ ˜ = ∑ij=1 m j (x), ˜ such that there are C1 embedded submanifold of M, where ki (x) numbers b > a ≥ 0 with λi (x) ˜ > b and λi+1 (x) ˜ < a, 0 < ε < min{1, (b − a)/200}, 0 < C1 ≤ 1 ≤ C2 , and for any y0 ∈ W0 (x) ˜ there exists a unique y˜ = {yn }n∈Z ∈ M f such that py˜ = y0 and d(y−n , x−n ) ≤ C1 e−n(b−3ε ), and

∀n ∈ N,

d(y−n , x−n ) ≤ C2 e−n(b−2ε )d(y0 , x0 ),

∀n ∈ N.

(V.30)

(V.31)

u,i We call W0 (x) ˜ a local ith -unstable manifold of f at x˜ and denote it by Wloc (x). ˜ Moreover, let  u,i f  u,i (x) ˜ y˜ satisfies (V.30) and (V.31) , (V.32) W loc ˜ = y˜ ∈ M : y0 ∈ Wloc (x),

which will be called a local ith -unstable set of f at x˜ in M f . Remark V.9. Let  Δ0 = x˜ ∈ Δ0 : λi (x) ˜ > 0 for some 1 ≤ i ≤ r(x) ˜ .

(V.33)

From Theorem V.4.3 we know that for any x˜ ∈ Δ0 there exists a ku (x)-dimensional ˜ C1 embedded submanifold of M, where ku (x) ˜ = ∑λi (x)>0 mi (x) ˜ such that there are ˜

V.4 Local Unstable Manifolds

71

numbers λ > 0, 0 < ε < λ /200, 0 < C1 ≤ 1 ≤ C2 , and for any y0 ∈ W0 (x) ˜ there exists a unique y˜ = {yn }n∈Z ∈ M f such that py˜ = y0 and d(y−n , x−n ) ≤ C1 e−nε , and

∀n ∈ N,

d(y−n , x−n ) ≤ C2 e−n(λ −ε )d(y0 , x0 ),

∀n ∈ N.

(V.34)

(V.35)

u (x). We call W0 (x) ˜ a local unstable manifold of f at x˜ and denote it by Wloc ˜ Moreover, let  u u loc W (x) ˜ = y˜ ∈ M f : y0 ∈ Wloc (x), ˜ y˜ satisfies (V.34) and (V.35) , (V.36)

which will be called a local unstable set of f at x˜ in M f . For any x˜ ∈ Δ˜ , put Exu˜ =

&

Ei (x), ˜

Excs ˜ =

λi (x)>0 ˜

&

E j (x) ˜

λ j (x)≤0 ˜

and Exi˜ =

i &

r−(i)

E j (x), ˜

Ex˜

j=1

=

r(x) ˜ &

, i = 1, 2, · · · , u + c,

j=i+1

where u, c and s denote respectively the number of positive, neutral and negative Lyapunov exponents of f at x, ˜ i.e. def

u = u(x) ˜ = #{λ j (x) ˜ > 0 : 1 ≤ j ≤ r(x)}, ˜ def

˜ = 0 : 1 ≤ j ≤ r(x)}, ˜ c = c(x) ˜ = #{λ j (x) def

s = s(x) ˜ = #{λ j (x) ˜ < 0 : 1 ≤ j ≤ r(x)}. ˜ From the above theorem and remarks, we have the following proposition which plays an important role in the discussions in Chapter VII. Proposition V.4.4 There is a countable number of compact subsets Δi , i ∈ N, of M f   with i∈N Δi ⊂ Δ0 and μ˜ (Δ0 \ i∈N Δi ) = 0 such that: (1) For each Δi , dim Exu˜ = const., written ki , for all x˜ ∈ Δi ; Exu˜ and Excs ˜ depend continuously on x˜ ∈ Δi ; 1,1 (2) For  uany Δi , there is a continuous family of C embedded ki -dimensional discs Wloc (x) ˜ x∈ in M together with positive numbers λi , εi < λi /200, ri < 1, γi , αi ˜ Δi and βi such that the following properties hold for each x˜ ∈ Δi : u (i) There is a C1,1 map hx˜ : Ox˜ → Excs ˜ , where Ox˜ is an open subset of Ex˜ which u contains ξ ∈ Ex˜ : |ξ | < αi , satisfying

72

V Unstable and Stable Manifolds for Endomorphisms

(a) hx˜ (0) = 0, Dhx˜ (0) = 0; (b) Lip(hx˜ ) ≤ βi ; Lip(Dh x˜ ) ≤ βi , where Dhx˜ (·) : ξ → Dhx˜ (ξ );  u (x) (c) Wloc ˜ = expx0 Graph(hx˜ ) ⊂ B(x0 , κ˜ (x) ˜ −1 ). u (x) (ii) For any y0 ∈ Wloc ˜ there is a unique y˜ ∈ M f such that py˜ = y0 , d(y−n , x−n ) ≤ ri e−εi n , and

∀n ∈ N,

d(y−n , x−n ) ≤ γi e−λi n d(y0 , x0 ),

∀n ∈ N,

u (x) u (x)) u (θ −1 x). Wloc ˜ ⊂ B(x0 , κ˜ (x) ˜ −1 ) and hx˜ (Wloc ˜ ⊂ Wloc ˜ (iii) Let  u u loc (x) ˜ = y˜ ∈ M f : y−n ∈ Wloc (θ −n x), ˜ ∀n ∈ N . W

It is clear that

u u loc (x) ˜ → Wloc (x) ˜ p :W

is bijective.  u (x), (iv) For any y, ˜ z˜ ∈ W loc ˜ dθu −n x˜ (y−n , z−n ) ≤ γi e−λi n dxu˜ (y0 , z0 ), u (θ −n x) ˜ for n ∈ Z+ . for all n ∈ Z+ , where dθu −n x˜ ( , ) is the distance along Wloc

Similarly we have the following proposition which plays an important role in the discussions in Chapter IX. (i) Proposition V.4.5 For fixed i, 1 ≤ i ≤ m0 with μ˜ (Δ˜0 ) > 0, there is a count (i) (i) (i) able number of compact subsets Δ , k ∈ N, of M f with k∈N Δ ⊂ Δ˜ and (i)  (i) μ˜ (Δ˜ 0 \ k∈N Δk ) = 0 such that:

k

0

k

(i)

r−(i)

(1) For each Δk , dim Exi˜ = const., written ki , for all x˜ ∈ Δi ; Exi˜ and Ex˜ continuously on x˜ ∈ (i)

depend

(i) Δk ;

(2) For any Δk , there is a continuous family of C1,1 embedded ki -dimensional discs  u,i  Wloc (x) ˜ ˜ > bk and (i) in M together with positive numbers bk > ak with λi (x) x∈ ˜ Δk

(i)

λi+1 (x) ˜ < ak for all x˜ ∈ Δk , εk < min{1, (bk − ak )/200}, rk < 1, γk , αk and βk (i) such that the following properties hold for each x˜ ∈ Δk : r−(i)

i (i) There is aC1,1 map hix˜ : Ox˜ →  Ex˜ , where Ox˜ is an open subset of Ex˜ which i contains ξ ∈ Ex˜ : |ξ | < αk , satisfying (a) hix˜ (0) = 0, Dhix˜ (0) = 0; (b) Lip(hix˜ ) ≤ βk ; Lip(Dhix˜ ) ≤ βk , where Dhix˜ (·) : ξ → Dhix˜ (ξ );   u,i (c) Wloc (x) ˜ = expx0 Graph(hix˜ ) ⊂ B(x0 , κ˜ (x) ˜ −1 ).

V.5 Global Unstable Sets

73

u,i (ii) For any y0 ∈ Wloc (x) ˜ there is a unique y˜ ∈ M f such that py˜ = y0 ,

d(y−n , x−n ) ≤ rk e−n(bk −3εk ) , and

∀n ∈ N,

d(y−n , x−n ) ≤ γk e−n(bk −2εk ) d(y0 , x0 ),

∀n ∈ N,

u,i u,i u,i −1 (x) ˜ ⊂ B(x0 , κ˜ (x) ˜ −1 ) and hix˜ (Wloc (x)) ˜ ⊂ Wloc (θ x). ˜ Wloc (iii) Let  u,i −n f  u,i (x) W ˜ = y ˜ ∈ M : y ∈ W ( θ x), ˜ ∀n ∈ N . −n loc loc

It is clear that

u,i  u,i (x) p:W ˜ loc ˜ → Wloc (x)

is bijective.  u,i (x), (iv) For any y, ˜ z˜ ∈ W loc ˜ dθu,i−n x˜ (y−n , z−n ) ≤ γk e−n(bk −2εk ) dxu,i ˜ (y0 , z0 ), u,i −n for all n ∈ Z+ , where dθu,i−n x˜ ( , ) is the distance along Wloc (θ x) ˜ for n ∈ Z+ .

V.5 Global Unstable Sets Let x˜ ∈ Δ0 and let λ1 (x) ˜ > · · · > λu(x) ˜ be the strictly positive Lyapunov exponents ˜ (x) u,1 u,u   at x. ˜ Define W (x) ˜ ⊂ · · · ⊂ W (x) ˜ by  ' 1  u,i (x) W ˜ = y˜ ∈ M f : lim sup log d(y−n , x−n ) ≤ −λi (x) ˜ , n→+∞ n  u,u (x)  u (x) for 1 ≤ i ≤ u(x). ˜ And W ˜ is denoted by W ˜ . Then  u,i (θ x)  u,i (x). W ˜ = θW ˜ Let i ∈ {1, 2, . . . , u} be fixed and let ki = dim Exi˜ ,

 u,i (x). and W u,i (x) ˜ = pW ˜

Let a, b, ε , C and r satisfy the following properties:   ˜ < a < b < λi (x) ˜ max 0, λi+1 (x)

(V.37)

74

V Unstable and Stable Manifolds for Endomorphisms

(we admit here λr(x)+1 (x) ˜ = −∞), and ˜  b − a 0 < ε < min 1, 200



C ,r and x˜ ∈ Δa,b,k, ε.

 +∞ Correspondingly, we have the sequence of norms  ·  x,n and numbers A and ˜ n=0 r0 as defined in Remark V.3 and the proof of Theorem V.4.3 respectively. Let  +∞ Wn (x) ˜ n=0 be the sequence of embedded ki -dimensional discs obtained by apply

C ,r ing Theorem V.4.3 to the set Δa,b,k, ε . Define

 1  (x) W ˜ = y˜ ∈ M f : lim sup log d(y−n , x−n ) ≤ −b + 2ε . n→+∞ n  (x), Lemma V.5.1 If y˜ ∈ W ˜ then there is n0 ∈ N such that y−n ∈ Wn (x) ˜ for all n ≥ n0 .  (x), Proof. From the definition, for y˜ ∈ W ˜ there exists n0 ∈ N such that d(y−n , x−n ) ≤ e(−b+5ε /2)n, and

∀n ≥ n0

e(−b+5ε /2)n0 ≤ A−1 r0 e−5ε n0 /2 .

Then we have for each l ∈ Z+ d(y−n0 −l , x−n0 −l ) ≤ A−1 r0 e−5ε n0 /2 e(−b+5ε /2)l which together with (V.21) and 3) of Remark V.3 yields that y−n0 ∈ Wn0 (x). ˜ Then ˜ for all n ≥ n0 .   by Theorem V.4.3 we have y−n ∈ Wn (x) Let b , ε , C

and r

be numbers such that b ≤ b < λi (x), ˜ ε ≤ ε , C

≥ C , r

≥ r , C

,r

C

,r

and x˜ ∈ Δa,b

,k,ε . Corresponding to Δ a,b ,k,ε , we obtain analogously the sequence of  +∞ norms  · 

˜ n=0 , the numbers A and r0 and the sequence of embedded discs  +∞ x,n Wn (x) ˜ n=0 . Define  1  (x) ˜ = y˜ ∈ M f : lim sup log d(y−n , x−n ) ≤ −b + 2ε . W n→+∞ n Obviously,

 u,i (x)  (x)  (x). W ˜ ⊂W ˜ ⊂W ˜

(V.38)

 (x), Lemma V.5.2 If y˜ ∈ W ˜ then there is n0 ∈ N such that y−n ∈ Wn (x) ˜ for all n ≥ n0 . Proof. Indeed, it is clear that for any n ∈ Z+ and v = vs + vu ∈ Eθi −n x˜ ⊕ Eθ −n x˜

r−(i)

u

vu  x,n ˜ = v x,n ˜

V.5 Global Unstable Sets

and

75

s

ε n/2 s

v x,n vs  x,n ˜ ≤ v x,n ˜ ≤Ae ˜ .

Hence





ε n/2 ζ  x,n ζ  x,n ˜ ≤ ζ x,n ˜ ≤Ae ˜ ,

ζ ∈ Tx−n M.

For each n ∈ Z+ , write Wn (x) ˜ = expx−n Graph(hx,n ˜ ), and define

Wn (x) ˜ = expx−n Graph(h x,n ˜ )

  −2ε n −2ε n  Un (x) . ˜ = ξ ∈ E i (θ −n x) ˜ : ξ 

x,n , r0 e ˜ ≤ min r0 e

Since for each ζ ∈ Graph(h x,n ˜ ) ˜ |Un (x) 

 l Gn (x) ˜ ζ x,n+l ≤ ˜ ≤ ≤

 l 

Gn (x) ˜ ζ x,n+l ˜ (−b +2ε )l ζ 

x,n ˜ e r0 e−2ε n e(−b+3ε )l

for all l ∈ Z+ , we have   Graph h x,n ˜ ) ˜ ⊂ Graph(hx,n ˜ |Un (x) and hence

    Graph h x,n ˜ |Un (x) ˜ = Graph hx,n ˜ . ˜ |Un (x)

(V.39)

+ For each ζ ∈ Graph(hx,n ˜ ) and any n ∈ Z , when l is large enough we have

 l 

 

Gn (x) ˜ ζ x,n+l ≤ A eε (n+l)/2 Gln (x) ˜ ζ x,n+l ˜ ˜

≤ A eε (n+l)/2 r0 e−2ε n e(−b+2ε )l

≤ min{r0 e−2ε (n+l) , r0 e−2ε (n+l) }. Hence by (V.39),

 

fθ−l−n x˜ Wn (x) ˜ ⊂ Wn+l (x). ˜

Thus by Lemma V.5.1 we have n0 ∈ N such that y−n ∈ Wn (x) ˜ for all n ≥ n0 . Corollary V.5.2.1

 (x)  (x).  u,i (x) ˜ =W ˜ =W ˜ W

Proof. As a consequence of Lemma V.5.2 and Theorem V.4.3,  (x)  (x), W ˜ ⊂W ˜ which together with (V.38) yields  (x)  (x). ˜ W ˜ =W

  (V.40)

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V Unstable and Stable Manifolds for Endomorphisms

Since the above identity holds for arbitrary b and ε satisfying b ≤ b < λi (x) ˜ and  (x) ε ≤ ε , for each y˜ ∈ W ˜ one has 1 lim sup log d(y−n , x−n ) ≤ lim (−b + 2ε ) = −λi (x). ˜

n→+∞ n b ↑ λi (x), ˜ ε ↓ 0 Consequently,

 (x)  (x).  u,i (x) ˜ =W ˜ =W ˜ W  

Theorem V.5.3 W u,i(x) ˜ is the union of an increasing sequence of images of C1,1  +∞ n embedded discs { f Wn (x) ˜ }n=0 and hence contains a ki -dimensional open disc u,i  u,i (x), Wloc (x) ˜ which is tangent to Exi˜ at x. ˜ In addition, if y˜ ∈ W ˜ then 1 lim sup log dθu,i−n x˜ (y−n , x−n ) ≤ −λi (x), ˜ n→+∞ n

(V.41)

u,i −n where dθu,i−n x˜ (y, z) is the distance along the submanifold Wloc (θ x) ˜ if y, z ∈ u,i −n u,i u,i −n ˜ or dθ −n x˜ (y, z) = 1 if y, z ∈ Wloc (θ x). ˜ Wloc (θ x)

Proof. Let W=

+∞

  f n Wn (x) ˜ .

(V.42)

n=0

According to Lemma V.5.1 and Corolary V.5.2.1,  u,i (x)  (x) ˜ = pW ˜ = pW ˜ ⊂ W. W u,i (x)

(V.43)

˜ On the other hand, for any y0 ∈ W , from (V.42), there is n0 ∈ N and y−n0 ∈ Wn0 (x) such that f n0 (y−n0 ) = y0 . From 3) of Theorem V.4.3, there is an orbit y˜ satisfying 1 lim sup log d(y−n , x−n ) ≤ −b + 2ε . n→+∞ n Therefore, Consequently,

 (x) pW ˜ ⊃ W.  u,i (x)  (x) ˜ = pW ˜ = pW ˜ =W W u,i (x)  

because of (V.43).



C ,r

Remark V.10. Let Δ = Δa,b,k, ε be a set as considered in Theorem V.4.3. For x˜ ∈ Δ let λ1 (x) > · · · > λi (x) be the Lyapunov exponents which are greater than b. As a result of Corollary V.5.2.1,

V.5 Global Unstable Sets

77



1  u,i (x) ˜ = y˜ ∈ M f : lim sup log d(y−n , x−n ) ≤ −b . W n→+∞ n

(V.44)

 u (x) Remark V.11. Given f and a point x˜ ∈ M f , the global unstable set W ˜ of f at x˜ f in M is defined by  1 def  u (x) W ˜ = y˜ ∈ M f : lim sup log d(y−n , x−n ) < 0 . n→+∞ n

(V.45)

˜ > · · · > λu(x) ˜ are the strictly From (V.44) we know that, if x˜ ∈ Δ0 and λ1 (x) ˜ (x) positive exponents at x, ˜ then ˜  u,u(x)  u (x) ˜ =W (x). ˜ W

Summarizing the above results, we have the following result concerning the global unstable set of f at x, ˜ which is defined by (V.45), and the global unstable set of f at x˜ in M defined by  u (x), ˜ = pW ˜ W u (x)

if x˜ ∈ Δ0 ,

(V.46)

and otherwise ˜ = {x0 }. W u (x) ˜ is an immersed submanifold in M which depends If f is a diffeomorphism, W u (x) on x0 only. Proposition V.5.4 For any x˜ ∈ Δ0 one has  1  u (x) ˜ = y˜ ∈ M f : lim sup log d(y−n , x−n ) ≤ −λu(x) ˜ , W ˜ (x) n→+∞ n where λu(x) ˜ is the smallest strictly positive Lyapumov exponent of θ at x, ˜ and ˜ (x) u 1,1 W (x) ˜ is the union of an increasing sequence of images of C embedded discs ˜ = W u (x)

+∞

 u −n  f n Wloc (θ x) ˜ .

n=0

 u (x), Moreover, if y˜ ∈ W ˜ then 1 lim sup log dθu −n x˜ (y−n , x−n ) ≤ −λu(x) ˜ ˜ (x), n→+∞ n ˜ for n ∈ Z+ , i.e. dθu −n x˜ (y, z) is where dθu −n x˜ ( , ) is the distance along W u (θ −n x) u −n ˜ if y z ∈ W u (θ −n x) ˜ and otherwise the distance between y and z along W (θ x) u dθ −n x˜ (y, z) = 1.

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V Unstable and Stable Manifolds for Endomorphisms

V.6 Local and Global Stable Manifolds Let [a, b], a < b ≤ 0 be a closed interval of R. Denote by Λa,b the subset of Γ (see (I.4) of Appendix) which consists of points x such that λi (x) ∈ [a, b], i = 1, 2, · · · , r(x). It is clear that f Λa,b ⊂ Λa,b . For x ∈ Λa,b and n, l ∈ Z+ we sometimes use the following notations:

E0 (x) =

V (i) (x),

H0 (x) = E0 (x)⊥

λi (x)
Tnl (x) = T f n x f l We now fix arbitrarily k ∈ {1, · · · , m0 } and 0 < ε ≤ min{1, (b − a)/200} and assume def

that the set Λa,b,k = {x ∈ Λa,b : dim E0 (x) = k} = 0/ . Analogous of Lemma V.2.1, one has the following lemma. Lemma V.6.1 Given ε > 0. There exists a measurable function Cε : Λa,b,k → [1, +∞) such that for any x ∈ Λa,b,k and n, l ∈ Z+ 1) 2) 3) 4)

|Tnl (x)ξ | ≤ Cε ( f n x)el(a+ε /2) |ξ |, ξ ∈ En (x); |Tnl (x)η | ≥ Cε (x)−1 e−ε n el(b−ε /2)|η |, η ∈ Hn (x); |ξ | + |η | ≤ Cε ( f n x)|ξ + η |, ξ ∈ En (x), η ∈ Hn (x); Cε ( f x) ≤ Cε (x)eε .

def

C

Let C be a number such that the set Λa,b,k, ε = {x ∈ Λa,b,k : Cε /4 (x) ≤ C } is not empty.

C Lemma V.6.2 E0 (x) and H0 (x) depend continuously on x ∈ Λa,b,k, ε.

C Let x ∈ Λa,b,k, ε and n ∈ Z. Lemma V.6.1 allows us to introduce an inner product < ·, · >x,n on T f n x M such that for any ξ , ξ ∈ En (x) and η , η ∈ Hn (x)

< ξ , ξ >x,n =

def

< η , η >x,n =

def

+∞

∑ e−2(a+ε )l < Tnl (x)ξ , Tnl (x)ξ >,

(V.47)

l l (x)]−1 η , [Tn−l (x)]−1 η > ∑ e2(b+ε )l < [Tn−l

(V.48)

l=0 n l=0

and En (x) and Hn (x) are orthogonal with respect to < ·, · >x,n . The norm induce by the inner product is denoted by  · x,n .

+∞ C Lemma V.6.3 Let x ∈ Λa,b,k, ε . Then the sequence of norms { · x,n }n=0 satisfies for + each n ∈ Z

1) Tn1 (x)ξ x,n+1 ≤ ea+ε ξ x,n , ξ ∈ En (x); 2) Tn1 (x)η x,n+1 ≥ eb−ε η x,n , η ∈ Hn (x); 3) √12 |ζ | ≤ ζ x,n ≤ Aeε n/2 |ζ |, ζ ∈ T f n x M, where A = 4C 2 /ε .

V.6 Local and Global Stable Manifolds

79

Let r ≥ max{1, 2ρ1−1} be a number such that the Borel set





C ,r r



ˆ Λa,b,k, ε = {x ∈ Λa,b,k : x ∈ Γ0 and κ (x) ≤ r } def



C ,r is not empty. We fix such a set Λa,b,k, ε and confine ourselves to it. We shall write



C ,r Λ = Λa,b,k, ε

for simplicity of notations. Keeping Theorem V.4.2 in mind, one can establish the following theorem about local stable manifolds of f . + 1,1 Theorem V.6.4 For each n ∈  Z , there exists a continuous family of C embedded k-dimensional discs Wn (x) x∈Λ in M and there exist numbers αn , βn and γn which depend only on a, b, k, ε ,C and r such that the followings hold for every x ∈ Λ :

1) There exists a C1,1 map hx,n : On (x) → Hn (x),

  where On (x) is an open subset of En (x) which contains ξ : |ξ | < αn , such that i) hx,n (0) = 0, Dhx,n (0) = 0; ii) Lip(hx,n ) ≤ βn , Lip(Dh x,n ) ≤βn ;  iii) Wn (x) = exp f n x Graph(hx,n ) which is tangent to En (x) at the point f n x;   2) Wn (x) ⊂ B f n x, r −1 e−ε n and fWn (x) ⊂ Wn+1 (x); 3) d sf n+l x ( f l y, f l z) ≤ γn e(a+2ε )l d sf n x (y, z), y, z ∈ Wn (x), l ∈ Z+ , where d sf m x ( , ) is the distance along Wm (x) for m ∈ Z+ ; 4) αn+1 = αn e−3ε , βn+1 = βn e3ε , γn+1 = γn eε . For any x ∈ Γ0 , Let

Exs =

V (i) (x),

Excu = (Exs )⊥ ,

λi (x)<0

and

def

s(x) = #{λi (x) < 0 : 1 ≤ i ≤ r(x)},

i.e., s(x) denotes the number of negative Lyapunov exponents at the point x. It is def

clear that λr(x)−s(x)+1 (x) < 0 and λr(x)−s(x) (x) ≥ 0, where λr(x)+1 (x) = −∞. From the above theorem, we have the following theorem which plays an important role in the discussions in Chapter VI. Theorem V.6.5 For any x ∈ {x ∈ Γ0 : s(x) ≥ 1}, there exists a sequence of C1,1 embedded dim Exs -dimensional discs {Wn (x)}+∞ n=0 satisfying the following: 1) For each n ∈ Z+ there exists a C1,1 map hx,n : On (x) → E cu ( f n x),

80

V Unstable and Stable Manifolds for Endomorphisms

where On (x) is an open subset of E sf n x , such that hx,n (0) = 0, Dhx,n (0) = 0 and   Wn (x) = exp f n x Graph(hx,n ) ;   2) Wn (x) ⊂ B f n x, κ ( f n x) and fWn (x) ⊂ Wn+1 (x); 3) Define  1 W s (x) = y ∈ M : lim sup log d( f n y, f n x) < 0 , n→+∞ n

if x ∈ Γ0 , s(x) ≥ 1

and otherwise W s (x) = {x}. Then for each x ∈ {x ∈ Γ0 : s(x) ≥ 1} W s (x) =

+∞

f −nWn (x).

n=0

In addition, if y ∈ W s (x), then 1 lim sup log d sf n x ( f n y, f n x) ≤ λr(x)−s(x)+1 (x), n→+∞ n where d sf n x (·, ·) is the distance along W s ( f n x) for n ∈ Z+ , i.e. d sf n x (y, z) is the distance between y and z along W s ( f n x) if y, z ∈ W s ( f n x) and otherwise d sf n x (y, z) = 1. 4) For each x ∈ {x ∈ Γ0 : s(x) ≥ 1} 1 W s (x) = {y ∈ M : lim sup log d( f n y, f n x) ≤ λr(x)−s(x)+1 (x)}. n→+∞ n Similarly we have the following theorem. Theorem V.6.6 For fixed i and any x ∈ {x ∈ Γ0 : s(x) ≥ i}, there exists a sequence of C1,1 embedded dimV (r(x)−i+1) (x)-dimensional discs {Wn (x)}+∞ n=0 satisfying the following: 1) For each n ∈ Z+ there exists a C1,1 map hix,n : On (x) → V (r(x)−i+1) ( f n x)⊥ , where On (x) is an open subset of V (r(x)−i+1) ( f n x), such that hix,n (0) = 0,   Dhix,n (0) = 0 and Wn (x) = exp f n x Graph(hix,n ) ;   2) Wn (x) ⊂ B f n x, κ ( f n x) and fWn (x) ⊂ Wn+1 (x); 3) Define  1 W s,i (x) = y ∈ M : lim sup log d( f n y, f n x) ≤ λr(x)−i+1 n→+∞ n

V.7 H¨older Continuity of Sub-bundles

81

if x ∈ Γ0 , s(x) ≥ i and otherwise W s,i (x) = {x} . Then for each x ∈ {x ∈ Γ0 : s(x) ≥ i} W s,i (x) =

+∞

f −nWn (x).

n=0

In addition, if

y ∈ W s,i (x),

then

1 n n lim sup log d s,i f n x ( f y, f x) ≤ λr(x)−i+1 (x), n→+∞ n s,i s,i n + where d s,i f n x (·, ·) is the distance along W ( f x) for n ∈ Z , i.e. d f n x (y, z) is the distance between y and z along W s,i ( f n x) if y, z ∈ W s,i ( f n x) and otherwise d s,i f n x (y, z) = 1. 4) For each x ∈ {x ∈ Γ0 : s(x) ≥ i}

1 W s,i (x) = {y ∈ M : lim sup log d( f n y, f n x) < λr−i (x)}. n→+∞ n

V.7 H¨older Continuity of Sub-bundles Here we consider the H¨older continuity of sub-bundles of T M constituted of tangent spaces of tangent spaces of the submanifolds W s,i (x). This kind of continuity is necessary for dealing with the absolute continuity of foliations formed by invariant manifolds. First we introduce the notion of H¨older continuity for subbundles of a trivial vector bundle. Definition V.7.1 Let Δ be a metric space, H a Hilbert space and {Ex }x∈Δ a family of subspace of H. The family {Ex }x∈Δ is called H¨older continuous in x with exponent α > 0 and constant L > 0, if for any x, y ∈ Δ d(Ex , Ey ) = max{Γ (Ex , Ey ), Γ (Ey , Ex )} ≤ L[d(x, y)]α , def

where we define for two subspaces E and F of H

Γ (E, F) =

sup

inf ξ − η 

ξ ∈Eξ =1 η ∈F

and call it the aperture between E and F. Now we turn to the case of subbundles of T M. As before, let ρ0 > 0 be as introduced in Section II.2. If x, y ∈ M with d(x, y) < ρ0 , we denote by P(x, y) the isometry from Tx M to Ty M defined by the parallel displacement along the unique shortest geodesic connecting x and y. Then for any x, y ∈ M, define

82

V Unstable and Stable Manifolds for Endomorphisms def

d(Ex , Ey ) =



1 if d(x, y) ≥ ρ0 /4, d(Ex , P(y, x)Ey ) if d(x, y) < ρ0 /4.

(V.49)

Analogously of Definition V.7.1 we introduce Definition V.7.2 Let Δ ⊂ M be a set. A family {Ex }x∈Δ of subspaces Ex ∈ Tx M is called H¨older continuous in x on Δ with exponent α > 0 and constant L > 0, if for any x, y ∈ Δ d(Ex , Ey ) ≤ L[d(x, y)]α . Let x, x , y, y ∈ M and let A : Tx M → Tx M, B : Ty M → Ty M be linear maps. We introduce the following distance:  d(A, B) =

|A| + |B|

|A − P(y , x ) ◦ B ◦ P(x, y)|

if max{d(x, x ), d(y, y )} ≥ ρ0 /4 otherwise.

(V.50)

And then for a differentiable map f : M → M and for a number σ ∈ (0, 1], we define d(Tx f , Ty f ) , σ x,y∈M [d(x, y)]

|T f |H σ = | f |C1 + sup

(V.51)

where | f |C1 is defined to be sup{|Tx f | : x ∈ M} as usual and we admit d(Tx f , Ty f )/[d(x, y)]σ = 1 if x = y. The following lemma follows from [51, Lemma III.4.1]. Lemma V.7.1 Let c% = log 2 + 2 log |T f |H σ + |σ log(ρ0 /4)|. Then for any x, y ∈ M and any n ∈ N d(Tx f n , Ty f n ) ≤ ec%n [d(x, y)]σ . (V.52) Proposition V.7.2 Fix C% ≥ 1 and a% < % b and let ΔC,% be the (maybe empty) set of % a,% b points x for which there exist splittings Tx M = Ex ⊕ Ex⊥ such that for any n ∈ N % a%n |ξ |, |Tx f n ξ | ≤ Ce %−1 %bn

|Tx f n η | ≥ C

e |η |,

ξ ∈ Ex , η ∈ Ex⊥ .

with constant L = Then the family {Ex } is H¨older continuous in x on ΔC,% % a,% b % b−% a 2 3C% e max{1, 4/ρ0} and exponents α = c%+ |% a|. From now on we keep the notations of Section 6. Fix ρ0 ∈ (0, ρ0 ) such that for every x ∈ M, if ξ ∈ Tx M and |ξ | ≤ ρ0 , then max{|Tξ expx |, |(Tξ expx )−1 |} ≤ [1 +

1 1 ]4 . 16A2

(V.53)

V.8 Absolute Continuity of Families of Submanifolds

And define r0 = min{(

83

1 1 − e−2ε 2 ρ0

) r0 , }. · 4A 1 + e2ε 4

(V.54)



C ,r Let Δ = Λa,b,k, ε be a set as considered in Theorem V.6.4. For point x in this set we put % (x) = expx Graph(hx,0 |{ξ ∈E (x):ξ  <ρ } ) W (V.55) 0 x,0 0

which is obviously an open subset of W0 (x). We then introduce the Borel set % (Δ ) = W



% (x). W

(V.56)

x∈Δ

% (Δ ) and assume that y ∈ W % (x). Denote by Ey the subspace of Ty M tangent Let y ∈ W % % (x ) then the tangent spaces of W % (x ) and to W (x). Notice that if y also lies in W % % (x) W (x) at the point y coincide, hence Ey is define independent of the choice of W which contains y. Theorem V.7.3 Let {Ey }y∈W% (Δ ) be defined as above. Then this family is H¨older % (Δ ) with constant 12A2 [b(ρ0 /2)]2 and exponent α = (a − b + continuous in y on W 20ε )/(a + 10ε − d), where d = c%+ |a + 10ε |.

V.8 Absolute Continuity of Families of Submanifolds



C ,r We keep here the previous notations. Let Δ = Λa,b,k, ε be the Borel set introduced in Section 7. Then we can choose a sequence of compact sets {Δ l }+∞ l=1 such that Δ l ⊂ Δ , Δ l ⊂ Δ l+1 and μ (Δ \ Δ l ) ≤ l −1 for every l ≥ 1. We now fix arbitrarily such a set Δ l . For x ∈ Δ and sufficiently small r > 0 we put

UΔ (x, r) = expx {ζ ∈ Tx M : ζ x,0 < r} and if x ∈ Δ l we put

VΔ l (x, r) = Δ l



UΔ (x, r).

(V.57)

(V.58)

From the formulation of Theorem V.6.4 together with the compactness of Δ l it follows immediately that there exists a number δΔ l > 0 such that for each x ∈ Δ l , if x ∈ UΔ l (x, q/2), 0 < q ≤ δΔ l , there is a C1 map ϕ : {ξ ∈ E0 (x) : ξ x,0 < q} → H0 (x) satisfying  %

UΔ (x, q)] = Graph(ϕ ) (V.59) exp−1 x [W (x ) and

1 sup{Tξ ϕ x,0 : ξ ∈ E0 (x), ξ x,0 < q} ≤ . 3

(V.60)

84

V Unstable and Stable Manifolds for Endomorphisms

Let x ∈ Δ l and 0 < q ≤ δΔ l . We denote by FΔ l (x, q) the collection of submani% (y) passing through y ∈ Δ l  UΔ (x, q/2). Set folds W

Δ%l (x, q) = y∈Δ l



% (y) W



UΔ (x, q).

(V.61)

UΔ (x,q/2)

Definition V.8.1 A submanifold W of M is called transversal to the family FΔ l (x, q) if the following hold true: 1 i) W ⊂ UΔ (x, q) and exp−1 x W is the graph of a C map ψ : {η ∈ H0 (x) : η x,0 < q} → E0 (x); % (y), y ∈ Δ l  UΔ (x, q/2), at exactly on point and this intersecii) W intersects any W % (y) = Tz M where z = W  W % (y). tion is transversal, i.e. TzW ⊕ TzW

For a submanifold W transversal to FΔ l (x, q) we define W  = sup ψ (η )x,0 + supTη ψ x,0

(V.62)

where the supremums are taken over the set {η ∈ H0 (x) : η x,0 < q} and ψ is defined as above. And we shall denote by mW the Lebesgue measure on W induced by the Riemannian metric on W inherited from M. Consider now two submanifolds W 1 and W 2 transversal to FΔ l (x, vq). Since, by % (y)}y∈Δ is a continuous family of C1 embedded discs, Theorem V.6.4 and (V.55), {W % 2 respectively of W 1 and W 2 such that % 1 and W there exist two open submanifolds W we can well define a so-called Poincar´e map %1 PW% 1 ,W% 2 : W by letting



%2 Δ%l (x, q) → W

%2 PW% 1 ,W% 2 : z → W





Δ%l (x, q)

% (y) W

% (y), y ∈ Δ l  UΔ (x, q/2), and moreover, P% 1 % 2 is a homeomorphism. % 1 W for z = W W ,W Definition V.8.2 The family FΔ l (x, q) is said to be absolutely continuous if there exists a number εΔ l (x, q) > 0 such that, for any two submanifolds W 1 and W 2 transversal to FΔ l (x, q) and satisfying W i  ≤ εΔ l (x, q), i = 1, 2, every Poincar´e map PW% 1 ,W% 2 constructed as above is absolutely continuous with respect to mW 1 and mW 2 . Besides Theorem V.6.4, which describes the existence of invariant families of local stable manifolds corresponding to Lyapunov exponents smaller than a fixed number a < 0, we have now the following other main result of this chapter which deals with absolute continuity of such families and which follows from [51, Theorem III.5.1]. As before, let m denote the Lebesgue measure on M. Theorem V.8.1 Let Δ l be given as above. There exist constants 0 < qΔ l ≤ δΔ l , εΔ l > 0 and JΔ l > 0 such that the following hold true for each x ∈ Δ l :

V.9 Absolute Continuity of Conditional Measures

85

1) The family FΔ l (x, qΔ l ) is absolutely continuous. 2) If m(Δ l ) > 0 and x is a density point of Δ l with respect to m, then for every two submanifolds W 1 and W 2 transversal to FΔ l (x, qΔ l ) and satisfying W i  ≤ εΔ l , i = 1, 2 any Poincar´e map PW% 1 ,W% 2 is absolutely continuous and the Jacobian J(PW% 1 ,W% 2 ) satisfies the inequality JΔ−1l ≤ J(PW% 1 ,W% 2 ) ≤ JΔ l % 1  Δ%l (x, q l ). for mW 1 -almost all points y ∈ W Δ Roughly speaking, in smooth ergodic theory of deterministic dynamical systems, presence in a smooth system of an absolutely continuous family of local stable manifolds ensures many important ergodic properties of the system, for expample, the positiveness of entropy and the Bernoulli property etc. (see [63]). It would be seen that, in ergodic theory for endomorphisms as considered in this book, the presence in such a system of an absolutely continuous family of local stable manifolds also plays a very important role. For example, it will enable us to estimate the entropy of the system through its Lyapunov exponents (see Chapter VI) and to obtain the generic property of the SRB measure and the ergodic property of Lyapunov exponents for an Axiom A attractor or nonuniformly completely hyperbolic attractor of a C2 endomorphism (see Chapter VIII).

V.9 Absolute Continuity of Conditional Measures The purpose of this section is to present an important theorem which is a consequence of the absolute continuity theorem (Theorem V.8.1) and of Fubini Theorem. Roughly speaking, this theorem asserts that the conditional measures induced on local stable manifolds of endomorphisms by an absolutely continuous measure on M are absolutely continuous on these submanifolds. We shall need the following basic proposition (see [51, Proposition III.6.1]) which is a straightforward corollary of the definition of conditional measures. Proposition V.9.1 Let (X, B, ν ) be a Lebesgue space and let α be a measurable partition of X. If ν% is another probability measure on B which is absolutely continuous with respect to ν , then for ν%-almost all x ∈ X the conditional measure ν%α (x) is absolutely continuous with respect to να (x) and d ν%α (x) g|α (x) =( d να (x) α (x) gd να (x)

(V.63)

where g = d ν%/d ν . Let Δ l be a set as introduced in section 7. Beginning from now, we suppose that the numbers qΔ l and εΔ l in Theorem V.8.1 satisfies qΔ l = εΔ l . Note that this last assumption does not present any restriction of generality.

86

V Unstable and Stable Manifolds for Endomorphisms

Henceforth, we confine ourselves to an arbitrarily fixed point x ∈ Δ l which is such that m(Δ l ) > 0 and x is a density point of Δ l with respect to m. We introduce now the following notations. % : UΔ (x, q l ); U Δ % B1 : {ξ ∈ E0 (x) : ξ x,0 < qΔ l }; B%2 : {η ∈ H0 (x) : η x,0 < qΔ l }; %2)} %1 of U; % β : the measurable partition of {expx ({ξ } × B ξ ∈B  %l I : β (x) Δ (x, qΔ l );  % % (y)  U} α : the partition {W of Δ%l (x, q l ); l y∈Δ



UΔ (x,qΔ l /2)

Δ

[N] : z∈N α (z) for N ⊂ I; βI : the restriction of β to [I]; mX : the normalized Lebesgue measure m/m(X) on a Borel subset X of M with m(X) > 0; β % induced by the inherited my : the normalized Lebesgue measure on β (y), y ∈ U Riemannian metric; β β β mz I : mz /mz (βI (z)) for z ∈ [I]; α mz : the normalized Lebesgue measure on α (z), z ∈ [I] induced by the inherited Riemannian metric. Remark V.12. Clearly the following statements hold true. 1) [I] = Δ%l (x, qΔ l ). 2) Since x is a density point of Δ l with respect to m, one has m(Δ l



UΔ (x, qΔ l /2)) > 0

and hence m([I]) > 0. In addition, one easily sees that α is a measurable partition % (y)} l is a continuous family of C1 embedded discs. of [I] since {W y∈Δ 3) From Proposition V.9.1, Fubini Theorem, Theorem V.8.1 and from the fact β m([I]) > 0 it follows clearly that mz (βI (z)) > 0 for every z ∈ [I]. Now we formulate our main result of this section as follows: [I]

Theorem V.9.2 Let {mα (z) }z∈[I] be a canonical system of conditional measures of m[I] associated with the measurable partition α . Then for m-almost every z ∈ [I] the [I] measure mα (z) is equivalent to mαz , moreover, the following estimate [I]

R−1 ≤ Δl

dmα (z) dmαz

≤ RΔ l

(V.64)

holds mαz -almost everywhere on α (z), where RΔ l > 0 is a number depending only on the set Δ l but not on individual x ∈ Δ l . Proof. See page 87 of [51].

 

Chapter VI

Pesin’s Entropy Formula for Endomorphisms

As we have seen in Chapters III and IV, Pesin’s entropy formula plays an important role in the smooth ergodic theory of dynamical systems. There are extensive results concerning Pesin’s entropy formula in both deterministic dynamical systems and random dynamical systems of diffeomorphisms. In [63], Pesin showed that if an invariant measure of a C2 diffeomorphism of a compact manifold is absolutely continuous with respect to the Lebesgue measure of the manifold, then it satisfies Pesin’s entropy formula. (See [55] for a simplified proof given by Ma˜ne´ .) The above results were successfully generalized to random dynamical systems of diffeomorphisms [44, 36, 51]. The analogous result for non-invertible transformations was first considered for expanding maps [27, 57], as we have seen in Chapter III, and then obtained for generic C1+α maps [90]. Afterwards, Liu [46] provided a simpler proof using stable foliations. In this chapter, we extend Pesin’s result to non-invertible C2 endomorphisms along the line of [46].

VI.1 Main Results In this chapter, we prove the following theorem, which is an extension of Pesin’s result to non-invertible transformations. Theorem VI.1.1 Let f be a C2 endomorphism and μ an f -invariant Borel probability measure on M. If μ  m, where m is the Lebesgue measure on M, then the entropy of f with respect to μ is given by hμ ( f ) =

 s(x)

∑ λi (x)+ mi (x) d μ (x).

(VI.1)

M i=1

M. Qian et al., Smooth Ergodic Theory for Endomorphisms, Lecture Notes in Mathematics 1978, DOI 10.1007/978-3-642-01954-8 VI, c Springer-Verlag Berlin Heidelberg 2009 

87

88

VI Pesin’s Entropy Formula

Remark VI.1. Under the conditions formulated in Theorem VI.1.1, it can be verified that log | det(Tx f )| ∈ L1 (M, μ ) (see Subsection 2.2), and hence λ (s(x)) (x) > −∞ for μ -a.e. x ∈ M by (I.I.2). Theorem VI.1.1 allows us to compute the entropy of a smooth map via its Lyapunov exponents. For example, the following two results, which have been proved in other ways, can be obtained as natural consequences of the theorem. Corollary VI.1.1.1 (Chapter III) Let f be a C2 expanding endomorphism on M and let μ be the unique f -invariant Borel probability measure which is absolutely continuous with respect to the Lebesgue measure on M. Then hμ ( f ) =

 s(x)

∑ λi (x)mi (x) d μ (x) =



M i=1

M

log | det(Tx f )|d μ .

Corollary VI.1.1.2 ([92, Section 8.4]) Suppose that A : K p → K p is a surjective (group) endomorphism of p-dimensional torus. If m is the Haar measure on K p , then hm (A) = ∑ log |λi | {i:|λi |>1}

 : R p → R p which where λ1 , . . . , λ p are the eigenvalues of the linear transformation A covers A.

VI.2 Preliminaries Let Γ0 denote the set of critical points of f . By Sard’s Theorem, m( f n Γ0 ) = 0 and hence μ ( f n Γ0 ) = 0 for all n ≥ 1 since μ  m. Put g= and

dμ , dm

(VI.2)

  Γ1 = x ∈ M : g(x) = 0 .

Write

Γ% =

+∞

k=1

f k Γ0

and Γ =

+∞

f −n (Γ%



Γ1 ).

n=0

Clearly μ (Γ) = 0 and f −1Γ ⊂ Γ. Choose a Borel set Λ ⊂ M \ Γ such that μ (Λ ) = 1, f Λ ⊂ Λ and every point x ∈ Λ is regular in the sense of Oseledec. It is easy to see that every x ∈ Λ is both a regular point and a regular value of f n for all n ≥ 1. Hence for every x ∈ Λ and an arbitrarily fixed natural number n, there is an open ball V centered at x such that ( f n )−1V has a finite number of connected components {Ui } and f n |Ui : Ui → V is a C2 diffeomorphism for each Ui . From this there follows the following simple fact.

VI.2

Preliminaries

89

Lemma VI.2.1 Let n ∈ N, 0 < k ≤ m0 and 0 ≤ θ ≤ 1. For any k-dimensional C1,θ embedded submanifold W of M, there exists a k-dimensional C1,θ embedded submanifold W of M such that W ⊂ ( f n )−1W and ( f n )−1W



Λ = W



Λ.

  Let I0 = x ∈ Λ : λi (x) ≥ 0 for all 1 ≤ i ≤ s(x) and I = Λ \ I0 . Clearly f I0 ⊂ I0  and f I ⊂ I. For x ∈ I, write E s (x) = λi (x)<0 V (i) (x) and define the stable manifold of f at x as   1 W s (x) = y ∈ M : lim sup log d( f n x, f n y) < 0 . n→+∞ n According to Theorem V.V.6.4, for μ -a.e. x ∈ I, there exists a sequence of C1,1 s embedded k(x)-dimensional discs {Wn (x)}+∞ n=0 (where k(x) = dim E (x)) such that fWn (x) ⊂ Wn+1 (x) for all n ≥ 0 and W s (x) =

+∞

( f n )−1Wn (x).

n=0

For x ∈ I0 , we define W s (x) = {x}. Lemma VI.2.2 There exists a measurable partition η of M with the following properties. (1) f −1 η ≤ η . (2) For μ -a.e. x ∈ Λ , there exists a k(x)-dimensional C1,1 embedded submanifold Wx of M such that Wx ⊂ W s (x), η (x) ⊂ Wx and η (x) contains an open neighborhood of x in Wx (with respect to the induced topology of Wx as a submanifold of M). (3) For any Borel set B ⊂ Λ , function    PB (x) = msx η (x) B

is measurable and μ almost everywhere finite, where msx is the Lebesgue measure on Wx induced by its inherited Riemannian structure as a submanifold of M (msx = δx if W s (x) = {x}). (4) Let { μxη }x∈Λ be a canonical system of conditional measures of μ associated with η . Then μxη  msx μ -a.e. x. This lemma is similar to [51, Proposition IV.2.1] (restricted to the deterministic case), which is a variant of [41, Proposition 3.1]. A detailed proof of that proposition in [51] is given in [51, Section IV.2] with the needed properties of local stable manifolds being worked out in [51, Chapter III] and stated now in Section V.6-9 of this book. The difference between our present situation and that of [51] lies in that we are dealing with a non-invertible endomorphism rather than a diffeomorphism. But one can check that this deficiency can be overcome by the local diffeomorphism

90

VI Pesin’s Entropy Formula

property of f (as far as points in Λ are concerned) and Lemma VI.2.1. That is to say, Lemma VI.2.2 can be proved by almost the same arguments as the corresponding proof in [51] with some slight modifications caused by applying the local diffeomorphism property and Lemma VI.2.1 instead of the diffeomorphism property. See also the proof of Proposition VII.VII.2.4 in the next chapter of this book, where we are constructing a measurable partition subordinate to W u -manifolds instead of W s manifolds. Here we omit presenting the long arguments and refer the reader to [51] for details. (3) of Lemma VI.2.2 allows one to define a σ -finite Borel measure m∗ on Λ by m∗ (B) =



   msx η (x) B d μ

for any Borel set B ⊂ Λ . From (4) of Lemma VI.2.2 it follows that μ  m∗ . Define h=

dμ . dm∗

(VI.3)

It is not difficult to obtain the following lemma. Lemma VI.2.3 For μ -a.e. x ∈ Λ , h=

d μxη dmsx

msx -a.e. on η (x).

Since Tx f is nondegenerate and g(x) > 0 (g = d μ /dm) for every x ∈ Λ , one can α = {Ai } of Λ such that f restricted easily choose a countable measurable partition  to each Ai , written fAi , is injective and μ fAi (B) = 0 if B is a Borel subset of Ai and μ (B) = 0 ( f (B) is clearly Borel if B is Borel). From this we can define a measure μAi on each Ai by   μAi (B) = μ fAi (B) for any Borel set B ⊂ Ai . Clearly μAi is equivalent to μi (μi = μ |Ai , the restriction of μ to Ai ). Define a measurable function J( f ) : Λ → R+ by J( f )(x) =

d μA i (x) d μi

for x ∈ Ai .

It is easy to see that J( f ) is independent of the choice of partition α . We call J( f ) the Jacobian of f . By Radon-Nikodym Theorem, one can easily see that J( f )(x) =

 g( f x)  det(Tx f ), g(x)

for x ∈ Λ .

Since μ is f -invariant, one has

μAi (B) = μ ( f −1 fAi B) ≥ μ (B) = μi (B)

(VI.4)

VI.3

Proof of Theorem VI.1.1

91

for any Borel set B ⊂ Ai . Hence J( f ) ≥ 1 μ -a.e. on each Ai , and therefore on Λ . Thus by (VI.4), for μ -a.e. x ∈ Λ log J( f )(x) = log

g( f x) + log | det(Tx f )| ≥ 0. g(x)

(VI.5)

This yields that log−

g( f x) ≥ − log+ | det(Tx f )|, g(x)

μ -a.e. x ∈ Λ ,

(VI.6)

which, by [51, Lemma I.3.1] and the integrability of log+ | det(Tx f )|, implies that 

log M

g◦ f d μ = 0. g

Consequently,  Λ

log | det(Tx f )| d μ (x) =

 Λ

log J( f )(x) d μ (x) ≥ 0.

We have the following result. Proposition VI.2.4 Let f : M → M be a C2 endomorphism and μ a smooth f invariant Borel probability measure. Then log | det(Tx f )| ∈ L1 (M, μ ).

VI.3 Proof of Theorem VI.1.1 In this section, we give the complete proof of Theorem VI.1.1. By virtue of Margulis-Ruelle inequality, it remains to prove hμ ( f ) ≥

 s(x)

∑ λi (x)+ mi (x) d μ (x).

(VI.7)

M i=1

Let η be a measurable partition of M as introduced by Lemma VI.2.2. Let −1 { μxf η }x∈Λ be a canonical system of conditional measures of μ associated with the partition f −1 η . Then   +∞  h μ ( f ) ≥ h μ ( f , η ) = Hμ η | f −n η n=1

= Hμ (η | f −1 η ) =−



Λ

log μxf

−1 η

(η (x)) d μ (x).

92

VI Pesin’s Entropy Formula

Therefore, in order to prove (VI.7), it suffices to show − and −



log μxf

I0

 I

log μxf

−1 η

−1 η

(η (x)) d μ (x) ≥

(η (x)) d μ (x) ≥



∑ λi (x)mi (x) d μ (x)

(VI.8)

∑ λi (x)+ mi (x) d μ (x).

(VI.9)

I0 i



I i

We now first prove (VI.8). Let Bμ (I0 ) denote the completion of the Borel σ -algebra of I0 with respect to μ . Then (I0 , Bμ (I0 ), μ ) is a Lebesgue space. Since f (I0 ) ⊂ I0 , and η (x) = {x} for μ -a.e. x ∈ I0 , by [61, Lemma 10.5] − log μxf

−1 η

(η (x)) = log J( f )(x),

μ -a.e. x ∈ I0 .

By (VI.6), it follows from [41, Proposition 2.2] and integrability of log+ | det(Tx f )| that  g◦ f d μ = 0. log g I0 Hence 0≤−

 I0

log μxf

−1 η

(η (x)) d μ (x) = =

 I

0

log | det(Tx f )| d μ (x)

∑ λi (x)mi (x) d μ (x)

I0 i

by Oseledec’s theorem. Now we proceed to the proof of (VI.9). Without loss of generality, we may assume that μ (I) = 1 and even I = Λ . So in what follows we take this assumption. We first introduce the following measurable functions on Λ : −1

W (z) = μzf η (η (z)), g(z) h( f z) · , X(z) = g( f z) h(z) | det(Tz f |E s (z) )| , Y (z) = | det(Tz f )| where h is defined by (VI.3). We now first present several claims, whose proofs will be given later on. Claim 1 W = XY , μ almost everywhere on Λ . Claim 2 logY ∈ L1 (Λ , μ ) and −

 Λ

logY d μ =



∑ λi (x)+ mi (x) d μ (x).

Λ i

VI.3

Proof of Theorem VI.1.1

93

(

Claim 3 log X ∈ L1 (Λ , μ ) and log Xd μ = 0. It is not hard to see that (VI.9) follows immediately from the above Claims. To prove Claim 1, we require the following two lemmas. Lemma VI.3.1 Let A ⊂ Λ be a Borel set such that μ (A) > 0 and f |A : A → f A is injective. Then for μ -a.e. x ∈ Λ one has  

( f −1 η )(x) A

J( f ) d μxf

−1 η

= μ ηf x ( f A).

(VI.10)

Proof. Let F(x) and G(x) denote respectively the function at the left hand and that at the right hand of equation (VI.10). By the uniqueness of canonical systems of conditional measures one has G(x) = μxf

−1 η



 f −1 ( f A) ,

μ -a.e. x ∈ Λ .

Let B( f −1 η ) denote the σ -algebra generated by f −1 η . Clearly F(x) and G(x) are both measurable with respect to B( f −1 η ). So, in order to prove (VI.10), it suffices to show that for any C ∈ B( f −1 η )  C

F(x) d μ (x) =

 C

G(x) d μ (x).

(VI.11)

It is seen that B( f −1 η ) = f −1 B(η ). Hence for any C ∈ B( f −1 η ), we have some B ∈ B(η ) such that C = f −1 B. Therefore  C

F(x) d μ (x) = =



χC

Λ

 

 ( f −1 η )(x)

( f −1 η )(x)

Λ



−1 η

d μ (x)

−1 η

d μ (x)

χA J( f ) d μxf

χC χA J( f ) d μxf

J( f ) d μ      = μ f (A C) = μ f (A f −1 B)    = μ B fA , =



A C



where the last equality holds true since by the choice of A, we have f (A



f −1 B) = f A



B;

On the other hand  C

G(x) d μ (x) = =

 Λ

χC

  Λ

 ( f −1 η )(x)

( f −1 η )(x)

−1 η

d μ (x)

−1 η

d μ (x)

χ f −1 ( f A) d μxf

χC χ f −1 ( f A) d μxf

94

VI Pesin’s Entropy Formula

      = μ C f −1 ( f A) = μ f −1 B f −1 ( f A)    = μ B f A) .

This proves (VI.11) and completes the proof of Lemma VI.3.1.

 

Lemma VI.3.2 Let A be as given in Lemma VI.3.1. Then for μ -a.e. x ∈ A one has −1 η

μxf



(B) =

1 d μ ηf x −1 f B J( f ) ◦ f A



for any Borel set B ⊂ ( f −1 η )(x) A, where fA = f |A : A → f A. Proof. Put ξ = η | f A and ζ = fA−1 ξ = ( f −1 η )|A . Write ν = μ |A , νˆ = μ | f A and let measure νA on A be defined by d νA /d ν = J( f ). It is easy to see that a canonical system of conditional (probability) measures of ν associated with ζ is given by 

νxζ

: νxζ (·) =

μxf

−1 η

f −1 η

μx

(·)

(A)

x∈A

.

Then, by [33, Proposition II.11.1], a canonical system of conditional measures of νA associated with ζ is  J( f )|ζ (x) ζ d ν . (νA )ζx : d(νA )ζx = ( x ζ x∈A ζ (x) J( f )d νx Since fA : (A, νA ) → ( f A, νˆ ) is measure-preserving, fA ζ = ξ and a canonical system of conditional measures of νˆ associated with ξ is given by  μyη (·) νˆ yξ : νˆ yξ (·) = η , μy ( f A) y∈ f A then, by the uniqueness of canonical systems of conditional measures, one has for μ -a.e. x ∈ A μ ηf x ( f B) ξ (νA )ζx (B) = νˆ f x ( f B) = η μ f x ( f A) if B ⊂ ζ (x) is a Borel set. Therefore, by Lemma VI.3.1 for μ -a.e. x ∈ A and any Borel B ⊂ ζ (x),

μxf

−1 η

(B) = μxf =

−1 η

(A) · νxζ (B)

−1 μxf η (A) ·





=



ζ (x)

J( f ) d νxζ

J( f ) d μxf

( f −1 η )(x) A μ ηf x ( f A)

−1 η

·



1 d(νA )ζx J( f) B

 B

1 d(μ ηf x ◦ fA ) J( f )

VI.3

Proof of Theorem VI.1.1

95



1 η d(μ f x ◦ fA ) B J( f )  1 = d μ ηf x . −1 f B J( f ) ◦ f A

=

This completes the proof of the lemma. Proof of Claim 1. It suffices to show that for μ -a.e. x ∈ Λ one has W (z) = X(z)Y (z),

 

μxη -a.e. z.

Let α = {Ai } be the partition of Λ introduced above. Then for μ -a.e. x ∈ Λ we have for any Borel set B ⊂ η (x) 1 −1 μ f η (B) W (x) x  −1 1 = μxf η (Ci ) (Ci = B Ai ) ∑ W (x) i

μxη (B) =

=

1 W (x) ∑ i

=

1 W (x) ∑ i

1 = W (x) ∑ i =

1 W (x)

  



fCi

1 η dμ f x J( f ) ◦ fA−1 i

fCi

1 h(z) dmsf x (z) J( f ) ◦ fA−1 (z) i

Ci

(by Lemma VI.3.2)

1 h( f z)| det(Tz f |E s (z) )| dmsx (z) J( f )(z)

1 h( f z)| det(Tz f |E s (z) )| dmsx (z) J( f )(z)

B

and, on the other hand,

μxη (B) =

 B

h(z) dmsx (z).

Since Borel set B is arbitrarily chosen, we have h(z) =

1 1 · h( f z)| det(Tz f |E s (z) )| W (x) J( f )(z)

for msx -a.e. z ∈ η (x). Since W (z) = W (x) for any z ∈ η (x), it follows that h( f z) 1 · | det(Tz f |E s (z) )| J( f )(z) h(z) = X(z)Y (z), μxη -a.e. z ∈ η (x).

W (z) =

Then there follows the claim.

 

96

VI Pesin’s Entropy Formula

Proof of Claim 2.

Noting that for μ -a.e. z ∈ Λ   Tz f |E s (z)  ≤ | f | 1 , C

  we have log+ Tz f |E s (z)  ∈ L1 (Λ , μ ). By Oseledec multiplicative ergodic theorem we have     logdet(Tz f ) d μ = ∑ λi (z)mi (z) d μ (z) (VI.12) Λ

and

 Λ

Λ i

log | det(Tz f |E s (z) )| d μ (z) =



∑ λi (z)− mi (z) d μ (z).

Λ i

(VI.13)

Since log | det(Tz f )| ∈ L1 (Λ , μ ), one has ∑i λi (z)mi (z) ∈ L1 (Λ , μ ) and hence ∑i λi (z)− mi (z) ∈ L1 (Λ , μ ). So, by (VI.13), it follows that log | det(Tz f |E s (z) )| ∈ L1 (Λ , μ ). Thus, logY is integrable and −



logY d μ = =

 

∑ λi (z)mi (z) d μ (z) − i

∑ λi (z)

+



∑ λi(z)− mi (z) d μ (z) i

mi (z) d μ (z),

i

 

which proves Claim 2.2. Proof of Claim 3. By Claim 1, logW = log X + logY ≤ 0,

μ -a.e. .

Hence, by Claim 2, log+ X(∈ L1 (Λ , μ ). Then, by [41, Proposition 2.2], we know that log X is integrable and log X d μ = 0. The whole proof of Theorem VI.1.1 is completed.  

Chapter VII

SRB Measures and Pesin’s Entropy Formula for Endomorphisms

A celebrated result conjectured by Ruelle and proved by Ledrappier, Strelcyn and Young in the smooth ergodic theory of diffeomorphisms is that an invariant measure of a C2 diffeomorphism satisfies Pesin’s entropy formula if and only if it has absolutely continuous conditional measures on unstable manifolds [41, 42]. The later property is known as SRB property of the invariant measure. This result was also successfully generalized to random dynamical systems of diffeomorphisms [51]. In this chapter, we present a formulation of the SRB property for invariant measures of C2 endomorphisms of a compact manifold via their inverse limit spaces, and then prove that this property is sufficient and necessary for the entropy formula. This is a non-invertible version of the main theorem of [42]. As a nontrivial corollary of this result, an invariant measure of a C2 endomorphism has this SRB property if it is absolutely continuous with respect to the Lebesgue measure of the manifold. Invariant measures having this SRB property also exist on Axiom A attractors of C2 endomorphisms (see Corollary VII.1.1.2). Comparing with the case of diffeomorphisms, one can see that two main difficulties arise while dealing with endomorphisms. One comes from singularities. However, a natural integrability condition (Condition (I.I.1)) will enable us to eliminate this difficulty easily. The other difficulty arises from the non-invertibility. To overcome this deficiency, the inverse limit space has to be introduced. One will see that, when the inverse limit space is introduced, the situation is very close to the case of random diffeomorphisms, for which a systematic treatment is now available in [51]. Therefore, with some necessary modifications, many ideas and techniques developed for the random diffeomorphisms can be applied to our present study. This is the first example which shows that a deterministic problem can be successfully solved by employing techniques developed for random dynamical systems. This chapter is a detailed account of [73] and organized as follows. In Section 1 we give our formulation of the SRB property and state the main result of this chapter. Some important consequences of this result will also be given. Section 2 consists of some technical preparations. The proof of the first part of the theorem,

M. Qian et al., Smooth Ergodic Theory for Endomorphisms, Lecture Notes in Mathematics 1978, DOI 10.1007/978-3-642-01954-8 VII, c Springer-Verlag Berlin Heidelberg 2009 

97

98

VII

SRB Measures and Pesin’s Entropy Formula for Endomorphisms

i.e. the sufficiency for the entropy formula, will be demonstrated in Section 3. We presents the completed proof of the second part of the theorem, i.e. the necessity for the entropy formula, in Sections 4–9.

VII.1 Formulation of the SRB Property and Main Results In this section, we give our formulation of the SRB property and state the main results of this chapter. In what follows, we always assume that f is a C2 endomorphism on M, μ is an f -invariant Borel probability measure on M satisfying condition (V.V.1). Let M f be the inverse limit space of (M, f ), and μ˜ the unique shift invariant Borel probability measure on M f with pμ˜ = μ . Moreover, we also assume that the σ -algebra associated with (M f , μ˜ ) is the completion of the Borel σ -algebra of M f with respect to μ˜ , denoted by Bμ˜ (M f ). u (x) ˜ be a local unstable manifold of f at x, ˜ and W u (x) ˜ the For x˜ ∈ M f , let Wloc global unstable set of f at x˜ in M, as defined in Sections V.4 and V.5. Recall that u if x˜ ∈ Δ0 , Wloc (x) ˜ is a k(x)-dimensional ˜ C1 embedded submanifold in M, where mi (x). ˜ k(x) ˜ = ∑λi (x)>0 ˜ Definition VII.1.1 A measurable partition η of M f is said to be subordinate to W u -manifolds of ( f , μ ) if for μ˜ -a.e. x˜ ∈ M f , η (x) ˜ has the following properties:   (1) p|η (x) ˜ → p η (x) ˜ is bijective; ˜ : η (x) (2) There exists a k(x)-dimensional ˜ C1 embedded submanifold Wx˜ of M such that u ˜ Wx˜ ⊂ W (x),   p η (x) ˜ ⊂ Wx˜ and p (η (x)) ˜ contains an open neighborhood of x0 in Wx˜ , this neighborhood being taken in the topology of Wx˜ as a submanifold (or a subset) of M. In Section 2, we will show that there always be measurable partitions subordinate to W u -manifolds of ( f , μ ). We are now prepared to give the definition of SRB property for f -invariant Borel probability measures. We first give a somewhat intuitive description of this property. Let f : (M, μ ) → (M, μ ) be as given above. Let V be a Borel subset of M f with  u -discs μ˜ (V ) > 0. Suppose that V is the disjoint union of a continuous family of W u     uof the type of Wloc (x) ˜ as introduced by (V.V.36). For x˜ ∈ V , let Vx˜ denote the W   disc in V containing x˜ and μ˜ x˜ the conditional probability measure of μ˜ on Vx˜ . We  as given above one has p (μ˜ x˜ )  λx˜u for say that μ has SRB property if for every V u μ˜ -a.e. x˜ ∈ V , where p(μ˜ x˜ ) is the projection of μ˜ x˜ under p|Vx˜ : Vx˜ → Wloc (x) ˜ and λx˜u u (x) is the Lebesgue measure on Wloc ˜ induced by its inherited Riemannian structure as a submanifold of M. The following is a precise (but equivalent) definition. Definition VII.1.2 We say that the f -invariant measure μ has SRB property, if for every measurable partition η of M f subordinate to W u -manifolds of ( f , μ ) we have for μ˜ -a.e. x˜ ∈ M f ,

VII.1

Formulation of the SRB Property and Main Results



99

η

p μ˜ x˜  λx˜u ,

  ˜ associated where μ˜ xη˜ x∈M f is a canonical system of conditional measures of μ ˜ u denotes the Lebesgue and λ with η , p(μ˜ xη˜ ) is the projection of μ˜ xη˜ under p|η (x) ˜ x˜ measure on Wx˜ induced by its inherited Riemannian structure as a submanifold of M (λx˜u = δx0 if W u (x) ˜ = {x0 }). Remark VII.1. The definition of the SRB property of μ is clearly independent of the choice of the submanifolds Wx˜ , x˜ ∈ M f . The main result of this chapter is the following theorem. Theorem VII.1.1 Let f be a C2 endomorphism on M with an invariant Borel probability measure μ satisfying the integrability condition (V.V.1). Then the entropy formula  hμ ( f ) =

∑ λi(x)+ mi (x) d μ (x)

(VII.1)

M i

holds if and only if μ has SRB property. Let f : M → M be as given above and let μ be an f -invariant Borel probability measure. It was proved in Chapter VI that, if μ is absolutely continuous with respect to the Lebesgue measure on M, then both integrability condition (V.V.1) and entropy formula (VII.1) hold. Then, as a corollary of Theorem VII.1.1, we have the following result. Corollary VII.1.1.1 Let f be a C2 endomorphism on M with an invariant Borel probability measure μ . If μ is absolutely continuous with respect to the Lebesgue measure on M, then μ has SRB property. Remark VII.2. In the diffeomorphism case, this corollary follows from the absolute continuity of the unstable foliation. But in the endomorphism case, it seems at least very difficult to prove the corollary along this line. As a corollary of Theorem IV.IV.1.1 and Theorem VII.1.1, invariant measures of Axiom A attractors of C2 endomorphisms have also SRB property. Corollary VII.1.1.2 Let Λ ⊂ O be an Axiom A attractor of f ∈ C2 (O, M), where O is an open subset of M, and assume that Tx f is nondegenerate for every x ∈ Λ . Then there exists a unique f -invariant Borel probability measure μ on Λ which is characterized by each of the following properties: (1) μ has SRB property. (2) Pesin’s entropy formula holds for the system f : (Λ , μ ) → (Λ , μ ). (3) When ε > 0 is small enough, 1 n−1 ∑ δ f kx = μ n→∞ n k=0 lim

 def  for Lebesque almost every x ∈ Bε (Λ ) = y ∈ M : d(y, Λ ) < ε .

100

VII

SRB Measures and Pesin’s Entropy Formula for Endomorphisms

VII.2 Technical Preparations for the Proof of the Main Result From the conclusion of Proposition V.V.4.4 one can easily see that the local unstable u (x) manifolds Wloc ˜ have some additional properties which we summarize in the next proposition. Let ρ2 be the number as defined in Section V.V.1.  u   u   (x) Proposition VII.2.1 Let Δi , Wloc (x) ˜ x∈ and W , i ∈ N, be as introloc ˜ x∈ ˜ Δi ˜ Δi duced in Proposition V.V.4.4. Then for any Δi there exist numbers ri ∈ (0, ρ2 /4), εi ∈ (0, 1) and Ri > 0 such that the following statements hold. (1) For any r ∈ [ri /2, ri ] and each x˜ ∈ Δi , if x˜ ∈ BΔi (x, ˜ εi r) = Δi u (x˜ )  B(x , r) is connected and the map Wloc 0 def

u (x˜ ) x˜ → Wloc





 x, B( ˜ εi r), then

B(x0 , r)

is a continuous map from BΔi (x, ˜ εi r) to the space of subsets of B(x0 , r) (endowed with the Hausdorff topology); ˜ εi r), then either (2) Let r ∈ [ri /2, ri ] and x˜ ∈ Δi . If x˜ and x˜

∈ BΔi (x, u Wloc (x˜ )



u B(x0 , r) = Wloc (x˜

)



B(x0 , r)

or the two terms in the above equation are disjoint. In the latter case, if it is  u (x˜ ), then assumed moreover that x˜

∈ W dxu˜ (y0 , z0 ) > 2ri 



u (x˜ ) B(x , r) and any z ∈ W u (x˜

) B(x , r), where d u (·, ·) for any y0 ∈ Wloc 0 0 0 loc x˜

is the distance along W u (x˜ ). u (x˜ )  B(x , r), then W u (x˜ ) (3) For each x˜ ∈ Δi , if x˜ ∈ BΔi (x, ˜ εi r) and y0 ∈ Wloc 0 loc contains the closed ball of centre y0 and dxu˜ radius Ri in W u (x˜ ).

Before we state and prove the main results of this section, we give some preliminary lemmas. First of all, we formulate a general lemma from measure theory (see also [41, 51]). For the purpose of completeness, a proof is included here. Lemma Let r0 > 0 and let ν be a finite Borel measure on R such that   VII.2.2  ν R\ 0, r0 = 0. Then for any α ∈ (0, 1) the Lebesgue measure of the set  Lα = r ∈ [0, r0 ] :

+∞

∑ν

k=0

is equal to r0 .

  r − α k , r + α k < +∞

VII.2

Technical Preparations for the Proof of the Main Result

101

Proof. Let α ∈ (0, 1) be given. For k ∈ N define     1  Nα ,k = r ∈ [0, r0 ] : ν r − α k , r + α k > 2 ν [0, r0 ] . k It is notk difficult  to see that Nα ,k can be covered by a finite number of intervals ri − α , ri + α k , 1 ≤ i ≤ s(k) such that every ri lies in Nα ,k and any point of R meets at most two of these intervals. Then it follows that   s(k)    s(k)  [0, r ν ] ≤ ∑ ν [ri − α k , ri + α k ] ≤ 2ν [0, r0 ] , 0 k2 i=1 which implies s(k) ≤ 2k2 . Thus

|Nα ,k | ≤ 2s(k)α k ≤ 4k2 α k ,

where |K| denotes the Lebesgue measure for Borel subset K of R. Consequently +∞

∑ |Nα ,k | < +∞.

k=1

According to Borel-Cantelli lemma [87], it follows that Lebesgue almost every r ∈ [0, r0 ] belongs only to a finite number of Nα ,k and thus satisfies +∞

∑ ν ([r − α k , r + α k ]) < +∞.

k=1

The proof is completed.   f to construct  u (x) We now prepare to use the local unstable sets W ˜ of f in M loc a suitable measurable partition of M f subordinate to W u -manifolds of f . Notice  u (x)} that the partition of M f into global unstable sets {W ˜ x∈M f is in general not ˜ measurable, but we may consider the σ -algebra consisting of measurable subsets of M f which are unions of some global unstable sets, i.e., the σ -algebra     u (x) B u = B ∈ Bμ˜ (M f )  x˜ ∈ B implies W ˜ ⊂B ,

(VII.2)

where Bμ˜ (M f ) is the completion of B(M f ) with respect to μ˜ . In addition, put    B I = A ∈ Bμ˜ (M f )  θ −1 A = A . We have B I ⊂ B u , μ˜ -mod 0. In general, we have the following useful fact.

(VII.3)

102

VII

SRB Measures and Pesin’s Entropy Formula for Endomorphisms

Lemma VII.2.3 Let (X, d) be a compact metric space. T is a homeomorphism with an invariant probability measure μ . Let  B I (X, T, μ ) = A ∈ Bμ (X) : T −1 A = A ,  B u (X, T, μ ) = B ∈ Bμ (X) : x ∈ B implies W u (x) ⊂ B .

and

 W u (x) = y ∈ X : lim d(T −n y, T −n x) = 0 .

where

n→+∞

Then

B I (X, T, μ ) ⊂ B u (X, T, μ ),

μ -mod 0.

Proof. For simplicity, write B u (X) = B u (X, T, μ ) and B I (X) = B I (X, T, μ ). Since X is a compact metric space, there exists a countable set F ⊂ C(X) which is dense in L2 (X, Bμ (X), μ ). For any g ∈ F , according to Birkhoff ergodic theorem and the general properties of conditional expectations, one has    1 n−1 g(T −k x) = E μ gB I (X) (x) ∑ n→+∞ n k=0

(VII.4)

lim



for every point x of a set Λg ∈ B I (X) with μ (Λg ) = 1. Denote ΛF = g∈F Λg . Since F is countable, we have μ (ΛF ) = 1. If two points y, z ∈ ΛF belong to the same unstable set, i.e. there exists x such that y, z ∈ W u (x), by (VII.4) we have for each g ∈ F ,       E μ gB I (X) (y) = E μ gB I (X) (z)    since d(T −n y, T −n z) → 0 as n → +∞. Therefore, each E μ gB I (X) Λ (the restricF    tion of E μ gB I (X) to ΛF ) is measurable with respect to B u (X)|ΛF and hence       E μ gB I (X) Λ : g ∈ F ⊂ L2 ΛF , B u (X)|ΛF , μ . F

(VII.5)

     Since F is a dense subset of L2 (X, Bμ (X), μ ), E μ gB I (X) : g ∈ F is dense in L2 X, B I (X), μ . Then from (VII.5) it follows that     L2 ΛF , B I (X)|ΛF , μ ⊂ L2 ΛF , B u (X)|ΛF , μ which implies

B I (X, T, μ ) ⊂ B u (X, T, μ ),

μ -mod 0,

as μ (ΛF ) = 1.   As a consequence of Lemma VII.2.3, we have the following fact, which plays a crucial role in this section and even in the whole proof of Theorem VII.1.1.

VII.2

Technical Preparations for the Proof of the Main Result

103

Corollary VII.2.3.1 B I ⊂ B u , μ˜ -mod 0. A measurable partition η of M f is said to be increasing if θ −1 η ≥ η . In what follows, we prove that there exists an increasing partition with a refinement subordinate to W u -manifolds of f and other good properties. Such kind of partitions are very useful in our proof of the main theorem. The main result of this section is the following proposition. Proposition VII.2.4 There exists a measurable partition η of M f which has the following properties:

θ −1 η ≥ η ; (1) ) −n (2) +∞ n=0  *θ+∞ ηn isequal uto the partition into single points; (3) B n=0 θ η = B , μ˜ -mod 0, where B(ξ ) is the σ -algebra consisting of all measurable ξ -sets for a measurable partition ξ of M f . (4) There exists a measurable partition η ≥ η such that η is subordinate to W u manifolds of f . Proof. We prove the proposition by constructing such a partition.   Let Δi ∈ Δi : i ∈ N be fixed arbitrarily. Since Δi is compact, the open cover   ˜ εi ri /2) x∈ has a finite subcover UΔi of Δi . Fix arbitrarily BΔi (x˜∗ , εi ri /2) ∈ BΔi (x, ˜ Δi UΔi . For each r ∈ [ri /2, ri ] put 

u loc y˜ ∈ W (x) ˜ : y0 ∈ B(x∗0 , r) , (VII.6) Sr = x∈ ˜ BΔi (x˜∗ ,εi r)

and assume that μ˜ (Sr ) > 0.   u (x) Let ξr denote the partition of M f into all the sets y˜ ∈ M f : y0 ∈ Wloc ˜  B(x∗0 , r) , x˜ ∈ BΔi (x˜∗ , εi r) and the set M f \Sr . From Proposition VII.2.1, it is easy to see that ξr is a measurable partition of M f . Define

ηr =

+∞ 

θ n ξr ,

S%r =

n=0

+∞

θ n Sr .

n=0

We claim that there exists r ∈ [ri /2, ri ] such that ηr has the following properties: (a) θ −1 ηr ≥ ηr ;    u (x) ˜ ⊂W ˜ and p ηr (y) ˜ contains an open neighborhood (b) For μ˜ -a.e. y˜ ∈ S%r , ηr (y) u (y). of y0 in Wloc ˜ In fact, ηr satisfies (a)–(b) for Lebesgue almost every r ∈ [ri /2, ri ]. We now prove this fact in several steps. Step 1. From the definition of ηr it follows clearly that ηr satisfies (a) for each r ∈ [ri /2, ri ].  u (y) Step 2. Let r ∈ [ri /2, ri ]. It is clear that for any y˜ ∈ S%r , one has ηr (y) ˜ ⊂W ˜ since u loc ηr (y) ˜ ⊂ θ nW (θ −n x) ˜ (VII.7) for some n ≥ 0 and some θ −n x˜ ∈ BΔi (x˜∗ , εi r).

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SRB Measures and Pesin’s Entropy Formula for Endomorphisms

On the other hand, we first claim that there exists a function βr : Sr → R+ such ˜ and d u (y0 , z0 ) ≤ βr (y) ˜ there is z˜ ∈ ηr (y) ˜ with p˜z = z0 . that for any y˜ ∈ Sr , z0 ∈ W u (y) Indeed, define for y˜ ∈ Sr  ' 1 r βr (y) ˜ = inf Ri , d(y−n , ∂ B(x∗0 , r))enλi , . n≥0 2γi γi  u (x). From the definition of Sr , there is x˜ ∈ BΔi (x˜∗ , εi r) such that y˜ ∈ W loc ˜ Since u u  ˜ with p˜z = z0 and d (y0 , z0 ) ≤ Ri , by (3) of Proposition VII.2.1, there is z˜ ∈ Wloc (x) hence for all n ≥ 0  1  d u (y−n , z−n ) ≤ γi e−nλi d u (y0 , z0 ) ≤ d y−n , ∂ B(x∗0 , r) 2 and

d u (y−n , z−n ) ≤ γi e−nλi ·

r ≤ r. γi

These together with (2) of Proposition VII.2.1 and with the definition of ξr imply easily that for all n ≥ 0 θ −n z˜ ∈ ξr (θ −n y), ˜ which proves what we claimed just above. We next claim that βr > 0 μ˜ -a.e. on Sr for Lebesgue almost every r ∈ [ri /2, ri ]. In fact, let ν be the finite non-negative Borel measure on [ri /2, ri ] defined by

ν (A) = μ

  x ∈ M : d(x, x∗0 ) ∈ A

for each Borel subset A of [ri /2, ri ]. According to Lemma VII.2.2, Lebesgue almost every r ∈ [ri /2, ri ] satisfies +∞

∑μ

  x ∈ M : |d(x, x∗0 ) − r| < e−nλi < +∞.

(VII.8)

n=0

  Let K0 = r ∈ [ri /2, ri ] : r satisfies (VII.8) and μ (∂ B(x∗0 , r)) = 0 . Clearly |K0 | = ri /2. Let r ∈ K0 . By the standard knowledge about Riemannian metrics, we have a constant D > 0 such that   d x, ∂ B(x∗0 , ρ ) < θ implies

|d(x, x∗0 ) − ρ | < Dθ

for ρ and θ satisfying 0 < θ < ρ ≤ ri . Thus from (VII.8) we obtain +∞

∑μ

n=0



x ∈ M : d(x, ∂ B(x∗0 , r)) < D−1 e−nλi



< +∞.

VII.2

Technical Preparations for the Proof of the Main Result

105

In other words, +∞

∑ μ˜



x˜ ∈ M f : d(x0 , ∂ B(x∗0 , r)) < D−1 e−nλi



< +∞.

n=0

This together with the θ -invariance of μ˜ yields that +∞

∑ μ˜



y˜ ∈ M f : d(y−n , ∂ B(x∗0 , r)) < D−1 e−nλi



< +∞.

n=0

Then, by the Borel-Cantelli lemma [16], we know that μ˜ -a.e. y˜ ∈ M f satisfies   d y−n , ∂ B(x∗0 , r) ≥ D−1 e−nλi when n is sufficiently large. Therefore, βr (y) ˜ > 0 for μ˜ -a.e. y˜ ∈ Sr . The second claim is then proved.   ˜ Let r ∈ K0 . The two claims above together imply that for μ˜ -a.e. y˜ ∈ Sr , p ηr (y) u (y). contains an open neighborhood of y0 in Wloc ˜ Furthermore, for every n ≥ 0 we have   n−1    ηr θ n Sr = θ k ξr θ n ηr  k=0

θ nS

= r

n−1  k=0

  θ k ξr 

 θ nS

r

   θ n ηr Sr .

(VII.9)

 Sinceμ (∂ B(x∗0 , r)) = 0 implies that μ˜ y˜ ∈ M f : y−l ∈ ∂ B(x∗0 , r) for some l ≥ 0 = 0, from (VII.9) it is easy to see that for μ˜ -a.e. y˜ ∈ θ n Sr , ηr (y) ˜ conu (y). ˜ Thus ηr satisfies the requirements tains an open neighborhood of y0 in Wloc in (b). +∞  k Put Ir = θ S%r . Obviously θ −1 Ir = Ir and μ˜ (Ir ) = μ˜ (S%r ) > 0. By Corollary k=0

VII.2.3.1 we may assume that Ir ∈ B u since otherwise there is Ir ∈ B u with θ −1 Ir = Ir and μ˜ (Ir Ir ) = 0 and we may verify that ηr restricted to Ir , written ηrI , satisfies the requirements (2) and (3). Put ηr− =

+∞ )

n=0

θ −n ηrI . Since θ : (M f , μ˜ ) → (M f , μ˜ ) restricted to Ir is also

measure-preserving, by Poincar´e’s recurrence theorem,for μ˜ -a.e. y˜ ∈ S%r there  exists infinitely many positive integers n j : j = 1, 2, · · · such that θ n j y˜ ∈ Sr for all j ≥ 1. Then, letting di be the maximum d u -diameter of the embedded  discs ˜ introduced in Proposition V.V.4.4 (2), we have that the d u -diameter of p ηr− (y) is less than γi di e−λi n j for all j ≥ 1 and hence is equal to 0. This proves that ηr− is equal to the partition of Ir into single points.  +∞ * n I In order to prove that B θ ηr ⊂ B u |Ir , μ˜ -mod 0, it suffices to ensure that n=0

 u (y), for μ˜ -a.e. y˜ ∈ Ir , if z˜ ∈ W ˜ then there exists k > 0 such that θ −k z˜ ∈ ηrI (θ −k y). ˜ In fact, for μ˜ -a.e. y˜ ∈ Ir , we first have

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1 lim sup log d u (y−n , z−n ) ≤ −λi < 0 n→+∞ n  u (y). if z˜ ∈ W ˜ Define now αr : Ir → R+ by 

αr (y) ˜ =

βr (y), ˜ 0,

if y˜ ∈ Sr ; otherwise.

According to Birkhoff’s ergodic theorem, one has for μ˜ -a.e. y˜ ∈ Ir , 1 n−1 ∑ αr (θ −k y)˜ = αr− (y)˜ n→+∞ n k=0 lim

exists and it is easy to see that αr− > 0 for μ˜ -a.e. y˜ ∈ Ir . Hence, for μ˜ -a.e. y˜ ∈ Ir , if  u (y), ˜ there will be some k > 0 such that z˜ ∈ W d u (y−k , z−k ) < αr (θ −k y) ˜ −k −k I −k ˜ On the other hand, it is clear which implies that  n θI  y˜ ∈ Sr and θ z˜ ∈ ηr (θ y). u that B |Ir ⊂ B θ ηr (μ˜ -mod 0) for all n ≥ 0 and hence

B u |Ir ⊂

+∞ 

+    +∞  B θ n ηrI = B θ n ηrI

n=0

(μ˜ − mod 0)

n=0

This proves that ηrI satisfies the requirement (3). Now notice that the treatment above holds for every element of

+∞  i=1

UΔi =

{U1 ,U2 , · · ·}. For each Un we denote by ηn the associated partition ηr satisfying (1)–(3) constructed above and by In the corresponding Ir . Set ηˆ n = ηn |In for n ≥ 1 and define a partition η of M f by ⎧ ˜ ⎨ ηˆ 1 (x), η (x) ˜ = ηˆ n (x), ˜ ⎩ {x}, ˜

if if if



x˜ ∈ I1 ;  x˜ ∈ In \ n−1 k=1 Ik ;  f x˜ ∈ M \ +∞ n=1 In . 

Since In ∈ B I B u for all n ≥ 1 and μ˜ (Δ0 \ +∞ n=1 In ) = 0, it is easy to see that η satisfies the requirements (1)–(3). To finish the proof of the proposition, for each Un , we denote by Sn the associated Sr as defined in (VII.6) for r in the corresponding K0 . Let ξn denote corresponding partition ξr of M f , and define a partition ξ of M f by ⎧ ˜ ⎨ ξ1 (x), ξ (x) ˜ = ξn (x), ˜ ⎩ {x}, ˜

if if if

x˜ ∈ S1 ;  x˜ ∈ Sn \ n−1 S; k=1 k x˜ ∈ M f \ +∞ n=1 Sn .

VII.2

Technical Preparations for the Proof of the Main Result

107

 u Since μ˜ (Δ0 \ +∞ n=1 Sn ) = 0, it is easy to see that ξ is subordinate to W -manifolds of

f . Now take η = η ∨ ξ , then η satisfies the requirement (4). The proof is completed.   ˜ -a.e. x˜ ∈ M f , Let η and η be as in the proof of Proposition VII.2.4. Clearly, for μ  ˜ xη˜ -mod 0) a countable partition. Let η (x) η |η (x) ˜ = i Ai , {A1 , A2 , · · ·} ⊂ η . ˜ is ( μ We know that for each Ai , there is a C1 embedded submanifold Wi of M such that p(Ai ) ⊂ Wi and for μ˜ -a.e. y˜ ∈ M f , if y˜ ∈ Ai , then p|Ai : Ai → p(Ai ) is bijective. Therefore there is a unique measure λ˜ i on Ai such that

λ˜ i (B) = λi (p(B)) for every Borel set B ⊂ Ai , where λi is the Lebesgue measure on Wi . Define λ˜ x˜η as

λ˜ x˜η (K) = ∑ λ˜ i (K



Ai ).

(VII.10)

i

Proposition VII.2.5 For every Borel subset B ⊂ M f , the function    ˜ = λ˜ x˜η η (x) ˜ B PB (x)

is measurable and μ˜ almost everywhere finite. Proof. Let r ∈ K0 and Sr defined as in Step 1 in the proof of Proposition VII.2.4. ˜ From (VII.7) it is easy to see that for any Borel subset B ⊂ M f the function PB (y) ) k ξ and S %rn = n−1 θ k Sr . is finite for μ˜ -a.e. y˜ ∈ S%r . Let n ∈ Z+ , and put ξrn = n−1 θ r k=0 k=0 From the definition of Sr it is clear that, if U is an open ball in M f , the function    ˜ = λ˜ y˜η ξrn (y) ˜ U PU,n (y)

is measurable on S%rn . Then the standard arguments from measure theory ensure that so is PB,n(y) ˜ for any B ∈ M f . It is also easy to see that for any Borel subset B ∈ M f PB,n (y) ˜ ≥ PB,n+1(y) ˜ for each y˜ ∈ S%rn , and

lim PB,n (y) ˜ = PB (y) ˜

n→+∞

for μ˜ -a.e. y˜ ∈ S%r . Therefore PB (y) ˜ is measurable and finite μ˜ -a.e. on S%r . Then by the definition of η , we conclude that PB (y) ˜ is measurable and finite μ˜ -a.e. on M f .   Let η be a partition as in Proposition VII.2.4. Proposition VII.2.4 allows us to define a Borel measure λ˜ ∗ on M f by

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VII

λ˜ ∗ (K) =



SRB Measures and Pesin’s Entropy Formula for Endomorphisms

PK (x) ˜ μ˜ (d x) ˜ =



   λ˜ x˜η η (x) ˜ K d μ˜ (x) ˜

(VII.11)

for each Borel subset K ⊂ M f . It is easy to see that λ˜ ∗ is σ -finite. Furthermore, by the definition of conditional measures we have

μ˜ (K) =



   μ˜ xη˜ η (x) ˜ K d μ˜ (x) ˜

(VII.12)

for each Borel subset K of M f , where μ˜ xη˜ is regarded as the conditional measure of μ˜ on η (x). ˜ If μ has SRB property, one easily has

μ˜  λ˜ ∗ . Define

d μ˜ . d λ˜ ∗ By a measure-theoretic observation one has the following proposition. g=

(VII.13)

Proposition VII.2.6 Suppose μ has SRB property. Let η be a partition as introduced in Proposition V II.2.4. Then for μ˜ -a.e. x˜ one has g=

d μ˜ xη˜ d λ˜ x˜η

λ˜ x˜η almost everywhere on η (x), ˜ where λ˜ x˜η is given by (V II.10). Proof. Let us first notice that (VII.11) can be written equivalently as 

χK d λ˜ ∗ =

 , η (x) ˜

χK (˜z) d λ˜ x˜η (˜z) d μ˜ (x) ˜

for any Borel set K ⊂ M f . Then using standard methods of measure theory we easily obtain   , η ∗ ˜ ˜ h dλ = h(˜z) d λ (˜z) d μ˜ (x) ˜ (VII.14) η (x) ˜

x˜

for any h ∈ L1 (M f , B(M f ), λ˜ ∗ ). Let A ∈ B(η ), B ∈ B(M f ) be two arbitrary sets. From (VII.11)-(VII.14) it follows that   , ˜∗ = ˜ η (˜z) d μ˜ (x) g d λ g(˜ z ) d λ ˜   x˜ A B

=



η (x) ˜ B

A



A B

d μ˜ =

 A

μ˜ xη˜ (η (x) ˜



(VII.15) B) d μ˜ (x). ˜

VII.2

Technical Preparations for the Proof of the Main Result

109

Since M f is a Borel subset of the Polish space M Z , by [51, Theorem 0.1.3], the measure space (M f , B(M f ), μ˜ ) is separable. Then by [51, Theorem 0.1.2], B(M f ) can be generated (μ˜ -mod 0) by a countable subalgebra {Bi }i∈N of B(M f ). Fixing i ∈ N, we apply (VII.15) to an arbitrary set A ∈ B(η ) and to B = Bi . As A is arbitrary, (VII.15) implies that there exists a measurable subset Zi ⊂ M f such that μ˜ (Zi ) = 1 and for each x˜ ∈ Zi one has  

η (x) ˜ Bi

g(˜z) d λ˜ x˜η (˜z) = μ˜ xη˜ (η (x) ˜



Bi ).

Then, according to [51, Theorem 0.1.1], we know that for μ˜ -a.e. x, ˜  

η (x) ˜ B

η

η

g(˜z) λ˜ x˜ (˜z) = μ˜ x˜ (η (x) ˜



B).

holds for any B ∈ B(M f ), and therefore g=

d μ˜ xη˜ d λ˜ x˜η

λ˜ x˜η almost everywhere on η (x). ˜   In the proof of the necessity for the entropy formula, we also need to consider a family of increasing partitions constructed in Proposition VII.2.4. We have the following lemma. Lemma VII.2.7 Let η1 and η2 be two partitions constructed in the proof of Proposition VII.2.4. Then hμ˜ (θ −1 , η1 ) = h μ˜ (θ −1 , η2 ). Proof. It suffices to( prove hμ˜ (θ −1 , η1 ∨ η2 ) = h μ˜ (θ −1 , η1 ). Using the fact that hμ˜ (θ −1 ) = hμ˜ (θ ) ≤ ∑i λi (x) ˜ + mi (x) ˜ μ˜ (d x) ˜ < +∞, by [92, Theorem 4.12] we have for every n ≥ 1, hμ˜ (θ −1 , η1 ∨ η2 ) = hμ˜ (θ −1 , θ n (η1 ∨ η2 )) = hμ˜ (θ −1 , η1 ∨ θ n η2 ) = Hμ˜ (η1 ∨ θ n η2 |θ η1 ∨ θ n+1 η2 ) = Hμ˜ (η1 |θ η1 ∨ θ n+1 η2 ) + Hμ˜ (η2 |θ η2 ∨ θ −n η1 ). By Proposition VII.2.4 (3), one has as n → +∞   +∞ θ η1 ∨ θ n+1 η2 ↓ θ η1 ∨ ∧ θ l η2 = θ η1 . l=0

Hence

Hμ˜ (η1 |θ η1 ∨ θ n+1 η2 ) → Hμ˜ (η1 |θ η1 )

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SRB Measures and Pesin’s Entropy Formula for Endomorphisms

as n → +∞. Also, by Proposition VII.2.4 (2), θ η2 ∨ θ −n η1 tends increasingly to the partition of M f into single points. Thus Hμ˜ (η2 |θ η2 ∨ θ −n η1 ) → 0 as n → +∞. This completes the proof.

 

VII.3 Proof of the Sufficiency for the Entropy Formula Assume that μ has SRB property. Our purpose in this section is to prove that the identity (VII.1) holds. Due to Margulis-Ruelle inequality, it suffices to show the inequality 

∑ λi (x)+ mi (x) d μ (x),

hμ ( f ) ≥

i

which is equivalent to hμ˜ (θ ) ≥



˜ ∑ λi (x)˜ + mi (x)˜ d μ˜ (x),

(VII.16)

i

because of the identity hμ˜ (θ ) = hμ ( f ) and (7) of Proposition 0.I.3.5. Let η be a partition of M f as introduced in Proposition VII.2.4. By the general properties of entropies and (1) of Proposition VII.2.4, one has h μ˜ (θ ) ≥ hμ˜ (θ , η ) = Hμ˜ (θ −1 η |η ) = −



  log μ˜ xη˜ (θ −1 η )(x) ˜ d μ˜ (x) ˜

(VII.17)



f Put I = +∞ n=1 In (see the proof of Proposition VII.2.4) and J = M \I. Note that the −1 restrictions of η and θ η to J is the partition into single points. Hence, for each x˜ ∈ J,   ˜ = 0. log μ˜ xη˜ (θ −1 η )(x)

On the other hand,



∑ λi (x)˜ + mi (x)˜ d μ˜ (x)˜ = 0.

J i

Therefore, without loss of generality, we may assume that μ˜ (I) = 1. For μ˜ -a.e. x˜ ∈ M f we may define   ˜ , X(x) ˜ = μ˜ xη˜ (θ −1 η )(x) g(x) ˜ , g(θ x) ˜    . Z(x) ˜ = det Tx0 f |E u (x) ˜

Y (x) ˜ =

It is easy to see that X, Y and Z are all measurable and μ˜ -a.e. finite functions on M f . We first make the following claims, whose proofs will be given later.

VII.3

Proof of the Sufficiency for the Entropy Formula

111

Claim 3.1 X = Y Z −1 , μ˜ -a.e. Claim 3.2 (a) log Z ∈ L1 (M f , μ˜ ); 

(b)

log Z d μ˜ =



˜ ∑ λi (x)˜ + mi (x)˜ d μ˜ (x). i

Claim 3.3 (a) logY ∈ L1 (M f , μ˜ ); 

(b)

logY d μ˜ = 0.

From these and (VII.17) one can immediately obtain (VII.16) and completes the proof of the sufficiency. ˜ xη˜ -mod 0) a ˜ (θ −1 η )|η (x) Proof of Claim 3.1 First notice that for μ˜ -a.e. x, ˜ is ( μ −1 countable partition. Then for μ˜ -a.e. x, ˜ for any y˜ ∈ (θ η )(x) ˜ and a sufficiently small neighborhood of y, ˜ denoted by B, we have

μ˜ xθ˜

−1 η

   1 μ˜ xη˜ (θ −1 η )(x) ˜ B X(x) ˜  1 z) d λ˜ x˜η (˜z); =  g(˜ X(x) ˜ (θ −1 η )(x) ˜ B

(B) =

on the other hand,

μ˜ xθ˜

−1 η

(B) = μ˜ θηx˜ (θ B) = =





η (θ x) ˜ (θ B)





g(˜z) d λ˜ θηx˜ (˜z)

(θ −1 η )(x) ˜ B

  g(θ z˜)det(D fz0 |E u (˜z) ) d λ˜ x˜η (˜z).

Since B is arbitrarily chosen, we obtain for μ˜ -a.e. x˜   1  g(y) ˜ = g(θ y) ˜ det(D fy0 |E u (y) ˜ ) X(x) ˜ for λ˜ x˜η − a.e. y˜ ∈ (θ −1 η )(x). ˜ This implies that for μ˜ -a.e. x˜ one has X(y) ˜ = Y (y)Z( ˜ y) ˜ −1 ,

μ˜ xθ˜

−1 η

− a.e. y˜ ∈ (θ −1 η )(x) ˜

˜ From this there follows Claim 3.1. as X(x) ˜ = X(y) ˜ for each y˜ ∈ (θ −1 η )(x).

 

Proof of Claim 3.2 By assumption (V.V.1) and the Multiplicative Ergodic Theorem, it is clear that log Z ∈ L1 (M f , μ˜ ) and 

log Z d μ˜ =



˜ ∑ λi (x)˜ + mi (x)˜ d μ˜ (x). i

 

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Proof of Claim 3.3 By Claim 3.1, one has log X = logY − log Z ≤ 0, which implies

log+ Y ≤ log+ Z,

μ˜ − a.e.

μ˜ − a.e.

+

Hence, by Claim 3.2, log Y ∈ L1 (M f , μ˜ ). Then this claim follows from the subadditive ergodic theorem.   Up to now, we have completed the proof of the first part of Theorem VII.1.1, i.e. if the invariant measure μ has SRB property, then entropy formula (VII.1) holds. From next section, we start the proof of the second part of the Theorem.

VII.4 Lyapunov Charts From now on in the rest of this chapter, we prove the second part of Theorem VII.1.1, i.e. the necessity for the entropy formula. Suppose that system (M f , θ , μ˜ ) is ergodic. Under this hypothesis we know that there exists a Borel set Δ1 ⊂ Δ˜ with μ˜ (Δ1 ) = 1 and θ (Δ1 ) = Δ1 such that for any ˜ λi (x) ˜ and mi (x) ˜ equal respectively constants r, λi and mi , 1 ≤ i ≤ r. x˜ ∈ Δ1 , r(x), For x˜ ∈ Δ1 , let Exu˜ =

. λi >0

Ei (x), ˜

Exc˜ = Ei0 (x), ˜ λi0 = 0, Exs˜ =

. λi <0

Ei (x), ˜

  λ + = min λi : λi > 0 ,

ku = dim Exu˜ ; kc = dim Exc˜ ; ks = dim Exs˜ ;   λ − = max λi : λi < 0 .

We now begin the proof by constructing in this section Lyapunov charts, which will be particularly useful in the subsequent sections. The construction needs the following two lemmas. Lemma VII.4.1 There exists a Borel set Δ2 ⊂ Δ1 with μ˜ (Δ2 ) = 1 and θ (Δ2 ) = Δ2 such that for any given δ > 0 one can define a Borel function Rδ : Δ2 → [1, +∞) with the following properties: 1) For each x˜ ∈ Δ2 the maps   x˜ def x : Tx M Rδ (x) H =H ˜ −1 → Tx1 M, 0 0   def  −1 x−1 H ˜ −1 → Tx−1 M ˜ = Hx−1 : Tx0 M Rδ (x)

VII.4

Lyapunov Charts

113

are well defined and x˜ (·)) ≤ Rδ (x), ˜ Lip(DH 2) The map

x−1 Lip(DH ˜ ˜ (·)) ≤ Rδ (x);

  −1 x−1 fx˜−1 = expx−1 ◦H ˜ −1 → M ˜ ◦ expx0 : B x0 , Rδ (x)

is well defined and

f ◦ fx˜−1 = id|B(x0 ,Rδ (x) ˜ −1 ) ;

˜ ≤ Rδ (x)e ˜ δ for all x˜ ∈ Δ2 . 3) Rδ (θ ±1 x) % x) ˜ = max{κ (x−1 ), κ (x0 )}, where function κ is as Proof. For any x˜ ∈ M f , let R( defined in Lemma V.V.1.3. By Lemma V.V.1.3 and V.V.2.4, for any x˜ ∈ Δ˜ , functions   x˜ = H x : Tx M R( % x) H ˜ −1 → Tx1 M 0 0 and

   −1 % ˜ −1 → Tx M x−1 H ˜ = Hx−1 : Tx0 M R(x) −1

are well defined, and x˜ (·)) ≤ R( % x), Lip(DH ˜

x−1 % ˜ Lip(DH ˜ (·)) ≤ R(x)

% x) on Tx0 M(R( ˜ −1 ). Since log κ ∈ L1 (M, μ ), one has log R% ∈ L1 (M f , μ˜ ). Hence by Birkhoff ergodic theorem, there is Δ2 ⊂ Δ1 with θ (Δ2 ) = Δ2 and μ˜ (Δ2 ) = 1 such that 1 % θ n x) lim log R( ˜ =0 n→±∞ n for any x˜ ∈ Δ2 . Given δ > 0, define Rδ : Δ2 → [1, +∞) as  % θ n x)e Rδ (x) ˜ = sup R( ˜ −|n|δ : n ∈ Z}. Then it is easy to verify that Δ2 and the function Rδ satisfy the requirements of the lemma.   From here on, we always denote the Euclidean norm by  · . For z = (zu , zc , zs ) ∈ Rku × Rkc × Rks we define  

  z = max zu , zc , zs  . Then we put

¯ u (ρ ) × R ¯ c(ρ ) × R ¯ s (ρ ) ¯ ρ) = R R(

¯ u (ρ ), R ¯ c (ρ ) and R ¯ s (ρ ) denote respectively the closed discs in for ρ > 0, where R k k k u c s R , R and R of radius ρ centered at 0. Put Δ λ = min{|λi − λ j |, λ + , −λ − : i = j}. (VII.18)

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This is the minimum  gap of different Lyapunov exponents and 0. Let 0 < ε < min 1, Δ λ /100m0 be given, and let ρ2 be a number as introduced in Section V.V.1. In what follows we define for each x˜ ∈ Δ2 a change of coordinates in some neighborhood of x0 in M. The size of the neighborhood, the local chart and the related estimates will vary with x˜ ∈ Δ2 . This is the following proposition. Proposition VII.4.2 There is a measurable function  : Δ2 → [1, +∞) satisfying ˜ ≤ (x)e ˜ ε for all x˜ ∈ Δ2 , and for each x˜ ∈ Δ2 there is a C∞ embedding (θ ±1 x) ¯ (x) Φx˜ : R ˜ −1 → M with the following properties: 1) Φx˜ (0) = x0 , DΦx˜ (0) maps Rmi , i = 1, · · · , r onto Ei (x), ˜ i = 1,  · · · , r respectively. ¯ (x) Furthermore, Φx˜ coincides with expx0 ◦DΦx˜ (0) on R ˜ −1 and (x) ˜ −1 z ≤ DΦx˜ (0)z ≤

√ m0 z,

for any z ∈ Rm0 ;

2) Let Hx˜ = Φθ−1 x˜ ◦ f ◦ Φx˜ be the connecting map between the chart at x0 and the def

−1 −1 chart at x1 , defined wherever it makes sense. Let Hx−1 ˜ = Φθ −1 x˜ ◦ f x˜ ◦ Φx˜ and def

def

Hxn˜ =



n Φθ−1 n x˜ ◦ f ◦ Φx˜ , if n > 0 −1 Φθ n x˜ ◦ fx˜n ◦ Φx˜ , if n ≤ 0

be defined similarly. Then (i) For each i, 1 ≤ i ≤ r and any z ∈ Rmi , eλi −ε z ≤ e−λi −ε z ≤

   x˜ (0)z ≤ eλi +ε z,  DH−1  DH (0)z ≤ e−λi +ε z. x˜

Hence eλ

  for all z ∈ Rku , z ≤ DHx˜ (0)z   for all z ∈ Rkc , e−ε z ≤ DHx˜ (0)z ≤ eε z   DHx˜ (0)z ≤ eλ − +ε z for all z ∈ Rks ;

+ −ε

(ii) −1 Lip(Hx˜ − DHx˜(0)) ≤ ε , Lip(Hx−1 ˜ − DHx˜ (0)) ≤ ε ,

˜ Lip(DHx˜ (·)) ≤ ε (x),

Lip(DHx−1 ˜ ˜ (·)) ≤ ε (x);

      ≤ eλ0 for all ξ ∈ R ¯ e−λ0 −ε (x) (iii) max DHx˜ (ξ ), DHx−1 ˜ −1 , where λ0 ˜ (ξ ) is a positive number depending only on ε and the exponents.

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115

  ¯ (x) 3) For any z, z ∈ R ˜ −1 , we have ˜ d(Φx˜ z, Φx˜ z ) K0−1 d(Φx˜ z, Φx˜ z ) ≤ z − z  ≤ (x) for some universal constant K0 > 0. def

Proof. Let  · x˜ be the norm induced by the inner product  , x˜ =  , x,0 ˜ on Tx0 M introduced in Section V.2 (see (V.V.10)-(V.V.12)). Then from Lemma V.V.2.3, we ˜ 1≤i≤r have for all ξ ∈ Ei (x), eλi −ε ξ x˜ ≤ T01 (x) ˜ ξ θ x˜ ≤ eλi +ε ξ x˜ and 1 ˜ ε )|ξ |, √ |ξ | ≤ ξ x˜ ≤ A(x, m0

∀ξ ∈ Tx0 M,

where A(x, ˜ ε ) = 4ε Cε /4 (x) ˜ 2. Let b = b(ρ2 /2) be a number introduced by Lemma II.II.2.1 and define a function  : Δ2 → [1, +∞) by  6m 9bm eε /2 0 0 (x) ˜ = max , A(x, ˜ ε )Rε /2 (x) ˜ ρ2 ε and let

 λ0 = max |λi | : 1 ≤ i ≤ r + 2ε ,

K0 =

√ m0 b.

Next, for each x˜ ∈ Δ2 take a linear transformation Lx˜ : Tx0 M → Rm1 × · · · × Rmr such that it maps Ei (x), ˜ i = 1, · · · , r onto {0} × · · ·× {0} × Rmi × {0} × · · ·× {0}, i = 1, · · · , r respectively and satisfies Lx˜ ξ , Lx˜ η  = ξ , η x˜ for all ξ , η ∈ Tx0 M, where  ,  is the usual Euclidean inner product. Then we set for each x˜ ∈ Δ2 Φx˜ = expx0 ◦L−1 ¯ x) x˜ |R(( ˜ −1 ) . With the entries defined above, one can easily check that 1)–3) of the proposition are satisfied.   satisfying From here on, we shall refer to any system of local charts {Φx˜ }x∈ ˜ Δ2 1)–3) in Proposition VII.4.2 as a system of (ε , )-charts, and λ0 and K0 will be defined as above.

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VII.5 Local Unstable Manifolds and Center Unstable Sets As we have already mentioned in the previous section, in order to prove the second part of Theorem VII.1.1 it is necessary to consider explicitly the role played by the zero exponent as well as by the positive Lyapunov exponents. In this section we use Lyapunov charts described in the previous section to introduce some nonlinear constructions related to these exponents. These constructions will be used in the next section to deal with some measurable partitions of M f which play central roles in the whole proof. For the sake of clearness of presentation, we divide this section into three subsections.

VII.5.1 Local Unstable Manifolds and Center Unstable Sets Let {Φx˜ }x∈ ˜ Δ 2 be a system of (ε , )-charts. Sometimes it is necessary to reduce the size of the charts. Let 0 < δ ≤ 1 be a reduction factor. For x˜ ∈ Δ2 , define    −n ¯ (x) Sδcu (x) ˜ = ζ ∈R ˜ −1 : Φθ−1 ◦ Φx˜ ζ  ≤ δ (θ −n x) ˜ −1 , n ∈ Z+ , −n x˜ ◦ f x˜ where fx˜−n = fθ −n+1 x˜ ◦ · · · ◦ fx˜−1 ; that is, Φx˜ Sδcu (x) ˜ consists of those points in a small neighborhood of x0 whose forward orbit under actions fx˜−n stays inside the domains of the charts at θ −n x˜ for all n ≥ 0. It is called the center unstable set of f at x˜ cu ˜ and for all n ≥ 1, one clearly has associated with ({Φx˜ }x∈ ˜ Δ 2 , δ ). On Sδ (x) −1 −n −1 −1 −1 Hx−n ˜ = Φθ −n x˜ ◦ f x˜ ◦ Φx˜ = Hθ −n+1 x˜ ◦ · · · ◦ Hθ −1 x˜ ◦ Hx˜ . def

We next introduce the local unstable manifold at x˜ ∈ Δ2 associated  with  u ˜ Φ R ¯ δ (x) ({Φx˜ }x∈ ˜ −1 that x˜ ˜ Δ 2 , δ ). It is defined to be the component of W (x) u ˜ contains x0 . The Φx−1 ˜ -image of this set is denoted by Wx, ˜ δ (x). Lemma VII.5.1 Let {Φx˜ }x∈ ˜ Δ 2 be a system of (ε , )-charts. 1) If 0 < δ ≤ e−(λ0 +ε ) and x˜ ∈ Δ2 , then i) Wx,˜uδ (x) ˜ is the graph of a C1 function     ¯ u δ (x) ¯ c+s δ (x) hx,˜ x˜ : R ˜ −1 → R ˜ −1 with hx,˜ x˜ (0) = 0 and Lip(hx,˜ x˜ ) < 1; ii) Wx,˜uδ (x) ˜ ⊂ Sδcu (x). ˜ 2) If 0 < δ ≤ e−2(λ0 +ε ) and x˜ ∈ Δ2 , then Hx˜Wx,˜uδ (x) ˜

 ¯ δ (θ x) ˜ −1 = Wθux,˜ δ (θ x). ˜ R

 

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117

Proof. Let 0 < δ ≤ e−(λ0 +ε ) and x˜ ∈ Δ2 . By Proposition VII.4.2 we have   DH −1 (0)ξ  ≤ e−λ + +ε ξ  x˜   DH −1 (0)η  ≥ e−ε η  x˜

for ξ ∈ Rku , for η ∈ Rkc +ks

and −1 ˜ · δ (x) ˜ −1 Lip(Hx−1 ˜ − DHx˜ (0)) ≤ ε (x)

= εδ   + + + < min e−λ +2ε − e−λ +ε , e−ε − e−λ +2ε ,   ¯ where Hx−1 ˜ −1 . Then by arguments analogous to those in ˜ is restricted to R δ (x) Steps 1–6 in the proof of Theorem V.V.4.3 one can easily see that there is a C1 function     ¯ c+s δ (x) ¯ u δ (x) hx,˜ x˜ : R ˜ −1 → R ˜ −1 −1 with gx,x ˜ (0) = 0 and Lip( f x˜ ) < 1 such that

Φx˜ Graph(hx,˜ x˜ ) ⊂ W u (x) ˜ and moreover      −1 −(λ + −2ε )n +  ≤ δ (x) ¯ δ (x) ˜ −1 : Hx−n ζ ˜ e , n ∈ Z . Graph(hx,˜ x˜ ) = ζ ∈ R ˜ Then 1) follows immediately. If 0 < δ ≤ e−2(λ0 +ε ) and x˜ ∈ Δ2 , then it is not hard to verify that ˜ ⊂ Hx˜Wx,˜uδ (x) ˜ ⊂ Wθux,˜ δ (θ x) ˜ Wθux,˜ δ (θ x) where δ = e−(λ0 +ε ) . This together with 1) yields 2).

 

Lemma VII.5.2 If 0 < δ ≤ e−2(λ0 +ε ) , then for μ˜ -a.e. x˜ ∈ Δ2 we have ˜ Sδcu (x)



u Φx−1 ˜ = Wx,˜uδ (x). ˜ ˜ W (x)

Proof. By definition and Lemma VII.5.1, it is easy to see ˜ Sδcu (x)



u Φx−1 ˜ ⊃ Wx,˜uδ (x). ˜ ˜ W (x)

Now it suffices to show that for μ˜ -a.e. x˜ ∈ Δ2 Sδcu (x) ˜ 



u Φx−1 ˜ ⊂ Wx,˜uδ (x). ˜ ˜ W (x)

u ˜ From Theorem V.V.5.3, there is a point y˜ ∈ W  (x) Let ζ ∈ Sδcu (x) ˜ Φx−1 ˜ such ˜ W (x). that y0 = Φx˜ ζ . By Theorem V.V.5.3 and Remark V.V.11 we have d u (y−n , x−n ) → 0

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as n → +∞, where d u ( , ) denote the Riemannian distance along W u -manifolds (see Theorem V.V.5.3 for the precise meaning). But according to the Poincar´e Recur˜ −1 does not tend to 0 as n → +∞ for μ˜ -a.e. x˜ ∈ Δ2 . This rence Theorem, (θ −n x) u −k ˜ implies that for μ˜ -a.e. x˜ ∈ Δ2 there is some k ≥ 0 such that Hx−1 ˜ ζ ∈ Wx, ˜ δ (θ x). Let k = k(x) ˜ be the smallest among such nonnegative integers. If k > 0, then by   −(k−1) ¯ δ (θ −k+1 x) 2) in Lemma VII.5.1, Hx˜ ζ ∈ R ˜ −1 , which in contradiction with ζ ∈ Sδcu (x). ˜ So k = 0, or equivalently, ζ ∈ Wx,˜uδ (x). ˜   Let x˜ ∈ Δ2 . Write  cu −n  cu (x) (VII.19) W ˜ ∀n ∈ Z+ . δ ˜ = y˜ ∈ Δ 2 : y−n ∈ Φθ −n x˜ Sδ (θ x), From Remark V.V.7, it is easy to see that  cu   (x)  cu ˜ → p W p|W cu (x) δ ˜ ˜ : Wδ (x) δ

is bijective. u  cu (x) Consider now y˜ ∈ W ˜ denote the Φx−1 ˜ -image δ ˜ where 0 < δ ≤ 1/4. Let Wx,2 ˜ δ (y) of the component of ˜ W u (y)



 u    ¯ 2δ (x) ¯ c+s 4δ (x) Φx˜ R ˜ −1 × R ˜ −1 .

u ˜ contains an open neighborhood of y0 in W u (y) ˜ that contains y0 . Then Φx˜Wx,2 ˜ δ (y) and is also referred to as a local unstable manifold of f at y˜ (although in general u u Φy˜Wy,2 ˜ = Φx˜Wx,2 ˜ A reduction factor 0 < δ ≤ 1/4 is taken because when ˜ δ (y) ˜ δ (y)). working in the charts of θ −n x˜ we cannot control the unstable manifolds of points −n to the boundaries of whose backward orbits    under the actions fx˜ come too close  −n −1  ¯ (θ x) Φθ −n x˜ R ˜ (y ) =  0. Aside . Another technical nuisance is that Φθ−1 −n x˜ −n from these, we have the following analogue of Lemmas VII.5.1 and VII.5.2.

Lemma VII.5.3 Let {Φx˜ }x∈ ˜ Δ 2 be a system of (ε , )-charts.  cu (x), 1) Let 0 < δ ≤ 14 e−(λ0 +ε ) and x˜ ∈ Δ2 . If y˜ ∈ W δ ˜ then u i) Wx,2 ˜ is the graph of a C1 function ˜ δ (y)

    ¯ u 2δ (x) ¯ c+s 4δ (x) hx,˜ y˜ : R ˜ −1 → R ˜ −1 with Lip(hx,˜ y˜ ) < 1; u ii) Wx,2 ˜ ⊂ S4cuδ (x); ˜ ˜ δ (y) cu  cu (x) ˜ then 2) Let 0 < δ ≤ 14 e−2(λ0+ε ) and x˜ ∈ Δ2 . If y˜ ∈ W δ ˜ with y1 ∈ Φθ x˜ Sδ (θ x), u ˜ Hx˜Wx,2 ˜ δ (y)



    ¯ u 2δ (θ x) ¯ c+s 4δ (θ x) R ˜ −1 × R ˜ −1 ⊂ Wθux,2 ˜ ˜ δ (θ y);

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119

 cu (x), 3) Let 0 < δ ≤ 14 e−2(λ0+ε ) . Then for μ˜ -a.e. x˜ ∈ Δ2 , if y˜ ∈ W δ ˜ one has S2cuδ (x) ˜



u u Φx−1 ˜ ⊂ Wx,2 ˜ ⊂ S4cuδ (x) ˜ ˜ W (y) ˜ δ (y)



u Φx−1 ˜ ˜ W (y).

−1  cu (x). Proof. Let 0 < δ ≤ 14 e−(λ0 +ε ) , x˜ ∈ Δ2 and y˜ ∈ W δ ˜ Denote ζy˜ = Φx˜ y0 . Write

Tζy˜ Hx−1 ˜ = rx,˜ y˜ = Hx−1 ˜ −

,

,

G11 G12 : Rku × Rkc +ks → Rku × Rkc +ks , G21 G22

-  G11 0 ˜ −1 → Ru+c+s . : ζ ∈ Ru+c+s : ζ − ζy˜ ≤ 3δ (x) 0 G22

By Proposition VII.4.2 one can easily verify the following estimates:   G11 ξ  ≤ (e−λ + +ε + εδ )ξ  for ξ ∈ Rku ,   G22 η  ≥ (e−ε − εδ )η  for η ∈ Rkc +ks ,   −λ + +3ε + + Lip(rx,˜ y˜ ) ≤ min e − (e−λ +ε + εδ ), (e−ε − εδ ) − e−λ +3ε . Then by arguments analogous to those in Steps 1–6 in the proof of Theorem V.V.4.3 one can prove that there is a C1 map     ¯ c+s 3δ (x) ¯ u 3δ (x) ˜ −1 → R ˜ −1 hˆ x,˜ y˜ : R

(VII.20)

with hˆ x,˜ y˜ (0) = 0 and Lip(hˆ x,˜ y˜ ) < 1 such that   Φx˜ ζy˜ + Graph(hˆ x,˜ y˜ ) ⊂ W u (y) ˜ and moreover

ζy˜ + Graph(hˆ x,˜ y˜ )      + −n  ¯ (x) ˜ −1 : Hx−n ˜ −1 e−(λ −3ε )n , n ∈ Z+ . = ζ ∈R ˜ ζ − Hx˜ ζy˜ ≤ 3δ (x) Define     ¯ c+s 4δ (x) ¯ u 2δ (x) ˜ −1 → R ˜ −1 , hx,˜ y˜ : R

ξ → hˆ x,˜ y˜ (ξ − ξy˜) + ηy˜

where ζy˜ = (ξy˜ , ηy˜ ) ∈ Rku × Rkc +ks . Then 1) follows immediately. Let now 0 < δ ≤ 1 −2(λ0 +ε ) . We have 4e   u Hx˜Wx,2 ˜ ⊂ Hx˜ ζy˜ + Graph(hˆ x,˜ y˜ ) ˜ δ (y) ⊂ ζθ y˜ + Graph(hˆ θ x,˜ θ y˜ )

(VII.21)

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where hˆ θ x,˜ θ y˜ is the function defined analogously to (VII.20), corresponding to δ = 1 −(λ0 +ε ) . But, if y1 ∈ Φθ x˜ Sδcu (θ x), ˜ then 4e 

      u ¯ 2δ (θ x) ¯ c+s 4δ (θ x) R ζθ y˜ + Graph(hˆ θ x,˜ θ y˜ ) ˜ −1 × R ˜ −1 = Wθux,2 ˜ ˜ δ (y).

This proves 2). The proof of 3) is almost identical to that of Lemma VII.5.2.   ˜ is a rather messy set. Among other things we We remark that in general Sδcu (x) think of it as containing pieces of local unstable manifolds (in view of Lemma VII.5.3 1) ii)). In the case when there is no zero exponent, Sδcu (x) ˜ is equal to Wx,˜uδ (x). ˜

VII.5.2 Some Estimates We now list some estimates which will be used in later sections. Let {Φx˜ }x∈ ˜ Δ2 be a system of (ε , )-charts. When working in charts, we use ζ u to denote the u-coordinate of the vector ζ ∈ Ru+c+s . Other notations such as ζ s and ζ cu are understood to have analogous meanings. Lemma VII.5.4 Let 0 < δ ≤ e−(λ0 +ε ) and x˜ ∈ Δ2 .   ¯ δ (x) ˜ −1 and ζ − ζ  = ζ u − ζ u , then 1) If ζ , ζ ∈ R     Hx˜ ζ − Hx˜ ζ  = (Hx˜ ζ )u − (Hx˜ ζ )u  ≥ eλ + −2ε ζ − ζ ;   ¯ δ (x) ˜ −1 and ζ − ζ  = ζ cu − ζ cu , then 2) If ζ , ζ ∈ R     Hx˜ ζ − Hx˜ζ  = (Hx˜ ζ )cu − (Hx˜ ζ )cu  ≥ e−2ε ζ − ζ ; ˜ then 3) If ζ , ζ ∈ Sδcu (x),  −1  H ζ − H −1 ζ  ≤ e2ε ζ − ζ . x˜ x˜ Proof. By Proposition VII.4.2 2) ii) one has   Lip (Hx˜ − DHx˜(0))|R( ¯ δ (x) ˜ −1 ) ≤ εδ , which together with Proposition VII.4.2 2) i) yields 1) and 2) by a simple calcula . In fact, if it is tion. We now prove 3). First we claim that ζ − ζ  = ζcu − ζcu −1 not true, by applying 1) to Hx˜ we have −1

−1 s −1 s −λ Hx−1 ˜ ζ − Hx˜ ζ  = (Hx˜ ζ ) − (Hx˜ ζ )  ≥ e

− −2ε

ζ − ζ .

Inductively, this gives  n    Gx˜ ζ − Gx˜n ζ  = (Gx˜n ζ )s − (Gx˜n ζ )s  ≥ e−(λ −+2ε )n ζ − ζ 

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121

for all n ≥ 0. This contradicts the fact that ζ , ζ ∈ Sδcu (x). ˜ Now this argument also ap−1

ζ and H ζ , since they are all elements in Sδcu (θ −1 x). ˜ Then it follows plies to Hx−1 ˜ x˜ from 2) that   ζ − ζ  ≥ e−2ε H −1 ζ − H −1ζ  x˜

x˜

 

which is the desired conclusion.

 cu (x), Lemma VII.5.5 Suppose that 0 < δ ≤ 14 e−2(λ0 +ε ) and x˜ ∈ Δ2 . Let y˜ ∈ W δ ˜ kc +ks  u

kc +ks  u η = ({0} × R ) Wx,2 ( y) ˜ and η = ({0} × R ) W ( θ y), ˜ where δ = ˜ δ θ x,2 ˜ δ

1 −(λ0 +ε ) . 4e

Then

η  ≤ e3ε η .

Proof. Since, by (VII.21), u ˜ ⊂ ζθ y˜ + Graph(hˆ θ x,˜ θ y˜ ) Hx˜Wx,2 ˜ δ (y)

(VII.22)

and     u

   ¯ 2δ (θ x) ¯ c+s 4δ (θ x) ˜ = ζy˜ + Graph(hˆ θ x,˜ θ y˜ ) ˜ −1 × R ˜ −1 , Wθux,2 R ˜ δ (y) (VII.23) one has     η  ≤ (Hx˜ η )cs  + (Hx˜ η )u . By Proposition VII.4.2 2),   (Hx˜ η )cs  ≤ (eε + ε )η  and Therefore,

  (Hx˜ η )u  ≤ ε η . η  ≤ (eε + 2ε )η  ≤ e3ε η .

 

VII.5.3 Lipschitz Property of Unstable Subspaces within Center Unstable Sets 1 −(λ0 +ε )

Let {Φx˜ }x∈ . Denote by ˜ Δ 2 be a system of (ε , )-charts, x˜ ∈ Δ 2 and δ = 4 e ku kc +ks ku kc +ks L(R , R ) the space of all linear maps from R to R . By 1) of Lemma ku kc +ks )  cu (x), VII.5.3 we know that, if y˜ ∈ W δ ˜ then there exists a unique Py˜ ∈ L(R , R with Py˜ < 1 such that u DΦx−1 ˜ (y0 )Ey˜ = Graph(Py˜ ).

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  cu (x) ˜ → L(Rku , Rkc +ks ), Lx˜ : y0 : y˜ ∈ W δ

Define by

y0 → Py˜ . In the sequel we show that the map Lx˜ is Lipschitz. For this purpose we make use of the following lemma:   Lemma VII.5.6 Let E, |·| be a Banach space with a decomposition into the direct 1 2 1 2 sum of two subspaces suppose that the norm | · |  E and  E , i.e. E = E ⊕ E , and satisfies |ξ | = max |ξ1 |, |ξ2 | for any ξ = ξ1 + ξ2 ∈ E 1 ⊕E 2 . Let L(E, E)(L(E 1 , E 2 )) denote the space of all bounded linear operators from E(E 1 ) to itself (E 2 ). Given def positive numbers λ , K and δ satisfying eλ < K and 0 < α = (eδ + δ )/(eλ −δ − δ ) < 1, we denote L1 (E, E) the subset of L(E, E) consisting of all bounded linear operators A which is invertible with |A| ≤ K and ,

A11 A12 A= : E1 ⊕ E2 → E1 ⊕ E2 A21 A22 satisfying |(A11 )−1 | ≤ e−λ +δ ,

|A22 | ≤ eδ ,

|A12 | ≤ δ

and |A21 | ≤ δ .

Now we define    = {An }n∈N : An ∈ L1 (E, E), n ∈ N , X= A and

  Y = P = {Pn }n∈N : Pn ∈ L(E 1 , E 2 ) and |Pn | ≤ α , n ∈ N .

Let X and Y endowed respectively with the following metrics:   A  = d A,

+∞

∑ α0n |An − A n|,

A  ∈ X, A,

n=1

   P = d P,

+∞

∑ α0n |Pn − Pn |,

 P ∈ Y, P,

n=1

where α0 = (1 + α )/2. Then we have the following results.    ∈ X, there exists a unique P = {Pn }n∈N def  ∈ Y such that (1) For any A = ϕ (A) A−1 n Graph(Pn ) = Graph(Pn+1 )

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123

 also satisfies for each ζ ∈ Graph(P1 ) and for all n ∈ N. This unique P = ϕ (A) n∈N  −n  A  ζ  ≤ γ n |ζ |, −1 λ −δ − δ )−1 . −n = A−1 where A n ◦ · · · ◦ A1 and γ = (e  → ϕ (A)  is Lipschitz and Lip(ϕ ) ≤ C with respect to the (2) The map ϕ : X → Y , A metrics defined above, where C is a number depending only on λ , K and δ .  and A  ∈ X, let P = ϕ (A)  and P = ϕ (A  ). Then (3) For A

    A  .  P ≤ Cd A, d P, Proof. Clearly, both (X, d) and (Y, d) are complete metric spaces. Define

θ : X × Y → Y,

 P)   → Q (A,

 = {Qn }n∈N is given by where Q  (n)  (n) −1 (n) (n) Qn = A21 + A22 Pn+1 A11 + A12 Pn+1 ,

n ∈ N.

 P)  ∈ X × Y . Since Let us first check that this definition makes sense. Suppose (A,     (n) A Pn+1  ≤ δ < eλ −δ ≤ (A(n) )−1 −1 , 12 11 (n)

(n)

A11 + A12 Pn+1 is invertible and  (n)  (A + A(n) Pn+1)−1  ≤ (eλ −δ − δ )−1 . 11 12 Thus Qn ∈ L(E 1 , E 2 ) and |Qn | ≤ (eδ + δ )/(eλ −δ − δ ) = α . So θ is well defined.  Note that for each Next, we verify that θ is a uniform contraction on the factor P.  A ∈ X and n ∈ N the map     ΓAn : S ∈ L(E 1 , E 2 ) : |S| < 1 → S ∈ L(E 1 , E 2 ) : |S| < 1  (n)  (n) (n) (n) −1 S → A21 + A22 S A11 + A12 S is a C1 map with  (n) (n)   (n) (n) −1 (DΓAn (S0 ))S = A22 − ΓAn (S0 )A12 S A11 + A12 S0 ,

S ∈ L(E 1 , E 2 )

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for each S0 ∈ L(E1 , E 2 ) with |S0 | < 1. By asimple calculation one has |DΓAn (S0 )| ≤ α for each S0 ∈ S ∈ L(E 1 , E 2 ) : |S| < 1 . ΓAn is therefore a Lipschitz map with  ∈ X, then Lip(ΓAn ) ≤ α . From this it follows that, if A      P),  P ) ≤ α d P,  θ (A,  P

d θ (A, α0  P ∈ Y , i.e. θ is an α /α0 -contraction on the second factor. for any P,   def  ∈ Y such that  ∈ X. Then there is a unique point P = Pn = ϕ (A) Let A n∈N  P)  = P,  θ (A, i.e.

A−1 n Graph(Pn ) = Graph(Pn+1 )

for all n ∈ N. Also if n ∈ N and (ξ , Pn+1 ξ ) ∈ Graph(Pn+1 ), then     An (ξ , Pn+1 ξ ) = (A(n) + A(n) Pn+1)ξ  11 12

  ≥ (eλ −δ − δ )|ξ | = (eλ −δ − δ )(ξ , Pn+1 ξ ).

This proves the first statement of the lemma. For the second statement, let X × Y endowed with the product metric d( , ) (see Lemma V.V.3.1). Then it is easy to verify that θ : (X ×Y, d) → (Y, d) is a Lipschitz map and there is a number l > 0 depending only on the parameters λ , K and δ such  → ϕ (A)  is that Lip(θ ) ≤ l. The it follows, from Lemma V.V.3.1, that ϕ : X → Y , A def

Lipschitz and Lip(ϕ ) ≤ (1 − α /α0 )−1 l = C.  and A  ∈ X, let P = ϕ (A)  = θ (A,  P)  ) = θ (A  , P ). Then  and P = ϕ (A For A        P),  P ) + d θ (A,  P ), θ (A  , P )  P ≤ d θ (A,  θ (A, d P,   α     A  . d P, P + ld A, ≤ α0 Then it follows that

    A 

 P ≤ Cd A, d P,

by the definition of C given above. The proof of the lemma is completed.

 

Lemma VII.5.7 Let {Φx˜ }x∈ ˜ Δ 2 be a system of (ε , )-charts with ε being small + +10ε − λ enough so that e + e5ε < 2. Then for each x˜ ∈ Δ2 , Lx˜ is a Lipschitz map and ˜2 Lip(Lx˜ ) ≤ D0 (x) where D0 > 0 is a number depending only on the exponents and ε .

VII.6

Related Measurable Partitions

125

Proof. Here we keep the notations introduced in Lemma VII.5.6. Put λ = λ + , −1  cu (x), K = eλ0 and δ = 2ε . Let now x˜ ∈ Δ2 . For y˜ ∈ W δ ˜ set ζy˜ = Φx˜ y0 and define   x˜ (y) ˜ = DFθ −n x˜ (Gx˜n ζy˜ ) n∈N . A     x˜ (y) x˜ (y) It is easy to see that A ˜ ∈ X. Write then ϕ A ˜ = Px,n ˜ n∈N . By Lemma ˜ (y) VII.5.6 we have also ˜ Lx˜ (y0 ) = Px,1 ˜ (y).  cu (x), Hence, for any z˜, z˜ ∈ W δ ˜     Lx˜ (z0 ) − Lx˜(z 0 ) = Px,1 z) − Px,1 z ) ˜ (˜ ˜ (˜ +∞   ≤ C ∑ α0n DFθ −n x˜ (Gx˜n ζz˜ ) − DFθ −nx˜ (Gx˜n ζz˜ )

(by Lemma VII.5.6)

n=1 +∞

  ≤ C ∑ α0n ε (θ −n x) ˜ Gx˜n ζz˜ − Gx˜n ζz˜ 

(by 2) of Proposition VII.4.2)

≤ C ∑ α0n ε (θ −n x)(e ˜ 2ε )n ζz˜ − ζz˜ 

(by 3) of Lemma VII.5.4)

n=1 +∞

n=1 +∞

≤ Cε ≤ Cε

˜ ζz˜ − ζz˜  ∑ (α0 e3ε )n (x)

n=1 +∞

∑ (α0 e3ε )n (x)˜ 2 d(z0 , z 0 )

n=1

= D0 (x) ˜ 2 d(z0 , z 0 ) where D0 = Cε /(1 − α0e3ε ) since α0 e3ε < 1.

 

VII.6 Related Measurable Partitions VII.6.1 Partitions Adapted to Lyapunov Charts In order to make use of the geometry of Lyapunov charts in the calculation of entropy, it is convenient to have partitions whose elements lie in charts. This leads to the following concept. If P is a measurable partition of M f , write P + = )+∞ n ˜ Δ 2 be a system of (ε , )-charts and 0 < δ ≤ 1. n=0 θ P. Let {Φx˜ }x∈ Definition partition P of M f is said to be adapted to the  measurable  VII.6.1 A + ˜ ⊂ Φx˜ Sδcu (x) ˜ for μ˜ -a.e. x˜ ∈ M f . charts {Φx˜ }x∈ ˜ Δ 2 , δ if pP (x)

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Our purpose in this subsection  that there always exists such a parti is to show tion P adapted to the charts {Φx˜ }x∈ ˜ Δ 2 , δ with Hμ˜ (P) < +∞. We shall use the following two preliminary lemmas. Lemma VII.6.1 If xn ∈ [0, 1] for all n ≥ 0, then +∞

+∞

n=0

n=1

− ∑ xn log xn ≤

∑ nxn + c0,

(VII.24)

1

where we admit 0 log 0 = 0 and c0 = 2[e(1 − e− 2 )]−1 . +∞ Proof. If ∑+∞ n=1 nxn = +∞, (VII.24) is obvious. Now suppose ∑n=1 nxn < +∞. Let S be the set of integers n such that xn > 0 and − log xn < n. Then +∞

− ∑ xn log xn = − ∑ xn log xn − ∑ xn log xn n=0

n∈S

n∈S

≤

∑ nxn − ∑ xn log xn . n∈S

n∈S

Note that n ∈ S implies xn ≤ e−n . On the other hand, √ 2 − t logt ≤ e for all t ∈ [0, 1]. Hence, we obtain − ∑ xn log xn ≤ n∈S

2√

∑e

n∈S

xn ≤

2 +∞ −n/2 ∑e . e n=0  

This completes the proof. Lemma VII.6.2 Let ρ : M f → (0, 1] be a measurable function with −∞. Then there exists a measurable partition P of M f such that:   (1) Diamp P(x) ˜ < ρ (x) ˜ for any x˜ ∈ M f ; (2) Hμ˜ (P) < +∞.

(

log ρ d μ˜ >

Proof. Take numbers C > 0 and r0 > 0 such that if 0 < r ≤ r0 there exists a measurable partition ξr of M which satisfies Diamξr (x) ≤ r for all x ∈ M and

|ξr | ≤ Cr−m0 ,

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127

where |ξr | denotes the number of elements in ξr . For each n ≥ 0, set   Un = x˜ ∈ M f : e−(n+1) < ρ (x) ˜ ≤ e−n . The integrability of function log ρ implies that for every N ≥ 1, N

∑ nμ˜ (Un) ≤

n=1

N

∑−



n=1

so that

Un

log ρ d μ˜ ≤ −

 Mf

log ρ d μ˜ ,

+∞

∑ nμ˜ (Un) < +∞.

(VII.25)

n=1

  Define a partition P of M f by demanding that P ≥ Un : n ≥ 0 and P|Un =  p−1 A : A ∈ ξrn :Un , where n ≥ 0 and rn = e−(n+1) . Then P is clearly a measurable partition satisfying requirement (1). We now verify that Hμ˜ (P) < +∞. In fact, we have 

Hμ˜ (P) =

+∞ 

∑

−

n=0

∑

 μ˜ (P) log μ˜ (P) .

P⊂Un

For each n ≥ 0, −

∑

μ˜ (P) log μ˜ (P)

P⊂Un

  ≤ μ˜ (Un ) log |ξrn | − log μ˜ (Un )   ≤ μ˜ (Un ) logC − m0 log rn − log μ˜ (Un ) ≤ μ˜ (Un ) logC + m0 (n + 1)μ˜ (Un ) − μ˜ (Un ) log μ˜ (Un ). Taking the sum in the above inequality over n ≥ 0, we obtain +∞

+∞

n=0

n=0

Hμ˜ (P) ≤ logC + m0 ∑ (n + 1)μ˜ (Un ) − ∑ μ˜ (Un ) log μ˜ (Un ), which, by Lemma 5.1, implies +∞

Hμ˜ (P) ≤ logC + m0 + c0 + (m0 + 1) ∑ nμ˜ (Un ). n=1

This together with (VII.25) yields (2).

 

Remark VII.3. From the proof, one can see that Lemma VII.6.2 also holds when ( f , μ ) is not ergodic.

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  Proposition VII.6.3 Let Φx˜ x∈ be a system of (ε , )-charts and 0 < δ ≤ 1. Then ˜ Δ2 there exists a measurable partition P of M f such that: (1) P is adapted to ({Φx˜ }x∈ ˜ Δ 2 , δ ); (2) Hμ˜ (P) < +∞.

   Proof. Let 0 > 1 such that the set Δ20 = x˜ ∈ Δ2 : (x) ˜ ≤ 0 has positive μ˜ measure. The ergodicity of (M f , θ , μ˜ ) implies that there is a Borel set Δ3 ⊂ M f with μ˜ (Δ3 ) = 1 such that for any x˜ ∈ Δ3 there is a k > 0 such that θ k x˜ ∈ Δ20 . For    x˜ ∈ Δ3 , put n+ (x) ˜ = min k : k > 0, θ k x˜ ∈ Δ20 . Let Δ20 ,∗ = Δ20 Δ3 and define ϕ : M f → (0, 1] by $

ϕ (x) ˜ = log ϕ is μ˜ -integrable since

(

δ, if −2 −(λ0 +ε )n+ (x) ˜ δ 0 e , if  ,∗

Δ 20

x˜ ∈ Δ20 ,∗ ; x˜ ∈ Δ20 ,∗ .

n+ (x) ˜ d μ˜ (x) ˜ = 1 ([66, Chapter 2, Theorem 4.6]).

By Lemma VII.6.2, there is a measurable partition P of M f such that Hμ˜ (P) < +∞ ˜ for all x˜ ∈ M f . and pP(x) ˜ ⊂ B(x0 , ϕ (x))   We now prove that this measurable partition P is adapted to {Φx˜ }x∈ ˜ Δ 2 , δ . In +∞ n 0 ,∗  +∞ n 0 0 ,∗  fact, it suffices to show that for all x˜ ∈ \ n=0 θ (Δ2 \Δ2 ) , n=0 θ Δ 2 0 ,∗ + −1 ¯ pP (x) ˜ ⊂ Φx˜ B(δ (x) ˜ ). First consider x˜ ∈ Δ2 . By the choice of P, we have pP + (x) ⊂ pP(x) ⊂ B(x0 , ϕ (x)), ˜ where B(x0 , ϕ (x)) ˜ is contained  ,∗ ¯ δ (x) ˜ −1 ) because ϕ (x) ˜ ≤ δ (x) ˜ −2 . Suppose now that x˜ ∈ Δ20 but in Φx˜ R( +∞ n 0 ,∗  +∞ n 0 0 ,∗  x˜ ∈ n=0 θ Δ2 \ n=0 θ (Δ2 \Δ2 ) . Let n > 0 be the smallest positive integer      ,∗ such that θ −n x˜ ∈ Δ20 . Then pP + (x) ⊂ f n pP + (θ −n x) ˜ ⊂ f n B x−n , ϕ (θ −n x) ˜ . Now   f n B x−n , ϕ (θ −n x) ˜  −n ˜  ˜ −1 e−(λ0 +ε )n+ (θ x) ⊂ Φx˜ Fθn−n x˜ B¯ δ (θ −n x)  −n ˜ λ n  ¯ δ (x) ˜ −1 enε e−(λ0 +ε )n+ (θ x) e 0 ⊂ Φx˜ R ¯ δ (x) ⊂ Φx˜ B( ˜ −1 ) ˜ Note that the computation above makes sense since for any as n ≤ n+ (θ −n x). −n ˜ ¯ δ (θ −n x) ¯ θ −n+k x) ˜ −1 e−(λ0 +ε )n+ (θ x) ) ⊂ B(( ˜ −1 e−(λ0 +ε ) ). The 1 ≤ k < n, Fθk−n x˜ R( proof is completed.  

VII.6.2 More on Increasing Partitions    Let Φx˜ x∈ be a system of (ε , )-charts. Fix 0 > 0 such that the set Δ20 = x˜ ∈ ˜ Δ 2 Δ2 : (x) ˜ ≤ 0 has positive μ˜ measure. Proposition V.V.4.4 asserts that there exist

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129

 a compact set Δc ⊂ Δ20 with μ˜ (Δc ) > 0 and a continuous family of C1 embedded u (x)} ˜ x∈ u-dimensional disks {Wloc ˜ Δ c such that the following five items hold: u (x) ˜ ⊂ Φx˜W u ˆ (x) ˜ for all x˜ ∈ Δc , where δˆ = 14 e−(λ0 +ε ) ; (i) Wloc x, ˜δ u ˜ z˜ ∈ Wloc (x), ˜ then (ii) There exist λˆ > 0 and γˆ > 0 such that for any x˜ ∈ Δc , if y, for all n ≥ 0 ˆ d u (y−n , z−n ) ≤ γˆe−nλ d u (y0 , z0 );

(iii) There exist numbers rˆ, εˆ and dˆ with 0 < rˆ < ρ2 /4, 0 < εˆ < 1 and dˆ >  def  x, 2ˆr such  that for any r ∈ (0, rˆ] and x˜ ∈ Δc , if x˜ ∈ BΔc (x, ˜ εˆ r), then ˜ εˆ r) = Δc B( u

u Wloc (x˜ ) B(x0 , r) is connected, its d -diameter is less than dˆ and the map u x˜ → Wloc (x˜ )



B(x0 , r)

is a continuous map from BΔc (x, ˜ εˆ r) to the space of subsets of B(x0 , r) (endowed with the Hausdorff topology); ˜ εˆ r), then either (iv) Let r ∈ [ˆr/2, rˆ] and x˜ ∈ Δc . If x˜ , x˜

∈ BΔc (x, u Wloc (x˜ )



u B(x0 , r) = Wloc (x˜

)



B(x0 , r)

or otherwise the two terms in the above equation are disjoint. In the latter case, if it  u (x˜ ), then is assumed moreover that x˜

∈ W d u (y0 , z0 ) > dˆ > 2ˆr 



u (x˜ ) B(x , r) and any z ∈ W u (x˜

) B(x , r); for any y0 ∈ Wloc 0 0 0 loc ˜ εˆ rˆ) and y0 ∈ (v) There exists Rˆ > 0 such that for any x˜ ∈ Δc , if x˜ ∈ BΔc (x, u (x˜ )  B(x , rˆ), then W u (x˜ ) contains the closed ball of center y and d u radius Wloc 0 0 loc Rˆ in W u (x˜ ). As in the proof of Proposition VII.2.4, we choose x˜∗ ∈ Δc such that BΔc (x˜∗ , εˆ rˆ/2) has positive μ˜ measure. For each r ∈ [ˆr /2, rˆ], put

Sr =



 u loc y˜ ∈ W (x) ˜ : y0 ∈ B(x∗0 , r) ,

x∈ ˜ BΔc (x˜∗ ,ˆε r)

  u (x) and let ξr denote the partition of M f into all the sets y˜ ∈ M f : y0 ∈ Wloc ˜  B(x∗0 , r) , x˜ ∈ BΔc (x˜∗ , εˆ r) and the set M f \Sr . We now define a measurable function βr : Sr → R+ by  '  ˆ 1 d y−n , ∂ B(x∗0 , r))enλˆ , r . βr (y) ˜ = inf R, n≥0 2γˆ γˆ By arguments completely analogous to those in the proof of Proposition VII.2.4, we know that there exists r ∈ [ˆr/2, rˆ] such that βr > 0 μ˜ -a.e. on Sr . This implies that

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VII

the partition η == ξr+ = Proposition VII.2.4. def

SRB Measures and Pesin’s Entropy Formula for Endomorphisms

)+∞

n=0 θ

nξ

r

is also a partition satisfies the requirements of

Remark VII.4. Each partition η we just constructed has the following additional characterization: Let Sˆ denote the set Sr introduced in the construction of η . If ) n ˆ f ˆ f ηˆ = +∞ ˜ if and only if y˜ ∈ ηˆ (x) ˜ n=0 θ {S, M \S}, then for every x˜ ∈ M , y˜ ∈ η (x) u −n ˆ ˆ and d (x−n , y−n ) ≤ d whenever θ x˜ ∈ S (we define d u (x−n , y−n ) = +∞ if y−n ∈ W u (θ −n x)). ˜

VII.6.3 Two Useful Partitions   Φx˜ x∈ be a system of (ε , )-charts with ε being small enough so that ˜ Δ2 −λ + +10ε 5ε e + e < 2. Let 0 , Sˆ and dˆ be as in the last subsection, η a partition of the type in Proposition VII.2.4 constructed in the last subsection. Fix 0 < δ ≤ 1 −2(λ0 +ε ) ˆ min{ 16 e , d/2K0 } and let P be a partition adapted to ({Φx˜ }x∈ ˜ Δ 2 , δ ) with ˆ M f \S} ˆ and {E, ˆ M f \E}, ˆ where Eˆ is a Hμ˜ (P) < +∞. We require that P refines {S, Borel set of positive μ˜ measure which will be specified in §VII.6.5. Define

Let

η1 = η ∨ P + , η2 = P + . These two partitions will play central roles in Section VII.7. Some of their properties are described in the following lemma. Lemma VII.6.4 (1) θ η1 ≤ η1 , θ η2 ≤ η2 ; (2) η1 ≥ η2 ; (3) pη2 (x) ˜ ⊂ Φx˜ Sδcu (x) ˜ and pη1 (x) ˜ ⊂ Φx˜Wx,˜uδ (x) ˜ for μ˜ -a.e. x; ˜ −1 −1 −1 −1 (4) hμ˜ (θ , η2 ) = hμ˜ (θ , P), h μ˜ (θ , η1 ) = hμ˜ (θ , η ). Proof. Properties (1) and (2) and the first half of (3) follow from the definitions of η1 and η2 . The second half of (3) is a consequence of Lemma VII.5.2. The first half of (4) is straightforward. We now prove the last assertion. In view of the fact that hμ˜ (θ −1 ) < +∞ and Hμ˜ (P) < +∞, by [92, Theorem 4.12] we have for every n ≥ 1 h μ˜ (θ −1 , η1 ) = hμ˜ (θ −1 , θ n η ∨ θ n P + ) = hμ˜ (θ −1 , ξ ∨ θ n P + ) = Hμ˜ (ξ ∨ θ n P + |θ ξ ∨ θ n+1 P + ) ≤ Hμ˜ (ξ |θ ξ ) + Hμ˜ (P + |θ −n η ∨ θ P + ). Hμ˜ (P + |θ −n η ∨ θ P + ) → 0 as n → +∞ since θ −n η increasingly tends to the partition into single points and Hμ˜ (P + |η ∨ θ P + ) ≤ Hμ˜ (P) < +∞. Hence

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Related Measurable Partitions

131

hμ˜ (θ −1 , η1 ) ≤ Hμ˜ (η |θ η ) = hμ˜ (θ −1 , η ). On the other hand, also by [92, Theorem 4.12], we have hμ˜ (θ −1 , η1 ) = h μ˜ (θ −1 , η ∨ P) ≥ hμ˜ (θ −1 , η ) n since Hμ˜ (η1 | ∨+∞ n=1 θ (η ∨ P)) < +∞ and Hμ˜ (η ∨ P|θ η ) < +∞. This completes the proof.  

VII.6.4 Quotient Structure Since η1 ≥ η2 , for each x˜ ∈ M f we can view η1 (x) ˜ restricted to η2 (x), ˜ written cu ˜ Reη1 |η2 (x) , as a subpartition of η ( x). ˜ Let x ˜ ∈ Δ such that η ( x) ˜ ⊂ Φ S x ˜ 2 2 2 ˜ δ (x). cu u call that for every y˜ ∈ Φx˜ Sδ (x) ˜ with y˜ ∈ Δ2 , Wx,2 ˜ is the graph of a function ˜ δ (y) u −1 c+s −1 ¯ ¯ from B (2δ (x) ˜ ) to R (4δ (x) ˜ ). The restriction of these graphs to Φx−1 ˜ ˜ η2 (x) η ( x). ˜ The next lemma says that gives, roughly speaking, a natural partition of Φx−1 2 ˜ this corresponds to η1 |η2 (x) ˜ . Lemma VII.6.5 For μ˜ -a.e. x˜ ∈ Δ2 , if y˜ ∈ η2 (x) ˜ with y˜ ∈ Δ2 and pη (y) ˜ ⊂ W u (y), ˜ then  u Φx˜Wx,2 ˜ pη2 (x) ˜ = pη1 (y). ˜ ˜ δ (y) u Proof. First consider z˜ ∈ η2 (x) ˜ with z0 ∈ Φx˜Wx,2 ˜ We shall prove that z0 ∈ ˜ δ (y). f + ˆ M \S} ˆ and z˜ ∈ P (y), ˜ Since P refines {S, ˜ in view of Remark VII.4, it pη (y). ˆ This is in fact true for all suffices to show that d u (y−n , z−n ) ≤ dˆ whenever θ −n y˜ ∈ S. n ≥ 0, since by Lemma VII.5.3 3) and Lemma VII.5.4 1) one has for all n ≥ 0

  −n −1 H Φ y0 − H −nΦ −1 z0  x˜ x˜ x˜ x˜   + −1  ≤ e−(λ −2ε )n Φx−1 ˜ y0 − Φx˜ z0   ≤ Φ −1 y0 − Φ −1 z0  x˜

x˜

≤ 2δ (x) ˜ −1  u which implies d u (y−n , z−n ) ≤ K0 2δ (x) ˜ −1 ≤ d.ˆ Thus Φx˜Wx,2 ˜ pη2 (x) ˜ ⊂ ˜ δ (y) pη1 (y). ˜ The reverse relation follows from pη (y) ˜ ⊂ W u (y) ˜ and Lemma VII.5.3 3). ˜ we complete the proof.   Noting that the above argument holds for μ˜ -a.e. x,  This lemma allows us to regard the factor-space p η2 (x)/( ˜ η1 |η2 (x) ) , or writ˜   ˜ η1 , as a subset of Rkc +ks via the correspondence pη1 (y) ˜ ↔ ten simply p η2 (x)/  u k +k c s Wx,2 ˜ ({0} × R ). If we identify pη1 (x) ˜ and η2 (x) ˜ with η1 (x) ˜ and η2 (x) ˜ re˜ δ (y) −1 (η (x)) spectively, the next lemma tells then that θ |θ −1 (η2 (x)) : θ ˜ → η ( x) ˜ acts like 2 2 ˜ a skew product with respect to the above quotient structure.

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Lemma VII.6.6 For μ˜ -a.e. x˜ ∈ Δ2 , if y˜ ∈ η2 (x) ˜ with y˜ ∈ Δ2 , pη (y) ˜ ⊂ W u (y) ˜ and −1 u −1 ˜ ⊂ W (θ y), ˜ then we have pη (θ y)

θ −1 (η1 (y)) ˜ = η1 (θ −1 (y)) ˜



θ −1 (η2 (x)). ˜

Proof. From the definitions of η1 and η2 it follows clearly that θ −1 (η1 (y)) ˜ ⊂  −1 −1 η1 (θ (y)) ˜ θ (η2 (x)) ˜ for every x˜ ∈ M f and any y˜ ∈ η2 (x). ˜ On the other hand, if both x˜ and θ −1 x˜ meet the requirement of Lemma VII.6.5 and y˜ is a point in η2 (x) ˜ such that y˜ ∈ Δ2 , pη (y) ˜ ⊂ W u (y) ˜ and pη (θ −1 y) ˜ ⊂ W u (θ −1 y), ˜ then the reverse relation follows from Lemma VII.6.5 and Lemma VII.5.3 2).  

VII.6.5 Transverse Metrics As we have mentioned at the beginning of this chapter, the first main point for the proof of necessity of the entropy formula is to prove that the entropy hμ ( f ) is determined by actions of f on the W u -manifolds of f , or more precisely, to prove that hμ˜ (θ ) = Hμ˜ (η |θ η ) where η is a certain increasing partition satisfying Proposition VII.2.4. In order to use the fact that all the expansion of f occurs along the W u manifolds to prove this assertion, we need to show that the action induced by θ on (θ −1 (η2 (x)))/ ˜ η1 → η2 (x)/ ˜ η1 does not expand distances. For this purpose we define in this subsection a metric on the factor-space η2 (x)/ ˜ η1 for μ˜ -a.e.x. ˜ This will be referred to as a transverse metric. We shall actually deal with η1 and η2 restricted to a certain measurable set of full μ˜ measure. Now we choose a measurable set Δ3 ⊂ Δ2 with μ˜ (Δ3 ) = 1 and θ Δ3 = Δ3 such that for each x˜ ∈ Δ3 , pη (x) ˜ ⊂ W u (x), ˜ pη2 (x) ˜ ⊂ Φx˜ Sδcu (x) ˜ and the requirements of Lemmas VII.6.5 and VII.6.6 are satisfied. We then put

η1 = η1 |Δ3 ,

η2 = η2 |Δ3 .

˜ η1 for μ˜ -a.e. x˜ ∈ Δ3 . In what follows we define a transverse metric on η2 (x)/ First we give a point-dependent definition. Let x˜ ∈ Δ3 . From Lemma VII.5.3 we u ˜ Wx,2 ˜ intersects {0} × Rkc +ks at exactly one point. know that for every y˜ ∈ η2 (x), ˜ δ (y)

We denote this point by ζy . For y, ˜ y˜ ∈ η2 (x), ˜ define ˜ y˜ ) = ζy − ζy . dx˜ (y, ˜ η1 , but in general, dx˜ ( , ) = By Lemma VII.6.5, dx˜ ( , ) induces a metric on η2 (x)/



dx˜ ( , ) for x˜ ∈ η2 (x) ˜ with x˜ = x. ˜ Now we need to rectify this situation to give a point-independent definition. To this end we shall first specify Eˆ (see Subsection VII.6.3) and then choose a reference plane T and standardize all measurements with respect   to T . By Proposition V.V.4.4, ˜ ≤ 0 with μ˜ (Δc ) > 0 and satisfying there exists a compact set Δc ⊂ x˜ ∈ Δ2 : (x) the following two requirements:

VII.6

Related Measurable Partitions

133

def

(i) Exu˜ and Exc+s = Exc˜ ⊕ Exs˜ depend continuously on x˜ ∈ Δc ; ˜ (ii) There exists a number 0 < α0 < ρ2 /8 and for each x˜ ∈ Δc there exists a C1 map   hx˜ : ξ ∈ Exu˜ : |ξ | < α0 → Exc+s ˜ def

such that hx˜ (0) = 0, Lip(hx˜ ) ≤ 1/3, Dx˜ = expx0 Graph(hx˜ ) ⊂ W u (x) ˜ and   1 is a continuous family of C embedded disks of dimension u. Dx˜ x∈ ˜ Δc  u ˜ ρ) = ξ ∈ For x˜ ∈ Δcand ρ > 0, we put Ex˜ (ρ ) = Exu˜ (ρ ) × Exc+s ˜ (ρ ) where E (x)( c+s Exu˜ : |ξ | < ρ and Exc+s : |η | < ρ . From the compactness of Δc ˜ (ρ ) = η ∈ Ex˜ it follows that there are positive numbers t0 and r with t0 ≤ α0 /2 such that for each ˜ r) the following (a) and (b) hold: x˜ ∈ Δc and any x˜ ∈ BΔc (x, ¯ δ (x) (a) expx0 Ex˜ (t0 ) ⊂ Φx˜ R( ˜ −1 ); ◦ expx0 : Ex˜ (α0 ) → Tx M is well defined with Lip(Ix˜ ,x˜ ) ≤ (b) The map Ix˜ ,x˜ = exp−1 x

def

0

0

x ∈ Ex˜ (t0 ) and exp−1 Dx˜ intersects {0} × Exc+s at exactly one point. 2. exp−1 ˜

x 0 0 x 0 Moreover,  Dx˜ Ex˜ (t0 ) = Graph(hx˜ ,x˜ ) exp−1 x

0

where hx˜ ,x˜ : Exu˜ (t0 ) → Exc+s ˜ (t0 ) is a C1 map with Lip(hx˜ ,x˜ ) ≤ 1/2, and Ix˜ ,x˜ ({0} × Exc+s ˜ (2t0 ))



Ex˜ (t0 ) = Graph(gx˜ ,x˜ )

u 1

where gx˜ ,x˜ : Exc+s ˜ (t0 ) → Ex˜ (t0 ) is a C map with Lip(gx˜ ,x˜ ) ≤ 1/100.

Choose now x˜∗ ∈ Δc such that BΔc (x˜∗ , r/2) has positive μ˜ measure. Then we define   Eˆ = BΔc (x˜∗ , r/2), T = expx0 {0} × Exc+s ˜∗ (2t0 ) .   ˆ M f \Eˆ (see With Eˆ and T thus specified, noting that P is required to refine E,  n ˆ

Subsection VII.6.3), we define now a metric on η2 (x)/ ˜ η1 for every x˜ ∈ +∞ n=0 θ E where  Eˆ = Eˆ Δ3 . c+s and define a function First take an isomorphism Ix˜∗ : Exc+s ˜∗ → R

p:

+∞

θ n Eˆ → Rkc +ks

n=0

as follows: for x˜ ∈ Eˆ , let p(x) ˜ = (Ix˜∗ ◦ exp−1 x0 ){T



Dx˜ }

134

VII

and in general, let

SRB Measures and Pesin’s Entropy Formula for Endomorphisms ˜ x) ˜ p(x) ˜ = p(θ −n(x)

˜ x˜ ∈ Eˆ . Then define where n(x) ˜ isthe smallest nonnegative integer such that θ −n(x) n ˆ

for each x˜ ∈ +∞ θ E n=0 dxT˜ (y, ˜ y˜ ) = p(y) ˜ − p(y˜ )

if y, ˜ y˜ ∈ η2 (x), ˜ where  ·  denotes the usual Euclidean distance. We now explain why dxT˜ ( , ) induces a metric on η2 (x)/ ˜ η1 . +∞ n

ˆ M f \E} ˆ and η2 = P + |Δ , for every n ≥ 0 Let x˜ ∈ n=0 θ Eˆ . Since P ≥ {E, 3 

−n



−n

Eˆ = 0. ˜ ⊂ Eˆ or θ (η2 (x)) ˜ / Moreover, when θ −n x˜ ∈ Eˆ , for either θ (η2 (x)) each y˜ ∈ η2 (x) ˜ one can inductively prove by using Lemma VII.6.6

θ −n (η1 (y)) ˜ = η1 (θ −n y) ˜



θ −n (η2 (x)) ˜

and, using Lemma VII.6.5, one can easily obtain

η1 (θ −n y) ˜ = p−1 Dθ −n y˜ hence

θ −n (η1 (y)) ˜ = p−1 Dθ −n y˜





η2 (θ −n x), ˜ θ −n (η2 (x)). ˜

This guarantees that dxT˜ ( , ) induces a genuine metric on η2 (x)/ ˜ η1 and that for any x˜ ∈ η2 (x), ˜ dxT˜ ( , ) = dxT˜ ( , ). Lemma VII.6.7 Let Eˆ and T be as introduced above. Then there is a number N = N(0 ) such that for all x˜ ∈ Eˆ , 1 dx˜ ( , ) ≤ dxT˜ ( , ) ≤ Ndx˜ ( , ). N Proof. Let x˜ ∈ Eˆ . We define the Poincar´e map    θ : ({0} × Exc+s Graph(hx,˜ x˜ ) : x˜ ∈ η2 (x) ˜ → exp−1 x0 T ˜ )

by sliding along Graph(hx,˜ x˜ ). Lemma VII.5.7 tells us that there is a number D = D (λ0 , K0 , λ + , ε ) such that   ˜2 max Lip(θ ), Lip(θ −1 ) ≤ D (x) where θ −1 is understood to be defined on the image of θ . Thus, if y, ˜ y˜ ∈ η2 (x) ˜ and ζy˜ and ζy˜ are respectively the points of intersection of Graph(hx,˜ y˜ ) and Graph(hx,˜ y˜ ) with exp−1 x0 T , then dexp−1 (ζy˜ , ζy˜ ) ≤ K0 b(ρ2 /2)D 20 dx˜ (y, ˜ y˜ ) x T 0

VII.7

Some Consequences of Besicovitch’s Covering Theorem

135

where dexp−1 ( , ) is the distance along the submanifold exp−1 x0 T . Therefore we have x T dxT˜ (y, ˜ y˜ )

= p(y) ˜ − p(y˜ ) = |Ix˜∗ ,x˜ ζy˜ − Ix˜∗ ,x˜ ζy˜ | ≤ 2K0 b(ρ2 /2)D 20 dx˜ (y, ˜ y˜ ).

The other inequality is proved similarly.   As is evident from the proof, the number N depends only on the charts and on 0 . It is independent of η1 and η2 , or the choice of Eˆ and T (provided of course that everything is as described before). Finally, what we have done in the last two subsections is, roughly speaking, to present M f /η1 as a subset of M f /η2 × Rkc +ks , and to define transverse metrics on η2 (x)/ ˜ η1 that correspond to the Euclidean distance on Rkc +ks . This Euclidean space geometry plays a role in some of the averaging arguments in the next section.

VII.7 Some Consequences of Besicovitch’s Covering Theorem ¯ r) For x ∈ Rn and r > 0, let B(x, r) denote the ball of radius r centered at x and let B(x, denote the correponding closed ball. All distances are the usual Euclidean ones in this section. The following Besicovitch’s covering theorem (BCT) (see [23]) is a valuable tool in the theory of differentiation and in many other fields of analysis. Theorem VII.7.1 (Besicovitch’s Covering Theorem) Let A be a bounded subset of ¯ Rn . For each  x ∈ A a closed ball B(x, r(x)) with center x and radius r(x) is given. ¯ Set A = B(x, r(x)) x∈A . Then there exists a subset A of A such that A covers A and no point in Rn lies in more than c(n) elements of A , where c(n) depends on n only. Remark VII.5. If in BCT subset A is not bounded but   sup r(x) : x ∈ A = R < +∞, the above covering theorem is still valid with the constant c(n) changed  conveniently. To show this, itis sufficient to partition Rn into disjoint sets Ai = x ∈ Rn : 3iR ≤ x < 3(i + 1)R , i ∈ Z+ and apply BCT to the intersection of A with each one of these sets Ai . In what follows we derive some useful lemmas from BCT. Now let m be a Borel probability measure on Rn . The next two lemmas are elementary if m is replaced by the Lebesgue measure. When working with arbitrary finite Borel measures, we use BCT instead of Vitali’s covering theorem. Let g ∈ L1 (Rn , m) and define for any δ >0

136

VII

SRB Measures and Pesin’s Entropy Formula for Endomorphisms

gδ (x) =

1 m(B(x, δ ))

 B(x,δ )

g dm,

for each x ∈ Rn (we admit here that 0/0 = 1). If g ≥ 0 m-a.e., we then further define g∗ = sup gδ δ >0

and g∗ = inf gδ . δ >0

  ∗ g∗ and  g∗ are Borelmeasurable functions since for each t ∈ R the sets x : g (x) > t and x : g∗ (x) < t are open. Lemma VII.7.2 (1) For any t ≥ 0, m({g∗ > t}) ≤

c(n) t



g dm;

(2) Let ν be defined by d ν = gdm. Then for any t ≥ 0,

ν ({g∗ < t}) ≤ c(n)t. 

Proof. Let A = {g∗ > t} C where C is an arbitrarily fixed bounded Borel set. For each x ∈ A we can choose δ (x) such that  ¯ δ (x)) B(x,

  ¯ δ (x)) . g dm > tm B(x,

  ¯ δ (x)) and choosing A as in BCT, we have Letting A = B(x, x∈A 1 m(A) ≤ ∑ m(B) ≤ ∑ t B∈A

B∈A



c(n) g dm ≤ t B

 Rn

g dm.

Then (1) follows since C is chosen arbitrarily. (2) can be obtained similarly.

 

Lemma VII.7.3 Let g ∈ L1 (Rn , m). Then gδ (x) → g as δ → 0 for m-a.e. x ∈ Rn . Proof. It is sufficient to verify that for each t > 0, the m-measure of the set    At = x ∈ Rn : lim sup δ →0

1 m(B(x, δ ))

 B(x,δ )

  g dm − g(x) > t

is zero. We prove m(At ) = 0 in the following way. Given ε > 0, we take a continuous function f such that h = g − f satisfies h1 ≤ ε where  · 1 is the L1 -norm in L1 (Rn , m). For f we have obviously at m-a.e. x ∈ Rn

VII.7

Some Consequences of Besicovitch’s Covering Theorem

1 δ →0 m(B(x, δ ))



lim

137

f dm = f (x),

B(x,δ )

and so m(At ) = m(Bt ) where    Bt = x ∈ Rn : lim sup δ →0

Notice that Bt ⊂ Bt1

1 m(B(x, δ ))

 B(x,δ )

  h dm − h(x) > t .

 2 Bt where

   Bt1 = x ∈ Rn : lim sup δ →0

and

1 m(B(x, δ ))



 t  hdm > 2 B(x,δ )

 t Bt2 = x ∈ Rn : |h(x)| > . 2

Since for each x ∈ Rn   lim sup δ →0

1 m(B(x, δ ))

 B(x,δ )

  h dm ≤ |h|∗ (x)

where |h|∗ is defined analogously to g∗ , by Lemma VII.7.2 (1) we have m(Bt1 ) ≤ m

 t  2c(n) 2c(n) |h|∗ > ε. ≤ h1 ≤ 2 t t

Also

2 2 m(Bt2 ) ≤ h1 ≤ ε . t t Since ε > 0 is arbitrary, we obtain m(At ) = 0.   The next lemma (see [42]) is usually stated in a slightly different way in the literature. For geometric reasons we take average over balls instead of taking conditional expectations with respect to fixed partitions. Lemma VII.7.4 Suppose that (X, B, m) is a Lebesgue space and π : X → Rn is a measurable map. Let {mξ }ξ ∈Rn be a canonical system of conditional measures of   m associated with the partition π −1 {ξ } ξ ∈Rn . Let α be a measurable partition of X with Hm (α ) < +∞. For ξ ∈ Rn and A ∈ α , define gA (ξ ) = mξ (A). Let gAδ and gA∗ be functions on Rn defined as above. Define g, gδ and g∗ : X → R by g(x) =

∑ χA(x)gA (π x),

A∈α

138

VII

SRB Measures and Pesin’s Entropy Formula for Endomorphisms

gδ (x) = g∗ (x) =

∑ χA(x)gAδ (π x),

A∈α

∑ χA(x)gA∗ (π x).

A∈α

Then gδ → g m-a.e. on X and 

− log g∗ dm ≤ Hm (α ) + log c(n) + 1,

where c(n) is as in BCT. Proof. First by Lemma VII.7.3 we have gAδ (ξ ) → gA (ξ ) as δ → 0 for π m-a.e. ξ ∈ Rn for each A ∈ α and hence gδ → g m-a.e. on X since Hm (α ) < +∞. Note also that the function h : R+ → R+ , s → m({− log g∗ > s}) is continuous almost everywhere (in the sense of Lebesgue) and hence is Riemannian integrable on any interval [0, b], b > 0. From this it follows that 

− log g∗ dm = =

Now for each A ∈ α

 +∞ 0

 +∞ 0

m({− logg∗ > s}) ds

∑m

A∈α

  A  A {g∗ ◦ π < e−s } ds.

   m A {gA∗ ◦ π < e−s } ≤ m(A)

and    m A {gA∗ ◦ π < e−s } = = =

 

χA χ{gA∗ ◦π <e−s } dm

  Em χA χ{gA∗ ◦π <e−s } |B({π −1 {ξ }}ξ ∈Rn ) dm

   gA ◦ π χ{gA ◦π <e−s } dm ∗

≤ c(n)e−s , where the last estimate follows from Lemma VII.7.2 (2). Thus by a simple calculation, there follows 

− log g∗ dm ≤

∑

 +∞

  min m(A), c(n)e−s ds

A∈α 0

≤ Hm (α ) + logc(n) + 1.   Another consequence of BCT is the following result.

VII.8

The Main Proposition

139

Lemma VII.7.5 Let m be a finite Borel measure on Rn . Then inf

0<ε ≤1

In particular, lim sup ε →0

m(B(x, ε )) >0 εn

for m-a.e. x ∈ Rn .

log m(B(x, ε )) ≤n log ε

for m-a.e. x ∈ Rn .

Proof. Let N be a positive integer. Define  m(B(x, ε )) AN = x ∈ Rn : x ≤ N and inf = 0 . 0<ε ≤1 εn It is sufficient to prove that m(AN ) = 0. Let δ > 0 be given arbitrarily. For any ¯ ε (x)) ≤ δ ε (x)n . Setting x ∈ A N there is anumber ε (x) ∈ (0, 1] such that m B(x,

¯ ε (x)) A = B(x, and choosing A ⊂ A as in BCT, we have x∈A N

m(AN ) ≤ ≤

∑

m(B)

B∈A

∑ δ nnλ (B)

B∈A

¯ N + 1)), ≤ δ nn c(n)λ (B(0, where λ is the Lebesgue measure on Rn . Since δ is arbitrary, letting δ → 0 yields m(AN ) = 0.  

VII.8 The Main Proposition Using the machinary developed in Sections 4–7 we can now complete the first step of the proof of the necessity of the entropy formula, i.e. we can now prove that hμ˜ (θ −1 ) is equal to the entropy of θ −1 with respect to certain partition subordinate to W u -manifolds of f . Proposition VII.8.1 Let f be a C2 endomorphism on M with an ergodic invariant Borel probability measure μ . Then for any β > 0, there exists a measurable partition ηβ of M f and of the type as constructed in Subsection VII.6.2 such that

β (c + s) ≥ (1 − β )[hμ˜ (θ −1 ) − hμ˜ (θ −1 , ηβ ) − β ]. Proof. The strategy is to construct ηβ as in Subsection VII.6.2 and to use it to construct η1 and η2 as in Subsection VII.6.2 with hμ˜ (θ −1 , η2 ) ≥ hμ˜ (θ −1 ) − β /3.

140

VII



SRB Measures and Pesin’s Entropy Formula for Endomorphisms

 and μ˜ η2 (x) be respectively (μ˜ -mod 0 unique) canonical systems Let μ˜ η1 (x) ˜ ˜ ˜ μ of conditionalmeasures of   2  associated with η1 and η2 and denote them re1 and μ ˜ x˜ for simplicity ˜ of notations. We shall prove that, if spectively by μ x˜   BT (x, ˜ ρ ) = y˜ ∈ η2 (x) ˜ : dxT˜ (x, ˜ y) ˜ < ρ , then 

β · lim sup ρ →0



 log μ˜ x2˜ BT (x, ˜ ρ) 2β  ≥ (1 − β ) hμ˜ (θ −1 , η2 ) − hμ˜ (θ −1 , η1 ) − log ρ 3

for μ˜ -a.e. x. ˜ The desired conclusion then follows immediately from this and Lemmas VII.6.4 and VII.7.5. We divide the proof into five parts. (A) We start by enumerating the specifications on ηβ , η1 and η2 . First fix + ε > 0 such that ε < min{β /3, λ +/100m0, −λ − /100m0} and e−λ +10ε + e5ε < 2 (see Lemma VII.5.7). Let {Φx˜ }x∈ ˜ Δ 2 be a system of (ε , )-charts as described in Section 4. Using these charts, we construct an increasing measurable partition ηβ as the same meaning in that subin the Subsection VII.6.2 0 , Sˆ and dˆ having   1 with −( λ + ε ) )2 , d/2K ˆ 0 section. Let δ0 = min ( 4 e 0 . Choose Eˆ and T as in Subsection ˆ VII.6.5. We assume that e−β ε N 4μ˜ (E) < 1 where N = N(0 ) is the number introduced P of M f adapted  in LemmaVII.6.7. Now we take a measurable partition f ˆ ˆ ˆ M f \E} ˆ and to {Φx˜ }x∈ ˜ Δ 2 , δ0 such that Hμ˜ (P) < +∞, P ≥ {S, M \S} ∨ {E, −1 −1 + + hμ˜ (θ , P) ≥ h μ˜ (θ ) − ε . Then we set η1 = ηβ ∨ P and η2 = P . Let Δ3 be a set as chosen in Subsection VII.6.5. Recalling that μ˜ (Δ3 ) = 1 and θ Δ3 Δ3 , for the sake of simplicity of presentation we may assume that Δ3 = M f since otherwise the discussions below also apply to the system θ : (Δ3 , μ˜ ) → (Δ3 , μ˜ ) and lead to the ˜ η1 has then a nice quotient same conclusion. With η1 and η2 so constructed, η2 (x)/ structure endowed with a transverse metric dxT˜ ( , ) for μ˜ -a.e. x. ˜ (B) Before proceeding with the main argument, we record some estimates derived from the results of Section 7. For δ > 0, define g, gδ and g∗ : M f → R by g(y) ˜ = μ˜ y1˜ (θ −1 η2 )(y), ˜  1 gδ (y) ˜ = 2 T μ˜ 1 (θ −1 η2 )(y) ˜ d μ˜ y2˜ (˜z) μ˜ y˜ B (y, ˜ δ ) BT (y,˜ δ ) z˜ g∗ (y) ˜ = inf gδ (y) ˜ δ ∈Q

  ˜ is also where Q = e−β l N 2 j : l, j ∈ Z+ . By Lemma VII.6.6 we know that g(y) 1 −1 equal to μ˜ y˜ (θ η1 )(y) ˜ for μ˜ -a.e. y. ˜ For each δ > 0, one can check that the functions T ˜ δ ) are measurable and μ ˜ y2˜ BT (y, y˜ → μ˜ y2˜ BT (y, ˜ δ ) and y˜ → μ˜ (θ −1 η2 )(y) ˜ δ ) > 0 for ˜ B (y, −1 2 −1 μ˜ -a.e. y. ˜ Since Hμ˜ (θ η2 |η2 ) < +∞, for μ˜ -a.e. y˜ one has μ˜ y˜ ((θ η2 )(y)) ˜ > 0 and hence 

μ˜ y2˜ (BT (y, ˜ δ ) (θ −1 η2 )y) ˜ gδ (y) ˜ = 2 T μ˜ y˜ B (y, ˜ δ)

VII.8

The Main Proposition

141

=

T ˜ δ) μ˜ (θ −1 η2 )(y) ˜ B (y,

μ˜ y2˜ BT (y, ˜ δ)

˜ · μ˜ y2˜ (θ −1 η2 )(y).

Therefore gδ is measurable for each fixed δ > 0. The measurability of g∗ is obvious. f ˜ ( We claim that gδ → g μ -a.e. on M when δ ∈ Q and δ → 0 and that − log g∗ d μ˜ < +∞. To see this, first consider one element of η2 at a time. Fix x. ˜ ˜ μ˜ x2˜ ) for (X, m) in Lemma VII.7.4, let p : η2 (x) ˜ → Rkc +ks be the Substitute (η2 (x), p defined in Subsection VII.6.5 and let α = (θ −1 η2 )|η2 (x) ˜ . Then we can conclude that gδ → g μ˜ x2˜ -a.e. as δ ∈ Q and δ → 0 and that 

− log g∗ d μ˜ (2x) ˜ ≤



− log( inf gδ y) ˜ d μ˜ (2x) ˜ ˜ (y) δ >0

≤ Hμ˜ 2 (θ −1 η2 ) + log c(n) + 1. x˜

Integrating over M f , this gives 

− log g∗ d μ˜ ≤ Hμ˜ (θ −1 η2 |η2 ) + logc(n) + 1 < +∞.

(C) In this step we study the induced action of θ on θ −1 (η2 (x))/ ˜ η1 → η2 (x)/ ˜ η1 with respect to the metrics dθT−1 x˜ ( , ) and dxT˜ ( , ). Consider x˜ ∈ M f . The point x˜ will be subjected to a finite number of a.e. assumptions. Let r0 < r1 < r2 < · · · be the ˜ successive times t when θ t x˜ ∈ Eˆ with r0 ≤ 0 < r1 . Note that r0 is constant on η2 (x). For large n and 0 ≤ k < n, define a(x, ˜ k) as follows: If r j ≤ k < r j+1 , then ˜ e−β (n−r j ) N 2 j ). a(x, ˜ k) = BT (θ k x, We now claim that a(x, ˜ k)



(θ −1 η2 )(θ k x) ˜ ⊂ θ −1 a(x, ˜ k + 1).

(VII.26)



˜ k) η2 (θ k+1 x) ˜ = a(x, ˜ k + 1) In fact, if k = r j − 1 for any j, then we have θ a(x, T T ˆ The automatically since dθ k x˜ ( , ) and dθ k+1 x˜ ( , ) are defined by pulling back to E. case when k = r j − 1 for some j reduces to the following consideration : Let y˜ ∈ Eˆ ˆ Let z˜ ∈ (θ −r η2 )(y). ˜ It suffices and let r > 0 be the smallest integer such that θ r y˜ ∈ E. to show that ˜ θ r z˜) ≤ N 2 erβ dyT˜ (y, ˜ z˜). dθTr y˜ (θ r y, ˜ z˜) ≤ NdyT˜ (y, ˜ z˜) (for the definition of dy˜ ( , ) see Subsection First we have dy˜ (y, ˜ θ i z˜) ≤ VII.6.5). Then for i = 1, 2, · · · , r, Lemma VII.5.5 tells us that dθ i y˜ (θ i y, eβ i dy˜ (y, ˜ z˜). We pick up another factor of N when converting back to the d T -metric at θ r y. ˜ What we claimed above is thus proved. (D) It is easy to see that there exists a Borel set  ⊂ M f with μ˜ () = 1 and θ  =  such that, if x˜ ∈ , then μ˜ x2˜ BT (x, ˜ δ ) > 0 for all δ ∈ Q. We now estimate

142

VII

SRB Measures and Pesin’s Entropy Formula for Endomorphisms

˜ )=μ ˜ x2˜ a(x, μ˜ x2˜ BT (x, ˜ e−β (n−r0(x)) ˜ 0) for x˜ ∈  which will be subjected to a finite number of a.e. assumptions. Write

μ˜ x2˜ a(x, ˜ 0) =

p−1

∏ μ˜ 2

k=0

μ˜ θ2k x˜ a(x, ˜ k)

θ k+1 x˜

a(x, ˜ k + 1)

˜ p) · μ˜ θ2 p x˜ a(x,

where p = [n(1 − ε )]. First note that the last term ≤ 1. For each 0 ≤ k < p,

μ˜ θ2k x˜ a(x, ˜ k)

μ˜ θ2k+1 x˜ a(x, ˜ k + 1)

˜ k) · = μ˜ θ2k x˜ a(x,

μ˜ θ2k x˜ θ −1 (η2 (θ k+1 x)) ˜ 2 −1 μ˜ θ k x˜ θ (a(x, ˜ k + 1))

by the θ -invariance of μ˜ and by uniqueness of conditional measures. This is ≤

μ˜ θ2k x˜ a(x, ˜ k)



μ˜ θ2k x˜ ((θ −1 η2 )(θ k x) ˜ a(x, ˜ k))

˜ · μ˜ θ2k x˜ (θ −1 η2 )(θ k x)

(VII.27)

by (VII.26). If gδ is defined as in (B), the first quotient in (VII.27) is equal to [gδ (x,n,k) (θ k x)] ˜ −1 ˜ where

˜ δ (x, ˜ n, k) = e−β (n−r j (x)) N2 j

and

ˆ j = #{0 < i ≤ k : θ i x˜ ∈ E}.

˜ Then the second term in (VII.27) is equal to Write I(x) ˜ = − log μ˜ x2˜ (θ −1 η2 )(x). k ˜ . Thus e−I(θ x) ˜ log μ˜ x2˜ BT (x, ˜ e−β (n−r0 (x)) ) p−1

p−1

k=0

k=0

(θ k x) ˜ − ≤ − ∑ log gδ (x,n,k) ˜

˜ ∑ I(θ k x).

Multiplying by − 1n and taking liminf on both sides of this inequality, we have

β · lim sup ρ →0

log μ˜ x2˜ BT (x, ˜ ρ) log ρ

≥ β · lim inf n→+∞

≥ lim inf n→+∞

˜ ) log μ˜ x2˜ BT (x, ˜ e−β (n−r0(x)) log e−β n

1 [n(1−ε )] 1 [n(1−ε )] k log g ( θ x) ˜ + lim ˜ ∑ δ (x,n,k) ˜ ∑ I(θ k x). n→+∞ n n n=0 n=0

VII.8

The Main Proposition

143

The last limit= (1 − ε )Hμ˜ (θ −1 η2 |η2 ) ≥ (1 − ε )(hμ˜ (θ −1 ) − ε ). Thus Proposition VII.8.1 is proved if we show that lim sup − n→+∞

1 [n(1−ε )] (θ k x) ˜ ≤ (1 − ε )(hμ˜ (θ −1 , η1 ) + 2ε ). ∑ log gδ (x,n,k) ˜ n n=0

(VII.28)

(E) We now prove this last assertion (VII.28). It follows from (B) that there is a for μ˜ -a.e.x, ˜ if δ ∈ Q and δ ≤ δ (x), ˜ then measurable function δ : M f → R+ such that ( ˜ − log gδ (x) ˜ ≤ − log g(x) ˜ + ε . Also, since − log g d μ < +∞, there is a number δ1 ∗ ( such that if A = {x˜ : δ (x) ˜ > δ1 } then M f \A − log g∗ d μ˜ ≤ ε . We claim that for μ˜ -a.e. x, ˜ if n is sufficiently large, then δ (x, ˜ n, k) ≤ δ1 for all k ≤ n(1 − ε ). First, by Birkhoff ergodic theorem, there is a positive integer N(x) ˜ ˆ ≤ 2nμ˜ (E). ˆ If n ≥ N(x), such that for n ≥ N(x), ˜ #{i : 0 ≤ i < n, θ i x˜ ∈ E} ˜ then for each k ≤ n(1 − ε ) ˜ δ (x, ˜ n, k) = e−β (n−r j (x)) N2 j

≤ e−β ε n N 2·2nμ˜ (E) . ˆ

˜ n, k) is less than δ1 for sufficiently large n. Thus Since e−β ε N 4μ˜ (E) < 1, δ (x, ˆ

[n(1−ε )]

∑

− log gδ (x,n,k) (θ k x) ˜ ˜

n=0

≤

[n(1−ε )]

[n(1−ε )]

k=0 ˜ θ k x∈A

k=0 θ k x˜∈A

∑

∑

(− log g(θ k x) ˜ + ε) +

− log g∗ (θ k x) ˜

and the limsup we wish to estimate in (VII.28) is bounded above by (1 − ε )

,

− log gd μ˜ + ε +

 M f \A

− log g∗ d μ˜ .

˜ for μ˜ -a.e. x, ˜ we have Recalling that g(x) ˜ = μ˜ x1˜ (θ −1 η1 )(x) 

− log g d μ˜ = h μ˜ (θ −1 , η1 ).

The right hand side n the above identity is equal to hμ˜ (θ −1 , ηβ ) by Lemma VII.6.4. This completes the proof.   Corollary VII.8.1.1 Let f be given with μ being ergodic. Then for any partition η of the type as constructed in the Subsection VII.6.2, we have h μ˜ (θ −1 , η ) = h μ˜ (θ −1 ).

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Proof. For any β > 0, by Lemma VII.2.7 we have hμ˜ (θ −1 , η ) = hμ˜ (θ −1 , ηβ ) where ηβ is as in Proposition VII.8.1. Letting β → 0, we obtain the desired conclusion.  

VII.9 Proof of the Necessity for the Entropy Formula In this section we complete the proof of the second part of Theorem VII.1.1.

VII.9.1 The Ergodic Case Without loss of generality, we assume that u > 0, since the result is completely trivial if u = 0. Let η be an increasing partition of M f of the type in Proposition VII.2.4 con+ structed in §VII.6.2 with the associated parameter ε satisfying e−λ +10ε + e5ε < 2. By Corollary VII.8.1.1, hμ˜ (θ −1 , η ) = Hμ˜ (η |θ η ) = hμ˜ (θ −1 ) = hμ˜ (θ ).   Let μ˜ xη˜ be the canonical system of conditional measures of μ˜ associated with the partition η and let λx˜u be the Lebesgue measure on Wx˜ induced by its inherited Riemannian structure as a submanifold of M. We shall prove that the identity Hμ˜ (η |θ η ) = implies that

μ˜ xη˜  λ˜ x˜η

∑ λ i mi

λi >0

μ˜ -a.e. x˜ ∈ M f ,

for

η

where λ˜ x˜ is defined by (VII.10).   ˜ = det(T f | u ) for μ˜ -a.e. x˜ ∈ M f . The idea of the proof is as follows. Put J u (x) ( x0 u Ex˜ Then by Oseledec multiplicative ergodic theorem, log J d μ˜ = ∑λi >0 λi mi . Supη η η η ˜ μ˜ x˜  λ˜ x˜ . Then d μ˜ x˜ = ρ d λ˜ x˜ , μ˜ -a.e. x˜ for some pose we know that for μ˜ -a.e. x, f + function ρ : M → R . This function must satisfy for μ˜ -a.e. x˜  η (x) ˜

ρ (y) ˜ d λ˜ x˜η (y) ˜ =1

and on η (x) ˜

ρ (y) ˜ =

1

μ˜ xη˜ ((θ −1 η )(x)) ˜

·

ρ (θ −1 (y)) ˜ , J u (θ −1 (y)) ˜

λ˜ x˜η -a.e. y˜

VII.9

Proof of the Necessity for the Entropy Formula

145

by the formula for change of variables (see the proof of Claim 3.1). From this one ˜ if y˜ ∈ η (x), ˜ can guess that for μ˜ -a.e.x, def

(x, ˜ y) ˜ =

+∞ u −k ρ (y) ˜ J (θ x) ˜ = ∏ u −k . ρ (x) ˜ ˜ k=1 J (θ y)

A candidate for ρ is then

ρ (y) ˜ = if y˜ ∈ η (x), ˜ where L(x) ˜ =

 η (x) ˜

(x, ˜ y) ˜ L(x) ˜

(x, ˜ y) ˜ d λ˜ x˜η (y). ˜

In the sequel we prove rigorously that all this makes sense. ˜ function y˜ → log (x, ˜ y) ˜ is well-defined Lipschitz Lemma VII.9.1 For μ˜ -a.e. x, 1 −(λ0 +ε ) u ˆ  continuous on W ˆ (x) ˜ where δ = 4 e and δ

  u  ˆu (x)  u (x) W ˜ = y ˜ ∈ W ˜ : y ∈ Φ ˜ . −n x˜W ˆ (x) −n θ x, ˜δ δ

(VII.29)

It follows that for μ˜ -a.e. x, ˜ y˜ → (x, ˜ y) ˜ is a well-defined function on η (x) ˜ and is uniformly bounded away from 0 and +∞ on η (x). ˜ Proof. By Proposition VII.4.2 and Lemma VII.5.7, it is clear that there is a number C0 > 0 such that for any x˜ ∈ 2 J u (y) ˜ ≤ C0 (x) ˜ m0 , and

 u (x) for any y˜ ∈ W ˜

  u J (y) ˜ − J u(˜z) ≤ C0 (x) ˜ 8 d(y, ˜ z˜).

Then due to Lemma VII.5.1 and 1) of Lemma VII.5.4 we have   J u (θ −k y) ˜   ∑ log J u (θ −k z˜) 

+∞ 

= ≤ ≤ ≤

k=1 +∞ 

 



∑ log J u (θ −k y)˜ − logJ u(θ −k z˜)

k=1 +∞





∑ C0 (θ −k x)˜ m0 J u (θ −k y)˜ − J u(θ −k z˜)

k=1 +∞

∑ C0 (θ −k x)˜ m0 C0 (θ −k x)˜ 8 d(y−k , z−k )

k=1 +∞

∑ C02 K0 (x)˜ m0 +9e[−λ

k=1

+ +(m +10)ε ]k 0

d(y0 , z0 ).

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SRB Measures and Pesin’s Entropy Formula for Endomorphisms

From this the first part of the lemma follows clearly. Since for μ˜ -a.e. x˜ there is  u (x˜ ) for some x˜ ∈ 2 , the second part follows ˜ ⊂W n > 0 such that θ −n (η (x)) immediately.   Lemma VII.9.2 There exists a measurable function ρ : M f → R+ such that for μ˜ -a.e. x, ˜ (x, ˜ y) ˜ ρ (y) ˜ = L(x) ˜ for each y˜ ∈ η (x). ˜ Proof. We construct a sequence of functions ρn : M f → R+ , n ≥ 1 as follows. Let  u (x) ˜ ⊂W ˜ for some k ≥ 0 and some x˜ ∈ 2 , then define for x˜ ∈ M f . If θ −k (η (x)) δˆ ˜ each n ≥ 1 and y˜ ∈ η (x) ˜ −1 ∏ni=1 J u (θ −i y) n ˜ η (y) u −i ˜ −1 d λ η( x) ˜ ∏i=1 J (θ y) x˜ ˜

ρn (y) ˜ = (

˜ Ju (θ −i x)

∏ni=1 Ju (θ −i y) ˜

= (

˜ n Ju (θ −i x) η( x) ˜ ∏i=1 Ju (θ −i y) ˜

Otherwise, we define

d λ˜ x˜η (y) ˜

ρn (y) ˜ =1

for each n ≥ 1 and y˜ ∈ η (x). ˜ From the construction of η it is easy to see that ρn is measurable on M f for all n ≥ 1 and, by Lemma VII.9.1, for each x˜ the limit def

ρ (y) ˜ = lim ρn (y) ˜ n→+∞

(VII.30)

exists for all y˜ ∈ η (x). ˜ Clearly function ρ : M f → R+ defined by (VII.30) satisfies the requirement of the lemma.   Suppose that ρ : M f → R+ is as defined above. We now define a measure ν on M f by    ν (A) = χA (y) ˜ ρ (y) ˜ d λ˜ η (y) ˜ d μ˜ (x) ˜ (VII.31) x˜

η (x) ˜

for any Borel set A ⊂ M f . By Proposition VII.6, using standard arguments from measure theory one can easily verify that ν is indeed a well-defined Borel  probability measure. Also, from the definition of ν it follows clearly that, if νxη˜ is a canonical system of conditional measures of ν associated with η , then d νxη˜ = ρ d λ˜ x˜η for μ˜ -a.e. x˜ and that ν coincides with μ˜ on B(η ) (the σ -algebra consisting of all measurable η -sets). 

Lemma VII.9.3

− log νxη˜ (θ −1 η )(x) ˜ d μ˜ =



log J u d μ˜ .

Proof. Define q(x) ˜ = νxη˜ ((θ −1 η )(x)). ˜ By the definition of ν , it is easy to varify that for μ˜ -a.e. x˜

VII.9

Proof of the Necessity for the Entropy Formula

q(x) ˜ =

147

1 ˜ L(θ x) · u . L(x) ˜ J (x) ˜

(VII.32)

Since, by Lemma VII.9.1, L(x) ˜ = lim



n→+∞

n

J u (θ −k x) ˜

∏ J u(θ −k y)˜ λ˜ x˜η (d y)˜ η (x) ˜ k=1

˜ it follows that L is a positive finite-valued measurable function on M f for μ˜ -a.e.x, with   L◦θ d μ˜ ≤ log+ J u d μ˜ < +∞. log+ L Thus, by Lemma I.3.1 of [51], 

log

L◦θ d μ˜ = 0. L

The lemma then follows from (VII.32) naturally.   In the above we have indicated that ν = μ˜ on B(η ). The next lemma and an induction will show that ν = μ˜ on B(θ −n η ) for all n ≥ 0 and hence they are equal −n η ) = B(∨+∞ θ −n η ) = B (M f ). on ∨+∞ μ˜ n=0 B(θ n=0 (

Lemma VII.9.4 Identity log J u d μ˜ = Hμ˜ (η |θ η ) implies ν = μ˜ on B(θ −1 η ). Proof. For y˜ ∈ M f , define P(y) ˜ =

νyη˜ ((θ −1 η )(y)) ˜

μ˜ yη˜ ((θ −1 η )(y)) ˜

.

P is well defined μ˜ almost everywhere since Hμ˜ (θ −1 η |η ) < +∞. Noting that for ˜ xη˜ -mod 0) a countable partition, by Jensen’s inequality we μ˜ -a.e. x, ˜ θ −1 η |η (x) ˜ is (μ have   log P d μ˜ ≤ log P d μ˜ ≤ 0 (

(

with log P d μ˜ = 0 if and only if P = 1 μ˜ -a.e. But we know that log Pd μ˜ = 0 since Lemma VII.9.3 says that −



log νxη˜ (θ −1 η )(x) ˜ d μ˜ =



log J u d μ˜

= Hμ˜ (θ −1 η |η ) =−



log μ˜ xη˜ ((θ −1 η )(x)) ˜ d μ˜ .

Thus ν = μ˜ on B(θ −1 η ).   Now let ξ be an arbitrary measurable partition of M f subordinate to W u  ξ manifolds of ( f , μ ) and let μ˜ x˜ be the canonical system of conditional measures

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SRB Measures and Pesin’s Entropy Formula for Endomorphisms

ξ of μ˜ associated with ξ . In order to prove that p(μ˜ x˜ )  λx˜u for μ˜ -a.e. x, ˜ we take a  ξ ∨η  partition η as dealt with previously. Suppose that μ˜ x˜ is the canonical system of conditional measures of μ˜ associated with partition ξ ∨ η . Noting that (ξ ∨ η )|η (x) ˜ η ˜ ˜ and (ξ ∨ η )|ξ (x) are countable partitions for μ -a.e. x, ˜ we have μ (( ξ ∨ η )( x)) ˜ > 0, ˜ x˜ ξ μ˜ x˜ ((ξ ∨ η )(x)) ˜ > 0 and

μ˜ xξ˜ ∨η (·) =

ξ μ˜ xη˜ (·) μ˜ x˜ (·) = ξ ∨ η )(x)) ˜ μ˜ x˜ ((ξ ∨ η )(x)) ˜

μ˜ xη˜ ((ξ

ξ for μ˜ -a.e. x. ˜ From this it follows clearly that p(μ˜ x˜ )  λx˜u for μ˜ -a.e. x. ˜ The proof of the ergodic case of the second part of Theorem VII.1.1 is completed.

VII.9.2 The General Case Finally we finish the proof of the second part of Theorem VII.1.1 by showing the general case using the result for the ergodic case obtained in the previous section. For that purpose we make use of the ergodic decomposition of a measure-preserving dynamical system (see [75]). It says that there exists a unique (μ˜ -mod 0) measurable partition ζ of M f fixed under θ (i.e. θ −1C = C for any C ∈ ζ ), such that the condi˜ -a.e. x˜ ∈ M f . Moreover, tional measures μ˜ ζ (x) ˜ associated with ζ are ergodic for μ these measures are also θ -invariant. In other words, there is a θ -invariant measurable set N1 of full μ˜ -measure such that for all x˜ ∈ N1 , μ˜ ζ (x) ˜ is θ -invariant and ergodic. Assume now that entropy formula (VII.1) holds. Then equivalently hμ˜ (θ ) =



˜ ∑ λi (x)˜ + mi (x)˜ d μ˜ (x). i

Since h μ˜ (θ ) = and



∑ λi(x)˜ + mi (x)˜ d μ˜ (x)˜ = i

 Mf



hμ˜ ζ (x)˜ (θ ) d μ˜ (x), ˜ 

Mf

∑ λi(y)˜ + mi (y)˜ d μ˜ ζ (x)˜ (y)˜ d μ˜ (x)˜ i

and by Margulis-Ruelle Inequality, hμ˜ ζ (x)˜ (θ ) ≤



∑ λi (y)˜ + mi (y)˜ d μ˜ ζ (x)˜ (y)˜ i

for any x˜ ∈ N1 , it follows that there is a θ -invariant measurable set N2 ⊂ N1 with μ˜ (N2 ) = 1 such that for all x˜ ∈ N2

VII.9

Proof of the Necessity for the Entropy Formula

hμ˜ ζ (x)˜ (θ ) =



149

∑ λi (y)˜ + mi (y)˜ d μ˜ ζ (x)˜ (y)˜ i

Now let ξ be an arbitrary measurable partition subordinate to W u -manifolds of ( f , μ ). Let A denote the set of those points x˜ satisfying the following two conditions:   (1) p|ξ (x) ˜ → p ξ (x) ˜ is bijective; ˜ : ξ (x) 1 (2) There ˜  exists a k(x)-dimensional  C embedded submanifold Wx˜ of M such that p ξ (x) ˜ ⊂ Wx˜ ⊂ W u (x) ˜ and p ξ (x) ˜ contains an open neighborhood of x0 in Wx˜ . Obviously μ˜ (A) = 1. Moreover, we have μ˜ ζ (x˜ ) (A) = 1 for μ˜ -a.e. x˜ by the definition of conditional measures. In other words, there is a θ -invariant set N3 ⊂ N2 with μ˜ (N3 ) = 1 such that for every x˜ ∈ N2 , μ˜ ζ (x˜ ) (A) = 1. This shows that ξ is also a measurable partition subordinate to W u -manifolds of ( f , p μ˜ ζ (x˜ ) ) for any x˜ ∈ N3 . By Corollary VII.2.3.1 ξ refines ζ (mod 0) and therefore, by the transitivity of conditional measures,  is a θ -invariant set N4 ⊂ N3 with μ˜ (N4 ) = 1 such that  there for every x˜ ∈ N4 , μ˜ ζ (x˜ ) is the family of conditional measures belonging to ξ in    the space N4 ζ (x˜ ), μ˜ ζ (x˜ ) . ξ

Now we can apply the result of the ergodic case and obtain that pμ˜ x˜  λx˜u for μ˜ ζ (x˜ ) -a.e. x. ˜ Let   ξ (VII.33) B = x˜ ∈ M f : pμ˜ x˜  λx˜u . We have shown that μ˜ ζ (x˜ ) (B) = 1 for μ˜ -a.e. x˜ . Consequently μ˜ (B) = 1 if B ∈ Bμ˜ (M f ). We now prove the measurability of B if ξ additionally satisfies the following   ξ ˜ = λ˜ x˜ ξ (x) ˜ C is property: for every Borel subset C ⊂ M f , the function PC (x) ξ measurable and μ˜ -a.e. finite, where λ˜ x˜ = λx˜u ◦ p|ξ (x) ˜ . This can be assumed without loss of generality since in order to prove the SRB property it is sufficient to show the absolute continuity for one measurable partition subordinate to W u -manifolds of ( f , μ ) (see the end of the last subsection). Lemma VII.9.5 Let ξ be a measurable partition subordinate to W u -manifolds of ( f , μ ) and satisfying the above property. Then B ∈ Bμ˜ (M f ).   Proof. We start by introducing measures on M f × M f , Bμ˜ (M f ) ⊗ B(M f ) defined by     λ˜ ∗ (D × C) = λ˜ ξ ξ (x) ˜ C d μ˜ (x), ˜ D

μ˜ ∗ (D × C) =



D

x˜

   μ˜ xξ˜ ξ (x) ˜ C d μ˜ (x), ˜

(M f )

and Borel subset C ⊂ M f and extended to all measurable sets for any D ∈ Bμ˜ in the σ -algebra Bμ˜ (M f )⊗ B(M f ). By the Lebesgue decomposition theorem, there is a measurable set E ⊂ M f × M f with (1) the measure F → μ˜ ∗ (F (2) λ˜ ∗ (E c ) = 0.



E) is absolutely continuous with respect to λ˜ ∗ ,

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SRB Measures and Pesin’s Entropy Formula for Endomorphisms

Now (2) can be written as  Mf

   λ˜ x˜ξ ξ (x) ˜ Exc˜ d μ˜ (x) ˜ = 0,

    ξ where Exc˜ = (Ex˜ )c and Ex˜ = E {x} ˜ Exc˜ = 0 μ˜ ˜ × M f . This implies λ˜ x˜ ξ (x) almost everywhere. ξ ξ We are looking for those x˜ ∈ M f with μ˜ x˜  λ˜ x˜ (which is equivalent to   ξ ξ pμ˜ x˜  λx˜u ). If x˜ has this property then it must satisfy μ˜ x˜ ξ (x) ˜ Exc˜ = 0, i.e.   ξ μ˜ x˜ ξ (x) ˜ Ex˜ = 1. We now show that this property is also sufficient for the absolute continuity.     ξ We set B = x˜ : μ˜ x˜ ξ (x) ˜ Ex˜ = 1 . Let D be a measurable subset of B and C ∈ B(M f ). Then by (1) there is a non-negative measurable function g with        μ˜ ∗ (D × C) E = μ˜ xξ˜ ξ (x) ˜ C d μ˜ (x) ˜

= =

D

D×C

g d λ˜ ∗

 



ξ g d λ˜ x˜ d μ˜ (x). ˜



ξ g d λ˜ x˜

D ξ (x) ˜ C

Since D is arbitrarily chosen, it follows that    ξ μ˜ x˜ ξ (x) ˜ C =

ξ (x) ˜ C

ξ ξ for μ˜ -a.e. x˜ ∈ B . This equation holds for all C ∈ B(M f ) and hence μ˜ x˜  λ˜ x˜ μ˜ -a.e.



on B . We have just shown that B coincides with B (mod 0) which immediately implies that the desired measurability.  

Chapter VIII

Ergodic Property of Lyapunov Exponents

Generic property of SRB measures was first investigated by Bowen [10]. Theorem 4.12 in [10] says that if Ω is a hyperbolic attractor of a C2 Axiom A diffeomorphism (M, f ) and m is the volume measure on the compact Riemannian manifold M induced by the Riemannian metric, then for m-almost all x in the basin of attraction W s (Ω ), 1 n−1 lim (VIII.1) ∑ δ f k x = μ+ , n→+∞ n k=0 where μ+ is the SRB measure for f on Ω . As μ+ is an ergodic measure, the Lyapunov exponents of system f : M ← are μ+ -almost everywhere constants. Recently, by exploiting a Ruelle’s perturbation theorem [79, Theorem 4.1] Jiang et al. [29] proved that m-almost all x ∈ W s (Ω ) is positively regular and the Lyapunov spectrum of the system (i.e., the Lyapunov exponents associated with their multiplicities) at x are the constants {(λ1 (μ+ , f ), m1 (μ+ , f )), · · · , (λr (μ+ , f ), mr (μ+ , f ))}. This is called the ergodic property of Lyapunov exponents. Similar results have also been obtained in [29] for nonuniformly completely hyperbolic attractors of C2 diffeomorphisms. The generic property of SRB measures of Axiom A endomorphisms has already been introduced in Chapter IV. In this chapter we study the ergodic property of Lyapunov exponents for C2 endomorphisms along the line of [28].

VIII.1 Introduction and Main Results Let M be a smooth, compact and connected Riemannian manifold without boundary and let m be the Lebesgue measure on M induced by the Riemannian metric. Let O be an open subset of M, and let Δ ⊂ O be an Axiom A attractor of an endomorphism M. Qian et al., Smooth Ergodic Theory for Endomorphisms, Lecture Notes in Mathematics 1978, DOI 10.1007/978-3-642-01954-8 VIII, c Springer-Verlag Berlin Heidelberg 2009 

151

152

VIII

Ergodic Property of Lyapunov Exponents

f ∈ C2 (O, M). As mentioned in Chapter IV, Qian and Zhang [72] showed that there exists a unique f -invariant Borel probability measure μ+ on Δ satisfying Pesin’s entropy formula: hμ+ ( f ) =

 r(x)

∑ λi (x)+ mi (x)d μ+ (x),

Δ i=1

where hμ+ ( f ) is the measure-theoretic entropy of f with respect to μ+ , and +∞ > λ1 (x) > λ2 (x) > · · · > λr(x) (x) ≥ −∞ are the Lyapunov exponents of ( f , T f ) at x with the multiplicities mi (x), 1 ≤ i ≤ r(x). The SRB measure μ+ is ergodic. If ε > 0 is small enough and the set of critical points C f = {x ∈ O : det(Tx f ) = 0} has zero Lebesgue measure, then for m-almost all x ∈ B(Δ , ε ), 1 n−1 ∑ δ f i x = μ+ , n→+∞ n i=0 lim

(VIII.2)

where B(Δ , ε ) = {y ∈ O : d(y, Δ ) < ε }. The Lyapunov spectrum of ( f , T f ) are μ+ almost everywhere equal to constants {(λi (μ+ , f ), mi (μ+ , f )) : 1 ≤ i ≤ r}. We can say more about the Lyapunov spectrum. In fact we have the following results. Theorem VIII.1.1 Let f ∈ C2 (O, M) and Δ ⊂ O be an Axiom A attractor of f , and suppose that Tx f is non-degenerate for every x ∈ Δ . There exists ε > 0 such that Lebesgue-almost every x ∈ B(Δ , ε ) is positively regular and the Lyapunov spectrum of ( f , T f ) at x are the constants {(λi (μ+ , f ), mi (μ+ , f )) : 1 ≤ i ≤ r}, where μ+ is the unique SRB measure of the system. Theorem VIII.1.2 Suppose that μ+ is an ergodic invariant measure of the C2 endomorphism (M, f ) satisfying that: 1) log | det(Tx f )| ∈ L1 (M, μ+ ); 2) μ+ is an SRB measure of (M, f ); 3) The Lyapunov exponents of ( f , T f ) are μ+ -almost everywhere nonzero. Moreover, the smallest Lyapunov exponent λr (μ+ , f ) < 0. Then there exists a Borel set Δ ⊂ M such that f Δ = Δ , μ+ (Δ ) = 1 and that for every def 

x ∈ W s (Δ ) =

y∈Δ W

s (y),

1 n−1 ∑ δ f i x = μ+ , n→+∞ n i=0 lim

where W s (y) is the global stable set of f at y, moreover, m(W s (Δ )) > 0. Every  −n point x ∈ W s (Δ ) \ +∞ n=0 f C f is positively regular and the Lyapunov spectrum of ( f , T f ) at x are the constants {(λi (μ+ , f ), mi (μ+ , f )) : 1 ≤ i ≤ r}, where the set

VIII.2

Lyapunov Exponents of Axiom A Attractors

153

of critical points C f = {y ∈ M : det(Ty f ) = 0}. If, in addition, m(C f ) = 0, then  −nC ) > 0. f m(W s (Δ ) \ +∞ f n=0 The above results justify that for uniformly or nonuniformly completely hyperbolic attractors of endomorphisms, the initial points can be chosen close to the attractors uniformly with respect to the Lebesgue measures, to compute the space averages of observables approximately via their time averages, or to compute approximately the Lyapunov exponents associated with the SRB measures. In applications, if the differentiable maps or the equations of motion that define dynamical systems are completely known, Lyapunov exponents are computed with straightforward techniques using a phase space plus tangent space approach (see [8, 84, 93, 17, 22]).

VIII.2 Lyapunov Exponents of Axiom A Attractors of Endomorphisms The purpose of this section is to prove Theorem VIII.1.1. But first we need to review some basic notions and results about hyperbolic sets of endomorphisms. Let Δ be a hyperbolic invariant set of f ∈ Cr (O, M)(r ≥ 1) and let E|Δ f = E s ⊕ E u be the hyperbolic splitting. For each x˜ ∈ Δ f and δ > 0, write Exs˜ (δ ) = {vs ∈ Exs˜ : |vs | < δ }, Exu˜ (δ ) = {vu ∈ Exu˜ : |vu | < δ }, and Ex˜ (δ ) = Exs˜ (δ ) ⊕ Exu˜ (δ ). For small ε > 0, x ∈ Δ and x˜ ∈ Δ f , the local stable and unstable manifolds are defined respectively as Wεs (x) = {y ∈ M : d( f n x, f n y) < ε , ∀n ∈ Z+ }, def

Wεu (x) ˜ = {y0 ∈ M : ∃y˜ ∈ M f such that y0 = py, ˜ d(x−n , y−n ) < ε , ∀n ∈ Z+ }. def

 u (x) For x˜ ∈ Δ f , the global unstable set W ˜ of f at x˜ in M f is defined by 1 def  u (x) W ˜ = {y˜ ∈ M f : lim sup log d(y−n , x−n ) < 0}, n→+∞ n def  u (x). ˜ = pW ˜ Let and the global unstable set of f at x˜ in M is defined by W u (x) u f +  Wε (x) ˜ = {y˜ ∈ M : d(y−n , x−n ) < ε , ∀n ∈ Z }, which is called a local unstable set εu (x) ˜ → Wεu (x) ˜ is bijective. For x ∈ Δ , the global stable of f at x˜ in M f . Then p : W s set W (x) of f at x in M is defined by

154

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Ergodic Property of Lyapunov Exponents

1 def W s (x) = {y ∈ M : lim sup log d( f n y, f n x) < 0}. n→+∞ n One has

+∞

W u (x) ˜ =

f n (Wεu (θ −n x)), ˜ ∀x˜ ∈ Δ f

n=0

and

+∞

W s (x) =

f −n (Wεs ( f n x)), ∀x ∈ Δ .

n=0

If Δ is an Axiom A attractor of f , then by Proposition IV.IV.2.5 there exists a suffi˜ ⊂ Δ , ∀x˜ ∈ Δ f , hence ciently small ε > 0 such that Wεu (x) W u (x) ˜ =

+∞

f n (Wεu (θ −n x)) ˜ ⊂ Δ , ∀x˜ ∈ Δ f .

n=0

Denote by B(x, δ ) the open ball on M of radius δ centered at x ∈ M, and by B(x, ˜ δ ) the open ball on M f of radius δ centered at x˜ ∈ M f . By the property of the continuous splitting E|Δ f = E s ⊕ E u , there exists a constant a > 0 such that for any v = vs ⊕ vu with vs ∈ E s , vu ∈ E u , one has a max{|vs |, |vu |} ≤ |v|. 2 The following proposition is a result of changing coordinates from the stable and unstable manifold theorems for hyperbolic invariant sets of endomorphisms (see Theorems IV.IV.2.1 and IV.IV.2.2). Proposition VIII.2.1 Let f ∈ Cr (O, M)(r ≥ 1) and Δ ⊂ O be a hyperbolic invariant set of f . Then there exists a number δ0 > 0 such that for each x˜ ∈ Δ f , if y˜ ∈  B(x, ˜ δ20 ) Δ f , then there are Cr maps φx,˜s y˜ : Exs˜ (aδ0 ) → Exu˜ and φx,˜u y˜ : Exu˜ (aδ0 ) → Exs˜ satisfying  Wδs0 (y0 ) = B(y0 , δ0 ) expx0 ◦p∗ ◦ (id, φx,˜s y˜ )Exs˜ (aδ0 ), ˜ = B(y0 , δ0 ) Wδu0 (y)



sup{|Tvs φx,˜s y˜ | : y˜ ∈ B(x, ˜

expx0 ◦p∗ ◦ (φx,˜u y˜ , id)Exu˜ (aδ0 ), 1 δ0  f s ) Δ , v ∈ Exs˜ (aδ0 )} ≤ , 2 2

and

δ0  f u 1 ) Δ , v ∈ Exu˜ (aδ0 )} ≤ . 2 2 We now state without proof the Ruelle’s perturbation theorem [79, Theorem 4.1]. This theorem plays a fundamental role in this section. ˜ sup{|Tvu φx,˜u y˜ | : y˜ ∈ B(x,

Theorem VIII.2.2 (Ruelle’s Perturbation Theorem) Let T = {Tn }n>0 be a sequence of real m × m matrices such that

VIII.2

Lyapunov Exponents of Axiom A Attractors

155

1 lim sup log Tn  ≤ 0. n→+∞ n Write T n = Tn ◦ · · · ◦ T2 ◦ T1 and assume the existence of 1

lim ((T n )∗ T n ) 2n = Λ

n→+∞

with det(Λ ) = 0. Denote by eλ1 > · · · > eλr the eigenvalues of Λ . Let η > 0 be given. For another sequence of real m × m matrices T˜ = {T˜n }n>0, Let T˜ − T  = sup T˜n − Tn enη n

and T˜ n = T˜n · T˜2 T˜1 . Then there exists δ > 0 such that, if T˜ − T  ≤ δ then lim ((T˜ n )∗ T˜ n ) 2n = Λ 1

n→+∞

exists and has the same eigenvalues as Λ (including the multiplicities). Let  corresponding to eλ1 , · · · , eλr , then 1 , · · · , U r be the eigenspaces of Λ U lim

n→+∞

1 log T˜ n v = λi n

i \ {0} for i = 1, · · · , r. when v ∈ U Exploiting Ruelle’s perturbation theorem for the spectrum of matrix products, along the line of the proof of Theorem 3.1 in Jiang et al. [29], we will prove the following fact: Proposition VIII.2.3 Let f ∈ C2 (O, M) and Δ ⊂ O be a hyperbolic invariant set of f . For each x ∈ Δ , if there exists some positively regular point y0 ∈ W s (x) and the smallest Lyapunov exponent of ( f , T f ) at y0 , λr(y0 ) (y0 ) > −∞, then every y ∈  W s (x) \ k≥0 f −kC f is positively regular and the Lyapunov spectrum of ( f , T f ) at y are the same as those at the point y0 , where C f = {y ∈ M : det(Ty f ) = 0}. Proof. Since M is a smooth compact Riemannian manifold, one can define a piecewise smooth map M × Rm0 → T M, (x, u) → ψx (u) such that it is a bijection of M × Rm0 onto T M, and ψx : Rm0 → Tx M is a linear operator with ψx  ≤ b and ψx−1  ≤ b for some constant b > 1. As f is uniformly continuous on M, we can choose β so small that the image by f ◦ expx ◦ψx of the ¯ β ) ⊂ Rm0 is contained in exp f x ◦ψ f x (B(0, ρ0 /b )) for all x ∈ M. Let closed ball B(0, ¯ β ) → M and Fx = φ −1 ◦ f ◦ φx . Put Fxn = Ff n−1 x ◦ · · · ◦ Ff x ◦ Fx φx = expx ◦ψx : B(0, fx for all n ∈ N. Now we define Tn = DFf n−1 y0 (0)

156

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Ergodic Property of Lyapunov Exponents

and T n = Tn ◦ · · · ◦ T2 ◦ T1 for all n ∈ N. Then we have n T n = DFyn0 (0) = D(φ −1 f n y0 ◦ f ◦ φy0 )(0) n n = Dφ −1 f n y0 ( f y0 ) ◦ Ty0 f ◦ Dφy0 (0).

By the compactness of M and the continuity of T· f , T· f  is bounded up, i.e. T· f  ≤ A for some constant A > 1. Hence 1 lim sup log Tn  ≤ 0. n→+∞ n

(VIII.3)

By Lemma II.II.2.1, we have n −1 −1 n

Dφ −1 f n y0 ( f y0 ) = Dψ f n y0 ◦ D exp f n y0 ( f y0 ) ≤ b b

and

Dφy0 (0) = D expy0 (0) ◦ ψy0  ≤ b b,

where b = b(ρ0 ). Thus by (VIII.3) the following limit 1

lim ((T n )∗ T n ) 2n = Λ

n→+∞

(VIII.4)

exists with det(Λ ) = 0. The eigenvalues of Λ are eλ1 (y0 ) > · · · > e

λr(y ) (y0 ) 0

with multiplicities m1 (y0 ), · · · , mr(y0 ) (y0 ). −1 Since Fx = φ −1 f x ◦ f ◦ φx , DFx (u) = Dφ f x ( f φx u) ◦ Tφx u f ◦ Dφx (u). As ψx (·) is a linear map and expx (·) is a smooth map, DFx (u) is a Lipschitz map of u and there exists a constant L > 0 such that Lip(DFx (·)) ≤ L ¯ β ) ⊂ Rm0 for any x ∈ M. holds in B(0, By Theorem V.V.6.4, there are constants c > 1 and η > 0 such that for any z ∈ Wεs (y0 ), d( f n y0 , f n z) ≤ ce−nη d(y0 , z) for all n ≥ 0. Suppose that ε ∈ (0, bβ c ), then for any fixed point z ∈ Wεs (y0 ), there is an u ∈ Rm0 such that φy0 (u) = expy0 ◦ψy0 (u) = z. And n −1 −1 n Fyn0 (u) = φ −1 f n y0 ◦ f ◦ φy0 (u) = ψ f n y0 ◦ exp f n y0 ( f z) n

n n ≤ b  exp−1 f n y0 ( f z) = b d( f y0 , f z)

≤ b ce−nη d(y0 , z).

VIII.2

Lyapunov Exponents of Axiom A Attractors

157

Let δ > 0 be the constant with which the result in Theorem VIII.2.2 holds true for T = {Tn }n>0 and the constant η . Let T˜n = DFf n−1 y0 Fyn−1 (u) and T˜ n = T˜n ◦ · · · ◦ T˜2 ◦ T˜1 for all n ∈ N. Then 0 T˜ − T  = sup T˜n − Tn enη n   = sup DFf n−1y0 Fyn−1 (u) − DFf n−1 y0 (0)enη 0 n

≤ sup LFyn−1 (u)enη 0 n

≤ Lb ceη d(y0 , z) ≤ Lb ceη ε . We can choose ε small enough to make Lb ceη ε < δ , so lim ((T˜ n )∗ T˜ n ) 2n = Λ 1

n→+∞

exists and has the same eigenvalues as Λ (including the multiplicities).  corresponding to eλ1 (y0 ) , . . . , eλr(y0 ) (y0 ) , r(y ) be the eigenspaces of Λ 1 , · · · , U Let U 0 then 1 lim log T˜ n v = λi (y0 ) n→+∞ n i \ {0} for i = 1, · · · , r(y0 ). when v ∈ U i . By Define Ui (y0 ) = (Dφy0 (u))U T˜ n = DFyn0 (u) n n = Dφ −1 f n y0 ( f ◦ φy0 (u)) ◦ Tz f ◦ Dφy0 (u)

  −1 n n = ψ −1 f n y0 ◦ D exp f n y0 ( f z) ◦ Tz f ◦ D expy0 ψy0 (u) ◦ ψy0

and the boundedness of ψ· , D exp· (·), ψ·−1 and D exp−1 · (·), the Lyapunov spectrum of ( f , T f ) at the point z = φy0 (u) are {(λi (y0 ), mi (y0 )) : 1 ≤ i ≤ r(y0 )}, and lim

n→+∞

1 log |Tz f n v| = λi (y0 ) n

when v ∈ Ui (y0 ) \ {0} for i = 1, · · · , r(y0 ).  Notice that W s (x) = W s (y0 ) if y0 ∈ W s (x). Furthermore W s (y0 ) = n≥0 f −nWεs ( f n y0 ) and the Lyapunov spectrum of ( f , T f ) at the point f n y0 is the same as that at the point y0 . Hence we obtain that the Lyapunov spectrum of ( f , T f ) at any point z ∈ W s (x) \ k≥0 f −kC f are {(λ1 (y0 ), m1 (y0 )), · · · , (λr(y0 ) (y0 ), mr(y0 ) (y0 ))}.  

158

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Ergodic Property of Lyapunov Exponents

Now we present the absolute continuity of the local stable manifolds of an Axiom A attractor of an endomorphism. Suppose that Δ is an Axiom A attractor of f ∈ C2 (O, M). For small ε > 0 and x˜ ∈ Δ f , we denote by F (x, ˜ ε ) the collection of local stable manifolds Wεs (y) passing through y ∈ Wεu (x) ˜ ⊂ Δ . Set U(x, ˜ ε) =



Wεs (y).

y∈Wεu (x) ˜

As we know, U(x, ˜ ε ) is an open neighborhood of x0 in M. Definition VIII.2.1 A submanifold W of M is called transversal to the family F (x, ˜ ε ) if the following hold true: 1 u s (i) W ⊂ U(x, ˜ ε ) and p∗x˜ ◦ exp−1 x0 W is the graph of a C map ψ : Ex˜ (ε ) → Ex˜ ; u s (ii) for any y ∈ Wε (x), ˜ W intersects Wε (y) at exactly one point and this intersection is transversal, i.e. TzW ⊕ TzWεs (y) = Tz M,

where z = W



Wεs (y).

We denote by mW the Lebesgue measure on W induced by the Riemannian metric on W inherited from M. Now consider two submanifolds W1 and W2 transversal to F (x, ˜ ε ). Since {Wεs (y)}y∈Δ is a continuous family of C2 embedded discs, there exist two submanifolds Wˆ 1 and Wˆ 2 respectively of W1 and W2 such that we can well define  ˆ 1 U(x, ˜ ε ) → Wˆ 2 U(x, ˜ ε ) by letting a so-called Poincar´e map PWˆ 1 ,Wˆ 2 : W PWˆ 1 ,Wˆ 2 : z → Wˆ 2



Wεs (y)



˜ and moreover, PWˆ 1 ,Wˆ 2 is a homeomorphism. for z = Wˆ 1 Wεs (y), y ∈ Wεu (x), Definition VIII.2.2 A map T : X →Y between two σ -finite measure spaces (X, A , μ ) and (Y, B, ν ) is said to be absolutely continuous if the following three conditions are satisfied: (1) T is injective; (2) if A ∈ A then TA ∈ B; (3) A ∈ A and μ (A) = 0 imply ν (TA) = 0. Proposition VIII.2.4 There exists a number ε0 > 0 such that for each x˜ ∈ Δ f and ε ∈ (0, ε0 ), the family of C2 embedded discs F (x, ˜ ε ) = {Wεs (y)}y∈Wεu (x) ˜ is absolutely continuous in the following sense: for every two submanifolds W1 and W2 contained in U(x, ˜ ε ) and transversal to the family F (x, ˜ ε ), the Poincar´e map PWˆ 1 ,Wˆ 2 constructed as above is absolutely continuous with respect to the Lebesgue measures mW1 and mW2 . A detailed proof of this proposition can be carried out by a completely parallel argument with that of Part II of Katok and Strelcyn [33]. We omit the details here. The proof of the absolute continuity of local stable manifolds for Anosov

VIII.2

Lyapunov Exponents of Axiom A Attractors

159

diffeomorphisms was given in [2]. (See also [57, Chapter III, Theorem 3.1].) For uniformly partially hyperbolic systems it was formulated in [14]. The case of nonuniformly partially hyperbolic systems was considered in [62, 33, 69]. Suppose that Δ is an Axiom A basic set of f ∈ C2 (O, M). Qian and Zhang exploited the local product structure of the Axiom A basic set Δ [72, Theorem 2.2] (see Theorem IV.IV.2.3, Chapter IV) to show that (Δ f , d1 , θ ) is a Smale space, where d1 is a suitable metric on Δ f . So Δ f has Markov partitions of arbitrarily small diameter. (For the definition of Smale space and its Markov partition, see Ruelle [78, Chapter 7].) One can obtain the symbolic representations of (Δ f , d1 , θ ) via its Markov partitions. Applying the thermodynamic formalism of symbolic spaces, one can prove that every H¨older continuous function on Δ f has a unique equilibrium state with respect to θ . Define φ u : Δ f → R as

φ u (x) ˜ = − log | det(T f |Exu˜ )|,

x˜ ∈ Δ f .

φ u is H¨older continuous and has a unique equilibrium state μ˜ φ u with respect to θ (see Chapter IV), i.e. hμ˜ φ u (θ ) +



φ u d μ˜ φ u =

sup μ˜ ∈Mθ

(Δ f )

  h μ˜ (θ ) + φ u d μ˜

= Pθ (φ u ),

where hμ˜ (θ ) is the measure-theoretic entropy of θ with respect to μ˜ , and Pθ (φ u ) is the topological pressure of φ u with respect to θ . In case of Δ being an Axiom A def attractor, Pθ (φ u ) = 0 and the measure μ+ = p μ˜ φ u is the unique Borel probability measure on Δ satisfying Pesin’s entropy formula hμ+ ( f ) =

 r(x)

∑ λi (x)+ mi (x)d μ+ (x).

Δ i=1

Now we review the SRB property of the measure μ+ constructed above. For any fixed ν ∈ M f (Δ ), let ν˜ be the unique θ -invariant Borel probability measure on Δ f so that p ν˜ = ν . Let du = dim E u . A measurable partition ξ of Δ f is said to be subordinate to W u -manifolds of f with respect to ν if for ν˜ -a.e. x˜ ∈ Δ f , the member of ξ which contains x, ˜ denoted by ξ (x), ˜ has the following properties:   1) p |ξ (x) ˜ → p ξ (x) ˜ is bijective; ˜ : ξ (x) 1 -dimensional submanifold Wx˜ of M such that Wx˜ ⊂ 2) There exists a d u    C embedded  u W (x), ˜ p ξ (x) ˜ ⊂ Wx˜ and p ξ (x) ˜ contains an open neighborhood of x0 = p x˜ in the submanifold topology of Wx˜ . We say that ν has SRB property if for every measurable partition ξ of Δ f subordiu ˜ -a.e. x˜ ∈ Δ f , where nate to W u -manifolds of f with respect to ν , p (ν˜ ξ (x) ˜ )  mx˜ for ν ˜ ˜ {νξ (x) ˜ }x∈ ˜ Δ f is a canonical system of conditional measures of ν associated with ξ , and mux˜ is the Lebesgue measure on Wx˜ induced by its inherited Riemannian metric

160

VIII

Ergodic Property of Lyapunov Exponents

as a submanifold of M. (We refer the readers to [74, 51] for the details of conditional measures given a measurable partition which is closely related to regular conditional probability distributions in probability theory.) The proposition below is Corollary 1.1.2 in [73]. Proposition VIII.2.5 Let f ∈ C2 (O, M) and Δ ⊂ O be an Axiom A attractor of f , and assume that Tx f is non-degenerate for every x ∈ Δ . Then the measure μ+ = p μ˜ φ u is the unique f -invariant Borel probability measure on Δ which is characterized by each of the following properties: 1) when ε > 0 is small enough, 1 n−1 ∑ δ f k x = μ+ n→+∞ n k=0 lim

def

for Lebesgue-almost every x ∈ B(Δ , ε ) = {y ∈ M : d(y, Δ ) < ε }; 2) Pesin’s entropy formula holds for the system (Δ , f , μ+ ); 3) μ+ has SRB property. In fact, if ξ is a measurable partition of Δ f subordinate to W u -manifolds of f with respect to μ+ , and let ρx˜ be the density of p (μ˜ φ u ,ξ (x) ˜ ) with respect to mux˜ , then for μ˜ φ u -almost all x˜ ∈ Δ f , there exists a countable number of ˜ n ∈ N of Wx˜ such that disjoint open subsets Un (x),

Un (x) ˜ ⊂ ξ (x), ˜

n∈N

  mux˜ ξ (x) ˜ \ Un (x) ˜ =0 n∈N

˜ ρx˜ is a strictly positive function satisfying and on each Un (x), −1 u y)) ρx˜ (y) +∞ exp(φ (p |ξ (x) ˜ =∏ , ρx˜ (z) i=1 exp(φ u (p |−1 z)) ξ (x) ˜

∀y, z ∈ Un (x). ˜

In particular, log ρx˜ restricted to each Un (x) ˜ is Lipschitz along Wx˜ . By the assumption det(Tx f ) = 0, ∀x ∈ Δ , the smallest Lyapunov exponent λr(x) (x) of ( f , T f ) is μ+ -almost everywhere not −∞. Since μ+ is f -ergodic, the Lyapunov spectrum of ( f , T f ) are μ+ -almost everywhere equal to constants {(λi (μ+ , f ), mi (μ+ , f )) : 1 ≤ i ≤ r}. Denote by Γμ+ the set of positively regular point x ∈ O such that the Lyapunov spectrum of ( f , T f ) at x are the constants {(λi (μ+ , f ), mi (μ+ , f )) : 1 ≤ i ≤ r}. Write Γμc+ = O \ Γμ+ . Let Γ˜μ+ be the set of positively regular point x˜ ∈ Δ f such that the Lyapunov spectrum of (θ , T f ) at x˜ are {(λi (μ+ , f ), mi (μ+ , f )) : 1 ≤ i ≤ r}, then m pΓ˜μ+ ⊂ Γμ+ . Denote by mB the normalized Lebesgue measure m(B) on a Borel subset B of M with m(B) > 0.

VIII.2

Lyapunov Exponents of Axiom A Attractors

161

Proof of Theorem VIII.1.1. Let R = {R1 , R2 , · · · , Rk0 } be a Markov partition of Δ f with diameter smaller than min(δ0 , ε0 )/2, where δ0 are the constants specified in Proposition VIII.2.1 and ε0 are the constants in Proposition VIII.2.4. The elements of R are closed proper rectangles, and some of its elements intersect with one another on the boundary. We can modify the elements of R appropriately on the boundary to make them not intersect with one another. Then R becomes a measurable partition of Δ f .  u  (y) Denote by ξ the measurable partition of Δ f into sets having the form Ri W δ0 ˜  for Ri ∈ R and y˜ ∈ Ri . Let ∂ R = ∂ + R ∂ − R be the boundary of R as defined in [78, Chapter 7]. As μ˜ φ u (∂ R) = 0 (see [78, Chapter 7]), the measurable partition ξ of Δ f is subordinate to W u -manifolds of f with respect to μ+ . Let {μ˜ φ u ,ξ (y) ˜ }y∈ ˜ Δf be a canonical system of conditional measures of μ˜ φ u associated with ξ . For each ξ ˜ induced by the y˜ ∈ Δ f , denote by my˜ the normalized Lebesgue measure on p (ξ (y)) inherited Riemannian metric. By the Oseledec multiplicative ergodic theorem,

μ˜ φ u (Δ f



Γ˜μ+ ) =

 Δf

    f ˜ ( μ˜ φ u ,ξ (y) Δ Γ ) ξ ( y) ˜ d μ˜ φ u (y) ˜ = 1. μ ˜ +

(VIII.5)

Hence for μ˜ φ u -almost all y˜ ∈ Δ f ,    ˜ μ˜ φ u ,ξ (y) Γ ξ ( y) ˜ = 1. μ+ ˜

(VIII.6)

By the SRB property of μ+ (Proposition VIII.2.5), for μ˜ φ u -almost all y˜ ∈ Δ f , ξ ˜ φ u (Ri ) > 0, there exists x˜i ∈ Ri p (μ˜ φ u ,ξ (y) ˜ ) is equivalent to my˜ . For each Ri ∈ R, as μ such that    μ˜ φ u ,ξ (x˜i ) Γ˜μ+ ξ (x˜i ) = 1 ξ

and p (μ˜ φ u ,ξ (x˜i ) ) is equivalent to mx˜i , therefore    ξ mx˜i Γμ+ p (ξ (x˜i )) = 1.

(VIII.7)

For the above point x˜i , let η be the measurable partition {expxi ◦p∗ ({vs } × Exu˜i (aδ0 ))}vs ∈E si (aδ0 ) 0

x˜

of (expxi ◦p∗ )(Ex˜i (aδ0 )), and η˜ the restriction of η to 0



def

Ui =

 y∈Wδu (x˜i ) p (Ri )

Wδs0 ∧ε0 (y),

0

where δ0 ∧ ε0 = min(δ0 , ε0 ). For each y ∈ Ui , denote by mηy the normalized Lebesgue measure on η˜ (y) induced by the inherited Riemannian metric. Let {mUη˜ i(y) }y∈Ui be ˜

162

VIII

Ergodic Property of Lyapunov Exponents

a canonical system of conditional measuresof mUi associated with the partition η˜ . Ui , and a simple argument, Then by the Fubini theorem, applied to Ex˜i p∗x˜i ◦ exp−1 xi 0

one can prove that for mUi -almost all y ∈ Ui , the measure mUη˜ i(y) is equivalent to mηy and there exists a number C > 1 such that C

−1

≤

dmUη˜ i(y) dmηy ˜

˜

≤C

(VIII.8)

˜ holds mηy -almost everywhere on η˜ (y). By Proposition VIII.2.1, for each y ∈ Ui , η˜ (y) is transversal to the family def F˜ (x˜i , δ0 ∧ ε0 ) = {Wδs0 ∧ε0 (z)}z∈W u (x˜i )  p (Ri ) . δ0

   p ξ (x˜i ) = Wδu0 (x˜i ) p (Ri ) is also transversal to the family F˜ (x˜i , δ0 ∧ ε0 ). We can assume that δ0 ∧ ε0 is small enough so that det(Tx f ) = 0 for any x ∈ Bδ0 ∧ε0 (Δ ). Then by Proposition VIII.2.3, we have Pp (ξ (x˜i )),η˜ (y) (Γμ+ and

Pp (ξ (x˜i )),η˜ (y) (Γμc+





p (ξ (x˜i ))) = Γμ+ p (ξ (x˜i ))) = Γμc+





η˜ (y)

(VIII.9)

η˜ (y),

where Pp (ξ (x˜i )),η˜ (y) is a Poincar´e map. Then from (VIII.7), (VIII.9) and the absolute continuity of F˜ (x˜i , δ0 ∧ ε0 ) (Proposition VIII.2.4), we have       mηy˜ Γμc+ η˜ (y) = mηy˜ Pp (ξ (x˜i )),η˜ (y) (Γμc+ p (ξ (x˜i ))) = 0.

(VIII.10)

By (VIII.8), for mUi -almost all y ∈ Ui , the measure mUη˜ i(y) is equivalent to mηy , therefore    mUη˜ i(y) Γμc+ η˜ (y) = 0 ˜

and

   mUη˜ i(y) Γμ+ η˜ (y) = 1.

Then we get mUi (Γμ+



Ui ) =

 Ui

mUη˜ i(y) (Γμ+





η˜ (y))dmUi (y) = 1.

(VIII.11)

0 Let G = ki=1 Ui and ε = δ0 ∧2 ε0 . Then G is an open neighborhood of Δ in M satisfying that def

Wεs (Δ ) = Wεs (x) ⊂ G

x∈Δ

VIII.3

and

Nonuniformly Completely Hyperbolic Attractors

m(Γμc



163

Wεs (Δ )) = 0.  

Remark VIII.1. Following the proof of Theorem VIII.1.1 and applying the Birkhoff ergodic theorem instead of the Oseledec multiplicative ergodic theorem, one can prove (VIII.2), the generic property of μ+ . Qian and Zhang [72] gave a different proof of this property following the main line of Sinai [88] and Bowen [10, Theorem 4.12] (see Chapter IV for details).

VIII.3 Nonuniformly Completely Hyperbolic Attractors This section is devoted to prove Theorem VIII.1.2. First we review some necessary concepts and results from the smooth ergodic theory for endomorphisms [73, 100]. Throughout this section, we assume that M is a smooth, compact and connected Riemannian manifold without boundary and f is a C2 endomorphism on M, i.e. f ∈ C2 (M, M). The inverse limit space M f of the system (M, f ), the natural projection p from M f to M, and the left-shift homeomorphism θ on M f can be defined as in Chapter I. By the Oseledec multiplicative ergodic theorem, there exists a Borel subset Γ ⊂ M with f Γ ⊂ Γ and μ (Γ ) = 1 for any μ ∈ M f (M), such that each point x ∈ Γ is positively regular and the Lyapunov spectrum of ( f , T f ) at x {(λi (x), mi (x)) : 1 ≤ i ≤ r(x)} (with λ1 (x) > λ2 (x) > · · · > λr(x) (x)) are well defined. From now on we fix a μ ∈ M f (M) satisfying the integrability condition log | det(Tx f )| ∈ L1 (M, μ ). Define

(VIII.12)

Γ∞ = {x ∈ Γ : Tx f is degenerate or λr(x) (x) = −∞}.

The integrability condition (VIII.12) implies that μ (Γ∞ ) = 0. Let

Γ =Γ \

+∞

f −n Γ∞ .

n=0

It is easy to see that f Γ ⊂ Γ , μ (Γ ) = 1 and for any x ∈ Γ , Tx f is an isomorphism and λr(x) (x) > −∞. Let

Δˆ = M f \

+∞

n=−∞

θ n (p−1Γ∞ ).

164

VIII

Ergodic Property of Lyapunov Exponents

Obviously θ Δˆ = Δˆ and for any x˜ = {xn }n∈Z ∈ Δˆ , we have xn ∈ M \ Γ∞ , ∀n ∈ Z. As a consequence of the integrability condition (VIII.12), μ˜ (Δˆ ) = 1, where μ˜ is the unique extension of μ in Mθ (M f ) such that p μ˜ = μ . By the Oseledec multiplicative ergodic theorem, there exists a Borel set Δ˜ ⊂ Δˆ such that θ Δ˜ = Δ˜ and μ˜ (Δ˜ ) = ˜ a measurable 1. Furthermore, for every x˜ = {xn }n∈Z ∈ Δ˜ , there is an integer r(x), splitting of the tangent space ˜ ⊕ · · · ⊕ E(r(x)) ˜ Tx0 M = E1 (x) ˜ (x), numbers

˜ > λ2 (x) ˜ > · · · > λr(x) ˜ > −∞, +∞ > λ1 (x) ˜ (x)

and integers mi (x), ˜ 1 ≤ i ≤ r(x), ˜ such that 1) r(x), ˜ λi (x), ˜ mi (x) ˜ are θ -invariant, dimEi (x) ˜ = mi (x); ˜ 2) The splitting is invariant under T f , i.e. Tx0 f (Ei (x)) ˜ = Ei (θ x), ˜ 1 ≤ i ≤ r(x); ˜ 3) For any m ∈ Z, let ⎧ if m > 0, ⎨ Tx0 f m , m if m = 0, ˜ = id, T (x) ⎩ (Txm f −m )−1 , if m < 0, then lim

m→±∞

1 log |T m (x) ˜ ξ | = λi (x), ˜ m

for all 0 = ξ ∈ Ei (x), ˜ 1 ≤ i ≤ r(x); ˜ ˜ = r(x0 ), λi (x) ˜ = λi (x0 ) and mi (x) ˜ = mi (x0 ) for all i = 1, · · · , r(x), ˜ 4) x0 ∈ Γ and r(x) where {(λi (x0 ), mi (x0 )) : 1 ≤ i ≤ r(x0 )} are the Lyapunov spectrum of ( f , T f ) at x0 . ˜ : 1 ≤ i ≤ r(x)} ˜ are called the Lyapunov exponents of (M f , θ , μ˜ ) The numbers {λi (x) at x, ˜ and mi (x) ˜ is called the multiplicity of λi (x). ˜ For simplicity, except for additional statement, we assume that μ is f -ergodic. Then the Lyapunov spectrum of ( f , T f ) are μ -almost everywhere equal to constants {(λi (μ , f ), mi (μ , f )) : 1 ≤ i ≤ r}, and so are μ˜ -a.e. the Lyapunov spectrum of (M f , θ , μ˜ ). As is stated in Theorem VIII.1.2, we assume that λi (μ , f ) = 0, ∀i. If λ1 (μ , f ) = max{λi (μ , f ) : 1 ≤ i ≤ r} < 0, then the support of μ contains only one attracting periodic orbit (see [79, Corollary 6.2]). The local stable manifold of f at each point x on this orbit is a neighborhood of x. The desired result is obvious in this trivial case, so we assume that λ1 (μ , f ) > 0. As is stated in Theorem VIII.1.2, we also assume that λr (μ , f ) = min{λi (μ , f ) : 1 ≤ i ≤ r} < 0, otherwise, the assumption λi (μ , f ) = 0, ∀i, implies that λi (μ , f ) > 0, ∀i, hence (M, f ) is an expanding map considered in Chapter III.

VIII.3

Nonuniformly Completely Hyperbolic Attractors

165

In Chapters V and VII we have developed a rigorous theory of stable and unstable manifold theorem for endomorphisms, borrowing the techniques given in Liu and  u (x) ˜ of f at x˜ in Qian [51]. Given f and a point x˜ ∈ M f , the global unstable set W f M is defined by 1 def  u (x) ˜ = {y˜ ∈ M f : lim sup log d(y−n , x−n ) < 0}, W n→+∞ n while the global unstable set of f at x˜ in M is defined by  u (x) W u (x) ˜ = pW ˜ def

if x˜ ∈ Δ˜ and λ1 (x) ˜ > 0, otherwise, by W u (x) ˜ = {x0 }. For x ∈ M, the global stable set s W (x) of f at x in M is defined by 1 def W s (x) = {y ∈ M : lim sup log d( f n y, f n x) < 0}. n→+∞ n We can assume that the Lyapunov spectrum of (M f , θ , μ˜ ) at each x˜ ∈ Δ˜ are the constants {(λi (μ , f ), mi (μ , f )) : 1 ≤ i ≤ r}. For each x˜ ∈ Δ˜ , put Exu˜ =

&

Ei (x), ˜

λi (x)>0 ˜

Exs˜ =

&

Ei (x), ˜

λi (x)<0 ˜

countable number d s = dimExs˜ and d u = dimExu˜ . As is shownin Chapter V, there is a  of compact subsets Δi , i ∈ N, of M f with i∈N Δi ⊂ Δ˜ and μ˜ (Δ˜ \ i∈N Δi ) = 0 such that: 1) Exu˜ and Exs˜ depend continuously on x˜ ∈ Δi . u 2) For each Δi , the local unstable manifolds Wloc (x) ˜ of f in M at x, ˜ x˜ ∈ Δi , con1,1 stitute a continuous family of C embedded d u -dimensional discs. There are positive numbers λiu , εiu < λiu /200, riu < 1, γiu , αiu and βiu such that the following properties hold for each x˜ ∈ Δi : i) There is a C1,1 map hux˜ : Ux˜ → Exs˜ , where Ux˜ is an open subset of Exu˜ which def

contains Exu˜ (αiu ) = {ξ ∈ Exu˜ : |ξ | < αiu }, satisfying (a) hux˜ (0) = 0, T0 hux˜ = 0; (b) Lip(hux˜ ) ≤ βiu , Lip(T· hux˜ ) ≤ βiu , where T· hux˜ : ξ → Tξ hux˜ ; u (x) ˜ = expx0 (Graph(hux˜ )). (c) Wloc u (x), ˜ there is a unique y˜ ∈ M f such that p y˜ = y0 , ii) For any y0 ∈ Wloc

d(y−n , x−n ) ≤ riu e−nεi , u

∀n ∈ N,

166

and

VIII

Ergodic Property of Lyapunov Exponents

d(y−n , x−n ) ≤ γiu e−nλi d(y0 , x0 ), u

∀n ∈ N.

f u −n x),  u (x) ˜ ∀n ∈ N}, which is called a local iii) Let W loc ˜ = {y˜ ∈ M : y−n ∈ Wloc (θ f u u (x)  ˜ → Wloc ˜ is bijective. unstable set of f at x˜ in M , then p : Wloc (x) s 3) For each Δi , the local stable manifolds Wloc (x) of f in M at x, x ∈ p Δi , constitute a continuous family of C1,1 embedded d s -dimensional discs. There are positive numbers λis , εis < λis /200, ris < 1, γis , αis and βis such that the following properties hold for each x˜ ∈ Δi : i) There is a C1,1 map hsx˜ : Ox˜ → Exu˜ , where Ox˜ is an open subset of Exs˜ which def

contains Exs˜ (αis ) = {ξ ∈ Exs˜ : |ξ | < αis }, satisfying (a) hsx˜ (0) = 0, T0 hsx˜ = 0; (b) Lip(hsx˜ ) ≤ βis , Lip(T· hsx˜ ) ≤ βis , where T· hsx˜ : ξ → Tξ hsx˜ ; s (x ) = exp (Graph(hs )). (c) Wloc 0 x0 x˜ s (x ), ii) For any y ∈ Wloc 0

d( f n y, f n x0 ) ≤ ris e−nεi , s

and

∀n ∈ N,

d( f n y, f n x0 ) ≤ γis e−nλi d(y, x0 ), s

One has W u (x) ˜ =

+∞

∀n ∈ N.

u f n (Wloc (θ −n x)) ˜

n=0

and W s (x0 ) =

+∞

s f −n (Wloc ( f n x0 )).

n=0

For each x˜ ∈ Δ˜ and sufficiently small q > 0, let U(x, ˜ q) = expx0 Ex˜ (q), where Ex˜ (q) = {(vs , vu ) ∈ p∗ Tx0 M : vs ∈ Exs˜ , vu ∈ Exu˜ , |vs | < q, |vu | < q}. The following proposition is a consequence of changing coordinates from the stable and unstable manifold theorems. Proposition VIII.3.1 For each Δi given above, there exists a number δi > 0 such   that for each x˜ ∈ Δi , y˜ ∈ Bx˜ ( δ2i ) Δi , y ∈ U(x, ˜ δ2i ) p Δi , there are C1 maps φx,˜u y˜ : s s u Exu˜ (δi ) → Exs˜ and φx,y ˜ : Ex˜ (δi ) → Ex˜ satisfying that u Wloc (y) ˜ s (y) Wloc



  U(x, ˜ δi ) = expx0 Graphφx,˜u y˜ U(x, ˜ δi ),



  s U(x, ˜ δi ) = expx0 Graphφx,y U(x, ˜ δi ), ˜

1 δi  sup{|φx,˜u y˜ (w)| + |Tw φx,˜u y˜ | : w ∈ Exu˜ (δi ), y˜ ∈ Bx˜ ( ) Δi } ≤ , 2 2

VIII.3

Nonuniformly Completely Hyperbolic Attractors

167

and

δi  1 ) p Δi } ≤ . 2 2 Now we review the absolute continuity of the local stable manifolds. For x˜ ∈ Δi and q ∈ (0, δi ], we denote byFΔi (x, ˜ q) the collection of local stable manifolds s (y) passing through y ∈ p Δ ˜ q2 ). Set Wloc i U(x, s s s sup{|φx,y ˜ ˜ (w)| + |Tw φx,y ˜ | : w ∈ Ex˜ (δi ), y ∈ U(x,

Δˆi (x, ˜ q) =

y∈ p Δ i



s Wloc (y)



U(x, ˜ q).

U(x, ˜ q2 )

Definition VIII.3.1 A submanifold W of M is called transversal to the family ˜ q) if the following hold true: FΔi (x, of a C1 map ψ : Exu˜ (q) → Exs˜ ; (i) W ⊂ U(x, ˜ q) and exp−1 x0 W is the graph  s ˜ q2 ), at exactly one point and this inter(ii) W intersects any Wloc (y), y ∈ p Δi U(x, section is transversal, i.e. s (y) = Tz M TzW ⊕ TzWloc

where z = W



s (y). Wloc

˜ q), we define For a submanifold W transversal to FΔi (x, |W | = sup |ψ (w)| + sup |Tw ψ |, w∈Exu˜ (q)

w∈Exu˜ (q)

where ψ is defined as above. We denote by mW the Lebesgue measure on W induced by the Riemannian metric on W inherited from M. Now consider two submans (y)} ifolds W1 and W2 transversal to FΔi (x, ˜ q). Since {Wloc y∈ p Δ i is a continuous 1 ˆ 1 and W ˆ 2 refamily of C embedded discs, there exist two open submanifolds W spectively of W1 and W2 such that we can well define a so-called Poincar´e map  ˆ 1  Δˆ i (x, PWˆ 1 ,Wˆ 2 : W ˜ q) → Wˆ 2 Δˆ i (x, ˜ q) by letting PWˆ 1 ,Wˆ 2 : z → Wˆ 2 



s Wloc (y)



ˆ 1 W s (y), y ∈ p Δi U(x, ˜ q2 ), and moreover, PWˆ 1 ,Wˆ 2 is a homeomorphism. for z = W loc The proposition below is a counterpart of Pesin’s absolute continuity theorem [62, Theorem 3.2.1] for diffeomorphisms. See also Theorem 4.1 in Part II of Katok and Strelcyn [33]. Proposition VIII.3.2 There exists a number qi ∈ (0, δi ] such that for every x˜ ∈ Δi and every two submanifolds W1 and W2 contained in U(x, ˜ qi ), transversal to the family FΔi (x, ˜ qi ) and satisfying |Wi | ≤ 12 , i = 1, 2, the Poincar´e map PWˆ 1 ,Wˆ b2 constructed as above is absolutely continuous with respect to the Lebesgue measures mW1 and mW2 .

168

VIII

Ergodic Property of Lyapunov Exponents

Now we assume that μ+ is an ergodic invariant measure of the C2 endomorphism (M, f ) satisfying the conditions in Theorem VIII.1.2: 1) log | det(Tx f )| ∈ L1 (M, μ+ ); 2) μ+ has SRB property; 3) the Lyapunov exponents of ( f , T f ) are μ+ -almost everywhere nonzero; the smallest Lyapunov exponent λr (μ+ , f ) < 0. u (x) Let mux˜ be the Lebesgue measure on Wloc ˜ induced by its inherited Riemannian u metric as a submanifold of M (mx˜ = δx0 if W u (x) ˜ = {x0 }). Proof of Theorem VIII.1.2. (a) We can assume that for every x˜ ∈ Δ˜ , x0 = p x˜ is generic with respect to μ+ , i.e.

1 n−1 ∑ δ f i x0 = μ+ , n→+∞ n i=0 lim

(VIII.13)

and the Lyapunov spectrum of (M f , θ , μ˜ + ) at x˜ are {(λi (μ+ , f ), mi (μ+ , f )) : 1 ≤ i ≤ r}. For every fixed x˜ ∈ Δ˜ and any y ∈ W s (x0 ), 1 lim sup log d( f n y, f n x0 ) < 0, n→+∞ n hence lim d( f n y, f n x0 ) = 0. Then from (VIII.13), one can easily get n→+∞

1 n−1 ∑ δ f i y = μ+ . n→+∞ n i=0 lim

s (x), d( f n y, f n x) ≤ γ s e−nλi d(y, x), ∀n ∈ N. For each x ∈ p Δi and any y ∈ Wloc i Exploiting this fact and followingthe line of the proof of Proposition VIII.2.3, we −n can prove that each y ∈ W s (x) \ +∞ n=0 f C f is positively regular with Lyapunov spectrum {(λi (μ+ , f ), mi (μ+ , f )) : 1 ≤ i ≤ r}. m (b) Denote by mB the normalized Lebesgue measure m(B) on a Borel subset B of M with m(B) > 0. Fix a compact set Δi such that μ˜ + (Δi ) > 0 and let qi be the constant specified in Proposition VIII.3.2. Then we can find a point x˜ ∈ Δi and a number   q ∈ (0, qi ] such that μ˜ + (B(x, ˜ q2 ) p−1U(x, ˜ q2 ) Δi ) > 0 and μ˜ + (p−1 ∂ U(x, ˜ q2 )) = 0. s u % = U(x, Let U ˜ q) and η be the measurable partition {expx0 ({w } × Ex˜ (q))}ws ∈Exs˜ (q) of U(x, ˜ q). Denote by mηy the normalized Lebesgue measure on η (y), y ∈ U(x, ˜ q), % induced by the inherited Riemannian metric, and let {mUη (y) }y∈U% be a canonical syss

%

tem of conditional measures of mU associated with the partition η . Then from the

VIII.3

Nonuniformly Completely Hyperbolic Attractors

169 %

% the measure mU is Fubini theorem, it follows clearly that for m-almost all y ∈ U, η (y) equivalent to mηy and there exists a number C > 1 such that %

C

−1

≤

dmUη (y) dmηy

≤C

(VIII.14)

holds mηy -almost everywhere on η (y). By Proposition VII.VII.2.4, we can construct a measurable partition ξ of M f subordinate to W u -manifolds of ( f , μ+ ). Let { μ˜ +,ξ (y) f be a canonical system of ˜ }y∈M ˜ conditional measures of μ˜ + associated with the partition ξ . Then by the assumption, ξ for μ˜ + -almost all y˜ ∈ M f , p (μ˜ +,ξ (y) ˜ ) is equivalent to my˜ , the normalized Lebesgue measure on p (ξ (y)) ˜ induced by the inherited Riemannian metric. As  q  q   ˜ ) p−1U(x, μ˜ + B(x, ˜ ) Δi 2 2   q  −1 q   B(x, ˜ ) p U(x, μ˜ +,ξ (y) ˜ ) Δi d μ˜ + (y) ˜ > 0, = ˜ 2 2 Mf there exists some point y˜ ∈ B(x, ˜ q2 )

(VIII.15)

 −1  p U(x, ˜ q2 ) Δi satisfying that

 q  −1 q   B( x, ˜ ) ) Δi > 0, μ˜ +,ξ (y) p U( x, ˜ ˜ 2 2 

p (ξ (y)) ˜ U(x, ˜ q2 ) contains an open neighborhood of y0 in the submanifold topology of Wy˜ (where Wy˜ is defined in Definition VII.VII.1.1 with η being replaced by ξ ), ξ and p (μ˜ +,ξ (y) ˜ ) is equivalent to my˜ , hence    q  ξ my˜ Δˆi (x, ˜ q) p (ξ (y)) ˜ U(x, ˜ ) 2    q   q  ξ ˜ ) Δi ξ (y) ˜ U(x, ˜ ) > 0. ≥ my˜ p B(x, 2 2

(VIII.16)

 s s s ˜ z∈ p Δ i Wloc (z) and Wloc (p Δ ) = z∈ p Δ˜ Wloc (z). By Proposi q % are transversal to the family tion VIII.3.1, p (ξ (y)) ˜ U(x, ˜ 2 ) and each η (y), y ∈ U, ˜ qi ). Then by the absolute continuity of FΔi (x, ˜ qi ) (Proposition VIII.3.2), for FΔi (x, s (p Δ ) = Let Wloc i



% each z ∈ U,



s mηz (Wloc (p Δi ) η (z))    q   ˆ ˜ q) p (ξ (y)) ˜ U(x, ˜ )) > 0. (VIII.17) ≥ mηz Pp (ξ (y)) ˜ U(x, ˜ q2 ),η (z) (Δ i (x, 2

% From (VIII.14) and (VIII.17), it follows that for m-almost all z ∈ U, %

s mUη (z) (Wloc (p Δi )



η (z)) > 0.

170

VIII

Ergodic Property of Lyapunov Exponents

Therefore %

s mU (Wloc (p Δi )



% = U)

 % U

   % % s mUη (z) Wloc (p Δi ) η (z) dmU (z) > 0,

(VIII.18)

and s s s m(Wloc (p Δ˜ )) ≥ m(Wloc (p Δi )) ≥ m(Wloc (p Δi )



% > 0. U)

(VIII.19)

s (p Δ˜ )) > 0 and every y ∈ W s (p Δ˜ ) is We have proved that m(W s (p Δ˜ )) ≥ m(Wloc generic with respect to μ+ . Moreover, we have shown that every z ∈ W s (pΔ˜ ) \ +∞ −n n=0 f C f is positively regular with Lyapunov spectrum

{(λi (μ+ , f ), mi (μ+ , f )) : 1 ≤ i ≤ r}. If, in addition, m(C f ) = 0, then one can inductively prove that m( f −nC f ) = 0, ∀n ∈  −n N. Hence m( +∞ n=0 f C f ) = 0, and  m W (p Δ˜ ) \ s

+∞

 f

−n

Cf

 ≥m

s (p Δ˜ ) \ Wloc

n=0

+∞

 f

−n

Cf

> 0.

n=0

It has been proved that the desired result holds for Δ = p Δ˜ .   By a simple argument, one can extend the result in Theorem VIII.1.2 to general f -invariant measures. Proposition VIII.3.3 Suppose that μ+ is an invariant Borel probability measure of a C2 endomorphism (M, f ) satisfying that: 1) log | det(Tx f )| ∈ L1 (M, μ+ ); 2) μ+ is an SRB measure of (M, f ); 3) The Lyapunov exponents of ( f , T f ) are μ+ -almost everywhere nonzero, moreover, for μ+ -almost every x ∈ M, the smallest Lyapunov exponent λr (x) of ( f , T f ) at x is less than zero. Then the following hold true: 1) Up to a set of zero measure w.r.t. μ+ , the support of μ+ is decomposed into a countable number of f -invariant measurable sets, Ak , k ∈ N; + is f -ergodic; 2) The normalization of μ+ on each Ak , μ+,k = μ+μ(A ) k

3) The basin of attraction of each Ak , W measure; For any y ∈ W s (Ak ),

s

def  (Ak ) = x∈Ak W s (x)

lim d( f n y, Ak ) = 0;

n→+∞

has positive Lebesgue

VIII.3

Nonuniformly Completely Hyperbolic Attractors

171

4) For each k and any y ∈ W s (Ak ), 1 n−1 ∑ δ f i y = μ+,k ; n→+∞ n i=0 lim



−n 5) For each k, the points in W s (Ak ) \ +∞ n=0 f C f are positively regular with constant Lyapunov exponents, where the set of critical points C f = {y ∈ M : det(Ty f ) = 0}. If, in addition, m(C f ) = 0, then

m(W s (Ak ) \

+∞

n=0

f −nC f ) > 0.

Chapter IX

Generalized Entropy Formula

The entropy and the Lyapunov exponents provide two different ways of measuring the complexity of the dynamical behavior of a C2 endomorphism f : M ← associated with an invariant measure μ . Generally speaking, the entropy of the system (M, f , μ ) is bounded up by the sum of positive Lyapunov exponents. This is the famous Ruelle inequality introduced in Chapter II. In some cases such as the invariant measure being absolutely continuous with respect to the Lebesgue measure (see Chapter VI), the inequality can become equality. The equality is the notable Pesin’s entropy formula. As we have shown in Chapter VII, Pesin’s entropy formula is equivalent to the SRB property of the invariant measure. Ledrappier and Young [43] presented a generalized entropy formula, which looks like and covers Pesin’s entropy formula, for any Borel probability measure invariant under a C2 diffeomorphism. This result is successfully generalized to random diffeomorphisms [70]. In this chapter we will extend Ledrappier and Young’s result to the case of C2 endomorphisms following the line of [71] (see Theorem IX.1.3).

IX.1 Related Notions and Statements of the Main Results Let M be an m0 -dimensional smooth and compact Riemannian manifold without boundary and f : M ← be a C2 endomorphism. Throughout this chapter, it is always assumed that the invariant measure μ of f satisfies the integrability condition (V.V.1). We will resume the settings in Chapter V and employ the results in Chapters V and VII in an essential way. Let λ1 (x) > λ2 (x) > · · · > λr(x) (x) be the Lyapunov exponents of f at x with multiplicities m1 (x), · · · , mr(x) (x). Since condition (V.V.1) holds, Tx f is invertible for μ -a.e. x. Denote by (M f , θ , μ˜ ) the system (M f , θ ) associated with μ˜ . We can apply the Oseledec multiplicative ergodic theorem to (M f , θ , μ˜ ) yielding the following. (See Appendix I, Proposition I.3.5.) There exists a θ -invariant Borel set Δ ⊂ M f

M. Qian et al., Smooth Ergodic Theory for Endomorphisms, Lecture Notes in Mathematics 1978, DOI 10.1007/978-3-642-01954-8 IX, c Springer-Verlag Berlin Heidelberg 2009 

173

174

IX

Generalized Entropy Formula

with full μ˜ -measure. For each x˜ ∈ Δ, there is a splitting (which depends measurably on x) ˜ of Tx0 M & & & ˜ E2 (x) ˜ ··· Er(x0 ) (x) ˜ (IX.1) Tx0 M = E1 (x) ˜ = mi (x0 ) for each i, such that with dimEi (x) lim

n→±∞

1 ˜ = λi (x0 ) log |T0n (x)v| n

˜ (i = 1, · · · , r(x0 )), where for any 0 = v ∈ Ei (x) def T0n (x) ˜ =

⎧ ⎨

Tx0 f n , if n > 0, id, if n = 0, ⎩ (Txn f −n )−1 , otherwise.

For each x˜ ∈ Δ, f is locally invertible along the full orbit x˜ = {xi }i∈Z . On M, the map fx˜−1 can be defined along a trajectory x˜ to be the “ inverse” map of f which maps x0 to x−1 , wherever it makes sense, i.e., fx˜−1 ◦ f = id and f ◦ fx˜−1 = id hold true on certain neighbors of x−1 and x0 respectively (see Chapter V.2). We write def −1 fx˜−n = fθ−1 −n+1 x˜ ◦ · · · ◦ f x˜ def

for n > 0 with fx˜0 = idM . Let u(x), c(x) and s(x) be the number of positive, neutral and negative Lyapunov exponents at x respectively, i.e., def

u(x) = #{1 ≤ j ≤ r(x) : λ j (x) > 0},

(IX.2)

def

c(x) = #{1 ≤ j ≤ r(x) : λ j (x) = 0},

(IX.3)

def

s(x) = #{1 ≤ j ≤ r(x) : λ j (x) < 0}.

(IX.4)

When μ is an ergodic measure, all numbers r(x), λi (x), mi (x), u(x), c(x), s(x) will be constants for μ -a.e. x. In this case, when writing them we will just omit x’s.

IX.1.1 Pointwise Dimensions and Transverse Dimensions def  i (x) Definition IX.1.1 W ˜ = {y˜ ∈ M f : lim sup n1 log d(x−n , y−n ) ≤ −λi (x0 )} is called n→+∞

 i (x) the ith -unstable set of f at x˜ in M f , where x˜ ∈ Δ and 1 ≤ i ≤ u(x0 ). W i (x) ˜ = pW ˜ th is called the i -unstable manifold of f at x˜ in M. def

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175

W i (x)’s ˜ are all C1,1 immersed submanifolds of M tangent at x0 to ⊕ij=1 E j (x) ˜ respectively (see Chapter V). Hence each W i (x) ˜ inherits a Riemannian structure from M. This gives rise to a Riemannian metric, written dxi˜ (·, ·) , on each leaf of ˜ W i (x). Definition IX.1.2 A measurable partition η of M f is said to be subordinate to W i ˜ η (x) ˜ has the following properties: manifolds if for μ˜ -a.e. x, ˜ → p η (x) ˜ is bijective; (1) p |η (x) ˜ : η (x) i

(2) There exists a ∑ mk (x0 )-dimensional C1 embedded submanifold Wx˜i of M with k=1

Wx˜i ⊂ W i (x) ˜ such that p η (x) ˜ ⊂ Wx˜i and p η (x) ˜ contains an open neighborhood of i x0 in Wx˜ , this neighborhood being taken in the topology of Wx˜i as a submanifold of M. We have included in Section IX.2.2 an outline of the construction of such partitions. See also [73] for a similar construction. Definition IX.1.3 A measurable partition η is said to be increasing, if θ −1 η > η , def and to be a generator , if B(η0n ) → B(M f ), (μ˜ − mod 0) as n → +∞, where η0n = n )

θ −k η and B(η ) denotes the σ -algebra generated by measurable η -sets.

k=0

In what follows we will define notions of transverse dimensions along unstable manifolds. Let ε > 0. For each x˜ ∈ Δ, define def  i (x) ˜ ε ) = {y˜ ∈ W ˜ : dxi˜ (x0 , y0 ) < ε }. Bi (x;

(IX.5)

Let η1 > η2 > · · · > ηu be a sequence of measurable partitions of M f with each ηi subordinate to the corresponding W i -manifolds. The canonical system of conditional measures of μ˜ associated with ηi is denoted by {μ˜ xη˜ i }. We define the lower and upper pointwise dimension of μ˜ along W i -manifolds at x˜ ∈ Δ with respect to partition ηi by   def δ i (x; ˜ ηi ) = lim inf log μ˜ xη˜ i Bi (x; ˜ ε ) / log ε , (IX.6) ε →0

  def δ i (x; ˜ ηi ) = lim sup log μ˜ xη˜ i Bi (x; ˜ ε ) / log ε . ε →0

(IX.7)

Sometimes we denote δ i (x; ˜ ηi ) by δ i (x; ˜ ηi , μ˜ ) to indicate the dependence of this quantity on μ˜ . Other notations have similar meanings. The following proposition tells us that the lower and upper pointwise dimension of μ˜ along W i -manifolds are coincident and in fact well defined on M. We call this common value the pointwise dimension of μ along W i -manifolds. It will be verified in Sections IX.4 and IX.5.

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Proposition IX.1.1 If ηi is an increasing generator subordinate to W i -manifolds, then δ i (x; ˜ ηi ) = δ i (x; ˜ ηi ), μ˜ − a.e. x; ˜ furthermore the common value, writing δi (x), ˜ is θ -invariant and depends μ˜ -a.e. only ˜ is simply denoted by δi (x0 ). on x0 , not on the choice of such ηi . Therefore δi (x) Proposition IX.1.2 Let δi (x), 1 ≤ i ≤ u(x) be introduced as above, define def

γi (x) = δi (x) − δi−1 (x) def

with δ0 (x) = 0 for μ -a.e. x and 1 ≤ i ≤ u(x), then 0 ≤ γi (x) ≤ mi (x), for i = 1, · · · , u(x). i−1 The number γi (x) is called the transverse dimension of μ on W i (x)/W ˜ at x.

IX.1.2 Statements of the Main Results The main results of this chapter are the following theorems.   Theorem IX.1.3 Let (M, f , μ ) be given such that logdet(Tx f ) ∈ L1 (M, μ ). Then entropy formula  hμ ( f ) =

∑ λi(x)+ γi (x)d μ

(IX.8)

i

holds true. Remark IX.1. Theorem II.II.1.1, the notable Margulis-Ruelle inequality, follows directly from Theorem IX.1.3 and Proposition IX.1.2. So does Theorem VII.VII.1.1, since the validity of Pesin’s entropy formula is equivalent to equations γi (x) = mi (x), i = 1, · · · , u(x), or equivalently δu (x) = ∑ui=1 mi (x) which is equivalent to the SRB property of the invariant measure μ . We will prove Theorem IX.1.3 in Sections IX.2–IX.5. Before we start the proof, we make in advance the assumption that μ (and hence μ˜ ) is ergodic for simplicity of presentation. It is used only in Sections IX.2–IX.4.

IX.2 Preliminaries In this section, we state some preliminary results which will be very useful in the subsequent sections. First we state in Section IX.2.1 some propositions about unstable manifolds, each of which is analogous of certain statement in Chapter VII (see

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177

also [51, Chapter VI, Sections 3–4]) and can be proved following the same line. Hence the proofs are omitted. In view of these propositions we can compare the ˜ with the original metric d(·, ·) on M via Lyapunov induced metric dxi˜ (·, ·) on W i (x) charts. Then we include in Section IX.2.2 an outline of the construction of measurable partitions subordinate to W i -manifolds. Some useful measurable partitions are also constructed. In Section IX.2.3 two types of transverse metrics are built on quotient spaces. Finally we present some entropy properties of the related partitions in Section IX.2.4.

IX.2.1 Some Estimations on Unstable Manifolds We write Rm0 = Rm1 × · · · × Rmr and for each z ∈ Rm0 , let (z1 , z2 , · · · , zr ) be its coordinates with respect to this splitting. The usual standard norm of Euclidean space will always be denoted by  · . Since we want to employ the Lyapunov charts introduced in Proposition VII.VII.4.2 to obtain estimations on the unstable and center unstable sets, for i = 1, · · · , c + u write ¯ (i) def = Rm1 +···+mi , R

¯ r−(i) def R = Rmi+1 +···+mr

and put ¯ (i) (ρ ) def R = {z ∈ Rm1 +···+mi : z ≤ ρ }, ¯ r−(i) (ρ ) def R = {z ∈ Rmi+1 +···+mr : z ≤ ρ }, def

¯ ρ ) = {z = (z1 , · · · , zr ) ∈ Rm0 : zi  ≤ ρ , 1 ≤ i ≤ r}. R( def

def

For each z ∈ Rm0 write z(i) = (z1 , · · · , zi ) and zr−(i) = (zi+1 , · · · , zr ); and define max¯ (i) × R ¯ r−(i) imum norms  ·  and  ·  i on Rm0 = Rm1 × · · · × Rmr and Rm0 = R respectively by z = max zi , def

1≤i≤r

def z i =

max(z(i) , zr−(i) ).

  Let 0 < ε < min 1, Δ λ /100m0 and e−λu +10ε + e5ε < 2, where Δ λ is defined def

by (VII.VII.18). Put λ0 = max{|λi | : 1 ≤ i ≤ r} + 2ε . By Proposition VII.VII.4.2, there exists a θ -invariant set Δ2 ⊂ M f of full μ˜ -measure such that {Φx˜ }x∈ ˜ Δ 2 is a system of (ε , )-charts. For i = 1, · · · , u, we introduce the local ith -unstable manifold of (M, f , μ ) at x˜ associated with ({Φx˜ }x∈ ˜ Δ 2 , δ ), where δ ∈ (0, 1]. It is defined to

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¯ δ (x) be the component of W i (x) ˜ Φx˜ R( ˜ −1 ) that contains x0 . The Φx−1 ˜ -image of i ˜ The following proposition characterizes Wx,˜i δ (x). ˜ See this set is denoted by Wx,˜ δ (x). Lemmas VII.VII.5.1 and VII.VII.5.2 and also [51, pp. 146-147] for the proof of similar results. Proposition IX.2.1 Let {Φx˜ }x∈ ˜ Δ 2 be a system of (ε , )-charts and 1 ≤ i ≤ u. (1) If 0 < δ ≤ e−λ0 −ε and x˜ ∈ Δ2 , then ˜ is the graph of a C1,1 function (i) Wx,˜i δ (x) ¯ (i) (δ (x) ¯ r−(i) (δ (x) gix˜ : R ˜ −1 ) → R ˜ −1 ) with gix˜ (0) = 0 and Lip(gix˜ ) < 1; ˜ ⊂ · · · ⊂ Wx,˜uδ (x) ˜ ⊂ Sδcu (x); ˜ (ii) Wx,˜1δ (x) (2) If 0 < δ ≤ e−2λ0 −2ε and x˜ ∈ Δ2 , then

 ¯ δ (θ x) (i) Hx˜Wx,˜i δ (x) ˜ R( ˜ −1 ) = Wθi x,˜ δ (x1 ); 

i (ii) Sδcu (x) ˜ Φx−1 ˜ = Wx,˜i δ (x). ˜ ˜ W (x)

 cu (x),  cu ˜ is deFix x˜ ∈ Δ2 . Let δ ∈ (0, 14 ]. Consider now y˜ ∈ W δ ˜ where Wδ (x) −1 i fined by (VII.VII.19). Let Wx,2 ˜ be the Φx˜ -image of the component of ˜ δ (y)  i (i) −1 r−(i) ¯ ¯ ˜ Φx˜ [R (2δ (x) ˜ )×R (4δ (x) ˜ −1 )] containing y0 . Then Φx˜W i (y) ˜ W (y) x,2 ˜ δ

contains an open neighborhood of y0 in W i (y) ˜ and is also referred to as a local i ith -unstable manifold of (M, f ) at y˜ along x˜ (although in general Φx˜Wy,2 ˜ δ (y0 ) = i Φx˜Wx,2 ˜ The following proposition holds analogue of Proposition IX.2.1 and ˜ δ (y)). can be proved following the line of the proof of Lemma VII.VII.5.3. Proposition IX.2.2 Let x˜ ∈ Δ2 and 1 ≤ i ≤ u.  cu (x), (1) Let 0 < δ ≤ 14 e−λ0 −ε . If y˜ ∈ W δ ˜ then i (i) Wx,2 ˜ is a graph of a C1 function ˜ δ (y)

¯ (i) (2δ (x) ¯ r−(i) (4δ (x) ˜ −1 ) → R ˜ −1 ) gix,˜ y˜ : R with Lip(gix,˜ y˜ ) < 1; 1 u ˜ ⊂ · · · ⊂ Wx,2 ˜ ⊂ S4cuδ (x); ˜ (ii) Wx,2 ˜ δ (y) ˜ δ (y) cu  cu (x) ˜ then (2) Let 0 < δ ≤ 14 e−2λ0 −2ε . If y˜ ∈ W δ ˜ with y1 ∈ Φθ x˜ Sδ (θ x), i ˜ Hx˜Wx,2 ˜ δ (y)



¯ (i) (2δ (θ x) ¯ r−(i)(4δ (θ x) [R ˜ −1 ) × R ˜ −1 )] ⊂ Wθi x,2 ˜ ˜ δ (θ y);

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179

(3) Let 0 < δ ≤ 14 e−2λ0 −2ε . For μ˜ -a.e. x˜ ∈ Δ2  cu (x), (i) if y˜ ∈ W δ ˜ then ˜ S2cuδ (x)



i i Φx−1 ˜ ⊂ Wx,2 ˜ ⊂ S4cuδ (x) ˜ ˜ W (y) ˜ δ (y)



i Φx−1 ˜ ˜ W (y);

u ˜ then  cu (x) (ii) if y˜ ∈ W δ ˜ with y0 ∈ Φx˜Wx, ˜ δ (x), i i+1 Wx,2 ˜ ⊂ Wx,2 ˜ i = 1, · · · , u − 1; ˜ δ (y) ˜ δ (y), u ˜ then either  cu (x) (iii) if y, ˜ z˜ ∈ W ˜ δ (x), δ ˜ with y0 , z0 ∈ Φx˜Wx, i i ˜ = Wx,2 z) Wx,2 ˜ δ (y) ˜ δ (˜

or otherwise the two terms in the above equation are disjoint. The following proposition describes the actions {Hxn˜ }n∈Z . Proposition IX.2.3 Let 0 < δ ≤ e−λ0 −ε . For each x˜ ∈ Δ2 and 1 ≤ i ≤ u ¯ −λ1 −3ε (x) ¯ θ x) ˜ −1 ), then Hx˜ z, Hx˜ z ∈ R(( ˜ −1 ) and (1) If z, z ∈ R(e Hx˜ z − Hx˜ z  ≤ eλ1 +2ε z − z  ; ¯ δ (x) ˜ −1 ) and z − z  i = z(i) − z (i) , then (2) If z, z ∈ R( Hx˜ z − Hx˜z  i = (Hx˜ z)(i) − (Hx˜ z )(i)  ≥ eλi −2ε z − z  i ; −1

−λi +2ε z − z  , ∀z, z ∈ W i (x); (3) Hx−1 ˜ z − Hx˜ z i ≤ e i x, ˜δ ˜ −1 −1

2 ε



˜ (4) Hx˜ z − Hx˜ z c+u ≤ e z − z c+u , ∀z, z ∈ Sδcu (x); u ˜  cu (x) (5) Let 0 < δ ≤ 14 e−2λ0 −2ε . For μ˜ -a.e. x˜ ∈ Δ2 , if y˜ ∈ W ˜ δ (x), δ ˜ with y0 ∈ Φx˜Wx, then −1

−λi +2ε i Hx−1 z − z  i , ∀z, z ∈ Wx,2 ˜ ˜ z − Hx˜ z i ≤ e ˜ δ (y).

The following lemma says that the metrics dxi˜ (·, ·) and d(·, ·) are locally equivalent within Φx˜Wx,˜i δ (x), ˜ a neighborhood of x0 in W i (x). ˜ ˜ then Lemma IX.2.4 If y, z ∈ Φx˜Wx,˜i δ (x), −1

d(y, z) ≤ dxi˜ (y, z) ≤ 2K0 Φx−1 ˜ z). ˜ y − Φx˜ zi ≤ 2K0 (x)d(y,

Proof. The first inequality is obvious. We prove the other two. By Proposition IX.2.1, we can assume y = Φx˜ (v0 , gix˜ (v0 )),

z = Φx˜ (v1 , gix˜ (v1 ))

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¯ (i) (δ (x)). ¯ (i) (δ (x)) for some v0 , v1 ∈ R ˜ Let {v(t)}0≤t≤1 be a smooth curve in R ˜ (i) ¯ ˜ Then connecting v0 and v1 . Let C be a collection of such curves in R (δ (x)). dxi˜ (y, z) = inf

 1 d

v(·)∈C 0

|

dt

Φx˜ (v(t), gix˜ (v(t)))|dt.

By Propositions VII.VII.4.2 and IX.2.1, we have inf

 1 d

v(·)∈C 0

|

dt

Φx˜ (v(t), gix˜ (v(t)))|dt ≤ K0 inf

 1

v(·)∈C 0

≤ 2K0 inf



 1

v(·)∈C 0

d (v(t), gix˜ (v(t)))dt dt 

d v(t)dt dt

−1

= 2K0 v1 − v0  = 2K0 Φx−1 ˜ y − Φx˜ zi . −1

˜ z) implies the last two inequalities This together with Φx−1 ˜ y − Φx˜ zi ≤ (x)d(y, in the lemma.  

IX.2.2 Related Partitions First we include here an outline of the construction of partitions subordinate to W i manifolds, which follows the same line presented in [73]. It is simpler because μ˜ is now assumed to be ergodic. ˜ ≤ Fix an i with 1 ≤ i ≤ u. Let l0 be large enough such that  = {x˜ ∈ Δ2 : (x) l0 } has positive measure. Then there exists an increasing sequence {k }k∈Z+ of  compact subsets of  such that μ˜ ( \ k k ) = 0. Fix a k with μ˜ (k ) > 0. Let ρ0 > 0 be as introduced in Chapter II.2. i (x)} 1,1 embedded ˜ x∈ Proposition IX.2.5 Let {Wloc k be a continuous family of C ˜ ∑ij=1 m j -dimensional disks described in Proposition V.V.4.5 with k in place of (i)

Δk and suitable αk . i (x) (1) For each x˜ ∈ k , Wloc ˜ ⊂ Φx˜Wx,˜i δ (x), ˜ where δ = 14 e−λ0 +ε ; i (x) (2) There exists Ak > 0 such that for all y, ˜ z˜ ∈ M f with y0 , z0 ∈ Wloc ˜ and n ≥ 0

dθi −n x˜ (y−n , z−n ) ≤ Ak e−n(λi −2ε ) dxi˜ (y0 , z0 ); (3) There exist rˆ ∈ (0, ρ0 /4), εˆ ∈ (0, 1) and dˆ ≥ 2ˆr such that for all ρ ∈ (0, rˆ] and def i (x˜ )  B(x , ρ ) ˜ εˆ ρ ) = {y˜ ∈ k : d(x, ˜ y) ˜ < εˆ ρ }, then Wloc x˜ ∈ k , if x˜ ∈ Bk (x; 0 is connected, its dxi˜ -diameter is less than dˆ and the map i x˜ → Wloc (x˜ )



B(x0 , ρ )

is a continuous map from Bk (x; ˜ εˆ ρ ) to the space of subsets of B(x0 , ρ ) (endowed with the Hausdorff topology);

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181

(4) Let ρ ∈ (0, rˆ] and x˜ ∈ k , if x˜ , x˜

∈ Bk (x; ˜ εˆ ρ ), then either i (x˜ ) Wloc



i B(x0 , ρ ) = Wloc (x˜

)



B(x0 , ρ )

or otherwise the two terms in the above equation are disjoint. In the later case, if it is assumed moreover that x

0 ∈ W i (x˜ ), then dxi˜ (y, z) > dˆ > 2ˆr 



i (x˜ ) B(x , ρ ) and z ∈ W i (x˜

) B(x , ρ ); for any y ∈ Wloc 0 0 loc (5) There exists R% > 0 such that for each x˜ ∈ k and y ∈ M, if x˜ ∈ Bk (x; ˜ εˆ ρ ) and  i

i

y ∈ Wloc (x˜ ) B(x0 , rˆ), then Wloc (x˜ ) contains the closed ball of center y and dxi˜ -radius R% in W i (x˜ ).

We now choose in k a density point x˜∗ . For each ρ ∈ [ˆr/2, rˆ], put

def

Sρ =

x∈B ˜ k (x˜∗ ;ˆε ρ )

i loc W (x) ˜



p−1 (B(x∗0 , ρ )),

def  u,i  i (x) where W (x) ˜ is defined by (V.V.32). Let ξρ denote the partition of M f loc ˜ = Wloc   i (x) into all sets W p−1 (B(x∗0 , ρ )), x˜ ∈ Bk (x˜∗ ; εˆ ρ ) and the set M f \ Sρ . We now loc ˜ define a measurable function βρ : Sρ → R+ by

% βρ (y) ˜ = inf {R, def

n≥0

ρ 1 d(y−n , ∂ B(x∗0 , ρ ))en(λi −2ε ) , }. 2Ak Ak

By arguments analogous to those in the proof of Proposition IV.2.1 in [51], we know def that there exists r ∈ [ˆr/2, rˆ] such that βr > 0 μ˜ -almost everywhere on S%i = Sr . Put +∞ 

ξi = ξr+ = def

θ n ξr .

n=0

Clearly, ξi is an increasing generator subordinate to W i -manifolds of (M, f , μ ). Let us introduce some more related partitions in order to make use of the geometry of Lyapunov charts in the evaluation of local entropy in Sections IX.3–IX.4. Let 1 −2(λ0 +ε ) ) be a reduction factor. {Φx˜ }x∈ ˜ Δ 2 be a system of (ε , )-charts and δ ∈ (0, 16 e f Then there exists a measurable partition D of M with Hμ˜ (D) < +∞ such that ) def +∞

(1) p (D + (x)) ˜ ⊂ Φx˜ Sδcu (x) ˜ for μ˜ -a.e. x, ˜ where D + = (2) (3)

n=0

θ n D;

{S%i , M f \ S%i } < D for i = 1, · · · , u; % M f \ E} % < D, where E% will be specified later in Section IX.2.3. {E,

We clearly have the following proposition.

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Proposition IX.2.6 The partition ξi is an increasing generator subordinate to W i manifolds of (M, f , μ ). Furthermore, if B i (M, f , μ ) denotes the σ -algebra of those   i ˜ then Borel sets A ⊂ M f such that A = x∈A ˜ W (x), B(

+∞ +

θ n ξi ) = B i (M, f , μ ),

μ˜ − mod 0.

n=0 def

Now define ηi = ξi

)

D + for i = 1, · · · , u. We have the following proposition.

Proposition IX.2.7 {ηi }ui=1 satisfy the following statements: (1) η1 > η2 > · · · > ηu ; ˜ ⊂ Φx˜Wx,˜i δ (x) ˜ for μ˜ -a.e. x; ˜ (2) ηi ’s are increasing generators and p(ηi (x))  − def def (3) hμ˜ (θ , ηi ) = hμ˜ (θ , ξi ) for i = 1, · · · , u, where hμ˜ (θ , η ) = Hμ˜ (η θ η ) with η − = k ∨+∞ k=0 θ η ; (4) For μ˜ -a.e. x˜ and 2 ≤ i ≤ u, if y˜ ∈ ηi (x) ˜ with y˜ ∈ Δ2 , then i−1 Φx˜Wx,2 ˜ ˜ δ (y)

and



p (ηi (x)) ˜ = p (ηi−1 (y)) ˜

θ −1 (ηi−1 (y)) ˜ = ηi−1 (θ −1 y) ˜



θ −1 (ηi (x)). ˜

The proof of item (3) in the above proposition is postponed later in subsection IX.2.4 (see the proof of Lemma IX.2.13); the other items are easy to be checked from the construction itself. We collect them all together for the convenience of the readers.

IX.2.3 Transverse Metrics on ηi (x)/ ˜ ηi−1 with 2 ≤ i ≤ u Let {Φx˜ }x∈ ˜ Δ 2 be a system of (ε , l)-charts. Fix a point x˜ ∈ Δ 2 . Let 1 ≤ i ≤ u and ¯ (i) , R ¯ r−(i) ) the space of all linear maps from R ¯ (i) to δ = 14 e−λ0 −ε . Denote by L(R  cu (x), ¯ r−(i) . By Proposition IX.2.2(1)(i) we know that, if y˜ ∈ W R δ ˜ then there exists a i (i) r−(i) i ¯ ¯ ) with Px,˜ y˜  < 1 such that unique Px,˜ y˜ ∈ L(R , R i i Ty0 Φx−1 ˜ y˜ ). ˜ Ey˜ = Graph(Px,

Define

¯ (i) ¯ r−(i) )  cu (x) Lx˜i : W δ ˜ → L(R , R y˜ → Px,˜i y˜ .

The following proposition says that the map Lx˜i is Lipschitz for all x˜ ∈ Δ2 .

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183

Proposition IX.2.8 For each x˜ ∈ Δ2 and 1 ≤ i ≤ u, Lx˜i is a Lipschitz map and ˜2 Lip(Lx˜i ) ≤ D0 (x) where D0 > 0 is a number depending only on the exponents and ε . The proof is similar to that of Lemma VII.VII.5.7 and hence is omitted. Let ηi ’s be introduced as above. For 2 ≤ i ≤ u, we now define two metrics on the factor-space ηi (x)/ ˜ ηi−1 for μ˜ -a.e. x. ˜ We shall actually deal with {ηi }i restricted to a certain measurable set with full μ˜ -measure. Now we choose a θ -invariant measurable set Δ0 ⊂ Δ2 with μ˜ (Δ0 ) = 1 such that for each x˜ ∈ Δ0 the requirements of Proposition IX.2.7 are satisfied. We then put

ηi = ηi |Δ , i = 1, · · · , u. def

0

˜ ηi−1 for μ˜ -a.e. x˜ ∈ Δ0 . In what follows we define two transverse metrics on ηi (x)/

First we give a point-dependent definition. Let x˜ ∈ Δ0 . From Proposition IX.2.7, i−1 ¯ r−(i−1) at exactly one we know that for every y˜ ∈ ηi (x), ˜ Wx,2 (y) ˜ intersects {0} × R ˜ δ th i m point. We denote the i coordinate of this point by ζy˜ ∈ R i . Clearly ζy˜i = the ith coordinate of the point (0, gi−1 ˜ y˜ ∈ ηi (x), ˜ define x, ˜ y˜ (0)). For y, def dˆxi˜ (y, ˜ y˜ ) = ζy˜i − ζy˜i 

By Proposition IX.2.7, dˆxi˜ (·, ·) induces a metric on ηi (x)/ ˜ ηi−1 for i = 2, · · · , u.



To introduce a second metric on ηi (x)/ ˜ ηi−1 for i = 2, · · · , u, we state the following lemma (straightening out lemma) without proof. Here d ≥ 2 is a fixed integer. Let positive integers n1 , · · · , nd and a number 0 < ρ < 1 be given. Dedef

note by Bi (ρ ) the closed disk centered at 0 of radius ρ in Rni . Consider B(ρ ) = B1 (ρ ) × · · · × Bd (ρ ) as a subset of Rn1 +···+nd .

Lemma IX.2.9 (See [43, Lemma 8.3.1 ].) For i = 1, · · · , d − 1, let Fi be a Lipschitz foliation with C1 leaves on some subset of Rn1 +···+nd containing B(ρ ). Assume that each leaf of Fi is the graph of a function gi : B1 (2ρ ) × · · · × Bi (2ρ ) → Rni+1 +···+nd with Dgi  ≤ 13 and that the function x → Tx Fi has Lipschitz constant smaller than some number C. Assume also that the Fi ’s are nested, i.e., if Fi (x) denotes the leaf of Fi containing the point x, then F1 (x) ⊂ F2 (x) ⊂ · · · ⊂ Fd−1(x),

∀x ∈ B(ρ )

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IX

Generalized Entropy Formula

Define O = (O1 , · · · , Od ) : B(ρ ) → Rn1 +···+nd as follows: for x = (x1 , · · · , xd ) ∈ B(ρ ), let O1 (x) = x1 , and let Oi (x) be the ith coordinate of the unique point of intersection of Fi−1 (x) and {0} × · · · × {0} × Rni+···+nd for i = 2, · · · , d. Then (1) O is a homeomorphism between B(ρ ) and its image; (2) For every x, y ∈ B(ρ ), O j (x) = O j (y) for j = i + 1, · · · , d if and only if y ∈ Fi (x); (3) Both O and O −1 are Lipschitz with Lipschitz constant depending only on C. In light of the above lemma, now we can express the metric dˆxi˜ (·, ·) in another way. Let ¯ (u) = Rm1 +···+mu p(u) : Rm0 = Rm1 +···+mr → R be the natural project map. Then for each x˜ ∈ Δ0 fixed, p(u) |W u (x) ˜ is a lipeomorx, ˜δ u phism between Wx,˜ δ (x) ˜ and its image. For i = 1, · · · , u − 1, define foliations by   def u  cu (x), Fx˜i = z ∈ p(u)Wx,˜i δ (y) ˜ : y˜ ∈ Δ0 W ˜ y ∈ Φ W ( x) ˜ , x ˜ 0 x, ˜ τδ δ where τ ∈ (0, 14 e−2(λ0 +ε ) ). By Proposition IX.2.8, these foliations satisfy the re˜ −1 and C = D0 (x) ˜ 2 , providing δ quirements of Lemma IX.2.9 with ρ = δ (x) ¯ (u) (ρ ) → and ε small enough. Hence there exists a map Ox˜ = (Ox1˜ , · · · , Oxu˜ ) : R ¯ (u) such that R ¯ (u) (ρ ) and its image; (1) Ox˜ is a homeomorphism between R j

(u) ¯ (ρ ), O (z) = O j (z ) for j = i + 1, · · ·, u iff z ∈ Fx˜i (z) and (2) For every z, z ∈ R x˜ x˜ −1 (3) Both Ox˜ and Ox˜ are Lipschitz with Lipschitz constant depending only on C. ¯ (u) be given by Let πx˜ = (πx1˜ , · · · , πxu˜ ) : Wx,˜uτδ (x) ˜ →R

πx˜ = Ox˜ ◦ p(u). def

˜ with 2 ≤ i ≤ u. We can conclude that Clearly ζy˜i = πxi˜ ◦ Φx−1 ˜ (y0 ) for each y˜ ∈ ηi (x) πx˜ is a lipeomorphism between Wx,˜uτδ (x) ˜ and its image with Lip(πx˜ ), Lip(πx−1 ˜ )≤ N(x), ˜ where N(x) ˜ depends only on (x) ˜ and the Lyapunov exponents. Moreover, i   ¯ (i) × {0} × · · · × {0}; πx˜ Wx,˜i δ (y) ˜ lies on a ∑ m j -dimensional plane parallel to R j=1     and if Wx,˜i δ (y) ˜ = Wx,˜i δ (y˜ ), then πx˜ Wx,˜i δ (y) ˜ and πx˜ Wx,˜i δ (y˜ ) lie on distinct planes. i



˜ ηi−1 , in general dˆxi˜ (·, ·) = dˆxi˜ (·, ·) for x˜ ∈ Though dˆx˜ (·, ·) is a metric on ηi (x)/



ηi (x) ˜ with x˜ = x. ˜ Now we need to rectify this situation to give a point-independent definition. Let x˜∗ be as introduced in Section IX.2.2 corresponding to k . Then there exist positive numbers τ0 and s0 with 0 < τ0 < 14 e−2(λ0 +ε ) and a set def E% = Δ0



Bk (x˜∗ ; s0 /2)

such that the following (a) and (b) hold true (where ρˆ = τ0 δ l0−1 ):

IX.2

Preliminaries

185

 

¯ ρˆ ). For i = 1, · · · , u − 1, if y˜ ∈ Δ W  cu (x) (a) Let x˜ ∈ k with x0 ∈ Φx˜∗ R( 0 δ ˜ with u ¯ ˜ Φx˜∗ R(ρˆ ), then there exists a map y0 ∈ Φx˜Wx,˜ τ δ (x) 0

¯ (i) (2ρˆ ) → R ¯ r−(i) (2ρˆ ) hix,˜ y˜ : R with Lip(hix,˜ y˜ ) <

1 3

and i ˜ Graph(hix,˜ y˜ ) = Φx−1 ˜∗ ◦ Φx˜Wx, ˜ δ (y)



¯ ρˆ ); R(2

(b) For each x˜ ∈ E% and 1 ≤ i ≤ u − 1, define a foliation by def

Fx˜i∗ ,x˜ =



p(u) (Graph(hix,˜ y˜ )) : y˜ ∈ Δ0

˜ with y0 ∈ Φx˜Wx,˜uτ0 δ (x)

 

 cu (x) W δ ˜

¯ ρˆ ) . Φx˜∗ R(

By Lemma IX.2.9, there exists a map ¯ (u) (ρˆ ) → R ¯ (u) Ox˜∗ ,x˜ : R satisfying the requirements analogous of the above (1)-(3) for Ox˜ . We then define a map π˜ = (π˜ 1 , · · · , π˜ u ) : 

+∞  n=0

¯ (u) as following: for each θ n E% → R

% suppose y0 ∈ Φx˜W u (x) ¯ ρˆ ) with x˜ ∈ k , put Φx˜∗ R( y˜ ∈ E, x, ˜ τ0 δ ˜

π˜ (y) ˜ = Ox˜∗ ,x˜ ◦ p(u) ◦ Φx−1 ˜∗ y0 def

def def ˜ y), % Thus and in general, let π˜ (y) ˜ = π˜ (θ −n(y) ˜ where n(y) ˜ = inf{k ≥ 0 : θ −k y˜ ∈ E}.



for i = 2, · · · , u we can define a point-independent metric on ηi (x)/ ˜ ηi−1 by def d˜xi˜ (y, ˜ y˜ ) = π˜ i (y) ˜ − π˜ i (y˜ ), ∀y, ˜ y˜ ∈ ηi (x). ˜

Clearly the above metrics satisfy the following two propositions. ˜ with 0 < τ < e−λ1 −3ε . Then for 1 ≤ i ≤ u Proposition IX.2.10 Let z ∈ Wx,˜uτδ (x) πθi x˜ ◦ Hx˜ z ≤ eλi +3ε πxi˜ z. Proposition IX.2.11 There exists N0 > 0 such that for all y, ˜ y˜ ∈ ηi (x) ˜ with x˜ ∈ E% N0−1 dˆxi˜ (y, ˜ y˜ ) ≤ d˜xi˜ (y, ˜ y˜ ) ≤ N0 dˆxi˜ (y, ˜ y˜ ).

186

IX

Generalized Entropy Formula

IX.2.4 Entropies of the Related Partitions About the entropies of the above partitions, we have the following Proposition IX.2.12 Let ξi and ξi be partitions subordinate to W i -manifolds constructed following the procedure presented in Section IX.2.2. Then hμ˜ (θ −1 , ξi ) = h μ˜ (θ −1 , ξi ). Proof. Let us first assume hμ ( f ) = hμ˜ (θ ) < +∞. It suffices to prove hμ˜ (θ −1 , ξi ∨ ξi ) = hμ˜ (θ −1 , ξi ). Noting that for every n ≥ 1

θ n (ξi ∨ ξi ) < ξi ∨ θ n ξi < ξi ∨ ξi

and  + Hμ˜ (ξi ∨ θ n ξi |θ θ n (ξi ∨ ξi ) ) = Hμ˜ (ξi ∨ θ n ξi |θ n+1 (ξi ∨ ξi )) = Hμ˜ (ξi |θ n+1 (ξi ∨ ξi )) + Hμ˜ (θ n ξi |ξi ∨ θ n+1 ξi ) ≤ Hμ˜ (ξi |θ n+1 ξi ) + Hμ˜ (θ n ξi |θ n+1 ξi ) = (n + 1)hμ˜ (θ −1 , ξi ) + hμ˜ (θ −1 , ξi ) ≤ (n + 2)hμ˜ (θ −1 ) < +∞ Hμ˜ (ξi ∨ ξi |θ (ξi ∨ θ n ξi )+ ) = Hμ˜ (ξi ∨ ξi |θ ξi ∨ θ n+1 ξi ) = Hμ˜ (ξi |θ ξi ∨ θ n+1 ξi ) + Hμ˜ (ξi |ξi ∨ θ n+1 ξi ) ≤ Hμ˜ (ξi |θ ξi ) + Hμ˜ (ξi |θ n+1 ξi ) = hμ˜ (θ −1 , ξi ) + (n + 1)h μ˜ (θ −1 , ξi ) ≤ (n + 2)hμ˜ (θ −1 ) < +∞, by [51, Theorem 0.5.2], we have hμ˜ (θ −1 , ξi ∨ ξi ) = h μ˜ (θ −1 , θ n (ξi ∨ ξi )) = hμ˜ (θ −1 , ξi ∨ θ n ξi ) = Hμ˜ (ξi ∨ θ n ξi |θ ξi ∨ θ n+1 ξi ) = Hμ˜ (ξi |θ ξi ∨ θ n+1 ξi ) + Hμ˜ (ξi |θ ξi ∨ θ −n ξi ). In view of Proposition IX.2.6, as n → +∞ we have k

θ ξi ∨ θ n+1 ξi " θ ξi ∨ (∧+∞ k=0 θ ξi ) = θ ξi .

IX.2

Preliminaries

Hence

187

Hμ˜ (ξi |θ ξi ∨ θ n+1 ξi ) → Hμ˜ (ξi |θ ξi ) = h μ˜ (θ −1 , ξi )

as n → +∞. Also by Proposition IX.2.6, θ ξi ∨ θ −n ξi tends increasingly to the partition of M f into single points. Thus Hμ˜ (ξi |θ ξi ∨ θ −n ξi ) → 0 as n → +∞. For the case hμ˜ (θ ) = +∞, since n := {x˜ ∈ M f : |λi (x)| M ˜ ≤ n, i = 1, · · · , r(x)} ˜ n the entropy hμ˜ n (θ ) ≤ nm < +∞ by Ruelle’s is θ -invariant and conditioned on M n ), we have inequality (where μ˜ n is the conditional measure on M hμ˜ n (θ −1 , ξi ) = hμ˜ n (θ −1 , ξi ) < +∞. The proof is finished by letting n → +∞.

 

Proposition IX.2.13 Let D be a measurable partition of M f with Hμ˜ (D) < +∞ and let ξi be partitions subordinate to W i -manifolds constructed following the procedure presented in Section IX.2.2. Then hμ˜ (θ −1 , ξi ∨ D + ) = h μ˜ (θ −1 , ξi ). Proof. The proof is similar to that of Proposition IX.2.12. One needs only to notice that θ n (ξi ∨ D + ) < ξ ∨ θ D + < ξ ∨ D + and

ξ < ξ ∨ D < ξ ∨ D+

and check the conditions Hμ˜ (ξ ∨ θ D + |θ n+1 (ξi ∨ D + )) < +∞, and

Hμ (ξ ∨ D + |θ (ξ ∨ θ D + )) < +∞

Hμ˜ (ξ ∨ D|θ ξ ) < +∞.

Hence the proof is omitted. The following proposition is just Corollary VII.8.1.1 restated here.

 

Proposition IX.2.14 For any partition ξu subordinate to W u -manifolds of the type as constructed following the procedure presented in subsection IX.2.2, we have hμ˜ (θ −1 , ξu ) = hμ˜ (θ −1 ) = hμ ( f ).

188

IX

Generalized Entropy Formula

IX.3 Definitions of Local Entropies along Unstable Manifolds In this section, we will define quantities named local entropy along unstable manifolds. These quantities play an important role in our arguments. In ergodic case, the notion of local entropy along W i -manifolds is described by a number hi which measures the amount of randomness along the leaves of W i -manifolds. As Ledrappier and Young have done in [43], though there are several equivalent definitions, we take a pointwise approach following [13]. Let ε > 0. For x˜ ∈ Δ2 and n ∈ Z+ , put  def  i (x) Bi (x; ˜ n, ε ) = y˜ ∈ W ˜ : dθi k x˜ (xk , yk ) < ε for 0 ≤ k ≤ n . Let η be a measurable partition of M f subordinate to W i -manifolds. Define 1 def ˜ ε , η ) = lim inf − log μ˜ xη˜ (Bi (x; ˜ n, ε )), hi (x; n→+∞ n 1 def hi (x; ˜ ε , η ) = lim sup − log μ˜ xη˜ (Bi (x; ˜ n, ε )). n n→+∞ One can easily show that these functions are indeed measurable. Furthermore, we define the lower and upper local entropy along W i -manifolds at x˜ with lrespect to η by def

hi (x; ˜ η ) = lim hi (x; ˜ ε , η ), ε →0

def

hi (x; ˜ η ) = lim hi (x; ˜ ε , η ). ε →0

˜ ε , η ) and hi (x; ˜ ε , η ) increase as ε ↓ 0. These limits exist because hi (x; Proposition IX.3.1 Let ξi be an increasing generator subordinate to W i -manifolds. Then ˜ ξi ) = hi (x; ˜ ξi ) =: hi = Hμ˜ (ξi |θ ξi ), μ˜ − a.e. x. ˜ hi (x; The proposition above tells us that the lower and upper local entropy along W i manifolds with respect to ξi are coincident. From the proof below and the ergodic decompositions of μ and μ˜ in Section 5, we know that in general hi depends only ξ on x0 and is an f -invariant function independent of the choice of ξi or {μ˜ x˜ i } (see Propositions IX.2.12–IX.2.14). So we write hi = hi (x0 ) and call it the local entropy along W i -manifolds at x0 . This completes the definition of hi . Let us first introduce some facts and postpone the proof of Proposition IX.3.1 at the end of this section. Lemma IX.3.2 Let α be a measurable partition of M f with Hμ˜ (α ) < +∞ and let ξ be an increasing generator. Then

IX.3

Definitions of Local Entropies along Unstable Manifolds

189

  1 ξ lim − log μ˜ x˜ ([α ξ ]n0 (x)) ˜ = Hμ˜ (ξ θ ξ ), μ˜ − a.e. x˜ ∈ M f . n

n→+∞

Lemma IX.3.3 There exists a measurable partition α of M f with Hμ˜ (α ) < +∞ such that  ˜ ⊂ Bi (x; ˜ n, δ ), ∀n ≥ n0 (x) ˜ [α0n ξi ](x) for μ˜ -a.e. x, ˜ where n0 : M f → Z+ is a measurable function.  def ξ Proof of Lemma IX.3.2. Define I(η ξ )(x) ˜ = − log μ˜ x˜ (η (x)). ˜ One has       1 1 1 n −1 I([α ξ ]n0 ξ )(x) ˜ = I(α ξ )(x) ˜ + ∑ I(α ξ θ ξ α−k )(θ k x), ˜ n n n k=0 def

l = where α−k

l )

def

θ − j α . Put In (x) ˜ = I(α

) 

ξ θ ξ

j=−k

(IX.9)

) −1 def α−n )(x) ˜ and I ∗ (x) ˜ = sup In (x). ˜ n≥1

One can prove that



0 and that {In , B(α−n

)

I ∗ (x)d ˜ μ˜ ≤ Hμ˜ (α

 

ξ θ ξ ) + 1

ξ )} is a supermartingale. Therefore L1 In −→ I∞ . μ˜ − a.e.

Hence the second term in the right side of equation (IX.9) tends μ˜ -a.e. to a θ -invariant Borel function F ∈ L1 . Then by the ergodicity of μ˜ , one has   1 I([α ξ ]n0 ξ )(x) ˜ = n→+∞ n

lim



Fd μ˜ =



I∞ d μ˜ .

So the limit function in (IX.9) is constant almost everywhere and is therefore equal to   1 lim Hμ˜ ([α ξ ]n0 ξ ) n→+∞ n which can be written as   1 1 Hμ˜ (ξ0n ξ ) + lim Hμ˜ (α0n θ −n ξ ). n→+∞ n n→+∞ n  The first term is equal to Hμ˜ (ξ θ ξ ). The second term goes to 0 since θ −n ξ generates.   Proof of Lemma IX.3.3. Without loss of generality, let ξi and S%i be as introduced in Section IX.2 and 0 < δ < 14 e−2λ0 −2ε . Let {Φx˜ }x∈ ˜ Δ 2 be a system of (ε , )-charts. def % 

Put S = Si {x˜ ∈ Δ2 : (x) ˜ ≤ l0 }, where l0 is large enough such that μ˜ (S ) > 0. lim

190

IX

Generalized Entropy Formula

First we define n+ , n− and n0 : S → Z+ by n+ (x) ˜ = inf{n > 0 : θ n x˜ ∈ S }, def

n− (x) ˜ = inf{n > 0 : θ −n x˜ ∈ S }, def

n0 (x) ˜ = inf{n ≥ 0 : θ n x˜ ∈ S }. def

Then let ψ : M f → R be given by $ def

ψ (x) ˜ =

δ −(λ0 +ε ) max(n+ (x),n ˜ − (x)) ˜ , 2K0 l0 e δ 2K0 l0 ,

if x˜ ∈ S , otherwise.

Finally we( define ψ+ by replacing max(n+ , n− ) in the definition of ψ by n+ . Since − log ψ d μ˜ < ∞, there exists a measurable partition α of M f with Hμ˜ (α ) < ∞ such that p (α (x)) ˜ ⊂ B(x0 , ψ (x)) ˜ for almost every x. ˜ ˜ and n0 = n0 (x) ˜ for simplicity of notation in Let x˜ ∈ S . We will write n+ = n+ (x) the rest of this section. We assert that

 i (x) Claim 1 If y˜ ∈ W ˜ with y0 ∈ Φx˜Wx,˜i δ (x) ˜ satisfies Φx−1 ˜ then ˜ y0 i ≤ l0 ψ+ (x),

dθi j x˜ (x j , y j ) < δ for 0 ≤ j ≤ n+ and yn+ ∈ Φθ n+ x˜Wθi n+ x,˜ δ (θ n+ x); ˜   i (x) ˜ with y0 ∈ p(α0n (x)) ˜ Φx˜Wx,˜i δ (x) ˜ for some n ≥ 0, then Claim 2 If y˜ ∈ W dθi j x˜ (x j , y j ) < δ for 0 ≤ j ≤ n; Claim 3 If y˜ ∈ [α0n0

)

ξi ](x), ˜ then dθi j x˜ (x j , y j ) < δ for 0 ≤ j ≤ n0

and yn0 ∈ Φθ n0 x˜Wθi n0 x,˜ δ (θ n0 x). ˜ Let’s postpone the proof of the above claims and first proceed the proof of Lemma IX.3.3. Consider now an arbitrary point x˜ with the property that θ n x˜ ∈ S

) infinitely often as n → ±∞. If y˜ ∈ [α0n ξi ](x) ˜ with n ≥ n0 , then by Claim 3 dθi j x˜ (x j , y j ) < δ for 0 ≤ j ≤ n0 and yn0 ∈ Φθ n0 x˜Wθi n0 x,˜ δ (θ n0 x). ˜ Then we can apply Claim 2 to θ n0 y˜ yielding that dθi j x˜ (x j , y j ) < δ for n0 ≤ j ≤ n, ) ˜ n, δ ) for all y˜ ∈ [α0n ξi ](x). ˜ which implies y˜ ∈ Bi (x;

 

IX.3

Definitions of Local Entropies along Unstable Manifolds

191

Proof of Claim 1. It follows from our assumptions on y˜ and Proposition i (θ j x) ˜ and (note that λ0 ≥ λ1 + 2ε ) IX.2.3(1) that Hx˜j Φx−1 ˜ y0 ∈ Wθ j x, ˜δ

j λ0

Hx˜ Φx−1 Φx−1 ˜ y0  i ≤ e ˜ y0 i for j > 0 j

−λ1 −3ε provided that Φx−1 l(θ k x) ˜ −1 for all 0 ≤ k < j. This is guar˜ y0 i exp(kλ0 ) ≤ e i j ¯ (i) (δ l(θ j x) ˜ is a graph over R ˜ −1 ) with slope< 1, anteed for j ≤ n+ . Since Wθ j x,˜ δ (θ x) one has dθi j x˜ (x j , y j ) ≤ 2K0 Φθ−1j x˜ y j  i < δ for 0 ≤ j ≤ n+ .2

˜ then Φx−1 ˜ ≤ l0 ψ+ (x). ˜ So we Proof of Claim 2. First if y˜ ∈ α (x), ˜ y0  ≤ l0 ψ (x) have the desired conclusion for 0 ≤ j ≤ n+ . Furthermore, if n ≥ n+ and y˜ ∈ α0n (x), ˜ then  yn+ ∈ p (α (θ n+ x)) ˜ Φθ n+ x˜Wθi n+ x,˜ δ (θ n+ x) ˜ and we can apply Claim 1 to θ n+ y˜ with θ n+ x, ˜ xn+ and yn+ in place of x, ˜ x0 and y0 respectively. An inductive argument completes the proof of Claim 2.   Proof of Claim 3. To prove this claim, let us assume that x˜ ∈ S and for simplicity of notation write k = n0 − n−(θ n0 x). ˜ That is, k is the largest integer< 0 such that θ k y˜ ∈ S . Clearly, n+ (θ k x) ˜ = n0 − k = n− (θ n0 x). ˜ Since θ −n ξ is increasing as k k i n → +∞, we have θ y˜ ∈ ξ (θ x) ˜ and yk ∈ Φθ k x˜Wθ k x,˜ δ (θ k x) ˜ by our choice of S%i . Also

ψ is chosen in such a way that p[θ − j (α (θ n0 x))] ˜ lies well inside the charts at xn0 − j for j = 1, 2, · · · , n0 − k. Hence

k+1 −1

Φθ k x˜ yk  i = Hxk˜ Φx−1 ˜ y0 i ≤ Hx˜ Φx˜ y0 i ≤ · · ·

−1

≤ Hx˜ 0 Φx−1 ˜ y0 i = Φθ n0 x˜ yn0 i n

≤ l0 ψ (θ n0 x) ˜ ≤ l0 ψ+ (θ k x). ˜ Therefore by Claim 1, Claim 3 holds true.   ˜ ξi ) Proof of Proposition IX.3.1. It follows directly from the definition of hi (x; ˜ ξi ) ≤ Hμ˜ (ξi |θ ξi ) for μ˜ -a.e. x. ˜ together with Lemmas IX.3.2 and IX.3.3 that hi (x; What remains is to verify hi (x; ˜ ξi ) ≥ Hμ˜ (ξi |θ ξi ), μ˜ − a.e.x˜ ∈ M f We know that Hμ˜ (ξi |θ ξi ) = Hμ˜ (θ −1 ξi |ξi ) =



ξ

− log μ˜ x˜ i ((θ −1 ξi )(x))d ˜ μ˜ ,

where the item behind log is a conditional measure of the denoted set. Put def

ξ

g(x) ˜ = − log μ˜ x˜ i ((θ −1 ξi )(x)), ˜ def f i  Aδ = {x˜ ∈ M : B (x; ˜ δ ) ⊂ ξi (x)}. ˜

(IX.10)

192

IX

Generalized Entropy Formula

Since ξi is an increasing generator subordinate to W i -manifolds, one has Aδ ↑ and μ˜ (Aδ ) ↑ 1 as δ ↓ 0. Hence given ε > 0, there exists δ > 0 such that for each δ ∈ (0, δ )  g(x)d ˜ μ˜ ≥ Hμ˜ (ξi |θ ξi ) − ε .

θ −1 Aδ



def

˜ n, δ ) = Define U i (x; Hence

0≤k≤n,θ k x∈A ˜

(θ −k ξi )(x). ˜ Then Bi (x; ˜ n, δ ) ⊂ U i (x; ˜ n, δ ). δ

ξ

ξ

− log μ˜ x˜ i (Bi (x; ˜ n, δ )) ≥ − log μ˜ x˜ i (U i (x; ˜ n, δ )).

(IX.11)

Furthermore, we will prove that for μ˜ -a.e. x˜ ξ

− log μ˜ x˜ i (U i (x; ˜ n, δ )) ≥

n−1

˜ ∑ (1θ −1Aδ · g)(θ k x),

(IX.12)

k=0

which combined with inequality (IX.11) implies 1 1 n−1 ξ − log μ˜ x˜ i (Bi (x; ˜ n, δ )) ≥ ∑ (1θ −1 Aδ · g)(θ k x). ˜ n n k=0 Then applying Birkhoff’s ergodic theorem, one has for each δ ∈ (0, δ ) hi (x; ˜ δ , ξi ) ≥

 θ −1 Aδ

g(x)d ˜ μ˜ ≥ Hμ˜ (ξi |θ ξi ) − ε .

Now we return to the proof of the assertion (IX.12). Let def

τ (x; ˜ n) = max{0 ≤ k ≤ n : θ k x˜ ∈ Aδ }, def

˜ n) is a finite number with sufficiently where max 0/ = +∞. For μ˜ -a.e. x˜ fixed, τ (x; large n by Poincar´e’s recurrence theorem. This together with the condition that ξi is an increasing generator yields ξ ˜ n, δ )) = − log μ˜ x˜ i (U i (x;

τ (x;n)−1 ˜

∑

θ − log μ˜ x˜

−k ξ i

((θ −k−1 ξi )(x)) ˜

k=0

≥ =

τ (x;n)−1 ˜

∑

k=0 n−1

(1θ −1 Aδ · g)(θ k x) ˜

˜ ∑ (1θ −1 Aδ · g)(θ k x).2

k=0

IX.4

Estimates of Local Entropies along Unstable Manifolds

193

IX.4 Estimates of Local Entropies along Unstable Manifolds In this section we will estimate local entropy along unstable manifolds through Lyapunove exponents and transverse dimensions. To be explicit, we will prove in this section via Propositions IX.4.1, IX.4.3, IX.4.6 and IX.4.7 that

δ i = δ i =: δi , 1 ≤ i ≤ u ˜ and δ i = δ i (x), ˜ and where δ i = δ i (x) def

γi = δi − δi−1 = (hi − hi−1)/λi ≤ mi , 1 ≤ i ≤ u

(IX.13)

with δ0 = 0 and h0 = 0. Thus identity u

h μ ( f ) = ∑ λi γi i=1

follows from hu = h μ˜ (θ ) = h μ ( f ).

IX.4.1 Estimate of Local Entropy h1 The local entropy h1 measures the amount of randomness along the leaves of W 1 manifolds and can be formulized as the following. Proposition IX.4.1 For μ˜ -a.e. x, ˜ δ 1 (x) ˜ = δ 1 (x) ˜ =: δ1 =: γ1 and h1 = λ1 γ1 . By Lemma VII.VII.7.5 and from the definition of δ 1 and δ 1 , it is clear that 0 ≤ γ1 ≤ m1 . Proof. Let δ > 0 be a sufficiently small number. We divide the proof into two parts. (i) First for each y˜ ∈ B1 (x; ˜ ρn ) with ρn = 12 K0−1 (x) ˜ −1 e−n(λ1 +2ε ) δ , one has y0 ∈ 1 (x) ˜ and dx1˜ (x0 , y0 ) < ρn . Hence by Proposition IX.2.3 and Lemma IX.2.4, for Wloc 1 (θ k x) each k ≤ n, yk ∈ Wloc ˜ and −1 −1 k k

k

dθ1 k x˜ (xk , yk ) ≤ 2K0 Φθ−1 k x˜ ◦ f x0 − Φθ k x˜ ◦ f y0 1 = 2K0 Hx˜ ◦ Φx˜ y0 1 k(λ1 +2ε ) ≤ 2K0 (x)d(x ˜ 0 , y0 )e ≤ 2K0 (x)d ˜ x1˜ (x0 , y0 )ek(λ1 +2ε ) < δ

Therefore B1 (x; ˜ ρn ) ⊂ B1 (x; ˜ n, δ ) for μ˜ -a.e. x˜ and all n ≥ 0. This implies h1 ≤ λ1 δ 1 . (ii) We then prove that h1 ≥ λ1 δ 1 . Let ξi ’s be as introduced in Section 3.3 and let α be as introduced in Lemma IX.3.3. By Lemma IX.3.3, one has [α0n



ξi ](x) ˜ ⊂ Bi (x; ˜ n, δ ),

∀n ≥ n0 (x). ˜

194

IX

Generalized Entropy Formula

n)

i (θ n x) Therefore for each y˜ ∈ (α0 ξi )(x) ˜ with n ≥ n0 (x), ˜ one has yn ∈ Wloc ˜ and i dθ n x˜ (xn , yn ) < δ . By Proposition IX.2.3(3)

−n −1

dxi˜ (x0 , y0 ) ≤ 2K0 Φx−1 ˜ y0 i = 2K0 Hθ n x˜ ◦ Φθ n x˜ yn 1

≤ 2K0 e−n(λi −2ε ) Φθ−1 n x˜ yn 1

n ˜ i (x , y ) < δ (x)e where by Lemma IX.2.4 Φθ−1 ˜ nε . Hence n x˜ yn i ≤ l(θ x)d θ n x˜ n n

dxi˜ (x0 , y0 ) < 2K0 δ e−n(λi −3ε ) (x). ˜ This implies for μ˜ -a.e. x˜ [α0n and hence



ξi ](x) ˜ ⊂ Bi (x; ˜ 2K0 δ (x)e ˜ −n(λi −3ε ) ),

[α0n



ξi ](x) ˜ ⊂ Bi (x; ˜ e−n(λi −4ε ) ),

∀n ≥ n0 (x) ˜

∀n ≥ n 0 (x) ˜

(IX.14)

def

with n 0 (x) ˜ = max{n0 (x), ˜ [ ε1 log(2K0 δ (x))] ˜ + 1}. By Lemma IX.3.2, equation   (IX.14) with i = 1 implies h1 ≥ λ1 δ 1 .

IX.4.2 Estimate of Local Entropy hi from Below with 2 ≤ i ≤ u Based on Proposition IX.4.1, we will then prove the coincidence of δ i and δ i for i = 2, · · · , u. For this end, let us assume from here on that we have proved inductively the coincidence of δ j and δ j for j = 1, 2, · · · , i − 1, i.e.,

δ j = δ j =: δ j , for j = 1, 2, · · · , i − 1. Then, in view of (IX.14), Proposition IX.3.1 and Lemmas IX.3.2 and IX.3.3, the following lemma holds. Lemma IX.4.2 For each sufficiently small ε > 0, let α be as introduced in Lemma IX.3.3 (with δ > 0 small enough). There exists a Borel function n% : M f → Z+ satisfying the following for μ˜ -a.e. x˜ (where 2 ≤ i ≤ u): (1) [ξi

) n α0 ](x) ˜ ⊂ Bi (x; ˜ e−n(λi −4ε ) ) for any n ≥ n%(x); ˜ ξ

˜ ≥ hi−1 − ε for any n ≥ n%(x); ˜ (2) − 1n log μ˜ x˜ i−1 (α0n (x)) ξi 1 n (3) − n log μ˜ x˜ (α0 (x)) ˜ ≤ hi + ε for any n ≥ n%(x); ˜ def i−1 −n( λ −4 ε ) i ˜e ) ⊂ ξi−1 (x) ˜ for any n ≥ n%(x); ˜ (4) L = B (x; / ξi−1 (5) log μ˜ x˜ (L) [−n(λi − 4ε )] ≤ δ i−1 + ε for any n ≥ n%(x); ˜ / ξi i −n( λ −4 ε ) i ˜ 2e )) [−n(λi − 4ε )] ≥ δ i − ε for infinitely many n ≥ n%(x). ˜ (6) log μ˜ x˜ (B (x;

IX.4

Estimates of Local Entropies along Unstable Manifolds

195

We then give a lower bound of the local entropy hi in terms of Lyapunov exponents and pointwise dimensions. ˜ (hi − hi−1 )/λi ≥ δ i − δ i−1 . Proposition IX.4.3 For 2 ≤ i ≤ u and μ˜ -a.e. x, Below is the famous Borel density lemma on a manifold M. Lemma IX.4.4 (Borel Density Lemma; See [7, Proposition 3] or [23].) Let m be a Borel probability measure on a manifold M and let A ⊂ M be a measurable set with m(A) > 0. Then for m-almost every x ∈ A 

m(A B(x, ρ )) = 1. lim ρ →0 m(B(x, ρ )) % ⊂ A with m(A) % > m(A) − δ and a Furthermore, for each δ > 0 there is a set A ∗ ∗ % number ρ such that for all x ∈ A and 0 < ρ < ρ one has m(A



1 B(x, ρ )) ≥ m(B(x, ρ )). 2

But in general, the inverse limit space M f is not a finite dimensional manifold. In the proof of Proposition IX.4.3, one has to overcome this deficiency of a Borel Density Lemma on M f ; Thus we establish the following slight variant of Density Lemma. Lemma IX.4.5 Let A ⊂ M f be a measurable set with μ˜ (A) > 0. Then for μ˜ -almost every x˜ ∈ A  ξ μ˜ x˜ i (A Bi (x, ˜ ρ )) lim = 1, (IX.15) ξ i ρ →0 μ (B i (x, ˜ ρ )) x˜

where 1 ≤ i ≤ u and each ξ is a measurable partition subordinate to the corre% ⊂ A with sponding W i -manifolds. Furthermore, for each δ > 0 there is a set A ∗ % % μ˜ (A) > μ˜ (A) − δ and a number ρ such that for all x˜ ∈ A and 0 < ρ < ρ ∗ one has

μ˜ (A



1 Bi (x, ˜ ρ )) ≥ μ (Bi (x, ˜ ρ )). 2

Proof. Let A be the set consisting of those points in A satisfying (IX.15). It is clearly measurable. Fix a point x˜ with the properties described in Definition IX.1.2 ˜ The induced distance on pC will be denoted as dC (·, ·), which and write C = ξi (x). is dxi˜ (·, ·) restricted on pC. We write a ball in pC centered at y with radius ρ as BC (y, ρ ) := {z ∈ pC : dC (y, z) < ρ }. In view of (IX.5), we have ˜ ρ ) = BC (y0 , ρ ), pC ∩ pBi (y;

∀y. ˜

196

IX

Generalized Entropy Formula

% on pC ⊂ M Since pC := p|C : C → pC is bijective, we can define a measure μ % ) becomes a measure preserving bijection. % := pC μ˜ C . pC : (C, μ˜ C ) → (pC, μ by μ Obviously % (pBi (y; % (BC (y0 , ρ ) ∩ pC (A)). μ˜ C (Bi (y; ˜ ρ ) ∩ A) = μ ˜ ρ ) ∩ pC (A)) = μ Therefore Borel Density Lemma on M [23] gives lim

ρ →0

% (BC (y0 , ρ ) ∩ pC (A)) μ =1 % (BC (y0 , ρ )) μ

(IX.16)

% -a.e. y0 ∈ pC (A). For such y0 define y˜ := pC−1 (y0 ) ∈ C. Then (IX.16) is equivafor μ lent to μ˜ (Bi (y; ˜ ρ ) ∩ A) lim C = 1. (IX.17)  ρ →0 μ ˜ (Bi (y; ˜ ρ )) C

Clearly each y˜ ∈ C ∩ A satisfying the above equation must be a point in A . Of % -a.e. y0 ∈ pC (A) satcourse each y˜ ∈ C ∩ A satisfies (IX.17). The assertion that μ isfies (IX.16) is equivalent to % (pC (A )) = μ % (pC (A)), μ which can be rewritten as μ˜ C (A ) = μ˜ C (A). Since this holds for μ˜ ξi -a.e. C, we have

μ˜ (A ) =



μ˜ C (A )d μ˜ ξi (C) =



μ˜ C (A)d μ˜ ξi (C) = μ˜ (A),

which implies the validity of (IX.15) for μ˜ -a.e. x˜ ∈ A. Proof of Proposition IX.4.3. Let

 

Γn := {x˜ : n%(x) ˜ ≤ n and x˜ satisfies the requirements (1)–(6) of Lemma IX.4.2}. Clearly we have μ˜ (Γn ) ↑ 1 as n tends to +∞. Therefore for any ε ∈ (0, 1), there is an integer N1 such that Γ := ΓN1 has μ˜ -measure≥ 1 − ε /2. Then by Lemma IX.4.5, there is another integer N2 ≥ N1 and a subset Γ% ⊂ Γ of μ˜ -measure ≥ 1 − ε such that for any x˜ ∈ Γ% fixed, we have ξ μ˜ x˜ i−1 (L



1 ξ Γ ) ≥ μ˜ x˜ i−1 (L) 2

for any n ≥ N2 , where L = Bi−1 (x; ˜ e−n(λi −4ε ) ). Then according to Lemma IX.4.2(5) one has  1 ξ μ˜ x˜ i−1 (L Γ ) ≥ exp(−n(λi − 4ε )(δ i−1 + ε )). 2

IX.4

Estimates of Local Entropies along Unstable Manifolds

197

By Lemma IX.4.2(2) it is clear that ξ ξ μ˜ x˜ i−1 (α0n (y)) ˜ = μ˜ y˜ i−1 (α0n (y)) ˜ ≤ exp(−n(hi−1 − ε ))



for each y˜ ∈ L Γ . Hence for any x˜ ∈ Γ% and n ≥ N2 #{α0n (y) ˜ : y˜ ∈ L



ξ Γ } ≥ μ˜ x˜ i−1 (L

≥



/ Γ ) exp(−n(hi−1 − ε ))

1 exp{n[hi−1 − ε − (λi − 4ε )(δ i−1 + ε )]}. 2

On the other hand, according to Lemma IX.4.2(1), one has [ξi



α0n ](y) ˜ ⊂ Bi (y; ˜ e−n(λi −4ε ) ),

∀y˜ ∈ Γ .

Clearly dxi˜ (x0 , y0 ) ≤ dxi−1 ˜ (x0 , y0 ) for any y˜ ∈ L. Therefore [ξi



α0n ](y) ˜ ⊂ Bi (x; ˜ 2e−n(λi −4ε ) ),

∀y˜ ∈ L



Γ .

Hence for any x˜ ∈ Γ% and n ≥ N2 ξ

log μ˜ x˜ i (Bi (x; ˜ 2e−n(λi −4ε ) )) ≥ log #{[ξi



α0n ](y) ˜ : y˜ ∈ L

≥ log #{α0n (y) ˜ : y˜ ∈ L





Γ } + logmin μ˜ xξ˜ i ([ξi y˜



α0n ](y)) ˜

Γ } + logmin μ˜ xξ˜ i (α0n (y)) ˜ y˜

≥ − log 2 − n[hi − hi−1 + 2ε + (λi − 4ε )(δ i−1 + ε )]. Comparing the above inequality with Lemma IX.4.2(6), we obtain (δ i − δ i−1 − 2ε )(λi − 4ε ) ≤

1 log 2 + hi − hi−1 + 2ε , n

which implies Proposition IX.4.3 by letting n → +∞, ε → 0 and finally ε → 0.  

IX.4.3 Estimate of Local Entropy hi from Above with 2 ≤ i ≤ u Let number N0 , map π˜ i : ηi (x) ˜ → Rmi and metrics d˜xi˜ (·, ·) and dˆxi˜ (·, ·) on ηi (x)/ ˜ ηi−1 be as introduced in Section IX.2. We denote μ˜ xη˜ i by μ˜ xi˜ for simplicity of notations. def ¯ (i) and put Write π˜i = (π˜ 1 , · · · , π˜ i ) : ηi (x) ˜ →R def ˜ ρ ) = {y˜ ∈ ηi (x) ˜ : d˜xi˜ (x, ˜ y) ˜ < ρ }. B%i (x,

Then we define transverse dimensions as the following.

198

IX

Generalized Entropy Formula

˜ i %i

˜ ρ )) log μx˜ (B (x; Definition IX.4.1 γ˜i (x) ˜ = lim inf is called the transverse dimension of log ρ def

ρ →0

ηi /ηi−1 at x. ˜

¯ (i) :zi −π˜ i (y)< ˜ ρ }) log ρ

Definition IX.4.2 For each y˜ ∈ ηi (x), ˜ γ%i (y; ˜ x) ˜ = lim inf log ν ({z∈R def

ρ →0

is

i−1 at y, called the transverse dimension of W i (x)/W ˜ ˜ where ν = μ˜ xi˜ ◦ π˜i−1 is a Borel (i) ¯ . probability on R def

Now we introduce the main results for this subsection. Proposition IX.4.6 Given β ∈ (0, 1). One has for sufficiently small δ and μ˜ -a.e. x˜ ˜ ≥ (1 − β )(hi − hi−1 − β ) (λi + β )γ˜i (x) ˜ ≤ mi . Hence and 0 ≤ γ˜i (x)

hi − hi−1 ≤ λi mi .

(IX.18)

˜ one has γ˜i (x) ˜ = γ%i (y; ˜ x) ˜ Proposition IX.4.7 Given β ∈ (0, 1). For μ˜ xi˜ -a.e. y˜ ∈ ηi (x), and (1 − β )(hi − hi−1 − β ) δ i − δ i−1 ≥ γ%i (y; ˜ x) ˜ ≥ . (IX.19) λi + β Hence by letting β → 0 we obtain (hi − hi−1 )/λi ≤ δ i − δ i−1 .

(IX.20)

In order to prove the above two proposition, we need the following results; since they are of pure measure theoretical nature and are simple consequences of the above Borel Density Lemma, we state them here without proof. Lemma IX.4.8 (See [43, Lemma 11.3.1].) Let μ be a probability measure on R p × Rq , π : R p × Rq → R p the natural projection. Let { μt } be a canonical system of conditional measures of μ associated with {{t} × Rq : t ∈ R p }. Define   log μ π −1 B p (t, ρ ) γ (t) = lim inf ρ →0 log ρ def

and let δ ≥ 0 be such that at μ -a.e. (t, x)   log μt Bq (x, ρ ) δ ≤ lim inf ρ →0 log ρ holds true, where B p (t, ρ ) is the open disk in R p centered at t of radius ρ . Then at μ -a.e. z = (t, x)   log μ B p+q(z, ρ ) δ + γ (t) ≤ lim inf ρ →0 log ρ holds true.

IX.4

Estimates of Local Entropies along Unstable Manifolds

199

Lemma IX.4.9 (See [43, Lemma 11.3.2].) Let (Ω , P) be an abstract probability space which is a Polish space. Let μ be a probability measure on Ω × Rq with marginal measure P on Ω . Let γ˜ ≥ 0 be such that at μ -a.e. (ω , x)   log μω Bq (x, ρ ) γ˜ ≤ lim inf ρ →0 log ρ holds. Then at μ -a.e. (ω , x)   log μ Ω × Bq(x, ρ ) γ˜ ≤ lim inf . ρ →0 log ρ % 4 μ˜ (E)

Proof of Proposition IX.4.6. Let e−(λi +β )ε N0 < 1 with β ∈ (0, 1) (this holds % small enough). We will prove that for μ˜ -a.e. x˜ true provided μ˜ (E) (λi + β ) lim inf ρ →0

log μ˜ xi˜ (B%i (x; ˜ ρ )) ≥ (1 − β )(hi − hi−1 − β ). log ρ

(IX.21)

The first conclusion then follows immediately from this together with Proposition IX.2.7. The second conclusion follows from the definition of γ˜i and Lemma VII.VII.7.5 since the point-independent metric d˜i (·, ·) on ηi (x)/ ˜ ηi−1 makes it isometric to a subset of (Rmi ,  · ). Now we come to the proof of (IX.21). First fix ε ∈ (0, β /3). Let Δ0 be a set as chosen in Section IX.2. Recalling that μ˜ (Δ0 ) = 1 and θ Δ0 = Δ0 , for the sake of presentation we may assume that Δ0 = M f . We divide the proof into four parts following the line presented in [51]. (A) Before proceeding with the main argument, we record some estimates analogous of those in [51, pp. 171]. For δ > 0, define g, gδ and g∗ : M f → R by −1 g(y) ˜ = μ˜ yi−1 ηi )(y)), ˜ ˜ ((θ  1 def gδ (y) ˜ = i μ˜ z˜i−1 ((θ −1 ηi )(y))d ˜ μ˜ yi˜ (˜z), i % B ˜μ (B%i (y; ( y; ˜ δ )) ˜ δ )) def

y˜

def

g∗ (y) ˜ = inf gδ (y), ˜ δ ∈Q

where Q = {e−(λi+β )l N02 j : l, j ∈ Z+ }. −1 η According to Proposition IX.2.7, one has g(y) ˜ = μ˜ yi−1 ˜ for μ˜ i−1 )(y)) ˜ ((θ i i % ˜ δ )) and a.e. y. ˜ For each δ > 0, one can check that the functions y˜ → μ˜ y˜ (B (y; θ −1 ηi %i i i y˜ → μ˜ (B (y; ˜ δ )) are measurable and μ˜ y˜ (B% (y; ˜ δ )) > 0 for μ˜ -a.e. y. ˜ Since def

y˜

Hμ˜ (θ −1 ηi |ηi ) < +∞, one has μ˜ yi˜ ((θ −1 ηi )(y)) ˜ > 0 for μ˜ -a.e. y˜ and

200

IX



gδ (y) ˜ =

Generalized Entropy Formula

−1

μ˜ yi˜ (B%i (y; ˜ δ ) (θ −1 ηi )(y)) ˜ μ˜ yθ˜ ηi (B%i (y; ˜ δ )) i −1 ˜ = · μ˜ y˜((θ ηi )(y)). μ˜ yi˜ (B%i (y; ˜ δ )) μ˜ yi˜ (B%i (y; ˜ δ ))

gδ is therefore measurable for each fixed δ > 0. The measurability of g∗ is obvious. We assert that gδ → g μ˜ -a.e. on M f when δ ∈ Q and δ → 0 and that ( − log g∗ d μ˜ < +∞. To see this, first consider one element of ηi at a time. Fix ¯ (i) ˜ μ˜ xi˜ ) for (X, m) in Lemma VII.VII.7.4, let π = π˜i : ηi (x) ˜ →R x. ˜ Substitute (ηi (x), −1 i and let α = θ ηi |ηi . Then we can conclude that gδ (·) → g(·) μ˜ x˜ -a.e. as δ ∈ Q and δ → 0 and that for μ˜ -a.e. x˜ −



log g∗ (˜z)μ˜ xi˜ (d z˜) ≤ −



log( inf gδ (˜z))μ˜ xi˜ (d z˜) δ >0

i

≤ Hμ˜ i (θ −1 ηi |ηi ) + logc( ∑ m j ) + 1 < +∞, x˜

j=1

where c(·) is the multiplicity defined in Theorem VII.VII.7.1. (B) The purpose of this step is to study the induced action of θ on

θ −1 (ηi (x))/ ˜ ηi−1 → ηi (x)/ ˜ ηi−1 with respect to the metrics d˜θi −1 x˜ (·, ·) and d˜xi˜ (·, ·). Consider x˜ ∈ M f . The point x˜ will be subjected to a finite number of μ˜ -a.e. assumptions. Let t0 < t1 < · · · be the successive times t when θ t x˜ ∈ E% with t0 ≤ 0 < t1 . Note that t0 is constant on ηi (x). ˜ For large n and 0 ≤ k < n, define a(x; ˜ k) as follows: if t j ≤ k < t j+1 , then def a(x; ˜ k) = B%i (θ k x; ˜ N02 j e−(λi +β )(n−t j ) ).

We now claim that a(x; ˜ k)



(θ −1 ηi )(θ k x) ˜ ⊂ θ −1 a(x; ˜ k + 1)

(IX.22)



In fact, if k = t j − 1 for any j, then θ a(x; ˜ k) ηi (θ k+1 x) ˜ = a(x; ˜ k + 1) automatii i ˜ ˜ % The case when cally since dθ k x˜ (·, ·) and dθ k+1 x˜ (·, ·) are defined by pulling back to E. k = t j − 1 for some j reduces to the following consideration: Let y˜ ∈ E% and let t > 0 % Let z˜ ∈ (θ −t ηi )(y). be the smallest integer such that θ t y˜ ∈ E. ˜ It suffices to show that d˜θi t y˜ (θ t y, ˜ θ t z˜) ≤ N02 et(λi +β ) d˜yi˜ (y, ˜ z˜). First dˆyi˜ (y, ˜ z˜) ≤ N0 d˜yi˜ ((y), ˜ z˜). Then for k = 1, 2, · · · ,t, Proposition IX.2.1 tells us that ˜ θ k z˜) ≤ ek(λi +β ) dˆyi˜ (y, ˜ z˜). dˆθi k y˜ (θ k y,

IX.4

Estimates of Local Entropies along Unstable Manifolds

201

We pick up another factor of N0 when converting back to the d˜i -metric at θ t y˜ (see Proposition IX.2.11). What we claimed above is thus proved.  ⊂ M f with full (C) It is easy to see that there exists a θ -invariant Borel set   then μ˜ i (B%i (x; μ˜ -measure such that, if x˜ ∈ , ˜ δ )) > 0 for all δ ∈ Q. We now estimate x˜ i i −( λ + β )(n−t ( x)) ˜ i  which will be subjected to a finite i 0 % μ˜ x˜ (B (x; ˜e )) = μ˜ x˜ (a(x; ˜ 0)) for x˜ ∈  number of a.e. assumptions. Write

μ˜ xi˜ (a(x; ˜ 0)) =

T −1

∏ μ˜ i

μ˜ θi k x˜ (a(x; ˜ k))

θ k+1 x˜

k=0

(a(x; ˜ k + 1))

˜ T )) · μ˜ θi T x˜ (a(x;

where T = [n(1 − ε )] (here [x] denotes the integer part of x). First note that the last term ≤ 1. For each 0 ≤ k < T , by the θ -invariance of μ˜ and by uniqueness of conditional measures one has

μ˜ θi k x˜ (a(x; ˜ k))

μ˜ θi k+1 x˜ (a(x; ˜ k + 1))

˜ k)) = μ˜ θi k x˜ (a(x;

μ˜ θi k x˜ (θ −1 (ηi (θ k+1 x))) ˜ . i −1 μ˜ θ k x˜ (θ a(x; ˜ k + 1))

This is ≤

μ˜ θi k x˜ (a(x; ˜ k))



μ˜ θi k x˜ ((θ −1 ηi )(θ k x) ˜ a(x; ˜ k))

˜ · μ˜ θi k x˜ ((θ −1 ηi )(θ k x))

(IX.23)

by (IX.22). If gδ is defined as in (A), the first quotient in (IX.23) is equal to [gδ (x;n,k) (θ k x)] ˜ −1 , ˜ ˜ where δ (x; ˜ n, k) = e−(λi +β )(n−t jk (x)) N0

2 jk

def % with jk = #{0 < i ≤ k : θ i x˜ ∈ E}.

def

Write I(x) ˜ = − log μ˜ xi˜ ((θ −1 ηi )(x)). ˜ Then the second term in (IX.23) is equal to k x) −I( θ ˜ e . Hence ˜ log μ˜ xi˜ (B%i (x; ˜ e−(λi +β )(n−t0(x)) )) ≤ −

T −1

T −1

k=0

k=0

(θ k x) ˜ − ∑ I(θ k x). ˜ ∑ log gδ (x;n,k) ˜

Multiplying by − 1n and taking lim inf on both sides of this inequality, (λi + β ) lim inf ρ →0

˜ )) log μ˜ xi˜ (B%i (x; ˜ ρ )) μ˜ i (B%i (x; ˜ e−(λi +β )(n−t0 (x)) = (λi + β ) lim inf x˜ −( λ + β )n n→+∞ log ρ log e i

≥ lim inf n→+∞

[n(1−ε )]

∑

log gδ (x;n,k) (θ k x) ˜ ˜

k=0

1 [n(1−ε )] ˜ ∑ I(θ k x). n→+∞ n k=0

+ lim

202

IX

Generalized Entropy Formula

The last limit = (1 − ε )Hμ˜ (θ −1 ηi |ηi ) = (1 − ε )hi by Birkhoff’s ergodic theorem. Hence Proposition IX.4.6 is proved if we show that lim sup − n→+∞

1 [n(1−ε )] (θ k x) ˜ ≤ (1 − ε )(hi−1 + 2ε ). ∑ log gδ (x;n,k) ˜ n k=0

(IX.24)

(D) We now prove the last assertion (IX.24). It follows from (A) that there is a measurable function δ : M f → R+ such (that for μ˜ -a.e. x, ˜ if δ ∈ Q and δ ≤ δ (x), ˜ ˜ < +∞, there is a number δ∗ ˜ ≤ − log g(x) ˜ + ε . Since − log g d μ then − log gδ (x) ∗ ( such that if A = {x˜ : δ (x) ˜ > δ∗ }, then M f \A − log g∗ d μ˜ ≤ ε . ˜ if n is sufficiently large, then δ (x; ˜ n, k) ≤ δ∗ for all We claim that for μ˜ -a.e. x, k ≤ n(1 − ε ). First by Birkhoff ergodic theorem, there is a positive integer N1 (x) ˜ % ≤ 2n μ˜ (E). % If n ≥ N1 (x), such that for n ≥ N1 (x), ˜ #{0 ≤ i < n : θ i x˜ ∈ E} ˜ then for each k ≤ n(1 − ε ) one has t jk (x) ˜ ≤ k ≤ n(1 − ε ) and % 4n μ˜ (E)

2j ˜ δ (x; ˜ n, k) = e−(λi +β )(n−t jk (x)) N0 k ≤ e−(λi +β )ε n N0 % 4 μ˜ (E)

Since e−(λi +β )ε N0 [n(1−ε )]

∑

.

< 1, δ (x; ˜ n, k) is less than δ∗ for sufficiently large n. Thus

− log gδ (x;n,k) (θ k x) ˜ ≤ ˜

k=0

[n(1−ε )]

∑

(− log g(θ k x) ˜ + ε) +

k=0 θ k x˜ ∈ A

[n(1−ε )]

∑

− log g∗ (θ k x) ˜

k=0 θ k x˜ ∈ A

and the lim sup we wish to estimate in (IX.24) is bounded above by 

(1 − ε )[ − log gd μ˜ + ε +

 M f \A

− log g∗ d μ˜ ].

−1 η ˜ for μ˜ -a.e. x, ˜ Recalling that g(x) ˜ = μ˜ xi−1 i−1 )(x)) ˜ ((θ



− log gd μ˜ = Hμ˜ (θ −1 ηi−1 |ηi−1 ) = Hμ˜ (ηi−1 |θ ηi−1 ) = hi−1 .

This completes the proof.   Proof of Proposition IX.4.7. It follows from the Lipschitz property of π˜i together with the definition of γ˜i and γ%i that for μ˜ -a.e. fixed x˜ and μ˜ xi˜ -a.e. y˜ ∈ ηi (x) ˜

γ%i (y; ˜ x) ˜ = γ˜i (x), ˜ if we assume μ˜ xi˜ (



n≥0

% = 1. Hence from Proposition IX.4.6 we may assume that θ n E)

γ%i (π˜i−1 z, x) ˜ ≥

(1 − β )[hi − hi−1 − β ] , λi + β

¯ (i) . ν − a.e. z ∈ R

IX.5

The General Case: without Ergodic Assumption

203

¯ (i) = R ¯ (i−1) × Rmi into planes of the Consider now the partition of π˜i (ηi (x)) ˜ ⊂R (i) ¯ form {z = (z1 , z2 , · · · , zi ) ∈ R : zi = const}. Using the Lipschitz property of π˜i and ¯ (i) the definition of δ i and ν , one can easily verify that at ν -a.e. z ∈ R

δ i−1 (x) ˜ ≤ lim inf ρ →0

¯ (i−1) : w − z(i−1) < ρ }) log νzi ({w ∈ R log ρ

and

δ i (x) ˜ = lim inf ρ →0

¯ (i) : w − z < ρ }) log ν ({w ∈ R , log ρ

¯ (i) : zi = c} (hence where νc is the conditional measure of ν on {z = (z1 , · · · , zi ) ∈ R (i−1) ¯ νc can be viewed as a Borel probability measure on R ). Lemma IX.4.8 then ˜   tells us that inequality (IX.19) holds for μ˜ xi˜ -a.e. y.

IX.5 The General Case: without Ergodic Assumption Now we prove Theorem IX.1.3 in the general case without ergodic assumption via ergodic decompositions of μ and μ˜ . If μ is not ergodic, then according to Rokhlin [75], there exists a (μ -mod 0) unique measurable partition ζ0 of M such that f −1C = C and f |C : (C, μ |C ) ← is ergodic for μζ0 -a.e. C ∈ ζ0 . Let ζ = p−1 ζ0 . Then θ C = C def def for μζ0 -a.e. C ∈ ζ0 , where C = p−1C ∈ ζ . μ˜ C = p−1 μC is an ergodic measure on C for μζ0 -a.e. C ∈ ζ0 . Since ξu is a partition of M f subordinate to W u -manifolds of f , ξu refines ζ by Corollary 3.1.1 in [73]. Hence the transitivity of conditional measures implies that ξ ξ μ˜ x˜ i = (μ˜ C)x˜i , for μ˜ ζ − a.e. C and μ˜ C − a.e. x. ˜

Then results in Sections IX.3 and IX.4 tell us that for μ˜ ζ -a.e. C ∈ ζ and μ˜ C-a.e. x˜ ∈ C hi (x; ˜ ξi , μ˜ ) = hi (x; ˜ ξi |C, μ˜ C) = hμ˜ C (θ |C, ξi |C) = hi (x; ˜ ξi |C, μ˜ C) = hi (x; ˜ ξi , μ˜ ) and

δ i (x; ˜ ξi , μ˜ ) = δ i (x; ˜ ξi |C, μ˜ C) = δ i (x; ˜ ξi |C, μ˜ C) = δ i (x; ˜ ξi , μ˜ ) =: δi (x; ˜ ξi , μ˜ ). def

˜ μ˜ ) = δi (x; ˜ ξi , μ˜ ) − δi−1 (x; ˜ ξi−1 , μ˜ ), it follows from equation (IX.13) that Since γi (x;

γi (x; ˜ μ˜ ) = γi (x; ˜ μ˜ C) =

h μ˜ C (θ |C, ξi |C) − hμ˜ C (θ |C, ξi−1 |C)

λi (x0 )

204

IX

Generalized Entropy Formula

 Hence γi ’s are a.e. θ -invariant funcholds true for μ˜ ζ -a.e. C ∈ ζ and μ˜ C-a.e. x˜ ∈ C. tions. It is easy to see that γi ’s are indeed a.e. functions well defined on M. Furthermore, the entropy map hμ˜ (θ ) is affine with respect to μ˜ , i.e. hμ˜ (θ ) =



 hμ˜ C (θ )d μ˜ ζ (C).

Hence by hμ ( f ) = hμ˜ (θ ) and hμ˜ C (θ ) = h μ˜ C (θ |C, ξu |C), Theorem IX.1.3 holds.

 

Chapter X

Exact Dimensionality of Hyperbolic Measures

In the survey article [17], Eckmann and Ruelle discussed various concepts of dimension and pointed out the importance of the so-called pointwise (local) dimension of invariant measures. For a Borel measure μ on a compact metric space M, the latter was introduced by Young in [95] and is defined by def

log μ (B(x, ρ )) ρ →0 log ρ

dμ (x) = lim

(X.1)

(provided the limit exists). As Barreira et al. indicated in [7], the notion is an important characteristic of the system. It plays a crucial role in dimension theory (see, for example, [19], the ICM address by Young [97, pp. 1232] and also [98, pp. 318]). In ergodic case the existence of the limit in (X.1) for a Borel probability measure μ on M implies the crucial fact that virtually all the known characteristics of dimension type of the measure (including the Hausdorff dimension, box dimension, and information dimension) coincide. The common value is called the fractal dimension of μ . In this chapter we would like to present a detailed proof of this famous dimension conjecture in two situations: one is the expanding maps’ case and the other is the diffeomorphisms’ case. The second one was solve by Barreira, Pesin and Schmeling [7]; the proof presented here is slightly different with theirs and seems more accessible. The first one is taken from [71]. The main results are the following two theorems. Theorem X.0.1 Let f : M ← be a C2 expanding map with respect to an invariant measure μ , then the limit in (X.1) exists and dμ (x) = δu (x) for μ -a.e. x, where δu (x) is defined in Chapter IX; hence d μ (x) is f -invariant, i.e., dμ ( f x) = dμ (x) for μ -a.e. x. Furthermore, if μ is an ergodic measure, then there is some constant d such that d μ (x) ≡ d for μ -a.e. x. Theorem X.0.2 Let f : M ← be a C2 diffeomorphism and let μ be an invariant measure of f . Suppose μ is weakly hyperbolic with respect to f , i.e., the Lyapunov exponents of the system ( f , μ ) are μ -almost everywhere non-zeroes. Then the limit M. Qian et al., Smooth Ergodic Theory for Endomorphisms, Lecture Notes in Mathematics 1978, DOI 10.1007/978-3-642-01954-8 X, c Springer-Verlag Berlin Heidelberg 2009 

205

206

X Exact Dimensionality of Hyperbolic Measures

in (X.1) exists and d μ (x) is f -invariant, i.e., dμ ( f x) = dμ (x) for μ -a.e. x. Furthermore, if μ is an ergodic measure, then there is some constant d such that dμ (x) ≡ d for μ -a.e. x.

X.1 Expanding Maps’ Case–Proof of Theorem X.0.1 In Chapter IX we have just proved the existence of unstable pointwise dimensions for endomorphisms (see Proposition IX.IX.1.1). As an application, we will prove Eckmann-Ruelle conjecture for expanding maps, i.e., Theorem X.0.1, in this section. This is the first attempt on Eckmann-Ruelle dimension conjecture for endomorphisms after Barreira et al [7] have proved Eckmann-Ruelle dimension conjecture for diffeomorphisms. Proof of Theorem X.0.1. Let f be a C2 expanding map and let ( f , μ ) be ergodic. The proof will be carried out in four steps. Step 1. First let us give some preparatory results. Since f is expanding on a compact manifold M, there exist positive numbers τ , λ and ε0 such that d( f x, f y) ≤ eλ d(x, y), and

d(x, y) ≤ δ e−nτ ,

if

∀x, y ∈ M

dn (x, y) ≤ δ ≤ ε0 ,

(X.2)

(X.3)

where dn (x, y) := max{d( f k x, f k y) : 0 ≤ k ≤ n}. Since f is an expanding map, one has Bu (x, ˜ ρ ) = p−1 B(x0 , ρ )



 u (x), W ˜ if 0 < ρ ≤ ε0 .

Moreover, since ηu is a measurable partition subordinate to W u -manifolds, p |ηu (x) ˜ : ηu (x) ˜ → pηu (x) ˜ is bijective. Therefore ˜ ρ) Bu (x,



ηu (x) ˜ = p−1 B(x0 , ρ )



ηu (x). ˜

Hence by Proposition IX.IX.1.1 log μ˜ xη˜ u (p−1 B(x0 , ρ )) = d μu˜ (x0 ) ≡ d u , μ˜ − a.e. x, ˜ ρ →0 log ρ lim

where dμu˜ (x0 ) := δu (x0 ) and d u is some constance due to the ergodicity of μ . Define the lower and upper pointwise dimension of μ at x ∈ M by log μ (B(x, ρ )) , log ρ log μ (B(x, ρ )) d μ (x) := lim sup . log ρ ρ →0

d μ (x) := lim inf ρ →0

(X.4)

X.1 Expanding Maps’ Case–Proof of Theorem X.0.1

207

The above quantities are bounded by the dimension of the manifold M. By the continuity of f , for any fixed point x and any δ > 0, there exists some 0 < ε = ε (x, δ ) < δ such that B(x, ε ) ⊂ f −1 B( f x, δ ). From this and the invariant property of μ , we clearly have d μ ( f x) ≤ d μ (x),

for μ − a.e. x ∈ M.

d μ ( f x) ≤ d μ (x),

Then it is clear that we should in fact have d μ ( f x) = d μ (x),

for μ − a.e. x ∈ M.

d μ ( f x) = d μ (x),

By the ergodictiy of μ , there exist constants d and d such that for μ -a.e. x ∈ M d μ (x) = d,

d μ (x) = d.

Therefore there is a number C1 ≥ 1 and a Borel set Γ ⊂ M with μ (Γ ) ≥ 1 − ε /2

+∞ + such that for each x ∈ Γ there are two subsequence {nk }+∞ k=1 and {nk }k=1 of Z satisfying μ (B(x, 2e−nk )) ≤ C1 e−nk (d−ε ) (X.5) and





μ (B(x, e−nk )) ≥ C1−1 e−nk (d+ε ) .

Step 2. Let us give some basic estimates. Let E be a maximum M and let β be a finite partition of M such that for any x ∈ E Int(β (x)) = {y ∈ M : d(y, x) < d(y, x ),

(X.6) ε0 2 -separate

for any x ∈ E \ {x}},

where Int(A) denotes the interior of a set A ⊂ M. It is easy to see that diam β (x) ≤ ε0 for any x ∈ M. Then we put

α = p−1 β . def

Clearly we have

  diam p α (x) ˜ ≤ ε0 ,

Let a be the integer part of 1 + 1τ and put def

qn =

an  k=0

θ −k α .

∀x. ˜

set of

208

X Exact Dimensionality of Hyperbolic Measures

Then by (X.2) and (X.3), one has   ˜ ≤ e−n diam p qn (x) for all x. ˜ Since f is an expanding map, β is a generator (with respect to f ), i.e. B(

+∞ )

f −k β ) = B(M). Then it is clear that for μ˜ -a.e. x˜

k=0

lim −

n→+∞

1 log μ˜ (qn (x)) ˜ = h μ˜ (θ , α ) = hμ ( f , β ) = hμ ( f ) =: h na

(X.7)

1 log μ˜ ηu (x) ˜ = h. ˜ (qn (x)) na

(X.8)

and lim −

n→+∞

For each ε , ε ∈ (0, 1/200] fixed, by (X.4), (X.7) and (X.8) there exists C2 > 1 and a set Λ1 ⊂ M f with μ˜ (Λ1 ) ≥ 1 − ε /2 such that for each x˜ ∈ Λ1 and n ≥ 0 C2−1 e−n(d

u −ε )

C2−1 e−na(h+ε ) C2−1 e−na(h+ε )

u +ε )

,

(X.9)

≤ C2 e

−na(h−ε )

,

(X.10)

≤ C2 e

−na(h−ε )

.

(X.11)

−n(d −1 −n ≤ μ˜ ηu (x) ˜ (p B(x0 , e )) ≤ C2 e

≤ ≤

μ˜ (qn (x)) ˜ μ˜ ηu (x) ˜ ˜ (qn (x))



Let Λ2 := Λ1 p−1Γ , then μ˜ (Λ2 ) ≥ 1 − ε . Step 3. Here we will prove d ≤ d u . Let μ˜ ηu be the induced measure on the factor space of M f with respect to the partition ηu , i.e., μ˜ ηu satisfies for each Borel subset A of M f  μ˜ (A) = μ˜ C (A)d μ˜ ηu (C). √ Put A := {C ∈ ηu : μ˜ C (Λ2 ) ≥ 1 − ε }, then

μ˜ (Λ2 ) =



√ √ μ˜ C (Λ2 )d μ˜ ηu ≤ 1 − ε + ε μ˜ ηu (A ),

√ %C := p μ˜ C , then μ %C is a Borel i.e., μ˜ ηu (A ) ≥ 1 − ε . For each C ∈ A fixed, let μ measure well defined on M. Define 1 KC := {y ∈ M : μ˜ C  p−1 {y} (Λ2 ) ≥ }. 2 √ √ %C (KC ) ≥ 1 − 2 ε . Hence Since μ˜ C (Λ2 ) ≥ 1 − ε , we have μ

μ˜ C (Λ2



 1 %C (KC B(y, e−n )). p−1 B(y, e−n )) ≥ μ 2

Then by Lemma IX.IX.4.4, there exists an integer n% = n%(C) and a set KC ⊂ KC of √ %C (KC ) ≥ 1 − 3 ε such that measure μ

X.1 Expanding Maps’ Case–Proof of Theorem X.0.1

%C (KC μ



209

1 %C (B(y, e−n )), ∀n ≥ n%, y ∈ KC . B(y, e−n )) ≥ μ 2

Thus we can define a measurable function n% : A → Z+ such that the above equation holds. Then there exists a number N1 such that the set A1 := {C ∈ A : n%(C) ≤ N1 } √   has measure μ˜ ηu (A1 ) ≥ 1 − 2 ε . Therefore if C ∈ A1 and y ∈ KC p (Λ2 C), then by (X.9) one has for each n ≥ N1

μ˜ C (Λ2



p−1 B(y, e−n )) ≥

1 −n(d u +ε ) e . 4C2

(X.12)

Hence if we define

Λ3 := {x˜ ∈ M f : μ˜ ηu (x) ˜ (Λ2





p−1 B(x0 , e−n )) ≥ 

1 −n(d u +ε ) e , ∀n ≥ N1 } 4C2



and put Λ4 := Λ2 Λ3 , then p−1 KC (Λ2 C) ⊂ Λ3 for any C ∈ A1 ; And we have        μ˜ C p−1 KC (Λ2 C) = μ˜ C p−1 KC Λ2     ≥ μ˜ C p−1 KC + μ˜ C Λ2 − 1 √ ≥ 1 − 4 ε .

Thus

μ˜ (Λ3 ) ≥

 A1

μ˜ C (Λ3 )d μ˜ ηu



    μ˜ C p−1 KC (Λ2 C) d μ˜ ηu A1 √ √ √ ≥ (1 − 4 ε ) · (1 − 2 ε ) ≥ 1 − 6 ε .

≥

√ Hence μ˜ (Λ4 ) ≥ 1 − 7 ε . Now fix a point x˜ ∈ Λ4 and n ≥ N1 + 1. Let N(n, A) be the number of atoms of the  partition qn intersecting a Borel set A ⊂ M f . We estimate N(n, Λ2 p−1 B(x0 , e−n ))   −1 ˜ Λ2 p B(x0 , e−n )). If and N(n, ηu (x) qn (y) ˜ then and



Λ2



p−1 B(x0 , e−n ) = 0, /

˜ ⊂ p−1 B(x0 , 2e−n ) qn (y)

μ˜ (qn (y)) ˜ ≥ C2−1 e−na(h+ε ).

210

X Exact Dimensionality of Hyperbolic Measures

Hence by (X.5) N(n, Λ2



p−1 B(x0 , e−n )) ≤ C1C2 e−n(d−ε )+2na(h+ε ). 



On the other hand, if z˜ ∈ qn (y) ˜ ηu (x) ˜ Λ2

(X.13)

 −1 p B(x0 , e−n )) = 0, / then

μ˜ ηu (x) ˜ = μ˜ ηu (˜z) (qn (˜z)) ≤ C2 e−na(h−ε ). ˜ (qn (y)) Therefore since x˜ ∈ Λ4 ⊂ Λ3 , one has ˜ N(n, ηu (x)



Λ2



u 1 p−1 B(x0 , e−n )) ≥ C2−2 e−n(d +ε )+na(h−ε ). 4

(X.14)

Clearly one has N(n, Λ2



p−1 B(x0 , e−n )) ≥ N(n, ηu (x) ˜



Λ2



p−1 B(x0 , e−n )).

Thus by (X.13) and (X.14), one can easily obtain d ≤ d u . Step 4. In this step we will prove that d ≥ d u . By (X.6), if we define K := {x ∈ M : μ˜ p−1 {x} (Λ2 ) ≥ 12 }, then μ (K) ≥ 1 − 2ε . Therefore

μ˜ (Λ2



p−1 B(x0 , e−n )) = ≥

 B(x0 ,e−n )

μ˜ p−1 {x} (Λ2 )μ (dx)

 1 μ (K B(x0 , e−n )). 2

Thus by Lemma IX.IX.4.4, one can choose an integer N2 and a set K1 ⊂ K of measure μ (K1 ) ≥ 1 − 3ε such that for every n ≥ N2 and x0 ∈ K1

μ (K



1 B(x0 , e−n )) ≥ μ (B(x0 , e−n )). 2



Let Λ5 := Λ2 p−1 K1 , then μ˜ (Λ5 ) ≥ 1 − 4ε . And for each x˜ ∈ Λ3 , there are infinitely many n satisfying

μ˜ (An (x)) ˜ ≥ (4C1 )−1 e−n(d+ε ) , 

where An (x) ˜ := Λ2 p−1 B(x0 , e−n ). On the other hand, by (X.9) one has ˜ : ηu (y) ˜ intersecting An (x)} ˜ max{ μ˜ ηu (y) ˜ (An (x)) −1 −n ≤ max{ μ˜ ηu (y) ˜ (p B(y0 , 2e )) : y˜ ∈ Λ2 }

≤ C2 e−(n−1)(d

u −ε )

.

(X.15)

X.2 Diffeomorphisms’ Case–Proof of Theorem X.0.2

211

Therefore

μ˜ (An (x)) ˜ =



μ˜ ηu (y) ˜ μ˜ (d y) ˜ ≤ C2 e−(n−1)(d ˜ (An (x))

u −ε )

.

Thus by (X.15) and (X.16) we have proved d ≥ d u .

(X.16)  

X.2 Diffeomorphisms’ Case–Proof of Theorem X.0.2 In this section, f : M ← is a C2 diffeomorphism and μ is an f -invariant measure. In this situation, the inverse limit space M f is just M f = {x˜ : xn = f n x0 , n ∈ Z} which is obviously Lipschitz equivalent to M itself with the obvious correspondence x˜ ←→ x0 . Bearing this in mind, we can recover from the foregoing chapter, i.e., Chapter IX, Ledrappier and Young’s entropy formula [43]. We would like to state below again for the convenience of the readers. The Oseledec multipicative ergodic theorem applied to f : (M, μ ) ← tells us that for μ -a.e. x ∈ M there exist numbers +∞ > λ1 (x) > λ2 (x) > · · · > λr(x) (x) > −∞ and a splitting of Tx M Tx M = E1 (x) ⊕ E2 (x) ⊕ · · · ⊕ Er(x) (x)

(X.17)

(all measurable in x) such that lim

n→±∞

1 log |Tx f n v| = λi (x) n

for all v ∈ Ei (x) \ {0}, 1 ≤ i ≤ r(x). The numbers λi (x), 1 ≤ i ≤ r(x) are called the Lyapunov exponents of the system ( f , μ ) at x, and mi (x) := dim Ei (x) is called the multiplicity of λi (x). Let u(x) be the number of positive Lyapunov exponents at x. Then for μ -a.e. x, if u(x) > 0, the set 1 W u,i (x) := {y ∈ M : lim sup log d( f −n x, f −n y) ≤ −λi (x)} n→+∞ n

212

X Exact Dimensionality of Hyperbolic Measures

is a C1,1 immersed ∑ik=1 mk (x)-dimensional submanifold of M for each 1 ≤ i ≤ u(x), and it is called the ith -unstable manifold of f at x. W u,u(x) (x) coincides with 1 W u (x) := {y ∈ M : lim sup log d( f −n x, f −n y) < 0} n→+∞ n and is called the unstable manifold of f at x. Define W u (x) = {x} if u(x) = 0. Let 1 ≤ i ≤ dim M and let η i be a measurable partition of M subordinate to the u,i W -manifolds of ( f , μ ); This means that for μ -a.e. x ∈ M, η i (x) has the following i

property: There exists a ∑ mk (x)-dimensional C1 embedded submanifold Wxi of M k=1

with Wxi ⊂ W u,i (x) such that η i (x) ⊂ Wxi and η i (x) contains an open neighborhood of x in Wxi , this neighborhood being taken in the topology of Wxi as a submanifold of M. The construction of η i is following the same line as that presented in Chapter IX; of course it is more simple now. Denote by μxi the conditional measure of μ on η i (x). Let dxu,i (·, ·) be the metric on W u,i (x) induced by its inherited Riemannian structure and put for ρ > 0 Bu,i (x; ρ ) = {y ∈ W u,i (x) : dxu,i (x, y) < ρ }.   log μxi Bu,i (x; ρ ) ρ →0 log ρ

Define

δi (x) := lim

(X.18)

provided the limit exists (the definition is independent of the choice of η i ) and call it the dimension of μ on the W u,i -manifold at x. The following theorem is the generalized entropy formula of Ledrappier and Young [43], which is the special case of our Theorem IX.IX.1.3 (combining with Proposition IX.IX.1.1). Theorem X.2.1 Let f : M ← be a C2 diffeomorphism and μ is an f -invariant measure. Then for μ -a.e. x the limit (X.18) exists for 1 ≤ i ≤ u(x) and the formula hμ ( f ) =



∑

i:λi (x)>0

λi (x)γi (x) d μ ,

(X.19)

holds true, where γi (x) := δi (x) − δi−1 (x) for i = 1, · · · , u(x) and δ0 (x) := 0. The above C2 assumption on f can be replaced by C1+α (0 < α ≤ 1), which can be checked by patient readers from the proof presented in Chapter IX (and/or [43]). By considering f −1 : (M, μ ) ←, Theorem X.2.1 can also be formulated by means of the negative Lyapunov exponents. That is, letting ( f , μ ) be as given in Theorem X.2.1 and letting 0 > λs(x) (x) > · · · > λr(x) (x) > −∞

X.2 Diffeomorphisms’ Case–Proof of Theorem X.0.2

213

be all the negative exponents of ( f , μ ) at x (if there are any), then one has hμ ( f ) =



∑

j:λ j (x)<0

−λ j (x)γ j (x) d μ ,

(X.20)

where γ j (x) := δ j (x) − δ j+1 (x) for s(x) ≤ j ≤ r(x) with the convention δr(x)+1 (x) := 0 and δ j (x) is the dimension of μ on the W s, j -manifold at x defined analogously to the unstable manifolds case by replacing W u,i (x) with the C1,1 immersed submanifold 1 W s, j (x) := {y ∈ M : lim sup log d( f n x, f n y) ≤ λ j (x)}. n→+∞ n We remark that W s,s(x) (x) coincides with 1 W s (x) := {y ∈ M : lim sup log d( f n x, f n y) < 0} n→+∞ n which is called the stable manifold of f at x. For μ -a.e. x, define W s (x) = {x} if there are no negative exponents at x. Let η s (resp. η u ) be a measurable partition of (M, μ ) subordinate to the W s manifolds (resp. W u -manifolds) of ( f , μ ). Let μxs be the conditional measure of μ on η s (x). Let dxs (·, ·) be the distance along W s (x) and put for ρ > 0 Bs (x; ρ ) = {y ∈ W s (x) : dxs (x, y) < ρ }. Define μxu and Bu (x; ρ ) similarly. Then, as asserted above, the limits log μxs (Bs (x; ρ )) , ρ →0 log ρ log μxu (Bu (x; ρ )) dμu (x) := lim ρ →0 log ρ dμs (x) := lim

(X.21) (X.22)

exist μ almost everywhere. As functions of x, they are f -invariant and hence, when μ is ergodic, dμs (x) = d s and dμu (x) = d u μ almost everywhere for some constants d s and d u . These facts play a crucial role in the proof of the main result stated in the next section. An invariant measure μ of f is said to be hyperbolic if all the Lyapunov exponents of f are nonzero μ almost everywhere. In the rest of this section, η s will be a measurable partition of (M, μ ) subordinate to the W s -manifolds of ( f , μ ) such that f η s ≥ η s , f n η s tends to the partition into single points as n → +∞, and Hμ ( f η s |η s ) = hμ ( f ); η u will be a measurable partition of (M, μ ) subordinate to the W u -manifolds of ( f , μ ) such that f −1 η u ≥ η u , f −n η s tends to the partition into single points, and Hμ ( f −1 η u |η u ) = hμ ( f ) .

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X Exact Dimensionality of Hyperbolic Measures

The main result of this section is the following theorem, which is a detailed reformulation of Theorem X.0.2. Theorem X.2.2 Let f : M ← be a C2 diffeomorphism and μ is an f -invariant measure. If μ is hyperbolic and ergodic, then the following properties hold: 1) for every ε > 0, there exists a set Λ ⊂ M with μ (Λ ) > 1 − ε and constants κ ≥ 1, ρ0 > 0 such that for every x ∈ Λ and every ρ ∈ (0, ρ0 )

ρ ρ ρ ε μxs (Bs (x; ))μxu (Bu (x; )) ≤ μ (B(x, ρ )) κ κ −ε s s ≤ ρ μx (B (x; κρ ))μxu (Bu (x; κρ )); 2) the limit dμ (x) := lim

ρ →0

log μ (B(x, ρ )) log ρ

(X.23)

(X.24)

exists for μ -a.e. x ∈ M and d μ (x) = d s + d u , μ −a.e. x ∈ M.

(X.25)

When μ is not ergodic, the limit (X.24) still exists and dμ (x) = d μs (x) + dμu (x)

(X.26)

for μ -a.e. x ∈ M. The proof of the above theorem is as the following. We first reconstruct a special partition with some special properties and two special requirements; and then present some preparatory lemmas. After that we start the proof under the ergodic assumption of μ and then get rid of this assumption by a suitable trick. The two special requirements just mentioned would be treated at the end of this section.

X.2.1 Reconstruction of a Special Partition For a measurable partition ξ of (M, μ ) and for all integers k, l ≥ 0, we define

ξkl :=

l 

f − jξ

j=−k

and put ξ + := ξ0∞ and ξ − := ξ∞0 . We have the following lemma which describes the exponential decay of the distance of two initial points whenever their orbits stay close enough in a long time interval [−n, +n].

X.2 Diffeomorphisms’ Case–Proof of Theorem X.0.2

215

Lemma X.2.3 Suppose that |λ j (x)| ≥ λ > 0, 1 ≤ j ≤ r(x) for μ -a.e. x and for a constant λ > 0. Let {Φx } be a system of (ε , )-charts similar to that introduced in 1 Proposition IX.IX.2.1 (see also [43, Section 8.1]) (where 0 < ε < 200m min{λ , 1}) 0

n and let Hxn := Φ −1 f n x ◦ f ◦ Φx , n ∈ Z be defined wherever they make sense. Then −1 ) ¯ for any 0 < δ ≤ 1, the following holds true for μ -a.e. x: if z = Φx−1 y ∈ R((x) satisfies Hxk z ≤ δ ( f k x)−1 for all |k| ≤ n,

then

z ≤ δ (x)−1 e−n(λ −3ε )

and therefore

d(x, y) ≤ 2K0 δ e−n(λ −3ε ),

where K0 is a universal constant as that introduced in Proposition IX.IX.2.1. A proof of the lemma can be carried out by employing Lyapunov charts (see Proposition IX.IX.2.1) and then using arguments similar to the uniformly hyperbolic case (see [47, Lemma 1.2]). The details are omitted here. We will need a measurable partition α of (M, μ ) so that it has finite conditional entropy Hμ (α ) < +∞ and the diameter of αnn (x) decays exponentially fast as n increases for μ -a.e. x. But the procedure presented in Chapter IX for the construction of the partition α in the current diffeomorphism situation (which is the corresponding construction procedure for a deterministic diffeomorphism in Ledrappier and Young [43]) can not guarantee α to have these properties. In what follows we reconstruct such a partition. The treatment depends heavily on similar results for diffeomorphisms of the previous chapter, to which we refer the reader for related notions and arguments (see also Ledrappier and Young [43], in which their notions are slightly different). In the rest of this subsection we assume that μ is ergodic and hyperbolic. Let λ := min{|λi | : i = 1, · · · , r}. Let {Φx } be a system of (ε , )-charts introduced in 1 Lemma X.2.3 with 0 < ε < 200m min{λ , 1}. Let l0 > 0 be a number so that the set 0

Δ := {x : (x) ≤ l0 } has positive μ -measure. Define n+ , n− , n+ and n− : Δ → Z+ by the following n+ (x) := inf{n ≥ 1 : f n x ∈ Δ },

n− (x) := inf{n ≥ 1 : f −n x ∈ Δ }, n+ (x) := inf{n ≥ 0 : f n x ∈ Δ }, n− (x) := inf{n ≥ 0 : f −n x ∈ Δ }, and then let ψ : M → R be given by 

ψ (x) =

δ · (2K0 l02 )−1 · e−2(λ0+ε ) max{n+ (x),n− (x)} , if x ∈ Δ , δ · (2K0 l02 )−1 , otherwise

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X Exact Dimensionality of Hyperbolic Measures

(where 0 < δ < 1) which is a modification of the corresponding definition in proof of Lemma IX.IX.3.3 (see also [43])1 by replacing S with Δ and by doubling the exponent of the exponential term in the expression of ψ (such a modification will accelerate the decay of the diameter of αnn (x) in n for the partition α constructed below). Let Uk := {x : e−k−1 < ψ (x) ≤ e−k }, k ∈ Z+ . For each k we choose a maximal 1 −k−1 -separated subset Ek of M and define a partition βk of M such that βk (x) ⊂ 2e int(βk (x)) and     int βk (x) = y : d(x, y) < d(z, y) for any z ∈ Ek \ {x} for all x ∈ Ek , where int(A) denotes the interior of a set A ⊂ M. We then define a countable measurable partition α of M by

α (x) := Uk ∩ βk (x), if x ∈ Uk for some k ≥ 0. Since, due to Kac’s lemma, − log ψ (x) ∈ L1 (M, μ ), the partition α has finite conditional entropy Hμ (α ) < +∞ and satisfies for μ -a.e. x   α (x) ⊂ B x, ψ (x) . Besides those similar properties described in Chapter IX, the partition α has the following additional properties given in Lemmas X.2.4 and X.2.5, where Hxk is as introduced in Lemma X.2.3 and ψ+ : M → Z+ is defined by 

ψ+ (x) :=

(2K0 l02 )−1 · e−(λ0+ε )·n+ (x) , if x ∈ Δ , (2K0 l02 )−1 , otherwise.

Lemma X.2.4 Let x ∈ Δ .   ¯ (x)−1 satisfies z ≤ δ l0 ψ+ (x), then (1) If z ∈ R  −1 Hxk z ≤ δ  f k x . k = 0, 1, · · · , n+ (x); (2) If y ∈ α0n (x) for some n ≥ 0, then  −1 Hxk z ≤ δ  f k x ,

k = 0, 1, · · · , n,

where z = Φx−1 y.

Here please skip over the minor difference that the α in Lemma IX.IX.3.3 is a partition of M f due to the non-invertible property of f in that context. 1

X.2 Diffeomorphisms’ Case–Proof of Theorem X.0.2

217

Lemma X.2.5 For μ -a.e. x, if y ∈ α0n (x) for some n ≥ n+ (x), then  −1 Hxk z ≤ δ  f k x ,

k = 0, 1, · · · , n,

where z = Φx−1 y. The proofs of Lemmas X.2.4 and X.2.5 follow the main lines of those of Claims 1–3 in Chapter IX (see also [43, Lemmas 9.3.2 and 9.3.3]). We present them below for completeness. For simplicity of notation, we will write n+ = n+ (x) and n+ = n+ (x) in the rest of this section. Proof of Lemma X.2.4. (1) We point out that a similar result to Proposition IX.IX.2.3 holds true in the current diffeomorphisms situation. What we need here ¯ −λ1 −3ε (x)−1 ), then Hx z, Hx z ∈ R(( ¯ f x)−1 ) is the following property: If z, z ∈ R(e and Hx z − Hx z  ≤ eλ1 +2ε z − z  , where 0 < δ ≤ e−λ0 −ε , x ∈ Δ and 1 ≤ i ≤ u. From our assumption on z and the above property, it follows that Hxj z ∈   j −1  ¯  f x R and (note that λ0 ≥ λ1 + 2ε ) Hxj z ≤ ze jλ0 for j > 0  −1 provided that zekλ0 ≤ δ  f k x e−λ1 −3ε for all 0 ≤ k < j. This is guaranteed for 0 ≤ j ≤ n+ . Hence the desired conclusion follows by the above property. (2) First observe that, if y ∈ α (x), then by the definition of α and the Lyapunov charts {Φx } (noting that a similar result to Proposition VII.VII.4.2 holds true)   ¯ δ l0 ψ+ (x) z = Φx−1 y ∈ R since (x) ≤ l0 and ψ (x) ≤ δ ψ+ (x). So we have the desired conclusion for 0 ≤ j ≤ n+ . Furthermore, if n > n+ and y ∈ α0n (x), then f n+ y ∈ α ( f n+ x) n+ and we can apply (1) with x and z being replaced by f n+ x and Φ −1 f n+ x ( f y) respectively. An inductive argument completes the proof of (2).   Proof of Lemma X.2.5. Consider a point x with the property that f n x ∈ Δ infinitely + often as n → ±∞. The collection of suchpoints is of full μ -measure. Let y ∈ α0n (x).  ¯ δ ( f j x)−1 for j = 0, 1, · · · , n+ . To prove the lemma, We claim that f j y ∈ Φ f j x R it suffices to prove this claim, since, if y ∈ α0n (x) for some n ≥ n+ , we can apply + + Lemma X.2.4(2) to f n y (noting that f n x ∈ Δ ) and this yields that

  ¯ δ ( f j x)−1 f j y ∈ Φ f j xR for j = n+ , · · · , n.

218

X Exact Dimensionality of Hyperbolic Measures +

To prove this claim, let us assume that x ∈ Δ and write k = −n− ( f n x) + n+ < 0, + i.e., k is the largest negative integer such that f k x ∈ Δ . Note that α ( f n x) lies well + + inside the chart at f n x since f n x ∈ Δ . Hence, by our choice of ψ , we can prove + + inductively that f − j+n y lies inside the chart at f − j+n x for j = 1, 2, · · · , n+ − k. Therefore Φ −1 ◦ f k (y) = H −1 ◦ Φ −1 ◦ f 1+k (y) f kx f 1+k x f 1+k x ◦ f 1+k (y) ≤ · · · ≤ eλ0 Φ −1 f 1+k x ≤ e(n

+ −k)λ

0

+

Φ −1 ◦ f n (y) n+ f

+

x

+

≤ e(n −k)λ0 l0 ψ ( f n x) ≤ δ l0 ψ+ ( f k x), the last inequality being guaranteed by +

+

2 max{n+ ( f n x), n− ( f n x)} − (n+ − k) ≥ n+ ( f k x), +

where n+ ( f k x) = n− ( f n x) = n+ − k since x ∈ Δ . Then, applying Lemma X.2.4 (1) to f k y yields the desired conclusion, since we have now n+ ( f k x) + k = n+ .   We can also apply the above arguments to the system f −1 : (M, μ ) ←, assuring that the partition α has the following additional properties. For μ -a.e. x, if y ∈ αn0 (x) for some n ≥ n− (x), one has  −1 Hx−k z ≤ δ  f −k x for 0 ≤ k ≤ n, where z = Φx−1 y. Then, by Lemma X.2.5 and the above equation, for μ -a.e. x we have  −1 for |k| ≤ n Hxk z ≤ δ  f k x provided Φx z ∈ αnn (x) for some n ≥ max{n+ (x), n− (x)}. Hence, by Lemma X.2.3, for μ -a.e. x we have   αnn (x) ⊂ B x, 2K0 δ e−n(λ −3ε ) when n is sufficiently large. In the current diffeomorphisms’ situation, Lemma IX.IX.2.4 reads Lemma X.2.6 If y, z ∈ ΦxWx,i δ (x), then d(y, z) ≤ dxi (y, z) ≤ 2K0 Φx−1 y − Φx−1 z i ≤ 2K0 (x)d(y, z), ˜ in Chapter IX, Section IX.2.1. where Wx,i δ (x) is defined analogously as Wx,˜i δ (x) From the above arguments together with Lemma X.2.6, it is also clear that for μ -a.e. x α + (x) ⊂ ΦxWx,s δ (x) ⊂ Bs (x; 2K0 δ )

X.2 Diffeomorphisms’ Case–Proof of Theorem X.0.2

and

219

α − (x) ⊂ ΦxWx,uδ (x) ⊂ Bu (x; 2K0 δ ).

Summing up the above results, we have proved the following Lemma X.2.7 Suppose that |λ j | ≥ λ > 0, 1 ≤ j ≤ r and fix 0 < δ ≤ 1 arbitrarily. Then there is a measurable partition α of M with Hμ (α ) < +∞ such that, for some measurable function n0 : M → Z+ , one has for μ -a.e. x and all n ≥ n0 (x)   αnn (x) ⊂ B x, 2K0 δ e−nλ /2 and [η s ∨ αn0 ](x) ⊂ Bs (x; n, δ ), [η u ∨ α0n ](x) ⊂ Bu (x; n, δ ),

  [η s ∨ αn0 ](x) ⊂ Bs x; 2K0 δ e−nλ /2 ,   [η u ∨ α0n ](x) ⊂ Bu x; 2K0 δ e−nλ /2 ,

where Bs (x; n, δ ) := {y ∈ W s (x) : d sf −k x ( f −k x, f −k y) < δ for 0 ≤ k ≤ n} and Bu (x; n, δ ) is defined analogously. Moreover, for μ -a.e. x one has

α + (x) ⊂ Bs (x; 2K0 δ ) and α − (x) ⊂ Bu (x; 2K0 δ ). Below we present some further properties of the partition α given above which will play an important role in our proof of Theorem X.2.2. Lemma X.2.8 Let α be the partition (depending particularly on 0 < δ ≤ 1) given above. Then one has for μ -a.e. x   1 log μ αnn (x) δ ↓0 n→+∞ 2n ∗   1 lim lim − log μxs αn0 (x) δ ↓0 n→+∞ n ∗   1 lim lim − log μxs αnn (x) n→+∞ n δ ↓0 ∗   1 lim lim − log μxu α0n (x) δ ↓0 n→+∞ n ∗   1 lim lim − log μxu αnn (x) δ ↓0 n→+∞ n   1 lim − log μxs α0n (x) n→+∞ n   1 lim − log μxu αn0 (x) n→+∞ n

lim lim −

= h μ ( f ),

(X.27)

= h μ ( f ),

(X.28)

= h μ ( f ),

(X.29)

= h μ ( f ),

(X.30)

= h μ ( f ),

(X.31)

= 0,

(X.32)

= 0,

(X.33)

220

X Exact Dimensionality of Hyperbolic Measures ∗

where the limits “ lim ” in (X.28)–(X.31) are understood as both “lim inf” and n→+∞

n→+∞

“lim sup”. n→+∞

We remark here that, (X.32) and (X.33) in fact hold true for any measurable partition α of finite entropy Hμ (α ) < +∞. These facts would be crucial in Subsection X.2.5. By the way, the cautious readers can also check that the limit symbols “limδ ↓0 ” in (X.28)–(X.31) can in fact be canceled by exploiting the tricks of switching of Lyapunov charts (with different parameters) presented in Subsection X.2.5. In order to prove the above lemma, we first give another lemma which is similar to Lemma IX.IX.3.2 or [43, Lemma 9.3.1]. Lemma X.2.9 Let α be a partition of M with Hμ (α ) < +∞. Then for μ -a.e. x   1 lim − log μxu [η u ∨ α ]n0 (x) = h μ ( f ) n→+∞ n   1 lim − log μxu [η u ∨ α ]nn (x) = h μ ( f ). n→+∞ n

(X.34) (X.35)

Proof. Equation (X.34) is just what Lemma IX.IX.3.2 reads in the current diffeomorphisms’ situation. Hence the proof is omitted. We now prove equation (X.35). Put Iμ (ξ )(x) := − log μ (ξ (x)),   Iμ (ξ η )(x) := − log μxη (ξ (x)) for any two measurable partitions ξ , η of M.       Clearly − log μxu [η u ∨ α ]nn (x) = Iμ [η u ∨ α ]nn η u (x) and we can write   u  1 n u n Iμ [η ∨ α ]n η (x) as     1  1  u   Iμ [η ∨ α ]n0 η u (x) + Iμ [η u ∨ α ]0n [η u ∨ α ]n0 (x), n n     whose first term, by (X.34), tends to Hμ (η u  f η u ) = Hμ ( f −1 η u η u ) = hμ ( f ) for μ -a.e. x. Hence it suffices to prove   1  u  Iμ [η ∨ α ]0n [η u ∨ α ]n0 (x) = 0 n→+∞ n lim

for μ -a.e. x. Clearly   n−1     −k   ( f (x)). Iμ [η u ∨ α ]0n [η u ∨ α ]n0 (x) = ∑ Iμ f α [η u ∨ α ]n+k 0 k=0

(X.36)

X.2 Diffeomorphisms’ Case–Proof of Theorem X.0.2

Put In (x) := Iμ



221

   f α [η u ∨ α ]n0 (x) and I ∗ (x) := supn≥1 In (x). One can prove that

( ∗ I (x)d μ ≤ Hμ (α )+ 1 and that {In , B( f α ∨[η u ∨ α ]n0 )} is a supermartingale. Since

B( f −n η u ) $ B(M), one then knows that In → 0

μ − a.e. and in L1  

Therefore (X.36) follows by a slight variant of Birkhoff ergodic theorem. Proof of Lemma X.2.8. We first claim that   h μ ( f , α ) = Hμ ( f −1 α α − ),   h μ ( f −1 , α ) = Hμ ( f α α + ).

(X.37) (X.38)

In fact, by the definition of conditional entropy we have (noting that f μ = μ ) 1 Hμ (α0n ) n  n−1 1  = lim [Hμ (α ) + ∑ Hμ ( f −1 α αk0 )] n→+∞ n k=0   = Hμ ( f −1 α α − )   = Hμ ( f −1 α − α − ).

hμ ( f , α ) := lim

n→+∞

Hence (X.37) holds true. (X.38) can be proved in a similar way. Now we turn to the proof of the lemma. Let us first prove (X.30). It is easy to see that     − log μxu α0n (x) ≤ − log μxu [(α ∨ η u )n0 ](x) . Hence, by (X.34), we have    1  lim sup − log μxu α0n (x) ≤ Hμ (η u  f η u ) = h μ ( f ). n n→+∞ On the other hand, by Lemma X.2.7, one has 1 1 lim lim inf − log μxu (α0n (x)) ≥ lim lim inf − log μxu (Bu (x; n, δ )) n n δ ↓0 n→+∞ δ ↓0 n→+∞ = lim hu (x; δ ) δ ↓0

= hμ ( f ), where the last two equalities follow from a similar result to Proposition IX.IX.3.1 (see also [43, Proposition 7.2.1 and Corollary 7.2.2]). Hence (X.30) holds true. The proof of (X.28) is similar.

222

X Exact Dimensionality of Hyperbolic Measures

Combining Lemmas X.2.7, X.2.9 and the similar result to Proposition IX.IX.3.1, one can easily prove (X.29) and (X.31) in a similar way. We now turn to the proof of (X.27). Obviously        Iμ αnn (x) = Iμ αn0 (x) + Iμ α0n αn0 (x). By Shannon-McMillan-Breiman theorem we have for μ -a.e. x 1  0 Iμ αn (x) = hμ ( f −1 , α ) = h μ ( f , α ) n→+∞ n lim

(the second equality is a standard fact). Write  n−1      0  k ( f x). Iμ α0n αn0 (x) = ∑ Iμ f −1 α αn+k k=0

Similarly to the proof of Lemma X.2.9, we have         Iμ f −1 α αn0 (x) → Iμ f −1 α α − (x)

μ -a.e. and in L1 as n → ∞. Hence a slight variant of Birkhoff ergodic theorem gives 1  n  0  1 n−1  −1  0  k Iμ α0 αn (x) = lim ∑ Iμ f α αn+k ( f x) n→+∞ n n→+∞ n k=0      = Iμ f −1 α α − ∨ σ (x)d μ   = Hμ ( f −1 α α − ) = h μ ( f , α ) lim

for μ -a.e. x. (X.27) is then proved since diam α → 0 as δ → 0 and hence lim hμ ( f , α ) = hμ ( f ). δ ↓0

  Finally we prove (X.33). In fact we can write − 1n log μxu αn0 (x) as  1  0  u  1 n   Iμ αn η (x) = ∑ Iμ α  f −k η u ∨ f −1 α0k−1 ( f −k (x)), n n k=0 where α0−1 := {M}. Similarly we have    Iμ α  f −n η u ∨ f −1 α0n−1 (x) → 0

μ − a.e. and in L1

as n → +∞ and then we have (X.33). (X.32) can be proved similarly.

 

X.2 Diffeomorphisms’ Case–Proof of Theorem X.0.2

223

Let 0 < ε < 1 be given sufficiently small and let α be the partition introduced above for a fixed sufficiently small δ = δ∗ > 0. The system of Lyapunov charts employed in the construction of the special partition α will be referred to (in Subsection 1 X.2.5) as a system of (ε∗ , ∗ )-Lyapunov charts (0 < ε∗ ≤ 200m min{λ , 1}) with 0 the corresponding universal constant K0 (given by a similar result to Proposition IX.IX.2.1) denoted by K∗ . By (X.21), (X.22) and Lemmas X.2.7 and X.2.8, one can find a set Γ ⊂ M of measure μ (Γ ) > 1 − ε /2 together with an integer n0 = n0 (ε ) ≥ 1 and a number C = C(ε ) > 1 such that, for every x ∈ Γ and any integer n ≥ n0 , the following statements hold: a) for all integers k ≥ 1 we have   C−1 e−k(h+ε ) ≤ μ α0k (x)   C−1 e−k(h+ε ) ≤ μ αk0 (x)   C−1 e−k(h+ε ) ≤ μxs [αk0 ](x)   C−1 e−kε ≤ μxs α0k (x)   C−1 e−k(h+ε ) ≤ μxu α0k (x)   C−1 e−kε ≤ μxu αk0 (x)

≤ Ce−k(h−ε ) ,

(X.39)

≤ Ce−k(h−ε ) ,

(X.40)

≤ Ce

−k(h−ε )

,

≤ 1, ≤ Ce

(X.41) (X.42)

−k(h−ε )

,

≤ 1,

(X.43) (X.44)

where h = h μ ( f ); b) for all integers k ≥ 1 we have   C−1 e−2k(h+ε ) ≤ μ αkk (x) ≤ Ce−2k(h−ε ) ,   C−1 e−k(h+ε ) ≤ μxs αkk (x) ≤ Ce−k(h−ε ) ,   C−1 e−k(h+ε ) ≤ μxu αkk (x) ≤ Ce−k(h−ε ) ;

(X.45) (X.46) (X.47)

c)   s ≤ μxs Bs (x; e−n ) ≤ e−n(d −ε ) ,   u u e−n(d +ε ) ≤ μxu Bu (x; e−n ) ≤ e−n(d −ε ) ; e−n(d

s +ε )

(X.48) (X.49)

d) an αan (x) ⊂ B(x, e−n ), Γ ∩ B(x, e−n ) ⊂ α (x), 0 [η s ∨ αan ](x) ⊂ Bs (x; e−n ) ⊂ η s (x), Bu (x; e−n ) ⊂ η u (x), [η u ∨ α0an ](x) ⊂

(X.50) (X.51) (X.52)

where a is the integral part of 2(1 + λ −1). (X.50)–(X.52) are clearly guaranteed by Lemma X.2.7.

224

X Exact Dimensionality of Hyperbolic Measures

Since the induced metrics on W s (x) and W u (x) are locally (i.e., in a neighborhood of x) equivalent to the metric on M, by increasing n0 if necessary, we may also assume the following: e) for each x ∈ Γ and any z, z ∈ Bs (x; e−n0 ) d(z, z ) ≤ dxs (z, z ) ≤ 2d(z, z ),

(X.53)

d(z, z )

(X.54)



≤ dxu (z, z )



≤ 2d(z, z );

f) for every x ∈ Γ and n ≥ n0 e−n ) ∩ η s (x) ⊂ Bs (x; e−n ) ⊂ B(x, e−n ) ∩ η s (x), 2 e−n B(x, ) ∩ η u (x) ⊂ Bu (x; e−n ) ⊂ B(x, e−n ) ∩ η u (x). 2 B(x,

(X.55) (X.56)

By Borel density lemma, Lemma IX.IX.4.4, one can choose an integer n1 ≥ n0 and a set Γ% ⊂ Γ with μ (Γ%) > 1 − ε such that, for every n ≥ n1 and x ∈ Γ%,   1 μ B(x, e−n ) ∩ Γ ≥ μ (B(x, e−n )), 2  s  1 s s s −n μx B (x; e ) ∩ Γ ≥ μx (B (x; e−n )), 2  u  1 u u u −n μx B (x; e ) ∩ Γ ≥ μx (B (x; e−n )). 2

(X.57) (X.58) (X.59)

Furthermore we can require that for every n ≥ n1 and x ∈ Γ%   s μxs α0an (x) ∩ Bs (x; e−n ) ∩ Γ ≥ e−n(d +ε ) ,  0  u μxu αan (x) ∩ Bu (x; e−n ) ∩ Γ ≥ e−n(d +ε ) .

(X.60) (X.61)

The requirements (X.60) and (X.61) seem easy to be satisfied but actually their proofs are very much involved with the techniques presented in [70] and are somehow lengthy. We will postpone the proofs to the last Subsection X.2.5.

X.2.2 Preparatory Lemmas To be more accessible, we use arguments slightly different from the corresponding ones of [7]. Fix x ∈ Γ and an integer n ≥ n0 . We consider the following classes Rn , an 0 Fns and Fnu of elements, respectively, of the partitions αan , αan and α0an (we call their elements “rectangles”):  an  an (y) ⊂ α (x) : αan (y) ∩ Γ = 0/ , Rn := αan  0  0 Fns := αan (y) ⊂ α (x) : αan (y) ∩ Γ = 0/ ,   Fnu := α0an (y) ⊂ α (x) : α0an (y) ∩ Γ = 0/ .

X.2 Diffeomorphisms’ Case–Proof of Theorem X.0.2

225

The rectangles in Rn carry all the μ -measure of the set α (x) ∩ Γ , i.e.,

∑

μ (R ∩ Γ ) = μ (α (x) ∩ Γ ).

R∈Rn

It is clear that the rectangles in Rn belong to Fns ∨ Fnu := {Rs ∩ Ru : Rs ∈ Fns , Ru ∈ Fnu , Rs ∩ Ru = 0}. / But, besides these elements, there are in general other ones in Fns ∨ Fnu . That is, the measure μ conditioned on α (x) ∩ Γ has a deviation from the “direct product structure” at the “level” n. Following [7], we will compare the numbers of rectangles in Rn , Fns and Fnu intersecting a given set. This will allow us to evaluate the deviation of the sample measure μ from the “direct product structure” at the level n. For each A ⊂ α (x), we define a series of subsets of Rn , Fns or Fnu by the following   N(n, A) := R ∈ Rn : R ∩ A = 0/ , N s (n, y, A) := N(n, η s (y) ∩ Γ ∩ A), N u (n, y, A) := % s (n, A) := N % u (n, A) := N

N(n, η u (y) ∩ Γ ∩ A),   R ∈ Fns : R ∩ A = 0/ ,   R ∈ Fnu : R ∩ A = 0/ .

One should bear in mind that, though we omit writing α (x)’s in the notations, all the sets defined above depend on α (x). From here on we always denote by #A the cardinality of a countable set A. Lemma X.2.10 For each x ∈ Γ and integer n ≥ n0 + 1, we have #N s (n, x, B(x, e−n )) ≤ μxs (Bs (x; 4e−n )) ·Cean(h+ε ), #N u (n, x, B(x, e−n )) ≤ μxu (Bu (x; 4e−n )) ·Cean(h+ε ) . Proof. Let R ∈ N s (n, x, B(x, e−n )) and let z ∈ R ∩ η s (x) ∩ B(x, e−n ) ∩ Γ . Then R = an (z). By (X.46) we have αan an μxs (R) = μzs (αan (z)) ≥ C−1 e−an(h+ε ).

By (X.51) one has R ∩ η s (x) ⊂ Bs (z; e−n ). But, by (X.55), we have z ∈ η s (x) ∩ B(x, e−n ) ⊂ Bs (x; e−n+1 ). Therefore and hence

R ∩ η s (x) ⊂ Bs (x; e−n + e−n+1) ⊂ Bs (x; 4e−n )

μxs (Bs (x; 4e−n )) ≥

∑

R∈N s (n,x,B(x,e−n ))

μxs (R).

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X Exact Dimensionality of Hyperbolic Measures

This implies

μxs (Bs (x; 4e−n )) ≥ #N s (n, x, B(x, e−n )) ·C−1 e−an(h+ε ). The proof of the second inequality is similar. Lemma X.2.11 For each (x) ∈ Γ% and n ≥ n1 , we have

 

μ (B(x, e−n )) ≤ #N(n, B(x, e−n ) ∩ Γ ) · 2Ce−2an(h−ε ). Proof. In view of (X.57) we have 1 μ (B(x, e−n )) ≤ μ (B(x, e−n ) ∩ Γ ) ≤ μ (R) ∑ 2 R∈N(n,B(x,e−n )∩Γ ) ≤ #N(n, B(x, e−n ) ∩ Γ ) · max{μ (R) : R ∈ N(n, B(x, e−n ) ∩ Γ )}. an (z) for a point z ∈ R ∩ B(x, e−n ) ∩ Γ , the Let R ∈ N(n, B(x, e−n ) ∩ Γ ). Since R = αan inequality in the lemma follows from (X.45).   We now estimate the numbers of rectangles in the classes Rn ,Fns and Fnu . Clearly we have the following lemma.

Lemma X.2.12 For each x ∈ Γ and each n ≥ n0 , % s (n, B(x, e−n ) ∩ Γ ) · #N % u (n, B(x, e−n ) ∩ Γ ). #N(n, B(x, e−n ) ∩ Γ ) ≤ #N Our next goal is to compare the growth rate in n of the number of rectangles % u (n, B(x, e−n ) ∩ Γ )) with that of the number of rect% s (n, B(x, e−n ) ∩ Γ ) (resp. N in N s −n angles in N (n, x, B(x, e )) (resp. N u (n, x, B(x, e−n ))). We start with the following auxiliary result. Lemma X.2.13 For each x ∈ Γ and integer n ≥ n0 , we have % s (n, α (x)) ≤ Cean(h+ε ), #N % u (n, α (x)) ≤ Cean(h+ε ). #N 0 (z), we have Proof. Since α (x) is the union of a collection of rectangles R = αan

1 ≥ μ (α (x)) ≥

∑

μ (R).

% s (n,α (x)) R∈N

Hence, by (X.40), we have % s (n, α (x)) ≤ #N

1 ≤ Cean(h+ε ). 0 (z)) : z ∈ Γ ∩ α (x)} min{ μ (αan

The second inequality in the lemma can be proved in a similar way.

 

X.2 Diffeomorphisms’ Case–Proof of Theorem X.0.2

227

% s (n, B(x, e−n )∩ Γ ) and The following two lemmas show that the cardinalities of N u −n % (respectively N (n, B(x, e ) ∩ Γ ) and N u (n, x, B(x, e−n ))) are of almost the same growth rate in n up to a small exponent. N s (n, x, B(x, e−n ))

Lemma X.2.14 For each (x) ∈ Γ , y ∈ α (x) and n ≥ n0 , we have % s (n, B(y, e−n ) ∩ Γ ) ≥ #N s (n, y, B(y, e−n )) ·C−2 e−2anε , #N % u (n, B(y, e−n ) ∩ Γ ) ≥ #N u (n, y, B(y, e−n )) ·C−2 e−2anε . #N % s (n, η s (y) ∩ B(y, e−n ) ∩ Γ ) ⊂ N % s (n, B(y, e−n ) ∩ Γ ), hence Proof. Obviously N % s (n, B(y, e−n ) ∩ Γ ). % s (n, η s (y) ∩ B(y, e−n ) ∩ Γ ) ≤ #N #N % s (n, η s (y) ∩ B(y, e−n ) ∩ Γ ) we can On the other hand, for each rectangle Rs ∈ N s −n 0 (z). For such z, R = α an (z) find a point z in η (y) ∩ B(y, e ) ∩ Γ such that Rs = αan an s −n s % s (n, η s (y) ∩ B(y, e−n ) ∩ Γ ), is a rectangle in N (n, y, B(y, e )). For any fixed R ∈ N there are at most An distinct rectangles R’s in N s (n, y, B(y, e−n )) with such a correspondence, where by (X.41) and (X.46) An ≤ =

min{ μys (R) :

μys (Rs ) R ∈ N s (n, y, B(y, e−n )), R ⊂ Rs }

0 μzs (αan (z))    s an



s min{ μy (αan (z )) : z ∈ η (y) B(y, e−n ) Rs Γ }

≤ Ce−an(h−ε ) /[C−1 e−an(h+ε )] = C2 e2anε . Therefore % s (n, η s (y) ∩ B(y, e−n ) ∩ Γ ) ≥ #N s (n, y, B(y, e−n )) ·C−2 e−2anε . #N This implies the first inequality in the lemma. The other one can be proved analogously.   Lemma X.2.15 For μ -a.e. y ∈ α (x) ∩ Γ%, we have % s (n, B(y, e−n ) ∩ Γ ) −5anε #N e = 0, s −n n→+∞ #N (n, y, B(y, e )) % u (n, B(y, e−n ) ∩ Γ ) −5anε #N e lim sup = 0. u −n n→+∞ #N (n, y, B(y, e )) lim sup

In order to prove Lemma X.2.15, we introduce the concept of Hausdorff dimension of a set and some related results without proof. For any subset F of the standard Euclidean space Rm , the λ -Hausdorff outer measure of F is defined by mH (F, λ ) := lim inf{ ε →0 G

∑ (diamU)λ },

U∈G

228

X Exact Dimensionality of Hyperbolic Measures

where the infimum is taken over all finite or countable covers G of F by open sets with diam G ≤ ε . Then the Hausdorff dimension of the set F is defined as dimH F := inf{λ : mH (F, λ ) = 0} = sup{λ : mH (F, λ ) = ∞}. The direct calculation of the Hausdorff dimension of a given set F ⊂ Rd based on its definition is usually very difficult. An effective method is to find a suitable Borel measure μ on Rm and relate the Hausdorff dimension dimH F with d μ (x), d μ (x) (where x ∈ F), the (upper and/or lower) pointwise dimension of μ . This is the results stated in [64, pp. 43]; We represented here for the convenience of the reader. (In [64, pp. 43] there are some clerical errors in symbol, we have corrected them here). Lemma X.2.16 Let F ⊂ Rm be a Borel set and let μ be a Borel measure on Rm such that 0 < μ (F) ≤ 1. (1) If there is a number d > 0 such that d μ (x) ≤ d for every x ∈ F, then dimH F ≤ d; (2) (Uniform mass distribution principle, [20]) If there are numbers d > 0,C > 0 such that μ (B(x, r)) ≤ C · rd , for μ − a.e. x ∈ F and r > 0, then dimH F ≥ d; (3) (Nonuniform mass distribution principle) If there is a number d > 0 such that d μ (x) ≥ d

for μ − a.e. x ∈ F,

i.e., for any ε > 0 and μ -a.e. x ∈ F one can find a constant C = C(x, ε ) satisfying for any r > 0 μ (B(x, r)) ≤ C · rd−ε , then dimH F ≥ d. Proof of Lemma X.2.15. Except applying our Lemma X.2.13 instead of [7, Lemma 4], the proof is the same as that of [7, Lemma 5], we present it here for completeness. By (X.55) and (X.58), for each y ∈ Γ% and n ≥ n1 , we have     1 μys B(y, e−n ) ∩ Γ ≥ μys Bs (y; e−n ) ∩ Γ ≥ μys (Bs (y; e−n )) 2 and

  μys B(y, e−n ) ∩ Γ ≤



∑

R∈N s n,y,B(y,e−n )



μys (R).

Then, by (X.41), (X.46), (X.48) and noting a > 1, one has   #N s n, y, B(y, e−n )   μys B(y, e−n ) ∩ Γ   ≥ an (z) : z ∈ η s (y) ∩ α (x) ∩ Γ } max{μys αan

(X.62)

X.2 Diffeomorphisms’ Case–Proof of Theorem X.0.2

≥

μys (Bs (y; e−n )) 1   · an (z) : z ∈ η s (y) ∩ α (x) ∩ Γ } 2 max{μzs αan

≥

1 e−n(d +ε ) 1 −n(d s −ah+2aε ) e ≥ . −an(h− ε ) 2C e 2C

229

s

(X.63)

For any integer k ≥ 1, we consider the set % s (n, B(y, e−n ) ∩ Γ ) −5anε 1 #N e ≥ }. Fk := {y ∈ α (x) ∩ Γ% : lim sup s −n k n→+∞ #N (n, y, B(y, e )) For each y ∈ Fk take an increasing sequence {m j = m j (y) ≥ n1 }∞j=1 of positive integers such that for n = m j , j = 1, 2, · · ·     % s n, B(y, e−n ) ∩ Γ ≥ 1 #N s n, y, B(y, e−n ) e5anε #N 2k 1 −n(d s −ah−3aε ) e ≥ . 4kC

(X.64)

We wish to show that μ (Fk ) = 0. Assume on the contrary that μ (Fk ) > 0. Let Fk ⊂ Fk be the set of points z ∈ Fk for which there exists the limit log μzs (Bs (z; ρ )) = ds. ρ →0 log ρ lim

(X.65)

Clearly μ (Fk ) = μ (Fk ) > 0. Then we can find a point y ∈ Fk such that   μys Fk ∩ η s (y) = μys (Fk ) = μys (Fk ) > 0. Since μzs = μys for each z ∈ Fk ∩ η s (y), it follows from Lemma X.2.16 that dimH (Fk ∩ η s (y)) = d s .

(X.66)

Let us consider the collection of balls   D := B(z, e−m j (z) ) : z ∈ Fk ∩ η s (y), j = 1, 2, · · · . By the Besicovitch covering lemma (see, for example, [23] or [51]), one can find a countable subcover D ⊂ D of Fk ∩ η s (y) of arbitrarily small diameter and finite multiplicity p (p depends only on the dimension of M). This means that for any L ≥ n1 one can choose a sequence of points {zi ∈ Fk ∩ η s (y)}∞ i=1 and a sequence of ∞ integers {ti }∞ , where t ∈ {m (z )} and t ≥ L for each i, such that the collection i j i i i=1 j=1 of balls   D = B(zi , e−ti ) : i = 1, 2, · · ·

230

X Exact Dimensionality of Hyperbolic Measures

comprises a cover of Fk ∩ η s (y) whose multiplicity does not exceed p. We write Bi = B(zi , e−ti ). The Hausdorff sum corresponding to this cover is S :=

∞

∑ (diam B)d −ε = 2d −ε ∑ e−ti (d −ε ) . s

s

s

i=1

B∈D

By (X.64), noting that a > 1, we obtain ∞

∞

∑ e−ti (d −ε ) ≤ ∑ #N% s (ti , Bi ∩ Γ ) · 4kCe−ati(h+2ε ) s

i=1

i=1

∞

∑ e−al(h+2ε ) ∑ #N% s (ti , Bi ∩ Γ )

≤ 4kC

l=n1

i:ti =l

Since the multiplicity of the subcover D is at most p, each set Bi appears in the sum % s (ti , Bi ∩ Γ ) at most p times. Furthermore, since ti ≥ n1 and zi ∈ α (x) ∩ Γ%, ∑i:ti =l #N by (X.50) we have Bi ∩ Γ = B(zi , e−ti ) ∩ Γ ⊂ α (zi ) = α (x). Hence

∑ #N% s (ti , Bi ∩ Γ ) ≤ p · #N% s(l, α (x)).

i:ti =l

From Lemma X.2.13 it follows that S ≤ 4kC2d

s −ε

∞

∑ e−al(h+2ε ) p · #N% s(l, α (x))

l=n1

≤ 4kpC2 2

d s −ε

= 4kpC2 2

d s −ε

∞

∑ e−al(h+2ε )+al(h+ε )

l=n1 ∞

∑ e−alε < +∞.

l=n1

Since L can be chosen arbitrarily large (and so can the numbers ti ), it follows that dimH (Fk ∩ η s (y)) ≤ d s − ε < d s . This contradicts (X.66). Hence μ (Fk ) = 0 for any k ≥ 1. This implies the first identity in the lemma. The other one can be proved in a similar way.   By Lemma X.2.15, for μ -a.e. x ∈ Γ% there exists an integer n2 (x) (measurable in x) such that for n ≥ n2 (x) we have % s (n, B(x, e−n ) ∩ Γ ) < #N s (n, x, B(x, e−n ))e5anε , #N % u (n, B(x, e−n ) ∩ Γ ) < #N u (n, x, B(x, e−n ))e5anε . #N

(X.67) (X.68)

X.2 Diffeomorphisms’ Case–Proof of Theorem X.0.2

231

Moreover, by Lusin’s theorem, there exists a subset Γ ε ⊂ Γ% of measure μ (Γ ε ) > μ (Γ%) − ε > 1 − 2ε such that nε := sup{n1 + 2, n2(x) : (x) ∈ Γ ε } < ∞ and the inequalities (X.67) and (X.68) hold for every n ≥ nε and x ∈ Γ ε . Furthermore, we can require that (X.57)–(X.59) hold true for any x ∈ Γ ε and n ≥ nε with Γ being replaced by Γ%.

X.2.3 Proof of Theorem X.2.2 Since Theorem X.2.2 2) follows directly from Theorem X.2.2 1) and (X.21), we will just prove Theorem X.2.2 1). We will divide it into the following two lemmas. Lemma X.2.17 For each x ∈ Γ ε and n ≥ nε ,       μxs Bs (x; e−n ) · μxu Bu (x; e−n ) ≤ μ B(x, 3e−n ) · 2C5 e7anε . Lemma X.2.18 For each x ∈ Γ ε and n ≥ nε ,

μ (B(x, e−n )) ≤ μxs (Bs (x; 4e−n ))μxu (Bu (x; 4e−n )) · 2C3 e14anε . Proof of Lemma X.2.17. Fix a point x ∈ Γ ε and an integer n ≥ nε . First we observe that B(z, e−n ) ⊂ B(x, 3e−n ),

∀z ∈ B(x, 2e−n ).

Consider rectangles in Rn which intersect B(x, 2e−n ) ∩ Γ and let R be such a rectanan (z). By (X.50) gle. Then there is a point z ∈ B(x, 2e−n ) ∩ Γ ⊂ α (x) such that R = αan we have R ⊂ B(z, e−n ) and hence R ⊂ B(x, 3e−n ). Therefore   μ B(x, 3e−n ) ≥ R∈N



∑

n,B(x,2e−n )∩Γ



μ (R)

  ≥ #N n, B(x, 2e−n ) ∩ Γ ·C−1 e−2an(h+ε ).

(X.69)

Then we observe that B(z, e−n ) ⊂ B(x, 2e−n ),

∀z ∈ B(x, e−n ).

(X.70)

0 (z) be a rectangle in F s which intersects B(x, e−n ) ∩ Γ %. We can Let now Rs = αan n   −n u 0 % assume z ∈ B(x, e ) ∩ Γ . Then, for each rectangle R ∈ N n, z, αan (z) ∩ B(z, e−n ) , −n R intersects B(z, e−n ) ∩ Γ and hence also  intersects−nB(x, 2e ) ∩ Γ by (X.70). This means that R is in fact a rectangle in N n, B(x, 2e ) ∩ Γ . For distinct rectangles

232

X Exact Dimensionality of Hyperbolic Measures

  0 (z) and α 0 (z ) in N 0 (z)  α 0 (z ) = 0 % s n, B(x, e−n ) ∩ Γ , since αan αan / there is an an    u n, z, α 0 (z) ∩ B(z, e−n ) and N u n, z , α 0 (z ) ∩ no common rectangle in both N an an  B(z , e−n ) . Thus   #N n, B(x, 2e−n ) ∩ Γ ∑

≥

0 (z)∈N % s n,B(x,e−n )∩Γ% αan



  0 #N u n, z, αan (z) ∩ B(z, e−n )

  % s n, B(x, e−n ) ∩ Γ% · ≥ #N   0 (z) ∩ B(z, e−n ) : z ∈ B(x, e−n ) ∩ Γ%}. inf{#N u n, z, αan

(X.71)

  0 (z) ∩ B(z, e−n ) for each z ∈ Γ %. By (X.56) we have We now estimate #N u n, z, αan 0 0 η u (z) ∩ αan (z) ∩ Bu (z; e−n ) ⊂ η u (z) ∩ αan (z) ∩ B(z, e−n ).

Therefore by (X.61) and noting a ≥ 2, one has 0 (z) ∩ B(z, e−n )) #N u (n, z, αan  0  μzu αan (z) ∩ Bu (z; e−n ) ∩ Γ   ≥ 0 (z) ∩ B(z, e−n ) } max{μzu (R) : R ∈ N u n, z, αan

≥ e−n(d

u +ε )

·C−1 ean(h−ε )

≥ C−1 ean(h−2ε ) μxu (Bu (x; e−n )).

(X.72)

Following the same line of the proof of Lemma X.2.14, we can prove that     % s n, B(x, e−n ) ∩ Γ% ≥ #N s n, x, B(x, e−n ) ∩ Γ% ·C−2 e−2anε . #N

(X.73)

Furthermore,   #N s n, x, B(x, e−n ) ∩ Γ%   μxs B(x, e−n ) ∩ Γ% ≥   max{μxs (R) : R ∈ N s n, x, B(x, e−n ) ∩ Γ% }   ≥ μxs Bs (x; e−n ) ∩ Γ% ·C−1 ean(h−ε ) ≥

1 an(h−ε ) s s e · μx (B (x; e−n )). 2C

(X.74)

Putting (X.69)-(X.74) together, we obtain the inequality in the lemma.   Proof of Lemma X.2.18. This lemma is a consequence of Lemmas X.2.10, X.2.11, X.2.12 and inequalities (X.67) and (X.68).  

X.2 Diffeomorphisms’ Case–Proof of Theorem X.0.2

233

X.2.4 The Case of Nonergodic Measures We show how to modify our arguments in the case when the measure μ is not ergodic. By an easy variant of Lemma X.2.8 for nonergodic measures, ( there exists an f -invariant (μ -mod 0) function hμ (·) : M → R+ such that hμ ( f ) = hμ (x)d μ and (X.27)-(X.31) hold with hμ ( f ) being replaced by hμ (x). Then, by Lusin’s theorem, given ε > 0, there is a set Γ ⊂ M with μ (Γ ) ≥ 1 − ε such that hμ (x), dμs (x), dμu (x), the smallest positive exponent λ u (x) := λu(x) (x) and the largest negative exponent λ s (x) := λs(x) (x) are all continuous on Γ . Let us fix 0 < δ < 1 and consider the sets

Γ (x) := {x ∈ M : |hμ (x) − hμ (x )| < δ , |dμs (x) − dμs (x )| < δ , |d μu (x) − dμu (x )| < δ , |λ u (x)−1 − λ u (x )−1 | < δ , |λ s (x)−1 − λ s(x )−1 | < δ } which are clearly f -invariant (μ -mod 0). These sets, restricted to Γ , form an open cover of Γ by our requirement. Since M is a Polish space, it is Lindel¨of and hence so is its subset Γ . Therefore there exists a countable sub-collection of sets {Γ i := Γ (xi )}i≥1 (with μ (Γ i ) > 0 for each i) which still covers Γ . Let μ i be the conditional measure of μ on Γ i and it is obviously f -invariant. Therefore the arguments presented in Chapter IX can be applied to μ i , yielding the existence of the local stable and unstable dimension, dμs i (x) and d μu i (x), of μ i . Since for μ -a.e. x ∈ Γ i hμ i (x) = h μ (x),

d μs i (x) = d μs (x),

d μu i (x) = d μu (x),

(X.75)

|d μu i (x) − diu | < δ ,

(X.76)

the following inequalities hold for μ -a.e. x ∈ Γ i |h μ i (x) − hi | < δ ,

|dμs i (x) − dis | < δ ,

where hi := h μ (xi ), dis := dμs i (xi ) and diu := d μu i (xi ). Then we can apply the arω

ω

guments in the proof of Theorem X.2.2 to the measure μ i , showing that for μ -a.e. x ∈ Γ i the following inequalities hold true d μ i (x) ≥ dμs i (x) + dμu i (x) − ci δ ,

(X.77)

d μ i (x) ≤ dμs i (x) + dμu i (x) + ci δ ,

(X.78)

where d μ i (x) and d μ i (x) are the lower and upper pointwise dimension of μ i , and ci is a constant which is independent of x or δ . In fact, one can take ci = 12 ai where ai is the integral part of 4 + 2 max{λ u (xi )−1 , |λ s (xi )|−1 }. To see this, one needs to go through slight modifications of the arguments in Section 4 and the proofs of Lemmas X.2.10–X.2.15, X.2.17 and X.2.18, made by replacing μ , a, h, d s and d u

234

X Exact Dimensionality of Hyperbolic Measures

respectively with μ i , ai , hi , dis and diu and taking δ into account due to (X.76). For examples, the first inequality of Lemma X.2.10 reads now     #N s n, x, B(x, e−n ) ≤ (μ i )sx Bs (x; 4e−n ) ·Ceai n(hi +δ +ε ) ; to modify Lemma X.2.15, replace (X.65) with the inequality   log(μ i )sz Bs (z; ρ ) = d μs i (z) ≥ dis − δ lim ρ →0 log ρ and apply Lemma X.2.16, yielding dimH (Fk ∩ η s (y)) ≥ dis − δ which takes the place of (X.66), then the limits in Lemma X.2.15 can be modified to   % s n, B(y, e−n ) ∩ Γ #N   e−ai n(5ε +4δ ) = 0, lim sup s −n n→+∞ #N n, y, B(y, e )   % u n, B(y, e−n ) ∩ Γ #N  e−ai n(5ε +4δ ) = 0.  lim sup n→+∞ #N u n, y, B(y, e−n ) From these it can be seen that one may take ci = 12 ai . It’s clear that for μ -a.e. x ∈ Γ i d μ i (x) = d μ (x),

d μ i (x) = d μ (x).

Noting that for μ -a.e. x ∈ Γ i |λ u (xi )|−1 < |λ u (x)|−1 + 1,

|λ s (xi )|−1 < |λ s (x)|−1 + 1,

by (X.75), (X.77) and (X.78), we have the following inequalities for μ -a.e. x ∈ Γ i and each i ≥ 1 d μ (x) ≥ d μs (x) + dμu (x) − c δ ,

(X.79)

d μs (x) + dμu (x) + c δ ,

(X.80)

d μ (x) ≤

where c = c(x) := 24 max{|λ u (x)|−1 + 3, |λ s(x)|−1 + 3}. Since {Γ i }i≥1 covers Γ , (X.79) and (X.80) hold true for μ -a.e. x ∈ Γ . Letting δ → 0, we know that for μ -a.e. x ∈Γ

d μ (x) = d μ (x) = d μs (x) + dμu (x). (X.81) Since ε > 0 is arbitrary and μ (Γ ) ≥ 1 − ε , (X.81) holds for μ -a.e. x ∈ M. This proves the last statement of Theorem X.2.2.

X.2 Diffeomorphisms’ Case–Proof of Theorem X.0.2

235

X.2.5 Proofs of Requirements (X.60) and (X.61) In this subsection we give proofs of requirements (X.60) and (X.61). We will inherit many similar notations introduced in Chapter IX without announcing beforehand. In fact we will prove Proposition X.2.19 Let μ be ergodic and let α , η s , η u and Γ be as introduced in Section X.2. Then for μ -a.e. x ∈ Γ one has 1 lim − log μxs (α0an (x) ∩ Bs (x; e−n ) ∩ Γ ) = d s , n 1 0 (x) ∩ Bu (x; e−n ) ∩ Γ ) = d u . lim − log μxu (αan n→+∞ n n→+∞

(X.82) (X.83)

We will only prove equation (X.83) since the other one can be obtained by applying the same arguments to the system f −1 : M ←. The proof goes essentially along the same line as Chapter IX and we set it into the following three parts (named (A), (B), (C) respectively). (A) Pseudo-pointwise Dimensions and Pseudo-local Entropies Let  := α ∨ {Γ , M \ Γ }. α  ) ≤ Hμ (α ) + log 2 < +∞. In view of (X.33) and (X.21) it is Clearly we have Hμ (α an easy exercise to check that (X.83) follows directly from the proposition below.  be as introduced above. Then for μ -a.e. x Proposition X.2.20 Let η u and α 0 1 η u ∨ αan lim − log μx (Bu (x; e−n )) = d u . n

n→+∞

The limit in the above proposition looks very much like the pointwise dimension of μ on W u -manifold at x defined analogous to that in Chapter IX and, for the sake of language, in this subsection we name it as the pseudo-pointwise dimension of μ on W u -manifold at x. The proposition states that the pseudo-pointwise dimension coincides with the usual pointwise dimension. We will prove the existence of pseudo-pointwise dimension on W u -manifolds and its coincidence with d u through quantities named pseudo-pointwise dimensions on W i -manifolds (Please bear in mind that in the rest of this subsection W u,i is denoted by W i for simplicity of presentation!) and pseudo-local entropies along W i -manifolds (i = 1, 2, · · · , u) which we will introduce below. We begin with the following notations. Let η 1 > η 2 > · · · > η u be a sequence of increasing measurable partitions of M with each η i being subordinate to the corresponding W i -manifolds. Let n : (0, 1) → Z+ and b : Z+ → Z+ be two monotonic functions such that 0 < lim − ρ ↓0

n(ρ ) < +∞, log ρ

0 < lim

n→+∞

b(n) < +∞ n

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X Exact Dimensionality of Hyperbolic Measures

(such functions will be called well rated functions in this subsection; in the subsequent arguments one may take n(ρ ) = [−a log ρ ] and b(n) = [bn] for some positive numbers a and b, where [t] denotes the integral part of a real number t). Put 0  η i ∨ αn( ρ)

 Bi (x; ρ ) / log ρ ,

δ i (x) := lim inf log μx ρ →0

0  η i ∨ αn( ρ)



 Bi (x; ρ ) / log ρ ,

δ i (x) := lim sup log μx ρ →0

η i ∨ α0   1 h i (x; ρ ) := lim inf − log μx b(n) Bi (x; n, ρ ) , n→+∞ n  η i ∨ α0  1

hi (x; ρ ) := lim sup − log μx b(n) Bi (x; n, ρ ) , n n→+∞

h i (x) := lim h i (x; ρ ),

(X.85) (X.86) (X.87) (X.88)

ρ ↓0



(X.84)



hi (x) := lim hi (x; ρ ),

(X.89)

ρ ↓0

where Bi (x; ρ ) and Bi (x; n, ρ ) are defined analogous to those in Chapter IX, i.e., Bi (x; ρ ) := {y ∈ W i (x) : dxi (x, y) < ρ }, B (x; n, ρ ) := {y ∈ W i (x) : d if k x ( f k x, f k y) < ρ for 0 ≤ k ≤ n}. i



We call δ i (x) and δ i (x), respectively, the lower and the upper pseudo-pointwise

dimension of μ on the W i -manifold at x, and hi (x), hi (x) the lower and the upper i pseudo-local entropy along W -manifolds at x.  ) := δ i (x) to indicate the dependence of this quantity Sometimes we write δ i (x; α  . Other similar notations have analogous meanings. on the partition α Remark X.1. By (X.33) and similar arguments presented in Chapter IX one can easily check



δ i (x) ≥ δ i (x) ≥ δi ,

hi (x; ρ ) ≥ h i (x; ρ ) ≥ hi (x; ρ ) and hence



hi (x) ≥ h i (x) ≥ hi

for μ -a.e. x and each i ≤ u, where δi , hi and hi are introduced analogous to those in Chapter IX.



It would be clear later that the quantities δ i , δ i , h i and hi are in fact independent of the choices of the well rated functions n(·) and b(·). We point out that for μ -a.e. x and i = 1, 2, · · · , u [η i ∨ α0n ](x) ⊂ Bi (x; n, δ∗ ),

∀n ≥ n0 (x)

(X.90)

X.2 Diffeomorphisms’ Case–Proof of Theorem X.0.2

237

and, for each y ∈ [η i ∨ α0n ](x) with n ≥ n0 (x), f k y stays well inside the (ε∗ , ∗ )-chart at f k x for k = 0, 1, · · · , n. For i = u, this has already been stated in Lemma X.2.7; for a general i, the conclusion can be proved analogously and the reader is referred to Chapter IX for details. These facts will be our starting point of many other basic estimates. (B) Coincidence of Pseudo and Usual Local Entropies  be the (fixed) partition introduced in the above part (A). For each sufficiently Let α small ε > 0, in particular, ε < ε∗ , we choose a system of (ε , )-Lyapunov charts {Φx } given by the analogous of Proposition VII.VII.4.2. First observe that ∗ (x) ≤ (x) and hence   ∗ f n x ≤ (x)e|n|ε ,

∀n ∈ Z

(X.91)

for μ -a.e. x. This can be deduced directly from the explicit formula of (x) (see, for example, Chapter VII or [51, pp. 142–145]). The above observation helps us switching from one system of Lyapunov charts with a fixed parameter ε∗ to another system of Lyapunov charts with an arbitrarily smaller parameter ε .

Then we will prove that hi (x) ≤ hi for any i ≤ u and hence the pseudo-local entropies coincide with the usual local entropies. Before doing this, we first prove the following fact. Lemma X.2.21 Let λ := min{|λi | : i = 1, 2, · · · , r} and let p ∈ (0, 1) be given. Let 0m for some m ∈ Z+ . Then for μ -a.e. x we have β := α [η i ∨ β0n ](x) ⊂ Bi (x; [np], e−m(λi −3ε∗ ) ),

∀n ≥ n 0 (x),

(X.92)

where n 0 : M → Z+ is a Borel function independent of m. Proof. In fact, by similar arguments presented in Chapter IX (see also the proof of [70, Prop. 5.1]) we can prove that   d if k x ( f k x, f k y) ≤ 2K∗ δ∗ ∗ f k x e−(n−k)(λi−3ε∗ ) ,

k = 0, 1, · · · , n

for any y ∈ [η i ∨ α0n ](x) with n ≥ n0 (x) and for μ -a.e. x. Let

ε0 := min{

ε∗ 1 − p , (λ − 3ε∗)} 2 3

and take a system of (ε0 , 0 )-Lyapunov charts (where 0 (x) := (x) is a function introduced analogous to that of Proposition VII.VII.4.2 corresponding to the parameter ε = ε0 ). Then, by (X.91), for k = 0, 1, · · · , n we have d if k x ( f k x, f k y) ≤ 2K∗ δ∗ 0 (x)e−(n−k)(λi −3ε∗ )+kε0 .

(X.93)

 ≥ α , (1 − p)(λi − 3ε∗ ) ≥ 3ε0 and Fix an integer m ≥ 0. In view of the facts α 0n+m ](x) with n ≥ n0 (x) we have (X.93), for any y ∈ [η i ∨ α

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X Exact Dimensionality of Hyperbolic Measures

d if k x ( f k x, f k y) ≤ 2K∗ δ∗ 0 (x)e−m(λi −3ε∗ )−2nε0 ,

k = 0, 1, · · · , [np].

This implies 0n+m ] ⊂ Bi (x; [np], 2K∗ δ∗ 0 (x)e−m(λi −3ε∗ )−2nε0 ) [η i ∨ α for μ -a.e. x and any n ≥ n0 (x). The conclusion (X.92) becomes then obvious with n 0 (x) being defined by n 0 (x) := max{n0 (x), 1 + [

  1 log 2K∗ δ∗ 0 (x) ]}. 2ε0  

0m for some m ∈ Z+ , then Lemma X.2.22 If β = α  η i ∨ α0   1  lim − log μx b(n) [η i ∨ β ]n0 (x) = Hμ (η i  f η i ) = hi n→+∞ n holds for μ -a.e. x, where b : Z+ → Z+ is a well rated function. 0m , for any n large enough we can write Proof. Since β = α η i ∨ α0   − log μx b(n) [η i ∨ β ]n0 (x)     0 b(n) = Iμ [η i ∨ β ]n0 η i ∨ α (x)  n      k  0 0 0 b(n) b(n) ∨ f kα = Iμ β η i ∨ α (x) + ∑ Iμ β ∨ η i  f η i ∨ f βk−1 ( f x) k=1

 n      k  0 m b(n) b(n)+k−1 = Iμ β η i ∨ α (x) + ∑ Iμ β ∨ η i  f η i ∨ f α ( f x) k=1

 n      k  0 0 b(n) (x) + ∑ Iμ β ∨ η i  f η i ∨ f βb(n)+k−1 ( f x). = Iμ β η i ∨ α k=1

Clearly we have     In (x) := Iμ β ∨ η i  f η i ∨ f βn0 (x) → I∞ (x)     in L1 and μ -a.e. as n tends to +∞ with I∞ (x) := Iμ β ∨ η i  f η i ∨ f β − (x). Note also that  ) < +∞ Hμ (β ) ≤ (m + 1)Hμ (α and



    I∞ (x)d μ = Hμ (β ∨ η i  f η i ∨ f β − ) = Hμ (η i  f η i ) = hi

X.2 Diffeomorphisms’ Case–Proof of Theorem X.0.2

239

which can be proved following the idea of the proofs of Propositions IX.IX.2.12 and IX.IX.2.13 (see also [51, Lemma VI.5.4]). Then a slight variant of Birkhoff ergodic theorem gives   η i ∨ α0  1 lim sup − log μx b(n) [η i ∨ β ]n0 (x) = I∞ (x)d μ = hi n n→+∞ for μ -a.e. x.   By means of the above lemmas, we can now prove the following proposition which tells the coincidence of the pseudo-local entropies with the usual local entropies. 

 Proposition X.2.23 hi (x) = hi (x) = Hμ (η i  f η i ) = hi for μ -a.e. x and 1 ≤ i ≤ u, where hi is the local entropy along W i -manifolds defined analogously to that of Chapter IX. Proof. Due to Remark X.1, one needs only to prove

hi (x) ≤ hi . Given δ > 0 arbitrarily, let m ≥ 1 be an integer so that e−m(λ −3ε∗ ) ≤ δ . Put β := m  α0 . Then, for any p ∈ (0, 1), by Lemma X.2.21 we have for μ -a.e. x for any n ≥ n 0 (x).

[(η i ∨ β )n0 ](x) ⊂ Bi (x; [np], δ )

Since p ∈ (0, 1), we have {[np] : n ∈ Z+ } = Z+ and hence  η i ∨ α0  1

hi (x; δ ) = lim sup − log μx b(n) Bi (x; n, δ ) n n→+∞ η i ∨ α0   1 log μx b([np]) Bi (x; [np], δ ) = lim sup − [np] n→+∞ η i ∨ α0 1 1 ≤ · lim sup − log μx b([np]) ([(η i ∨ β )n0 ](x)), p n→+∞ n

(X.94)

where b(·) is a well rated function. Clearly b (n) := b([np]) is still well rated for any fixed p ∈ (0, 1). Thus, by Lemma X.2.22, the limit in (X.94) is hi for any fixed p > 0. Hence we have proved

hi (x; δ ) ≤ hi /p

  for any δ > 0 and any p ∈ (0, 1). This proves hi (x) ≤ hi . For the need of the next subsection, we deduce another property of the parti. tion α

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X Exact Dimensionality of Hyperbolic Measures

0m and ε ∈ (0, ε∗ ) be given. Then there exists a Lemma X.2.24 Let m ≥ 1, β := α

+ Borel function n0 : M → Z (independent of m) such that for μ -a.e. x [η i ∨ β0n ](x) ⊂ Bi (x; e−n(λi −9ε ) ),

∀n ≥ n

0 (x).

Furthermore, for any well rated function b(·) and μ -a.e. x we have  η i ∨ α0  1 lim − log μx b(n) β0n (x) = hi . n→+∞ n Proof. From the proof of Lemma X.2.21 we have   d if k x ( f k x, f k y) ≤ 2K∗ δ∗ ∗ f k x e−(n+m−k)(λi −3ε∗ ) for any y ∈ [η i ∨ β0n ](x) and k = 0, 1, · · · , n + m with n ≥ n0 (x). By (X.91) one has d if k x ( f k x, f k y) ≤ 2K∗ δ∗ (x)e−(n+m−k)(λi−3ε∗ )+kε for k = 0, 1, · · · , n + m since ε < ε∗ . Then, noting that λi ≥ 200ε∗ ,we have p := 1− 4ε 1

3 λi −3ε∗ ∈ (0, 1). Hence, for any n ≥ n (x) := max{n0 (x), 4 + [ ε log 2K0 K∗ δ∗ (x) ]} (where K0 is a universal constant introduced analogous to that of Proposition VII.VII.4.2), we have  −2 d if k x ( f k x, f k y) ≤ K0−1  f k x ,

k = 0, 1, · · · , [np] + 1

for any y ∈ [η i ∨ β0n ](x). This means that f k y, k = 0, 1, · · · , [np] + 1 stay well inside the (ε , )-charts. Therefore we can employ (ε , )-charts for these points instead of 4ε (ε∗ , ∗ )-charts, yielding that (noting p = 1 − λi −3 ε∗ and λi ≥ 200ε∗ ) dxi (x, y) ≤ 2K0 δ∗ (x)e−([np]+1)(λi−3ε ) ≤ 2K0 δ∗ (x)e−np(λi−3ε )   2λ −18ε −n λi −9ε + λi −3ε ∗ ε ∗ i ≤ 2K0 δ∗ (x)e ≤ 2K0 δ∗ (x)e−nε · e−n(λi−9ε ) for any y ∈ [η i ∨ β0n ](x) with n ≥ n (x). Then one obtains the first conclusion with n

0 (x) being defined by   1 n

0 (x) := max{n (x), 1 + [ log 2K0 δ∗ (x) ]}. ε By Lemma X.2.22, we clearly have η i ∨ α0   1 lim sup − log μx b(n) β0n (x) ≤ hi . n n→+∞

X.2 Diffeomorphisms’ Case–Proof of Theorem X.0.2

241

On the other hand, by Lemma X.2.21 and Proposition X.2.23 we have for each p ∈ (0, 1)   η i ∨ α0  η i ∨ α0  1 1 0n+m (x) lim inf − log μx b(n) β0n (x) = lim inf − log μx b(n) α n→+∞ n→+∞ n n η i ∨ α0  n+km  1 0 log μx b(n+(k−1)m) α (x) = lim lim inf − k→+∞ n→+∞ n + (k − 1)m   η i ∨ α0 1 ≥ lim lim inf − log μx b(n+(k−1)m) Bi (x; [np], e−km(λi −3ε∗ ) ) k→+∞ n→+∞ n = p · lim h i (x; e−km(λi −3ε∗ ) ) = p · hi . k→+∞

Letting p → 1, we obtain the second conclusion. (C) Coincidence of Pseudo and Usual Pointwise Dimensions Based on Proposition X.2.23, we will prove inductively that for μ -a.e. x

δ i (x) = δ i (x) = δi ,

 

i = 1, 2, · · · , u

following the same line of Chapter IX, Section IX.4, where the constant δi for each i is the pointwise dimension of μ on W i -manifolds defined analogously. Let γi = δi − δi−1 be the transverse dimension of μ on W i /W i−1 introduced analogous to that of Chapter IX for i = 1, 2, · · · , u with the convention δ0 := 0. First we have

Proposition X.2.25 δ 1 (x) = δ 1 (x) = δ1 = γ1 for μ -a.e. x. Proof. We have already known that

δ 1 (x) ≥ δ1 = γ1 . 0m with m ≥ 1, by Lemmas X.2.24 and X.2.22 it is clear that For β = α

h1 = h 1 ≥ λ1 δ 1 ,

  i.e., δ 1 ≤ h1 /λ1 = δ1 .



Based on Proposition X.2.25, we will then prove the coincidence of δ i and δ i with δi for i = 2, · · · , u by induction. Let 2 ≤ i ≤ u and assume that we have already

proved the coincidence of δ i−1 and δ i−1 with δi−1 , i.e., for μ -a.e. x

δ i−1 (x) = δ i−1 (x) = δi−1 . In view of the facts

δ i (x) ≥ δi

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X Exact Dimensionality of Hyperbolic Measures

and

γi = δi − δi−1 = (hi − hi−1)/λi ,

the coincidence of δ i (x) and δ i (x) with δi can be deduced directly from the following proposition.



Proposition X.2.26 δ i − δ i−1 ≤ (hi − hi−1)/λi for μ -a.e. x. In order to prove Proposition X.2.26, we present some basic estimates in the following lemma, which follows readily from the above part (B) and Lemma X.2.24 and hence whose proof is omitted here. 0m for some integer m ≥ 1. There Lemma X.2.27 Let ε ∈ (0, ε∗ ) be given. Let β = α + exists a Borel function n% : M → Z such that the following hold for μ -a.e. x:   (1) [η i ∨ β0n ](x) ⊂ Bi x; e−n(λi −9ε ) for any n ≥ n%(x); 0   η i−1 ∨ αb(n) (2) − 1n log μx β0n (x) ≥ hi−1 − ε for any n ≥ n%(x);  η i ∨ α0  (3) − 1n log μx b(n) β0n (x) ≤ hi + ε for any n ≥ n%(x);  (4) L := Bi−1 x; e−n(λi −9ε ) ) ⊂ η i−1 (x) for any n ≥ n%(x); 0 η i−1 ∨ αb(n) /

(5) [log μx (L)] [−n(λi − 9ε )] ≤ δ i−1 + ε for any n ≥ n%(x); / η i ∨ α0 

(6) [log μx b(n) Bi (x; 2e−n(λi −9ε ) ) ] [−n(λi − 9ε )] ≥ δ i − ε for infinitely many n ≥ n%(x). Proof of Proposition X.2.26. The proof follows the main line of that of Proposition IX.IX.4.3 (see also [70, Prop. 5.3]) with some slight modifications. We present it here for completeness. Let

Λ n := {x : n%(x) ≤ n and x satisfies the requirements (1)–(6) of Lemma X.2.27}. Clearly μ (Λ n ) ↑ 1 as n → +∞. Hence for any ε ∈ (0, 1) there is an integer N1 such that Λ := Λ N1 has μ -measure not less than 1 − ε /2. Then we can assume that, for

 ∨ {Λ , M \ Λ }, i.e., μ -a.e. x, δ i−1 (x; α¯ ) = δ i−1 (x; α¯ ) = δi−1 where α¯ := α 0 η i−1 ∨α¯ n( ρ)

lim

ρ →0

log μx

(Bi−1 (x; ρ ))

= δ i−1 = δi−1 log ρ



where δ i−1 = δi−1 is constant and n(ρ ) is any a well rated function n : (0, 1) → Z+ . Hence for any well rated function b : Z+ → Z+ one has 0 η i−1 ∨ αb(n) 1

lim − log μx (Bi−1 (x; e−n(λi −9ε ) ) ∩ Λ ) = (λi − 9ε ) · δ i−1 n→+∞ n

X.2 Diffeomorphisms’ Case–Proof of Theorem X.0.2

243

for μ -a.e. x ∈ Λ . There is thus another integer N2 ≥ N1 and a subset Λ¯ ⊂ Λ of μ -measure not less than 1 − ε such that, for any x ∈ Λ¯ and any n ≥ N2 , 0 η i−1 ∨ αb(n)

μx



(L ∩ Λ ) ≥ e−n(λi −9ε )(δ i−1 +ε )

with L = Bi−1 (x; e−n(λi −9ε ) ). Now we fix arbitrarily a point x ∈ Λ¯ . By Lemma X.2.27 one has 0  η i−1 ∨ αb(n) η i−1 ∨ α0    μx β0n (y) = μ(ω ,y) b(n) β0n (y) ≤ e−n(hi−1−ε ) 0 ](x) ⊂ [η i−1 ∨ α 0 ](x) and n ≥ N . Hence, by Lemma b(n) b(n) for any y ∈ L ∩ Λ ∩ [α 2 X.2.27(5), 0 b(n) ](x)} #{β0n (y) : y ∈ L ∩ Λ ∩ [α 0 η i−1 ∨ αb(n)

≥ μx

/ (L ∩ Λ ) e−n(hi−1−ε )

≥ exp{n[hi−1 − ε − (λi − 9ε )(δ i−1 + ε )]}. On the other hand, according to Lemma X.2.27(1), we have   [η i ∨ β0n ](y) ⊂ Bi y; e−n(λi −9ε ) ,

∀n ≥ N2

for any (ω , y) ∈ Λ and hence also for any y ∈ L ∩ Λ . Clearly dxi (x, y) ≤ dxi−1 (x, y) for any y ∈ L. Therefore [η i ∨ β0n ](y) ⊂ Bi (x; 2e−n(λi −9ε ) ) ∀y ∈ L ∩ Λ and n ≥ N2 . Thus, noting that L ⊂ η i−1 (x) ⊂ η i (x), we have 0  η i ∨ αb(n)

 Bi (x; 2e−n(λi −9ε ) )

log μx

0 b(n) ≥ log #{[η i ∨ β0n ](y) : y ∈ L ∩ Λ ∩ [α ](x)} 0  η i ∨ αb(n)

 [η i ∨ β0n ](y)

+ log min μx y

0 b(n) ](x)} ≥ log #{β0n(y) : y ∈ L ∩ Λ ∩ [α 0  η i ∨ αb(n)

 β0n (y)

+ log min μx y



≥ −n[hi − hi−1 + 2ε + (λi − 9ε )(δ i−1 + ε )]. Then by Lemma X.2.27(6) we obtain for any x ∈ Λ¯



(λi − 9ε )(δ i − δ i−1 − 2ε ) ≤ hi − hi−1 + 2ε , which implies Proposition X.2.26 by letting ε → 0 and then ε → 0.

 

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X Exact Dimensionality of Hyperbolic Measures

X.3 Comments on Further Researches Theorem X.0.2 states the validity of Eckmann-Ruelle conjecture for diffeomorphisms. As has been mentioned already, Eckmann-Ruelle conjecture can also be proposed for endomorphisms. And Theorem X.0.1 inspires us to have some confidence of Eckmann-Ruelle conjecture for endomorphisms associated with hyperbolic measures, if we can further prove the existence of the stable pointwise dimensions; Then following the idea presented in [7] or Section X.2 of this chapter (see also [54] for a proof for the setting of random diffeomorphisms), what remains is to obtain estimates of the poinwise dimension of μ from below and above dominated by the sum of stable and unstable pointwise dimensions. This might encounter the relation between entropy, folding entropy and negative Lyapunov exponents, where the result in [50] may come in force 2 .

2 In December 2008 Dr. L. Shu proved first an entropy formula relating entropy, folding entropy and negative Lyapunov exponents for non-degenerate endomorphisms [85], then this conjecture for hyperbolic measures of non-degenerate endomorphisms [86].

Appendix A

Entropy Formula of Pesin Type for One-sided Stationary Random Maps

In this appendix we consider random dynamical systems (abbreviated as RDS’s) generated by compositions of one-sided stationary random endomorphisms of class C2 of a compact manifold following the line of [52]. We will first introduce the notions of entropy and Lyapunov exponents for such RDS’s and then prove that the entropy formula of Pesin type holds if the sample measures of an invariant measure are absolutely continuous with respect to the Lebesgue measure on the manifold. This result covers those obtained by Ledrappier and Young [44] and Liu [48] for i.i.d. (independent and identically distributed) random diffeomorphisms or (noninvertible) endomorphisms and that obtained by [48] for two-sided stationary random endomorphisms. As far as the phase spaces are compact and finite dimensional manifolds without boundary, this result may be considered as the almost final form of Pesin entropy formula for RDS’s with absolutely continuous invariant or sample measures.

A.1 Basic Notions A.1.1 Set-up In this appendix M will always be a compact Riemannian manifold without boundary. Let Cr (M, M)(r ≥ 1 integer) be the space of all Cr endomorphisms + on M endowed with the Cr topology and the Borel σ -algebra. Put Ω = Cr (M, M)Z and let it have the product σ -algebra A . Assume that P is a probability on (Ω , A ) which is invariant under the left shift operator θ on Ω . For each ω ∈ Ω we write ω = ( f0 (ω ), f1 (ω ), · · ·) and define  fωn

=

id if n = 0 fn−1 (ω ) ◦ · · · ◦ f0 (ω ) if n > 0.

(A.1)

245

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A Entropy Formula of Pesin Type

We are concerned with the asymptotic behavior of these composed maps for P-a.e. ω . This set-up will be referred to as X + (M, P) in the rest of this appendix and it falls into the general framework of the theory of RDS’s, for which we refer the reader to Kifer [35], Arnold [3] and Liu [49].

A.1.2 Invariant Measures and Sample Measures Let X + (M, P) be given. In what follows we will use B(X) to denote the Borel σ -algebra of a topological space X. Definition A.1.1 An invariant measure of X + (M, P) is defined as a probability μ on (Ω × M, A × B(M)) which has marginal P on Ω and which is invariant under the skew-product transformation

Θ : Ω × M → Ω × M,

(ω , x) → (θ ω , f0 (ω )x)

associated with X + (M, P). Such invariant measures always exist (see Arnold [3]), and let now μ be such an invariant measure. Since Cr (M, M) is a Polish space (see Hirsch [24]), (Ω × M, μ ) with the μ -completion of A × B(M) constitutes a Lebesgue space. According to Rokhlin [74], one can speak of the conditional measure μω of μ on {ω } × M (identified with M) for P-a.e. ω . {μω }ω ∈Ω is P-mod 0 uniquely defined and is called the family of sample measures of μ . It is easy to see that, for any n ≥ 1, the sample measures have the invariance property  θ −n {ω }

(n)

fωn μω dPω (ω ) = μω ,

P − a.e. ω

(A.2)

which will play an important role in the treatment of this appendix, where (n) {Pω }ω ∈Ω is a canonical system of conditional measures of P associated with the partition {θ −n {ω } : ω ∈ Ω } of Ω .

A.1.3 Entropy Given an invariant measure μ , there are some quantities to describe the complexity of the dynamical behavior of X + (M, P) with respect to μ . One is the entropy hμ (X + (M, P)), among the others are the Lyapunov exponents λi (ω , x), 1 ≤ i ≤ r(ω , x) which will be introduced in the next subsection. The entropy hμ (X + (M, P)) is defined as follows and it describes the average (on ω ) information creation rate of the time evolution of the system (X + (M, P), μ ).

A.1 Basic Notions

247

Proposition A.1.1 For any finite partition ξ of M the limit 1 hμ (X (M, P), ξ ) = lim n→∞ n +

def

exists.





Hμω



n−1 

( fωk )−1 ξ

dP(ω )

(A.3)

k=0



) x log x if x > 0 k −1 , write ξωn = n−1 k=0 ( f ω ) ξ and put an = 0 if x = 0 ( Hμω (ξωn )dP(ω ). Then for all n, m ∈ Z+

Proof. Define k(x) =

an+m = ≤

 

Hμω (ξωn+m )dP(ω ) Hμω (ξωn )dP(ω ) +

= an + = an − ≤ an −





Hμω (( fωn )−1 ξθmn ω )dP(ω )

H fωn μω (ξθmn ω )dP(ω )

  Ω



∑m k(( fωn μω )(C))dPω (ω )dP(ω ) (n)

θ −n {ω }

∑

Ω C∈ξ m

C∈ξω

 k

ω

(n)

θ −n {ω }

( fωn μω )(C)dPω (ω ) dP(ω )

(by the convexity of k(x) on [0, +∞)) = an + =

Hμω (ξωm )dP(ω ) (by (A.2))

Ω an + am .

1 an n→+∞ n

The limit lim



thus exists and equals infn≥1 n1 an . def

Definition A.1.2 The number hμ (X + (M, P)) = supξ hμ (X + (M, P), ξ ) is called the entropy of (X + (M, P), μ ), where the supremum is taken over the set of all finite partitions ξ of M. By the same argument as Bogensch¨utz [9, Theorem 3.1] one has h μ (X + (M, P)) = hF μ (Θ ),

(A.4) def

where hF μ (Θ ) is the conditional entropy of Θ : (Ω × M, μ ) ← with respect to F = {A × M : A ∈ A }.

A.1.4 Lyapunov Exponents Let X + (M, P)( be of class C1 (i.e., r = 1) and let μ be an invariant measure of X + (M, P). If log+ |Tx f0 (ω )|d μ (ω , x) < +∞, the Oseledec multiplicative ergodic

248

A Entropy Formula of Pesin Type

theorem applied to Θ : (Ω × M, μ ) ← yields that, for μ -a.e. (ω , x) ∈ Ω × M, there are measurable (in (ω , x)) numbers +∞ > λ1 (ω , x) > λ2 (ω , x) > · · · > λr(ω ,x) (ω , x) ≥ −∞ and an associated sequence of subspaces of Tx M V (0) (ω , x) = Tx M ⊃ V (1) (ω , x) ⊃ · · · ⊃ V (r(ω ,x)) (ω , x) = {0} (all measurable in (ω , x)) such that 1 log |Tx fωn ξ | = λi (ω , x) n→+∞ n lim

for all ξ ∈ V (i−1) (ω , x)\V (i) (ω , x), 1 ≤ i ≤ r(ω , x). The numbers λi (ω , x), 1 ≤ i ≤ def

r(ω , x) are called the Lyapunov exponents of X + (M, P) at (ω , x), and mi (ω , x) = dimV (i−1) (ω , x) − dimV (i) (ω , x) is called the multiplicity of λi (ω , x).

A.2 Statement of the Main Result A.2.1 Ruelle Inequality Let X + (M, P) be given and let μ be an X + (M, P)-invariant measure. Roughly speaking, the entropy and the Lyapunov exponents provide two different ways of measuring the complexity of the dynamical behavior of (X + (M, P), μ ). The entropy does it from the point-view of information, and the positive exponents measure geometrically how fast nearby orbits diverge (via the corresponding unstable manifolds theory formulated on the inverse limit space of Θ : (Ω × M, μ ) ←). Concerning the relationship between these two kinds of quantities there is first the following result. Proposition A.2.1 (Ruelle Inequality) Assume X + (M, P) is of class C1 (i.e., r = 1) and log+ | f0 (ω )|C1 ∈ L1 (Ω , P), where | f |C1 = supx∈M |Tx f | for f ∈ C1 (M, M). Then for any X + (M, P)-invariant measure μ one has def

hμ (X + (M, P)) ≤



∑ λi (ω , x)+ mi (ω , x)d μ .

(A.5)

i

This was first proved by Ruelle [77] (also by an unpublished work of Margulis) for a single C1 map and a similar result was proved by Bahnm¨uller and Bogeusch¨utz [4] for RDS’s over ergodic and invertible measure-preserving “noise” systems (see that paper for previous works by others). To prove the present result, let Ω ∗ = Cr (M, M)Z (with the product σ -algebra A ∗ ) and define

A.2 Statement of the Main Result

249

Θ ∗ : Ω ∗ × M → Ω ∗ × M,

(ω ∗ , x) → (θ ∗ ω ∗ , f0 (ω ∗ )x)

where θ ∗ is the left shift operator on Ω ∗ and (· · · , f−1 (ω ∗ ), f0 (ω ∗ ), f1 (ω ∗ ), · · ·) is the sequence of maps corresponding to ω ∗ . It is easy to show that there is a unique probability μ ∗ on (Ω ∗ ×M, A ∗ ×B(M)) such that Θ ∗ μ ∗ = μ ∗ and Π μ ∗ = μ , where Π : Ω ∗ × M → Ω × M, (ω ∗ , x) → (( f0 (ω ∗ ), f1 (ω ∗ ), · · ·), x) is the natural projection. By arguments similar to Liu [48, Prop.2.2] one has ∗

F ∗ hF μ (Θ ) = h μ ∗ (Θ )

(A.6)

where F ∗ = {A∗ × M : A∗ ∈ A ∗ }. By [4], ∗

∗ hF μ ∗ (Θ ) ≤



∑ λi (ω ∗ , x)+ mi (ω ∗ , x)d μ ∗ i

if log+ | f0 (ω ∗ )|C1 ∈ L1 (Ω ∗ , P∗ ), where P∗ is the marginal of μ ∗ on Ω ∗ ((Ω ∗ , θ ∗ , P∗ ) is clearly the natural extension of (Ω , θ , P)) and {(λi (ω ∗ , x), mi (ω ∗ , x)) : i = 1, · · · , r(ω ∗ , x)} is the Lyapunov spectrum of Θ ∗ : (Ω ∗ × M, μ ∗ ) ← at μ ∗ -a.e. (ω ∗ , x). This together with (A.6) proves Proposition A.2.1 since 



∑ λi(ω ∗ , x)+ mi (ω ∗ , x)d μ ∗ = ∑ λi (ω , x)+ mi (ω , x)d μ . i

i

A.2.2 Pesin (Entropy) Formula Our main result of this appendix is the following theorem, where | f |C2 is the C2 def

norm of f ∈ C2 (M, M) (see [51] for the definition) and D( f ) = inf | det Tx f |. x∈M

+ (M, P)

Theorem A.2.2 (Pesin Formula) Let X be of class C2 and assume + 1 log | f0 (ω )|C2 + log D( f0 (ω )) ∈ L (Ω , P). Let μ be an X + (M, P)-invariant measure. If μω  Leb for P-a.e. ω , then there holds the equality hμ (X + (M, P)) =



∑ λi (ω , x)+ mi (ω , x)d μ .

(A.7)

i

The proof of this theorem will be given in Section 3. Some results in this direction have been obtained previously and let us indicate how they can be covered by Theorem A.2.2. The formula (A.7) was first proved by Pesin [63] for a single diffeomorphism and later by Thieullen [90] for a single noninvertible map (along a line different from Pesin’s), and these results correspond to the situation of P being supported by a single point ( f , f , · · ·) ∈ Ω for some f ∈ C2 (M, M) (we remark that [63] and [90] made a weaker smoothness

250

A Entropy Formula of Pesin Type

assumption, i.e., a C1+α one on the map under consideration, and the map considered in [90] is allowed to have singularities; see Remark A.1 about the C1+α assumption in the random case). Ledrappier and Young [44] considered the case of X + (M, P) where fi (ω ), i = 0, 1, 2, · · · are assumed to be independent and iden+ tically distributed random diffeomorphisms (i.e., P = ν Z for some probability ν 2 on Diff (M)) and proved (A.7) for a probability ρ on M which (is stationary for the Markov process induced by X + (M, P) on M (or equivalently f ρ d ν ( f ) = ρ ) and which satisfies ρ  Leb, and this result was extended by Liu [48, Sect.2.2] + to i.i.d. random endomorphisms case. Since in these cases ν Z × ρ constitutes an invariant measure of X + (M, P), these results are covered by Theorem A.2.2. Getting rid of the i.i.d. assumption, Liu [48, Sect.2.1] considered the two-sided model X (M, P∗ ) defined similarly to X + (M, P) but over Ω ∗ = C2 (M, M)Z and proved a result similar to Theorem A.2.2. This result can also be easily deduced from Theorem A.2.2 (apart from some slight changes concerning integrability conditions) by projecting (X (M, P∗ ), μ ∗ ) (μ ∗ an X (M, P∗ )-invariant measure) to a one-sided RDS (X + (M, P), μ ) with μ = Π μ ∗ , since the projecting procedure preserves the smoothness of the sample measures. Note however that extending a one-sided RDS (X + (M, P), μ ) to a two-sided one (X (M, P∗ ), μ ∗ ) will generally destroy the smoothness of the sample measures (but can only guarantee the SRB property of them, see Liu [49, Remark 2.9] and Corollary A.2.2.1 below), hence Theorem A.2.2, especially the “i.i.d.” results in [44] and [48, Sect.2.2], can not be covered by the “two-sided” result in [48, Sect.2.1]. Theorem A.2.2 together with the main result of Bahnm¨uller and Liu [5] (see also [44] for the i.i.d. case) yields the following corollary. See [5] for the definition of SRB measures of RDS’s. Corollary A.2.2.1 Assume X + (M, P) is given such that f0 (ω ) ∈ Diff2 (M) for P-a.e. ω and log+ | f0 (ω )C2 + log+ | f0 (ω )−1 |C2 ∈ L1 (Ω , P). Let μ be an X + (M, P)invariant measure and let (Ω ∗ × M, Θ ∗ , μ ∗ ) be the natural extension of (Ω × M, Θ , μ ). If μω  Leb for P-a.e. ω , then μ ∗ is an SRB measure. Remark A.1. Since we need to work with Lebesgue spaces, we consider here C2 rather than C1+α (0 < α ≤ 1) endomorphisms for the reason that C2 (M, M) is Polish whereas C1+α (M, M) is in general not separable (see [38]). It is not clear to the authors if the C2 assumption could be reduced to a C1+α one by a suitable trick for the purpose of this appendix (note however that a C1 assumption is usually not sufficient, see Pugh [68] for a counterexample concerning the absolute continuity property of the stable manifolds). Remark A.2. One may consider the following slightly more general framework of RDS’s. Let W be a Polish space, P˜ a probability on (W, B(W )) and ˜ ← a measure-preserving transformation (take the classical Winner τ : (W, B(W ), P) space as an example). Assume that G : W → Cr (M, M)

A.2 Statement of the Main Result

251

˜ (r ≥ 0 integer) is a measurable map and one considers for P-a.e. w the composed maps  id for n = 0 def gnw = g(τ n−1 w) ◦ · · · ◦ g(w) for n > 0, def

where g(w) = G (w). This defines an RDS and we will denote it still by the notation G . Various notions defined for X + (M, P) can be adapted verbatim to the case of G by considering the skew-product transformation G : W × M → W × M, (w, x) → (τ w, g(w)x). The map Σ : W → Ω , w → (g(w), g(τ w), · · ·) projects P˜ to a probability P on Ω and this gives rise to an RDS X + (M, P). Clearly any invariant measure μ˜ of G can deduce an invariant measure μ of X + (M, P) via the map Σ˜ : W × M → Ω × M, (w, x) → (Σ w, x). By arguments similar to the proof of Proposition A.1.1 one can see that the entropy hμ˜ (G ) of (G , μ˜ ) is equal to hμ (X + (M, P)) and then has Theorem 2.2 Let the RDS G be as introduced above and let it be of class C2 . ˜ and let μ˜ be an invariant measure Assume log+ |g(w)|C2 + log D(g(w)) ∈ L1 (W, P) ( ˜ w, or more generally, if Σ −1 ω μ˜ w d P˜ω (w)  Leb where of G . If μ˜ w  Leb for P-a.e. P˜ω is the conditional measure of P˜ on Σ −1 ω , then hμ˜ (G ) =



∑ λi (w, x)+ mi (w, x)d μ˜ . i

˜ one can have a result similar to Corollary Using the natural extension of (W, τ , P), A.2.2.1.

A.2.3 Pesin Formula for Some Particular RDS’s Expanding in average RDS’s. Let X + (M, P) be of class Cr (r ≥ 1 integer) and assume that log | f0 (ω )|C−1 ∈ L1 (Ω , P), where | f |C−1 = infξ ∈T M,|ξ |=1 |T f ξ | for f ∈ Cr (M, M). If P-a.e. def

1 n−1 ∑ log | fk (ω )|C−1 =: a(ω ) > 0 n→+∞ n k=0 lim

(the limit exists for P-a.e. ω by Birkhoff theorem), X + (M, P) is then said to be expanding in average. The two-sided version of this model was introduced by Khanin and Kifer [34] and thermodynamic formalism was developed there for this twosided version. Particularly, existence of invariant measures whose sample measures are absolutely continuous with respect to Lebesgue is assured there under suitable smoothness and integrability conditions on the random maps. Using the extension technique in Subsection 2.1, this implies that for a C2 expanding in average RDS

252

A Entropy Formula of Pesin Type

X + (M, P) with log+ | f0 (ω )|C2 ∈ L1 (Ω , P) there is an (in fact unique) invariant measure μ which satisfies μω  Leb for P-a.e. ω . By Theorem A.2.2, this μ satisfies hμ (X + (M, P)) = =

 

∑ λi(ω , x)(ω , x)+mi (ω , x)d μ i

∑ λi(ω , x)mi (ω , x)d μ =



log | det Tx f0 (ω )|d μ .

i

This formula can also follow from the equilibrium state arguments in [34] together with the extension technique in Subsection 2.1, but now we obtain it directly. This can be regarded as a particular result of thermodynamic formalism of one-sided expanding in average RDS’s, a fairly full study of which is still lacking (see Baladi [6] for related results). + Markov RDS’s. Let Ω = { f1 , · · · , fN }Z , where f1 , · · · , fN are a finite number of Cr (r ≥ 0) maps on M, and let P be a probability on Ω . If the coordinate process fn (ω ), n = 0, 1, · · · on (Ω , P) constitutes a time homogeneous stationary Markov process, namely, if P( fn (ω ) = fin | f0 (ω ) = fi0 , · · · , fn−1 (ω ) = fin−1 ) = pin−1 in , ∀n ≥ 1, ∀i0 , i1 , · · · , in ∈ {1, · · ·, N} and P( f0 (ω ) = fi ) = πi , 1 ≤ i ≤ N for some transition matrix (pi j ) and some probability vector (π1 , · · · , πN ) with ∑Ni=1 πi pi j = π j , 1 ≤ j ≤ N, X + (M, P) is then called a finite-stated one-sided Markov RDS. Let now X + (M, P) be such an RDS and assume πi > 0 for all i. In this case there is a special kind of X + (M, P)-invariant measures μ such that at P-a.e. ω , μω depends only on f0 (ω ), i.e., there are probabilities μ1 , · · · , μN on M such that μω = μi whenever f0 (ω ) = fi for P almost all ω . In fact, for any μ1 , · · · , μN ∈ Prob(M) (Prob(X) is the space of all Borel probabilities on a topological space X) there is clearly μ ∈ Prob(Ω × M) with marginal P on Ω such that μω = μi whenever f0 (ω ) = fi , and the X + (M, P)-invariance of μ is equivalent to the π p equations μi = ∑Nj=1 jπi ji f j μ j . It is easy to see that the map L : Prob(M)N → π p π p Prob(M)N , (μ1 , · · · , μN ) → (∑Nj=1 jπ1j1 f j μ j , · · · , ∑Nj=1 jπNjN f j μ j ) has fixed points. This proves the existence of the special invariant measures μ . By Theorem A.2.2, to obtain Pesin formula for X + (M, P) with such an invariant measure μ , it is sufficient to check μi  Leb for i = 1, · · · , N if fi ’s are C2 and have no singularities.

A.3 Proof of Theorem A.2.2 By Proposition A.2.1, it remains to prove h μ (X + (M, P)) ≥



∑ λi (ω , x)+ mi (ω , x)d μ .

(A.8)

i

We will follow essentially the line of Ledrappier and Young [44] and especially that of Liu [48]. We first introduce the stable manifolds of (X + (M, P), μ ) and the proof of (A.8) consists in analysis along these manifolds.

A.3 Proof of Theorem A.2.2

253

Let Γ ⊂ Ω × M be a μ -full set such that ΘΓ ⊂ Γ and each point in Γ is regular in the sense of Oseledec as described in Subsection 1.4. Set I = {(ω , x) ∈ Γ : λi (ω , x) ≥ 0, 1 ≤ i ≤ r(ω , x)} and Δ = Γ \I. It is clear that Θ I ⊂I and Θ Δ ⊂ Δ . For (ω , x) ∈ Δ , put E s (ω , x) = λi (ω ,x)<0 V (i) (ω , x) and define the (global) stable manifold of X + (M, P) at point (ω , x) by 1 def W s (ω , x) = {y ∈ M : lim sup log d( fωn x, fωn y) < 0}. n→+∞ n Then for μ -a.e. (ω , x) ∈ Δ there exists a sequence of C1,1 embedded dim E s (ω , x)dimensional discs {Wn (ω , x)}+∞ n=0 such that f n (ω )Wn (ω , x) ⊂ Wn+1 (ω , x) for all n ≥ 0 and W s (ω , x) =

+∞

( fωn )−1Wn (ω , x)

n=0

(see [51, Chapter III] for a proof). The integrability condition log D( f0 (ω )) ∈ L1 (Ω , P) implies that for P-a.e. ω the maps fωn have no singularities for all n ≥ 0 and hence W s (ω , x) is a C1,1 immersed submanifold of M for μ -a.e. (ω , x) ∈ Δ . We define W s (ω , x) = {x} for any (ω , x) ∈ I. Since μω  Leb for P-a.e. ω , there exists a measurable partition η of Ω × M which has the following properties: def

i) Θ −1 η ≤ η , σ = {{ω } × M : ω ∈ Ω } ≤ η ; ii) η is subordinate to W s -manifolds of (X + (M, P), μ ) , i.e., for μ -a.e. (ω , x) def

ηω (x) = {y : (ω , y) ∈ η (ω , x)} ⊂ W s (ω , x) and it contains an open neighborhood of x in W s (ω , x), this neighborhood being taken in the submanifold topology of W s (ω , x);  iii) For every B ∈ B(Ω × M) the function (ω , x) → λ(sω ,x) (ηω (x) Bω ) is measurable and μ -a.e. finite, where Bω = {y : (ω , y) ∈ B} and λ(sω ,x) is the Lebesgue measure on W s (ω , x) induced by its inherited Riemannian structure as a submanifold of M (λ(sω ,x) = δx if W s (ω , x) = {x}); η η iv) (μω )x ω  λ(sω ,x) for μ -a.e. (ω , x), where (μω )x ω is the conditional measure of μω on ηω (x). A similar proof of the existence of such a partition η can be found in [51, IV.2] and [48]. It is worth mentioning that Lemma IV.2.2. in [51] remains true in the present case, i.e., B I ⊂ B s μ -mod(0), where B I = {B ∈ Bμ (Ω × M) : Θ −1 B = B} (Bμ (Ω , M) the completion of B(Ω × M) with respect to μ ) and B s = {B ∈  Bμ (Ω × M) : B = (ω ,x)∈B {ω } × W s (ω , x)} (see [5] for a proof). Let η be as given above. By a computation similar to [44, (4.8)] one has ) limn→+∞ n1 Hμ (η |Θ −n η σ ) ≤ hμ (X + (M, P)) if Hμ (η |Θ −n η ∨ σ ) < +∞

(A.9)

254

A Entropy Formula of Pesin Type

for all n ≥ 1. Hence, in order to prove (A.8), it is sufficient to prove that for every n ≥ 1 there holds (A.9) and 1 Hμ (η |Θ −n η ∨ σ ) ≥ n



∑ λi (ω , x)+mi (ω , x)d μ .

(A.10)

i

Now we fix n ≥ 1 arbitrarily. By the definition of conditional entropies one has Hμ (η |Θ −n η ∨ σ ) = − =−



−n

log μ(Θω ,x)η ∨σ (η (ω , x))d μ (ω , x)

 

Ω M

( f n )−1 ηθ n ω

log(μω )x ω

(ηω (x))d μω (x)dP(ω )

ξ

where {νz }z∈X denotes a canonical system of conditional measures of ν associated with a measurable partition ξ of a Lebesgue space (X, A , ν ). In what follows we write λ = Leb. Since μ  λ × P we can define

ϕ=

dμ d(λ × P)

which implies ϕω (·) = ϕ (ω , ·) = ddμλω (·) for P-a.e. ω . Put Λ = {(ω , x) : ϕ (ω , x) > 0}. By assumption, fωn : M → M has no singularities for P-a.e. ω . From this together with the assumption μω  λ , P-a.e. ω it follows easily that P-a.e. fωn μω  μθ n ω . We now fix arbitrarily an ω with these above properties. n def n n Choose a countable Borel partition {Anω ,i }+∞ i=1 of M such that f ω ,i = f ω |Aω ,i is injective for each i, and define Borel Probabilities ρω ,n and νω ,n on M such that for any Borel set B ⊂ Anω ,i def

ρω ,n (B) = ( fωn μω )( fωn ,i B),

νω ,n (B) = μθ n ω ( fωn ,i B).

We now state some preliminary facts 1)-4) as follows 1) For any Borel set B ⊂ Λθ n ω , ( fωn μω )(B) = μω (( fωn )−1 B) = =

 B

 ( f ωn )−1 B

(L fωn ϕω )(x)d λ (x) =

ϕω (x)d λ (x)



(L fωn ϕω )(x) d μθ n ω (x), ϕθ n ω (x) B

where L f is the Ruelle transfer operator of f ∈ C1 (M, M) defined by (L f l)(x) = l(y) ∑y∈ f −1 {x} | det Ty f | for measurable function l : M → R. Hence (L fωn ϕω )(x) d fωn μω =: Ψn (ω , x), μθ n ω −a.e. x. (x) = n d μθ ω ϕθ n ω (x)

A.3 Proof of Theorem A.2.2

255

Furthermore, for any Borel set B ⊂ M one has  

(n)

B θ −n {ω }

(L f n ϕω )(y)dPω (ω )d λ (y) = ω

= = =

   

 



(L f n ϕω )(y)d λ (y)dPω (ω ) ω

(n)

θ −n {ω } ( f ωn )−1 B

ϕω (y)d λ (y)dPω (ω ) (n)

θ −n {ω }

μω (( fωn )−1 B)dPω (ω ) (n)

θ −n {ω }

( fωn μω )(B)dPω (ω )

= μω (B) = Hence

(n)

θ −n {ω } B



B

ϕω (y)d λ (y).

(n)

θ −n {ω }

(L f n ϕω )(y)dPω (ω ) = ϕω (y)

(A.11)

ω

holds for λ -a.e. y and then holds for μω -a.e. y.  2) For any Borel set B ⊂ ( fωn )−1Λθ n ω , putting Bi = B Anω ,i one has

ρω ,n (B) = ∑ fωn μω ( fωn ,i Bi ) = ∑ i

=∑ i

which yields



(L fωn ϕω )(x) d μθ n ω ϕθ n ω (x)

f ωn ,i Bi i  n (L fωn ϕω )( fω x) d ν = ω ,n ϕθ n ω ( fωn x) Bi B



(L fωn ϕω )( fωn x) d νω ,n ϕθ n ω ( fωn x)

(L fωn ϕω )( fωn x) d ρω ,n = Ψn (ω , fωn x) (x) = d νω ,n ϕθ n ω ( fωn x)

(A.12)

for νω ,n -a.e. x ∈ ( fωn )−1Λθ n ω . 3) For any Borel set B ⊂ ( fωn )−1Λθ n ω one has

μω (B) =

∑ i

=

∑ i

=

∑ i

=



  

Bi

ϕω (x)d λ = ∑ i

f ωn ,i Bi

Bi



f ωn ,i Bi

ϕω (( fωn ,i )−1 y)| det Ty ( fωn ,i )−1 |d λ

ϕ (ω , ( fωn ,i )−1 y)| det T( f n

ω ,i )

−1 y

fωn |−1

1

ϕ (θ n ω , y)

ϕ (ω , x) | det Tx fωn |−1 d νω ,n ϕ (θ n ω , fωn ,i x)

ϕ (ω , x) | det Tx fωn |−1 d νω ,n n B ϕ ◦ Θ (ω , x)

which yields d μω ϕ (ω , x) | det Tx fωn |−1 =: Φn (ω , x) (x) = d νω ,n ϕ ◦ Θ n(ω , x) for νω ,n -a.e. x ∈ ( fωn )−1Λθ n ω .

d μθ n ω

256

A Entropy Formula of Pesin Type



It is easy to see that μω is equivalent to νω ,n on Λω ( fωn )−1Λθ n ω and  n −1 d νω ,n 1 n d μω = Φn (ω ,x) for μω -a.e. x ∈ Λω ( f ω ) Λθ ω . This together with (A.12) yields d ρω ,n d μω (x)

=

Ψn (ω , f ωn x) Φn (ω ,x)



for μω -a.e. x ∈ Λω ( fωn )−1Λθ n ω . Hence

fωn : (Λω



( fωn )−1Λθ n ω , μω ) → ( fωn Λω

has the Jacobian J( fωn )(x) =



Λθ n ω , fωn μω )

Ψn (ω , fωn x) Φn (ω , x)

(see Parry [61] orsee [48, 3.1.1] for the definition). Thus, if ξ is a measurable  partition of fωn Λω Λθ n ω , for μω -a.e. x ∈ Λω ( fωn )−1Λθ n ω one has ( f n )−1 ξ

(μω )x ω

(B) =

 f ωn B

1 ξ d( fωn μω ) f n x n ω J( fω ) ◦ ( fωn ,i )−1

(A.13)



for any Borel set B ⊂ (( fωn )−1 ξ )(x) Anω ,i (see [48, Lemma 3.1]). Since   μω (Λω ( fωn )−1Λθ n ω ) = 1 and ( fωn μω )( fωn Λω Λθ n ω ) = 1 for any measurable partition ξ of M (A.13) holds for μω -a.e. x ∈ M and for any Borel set  B ⊂ (( fωn )−1 ξ )(x) Anω ,i . 4) One can define a Borel measure λ ∗ on Ω × M by

λ ∗ (B) =



λ(sω ,x) (ηω (x)



Bω )d μ (ω , x) for Borel set B ⊂ Ω × M.

(



Recalling that μ (B) = (μω )ηx ω (ηω (x) Bω )d μ (ω , x) and (μω )ηx ω  λ(sω ,x) , we have μ  λ ∗ . Put g(ω , x) = g(ω , y) =

dμ d λ ∗ (ω , x).

We then have for μ -a.e. (ω , x)

d(μω )ηx ω (y), λ(sω ,x) −a.e. y ∈ ηω (x) d λ(sω ,x)

(see [51, Proposition IV.2.2]). Put for μ -a.e. (ω , x) ∈ Ω × M ( f n )−1 η n

θ ω Wn (ω , x) = (μω )x ω (ηω (x)), ϕ (ω , x) g ◦ Θ n(ω , x) , Xn (ω , x) = ϕ ◦ Θ n(ω , x) g(ω , x) $ | detT f n | | x ω Es | det Tx f ωn | if(ω , x) ∈ Δ , Yn (ω , x) = 1 if(ω , x) ∈ I, | detTx f n |

Zn (ω , x) =



ω

η

ηθ n ω ( f ωn x)

Ψn (ω , y)d(μθ n ω ) fωnθ xω (y). n

(A.14)

A.3 Proof of Theorem A.2.2

257

It is easy to see that Wn , Xn ,Yn , Zn are all measurable and μ -a.e. finite. We now present several claims, whose proofs will be given a bit later. Claim 1. Wn = XZnYn n μ -a.e. on Ω × M; ( ( Claim 2. logYn ∈ L1 (Ω × M, μ ) and − 1n logYn d μ = ∑i λi (ω , x)+ mi (ω , x)d μ . ( Claim 3. log Zn ∈ L1 (Ω × M, μ ) and ( log Zn d μ ≥ 0. Claim 4. log Xn ∈ L1 (Ω × M, μ ) and log Xn d μ = 0. By these claims one can easily see that logWn ∈ L1 (Ω × M, μ ) and 1 − n



logWn d μ ≥



∑ λi (ω , x)+ mi (ω , x)d μ . i

 

This proves (A.9) and (A.10) and thus proves Theorem A.2.2. Proof of Claim 1. Clearly one has ( f n )−1 ε

(μω )x ω

({x}) =

1 , μ −a.e. (ω , x) J( fωn )(x)

(A.15)

where ε is the partition of M into single points. Notice that if (ω , x) ∈ I, then the left side of (A.15) is Wn (ω , x) and Zn (ω , x) = Ψn (ω , fωn x). Thus Wn = XZnYn n holds μ -a.e. on I. Now we consider μ -a.e. (ω , x) ∈ Δ . By fact 1), one has η

n

η

n

d( fωn μω ) fωnθ xω d(μθ n ω ) fωnθ xω

(·) =

Ψn (ω , ·) . Zn (ω , x)

(A.16)

Then, for any Borel set B ⊂ ηω (x) ( f n )−1 η n

θ ω (B) (μω )x ω Wn (ω , x)  1 1 η n d( fωn μω ) fωnθ xω (y) = ∑ n n Wn (ω , x) i fω ,i Bi J( fω ) ◦ ( fωn ,i )−1 (y)

(μω )ηx ω (B) =

1 = Wn (ω , x) ∑ i



f ωn ,i Bi

(by (A.13))

1 Ψn (ω , y) η n d(μθ n ω ) fωnθ xω (y) J( fωn ) ◦ ( fωn ,i )−1 (y) Zn (ω , x) 

(by (A.16))

Ψn (ω , y) g(θ n ω , y)d λΘs n (ω ,x) (y) n J( fω ) ◦ ( fωn ,i )−1 (y)  Ψn (ω , fωn y) g(θ n ω , fωn y)| det(Ty fωn |E s (ω ,y) )|d λ(sω ,x) (y) n )(y) J( f B i ω i  Ψn (ω , fωn y) g ◦ Θ n(ω , y)| det(Ty fωn |E s (ω ,y) )|d λ(sω ,x) (y). n B J( f ω )(y)

=

1 Wn (ω , x)Zn (ω , x) ∑ i

=

1 Wn (ω , x)Zn (ω , x) ∑

=

1 Wn (ω , x)Zn (ω , x)

f ωn ,i Bi

258

A Entropy Formula of Pesin Type

Since B is arbitrary, this together with (A.14) yields

Ψn (ω , fωn y) 1 g ◦ Θ n(ω , y)| det Ty fωn |E s (ω ,y) | = g(ω , y) Wn (ω , x)Zn (ω , x) J( fωn )(y) for λ(sω ,x) -a.e. y ∈ ηω (x), and hence Wn (ω , y) =

Xn (ω , y)Yn (ω , y) for (μω )ηx ω -a.e. y ∈ ηω (x) Zn (ω , y)

since (μω )ηx ω  λ(sω ,x) and Wn (ω , y) = Wn (ω , x) and Zn (ω , y) = Zn (ω , x) for any

y ∈ ηω (x). This shows that Wn = XZnYn n holds for μ -a.e. (ω , x) on Δ .   The proof of Claim 2 is almost the same as that of [48, Claim 4.2] and is omitted here. Proof of Claim 3 and Claim 4. For f ∈ C1 (M, M) with no singularities, denote by deg( f ) the number of elements of {y : f (y) = x} (it is finite and independent M (see [48]), from | f | of x ∈ M). Since deg( fω1 ) ≤ λ (M)| fω1 |Cdim 1 C1 ≤ | f |C2 and + log | f0 (ω )|C2 ∈ L1 (Ω , P) it follows that log deg( f0 (ω )) ∈ L1 (Ω , P) (recall that f0 (ω ) has no singularities by assumption). Then one has ∞> ≥

 Ω



Ω

=− = =

log deg( fωn )dP Hμω (ε |( fωn )−1 ε )dP

 

 Ω M Ω ×M



Ω ×M

( f n )−1 ε

log(μω )x ω

({x})d μω dP

log J( fωn )(x)d μ

(log Ψn (ω , fωn x) + log

ϕ ◦ Θ n(ω , x) + log| det Tx fωn |)d μ ϕ (ω , x)

Noting that k(x) ≥ e−1 log e−1 , x ∈ [0, +∞) (k(x) is as introduced in the proof of Proposition A.1.1), hence   Ω M

= = ≥

 

Ω M

 

Ω M

=e

Ω M −1

log− Ψn (ω , fωn x)d μω (x)dP(ω ) log− Ψn (ω , y)d fωn μω (y)dP(ω ) k− (Ψn (ω , y))d μθ n ω (y)dP(ω ) e−1 log e−1 d μθ n ω (y)dP(ω )

log e−1 > −∞

one has log− Ψn (ω , fωn x) is μ -integrable.

A.3 Proof of Theorem A.2.2

259

Since log+

ϕ ◦ Θ n(ω , x) ( f n )−1 ε ≤ − log(μω )x ω ({x}) − log− Ψn (ω , fωn x) − log− | det Tx fωn |, ϕ (ω , x)

we know that log+ ϕ ◦ϕΘ is μ -integrable. Then, by [41, Prop.2.2], log ϕ ◦ϕΘ is inten

grable and

(

Ω ×M log

n

ϕ ◦Θ n ϕ dμ

= 0. On the other hand, ( f n )−1 ε

log+ Ψn (ω , fωn x) ≤ − log(μω )x ω

− log−

ϕ ◦ Θ n (ω , x) − log− | det Tx fωn | ϕ (ω , x)

and hence log+ Ψn (ω , fωn x) is μ -integrable. In what follows, given a probability space (X, A , m) and a sub-σ -algebra H of A , we use Em (·|H ) to denote the corresponding conditional expectation operator. By Bn,ω we will denote the σ -algebra generated by the partition ηθ n ω of M. Then   Ω M

= = = =

 

Ω M

 

Ω M

 

Ω M

 

Ω M

log+ Zn (ω , x)d μω dP +



log (



log+ (

η

ηθ n ω ( f ωn x)

η

ηθ n ω (y)

≤ = = =

  Ω M

 

Ω M

 

Ω M

 

Ω M

 

Ω M

n

Ψn (ω , z)d(μθ n ω )y θ ω ))d fωn μω (y)dP 

Ψn (ω , y) log+ (

n

η

ηθ n ω (y)

Ψn (ω , z)d(μθ n ω )y θ ω ))d μθ n ω (y)dP n

+

k (Vω ,n (y))d μθ n ω (y)dP

(where Vω ,n (y) = =

Ψn (ω , z)d(μθ n ω ) fωnθ xω ))d μω (x)dP



ηθ n ω (y)

η

Ψn (ω , z)d(μθ n ω )y θ ω ) n

k+ (Eμθ n ω (Ψn (ω , ·)|Bn,ω )(y))d μθ n ω (y)dP Eμθ n ω (k+ (Ψn (ω , ·))|Bn,ω )(y)d μθ n ω (y)dP k+ (Ψn (ω , y))d μθ n ω (y)dP log+ Ψn (ω , y)d fωn μω (y)dP log+ Ψn (ω , fωn x)d μω (y)dP < ∞.

This shows that log+ Zn ∈ L1 (Ω × M, μ ) which together with Claim 1 and Claim ( 2 yields log+ Xn ∈ L1 (Ω × M, μ ) since logWn ≤ 0. Hence log Xn d μ = 0 and then

260

A Entropy Formula of Pesin Type

log− Zn ∈ L1 (Ω × M, μ ) since log− Zn ≥ log− Xn + log− Yn μ -a.e.. We have thus proved Claim 4 and log Zn ∈ L1 (Ω × M, M). Finally, 

log Zn d μ =

 

= k(

Ω M

  Ω M



= k(

Ω M Ω

= 0.2

Ω M

Vω ,n (y)d μθ n ω dP)

Ψn (ω , y)d μθ n ω dP)

  

= k(

 

k(Vω ,n (y))d μθ n ω dP ≥ k(

{ω :θ n ω =θ n ω }

(L fωn ϕω )(y) (n) dPθ n ω (ω )d μθ n ω (y)dP(ω )) ϕθ n ω (y)

μθ n ω (M)dP(ω )) (by (A.11))

Appendix B

Large Deviations in Axiom A Endomorphisms

Here we present some large deviation estimates for Axiom A endomorphisms by applying a general large deviation theorem in Kifer [37] and Ruelle’s Smale space technique in [78]. This is the work of Liu et al [53].

B.1 Introduction and Statement of Main Results Consider a discrete time dynamical system generated by a measurable self-map f : X ← of some measurable space (X, B). Let P be a reference probability measure k on (X, B) and let ψ : X → R be an observable. If 1n ∑n−1 k=0 ψ ◦ f converges to some ∗ constant ψ P-a.e. as n → ∞, then, for given ε > 0, Qn (ε ) := {x ∈ X : |

1 n−1 ∑ ψ ( f k x) − ψ ∗| > ε } n k=0

satisfies P(Qn (ε )) → 0 as n → +∞. Large deviation theory in this set-up deals with estimates of the exponential speed of this last convergence to zero. More precisely and more generally, large deviation questions concern estimates of the following form: 1 n−1 1 (B.1) lim sup log P{x ∈ X : ∑ ψ ( f k x) ∈ K} ≤ − inf I(z) z∈K n k=0 n→+∞ n for any closed set K ⊂ R and 1 n−1 1 lim inf log P{x ∈ X : ∑ ψ ( f k x) ∈ G} ≥ − inf I(z) n→+∞ n z∈G n k=0

(B.2)

for any open set G ⊂ R, where I : R → [0, +∞) is a lower semi-continuous function and is called a rate function. Such questions have been well studied by Orey and Pelikan [59] for Anosov diffeomorphisms and by Young [96], among other things, 261

262

B

Large Deviations in Axiom A Endomorphisms

for Axiom A attractors. Developing ideas of [21, 89, 18, 1], Kifer [37] presents a unified approach to large deviations of dynamical systems and stochastic processes based on the existence of a pressure function and on the uniqueness of equilibrium states for certain potentials, and this approach enables one to generalize results from [59] and [96] and to recover the large deviation estimates in Donsker and Varadhan [15]. In this appendix we apply Kifer’s results in [37], together with Ruelle’s Smale space technique in [78], to give some large deviation estimates for Axiom A endomorphisms. Our set-up and main results are as follows. Let M be a Riemannian manifold without boundary, O an open subset of M with compact closure and f : O → M a Cr (r ≥ 1) map. Let Λ = f (Λ ) ⊂ O be a compact invariant set of f and let

Λ f := {x˜ = (xi )+∞ −∞ : xi ∈ Λ , f (xi ) = xi+1 , i ∈ Z} be the orbit space of (Λ , f ) with θ : Λ f → Λ f denoting the left shift operator on Λ f . Write E = p∗ TΛ M for the pull-back bundle of TΛ M via the natural projection p p : Λ f → Λ , x˜ → x0 , and write Ex˜ = p∗x˜ Tx0 M  p∗∗ Tx0 M for the natural isomorphisms x˜ between the fibres Ex˜ and Tx0 M. A fibre-preserving map on E which covers θ can be defined by p ∗θ x˜ ◦ T f ◦ p∗ : Ex˜ → Eθ x˜ for all x˜ ∈ Λ f , and for simplicity of notation we will denote it still by T f . Definition B.1.1 Λ is called a hyperbolic set of f if there is a continuous splitting E = E s ⊕ E u together with constants C > 0 and 0 < λ < 1 such that T f E s ⊂ E s,

T f Eu = Eu

and for all n ≥ 0 |T f n ξ | ≤ Cλ n |ξ | for ξ ∈ E s , |T f n η | ≥ C−1 λ −n |η | for η ∈ E u . Via a change of Riemannian metric we may—and will—assume that C = 1. Note that there may be points in Λ at which T f is degenerate, and that the splitting Ex˜ = Exs˜ ⊕ Exu˜ may depend on the past of x, ˜ i.e., it may happen that p∗ Exu˜ = p∗ Eyu˜ while p(x) ˜ = p(y). ˜ In what follows we denote by C f the set of points in O at which T f is degenerate, and by m the Lebesgue measure on M. A hyperbolic set Λ is said to be an Axiom A  basic set if Λ is locally maximal (i.e., n there exists a neighborhood U of Λ such that +∞ n=−∞ f U = Λ ) and f is positively topologically transitive on it (i.e., ( f n x0 )n≥0 is dense in Λ for some x0 ∈ Λ ). (It can be shown that periodic points are dense in an Axiom A basic set.) If an Axiom A basic set Λ has arbitrarily small open neighborhood U such that f U¯ ⊂ U and +∞ n n=0 f U = Λ , it is then called an Axiom A attractor, and U is called a basin of attraction of Λ . Applying Ruelle’s Smale space technique, Qian and Zhang [72] presents an ergodic theory of such an Axiom A basic set Λ . In particular, they proved that (Λ , f ) admits a unique equilibrium state μφ for each H¨older continuous φ : Λ → R and, in case of Λ being an attractor of f ∈ C2 (O, M) with basin of attraction U and m(C f ) = 0, Λ supports a unique f -invariant measure ρ , called the SRB measure, which is generic with respect to Lebesgue measure in the following sense: for m-a.e. x ∈ U¯ one has

B.1 Introduction and Statement of Main Results

1 n−1 ∑ ψ ( f k x) = n→∞ n k=0 lim

 Λ

263

¯ ψ d ρ for all ψ ∈ C(U).

Our main results of this note are as follows, where P(X) denotes the space of Borel probability measures on a compact metric space X endowed with the topology of weak convergence. Theorem B.1.1. (1) Let Λ be an Axiom A basic set of f ∈ C1 (O, M), let φ : Λ → R be H¨older continuous and let μφ be the unique equilibrium state. Then there hold 1 1 n−1 lim sup log μφ {x ∈ Λ : ∑ δ f k x ∈ K} ≤ − inf{J(ν ) : ν ∈ K} n k=0 n→+∞ n

(B.3)

for any closed K ⊂ P(Λ ) and 1 1 n−1 lim inf log μφ {x ∈ Λ : ∑ δ f k x ∈ G} ≥ − inf{J(ν ) : ν ∈ G} n→+∞ n n k=0

(B.4)

for any open G ⊂ P(Λ ), where  J(ν ) =

(

Pf (φ ) − φ d ν − hν ( f ) +∞

if ν ∈ P f (Λ ) otherwise,

(B.5)

P f (Λ ) is the set of f -invariant measures on Λ , Pf (φ ) is the pressure of f for φ and hν ( f ) is the entropy of ( f , ν ). (2) Let Λ be an Axiom A attractor of f ∈ C2 (O, M) and let ρ be the SRB measure on Λ . Then (B.3) and (B.4) hold true with μφ being replaced by ρ and with J(·) being defined by $( ∑ λi (x)+ mi (x)d ν − hν ( f ) if ν ∈ P f (Λ ) i (B.6) J(ν ) = +∞ otherwise where λi (x), 1 ≤ i ≤ r(x) are the Lyapunov exponents of f at x, mi (x) is the multiplicity of λi (x) and a+ := max{a, 0}. (3) Assume the circumstances of (2). Let U be a sufficiently small basin of at¯ Then (B.3) and traction of Λ and let m¯ be the normalized Lebesgue measure on U. (B.4) hold true with μφ and Λ being replaced by m¯ and U¯ respectively and with J(·) being given by (B.6). The proof of this theorem will be given in Section 2. From Theorem B.1.1 and the contraction principle there follows Corollary B.1.1.1. Let ψ : O → R be a continuous function and let us in the circumstances of Theorem B.1.1 (3). For J(·) given by (B.6) put I(z) = inf{J(ν ) :



ψ d ν = z}.

264

B

Large Deviations in Axiom A Endomorphisms

¯ Then (B.1) and (B.2) hold true with P and X being taken respectively as m¯ and U. In particular, for ε > 0 there exists h > 0 such that n−1

1 m{x ¯ ∈ U¯ : | ∑ ψ ( f k x) − n k=0



ψ d ρ | ≥ ε } ≤ e−hn

when n is sufficiently large. Similar things hold true in the circumstances of Theorem B.1.1 (1) or (2).

B.2 Proof of Theorem B.1.1 B.2.1 A Large Deviation Theorem from Kifer [37] Let (X, d) be a compact metric space, f : (X, d) ← a continuous map, and, as before, P(X) the space of Borel probabilities on X endowed with the weak convergence topology , and P f (X) the set of those elements in P(X) which are f -invariant. Put for x ∈ X, ε > 0 and n ∈ N B f (x, ε , n) = {y ∈ X : d( f k x, f k y) ≤ ε , 0 ≤ k ≤ n − 1}. The following theorem is a special case of the general large deviation results of [37], and we will apply it to Axiom A endomorphisms in this appendix. Theorem B.2.1. Suppose that μ ∈ P(X), the support of μ is the whole X, and there is φ ∈ C(X) such that for any given small ε > 0 and for all n ≥ 1, x ∈ X  −1

Aε (n)

n−1



≤ μ (B f (x, ε , n)) exp − ∑ φ ( f x) k

≤ Aε (n)

(B.7)

k=0

where Aε (n) > 0 is a constant satisfying 1n log Aε (n) → 0 as n → +∞. Then for any ψ ∈ C(X) there holds 1 lim log n→∞ n





exp

n−1

∑ ψ( f

 k

x) d μ (x) = Pf (φ + ψ ) = Pf |Y (φ + ψ )

(B.8)

k=0



where Pf (·) denotes the pressure of f and Y is the closure of ν ∈P f (X) suppν . Suppose further that the entropy hν ( f ) is upper semicontinuous at all ν ∈ P f (X) and define  ( − φ d ν − hν ( f ) if ν ∈ P f (X) J(ν ) = (B.9) +∞ otherwise.

B.2 Proof of Theorem B.1.1

265

Then the above conclusion implies n−1 1 lim sup log μ {x : ∑ δ f k x ∈ K} ≤ − inf{J(ν ) : ν ∈ K} n→+∞ n k=0

(B.10)

for any closed set K ⊂ P(X). If, moreover, there exist a countable number of functions ψ1 , ψ2 , · · · ∈ C(X) such that their span Γ = {∑ni=1 βi ψi : βi ∈ R, n ∈ N} is dense in C(X) with respect to the supremum norm and that for each ψ ∈ Γ there is a unique νψ ∈ P(X) satisfying Pf (φ + ψ ) =



ψ d νψ − J(νψ ),

(B.11)

then one has for any open G ⊂ P(X) n−1 1 lim inf log μ {x : ∑ δ f k x ∈ G} ≥ − inf{J(ν ) : ν ∈ G}. n→+∞ n k=0

(B.12)

B.2.2 Smale Spaces Here we recall the notion and some properties of Smale spaces from Ruelle [78]. Definition B.2.1 Suppose that (X, d) is a compact metric space and f : X → X a homeomorphism. (X, d, f ) is said to be a Smale space if for suitable ε > 0, 0 < δ < ε , 0 < λ < 1 there exists a continuous map [·, ·] : {(x, y) ∈ X × X : d(x, y) < ε } → X with the following properties: (1) [x, x] = x and [[x, y], z] = [x, z], [x, [y, z]] = [x, z] when the two sides of these relations are well defined. (2) f [x, y] = [ f x, f y] when both sides are well defined and d( f n y, f n z) ≤ λ n d(y, z) d( f −n y, f −n z) ≤ λ n d(y, z)

for y, z ∈ Vx+ (δ ), n > 0, for y, z ∈ Vx− (δ ), n > 0,

where Vx+ (δ ) = {u : u = [u, x], d(x, u) < δ } and Vx− (δ ) = {v : v = [x, v], d(x, v) < δ }. Here are some properties of a Smale space (X, d, f ). Define C f (X) to be the space of functions φ ∈ C(X) which satisfy the following conditions: There exist δ > 0 and K ≥ 0 such that if d( f k x, f k y) < δ for k = 0, 1, · · · , n then    n  n  k k  (B.13)  ∑ φ ( f x) − ∑ φ ( f y) ≤ K k=0  k=0 (φ ∈ C f (X) if it is H¨older continuous, see Ruelle [78, pp. 136]). If (X, f ) is positively topologically transitive, then it has a unique equilibrium state μφ for each

266

B

Large Deviations in Axiom A Endomorphisms

φ ∈ C f (X). (X, d, f ) is expansive. In the case of (X, d, f ) being topologically mixing, it has the specification property, and hence there is the following proposition which follows from Katok and Hasselblatt [32, Lemma 20.3.4 and Theorem 20.3.7]. For the transitive case, the proposition can be reduced to the mixing case by the spectral decomposition of Smale spaces (see also [78]) and by considering i ( f l , ∑l−1 i=0 φ ◦ f ) restricted to one of the basic sets, say X0 , in the spectral decomi fl position, where l is the number of the basic sets (note that ∑l−1 i=0 φ ◦ f ∈ C (X0 ) if i φ ∈ C f (X); when f is not H¨olderian, nor is ∑l−1 i=0 φ ◦ f even if φ is). Proposition B.2.2 Assume that (X, d, f ) is a positively topologically transitive Smale space and φ ∈ C f (X). Let μφ be the unique equilibrium state of (X, d, f ) for φ . Then, for small ε > 0 there exist Aε , Bε > 0 such that for x ∈ X and n ∈ N one has n−1

Aε ≤ μφ (B f (x, ε , n)) exp{− ∑ φ ( f k x) + nPf (φ )} ≤ Bε . k=0

B.2.3 Smale Space Property of Locally Maximal Hyperbolic Sets We first collect in the following proposition some properties of local stable and unstable manifolds of a hyperbolic set, for details see Qian and Zhang [72] which is the first paper to apply the Smale space technique in the study of Axiom A endomorphisms. Proposition B.2.3 Let Λ be a hyperbolic set of f ∈ Cr (O, M) (r ≥ 1). Then there exist in M a continuous family of Cr embedded dimE s -dimensional discs s {Wloc (x0 )}x0 ∈Λ and a continuous family of Cr embedded dimE u -dimensional discs u ˜ x∈ {Wloc (x)} ˜ Λ f with the following properties: s (x ) and W u (x) (1) For each x˜ = (xi )i∈Z ∈ Λ f , both Wloc 0 loc ˜ contain x0 , and s s u u ˜ ⊃ Wloc (θ x). ˜ fWloc (x0 ) ⊂ Wloc ( f x0 ), fWloc (x) (2) There is 0 < λ¯ < 1 such that, for any x˜ ∈ Λ f s d( f y, f z) ≤ λ¯ d(y, z) if y, z ∈ Wloc (x0 ) u (x) u (θ −1 x) ˜ there is a unique y−1 ∈ Wloc ˜ with f y−1 = y0 , and for each y0 ∈ Wloc u and this y−1 and similar z−1 for z0 ∈ Wloc (x) ˜ satisfy

d(y−1 , z−1 ) ≤ λ¯ d(y0 , z0 ). s (3) There is δ > 0 such that, for any x0 ∈ Λ , y˜ ∈ Λ f with d(x0 , y0 ) < δ , Wloc (x0 ) u ˜ at a unique point [x0 , y] ˜ which depends conintersects transversely with Wloc (y) tinuously on (x0 , y) ˜ ∈ {(u0 , v) ¯ ∈ Λ × Λ f : d(u0 , v0 ) < δ }, and, if furthermore Λ is locally maximal, then there is a unique z˜ ∈ Λ f satisfying z0 = [x0 , y] ˜ and u (θ −i y). ˜ zi ∈ Wloc

B.2 Proof of Theorem B.1.1

267

s (x ), x ∈ Λ can be constructed by We remark that the local stable manifolds Wloc 0 0 the usual standard argument, but, since Λ may contain degenerate points, the local u (x), unstable manifolds Wloc ˜ x˜ ∈ Λ f can not be constructed similarly. However, one u may construct Wloc (x) ˜ in the following (standard as well) way: Let x˜ = (xi )i∈Z ∈ Λ f . For small r > 0 let hx˜ : Exu˜ (r) → Exs˜ (r) be a Lipschitz map with hx˜ (0) = 0 and Lip(hx˜ ) ≤ 1, where Exa˜ (r) = {ξ ∈ Exa˜ : |ξ | < r}, a = u, s. Then one can show that there is a similar map hθ x˜ : Eθux˜ (r) → Eθs x˜ (r) such that

(exp−1 x1 ◦ f ◦ expx0 )Graph(hx˜ ) ⊃ Graph(hθ x˜ )

(B.14)

(see Liu [45, Proposition 2.6] for details). Starting from hθ −n x˜ : Eθu−n x˜ (r) → Eθs −n x˜ (r) ξ → 0 (n)

via the relation (B.14) one ends by succession with a Cr function hx˜ : Exu˜ (r) → (n) (n) (n) Exs˜ (r) with hx˜ (0) = 0 and Lip(hx˜ ) ≤ 1. It is easy to show that hx¯ converges as (∞) n → +∞ uniformly to a similar function hx˜ : Exu˜ (r) → Exs˜ (r) whose graph gives u (x) ˜ under the exponential map. Wloc Assume in what follows that Λ is locally maximal. Let 0 < λ¯ < 1 be as given in Proposition B.2.3 and define a metric on Λ f by  d f (x, ˜ y) ˜ =

1

+∞

∑

2

−|i|

N

d(xi , yi )

N

,

x, ˜ y˜ ∈ Λ f

i=−∞

where N > 0 is an integer such that λ¯ N < 12 . For sufficiently small ε > 0 define ˜ y) ˜ < ε } → Λ f , (x, ˜ y) ˜ → z˜ [·, ·] : {(x, ˜ y) ˜ ∈ Λ f × Λ f : d f (x, where z˜ is the unique point in Λ f given in Proposition B.2.3 (3) corresponding to x0 and y. ˜ It is then easy to have the following Proposition B.2.4 (Λ f , d f , θ ) is a Smale space. Clearly, when (Λ , f ) is positively topologically transitive, so is (Λ f , θ ).

B.2.4 Proof of Theorem B.1.1 In what follows we will always endow Λ f with the metric d f (·, ·). Proof of Theorem B.1.1 (1). Each H¨older continuous φ : Λ → R gives a H¨older continuous φ¯ = φ ◦ p : Λ f → R which hence belongs to Cθ (Λ f ). By results in the

268

B

Large Deviations in Axiom A Endomorphisms

last two subsections, there is a unique equilibrium state μ¯ φ¯ of θ for φ¯ and μφ = p μ¯ φ¯ gives the unique equilibrium state of f for φ . Noting that there is a countable set of H¨older continuous functions which is dense in C(Λ f ) and the entropy map of (Λ f , θ ) is upper semicontinuous, by Theorem B.2.1 and Proposition B.2.2 one has (B.3) and (B.4) for (Λ f , θ , μ¯ φ¯ ) with rate function  ¯ ν¯ ) = J(

(

Pθ (φ¯ ) − φ¯ d ν¯ − hν¯ (θ ) +∞

if ν¯ ∈ Pθ (Λ f ) otherwise.

(B.15)

Since Pθ (φ¯ ) = Pf (φ ) and for any ν ∈ P f (Λ ) there is a unique ν¯ ∈ Pθ (Λ f ) such that p ν¯ = ν and this ν¯ satisfies hν¯ (θ ) = hν ( f ), one has for any ν ∈ P(Λ ) ¯ ν¯ ) = J(ν ) inf J(

pν¯ =ν

where J(ν ) is given by (B.5). One obtains then Theorem B.1.1 (1) by the contraction principle.   Proof of Theorem B.1.1 (2). Define

φ¯ u (x) ˜ = − log |det(Tx0 f |Exu˜ )|, x˜ ∈ Λ f .

(B.16)

It is H¨older continuous (see [72]) and the unique equilibrium state μ¯ φ¯ u of θ for φ¯ u projects under p to the SRB measure ρ , and in this case $( inf J (ν¯ ) = p ν¯ =ν ¯u

∑ λ i (x)+ mi (x)d ν − hν ( f ) i

+∞

if ν ∈ P f (Λ ) otherwise

since Pθ (φ¯ u ) = 0, where J¯u (ν¯ ) is given by (B.15) corresponding to φ¯ u . This proves Theorem B.1.1 (2).   In order to prove Theorem B.1.1 (3), we need the following result. Lemma B.2.5 Let Λ be a hyperbolic set of f ∈ C2 (O, M). Then each H¨older con˜ = φˆ (y) ˜ tinuous φ¯ : Λ f → R is homologous to some φˆ ∈ C(Λ f ) which satisfies φˆ (x) whenever xi = yi for i ≤ 0, i.e., there is u¯ ∈ C(Λ f ) such that

φ¯ = φˆ + u¯ − u¯ ◦ θ . Proof. For each x0 ∈ Λ pick (zi,x0 )i∈Z ∈ Λ f with z0,x0 = x0 . Define r : Λ f → Λ f by r(x) ˜ = x˜∗ = (x∗i )i∈Z where $ x∗i =

xi

for i ≥ 0

zi,x0

for i < 0.

B.2 Proof of Theorem B.1.1

269

Let u¯ : Λ f → R be defined by u( ¯ x) ˜ =

+∞

˜ ∑ [φ¯ (θ j x)˜ − φ¯ (θ j r(x))]

j=0

Then φˆ = φ¯ + u¯ ◦ θ − u¯ satisfies the requirements (see Bowen [10, Lemma 1.6] for a similar argument).   Proof of Theorem B.1.1 (3). Let φ¯ u be given by (B.16). Let now Λ be an Axiom A attractor of f ∈ C2 (O, M). Lemma B.2.5 tells that φ¯ u is homologous to φ u ◦ p for some φ u ∈ C(Λ ). Extend φ u to a continuous function Φ u : V → R where V is a neighborhood of Λ . Take ε0 > 0 such that Proposition IV.IV.3.8 holds for ε = ε0 and for all small δ > 0. Let now U be a basin of attraction of Λ such that

 s U¯ ⊂ [Wloc (x0 ) B(x0 , ε0 )] x0 ∈Λ

(this union contains an open neighborhood of Λ , see Chapter IV; see also [72]), U¯ ⊂ V and, moreover, U1 ⊃ U¯ ⊃ U ⊃ U¯ 2 for two other basins of attraction U1 ,U2 of Λ . Put ¯ d(x, y) ≤ ε0 λ¯ n }. an = sup{|Φ u (x) − Φ u (y)| : x, y ∈ U, ¯ small δ > 0 and all n ≥ 0 From Proposition IV.IV.3.8 it follows that for any y0 ∈ U, one has   n−1

Aδ (n)−1 ≤ m(B f (y0 , δ , n)) exp − ∑ Φ u ( f k y0 )

≤ Aδ (n)

k=0

1 where Aδ (n) = Aε0 ,δ exp(∑n−1 k=0 ak ) which clearly satisfies n log Aδ (n) → 0 as n → ∞. Then, by a minor modification of the proof of Kifer [37, Proposition 3.2], one can ¯ prove that for any ψ ∈ C(U)    n−1 1 u k Pf |U (Φ + ψ ) ≤ lim log exp ∑ ψ ( f x) d m(x) ¯ ≤ Pf |U (Φ u + ψ ) 2 1 n→∞ n k=0

which implies 1 lim n→+∞ n





n−1

∑ ψ( f

exp

k=0

 k

x) d m(x) ¯ = Pf |Λ (φ u + ψ ),

where m¯ is the normalized Lebesgue measure on U¯ and, when working on U1 , one may take continuous extensions of Φ u and ψ . Noting that for each H¨older continuous ψ : U¯ → R there is a unique equilibrium state of f |U¯ for Φ u + ψ and the entropy map of f |U is upper semicontinuous, one obtains Theorem B.1.1 (3) by ¯ μ = m¯ and by applying Theorem B.2.1 to X = U, 

Φ dν = u



∑ λi(x)+ mi (x)d ν (x) i

¯  for all ν ∈ P f (U). 

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Index

(ε , )-charts, 115, 120, 177, 214 (n, ε )-ball, 16, 17, 18, 23 B(x, ε ), 15 Bn (x, ε ), 16 < ·, · >x,n , 78

·, ·x,n ˜ , 54 ·, ·x,n ˜ , 53

 ·  , 177, 179

 · i , 177, 179  · x,n ˜ , 53 Bμ˜ (M f ), 98 ˜ εˆ r), 129 BΔc (x, ˜ εi r), 100 BΔi (x, Δ˜ , 7 Δ, 7 Δ λ , 113 Δ , 64 Δ0 , 55 Δ0 , 70 (i)

Δ0 , 70 Δ1 , 112 Δ2 , 112 C ,r

Δa,b,k, ε , 64

C Δa,b,k, ε , 52 Δa,b,k , 52 Δa,b , 52 , 28 Ei (x), ˜ 7 Γ, 2 Γ , 3 Γ0 , 48 Γ∞ , 3 Hx˜ , 114, 178, 179 Hxn˜ , 114 Hx−1 ˜ , 114, 179

x˜ , 112 H x−1 H ˜ , 112 χi (A), 10, 12, 13 Λ , 27 Λ f , 27 C ,r

Λa,b,k, ε , 79

C Λa,b,k, ε , 78 Λa,b,k , 78 Λa,b , 78 Lip( · ), 46 (M, f ), 7 (M, f , μ ), 2, 45 (M f , θ , μ˜ ), 8, 128 M f , 4, 7 MT (X), 1, 4, 5 Mθ (X T ), 4, 5 M f (M), 2 Φx˜ , 114 Sδcu (x), ˜ 116 Θ , 246 Tnm (x), ˜ 8 V (i) (x), 2 i Wx,2 ˜ 178 ˜ δ (y), W s (x), 80 W u (x), ˜ 77, 98 u (x), ˜ 71, 98 Wloc W s,i (x), 81 u,i Wloc (x), ˜ 70 Wn (x), ˜ 64, 67 Wn (x), 79 ˜ 116 Wx,˜uδ (x),  u (x), W ˜ 73  u (x), W loc ˜ 71  Wδcu (x), ˜ 118, 178  u,i (x), W ˜ 73

275

276  u,i (x), W loc ˜ 70 ¯ ρ ), 177 R( ¯ c (ρ ), 113 R ¯ s (ρ ), 113 R ¯ u (ρ ), 113 R ¯ (i) , 177 R ¯ (i) (ρ ), 177 R ¯ r−(i) (ρ ), 177 R L f , 254 χi (A), 10 (x), ˜ 114 λ0 , 177 λi (x), ˜ 8 λi (ω , x), 248 λi (x), 2, 9, 15 μω , 246 mi (x), 2, 9, 15 ˜ 8 mi (x), ρ0 , 9 ρ1 , 46 ρ2 , 46 θ ∗ , 249 dxi˜ (·, ·), 175 f x˜−1, 55 f x˜−n, 56 f 0 (ω ), 245 f ωn , 245 f x˜−1, 113 hμ ( f , x), 24 ∗ ∗ hF μ ∗ (Θ ), 249 hμ (T ), 6 hF μ (Θ ), 247 hμ ( f ), 9, 15 hμ˜ (θ ), 6 ith -unstable set, 174 mi (ω , x), 248 X + (M, P), 246

absolutely continuous, 15, 16, 84, 99, 158 Axiom A, 28, 97 attractor, 28 basic set, 28 endomorphism, 28

Besicovitch’s covering theorem, 135 Borel-Cantelli lemma, 101, 105

center unstable set, 116 conditional measure, 19, 24, 97 continuous family of C1 embedded discs, 64

Index diffeomorphism, 20

endomorphism, 1, 15 Axiom A, 28 entropy, 1, 246 conditional entropy, 247 local entropy, 24 metric entropy, 1, 9, 15 equilibrium state, 26 Euclidean space, 10 expanding map, 15, 16 expanding set, 16 exponential map, 9

generator, 175 incresing, 175

H¨older continuous, 81, 82 Hahn-Banach Theorem, 5 hyperbolic set, 28

invariance property, 246 inverse limit space, 4, 27, 98

Jacobian, 90

large deviation, 261 Lebesgue measure, 16 left shift operator, 245 local ith -unstable manifold, 70 local ith -unstable set, 70 local product structure theorem, 31 local unstable manifold, 71, 116 local unstable set, 71 Lyapunov charts, 112 (ε , )-charts, 115, 177, 214 Lyapunov exponent, 2, 8, 9, 15, 248

Margulis-Ruelle inequality, 9, 15, 19, 24, 110, 248 Multiplicative Ergodic Theorem, 2 inverse limit space case, 7 random endomorphisms’ case, 248 multiplicity, 2, 8, 9, 248

Oseledec’s theorem, 2, 248

Index partition adapted to Lyapunov charts, 125 increasing, 103 measurable, 19, 25 Pesin’s entropy formula, 15, 17, 25, 28, 40, 97 Polish space, 246 pseudo-orbit-shadowing lemma, 31 pull back bundle, 7, 27

quasi-expanding map, 16

random diffeomorphisms, 9 random dynamical systems, 97 RDS, 245 Expanding in average, 251 Markov, 252 one-sidede, 246 two-sided, 248 Riemannian structure, 20 Rieze Representation Theorem, 5 Ruelle transfer operator, 254

277 sample measures, 246 separated set, 6, 11, 23, 40, 216 (n, ε )-separating set, 6, 23, 40 maximal separated set, 6, 11, 216 Shannon-McMillan-Breiman Theorem, 17 skew-product transformation, 246 Smale space, 38 SRB measures, 250 SRB property, 97, 98 subordinate to W s -manifolds, 253 W i -manifolds, 175 W u -manifolds, 98

transversal, 84, 158, 167 transverse dimension, 176, 198 transverse metric, 132

volume lemma, 36

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Recent Reprints and New Editions Vol. 1702: J. Ma, J. Yong, Forward-Backward Stochastic Differential Equations and their Applications. 1999 – Corr. 3rd printing (2007) Vol. 830: J.A. Green, Polynomial Representations of GLn , with an Appendix on Schensted Correspondence and Littelmann Paths by K. Erdmann, J.A. Green and M. Schoker 1980 – 2nd corr. and augmented edition (2007) Vol. 1693: S. Simons, From Hahn-Banach to Monotonicity (Minimax and Monotonicity 1998) – 2nd exp. edition (2008) Vol. 470: R.E. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. With a preface by D. Ruelle. Edited by J.-R. Chazottes. 1975 – 2nd rev. edition (2008) Vol. 523: S.A. Albeverio, R.J. Høegh-Krohn, S. Mazzucchi, Mathematical Theory of Feynman Path Integral. 1976 – 2nd corr. and enlarged edition (2008) Vol. 1764: A. Cannas da Silva, Lectures on Symplectic Geometry 2001 – Corr. 2nd printing (2008)

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