Dimension Theory in Dynamical Systems: Contemporary Views and Applications 9780226662237

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Dimension Theory in Dynamical Systems

Chicago Lectures in Mathematics Series Robert J. Zimmer, series editor J. Peter May, Spencer J. Bloch, Norman R. Lebovitz, William Fulton, and Carlos Kenig, editors

Other Chicago Lectures in Mathematics titles available from the University of Chicago Press: Simplicia1 Objects in Algebraic Topology, by J. Peter May (1967) Fields and Rings, Second Edition, by Irving Kaplansky (1969, 1972) Lie Algebras and Locally Compact Groups, by Irving Kaplansky (1971) Several Complex Variables, by Raghavan Narasimhan (1971) Torsion-Free Modules, by Eben Matlis (1973) Stable Homotopy and Generalised Homology, by J . F. Adams (1974) Rings with Involution, by I. N. Herstein (1976) Theory of Unitary Group Representation, by George V.Mackey (1976) Commutative Semigroup Rings, by Robert Gilmer (1984) Infinite-Dimensional Optimization and Convexity, by Ivar Ekeland and Thomas Turnbull (1983) Navier-Stokes Equations, by Peter Constantin and Ciprian Foias (1988) Essential Results of Functional Analysis, by Robert J. Zimmer (1990) Fuchsian Groups, by Zvetlana Katok (1992) Unstable Modules over the Steenrod Algebra and Sullivan's Fixed Point Set Conjecture, by Lionel Schwartz (1994) Topological Classijication of Stratijied Spaces, by Shmuel Weinberger ( 1994) Lectures on Exceptional Lie Groups, by J . F. Adams (1996) Geometry of Nonpositively Curved Manifolds, by Patrick B. Eberlein (1996)

Yakov B. Pesin

DIMENSION THEORY IN DYNAMICAL SYSTEMS: Contemporary Views and Applications

The University of Chicago Press Chicago and London

Yakov B. Pesin is professor of mathematics at Pennsylvania State University, University Park. He is the author of The General Theory of Smooth Hyperbolic Dynamical Systems and co-editor of Sinai's Moscow Seminar on Dynamical Systems.

The University of Chicago Press, Chicago 60637 The University of Chicago Press, Ltd., London 0 1997 by The University of Chicago All rights reserved. Published 1997 Printed in the United States of America 06050403020100999897 1 2 3 4 5 ISBN: 0-226-66221 -7 (cloth) ISBN: 0-226-66222-5 (paper) Library of Congress Cataloging-in-PublicationData Pesin, Ya. B. Dimension theory in dynamical systems : contemporary views and applications / Yakov B. Pesin. p. cm. - (Chicago lectures in mathematics series) Includes bibliographical references (p. - ) and index. ISBN 0-226-66221-7 (alk. paper). - ISBN 0-226-66222-5 (pbk.: alk. paper) 1. Dimension theory (Topology) 2. Differentiabledynamical systems. I. Title. 11. Series: Chicago lectures in mathematics. QA611.3.P47 1997 5 15'.352--d~2 1 97- 16686 CIP

0 The paper used in this publication meets the minimum requirements of the American National

Standard for Information Sciences--Permanenceof Paper for Printed Library Materials, ANSI 239.48-1984.

For my wife, Natasha, daughters, Ira and Lena, and my fathel; Boris, who make it all worthwhile

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Contents

PREFACE INTRODUCTION

ix 1

Part I: Carathkodory Dimension Characteristics CHAPTER1. GENERALCARATHEODORY CONSTRUCTION 11 12 1. CarathCodory Dimension of Sets 2. CarathCodory Capacity of Sets 16 21 3. CarathCodory Dimension and Capacity of Measures 4. Coincidence of CarathCodory Dimension and Carathbodory Capacity of Measures 28 5. Lower and Upper Bounds for CarathCodory Dimension of Sets; Carathkodory Dimension Spectrum 31 CHAPTER 2. C-STRUCTURES ASSOCIATED WITH METRICS:HAUSDORFF DIMENSION AND BOX DIMENSION 34 35 6. Hausdorff Dimension and Box Dimension of Sets 7. Hausdorff Dimension and Box Dimension of Measures; Pointwise Dimension; Mass Distribution Principle 41 CHAPTER3. C-STRUCTURES ASSOCIATED WITH METRICSAND MEASURES: DIMENSION SPECTRA 48 8. q-Dimension and q-Box Dimension of Sets 48 57 9. q-Dimension and q-Box Dimension of Measures APPENDIXI: HAUSDORFF(Box) DIMENSION AND Q-(Box) DIMENSION 61 OF SETS AND MEASURES IN GENERAL METRICSPACES ASSOCIATED WITH DYNAMICAL SYSTEMS: CHAPTER4. C-STRUCTURES 64 THERMODYNAMIC FORMALISM 10. A Modification of the General Carathkodory Construction 65 11. Dimensional Definition of Topological Pressure; Topological and 68 Measure-Theoretic Entropies 83 12. Non-additive Thermodynamic Formalism PRESSURE; APPENDIX11: VARIATIONAL PRINCIPLE FOR TOPOLOGICAL 87 SYMBOLIC DYNAMICAL SYSTEMS;BOWEN’SEQUATION APPENDIX111: A N EXAMPLE O F CARATHEODORY STRUCTURE GENERATED BY DYNAMICAL SYSTEMS 110 Part 11: Applications to Dimension Theory and Dynamical Systems CHAPTER 5. DIMENSION OF CANTOR-LIKE SETS AND SYMBOLIC DYNAMICS

117 13. Moran-like Geometric Constructions with Stationary (Constant) Ratio Coefficients 123 vii

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14. Regular Geometric Constructions 133 15. Moran-like Geometric Constructions with Non-stationary Ratio 140 Coefficients 16. Geometric Constructions with Rectangles; Non-coincidence of Box Dimension and Hausdorff Dimension of Sets 153 CHAPTER6. MULTIFRACTAL FORMALISM 170 17. Correlation Dimension 174 18. Dimension Spectra: Hentschel-Procaccia, R h y i , and f(a)-Spectra; Information Dimension 182 19. Multifractal Analysis of Gibbs Measures on Limit Sets of Geometric 189 Constructions OF SETS AND MEASURESINVARIANT UNDER CHAPTER 7. DIMENSION HYPERBOLIC SYSTEMS 196 20. Hausdorff Dimension and Box Dimension of Conformal Repellers for Smooth Expanding Maps 197 21. Multifractal Analysis of Gibbs Measures for Smooth Conformal Expanding Maps 209 22. Hausdorff Dimension and Box Dimension of Basic Sets for Axiom A DXeomorphisms 227 238 23. Hausdorff Dimension of Horseshoes and Solenoids 24. Multifractal Analysis of Equilibrium Measures on Basic Sets of Axiom A Diffeomorphisms 247 APPENDIXIV: A GENERALCONCEPTOF MULTIFRACTAL SPECTRA; MULTIFRACTAL RIGIDITY 259 CHAPTER 8. RELATIONS BETWEEN DIMENSION, ENTROPY, AND LYAPUNOV EXPONENTS 270 25. Existence and Non-existence of Pointwise Dimension for Invariant Measures 271 26. Dimension of Measures with Non-zero Lyapunov Exponents; The Eckmann-Ruelle Conjecture 279 293 APPENDIXV: SOMEUSEFULFACTS BIBLIOGRAPHY 295 INDEX 301

Preface

This is neither a book on dimension theory nor a book on the theory of dynamical systems. This book deals with a new direction of research that lies at the interface of these two theories. One would presumably start writing a book about a new area of research when this area is matured enough to be considered as an independent discipline. As indicators of the maturity, one can use, perhaps, the influence that the new discipline has on other areas and the presence of intriguing new ideas and profound methods of study, as well as exciting applications to other fields. One can find all these features in the dimension theory of dynamical systems. Although this new discipline was formed only in the last 10-15 years, its impact on both “parents” - dimension theory and the theory of dynamical systems - is quite strong and fruitful and its concepts and results are widely used in many applied fields. The goal of the book is to lay down the mathematical foundation for the new area of research as well as to present the current stage in its development and systematic exposition of its most important accomplishments. I believe this will help to shape the new area as an independent discipline: establish its own language, isolate basic notions and methods of study, find its origin, and trace the history of its development. I also hope this will stimulate further active study. I would like to point out another important circumstance: until recently physicists and applied mathematicians were the main creators of the dimension theory in dynamical systems. They developed many new concepts and posed a number of challenging problems. Although most “statements” were not rigorous but only intuitively clear and the “proofs” were based upon heuristic arguments, they essentially built up a new building and designed its architecture. Mathematicians came to lay up its foundation and to decorate the building. Their work paid off: they not only enjoyed some very interesting problems but also revealed some new and unexpected phenomena which did not fit in with the “physical” intuition and were not forestalled by physicists. Hardly any book alone can cover such a diverse and broad area of research as the dimension theory in dynamical systems. This is why I had to select the material following my own interest and my (certainly subjective) point of view. I am mainly concerned with the general concept of characteristics of dimension type and with the “dimension” approach to the theory of dynamical systems. Although this book is not designed as an introductory textbook on dimension theory or on the theory of dynamical systems some of its parts can be used for a special topics course since it contains all preliminary information from the “parent” disciplines. I suggest the following courses (they may also be considered

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Preface

as logically connected sequences of chapters which are recommended in the first reading): (1) Dimension of Cantor-like Sets and Symbolic Dynamics - Chapters 1, 2, and 4, Appendix I1 (proofs are optional), and Chapter 5 (Section 15 is optional) ; (2) Dimension and Hyperbolic Dynamics - Chapters 1, 2, and 4 (Sections 10 and l l ) , Appendix I1 (proofs are optional), Chapter 7 (Sections 20, 22, and 23) and Chapter 8 (Section 26 is optional); (3) Multifractal Analysis of Dynamical Systems - Chapters 1, 2, and 3 (Section 9 and Appendix I are optional), Chapter 4 (Sections 10 and ll), Appendix I1 (proofs are optional), Appendix I11 (optional), Chapter 6 (Sections 17 and 18), and Chapter 7 (Sections 21 and 24 and Appendix IV is optional). Let us point out that we number Theorems, Propositions, Examples, and Formulae in such a way that the numerals before the point indicate the number of the corresponding Section. Throughout the book the reader will find numerous Remarks; they contain the material which is “aside” from the mainstream of the book but provide useful additional information. There are five Appendices in the book. Appendices I-IV are brief (but sufficiently detailed) surveys on topics which are closely related to the main exposition and help extend reader’s vision of the area. Appendix V provides the reader with some useful information from Analysis and Measure Theory and thus makes the book a little more self-contained.

Acknowledgments I wish to acknowledge the invaluable assistance of several friends and collaborators including Valentine Afraimovich, Luis Barreira, Joerg Schmeling, and Howie Weiss, with whom I enjoyed numerous hours of fruitful discussions while this book was taking shape. They also reviewed the entire manuscript and made pany cogent comments which helped me improve the exposition of the book significantly, crystallize its content, and avoid some mathematical and stylistical errors and misprints. I am particularly indebted to Luis Barreira for his devoted assistance in producing the figures and editing the text and to Howie Weiss for his help in polishing the text. I sincerely thank Boris Hasselblatt who worked long hours modifying his TeX macros and thus enabling me to create a camera-ready copy of the manuscript. In 1995, Valentine Afraimovich designed and taught a course on dimension theory in dynamical systems at Northwestern University. It was the first course of this type anywhere in the world that I am aware of. His comments helped me clarify the presentation of some parts of the book so that it can be used (at least partly) as a textbook for students, and for this I thank him. In the Fall of 1996, I taught a graduate course on Dimension Theory and Dynamical Systems at The Pennsylvania State University based on a draft of this book. I would like to thank my students for their enthusiasm, patience, and assistance with numerous rough spots in my exposition. Our collaborative

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efforts that semester resulted in many additions and changes to the book. Special thanks go to Serge Ferleger, Boris Kalinin, Misha Guysinsky, and Serge Yaskolko. My thanks also go to Kathy Wyland and Pat Snare at The Pennsylvania State University who typed the first draft of most chapters in TeX. In the Spring of 1990, I enjoyed a wonderful and productive visit to the University of Chicago. While at Chicago, Robert Zimmer persuaded me to write a book for the Chicago Lectures in Mathematics Series, and this was a partial impetus for writing this book. I had the great fortune to have top-notch editorial assistance at the Chicago University Press throughout the entire writing and editing process. Particular thanks go to Penelope Kaiserlian, Vicki Jennings, Michael Koplow, Margaret Mahan, and Dave Aftandilian. Last but not least, I wish to thank Natasha Pesin, an experienced editor, who spent a great amount of time helping me edit and design the book in its present form. Most of all I thank her for her constant encouragement and inspiration. State College, Pennsylvania April, 1997

Yakov B. Pesin

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Introduction

The dimension of invariant sets is among the most important characteristics of dynamical systems. The study of the dimension of these sets has recently spawned a new and exciting area in dynamical systems. This book presents a comprehensive and systematic treatment of dimension theory in dynamical systems. The model for the notions of dimension we will consider is the Hausdorff dimension. Unlike the classical topological concept of dimension it is not of a purely topological nature. Carathkodory and Hausdorff originated this notion at the beginning of the twentieth century and later it has become a subject of intensive study by specialists in function theory, mainly by Besicovitch and his school. These investigators used the Hausdorff dimension as an appropriate quantitative characteristic of the complexity of topological structure of subsets in metric spaces; these sets are similar to the well-known Cantor set. Soon after the discovery of strange attractors - invariant sets of a special type -specialists in dynamical systems became interested in studying the Hausdorff dimension of these attractors, and in relating the dimension to other invariants of dynamical systems. Local topological structure of a strange attractor is often the product of a submanifold and a Cantor-like set. The local submanifold is unstable with respect to the dynamics and corresponds to the directions “along” the attractor. The Cantor-like set lies in the directions “transverse” to the attractor that are stable with respect to the dynamics. The dimension can be interpreted as a quantitative characteristic of the complexity of the topological siructure of the attractor in the transverse directions. A classical example of a strange attractor is the Lorenz attractor. There are many other examples of invariant sets with “wild” topological structure. Among them are the well-known Smale-Williams solenoid and Smale horseshoe. The latter is an example of a hyperbolic invariant set whose local topological structure is the product of different Cantor-like sets. Strange attractors are always associated with trajectories having extremely irregular behavior and are thought of as the origin of “dynamical” chaos. Specialists in dynamical systems strongly believe that there is a deep connection between the topology of the attractor and properties of the dynamics acting on it. This is a source for exciting relationships between dimension, as a characteristic of complexity of the topological structure, and invariants of the dynamics such as Lyapunov exponents and entropy which characterize instability and stochasticity of dynamical systems (see discussion in [GOY]). Revealing these interrelationships is one of the main goals in studying dimension in dynamical systems.

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2

Introduction

The great interest to this area among specialists in applied fields was inspired by the works of Mandelbrot and especially by his famous book Fractal Geometry of Nature [Ml]. This book introduces the reader to a range of ideas connected with dimension. It also gives a convincing heuristic description of the way in which the complicated topological structure of invariant sets can influence the qualitative behavior of dynamical systems generating irregular “turbulent” regimes. Mandelbrot also revealed another important aspect of this phenomenon. Assume that a physical system admits a group of scale similarities, i.e., it “reproduces” itself on smaller scales. From a mathematical point of view this means that the dynamical system, which describes the physical phenomenon, possesses invariant sets of a special self-similar structure known as fractals - a word coined by Mandelbrot in 1975 (see Chapters 5 and 6, where various types of fractals are considered). The works of Hausdorff, Besicovitch, and Mandelbrot shaped a new field in mathematics called fractal geometry. Many areas of science have adopted and widely used the methods and results of fractal geometry. The important feature of fractals is their independence of scaling. The rate of this scaling can be characterized quantitatively by a fractal dimension. Using many (often infinitely many) single fractals one can build a multifractal. Its topological structure is much more complicated and is the result of the “interaction” of topological structures of single kactals on different scales. Multifractals are used in many applied sciences to cope with phenomena associated with intricate structures involving more than one scaling exponent (see Chapter 6). The important conclusion of fractal geometry, based on selfsimilar properties of multifractals, is that one can produce complex shapes with “highly unusual” properties starting from some simple ones and using simple iteration schemes (see [Ba]). During the past 10 years the concepts of fractal geometry have become enormously popular among specialists in most natural sciences (see [Sc]). The “natural” popularity of fractal geometry has caused “natural” problems: the rigorous mathematical study was far behind applications. The “empty space” was immediately filled up with numerous “notions” and “results” obtained in studying fractals by a computer. Plausibility was the only criterion for immediate adoption of these notions and results into the theory. Unfortunately, Mandelbrot’s book, being directed at specialists in applied fields, hardly contains any rigorous general definitions of dimension and related rigorous results. In particular, the book does not reflect the crucial fact that, in a variety of ways, characteristics of dimension type can be quite “treacherous” and have some “pathological” properties. These properties may not fit in with the intuition that physicists may have developed in working with other objects of research. Undoubtedly, researchers were not quite satisfied with the notion of Hausdorff dimension which is not quite adopted to the dynamics. Moreover, in many cases the straightforward calculation of the Hausdorff dimension was very difficult. This prompted researchers to introduce other characteristics that, to a greater or lesser degree, aspire to be called dimensions. Among them are correlation dimension, information dimension, similarity dimension, etc. In many

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Introduction

cases both the motivation for the introduction of these characteristics and their definitions were vague and could be understood in a variety of ways. Nevertheless, most researchers were convinced that in “sufficiently good cases”, all these characteristics, if correctly interpreted, would coincide and determine what they called “the fractal dimension”. Moreover, a number of conjectures connecting this dimension with other invariants of dynamics have been put forward. Although not all of these conjectures have been confirmed, on the whole, intuition did not lead the researchers astray. The goal of this book is to study interrelations between dimension theory (and, in particular, fractal geometry) and the theory of dynamical systems. During the past 10-15 years these interrelations have grown from some isolated special results into a cohesive new area in the theory of dynamical systems with its own intrinsic structure. Its current state is characterized by an abundance of notions of dimension and exciting nontrivial relations between them and other invariants of dynamics, new promising methods of study, and growing interest from specialists in different fields. This book provides a rigorous mathematical description of the notions, methods, and results that shape the new area, and extends some concepts of fractal geometry by developing general ways to introduce various characteristics of dimension type. Let us describe a general scheme for introducing the notion of dimension. Let X be a set and m(.,a ) a family of cr-sub-additive set functions on X depending on a real parameter a. Assume that for each 2 c X there exists a critical “overchanged” value 00 of a such that m(2,a ) = 0 for a > 00 and m(2,a ) = m for o < oo while m(2,(YO) can be any number in the interval [0, m]. The number 00 is called the dimension characteristic of the set 2. Of course, from such a general standpoint, one cannot expect to obtain sufficiently meaningful results on dimension. Therefore, we propose a more restricted (but still sufficiently general) approach to construct a family of set functions with the above property. Our construction is an elaboration of the well-known Carathe‘odory construction in general measure theory [C].We generalize it to include new phenomena associated with the use of dimension in dynamical systems. Let us outline our approach. One can introduce the notion of dimension in a space X which is endowed with a special structure that we call a C-structure. The latter is given when one chooses a collection 3 of subsets in X and three 3 that satisfy some conditions. The non-negative set functions [ , q , $ : F + E set function $ is used to characterize the “size” of sets in E a set U E 3 is “small” if $ ( U ) is small. The role of the set functions [ and q can be understood using concepts of statistical physics. In this context the set X is viewed as a configurations space of a given physical system. To any cover G = { U l , . . . ,Un} of 2 by sets Ui E 3 one can associate the free energy

where [ is the weight function, q is the potential, and is the temperature of the physical system. One can now define the family of set functions m(2,a ) by

m(2,a ) = lim inf F ( G ) , €--to g

Introduction

4

where the infimum is taken over all finite or countable covers 6 c 7 of Z by sets U of “size” $(U)5 e (U E G). In Chapter 1 we will show that the family of set functions m(Z,a), 2 c X has a critical value a0 = ao(2).We call it the Carathe‘odory dimension of the set Z and denote it by dimc 2. Another procedure, when one uses covers of 2 by sets U with $(U)= e , leads to the definition of two other basic characteristics of dimension type - the lower and upper Carathe‘odory capacities of the set Z. We denote them by Cap,Z and G c Z respectively. The basic relationship between CarathCodory dimension and lower and upper Carathkodory capacities is the following inequality: dimc Z 5 Cap,Z 5 G c Z . One can generate a C-structure on X using other structures on X . For example, if X is a complete metric space then one can choose 7 to be the collection of = 1, 9(U) = $ ( U ) = diam U for U E 7. In this open subsets in X , and set ((U) case the CarathCodory dimension of a set Z is its Hausdorff dimension, dimH Z, and the lower and upper CarathCodory capacities of Z coincide with the lower and upper box dimensions of Z, b B Z and d i m ~ Z(see Section 6). We will be mostly interested in C-structures associated in one way or another with a dynamical system f acting on X . In this case the choice of the collection of subsets 7 and set functions [ , 9 , $ is determined by f in some “natural” way. The CarathCodory dimension and CarathCodory capacities of a set Z, which is invariant under f, turn out to be invariants of the dynamical system flZ. Examples are: (1) q-dimension of Z, dim,Z (q 2 -1, q # 0 is a parameter) that is used to characterize the multifractal structure of Z generated by f (see Chapter 3); (2) topological pressure of a function cp on 2, Pz(cp)and topological entropy on 2,h Z ( f ) (see Chapter 4); thus our approach exposes a “dimensional” nature of these well-known topological invariants of dynamical systems. The study of C-structures generated by dynamical systems leads to another class of characteristics of dimension type specified by a measure p on X . The formal definition does not involve any dynamics on X and is given in Chapter 1 (see Section 3). We call these characteristics the Carathe‘odory dimension of p and lower and upper Carathe‘odory capacities of p and denote them respectively by dimc p, Cap,p, and G c p . When p is invariant under a dynamical system f these characteristics are invariants of f associated with p. Among several examples given in Chapters 2, 3, and 4 let us mention the measure-theoretic entropy off, h,(f) (see Section 11). Thus our approach provides a “dimensional” interpretation of this important metric invariant of dynamical systems. The basic relationship between the CarathCodory dimension of p and lower and upper CarathCodory capacities of p is the following:

Introduction

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A challenging problem is to find s a c i e n t conditions that would guarantee equalities in (0.1) and (0.2). We stress that the coincidence of the CarathCodory dimension and CarathCodory capacities relative to a set 2 is a rare phenomenon and requires a special somewhat homogeneous structure of Z (see Chapter 5). As far as CarathCodory dimension and CarathCodory capacities relative to a measure p are concerned we present a powerful criterion that guarantees their coincidence (see Section 4). We believe that in “good” cases the conditions of this criterion hold and, thus, the common value represents the dimension of p defined by the C-structure on X . Another aspect of the notions of dimension and capacity type characteristics relative to a measure has to do with the above-mentioned phenomenon of selfsimilarity of an invariant set. In “real” situations, self-similarity is hardly ever exact. However, the invariant set sometimes can be “broken into pieces” each of which turns out to be “asymptotically self-similar” in a way. Such pieces are the supports of invariant ergodic measures, and “self-similarityscales” can be expressed in terms of the Lyapunov exponents of these measures. This is the clue to reveal the fundamental relation between dimension, Lyapunov exponents, and metric entropy (see Chapter 8). The book consists of two parts. In the first part we develop the general theory of CarathCodory dimension. In Chapter 1 we describe a generalized version of the classical Carathkodory construction in a space X and introduce the notions of Cstructure and CarathCodory dimension characteristics: CarathCodory dimension and lower and upper CarathCodory capacities of subsets of X and measures on X . In Chapter 2 we study C-structures generated by metrics on Euclidean spaces. We introduce the notions of Hausdorff dimension and lower and upper box dimensions of sets and measures and describe their basic properties. In Chapter 3 we deal with C-structures generated by both metrics and measures and introduce the notions of q-dimension and lower and upper q-box dimensions of sets and measures. This leads to an important application of the general CarathCodory construction developed in Chapter 1: the q-box dimension is closely related to dimension spectra of dynamical systems which are widely used in’numerical study of dynamical systems. We describe these spectra in detail in Chapter 6 . In Appendix I we extend results of Chapters 2 and 3 to arbitrary complete separable metric spaces. Our main example of a C-structure is given in Chapter 4,where we consider C-structures generated by dynamical systems acting on compact metric spaces and continuous functions. This example is one of the main manifestations of the general CarathCodory construction: we demonstrate how the “dimension” approach can be used to introduce a general concept of topological pressure and topological entropy for arbitrary subsets of the space as well as a concept of measure-theoretic entropy. In Appendix I1 we use the “dimension” approach to discuss various versions of the thermodynamic formalism (including the classical thermodynamic formalism of dynamical systems created by Bowen, Ruelle, Sinai, and Walters). Although this lies not strictly along the line of the main exposition, it is an important addition to Chapter 4 and is crucial for results in the second

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Introduction

part of the book. In Appendix I11 we describe an example of C-structure which plays a crucial role in studying some “weird” sets for dynamical systems. The second part of the book is devoted to applications of results in Part I to dimension theory and the theory of dynamical systems. In Chapter 5 we describe various geometric constructions - one of the most popular subjects in dimension theory. We demonstrate that the theory of dynamical systems grants powerful methods to study geometric constructions with complicated geometry of basic sets and essentially arbitrarily symbolic representation. Our main tool is the thermodynamic formalism developed in Chapter 4 that we apply to the symbolic dynamical system, associated with the geometric construction. In Chapter 6 we study another popular subject of dimension theory intimately connected with fractals and multifractals. We introduce various dimension spectra (the Hentschel-Procaccia spectrum for dimensions, RCnyi spectrum for dimensions, and the spectrum of so-called pointwise dimensions) and describe their relations to some well-known dimension characteristics of dynamical systems such as the correlation dimension and information dimension. We also use dimension spectra to discuss the mathematical content of the notion of multifractality, and we effect a complete multifractal analysis of Gibbs measures supported on the limit sets of Moran geometric constructions. The interrelation between dimension theory and the theory of dynamical systems is of benefit to both sides. In the last two chapters we show how methods of dimension theory can be applied to study various characteristics of dimension type of sets and measures invariant under hyperbolic dynamical systems. We consider repellers for smooth expanding maps (including hyperbolic Julia sets, repellers for one-dimensional Markov maps, and limit sets for Schottky groups), basic sets of Axiom A diffeomorphisms (including Smale horseshoes), and S m a l e Williams solenoids. We obtain most definite results in the case when dynamics is conformal and sharp “dimension estimates” in the non-conformal case. The approach we use demonstrates the power of the general CarathCodory construction which allows one to extend and unify many results on dimension of invariant sets for dynamical systems with hyperbolic behavior. A significant part of Chapter 7 is to develop a complete multifractal analysis of Gibbs measures for smooth conformal dynamical systems. In particular, we obtain a complete and surprisingly simple description of a highly non-trivial and intricate multifractal structure of conformal repellers and conformal hyperbolic sets associated with the pointwise dimension, local entropy, and Lyapunov exponent. The approach is built upon the general concept of multifractal spectra - a recent new direction of research in the theory of dynamical systems which we sketch in Appendix IV. Multifractal spectra provide refined information on some ergodic properties of dynamical systems. For example, multifractal spectra for local entropies describe their deviation from the mean value provided by the Shannon-McMillan-Breiman Theorem while multifractal spectra for Lyapunov exponents describe their distribution around the mean value given by the Multiplicative Ergodic Theorem. In Chapter 8 we deal with the dimension of invariant measures. In particular, we discuss the recent achievement in dimension theory of smooth hyperbolic

Introduction

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dynamical systems - the affirmative solution of the Eckmann-Ruelle conjecture obtained in [BPSl]. It establishes the existence of pointwise dimension for almost every point with respect to a hyperbolic invariant measure. This implies that all characteristics of dimension type of the measure coincide and thus, it justifies the strong opinion among experts that in “good cases”(and hyperbolic measures are “good” ones) all known methods of computing the dimension of a measure lead to the same quantity. Since hyperbolic measures are “responsible” for chaotic regimes generated by dynamical systems, this quantity stands in a row of most fundamental characteristics of such complicated motions.

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Part I

Carath6odory Dimension Characteristics

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Chapter 1

General Carathitodory Construction

The classical CarathCodory construction in the general measure theory was originated by Carathkodory in [C] (a contemporary exposition can be found in [Fe]). It was designed to produce a family of a-measures on a metric space X given by

where the infimum is taken over all finite or countable covers B = {Ui} of Z by open sets Ui with diamUi 5 E (one can easily see that the limit exists). Here 77 is a positive set function. One can also use an arbitrarily chosen collection of subsets of X to make up covers of Z . In this chapter we introduce a construction which is a generalization of the classical Carathkodory construction. It was elaborated by Pesin in [PZ] to produce various characteristics of dimension type. The starting point for the construction is a space X which is endowed with a special structure. We introduce this structure axiomatically by describing its basic elements and relations between them, and we call it the CarathCodory dimension structure (or, briefly, C-structure). This structure enables us to yield two types of quantities that are the dimension of subsets of X and the dimension of measures on X. We call these quantities the CarathCodory dimension characteristics (of sets and measures). They include the Carathhodory dimension and lower and upper Carathhodory capacities. We study some of their fundamental properties which will be widely used in the book.

-

C-structures can be generated by some other structures on X associated with metrics, measures, functions, etc. In Chapters 2 and 3, we will give some examples of C-structures and will illustrate general properties of CarathCodory dimension characteristics established in this chapter. We will be mainly interested in C-structures generated by dynamical systems acting on X. Let us notice that CarathCodory dimension characteristics are invariants of an isomorphism which preserves the C-structure. In the case when the Cstructure is generated by a dynamical system on X, the CarathCodory dimension and lower and upper CarathCodory capacities become invariants of the dynamics and can be used to characterize invariant sets and invariant measures. There are deep relations between them and other important invariants of dynamics which we will consider in the second part of the book. The original CarathCodory construction was created within the framework of the classical function theory where the dimension of subsets was a “natural”

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subject of study. The dimension of measures was introduced within the theory of dynamical systems in order to characterize invariant subsets and measures concentrated on them. The most exciting applications of the general Carathkodory construction can be obtained in this case and are related to the dimension characteristics of measures. It is worth emphasizing that the general Carathkodory construction does not require any dynamics on X and thus, its applications go far beyond the theory of dynamical systems and include such fields as function theory, geometry, etc.

1. Carathhodory Dimension of Sets Let X be a set and 3 a collection of subsets of X . Assume that there exist two set functions r ] , $: 3+ R+(= [0,m)) satisfying the following conditions: A l . 0 E 3; ~ ( 0=)0 and $(a) = 0; r](U) > 0 and $ ( U ) > 0 for any U E 3, UZ0; A2. for any 6 > 0 one can find E > 0 such that r](U)5 6 for any U E 3 with

$(U) I E.

A3. for any E > 0 there exists a finite or countable subcollection B c 3 which E. covers X (i.e., UuEgU3 X ) and $(B) gfsup{$(U) : U E 8) I Let [:3 R+ ibe a set function. We say that the collection of subsets 3 and the set functions [ , r ] , $ , satisfying Conditions Al,A2, and A3, introduce the Carathhodory dimension structure or C-structure r on X and we (,77, $). C-structures on X can be generated by other structures write 7 = (3, associated with metrics and measures on X , maps acting on X , etc. We illustrate this by the following examples; more general setups are given in Chapters 2, 3, and 4.

Examples. (1) Define the C-structure on the Euclidean space R" as follows. Let 3 be the collection of open sets, ( ( U ) = 1, r](U)= diamU, $ ( U ) = diamU for U E 3. It is easy to see that the collection of subsets 3 and set functions (,r ] , $ satisfy Conditions A l l A2, and A3 and hence introduce the C-structure 7 = (3, [, r ] , $) on X . (2) Let ,u be a Bore1 probability measure on the Euclidean space Rm. Fix any y > 0 and q 2 0. Define 3 to be the collection of open balls, B ( z ,E ) (z E X , E > 0) and set [ ( B ( z E, ) ) = ,u(B(z,y~))Q, r](B(z, E ) ) = $(B(z,E ) ) = E. One can easily check that the structure T ~ , -= , (3, [, r ] , $) is a C-structure on X . (3) Let X be a compact metric space endowed with a metric p and f : X -+ X a continuous map. Given 6 > 0, n 2 0, and z E X we denote &(z, 6 ) = {y E

x : p ( f ' ( z ) , fi(y))

2 6 for o 5 i I n).

x

w e define the C-structure 76 = (36,[,7],$) on by setting 3 6 = {Bn(z,6): z E X , n 2 0}, [(B,(z,S))= 1, r](B,(z,6)) = e-", and $(Bn(zl6)) = (Remark: we assume, for simplicity, that the map f is such that B,(z,6)# B,(y, 6) if n # m ;the general case is considered in Section 11).

i.

General Carathbodory Construction

13

Consider a set X endowed with a C-structure r = (3, t,77, $). Given a set 2 c X and numbers a E R, E > 0, we define

a ,4 =

"Ef{ C F ( U ) r l ( U ) a ) ,

(1.1)

U€G

where the infimum is taken over all finite or countable subcollections B c 7 covering 2 with $(G) I E . By Condition A3 the function M c ( 2 ,a, E ) is correctly defined. It is non-decreasing as E decreases. Therefore, there exists the limit

mc(2,a)= E-+O lim M c ( Z ,a, E ) .

(1.2)

We shall study the function rnc(2,a ) .

Proposition 1.1. For any a E R the set function mc(.,a ) satisfies the following properties: (1) mc(0,a)= 0 for a > 0; (2) mc(Z1,a) I mc(Zz,a) if z 1 c zz c x ; (3) m c ( . U Zi,a) I Cmc(Zi,a),where Zi c X , i = 0 , 1 , 2 , . . . . a20

i20

Proof. The first two statements follow directly from the definitions. We shall , ~i I E prove the third one. Given 6 > 0, E > 0, and i 2 0 one can find ~ 0i < and a cover B, = {Uij E 3,j 2 0) of the set Zi with $(Bi) 5 E, such that

The collection B of sets { U i j , i 2 0 , j 2 0) covers 2 = Ui20Zi and satisfies $(B) 5 E. Now we have that

-

Since E and 6 can be chosen arbitrarily small this implies the desired result.

W

If m c ( 0 , a )= 0 (this holds true for a > 0 but can also happen for some negative a ) the set function m ~ ( a. ), becomes an outer measure on X (see definition of outer measures and other relevant information in Appendix V). We call it the a-Carathhodory outer measure (specified by the collection of subsets 3 and the set functions [,q,$). According to the general measure theory the outer measure induces a a-additive measure on the a-field of measurable sets in X (see for example, [Fe] and also Appendix V). We call this measure the a-Carathhodory measure. Note that this measure is not necessarily a-finite. We shall now describe a crucial property of the function mc(2,.) for a fixed set 2.

Proposition 1.2. There exists a critical value ac, -co 5 CYC5 +co such that mc(2,a ) = 00 for a < ac and mc(2,a ) = 0 for a > ac. Proof. It follows from A2 that if co > mc(2,a) 2 0 for some a E R then mc(2,p) = 0 for any p > a and if rnc(Z,a ) = 00 for some a E R then mc(2,p) = 00 for any p < a. This proves the desired statement. W

14

Chapter 1 Let us remark that mc(2,ac)can be 0, 00, or a finite positive number. We define the Carath6odory dimension of a set 2 c X by dimc 2 = ac = inf{a : mc(2,a ) = 0) = sup{a : mc(Z,a)= m}.

(1.3)

The CarathCodory dimension clearly depends on the choice of C-structure

T

= (7,[, 1,Ijf) on X . We will use the more explicit notation dimc,, X when we

want to emphasize the C-structure we are dealing with. Let us point out that the definition of the CarathCodory dimension does not require any assumptions on the set function [. The use of this function allows one to broaden remarkably applications of the above construction. One can obtain different examples of CarathCodory dimension of sets by using the C-structures in Examples 1, 2, and 3 above. Thus, we have respectively: (1) the Hausdorff dimension of the set 2 which we denote by d i m H 2 (see Section 6); (2) the (q, 7)-dimension of the set 2 which we denote by dim,,, 2; we define the q-dimension of the set 2 by dim, 2 = sup,,1 dim,,, 2 (see Section 8) ;

(3) the 6-topological entropy of the map f on the set 2 which we denote by -hz(f,S); we define the topological entropy of f on 2 by hz(f) = limhz(f, 6); (we show in Section 11 that if 2 is f-invariant and compact 6-0 then this definition is equivalent to the well-known definition of topological entropy).

We now state some basic properties of the CarathCodory dimension.

Theorem 1.1. (1) d i m c 0 < 0. (2) di-21 5 dimcZz i f 2 1 c 2, c X . (3) dime( ,U 2,) = supdimc Zi, where Zi c X,i = 0 , 1 , 2 , . . . . a20

i>O

Proof. The first two statements follow directly from Proposition 1.1. In order to prove the third statement assume that dimc Zi < a for all i = 0 , 1 , 2 , .. .. It follows that mc(Zi,a ) = 0 and hence by Proposition 1.1, mc(Ui>oZi) = 0. Therefore, dimc(Ui&?i) a. This implies that dimc(Ui&) supi>, dimc Zi. The opposite inequality immediately follows from the second statement of the theorem.


0, there exists EO > 0 such that for any E , 0 < E I EO one can find a cover Q of 2 with $(Q) 5 E and U€G

6(u)v(u)a 5 Mc,T(z,(Y>E) + 6.

For each U E g we choose a set U’ which satisfies Conditions (a) and (b) above. The sets U‘ comprise the cover Q’ of 2 with $ I @ ‘ ) 5 K E .We have now Mc,T’(z,

KE)

E’(U’)v’(U’)a

I U‘€P

5 Kl+O C t ( U ) v ( U ) OI Xl+=(Mc,,(Z, a , € )+ 6). U€G

Taking the limit as E

+ 0 we obtain from here that mc,T’(z, _< K1+’2(mC,T(Z,

+ &)$

Since 6 can be chosen arbitrarily small this implies that

a ) 5 xl+amc,T(z, The desired result follows immediately. We show that the Carathkodory dimension is invariant under an isomorphism which preserves the C-structure. Let X and X’ be sets endowed respectively with C-structures r = (3,[, 17, $J) and r’ = (F’, 0 ,v’,4’). Let also x:X + X’ be a bijective map. mC,T’(zI

Theorem 1.3. Assume that there exists a constant K 1 1 such that for any U E 3 one can find sets U;, Ui E 3’ satisfying (1)

u; c X(U)c u;;

(2) K-’E’(U;) I E(U) I KE’(Ui), K-’v’(U;) K-l$J’w;) I $J(U)5 K$J’(ULJ. Then dime,,, x(Z)= dimc,T2 for any 2 c X .

I v ( U ) 5 Kv’(Ui),

and

Proof. Define

3”= {X(U) : u E 3}, El’

= ( 0 x-1, 77’’ = 7 0 x-1,

$’I

= $J 0 x-1.

It is easy to see that 7’’= (F”, E l ’ , v”,$J”) is a C-structure on X’ and that r’ < r” and r” < 7’. The desired result follows from Theorem 1.2.

16

Chapter 1

2. Carathkodory Capacity of Sets Let X be a set endowed with a C-structure r = (3, E,q, +). We now modify the above construction to produce another type of Carathhodory dimension characteristics. We shall assume that the following Condition A3' holds which is stronger than Condition A3: A3'. there exists E > 0 such that for any E 1 E > 0, one can find a finite subcollection 9 c 3 covering X such that $ ( U ) = E for any U E 9. Given a E R, E > 0, and a set Z c X , define

where the infimum is taken over all finite or countable subcollections 9 C 3 covering Z such that + ( U ) = E for any U E 9. According to A3', Rc(Z,a ,E ) is correctly defined. We set Tc(z,(Y) = @ &(Z, a , & ) , P C ( z , a ) = i 6 &(Z, a,&). E-+O E-+O

The following statement describes the behavior of the functions %(Z,.) and

Pc(Z,.). The proof is analogous to the proof of Proposition 1.2.

Proposition 2.1. F o r a n y Z c X , there ezist cxc, ZC E R such that (I) %(Z,a)= co for a < cxc and rc(Z, a ) = O for a > % (while ~ ~ ac) ( 2 , can be 0 , co, O F a finite positive number); (2) P c ( ~ a, ) = co f o r a < ZC and Vc(z,a ) = 0 for (Y > E c (while P c ( Z , a c ) can be 0 , co, OT a finite positive number). Given Z c X , we define the lower and upper CarathQodory capacities of the set Z by = % = inf{a : %(Z,a ) = 0) = sup{a : ~ ~ a( ) 2 = co}, ,

CapcZ = ZC

= inf{a : ~ c ( Z , c r=) 0) = sup{a : cc(Z, a ) = cm}.

(2.1)

These quantities clearly depend on the choice of the collection 3 and the set functions (,q,$. We will use the more explicit notation -7 Cap Z and G c , , Z when we want to emphasize the structure 7 we are dealing with. One can obtain examples of the lower and upper Carathhodory capacities by exploiting the C-structures in Examples 1, 2 , and 3 in Section 1 (these Cstructures clearly satisfy Condition A3'). Thus, we have respectively: (1) the lower and upper box dimensions of the set Z which we denote by h , Z , d i m ~ Z(see Section 6); (2) the lower and upper (q,y)-box dimensions of the set Z which we denote we define the lower and upper q-box dimensions by -,?Z and dim&;

General Carathkodory Construction

17

-

of Z by dim,Z = sup7>l dim,,,Z and dim,Z = SUP^,^ dim,,?Z (see Section 8); in Section 18 we establish relations between the lower and upper q-box dimensions and dimension spectra of dynamical systems; (3) the lower and upper &capacity topological entropies of f on Z which we denote by a , ( f,6) and C h ~ ( f6); , we define the lower and upper capacity topological entropies o f f on Z by setting CJzz(f) = K g + o CJz,(f, 6) and C h z ( f ) = G g + o Chz(f,6);we show in Section 11 that if Z is invariant under f then Ch,(f) = C h z ( f ) and if Z is invariant and compact then Ch,(f)= C h z ( f ) = hz(f).

Remark. The set functions rC(.,a ) and Pc(.,a ) satisfy: (a) ~ ~ a( ) 2 0 0 ,and Fc(0,a ) 2 0 for a > 0; (b) GA&, a ) I r C ( Z 2 , a ) ,and Pc(Z1, a ) I:Tc(Z2,a ) if 21 c 2 2 c X . In general, the set function rC(.,a)is not finitely sub-additive while the set function Fc(.,a) does have this property: one can show that for any finite number of sets 2,c X, i = 1 , . . . ,n \

/ n

n

ef

Note that if k ( 2 ,a ) = Fc(Z,a ) r c ( 2 ,a ) for any Z c X then the set function r c ( . ,a ) is a finite sub-additive outer measure on X provided r c ( 0 ,a ) = 0. We now state some basic properties of the lower and upper Carathkodory capacities of sets. The proofs follow directly from the definitions.

Theorem 2.1. (1) dimc Z 5 Cap,Z 5 G c Z for any 2 c X. for any (2) caP,Z1 5 Cap,& and E C Z l I (3) For any sets Zi c X , i = 1,2,. . .

For any

E

21 c 2 2

cX.

> 0 and any set Z C X let us set

where the infimum is taken over all finite or countable subcollections B C 7 covering Z for which $ ( U ) = E for all U E 8. Let us assume that the set function 17 satisfies the following condition: A4. q(U1) = 17(U2)for any U l , U2 E 7 for which $(U1) = $(Uz). Provided this condition holds the function V ( E ) = v ( U ) if $ ( U ) = E is correctly defined and the lower and upper Carathkodory capacities admit the following description.

18

Chapter 1

Theorem 2.2. If the set function 77 satisfies ConditionA4 then for any Z

cX,

Proof. We will prove the first equality; the second one can be proved in a similar fashion. Let us put Given y > 0, one can choose a sequence E, --t 0 such that O = k ( Z ,a 7)= lirn &(Z, a 7,En).

+

It follows that Rc(Z,a such numbers n ,

+

n+w

+ 7 , ~ 5~ 1) for all sufficiently large n. A(Z, En)v(En)a+7 5 1.

By virtue of Condition A2 one can assume that n. For such n we obtain from (2.4) that

Therefore, for (2.4)

< 1 for all sufficiently large

v ( E ~ )

Therefore. Hence, aIP-7.

Let us now choose a sequence E; such that

We have that

lirn Rc(Z,a - y,EL) 1 r c ( Z ,a - y) = co.

n+m

This implies that &(Z, a - 7,EL) 2 1 for all sufficiently large n. Therefore, for such n, w,E;)v(E;)a-7 z 1, and hence, Taking the limit as n

and consequently,

-+

co we obtain that

aIP+r.

(2.6)

Since y can be chosen arbitrarily small the inequalities (2.5) and (2.6) imply that a = p. Using Theorem 2.2 one can obtain a simple but useful criterion that guarantees the coincidence of the lower and upper Caratheodory capacities of a set

zc x.

General Carath6odory Construction

19

Theorem 2.3. Assume that the set functions ( and q satisfy Condition A4 and the following conditions: (1) for any set Z c X the function A(Z,E)increases monotonically while E decreases; (2) q ( ~decreases ) monotonically with E; (3) there exists a monotonically decreasing sequence E , + 0 and a number C > 0 such that q(~,+l) 2 C ~ ( E , ) ; (4) there exists the limit

w,

1% En) *f log(l/q(En))

Then, caP,Z =

= d.

Proof. Given E > 0, choose n such that q ( ~ , + 5~ q) ( ~ 0 then &,r,o(x) I /3 and i f @< 0 then & r , a ( x ) 2 p; (2) there exists E ( Z ) > 0 such that t(U(x,~))v(U(x,~))" < 1 for any set U ( x ,E ) E 3';moreover, the function E ( X ) is measurable; (3) there exist K > 0 and EO > 0 such that for any measurable set 2 c X of positive measure and any 0 < E I EO one can find a cover = { U ( x ,E ) } c 3' of 2 for which

c

U(2,E)EP

P(U(Z,E)) 5

K.

(3.7)

26

Chapter 1

Then

p

5 p.

Remark. Condition (3.7) establishes the relation between the C-structure in X and the structure induced by the measure p. Roughly speaking it means that sets of positive measure in X admit “finite multiplicity covers” comprised from sets U ( z ,E ) E 3’. In Chapters 2-4 we will show that this condition holds for a broad class of measures.

Proof of the theorem. We consider again only the case p > 0. Fix any y E (0,pz - p) and set a = p y. Let A be the set of points z E X for which Condition A5 and Conditions 1 and 2 of the theorem hold. It follows from (3.5) and the first condition of the theorem that for any z E A there exists &I(%), 0 < E ~ ( z )I ~ ( zsuch ) that if 0 < E 5 E ~ ( z )then

+

Using the second condition of the theorem we obtain that

Given p > 0, define A,, = {z E A : E ~ ( z )2 p } . It is easy to see that Apl c A,, if p1 2 pz and A = Up>0A,. Therefore, given 6 > 0 there exists po > 0 such that p ( A p ) 2 1 - 6 for any 0 < p I po. Let us fix 0 < p 5 po and choose K and EO in accordance with Condition 3 of the theorem. Furthermore, for E such that 0 < E I rnin(E0, p } let us choose a cover = { U ( z ,E ) } c F’of the set A, which satisfies (3.7). Since for each element of this cover condition (3.8) holds we obtain

It follows that &(A,,, /3 + 7, E ) I K . Taking the upper limit as E -+ 0 yields Tc(A,,P + y) 5 K . This implies that G c A , 5 p + y. Remembering that p ( A p ) 2 1 - 6 and that 6 was arbitrarily chosen (with p = p(6) -+ 0 as 6 -+ 0) we see that Since y was also arbitrarily chosen it follows that S c p 5 0. We consider the particular case [ ( U ) = 1 for any U E are independent of a, i.e.,

F. Obviously,

dc,p,a(z) and &,,,,(z)

Moreover,

0

I dc,Jz) I & , p ( z ) .

One can now reformulate Theorems 3.1 and 3.2 in the following way.

General Carathkodory Construction

27

Theorem 3.3. Assume that ((U)= 1for any U E 3. We have (1) if there mists P 2 0 such that cjc,,(x) 2 P for p-almost every x E X and Condition 3 of Theorem 3.1 holds then dimc p 2 P; (2) if there exists ,f3 2 0 such that &,,(z) 5 P for p-almost every x E X and Condition 3 of Theorem 3.2 holds then G c p I P.

Proof. We remark that in the case we consider, the second conditions of Theorems 3.1 and 3.2 hold in view of Condition A2. Thus, if P > 0 the result follows from these theorems. If = 0 the first statement is obvious since d i m c p 2 0. The second statement follows from the observation that for any E > 0 we have dc,& I E. We now obtain lower bounds for the Carathkodory dimension of the measure and upper bounds for the upper Carathkodory capacity of the measure regardless of the choice of the subcollection 3’. Given (Y E W, set

According to Condition A5 these quantities are correctly defined at least for p-almost every x E X. If 7’ = { U ( x ,E ) } is a subcollection of 3 then

Therefore, the following theorems are immediate consequences of Theorems 3.1 and 3.2.

Theorem 3.4. Assume that there are a number b ,’ # 0 and an interval [PI, 021 such that E (01,Pz)and for p-almost every x E X and any (Y E [PI,Pz] (1) if P > 0 then ( x ) 2 P and if P < 0 then Dc,,,~( x ) 5 P; (2) the second condition of Theorem 3.1 holds. Then dimc p 2

P.

Theorem 3.5. Assume that there are a number /3 # 0 and an interval [Pl,Pz] such that /3 E (PI, Pz) and f o r p-almost every x E X and any (Y E [PI, ,821

> 0 then Dc,,,,(x) 5 P and ZfP < 0 then a , , , a ( x ) 2 P; (2) the second and third conditions of Theorem 3.2 hold. (1) i f P

Then capc p 5 0.

28

Chapter 1

4. Coincidence of Carathkodory Dimension and Carathhodory Capacity of Measures

Using results of the previous section we obtain now some sufficient conditions that guarantee the coincidence of the CarathCodory dimension characteristics relative to measures. Let us point out that the CarathCodory dimension characteristics of sets are more “sensitive”: the coincidence of them is a rare phenomenon although it can happen in some specific rigid situations (see Sections 13, 14, and 15). We fix a subcollection 3’ = { U ( Z , E )E 3 : z E X , 0 < E 5 E } as in Section 3. Let p be a probability measure on X satisfying Condition A5. Consider the lower and upper a-pointwise CarathCodory dimensions of p specified by the subcollection 3’. We first consider the case when the set function ( is trivial, i.e., ( ( U ) = 1 for any U E 3.Then according to (3.5) the quantities &,p,a(z) and &,p,a(z) do not depend on a E W but may depend on z E X and may also be different. They are also non-negative. If they are “essentially” the same and are constant, Theorem 3.3 gives us sufficient conditions for coincidence of the CarathCodory dimension characteristics of measures.

Theorem 4.1. Assume that ( ( U ) = 1 for any U E 3. Assume also that there exists /3 2 0 such that for p-almost every x E X (1) &,p,a(z) = &,p,a(z) = B; (2) Condition 3 of Theorem 3.1 and Condition 3 of Theorem 3.2 hold. Then dimc p = caP,p = Capcp = /3. We now turn to the case of non-trivial set function (. The quantities and z ~ , ~ , ~“essentially” ( x ) depend on a E W and Condition 1 of Theorem 4.1 cannot be satisfied (although Condition 1 of Theorems 3.1 and 3.2 can work well so that these theorems still give us some estimates for the CarathCodory dimension and upper CarathCodory capacity). There are also some general conditions in this case that can be used to obtain the coincidence of the CarathCodory dimension characteristics .

Theorem 4.2. Assume that there are numbers /31,/32 E R, ,f31 < /3z such that for p-almost every z E X (1) &c.rr.a(x) =

[PI’;$21

-

2‘ d(a) for

&+,a(”)

(2) the equation d ( a ) = moreover,

(Y

any a E

has a unique root

0 < d’(@) < 1 if

cy

[/31,/32] and

= ,f3

E

d ( a ) E C1 on

(/31,/32)

and

/3 # 0;

/3 > 0, d’(@) > 1 if /3 < 0;

(3) there ezists E ( Z ) > 0 such that ( ( U ( Z , E ) ) ~ ( U ( ~ 0 if p > 0 and < 0 if p < 0. We can also assume that for all a E [pl,p2], 0 < d’(a) < 1 if p > 0 and d’(a) > 1 if p < 0 (otherwise the interval [pl,p2]can be replaced by a smaller subinterval [p;,p;] 3 p for which this assumption holds). Let A be the set of points x E X for which Condition A5 and Conditions 1 and 3 of the theorem are satisfied. Denote ,82

s=

{d’(a) { max min {d’(a)

:

a E [p1,p2]} if p > 0

:

(Y

E

[pl,p2]}

if

p 0 and s > 1 if p < 0. It follows from Condition 1 of the theorem that given 7 > 0, a E [p1,@2], and x E A, there exists E~(z), 0< E ~ ( z5) ~ ( x such ) that for any E , 0 < E 5 E~(z),

Let us fix x E A, 0 < E 5 E~(z), and a number y satisfying S

O < y < - min{P-Pl,P2-P}

2 1 O < y < - min{P-P1,P2-/3} 2

Considering a = /3 - E [pl,p2]in (4.1) and putting 7 = (1 - a)y if p < 0 we obtain

ifp>O ifp 0 and

Consider the function a ( y ) = d(P We have a(0) = d(0) = ,B and a’(y) = d’(p - :)(-$). Therefore, Condition 2 of the theorem implies that for all y satisfying (4.2) a’(?)

This gives us that

2 -1

if /3 > 0 and a’(y) 5 -1

if

p < 0.

30

Chapter 1

Combining this with (4.3) and taking into account that p - $

0 - $ < 0 if /3 < 0 we have

> 0 if ,8 > 0 and

z 1.

log cL(U(Z, E N / 1% (E(U(z,E ) ) V ( U ( Z ,E ) ) P - : ) In view of Condition 3 of the theorem it follows that

Cl(U(Z,E l )

w:.

I E(Wz7 E))V(U(X,

(4.4)

Repeating the argument, presented at the end of the proof of Theorem 3.1, and using (4.4) instead of (3.6), we obtain the lower bound dimc p 2 p. We now proceed with the upper bound and prove that G c p 5 p. It follows from Condition 1 of the theorem that given > 0, a E [p1,p2], and z E A, there 0 < E ~ ( z )5 ~ ( zsuch ) that for any E , 0 < E 5 E~(z), exists E~(z),

r

Let us fix x E A, 0 < E 5 &I(%) and choose a number y satisfying (4.2). Set a =p E [PI, p2]. By virtue of (4.1) we have that if p < 0 then

+z

and if

p > 0 then

Considering the function c(y) = d(P+ $) and repeating the arguments presented above one can show that c i ( ~ + aI )

P+~ if P > O

d ( p + S ) > p + y if P < O . Using these inequalities, (4.5) and (4.6), and the fact that and p + z < 0 if p < 0 we obtain log P(U(x,E))/log (E(U(Z,E))V(U(Z, E))P++)

p + $ > 0 if p > 0

I 1.

(4.7)

It follows from (4.7) and Condition 3 of the theorem that

4Y+$I p(U(x7E l ) .

W ( Z , 4)V(U(zY

(4.8)

Now, in order to obtain the inequality G C p 5 /3 we need only to repeat the arguments presented at the end of the proof of Theorem 3.2 and to use (4.8) instead of (3.8).

31

General Caratheodory Construction

5. Lower and Upper Bounds for Carathbodory Dimension of Sets; Carathbodory Dimension Spectrum We shall use results from the previous section to produce sharp lower and upper estimates for the Carathbodory dimension of a set by considering measures supported on the set. We begin with the upper bound. Let X be a separable topological space, 3 a collection of Borel subsets of X . Assume that 7, $: 3 -+ R+ are set functions satisfying Conditions A l , A2, and A3', and E : 3 -+ R+ is a set function. Let p be a Borel probability measure on X satisfying Condition A5. We fix a subcollection 3' = { U ( x , & )E 3 : x E X , 0 < E 5 E } as in Section 3 (i.e., x E U ( Z , E )and $ ( U ( Z , E ) )= E ) and consider the lower and upper a-pointwise Carathkodory dimensions of the measure p specified by the subcollection 3'. Given a Borel set 2 and 6 > 0, we call a cover B c 3' of 2 a Besicovitch cover if for any x E 2 there exists 0 < E = ~ ( x5) 6 such that the set U ( x ,E ) E 8.

Theorem 5.1. Assume that there exist numbers 0, PO E R, 00> 0 such that for p-almost every x E X i f p 2 0 then & p , a ( x ) 5 /3 and i f /3 < 0 then &,,+(x) 2 0; there exists ~ ( x>) 0 such that E(U(x,~ ) ) q ( U (E))'" x , < 1 for any U ( x ,E ) E 3' with E 5 ~ ( xand ) any a E [p,/3 PO]if p 2 0 and a E [p - PO,p] if ,B < 0; moreover, the function E ( Z ) is measurable; there exist K > 0 and EO > 0 such that for any measurable set 2 c X of positive measure and any Besicovitch cover B c 3' of 2 with $(Q) I EO one can find a subcover 5 c B of 2 for which

+

If A is the set of points for which Condition A5 and Conditions 1 and 2 hold then dimc A 5 /3. Proof. Let us choose numbers 6 > 0 and 0 < y 5 find a set U ( ~ , EE )3' with E = ~ ( x5) 6 such that

PO. Given x

E A, one can

32

Chapter 1

if /3 < 0. The sets { U ( ~ , E: x) E X , E = ~(x)}comprise a cover 6 of A which is clearly a Besicovitch cover. Therefore, if 6 is sufficiently small then by the third condition of the theorem, one can find a subcover 5 C 6 satisfying (5.1). In the case p 2 0 it follows that

+

Hence M c ( A , p y , ~ )5 K . Passing to the limit as E + 00 we obtain that mc(A, y) 5 K . This implies that dimc A I:/3 y. Since y is arbitrary the result follows. The case /3 < 0 is considered in a similar fashion. rn

+

+

We shall now obtain a lower bound for the Carathhodory dimension of a set

2

c X in the special case [ ( U ) = 1 for any U E 7.

Given a Borel measure p on X , denote by A, the set of points for which Condition A5 holds. In this case, &,r,a(x) 2 0 for any x E A, and it does not

ef

d ~ , , ( x ) .Denote also by M ( 2 ) the set of Borel depend on a , i.e., c i c , p , a ( ~ ) measures p on X with p ( 2 ) = 1 (implicitly, we assume that 2 is measurable with respect to p ) . Put

p = sup

inf dc,,(z).

p€M(Z)x E h ~

Theorem 5.2. dimc 2 2 0. Proof. It follows from the definition of p that for any E > 0 one can find a measure p on X with p ( 2 ) = 1 such that &,,(x) 2 /? - E for p-almost every x E 2. Theorem 3.3 implies now that dimc 2 1 dimc p 2 p - E and the desired result follows. rn We still assume that [ ( U ) = 1 for any U E 3.The previous results give rise to the following notion. Given a 2 0, define

Va = {Z E A,

: d c + ( ~ c )= a}.

The function f,(a) = dimcV, is called the Carath6odory dimension spectrum specified by the measure p. Some particular examples will be given in Chapters 6 and 7. Let us notice that if p ( V a ) > 0 for some a 2 0 then f&) = a. Let M c M ( 2 ) be a subset. For any p E M ( 2 ) we have dimc p 5 dimc 2. Hence, sup dimc p 5 dimc 2.

P€M

General Carathkodory Construction

33

We say that the CarathCodory dimension of 2 admits the variational prin-

ciple (with respect to M ) if

sup dimc p = dimc 2.

PEM

We say that a measure

I/

dimension (specified by M ) if

E M is the measure of full Caratheodory

dimc v = dimc 2.

(5.4)

In the following chapters of the book we will present explicit versions of the variational principle for Carathbodory dimension when the parameters of and 11, are fixed. Examples will include the the general construction F,E,q, classical variational principles for topological pressure and topological entropy in statistical physics (see Theorem A2.1 in Appendix 11). In this case the class of measures M c M ( 2 ) consists of measures invariant under the underlying dynamical system. We will see that any equilibrium measure is the measure of full CarathCodory dimension. Therefore, the above approach enables us to obtain the “dimension” interpretation of the thermodynamic formalism of statistical physics. Another example is the variational principle f o r the Hausdorff dimension of a set invariant under a dynamical system (with M to be the collection of all invariant measures). We will establish this principle and the existence of an invariant measure of full (CarathCodory) dimension for the limit sets of some geometric constructions (see Theorem 13.1) and conformal repellers (see Theorem 20.1).

Chapter 2

C-Structures Associated with Metrics: Hausdorff Dimension and Box Dimension

We begin now to exploit the general construction described in Chapter 1 in order to produce different Carathkodory dimension characteristics. This means that we should make a choice of the parameters of the construction, i.e., the 11, which satisfy Conditions collection of subsets .F and the set functions A l , A2, A3 (or A3‘).

c,~,

In this chapter we consider the C-structure induced by the standard metric (or an equivalent one) in the Euclidean space Rm (in Appendix I we consider C-structures which are induced by metrics in general metric spaces). This structure generates Carathkodory dimension characteristics that are well-known in dimension theory, i.e., the Hausdorff dimension and lower and upper box dimensions. We describe some of their properties and present some methods for their calculation. Our aim is to demonstrate how they can be used to characterize sets and measures invariant under dynamical systems. There are two aspects of this problem. One is to use some geometric constructions in order to estimate the Hausdorff dimension and box dimension of invariant sets and measures (see Chapters 7 and 8). The second is, conversely, to use some ideas in the theory of dynamical systems in order to compute the Hausdorff dimension and box dimension of “geometrically constructed” sets (see Chapters 5 and 6 ) . One of the challenging problems in dimension theory is the problem of coincidence of the Hausdorff dimension and lower and upper box dimensions of sets. It is now accepted by most experts in the field that the coincidence is a relatively rare phenomenon and can occur only in some “rigid” situations. One well-known class of sets for which the coincidence usually takes place is the class of limit sets for some geometric constructions (see Chapter 5).

For subsets which are invariant under a dynamical system one can pose another problem of the coincidence of the Hausdorff dimension and box dimension of invariant measures. In order to explain this let us consider a map f:U + R”’ , where U c R”’ is an open domain. Assume that f preserves an ergodic Bore1 probability measure p with compact support 2 C U (i.e., 2 is a compact invariant set and p ( 2 ) = 1). The stochastic properties of the map f l Z are closely related to the topological structure of the set 2 that, in many “physically” interesting situations, resembles a Cantor-like set. The relevant quantitative characteristics, which can be used to describe complexity of the topological structure of 2, are the Hausdorff dimension and box dimension of the measure p. If they coincide the common value can be used to characterize the “fractal” structure of the invariant 34

C-Structures Associated with Metrics: Hausdorff Dimension and Box Dimension

35

set, more precisely, the part of the set where the measure is concentrated; this also gives rise to the problem of multifractality which we consider in Chapter 6. The notion of Hausdorff dimension was introduced by Hausdorff [HI in 1919 to characterize sets with “pathologically complicated” topological structures (like the famous middle-third Cantor set) but some very basic ideas behind this nction were formulated earlier by Carathkodory [C] in 1914. The properties of the Hausdorff dimension have been intensively studied for a long time, within the framework of function theory, by Besicovitch and his students and collaborators, who have made significant progress and obtained several fundamental results. (Different aspects of their study are described in [Fe, Ro]; the contemporary exposition and further references can be found in [F5]). Apparently, they knew about box dimension but did not find it sufficiently useful. In 1932, Pontryagin and Shnirel’man [PSI proved a fundamental result involving the lower box dimension (which they called the metric order). Namely, the infimum of lower box dimensions of a compact subset Z in a compact metric space X , taken over all possible metrics on X , coincides with the topological dimension of Z . Il’yashenko [Ill considered the lower box dimension (under the name entropy dimension) in connection with the study of dimension of maximal attractors for some partial differential equations. Bakhtin [B] examined some properties of the lower box dimension. Furstenberg [Fu] obtained some relations between box dimension and topological entropy for symbolic dynamical systems. The general definition of lower and upper box dimensions was given by Young [Yl,Y2], who also studied relations between the Hausdorff dimension and lower and upper box dimensions. The significance of lower and upper box dimensions is acknowledged now by experts in the field (see, for example, [F5]; the lower and upper box dimensions have several other names: lower and upper box-counting dimensions, capacities, Minkowski dimensions; there is no special reason to prefer any of them, but the name “box dimension” seems most appropriate). One of the most successful methods in dimension theory for estimating the Hausdorff dimension of sets was suggested by Frostman [Fr] and is known as the mass distribution principle. A similar (but, in a way, more general) method for estimating the Hausdorff dimension of measures was formulated by Young [Y2]. Theorem 4.2 is an extension of her result to other characteristics of dimension type.

6. Hausdorff Dimension and Box Dimension of Sets We introduce a C-structure on the Euclidean space Rm endowed with a metric p which is equivalent to the standard metric. Let F be the collection of all open subsets of E m . For U E F define

~ ( u= )1,

q ( U ) = $ ( U ) = diamU.

(6.1)

A direct verification shows that the collection of subset F and set functions q and $ satisfy Conditions Al, A2, A3, and A3‘. Hence they determine a C-structure T = (F, I ,q, $) on Rm. The corresponding CarathCodory set function r n ~ ( .a, ) (see (1.1) and (1.2)) is an outer measure for any a 2 0. It is known as the

36

Chapter 2

a-Hausdorff outer measure of 2 and is denoted by mH(2, a ) . We have for any 2 c Rm and a 2 0, mH(2, a ) = lim inf E+O

p

where the infimum is taken over all finite or countable covers B of 2 by open sets with diamB 5 E. The set function mH(., a)is an outer measure on Rm and hence it induces a a-additive measure on R"' called the a-Hausdorff measure. This measure can be shown to be Borel (i.e., all Borel sets in Rm are measurable). Moreover, it is also a regular measure (see Appendix V). Further, for a subset 2, the above C-structure generates the CarathCodory dimension of 2 called the Hausdorff dimension of the set 2. We denote it by dimH 2. According to (1.3), we have d i m H Z = i n f { a : m H ( 2 , a ) =O}=sup{a: m H ( 2 , a ) =m}. (6.3) Moreover, the above C-structure generates also the lower and upper CarathCodory capacities of 2 called the lower and upper box dimensions of the set 2. We denote them by h B Z and d i m ~ respectively. 2 According to (2.1) we have d B Z = inf{a : rB(Z,a)= 0 } = SUP{Q : K B ( Z a) , = m}, (6.4) d i m e 2 = inf{a : ? ~ ( ( z a , ) = 0) = sup{a : ? i ~ (a2),= CO}, where

and the infimum is taken over all finite or countable covers of 2 by open sets of diameter E. The set function f g ( Z , . ) can be shown to be a finite sub-additive outer measure on R"' while the set function ~ ~ .)( may 2 , not have this property (see Remark in Section 2; Example 6.2 below provides a counterexample). In the above definition of the C-structure r one can choose 3 to be the collection of all closed subsets of Rm or even all subsets of R"' to obtain the same value of the a-Hausdorff measure. If, instead, one chooses 3 to be the collection of all open or closed balls in Rm then the value of the corresponding aHausdorff measure can change but the Hausdorff dimension and lower and upper box dimensions of 2 remain the same. Obviously, the set functions q and $J satisfy Condition A4. Therefore, Theorem 2.2 produces another equivalent definition of the lower and upper box dimensions of a set 2 c Rm; namely,

where, in accordance with (2.3), N ( 2 ,E ) is the least number of balls of radius E needed to cover 2. We formulate the main properties of the Hausdorff dimension and lower and upper box dimensions of sets. They are immediate corollaries of the definitions and Theorems 1.1, 2.1, and 2.4.

C-Structures Associated with Metrics: Hausdorff Dimension and Box Dimension

/ n

37

\

(6) If the set Z is finite then dimB-Z = d-i m ~ Z = 0. = dimaz’, w h e r e 2 is the closure of the set Z. (7) b B Z = b B Z , d i m ~ Z We remark that any Lipschitz continuous homeomorphism of Rm with a Lipschitz continuous inverse preserves the C-structure 7. This fact and Theorems 1.3 and 2.5 imply the following statement. Theorem 6.3. The Hausdorff dimension and lower and upper box dimensions

are invariant with respect to a Lipschitz continuous homeomorphism with a Lipschitz continuous inverse.

We now illustrate that Statements 2, 4, 5, and 6 of Theorem 6.2 cannot be improved. The set of rational points on [0,1] is countable and hence it has Hausdorff dimension 0. By Statement 7 of Theorem 6.2 its lower and upper box dimensions equal 1. This shows that Statement 6 may not be true for a countable set of points and strict inequalities may occur in Statement 4. We present now an example which demonstrates that Statement 2 of Theorem 6.2 may fail for a set of a very simple topological structure. In this example the Hausdorff dimension of the set is zero while the lower and upper box dimensions are positive and distinct. A more sophisticated example where all three characteristics are positive and distinct is constructed in Section 16.2. In fact, one can expect the non-coincidence of the Hausdorff dimension and lower and upper box dimensions to be a typical phenomenon in a sense. Example 6.1. For given 0 < a 5 p < 1 there is a closed countable set Z C [0,1] such that h B Z = a,d i m ~ Z = /3 while dimH Z = 0.

Proof. First we need the following lemma.

38

Chapter 2

Lemma. There exist sequences of positive numbers {a,} and {b,}, n 2 1 such that ( 1 ) a, decrease monotonically and a, -+ 0; b, increase monotonically and b, 00;

(3) a =

lirn l o g s n / l o g ~ , ~n+m lim = logs,/log&, n+cu

where&=

( 4 ) there exists C > 0 such that for any n > 0

Proof of the lemma. Set a, = an, where 0 < a 5 such that a+ < M < a-l. Let

nk

n

xbk;

k=l

$ and choose a number M (6.6)

be a growing sequence of integers. Define

I n < n4k+l if n4k+i I n < W + Z Mbn-1 b n = { a-Pn if W + Z I n < n4k+3 bn4~+3-1 if n4k+3 5 n < n4k+4 a-an

We now specify the sequence choose n4k+1 so large that

Let

n4k+Z

nk.

if n 4 k

Set no = 0. Assuming that

n4k

is given we

be the smallest integer such that a-an4k+1Mn4k+2-n4k+1

> - a-pn4k+2.

This inequality is equivalent to

(aaM),4k+l 5 (aPM)n4k+2 and can be solved by an appropriate choice of

a"M Now we choose n4k+3 so large that

n4k+z

> a P M > 1.

since by (6.6),

(6.8)

C-Structures Associated with Metrics: Hausdorff Dimension and Box Dimension Let

n4k+4

39

be the smallest integer such that a-pn4k+3

5 ,-,,-an4k+4.

(6.10)

This inequality can be solved since 1 > aa > up. The definition of the numbers bn can be better understood if one imagines the motion of the point (n,bn) when n grows starting from n = n 4 k . This point first moves along the graph of the function y = a-ax until n = n 4 k + l ; then it jumps up ( n 4 k + 2 - n 4 k + l ) times with the coefficient M until it reaches the graph of the function y = e-Pz (by virtue of (6.8) this happens when n = n 4 k + 2 ) ; then it moves along this graph until n = n 4 k + 3 ; then it stays on the line y = bn4,+, until this line crosses the graph of the function y = u p a x (by virtue of (6.10) it happens when n = n 4 k + 4 ) ; then the process is repeated. Clearly, the sequences a, and b, satisfy Conditions 1-2. Condition 3 follows from (6.7) and (6.9). We have according to (6.6) that

where C > 0 is a constant. This implies Condition 4 and completes the proof of the lemma. w In order to construct the required set Z we define by induction the sets Namely, set ZO = (0) and then

Zn = Zn-1 U { p n - 1 + an,. where PO= 0, P, =

n

ak[bk] k=l

and

.

[ b k ] denotes

pn-1+

Z,.

[bnlan},

the entire part of

bk.

Let

z=n20 u ZnU{P,}, where P, =

03

xak[bk]. k=l

w e wish to show that b B Z =

(Y

and

dim~z= p.

Giyen E > 0, consider the sets

where m ( ~is)the smallest integer for which urn(c) 5 E. These sets are disjoint and Iz1- z21 2 E for any distinct z1, z 2 E T I . It is easy to see that N(T1,~ / 2 = ) Sm(,) and

where C1 > 0 is a constant. Condition 4 of the lemma implies that "T2,E/2)

I C2Srn(E),

where C 2 > 0 is a constant. Now the desired result follows from Condition 3 of the lemma. w We now show that Statement 5 of Theorem 6.2 cannot be improved.

40

Chapter 2

Example 6.2. There are two closed countable disjoint sets Z1, Zz d&B(ZI u Zz)

c W such that

> m a x { h B Z 1 , d&BZz}.

Proof. By analyzing the construction of Example 6.1 one can show that there are countable closed sets Z1 C [O,11 and Zz c [2,3] satisfying the following conditions: (a) h B Z l = b B Z z = a , where a E (0,l) is a given number; (b) given i = 1 or 2 and a sequence E~ -+0 for which the limit

exists, we have

where j = 2 if i = 1 and j = 1 if i = 2. Indeed, defining the sequence b, in the proof of the lemma, we can make the point (n,b,) start moving along the graph of the function y = while constructing the set Z1 and start moving along the graph of the function y = a-PS while constructing the set Zz. Obviously, Z1 and Zz have the required properties. W We formulate a simple but useful criterion that guarantees the coincidence of the lower and upper box dimensions of a set 2 c Wm. It is a simple corollary of Theorem 2.3.

Theorem 6.4. Assume that E, is a monotonically decreasing sequence of numbers, E, + 0, and 2 CE, for some C > 0 which is independent of n. Assume also that there exists the limit

= d. Then, a B Z = d i m ~ Z

Let E c U c Wn and F c V c W m be two Borel sets (Uand V are bounded open subsets). It was a long-standing problem in dimension theory whether the Hausdorff dimension of the Cartesian product of E and F is equal to the sum of their Hausdorff dimensions. In general, this is false, and we state some positive results in this directions (see [F5] for detailed discussion and proofs).

Theorem 6.5. (1) dimH (E x F ) 2 dimH E dimH F . (2) dimH(E x F ) 5 dimH E d i m ~ F . (3) dim~(E x F ) 5 =BE dim~F. (4) If dimH E = =BE then dimH(E x F ) = dimH E Borel set F .

+ + +

+ dimH F

for any

The C-structure defined by (6.1) (which induces the Hausdorff measure on Rm) has a trivial weight function 0 there is a finite set Z6 c Q for which p ( z 6 ) 2 1 - 6. We have dim,Za = d i m ~ Z 6= 0. Thus, dim_~p= d i m ~ p = 0.

42

Chapter 2

It follows from definitions (7.1) that

Below we will construct examples where strict inequalities occur. On the other hand, using Theorem 4.1 we will obtain a powerful criterion that guarantees the coincidence of the Hausdorff dimension and lower and upper box dimensions of measures. Consider the collection of balls

F’= { B ( x , r ): z E R”,

T

> 0).

In accordance with Section 3 (see (3.5)) we define the lower and upper pointwise dimensions of p at x (specified by F’)by

Clearly, cip(x) 5 &(x). The following statements are consequences of Theorems 3.3 and 4.1. They were first established by Young in [Y2].

Theorem 7.1. Let p be a Borel finite measure on R”. Then the following statements hold: (1) i f d,(x) 2 d for p-almost every x then dimHp 2 d ; (2) i f & ( x ) 5 d for p-almost every x then d i m ~ p5 d ; (3) zfd,(x) = &(x) = d for p-almost every x then dimHp = b B p = dimBp = d. Proof. Assume first that p is a non-atomic measure. For any set 2 of positive measure and any E > 0 there exists a finite multiplicity cover of Z by balls of radius E (see Appendix V). This implies (3.7). The validity of the second conditions of Theorems 3.1 and 3.2 is obvious. The desired result follows now from Theorems 3.3 and 4.1. If p is an atomic measure then, obviously, &(x) = d,(x) = 0 and dimH p = dimap = dimBp = 0. rn The direct calculation of the Hausdorff dimension of a set Z c R” based on its definition is usually very difficult. In order to obtain an upper estimate of the Hausdorff dimension one must present a specific “good” cover U = {Ui} of 2 with elements of small diameter for which C(diamUi)d is finite (so that d provides an upper bound for the Hausdorff dimension). In many cases a keen-minded person can guess how to choose a “good” cover with an appropriate value d (which often turns out to be the actual value for the Hausdorff dimension). Another method for obtaining an upper estimate for the Hausdorff dimension of Z is based on Theorem 5.1.

Theorem 7.2. Let p be a Borel finite measure on Rm . Assume that there exists a number d > 0 such that &(x) 5 d for every x E 2. Then dimH 2 5 d. Proof. One can easily check that the first two conditions of Theorem 5.1 hold while the third condition follows directly from the Besicovitch Covering Lemma (see Appendix V). The result follows from Theorem 5.1.

C-Structures Associated with Metrics: Hausdorff Dimension and Box Dimension

43

Theorem 7.2 is an effective tool whieh allows one to establish an efficient upper estimate of the Hausdorff dimension and it is useful in applications (see, for example, Theorem 18.3). Moreover, in view of Theorems 7.1 and 7.2, one can actually compute the Hausdorff dimension of a set Z if one can find a measure p and a number d such that p ( Z ) > 0, &(x) 1 d for p-almost every x E 2, and &(x) 5 d for every x E 2 (we will use this fact in the proof of Lemma 2 in Theorem 21.1 and Lemma 1 in Theorem 24.1). There is a stronger version of Theorem 7.2 which is often used in applications. It claims the following: Assume that there are numbers d > 0 , C > 0 , and a Borel finite measure p on Z such that for every x E Z and r > 0 p(+,

r ) )2 Crd;

then dimH Z 5 d. In order to obtain a lower estimate of the Hausdorff dimension one must work with all open covers of Z with elements of small diameter. There is an intelligent way of producing lower bounds which is based on the first statement of Theorem 7.1. Namely, assume that one can construct a Borel finite measure p concentrated on Z such that &(x) 2 d for p-almost every x E Z and some d 2 0 then dimH 2 dimHp 2 d. This simple corollary of Theorem 7.1 can be reformulated as the nonuniform mass distribution principle: Assume that there are a number d > 0 and a Borel finite measure p on Z such that for any E > 0 and p-almost every x E Z one can find a constant C ( x ,E ) > 0 satisfying for any r > 0 p ( B ( x ,r ) ) 5 C ( X , E)rd-€;

(7.3)

then dimH Z 2 d. * It is surprising that in many interesting cases one can construct a measure that satisfies a stronger version of (7.3) known as the (uniform) mass distribution principle (see [Fr]): Assume that there are numbers d > 0 , C > 0 , and a Borel finite measure p on Z such that for p-almost every x E Z and r > 0 p ( B ( x ,r)) 5 Crd;

(7.4)

then dimH 2 d. It is easier to establish the non-uniform mass distribution principle than the uniform one. On the other hand, as soon as the uniform mass distribution principle is established for a given set 2, it is usually more effective and allows one to obtain more information about the Hausdorff dimension of 2. For example, (7.4) immediately produces the positivity of the d-Hausdorff measure of 2, namely mH(2, d ) 2 1/C.

Chapter 2

44

One can construct an example of a set Z of Hausdorff dimension d for which the non-uniform mass distribution principle (7.3) holds but the d-Hausdorff measure of Z is zero (see [PWI]). In [Fr], Frostman developed another method of estimating the Hausdorff dimension of a subset Z of R"' using measures supported on 2. It is known as the potential theoretic method. Let p be a Bore1 finite measure on Rm.For s 2 0 the s-potential a t a point x E Rm,due to the measure p, is defined as

The integral

Lb.4 =

/W"

%(.)

dp(z) =

/-

W"XR"

Iz -

Y r 8 d p b ) x dp(y)

(7.5)

is known as the s-energy of the measure p. The potential principle claims (see [F5]) that for any Z c R"' (a) if there is a measure p on Z with I s ( p ) < cy) then r n ~ ( Zs), = 00; (b) if r n ~ ( 2s), > 0 then there exists a measure p on 2 with & ( p ) < 00 for all t < s. This principle can be restated as follows. Define the s-capacity of a set Z by C S ( Z )= SUP 1 : P(Z) = I } .

{

P

Then

dimH Z = inf {s : Cs(Z) = 0) = sup {s : Cs(Z)> 0).

(7.6) We now proceed with Statement 3 of Theorem 7.1. A measure p is said to be exact dimensional if it satisfies (7.3) for p-almost every x. This notion was introduced by Cutler (see [Cu]). Let us emphasize that exact dimensionality includes two conditions: (1) the limit exists almost everywhere; (2) d,(z) is constant almost everywhere.

We shall discuss the relationships between these two conditions. Let us notice that for any x E Rm the function + ( z , r )= logp(B(x,r))/logr is rightcontinuous in r for every x. Therefore, the functions d,(x) and z P ( x ) are measurable. We consider the case when p is an invariant measure for a diffeomorphism f of a smooth Riemannian manifold M .

Theorem 7.3. The functions &(x)

d J f ( 4 ) = d P ( 4a,(.)

and z P ( x ) are invariant under f, i.e.,

= ;i,(f(x)).

Proof. The statement follows immediately from the obvious implications: for any x E M and r > 0,

B(fb)lC z . 1 c f(B(xc, c B(f(x),G r ) , where C1 > 0 and Cz > 0 are constants independent of x and r .

C-Structures Associated with Metrics: Hausdorff Dimension and Box Dimension def

45 def

Thus, if p is ergodic we have &(x) = const = d ( p ) and &(x) = const = d ( p ) almost everywhere. As Example 25.3 illustrates one can construct a smooth

map of the unit interval with &(x) < &(x) for almost all x (indeed, these functions are constant almost everywhere). In Section 26 we will show that an ergodic Borel measure with non-zero Lyapunov exponents invariant under a C1+a-diffeomorphismof a smooth Riemannian manifold is exact dimensional. The assumption, that the map is smooth, is crucial: as Example 25.2 demonstrates there exists a Holder continuous homeomorphism (whose Holder exponent can be made arbitrarily close to one) with positive topological entropy and unique measure of maximal entropy whose lower and upper pointwise dimensions are different at almost every point. Theorem 7.3 is not, in general, true for continuous maps on compact metric spaces. As Example 25.1 shows there exists a Holder continuous map f preserving an ergodic Borel probability measure p such that the limit (7.7) exists almost everywhere but is not essentially constant.

Remarks. (1) As we saw above the (lower and upper) pointwise dimension allows one to estimate (and sometimes to determine precisely) the value of the Hausdorff dimension and box dimension of sets. There is an alternative approach based on the notion of density of a measure at a point which simulates the notion of the Lebesgue density (see [Fe]). Let p be a finite Borel measure on R"'. Given a point z E R"', we call the auantities

the lower and upper a-densities of the measure p at the point x. It is straightforward to check that if D,(a, x) c for every point x of a set Z c R"' of positive Geasure ( c > 0 is a constant) then &(x) 5 a. Similarly, if D,(a, x) 5 c for every point x E Z ( c > 0 is a constant) then &(x) 2 a. Thus, in view of Theorems 7.1 and 7.2, estimating the lower and upper densities provides upper and lower bounds for the Hausdorff dimension and box dimension of Z . In fact, it gives more and allows one to obtain lower and upper bounds for the a-Hausdorff measure of Z .

>

Theorem 7.4. Let Z be a Borel subset of positive measure. (1) IfD,(a,z) 5 c for p-almost every x E z then ~ H ( z a, ) 2 p ( z ) / c> 0. (2) IfD,(a,x) 2 c for every x E Z then r n ~ ( Z , a 0, consider the set z6 which consists of points x E 2 satisfying

46

Chapter 2

for all 0 < r 5 6 and some E density that

> 0. It follows from the definition of the upper

z=UZ6. 6>0

Let {U,} be a cover of Z by open sets of diameter 5 6. For each Uj containing a point x E 2 6 , consider the ball B ( x ,IUil). Since Ui c B ( x , IUil), by the definition of the set 2 6 , we obtain that

It follows that

+

Z , Since this inequality holds for all 6 This implies that p(Z6) 5 ( c & ) r n ~ (a). and E the first statement of the theorem follows. We proceed with the second statement of the theorem. Note that it is sui€icient to assume that the set Z is bounded. Given 6 > 0 and E > 0, let A be the collection of balls B ( x ,r ) centered at points x E Z of radius 0 < T 5 6 for which

It follows from the definition of the upper density that

Applying the Vitali Covering Lemma (see Appendix V) we can find a subcollection B = {Bit c A consisting of disjoint balls for which

where hi denotes the closed ball concentric with Bi whose radius is four times the radius of B;. We have that

~ ) . E is chosen arbitrarily This implies that r n ~ ( Za,) 5 Sa(c - E ) - ~ ~ ( RSince the second statement of the theorem follows.

C-Structures Associated with Metrics: Hausdorff Dimension and Box Dimension

(2) measure densities referring

47

A subset Z C Rm is called an a-set (0 2 (Y 5 rn) if its a-Hausdorff is positive and finite. One can now be interested in estimating the of the measure at points in 2. We describe a result in this direction the reader to [F5].

Theorem 7.5. Let Z be a Borel a-set. Then (1) D p ( a , z )= o for mH(.,a)-aZmost all z outside Z ; (2) 1 2 O , ( a , z )5 2a for rnH(., a)-a~mostall z E Z . Let Z be an a-set. A point z E Z is called regular if Q p ( a , z ) = D p ( a , z ) = 1. Otherwise z is called irregular. An a-set 2 is called regular if rn~(.,a)-almostevery point z E 2 is regular; otherwise 2 is called irregular. Characterizing a-sets is one of the main directions of study in the dimension theory. One of the main results claims that an a-set cannot be regular unless a is an integer (see [F5]).

-

( 3 ) Let p be the measure that is constructed in Example 7.1. We have that -

0 = dimH p = b B p = dimBp < 1 = dimH S ( p ) = b B S ( p )= d i m B ~ ( p ) ,

where S ( p ) is the support of p. Another example of a measure (which is invariant under a dynamical system), whose Hausdorff dimension is strictly less than the Hausdorff dimension of its support, is given in Section 23 (see baker’s transformations).

Chapter 3

C-Structures Associated with Metrics and Measures: Dimension Spectra

In this Chapter we introduce and study C-structures on the Euclidean space Rm induced by measures and metrics on Rm (in Appendix I we consider C-structures which are generated by measures and metrics on general metric spaces). Namely, given a measure p , we define one-parameter families of CarathCodory dimension characteristics that we call q-dimension and lower and upper q-box dimensions of the sets or measures respectively. In Chapter 6 we will demonstrate an important role that these characteristics play in the theory of dynamical systems. In particular, we will show in Section 18 that the lower and upper q-box dimensions of the support of p are intimately related to the well-known dimension spectra specified by the measure p (i.e., the Hentschel-Procaccia spectrum for dimensions and the RCnyi spectrum for dimensions). In the case when p is an invariant measure for a dynamical system f acting on Rm, the q-dimension and lower and upper q-box dimensions are invariants of f which are completely specified by p. Among them the most valuable are the correlation dimensions of order q (see Section 17).

8. q-Dimension and q-Box Dimension of Sets Let p be a Bore1 finite measure on the Euclidean space Rm . We assume that Rm is endowed with a metric p which is equivalent to the standard metric. Given numbers q 2 0 and y > 0, we introduce a C-structure on Rm generated by p and p . Namely, let 7 be the collection of open balls in Rm. For any ball B ( x ,E ) E 7 , we define

It is easy to verify that the collection 7 and the set functions 6, 77, and $ satisfy conditions A l , A2, A3, and A3'. Hence they define a C-structure in Rm, T ~= ,(7, ~t, 17, $). The corresponding CarathCodory set function rnc(2,a ) , (where 2 c Rm and a E R) is called the (q,y)-set function. We denote it by w ~ ~ , ~ ( ZBy , avirtue ). of (1.1) and (1.2) this function is given as follows:

48

C-Structures Associated with Metrics and Measures: Dimension Spectra

49

where the infimum is taken over all finite or countable covers B c F of Z by balls B(zi,~ i with ) ~i 5 E . If m,,,(@,a) = 0 (this holds true for a > 0 but can also happen for some negative a ) the set function m,,,(.,a) becomes an outer measure on E P (see Appendix V). Hence it induces a a-additive measure on Rm that we call the (q, 7)-measure. This measure can be shown to be Borel (i.e., all Borel sets are measurable; see Appendix V) . Further, the above C-structure produces the Carathkodory dimension of Z. We call it the (q,r)-dimensionof the set Z and denote it by dim,,,Z. According to (1.3), we have dim,,,Z = inf{a : m,,,(Z, a ) = 0) = sup{a : m,,,(Z, a ) = co}.

(8.3)

Moreover, the above C-structure generates also the lower and upper Carathkodory capacities of 2. We call them the lower and upper (q, 7)-box dimensions of the set Z and denote them by dim,,,Z and dim,,,Z respectively. According to (2.1) we have

dim,,,Z = inf{a : %,,(Z,a) = 0) = sup{a : r,,,(Z,a) = co),

dim,,,^ = inf{a : F,,,(z,a ) = 0) = sup{a : F,,,(Z, a ) = co),

(8.4)

where

and the infimum is taken over all finite or countable covers 0 of Z by balls of radius E . It is obvious that the set functions 17 and II,satisfy Condition A4. Therefore, Theorem 2.2 gives us another equivalent definition of the lower and upper ( 4 , ~ ) boz dimensions of sets:

where in accordance with (2.3),

(here the infimum is taken over all finite or countable covers 9 of Z by balls of radius e). The main properties of the (q, y)-dimension and lower and upper (q, y)-box dimensions of sets are listed below. They follow immediately from the definitions and Theorems 1.1, 2.1, and 2.4.

50

Chapter 3

Theorem 8.1. (1) dim,,,& 5 dim,,,& (2)

if21 c 2 2 .

( 3 ) If 2 is a finite or countable set then dim& then dim,,,{z} = 0.

5 0; i f x is a n atom of

p

Theorem 8.2. (1) dim& < dim& 5 dimq,,Z f o r any 2 c Rm . (2) dim,,,& < dim,,,& and dirn,,,Zl 5 dim,,,& if21 c 2,. (3)

( 5 ) If 2 is a finite set then dim,,,Z 5 dim,,,Z 5 0; if x is a n atom of p then dim,,,{x} = dim,,,{z} - -= 0. ( 6 ) dim,,,Z = dim,,,Z, dimq,,Z = dim,,,Z, where Z is the closure of 2.

We remark that any Lipschitz continuous homeomorphism of Rm with a Lipschitz continuous inverse that moves the measure p into an equivalent measure is an isomorphism of the C-structure T,,,. This fact and Theorems 1.3 and 2.5 imply the following statement. Theorem 8.3. The (q, 7)-dimension and lower and upper (q, 7 ) - b o x dimensions are invariant with respect to any Lipschitz continuous homeomorphism with a Lipschitz continuous inverse that moves the measure p into an equivalent measum. We describe the behavior of the functions dim,,,Z, w , , Z , and dimq,,Z over y > 0 for a fixed set 2 C Rm and a number q 2 1. First let us notice that these functions are non-decreasing, i.e., for any 0 < y1 5 y2 dim,,,,Z 5 dim,,,,Z,

dim,,,lZ < dim,,,2Z,

dim,,,,Z

-

5 dim,,,,Z.

C-Structures Associated with Metrics and Measures: Dimension Spectra

51

We now obtain a formula that allows one to compute the lower and upper 2 1. Note that the function 2 p(B(2,~ ) ) q is measurable (this can easily be seen by decomposing p into discrete and continuous parts). Since it is bounded, it is integrable. For any measurable set Z c Rm, let us set (q, 7)-box dimensions in the case q

Theorem 8.4. The following statements hold. For any q 2 1, y > 1, and any measurable set Z

For any q

2 1, y 2 1, and any measurable set

where (Z), =

Z

~

c Rm

c Rm,

u B ( z , E )is the &-neighborhood o f 2

XEZ

Proof. For any S such that

> 0 and E > 0 one can find

c

P(B(%

B ( z i ,E)EP

a cover 4 of Z by balls B ( z ~ , E )

r&))q 5 Aq,,(Z, E ) + 6.

We have that

We use here the fact that B ( z i , y ~2) B ( z ,(y - 1 ) ~for ) any z E B ( z ~ , ESince ). 6 can be chosen arbitrarily small it follows that

Aq,,(Z,4 2 PqV, (7 - 1)4. This implies the first statement. We now prove the second statement. Given E > 0, one can choose a cover 4 = { B ( z i , ~ of ) } Z by balls of the same radius E which has finite multiplicity independent of E , q, and y (see Appendix V). Since the metric in Rm is equivalent to the standard metric the cover of Z by the balls { B ( x i , y ~ )has } finite

52

Chapter 3

The desired result now follows.

As an immediate consequence of Theorem 8.4 we obtain that for any set 2 of full measure, any q 2 1, and y > 1,

(since one can replace integral over the &-neighborhoodof Z by the integral over the set Z). This gives rise to the notion of q-dimension of the set 2 and lower and upper q-box dimensions of the set Z. Namely, for q 2 0 and any set 2 c Rm, we define dim,Z = inf dim,,,Z = lim dim,,,Z, 7>l

dim,Z = inf dim 7>1-17

711

2 = lim dim 711

2,

-17

dim,^ = inf dim,,,^ = lim dim,,7~. 7 >1

(8.9)

711

These quantities have the properties established in Theorems 8.1, 8.2, and 8.3. Moreover, if one considers these quantities as functions over q 2 0 then (1) they are non-increasing: dim,,Z 2 dim,,Z, dim,,Z > dim,,Z, and dim,, Z 2 dim, Z for any 0 5 q1 I q2; (2) dimoZ = dimHZ 2 0, and h Z = a B Z 2 0, dimoZ = d i m ~ 2 0; (3) if p ( 2 ) = 0 then dim12 < dim,Z 5 dim12 I 0 and hence, dim,Z 5 -2 5 dim,Z 5 0 for any q 2 1; (4) ifp(2) > 0 then dim12 = b , Z = dim12 = 0 and hence,

0 5 dim,Z < dim,Z 5 dim,Z and dim$

< dim,Z 5 dim,2 5 0

if 0 5 q 5 1 if q

2 1.

C-Structures Associated with Metrics and Measures: Dimension Spectra

53

Properties (1) and (2) are obvious. By (8.6) we obtain for q = 1 and every y > 1 that P(Z) 5 AI,7(Z,&)5 P ( ( Z M 5 1. This implies Properties (3) and (4). One can obtain formulae for computing the lower and upper q-box dimensions of sets using equalities (8.5): for q 2 0 and 2 C Wm,

where A,,,(Z,&) is given by (8.6). Given a set 2 and a E W, we set m,(Z,a) = inf mq,y(Z,a), where 7>1

m,,,(Z,a) is defined by (8.2). The set function mq(.,a)(a is fixed) has the properties described by Proposition 1.1 and the function m,(Z, .) (Z is fixed) has the properties described by Proposition 1.2 with the critical value CYC= dim,Z. The values dim,Z, and dim,Z can be obtained in a similar fashion. Another description of the lower and upper q-box dimensions of sets for q 2 1 is based on Theorem 8.4 (see also (8.8)). Namely, for any set 2 of full measure log JRm P ( B ( ~ , & ) ) ~ - ' ~ P ( x ) > log(l/E) -log JRm P ( B ( X , E ) ) ~ - ' ~ C L ( Z ) dim,Z = lim E+O log(l/4

dim,Z=l&l E+O

q

(8.10)

One can easily see that dim,Z 5 dim,Z for every compact set 2 and every 2 1. We construct an example of a measure p for which the strict inequality

occurs on an arbitrary interval in q.

Example 8.1. For any Q > 0 there exists a finite Bore1 measure p on the interval I = [O,p],for some p > 0, such that (1) p is equivalent to the Lebesgue measure; (2) dim,I < dim,I for any 1 < q 5 Q. Proof. We first choose any three numbers a,p, y such that 0 < a < p < y < 1. Let nk be an increasing sequence of integers. Define a, = a" and

bn =

{

r-, Mbn-1

p-n

bn4k+a-l

ifn4k 5 n < n4k+l if n4k+1 5 n < n4k+2 if n4k+2 5 n < n4k+3 if n4k+3 5 < n4k+4

where M > 0 is a number satisfying M > p-l. One can choose a sequence nk such that for any Q > q > 1,

Chapter 3

54 n

where A , = C a ; b f . Since ay-' < ap-' < 1 we have that p def = i=l

C anbn < 00. O0

n=l

Let L,L be the measure on the interval I that is absolutely continuous with respect to the Lebesgue measure with the density function f(x) given by f(x) = b, if

where

n

Tn

= xu, = i=l

&.

T,

5 x 1E+O

I,

p(B(z;,ra))Q

B(Zi,E)EO

hdl/E)

where the infimum is taken over all finite or countable covers Q of Z by balls of radius E . Proof. Let us fix qo > 1. Given 0 < a < 1, let p be the measure on [0,1] which is absolutely continuous with respect to the Lebesgue measure with the density function h ( x ) given by -a

h ( x ) = (i-z) Set Z =

1 1 ifO 0 is a constant (we recall that (Z), is the &-neighborhoodof the set 2). In view of (8.5) this implies that dim,Z 5 (d - 2 a ) ( l - q ) and hence dim,p> (d - 2 a ) ( l - q). Since a can be chosen arbitrarily small it follows that dim,p 5 d(1 - q). We can choose a set Z with p ( Z ) 2 1 - 6 for which -p > dim,Z - a. If 6 is sufficiently small the set

Y has positive measure. Let any y > 1 we have

c

= zn

be a cover of Y by balls B(z;,r)with r 5

p(B(xci,rr))Q =

B(xi,r)EB

x,,@n z,

c

p. For

P ( B ( Q , yr))q-'CL(B(xi, 7r))

B(ii,r)EG

- ,(Z+-a4(9-1) P(Y). >

This implies that dim,p

2 dim,.?? - a > dim,Y - a 2 ( a + 2 a ) ( l -

q ) - a.

Since a can be taken arbitrarily small the last inequalities yield dim,p 2 a(l - q ) and the desired result follows.

60

Chapter 3

As an immediate consequence of Theorem 9.2 we have: if the measure p satisfies

dJ4

=

a,(.)

= d,(z)

p-almost everywhere then

-

dim,p = dim,p = (1 - q)essinf XEW* d,(z).

(9.4)

Appendix I

Hausdorff (Box) Dimension and q-(Box) Dimension of Sets and Measures in General Metric Spaces

Hausdorff Dimension and Box Dimension of Sets and Measures Let X be a complete separable metric space endowed with a metric p. Consider the collection of all open subsets of X and set functions t ,71, and $ defined by (6.1). They satisfy Conditions A l , A2, A3, and A3’ and hence determine a Cstructure T = (3, [, 71, Ijl) on X . For any a 2 0, the corresponding Carathkodory set function r n ~ ( . , a(see ) (1.1) and (1.2)) is an outer measure on X and is given by (6.2). It is known as the a-Hausdorff outer measure on X . This outer measure induces a a-additive measure on X called the a-Hausdorff measure. The latter can be shown to be a Borel regular measure (see Appendix V). Further, for a subset Z c Rm, the C-structure T generates the Carathkodory dimension of 2 called the Hausdorff dimension of the set Z as well as the lower and upper Carathkodory capacities of 2 called the lower and upper box dimensions of the set 2. We denote them by dimH Z, h B Z , and d i m ~ Z respectively. They obey (6.3) and (6.4). Obviously, the set functions 71 and Ijl satisfy Condition A4 and thus, the lower and upper box dimensions of Z satisfy (6.5). The main properties of the Hausdorff dimension and lower and upper box diGensions of sets are described in Theorems 6.1, 6.2, and 6.3. Let p be a Borel finite measure on X . The C-structure T produces, in accordance with (3.1) and (3.2), the Carathkodory dimension of p and lower and upper Carathkodory capacities of p. They are called respectively the Hausdorff dimension of the measure p and lower and upper box dimensions of the measure p and are denoted by dimHp, h B p , and d i m ~ p They . satisfy (7.1). Consider the collection of balls 3’ = { B ( x ,r) : x E X , T > 0) and define in accordance with (3.5) the lower and upper pointwise dimensions of p at x by (7.2). We say that X is a metric space of finite multiplicity if the following condition holds: H1. there exist K > 0 and EO > 0 such that for any E , 0 < E 5 EO one can find a cover of X by balls of radius E of multiplicity K . The analysis of the proof of Theorem 7.1 shows that it holds for any complete separable metric space X of finite multiplicity and any Borel finite measure p 61

62

Appendix I

on X . In particular, i f &(z) = &(z) = d f o r p-almost every z then dimHp = dimBp = d i m ~ p = d. Moreover, the uniform and non-uniform mass distribution principles can be used to obtain lower bounds for the Hausdorff dimension of sets. Further, we say that a complete separable metric space X is a Besicovitch metric space if the following condition holds: H2. there exist K > 0 and EO > 0 such that for any subset Z c X and any cover { B ( z , E ( z ) :) z E Z, 0 < ~ ( z 5 ) E O } one can find a subcover of Z of multiplicity K (in other words, the Besicovitch Covering Lemma holds true with respect to the metric on X ; see Appendix V). One can show that Theorem 7.2 holds for any Besicovitch metric space and any Borel finite measure on it.

q-Dimension and q-Box Dimension of Sets and Measures Let p be a Borel finite measure on a complete separable metric space X . Given numbers q 2 0 and 7 > 0, consider the collection F of open balls in X and define set functions 5, 9, and $J by (8.1). They satisfy conditions A l , A2, A3, and A3' and hence define the C-structure rq,,in X . The corresponding Carathkodory set function rnc(Z, a ) (where Z c X and a E R) is called the (q, r)-set function. It is denoted by rnq,,(Z, a ) and is given by (8.2). Further, the C-structure r,,, produces the Carathkodory dimension of 2 as well as the lower and upper Carathkodory capacities of 2. We call them the (q, 7)-dimension of the set Z and lower and upper (q, ?)-box dimensions of the set Z respectively and denote them by dim,,,Z, dim,,,Z, and dim,,,Z. They obey (8.3) and (8.4). Moreover, since the set functions 9 and $J satisfy Condition A4 the lower and upper (q, 7)-box dimensions of sets can be computed using (8.5) and (8.6). The main properties of the (q, 7)-dimension and lower and upper (q, 7)-box dimensions of sets are stated in Theorems 8.1, 8.2, and 8.3. Finally, for any q 2 0 and any set Z c X , we define the q-dimension of the set 2 and lower and upper q-box dimensions of the set Z by formulae (8.9). We say that a complete separable metric space X is isotropic if the following condition holds: H3. for every A > 0 there exists B > 0 such that for any set Z c X and any cover of Z of finite multiplicity K by balls of radius E , the cover of 2 by concentric balls of radius AE is of finite multiplicity BK. Repeating arguments in the proof of Theorem 8.4 one can show that i f X is a complete separable isotropic metric space of finite multiplicity (see Conditions (Hl) and (H3)) then f o r any set Z of full measure the q-dimension of 2 and lower and upper q-box dimensions of Z can be computed by formulae (8.10). A Borel finite measure p on X is called diametrically regular if it satisfies Condition (8.15). It is easy to see that if p is a diametrically regular measure on X , then (8.16) holds. In [GY],Guysinsky and Yaskolko proved that if a complete separable metric space X admits a Borel finite diametrically regular measure v

Hausdorff (Box) Dimension and q-(Box) Dimension in Metric Spaces

63

then X is isotropic and of finite multiplicity. In this case for any Borel finite measure p and any set Z of full measure equalities (8.8) and (8.10) hold for any y > 1 and q 2 1. If v is a Borel finite measure on X then the C-structure T,,, defined above (and specified by the measure p ) yields, in accordance with (9.1), the (q,y)dimension of the measure v and the lower and upper (q,y)-box dimensions of the measure v. We denote them by dim,,,v, dim,,,^, and dim,,,v respectively. One can show that if X is a complete separable metric space of f i nite multiplicity and p is a Borel finite measure on X satisfying Condition (9.3) then f o r every q 2 0, dim,,, p = dim,,,p = dim,,+ = d ( l - q). Moreover, f o r every q 2 1, the conclusion of Theorem 9.2 holds.

Chapter 4

C-Structures Associated with Dynamical Systems: Thermodynamic Formalism

Let f be a continuous map acting on a compact metric space ( X , p ) with metric p and cp a continuous function on X . In this chapter we discuss the notion of topological pressure of cp on a subset 2 C X (specified by f). This notion was brought to the theory of dynamical systems by Ruelle [Rl], who was inspired by the theory of Gibbs states in statistical mechanics. Ruelle considered only the case when the set 2 is compact and f-invariant (he also assumed that f is a homeomorphism which separates points; the case of general continuous maps was later studied by Walters [W]). The fact that the topological pressure is a characteristic of dimension type was first noticed (implicitly) by Bowen [Boll. Pesin and Pitskel’ [PP]further developed his approach and extended the notion of topological pressure to arbitrary subsets of X which are not necessarily invariant or compact. In this chapter we systematically use the “dimensional” approach to the notion of topological pressure which is based on a modification of the general Carathkodory construction (we describe the modified version in Section 10). The topological pressure is a key notion in the thermodynamic formalism (see Appendix 11) which is the main tool in studying dimension of invariant sets and measures for dynamical systems and dimension of Cantor-like sets in dimension theory. The “dimension” approach that is faithful to the general Carathkodory construction gives us a new insight on the thermodynamic formalism and allows us to extend the classical notion of topological pressure to non-compact or noninvariant sets. This is an important advantage which we will use in studying dimension. Furthermore, we will use the “dimension” approach to obtain a more general non-additive version of the topological pressure. It was introduced by Barreira in [Bar2]. The associated non-additive thermodynamic formalism is a powerful tool to study dimension of Cantor-like sets with extremely complicated geometric structure where other methods of study failed to work. Let us outline the “dimension” approach. Given a finite open cover U of X , we introduce a C-structure T on X which is specified by the map f , metric p, continuous function cp, and cover U. According to Section 10, for any subset Z c X , this C-structure generates the Carathkodory dimension and lower and upper Carathkodory capacities of Z. We denote them respectively by Pz(cp,U),Cpz(cp,U), and mz(cp,U). We then show that these quantities have limits as diamU tends to zero which we call the topological pressure and lower and upper capacity topological pressures of cp on 2 and denote them by Pz(cp),Cpz(cp),and mz(cp) respectively. We show that, if Z is f invariant, then Cpz(cp)= mz(cp) and if, in addition, 2 is compact, then 64

C-Structures Associated with Dynamical Systems: Thermodynamic Formalism

65

Pz(’p) = Cp,(cp)

= -z(cp). In the latter case, the common value coincides with the “classical” topological pressure introduced by Bowen, Ruelle, and Walters.

We stress that for an arbitrary subset Z , one has three, in general distinct, quantities Pz(cp),Cp,(cp), and m z ( ( p )to be used as a generalization of the classical notion of topological pressure. In view of the variational principle (see Appendix I1 below) a crucial role is played by the quantity Pz(cp)while lower and upper capacity topological pressures are also often used in computing dimension of invariant sets for dynamical systems (see Chapters 5 and 7). There is an important particular case when cp = 0. We call the quantities Pz(O),Cp,(O), and m z ( 0 ) the topological entropy and lower and upper capacity topological entropies and we use the generally accepted notations h f ( Z ) ,C&(Z), and C h f ( 2 ) .The topological entropy is a well-known invariant of dynamical systems and plays a key role in topological dynamics. For example, for subshifts of finite type (and hence for Axiom A diffeomorphisms) the topological entropy is the exponential growth rate of the number of periodic points. The first definition of the topological entropy for compact invariant sets was given by Adler, Konheim, and McAndrew [AKM]. In [Boll, Bowen extended it to non-compact invariant sets and pointed out the “dimensional” nature of this notion. We emphasize that the straightforward generalization of the AdlerKonheim-McAndrew definition of the topological entropy for non-compact sets leads to the quantities mf(Z)and C h f ( Z ) . On the other hand, we show that Bowen’s topological entropy coincides with h f ( 2 ) . Let p be a Bore1 probability measure on X . In Section 10 we will show that the C-structure 7 generates the CarathCodory dimension and lower and upper CarathCodory capacities of p. We denote them by Pp(cp,U),Cp,(cp,U), and CP,(cp,U) respectively. We show that these quantities have limits as diamU cpdp, where tends to zero and these limits coincide and are equal to h,(f) h,(f) is the measure-theoretic entropy of f. The expression h,(f) cpdp is the potential function in the variational principle (see below Appendix 11). In the case cp = 0 our approach produces, in particular, a “dimension” definition of the measure-t heoretic entropy.

+ sx + sx

10. A Modification of the General Carathhodory Construction We describe a modification of a general CarathCodory construction. Let X and S be arbitrary sets and 3 = {Us: s E S} a collection of subsets in X . We assume that there exist two functions 77, $: S + IK+ satisfying the following conditions:

A l . there exists so E S such that Us,= 0 ; if Us = $(s) = 0; if Us# 0 then ~ ( s > ) 0 and $(s) > 0;

A2. for any 6 > 0 one can find $(s)

IE;

E

> 0 such

that q ( s )

0

then ~ ( s )= 0 and

5 6 for any

s

E S with

66

Chapter 4

A3. for any E > 0 there exists a finite or countable subcollection B covers X (i.e., U Us3 X ) and $@) Ef sup{$(s) : s E S} 5 E .

c S which

S€G

Let (:S+ R+ be a function. We say that the set S, collection of subsets 7 , and the functions El r], $, satisfying Conditions A l , A2, and A3, introduce the Carathhodory dimension structure or C-structure r on X and write

= (S17,E, r ] , $1. If the map s H Usis one-to-one then the functions E, r ] , and $ can be considered as being defined on the set F and thus, the above C-structure coincides with the C-structure introduced in Section 1. In the general case one can still follow the approach, described in Chapter 1, to define the CarathCodory dimension and 7-

lower and upper CarathCodory capacities generated by the C-structure. We shall briefly outline this approach. Given a set 2 c X and numbers a E R, E > 0, we define

where the infimum is taken over all finite or countable subcollections B C S covering 2 with $(P) 5 E . By Condition A3 the function M c ( 2 ,a,E ) is correctly defined. It is non-decreasing as E decreases. Therefore, the following limit exists:

One can show that the function mc(2,a)satisfies Propositions 1.1 and 1.2. We define the Carathhodory dimension of the set 2 by (1.3). It has prop erties stated in Theorem 1.1. Let X and X' be sets endowed with C-structures 7- = (S,7 ,El r ] , $) and r' = (S,7',E', r]', $J') respectively. One can show that the CarathCodory dimension of sets is invariant with respect to a bijective map x:X + X' which preserves the C-structures T and 7' (compare to Theorem 1.3). We shall now assume that the following condition holds: A3'. there exists E > 0 such that for any 0 < E 5 E there exists a finite or countable subcollection 0 c S covering X such that +(s) = E for any s E 8. Given a E R and E > 0, let us consider a set 2 c X and define

where the infimum is taken over all finite or countable subcollections B C S covering 2 such that $(s) = E for any s E 0. According to A3', Rc(2,a , E ) is correctly defined. We set

& ( Z , a ) = l&Rc(Z,a,E), E-+O

Pc(2,a)= ~ o R c ( Z , a , E )

C-Structures Associated with Dynamical Systems: Thermodynamic Formalism

67

One can show that the functions rC(Z, a ) and Fc(2,a ) satisfy Proposition 2.1. We define the lower and upper Carathdodory capacities of the set 2, =,Z and =,Z, by (2.1). They have properties stated in Theorem 2.1. For any E > 0 and any set Z c X , let us put

where the infimum is taken over all finite or countable subcollections covering Z for which $(s) = E for all s E P. Let us assume that the function 17 satisfies the following condition:

B cS

any s1, s2 E S for which $(sl) = $(s~). One can now correctly define the function V ( E ) of a real variable E by setting V ( E ) = ~ ( s if ) $(s) = E . One can prove that, provided Condition A4 holds, the lower and upper Carathdodory capacities of sets satisfy Theorem 2.2, 2.3, and 2.4. Let X and X‘ be sets endowed with C-structures r = ( S , F , [ , q , $ )and r’ = (S,F’,t‘,q’,$’) respectively. One can show that the lower and upper Carathkodory capacities of sets are invariants with respect to a bijective map x:X -+ X’which preserves C-structures 7 and 7’(compare to Theorem 2.5). Let ( X , p ) be a Lebesgue space with a probability measure p endowed with a C-structure 7 = (7,[, v,$). Assume that any set UsE 7 is measurable. We define the Carathdodory dimension of the measure p , dimH p , andlower and upper Carath6odory capacities of the measure p , c a P , p and Cap,p, by (3.1) and (3.2) respectively. We have that

A4.

~(51) = ~ ( s 2 )for

We shall now assume that the following condition holds:

A5. for p-almost every x E X and any E > 0, if s E S and Us3 x is a set with I E then p(Us) > 0 and [(s) > 0. For each point x E X and a number E , 0 < E 5 E , we choose s = S ( X , E ) E S

$(s)

such that x E Usand $(s) = E (this is possible in view of Condition A3‘). Once this choice is made we obtain the subcollection

S’ = { S ( X , E ) E S : x E x,0 < E 5

E}.

Given a E W and x E X , we define now the lower and upper a-Carath6odory pointwise dimensions of p at x by

68

Chapter 4

(see (3.5)). We have that &,p,a(z) 5 &,p,a(z) for any z E X. It is a simple exercise to prove that the conclusion of Theorems 3.1, 3.2, 3.3, 4.1, 4.2, 5.1, and 5.2 hold (with obvious modifications in the formulations). We also define

(10.1) +(a)=€

It is easy to check that conclusions of Theorem 3.5 and 3.6. hold.

11. Dimensional Definition of Topological Pressure; Topological and Measure-Theoretic Entropies

Topological Pressure Let (X, p ) be a compact metric space with metric p, f : X + X a continuous map, and 'p: X -+R a continuous function. Consider a finite open cover U of X and denote by Sm(U)the set of all strings U = {Ui, . . .Vim-, : Uij E U} of length m = m ( U ) . We put S = S(U)= Um~oSm(U). To a given string U = {Ui, . . .Ui,,,-l} E S ( U ) we associate the set

X(U)= {z

E

x : fj(z) E uij f o r j = 0 , . . . , m ( ~-)1).

(11.1)

Define the collection of subsets

3 = F ( U ) = { X ( U ):

u E S(U)}

(11.2)

and three functions E, 7,qb: S ( U ) + R as follows

(11.3)

It is straightforward to verify that the set S,the collection of subsets 3,and the functions rl, E , and 11, satisfy Conditions A l , A 2 , A 3 , and A3' in Section 10 and hence they determine a C-structure T = T(U)= (S,F ,E , 77, qb) on X. The corresponding Carathkodory function mc(Z,a) (where Z c X and a E W; see Section 10) depends on the cover U (and the function 'p) and is given by m c ( 2 ,a ) = lim M ( 2 , a, 'p,U,N ) , N-tm

C-Structures Associated with Dynamical Systems: Thermodynamic Formalism

69

and the infimum is taken over all finite or countable collections of strings G c S ( U ) such that m ( U ) 2 N for all U E G and 9 covers 2 (i.e., the collection of sets { X ( U ): U E G } covers 2). Furthermore, the CarathCodory functions rc(Z, a ) and T c ( 2 , a ) (where 2 c X and a E R; see Section 10) depend on the cover U and are given by zc(2,a) = Nlim R(Z,a,cp,U,N), Fc(2,C-u) = N+CC lim R(Z,a,cp,U,N), -rm

where

and the infimum is taken over all finite or countable collections of strings G C S ( U ) such that m ( U )= N for all U E G and 0 covers 2. According to Section 10, given a set 2 c X, the C-structure T gener-

ates the CarathCodory dimension of 2 and lower and upper CarathCodory capacities of 2 specified by the cover U and the map f. We denote them by Pz(cp,U),C p z ( c p , U ) ,and cpz(cp,U) respectively. We have that (compare to (1.3) and (2.1))

Pz(cp,U)= inf{a : m c ( Z , a )= 0} = sup{a: m c ( 2 , a )= co}, Cp,(cp,U) = inf{a : ~ ~ a( ) = 2 0,} = sup{a : ~ ~ a( ) = 2 co}, ,

-

CPz(cp,U)= inf{a

Let

: FC(Z, a ) = 0} = sup{a : f c ( 2 ,a ) = co}.

IUI = max{diamUi : Ui c U} be the diameter of the cover U.

Theorem 11.1. For any set 2 C X the following limits exist:

Proof. Let V be a finite open cover of X with diameter smaller than the Lebesgue number of U. One can see that each element V E V is contained in some element U ( V ) E U.To any string V = {KO. . .K,} E S ( V ) we associate the string U(V)= { U ( K o ) .. .U(K,,,)} E S ( U ) . If G c S ( V ) covers a set 2 c X then U(G)= {U(V) : V E G } c S ( U ) also covers 2. Let

y = y(U) = sup{ Icp(z) - cp(y)l : 2,y One can verify using (11.4) that for every a E

E

U for some U

E U}.

lR and N > 0 (11.5)

Chapter 4

70 This implies that

Pz(cp,U)- Y I Pz(cp,V ) .

Since X is compact it has finite open covers of arbitrarily small diameter. Therefore, Pz(cp,U) - Y I lim Pz(cp,V). IVI+O

If JUI-+ 0 then y(U)-+ 0 and hence

This implies the existence of the first limit. The existence of two other limits can be proved in a similar fashion by using the inequality which is an analog of (11.5) (11.6) 1" a,cp, U ,N ) I WZl - Yl cp, Vl N ) . rn This completes the proof of the theorem. We call the quantities Pz(cp),Cp,(cp), and m z ( p ) , respectively the topological pressure and lower and upper capacity topological pressures of the function cp on the set Z (with respect to f). Sometimes more explicit notations Pz,f(cp),Cpz,f(cp), and m z , f ( ' p )will be used to emphasize the dependence on the map f . We emphasize that the set 2 can be arbitrary and need not be compact or invariant under the map f. If f is a homeomorphism then for any set Z C X its topological pressure coincides with topological pressure on the invariant hull of Z (i.e., the set f n ( Z ) ; this follows from Theorem 11.2 below). However, this

u

nEZ

may not be true for lower and upper capacity topological pressures (see Example 11.2 below). We formulate the basic properties of topological pressure and lower and upper capacity topologital pressures. They are immediate corollaries of the definitions and Theorems 1.1 and 2.1.

Theorem 11.2. (1) P0(cp) 5 0 . (2) Pz,( 9 )I Pz,(cp) if 21 c z2 c (3) Pz(cp)= supi2l Pz,(cp), where 2 = Ui21Zi and Zi c X , i = 1,2,. . . . (4) Iff is a homeomorphism then Pz(cp)= Pfp)(cp).

x.

Theorem 11.3. (I) Cp0(cp) 5 0 , cP0((0)5 0 (2) c p z l (cp) I Cpz, (cp) and CPZ, ( 9 )I m z ,(cp) if z1 c 2 2 c (3) Cpz(cp) 2 SUPi2l cpz,(cp) a n d C P z ( c p ) 1 SUP,21 cpz,(cp), where z = UillZi and Zi c X , i = 1 , 2 , . . . . (4) If h : X -+ X is a homeomorphism which commutes with f (i.e., f o h = h o f ) then

x.

PZ(cp) = Ph(Z)(P0 h-IL Cpz(cp) = c p , ( Z ) ( c p O

h-l),

cpz(cp) = ~

h ( Z ) ( ' p h-l) O

C-Structures Associated with Dynamical Systems: Thermodynamic Formalism

71

Obviously, the functions 11 and $ satisfy Condition A4 in Section 10. Therefore, by Theorems 2.2 and 11.1, we have for any 2 C X that

1 CpZ(p)= lim lim -log IuI-+o N z N

A(Z,cp,U,N), (11.7)

where in accordance with (2.3), ( l l . l ) , and (11.3)

and the infimum is taken over all finite or countable collections of strings

S(U)such that m ( U ) = N for all U E B and 0 covers 2.

Bc

We also point out the continuity property of the topological pressure and lower and upper capacity topological pressures.

Theorem 11.4. For any two continuous functions cp and $ on X

where 11. 1 1 denotes the supremum norm in the space of continuous functions on X .

Proof. Given N

> 0, we have that N- 1

It follows that

This implies that

and concludes the proof of the first inequality. The proof of the other two inequalities is similar.

Chapter 4

72

One can easily see that

Below we will give an example where the strict inequalities occur (see Examples 11.1 and 11.2). The situation for invariant and compact sets is different. Theorem 11.5. (1) For any f-invariant set 2 C X we have Cpz(cp)= mz(cp);moreover, for any open coverU of X , we have c P , ( c p , U ) = c P z ( p , U ) . (2) For any compact invariant set 2 C X we have Pz(9) = Cpz(cp)= mz((p); moreover, for any open cover U of X , we have Pz(cp,U) = Cpz(rP,U) = c P z ( c p , U ) .

Proof. Let 2 c X be an f-invariant set. Choose two collections of strings G, c Sm(U)and Gn c Sn(U)which cover 2 and consider

Since 2 is f-invariant the collection of strings G,, estimate A ( 2 ,p , U , m n) using (11.8). We have

+

/

m-1

also covers 2. We wish to

/

\

n-1

This implies that

A ( 2 ,cp, U ,m + n) I h(2,cp, U ,m) x N Z , cp, U ,n). Let a , = logh(2, cp,U, m). Note that h(2,cp,U, m) 2 e-mllpll. Therefore, inf,?l 2 -llcpll > -m. The desired result is now a direct consequence of (11.7) and the following lemma (we leave its proof to the reader; see Lemma 1.18 in [ B o ~ ] ) .

2

2

Lemma. Let a,, m = 1 , 2 , . . . be a sequence of numbers satisfying inf,21 > -m and am+, 5 a , a, for all m, n 2 1. Then the limit limm+co exists and coincides with inf,>l

+

2.

Choose any a > Pz(cp,U). There exist N covers 2 and

2

> 0 and Q c S(U)such that G

C-Structures Associated with Dynamical Systems: Thermodynamic Formalism

73

Since 2 is compact we can choose 6 to be finite and hence

for some M 2 1. Put

UG". 00

B n = { U l . . .Un : U i ~ G } a n d I ' =

n=l

Since 2 is invariant r covers 2. It is a simple exercise (which we leave to the reader) to check that for every n > 0,

Q(z,a , Gn)

5 Q ( Z ,a , 6)".

Hence, n=l

Let us fix some N > 0 and consider a point x E 2. Since r covers 2 there exists a string U E I? such that x E X ( U ) and N 5 m(U) < N M . Denote by U' the substring that consists of the first N symbols of the string U . We have that

+

N-1

SUP

YEX(U') k=O

If

I'N

c cp(fk(Y)) + Mllcpll.

m(U)-l

Ccp(f"Y)) 5

SUP

k=O

denotes the collection of all substrings U* constructed above then N-1

By (11.7) we obtain that a (11.9).

> m z ( c p ) ,and hence the desired result follow

"

1

Theorem 11.5 shows that for a compact invariant set Z the topological pressure and lower and upper capacity topological pressures coincide and the common value yields the classical topological pressure (see, for example, [ B o ~ ] ) .It is worth pointing out that this common value is a topological invariant (i.e., Px(cp) = Px(p o h ) , where h is a homeomorphism which commutes with f ) . This means that the pressure does not depend on the metric on X . If a set 2 is neither invariant nor compact one has three, in general distinct, quantities: the topological pressure, Pz(cp),and lower and upper capacity topological pressures, Cpz(cp)and mz(cp). The latter coincide if the set 2 is invariant and may not otherwise (see Example 11.1 below). Furthermore, they are defined by formulae (11.7) and (11.8) which are a straightforward generalization of the classical definition of the topological pressure. In view of the

74

Chapter 4

variational principle, the topological pressure PZ(9) seems more adapted to the case of non-compact sets and plays a crucial role in the thermodynamic formalism (see Appendix 11).

Remarks. (1) We describe another approach to the definition of topological pressure. Let ( X , p ) be a compact metric space with metric p , f : X X a continuous map, and 'p:X -+ R a continuous function. Fix a number 6 > 0. Given n > 0 and a point x E X , define the (n, 6)-ball at x by

B,(z,6) = {y E x : p(fi(x),fi(y )) 5 6, for o 5 i 5 n),

(11.10)

Put S = X x N. We define the collection of subsets

F = {B,(z,6) : 2 E x,n E N} and three functions c,r],+t!x S --t R as follows

r](x,n ) = exp(-n),

+(x, n) = n-I

One can directly verify that the set S, the collection of subsets F ,and functions 7,6, and $ satisfy Conditions A l , A2, A3, and A3' in Section 10 and hence der ] , $) on X . According to Section 10, given termine a C-structure -r = (S,F,E, a set 2 c X , this C-structure generates the Carathbodory dimension of 2 and lower and upper Carathkodory capacities of 2 which depend on 6. We denote 6) respectively. them by Pz('p,6), CP,(p, 6 ) , and mz('p, Let U be a finite open cover of X and 6(U)its Lebesgue number. It is easily seen that for every x E X , if z E X ( U ) for some U E S(U) then

It follows now from Theorem 11.1 that

(2) If the map f : X -+ X is a homeomorphism we can consider the topological pressure and lower and upper capacity topological pressures for the map f as well as for the inverse map f - l . If Z is an invariant subset of X then for any continuous function 'p: X -+ R,

C-Structures Associated with Dynamical Systems: Thermodynamic Formalism

75

These equalities hold no matter whether Z is compact or not but may fail to be true if Z is not invariant (see Example 11.3 below). If 2 is invariant and compact then in addition we have that

and this may fail if Z is not compact (although still invariant; see Example 11.3 below).

Topological Entropy We consider the special case cp = 0. Given a set Z

c X , we call the quantities

respectively, the topological entropy and lower and upper capacity topological entropies of the map f on Z. We stress again that the set Z can be arbitrary and need not be compact or invariant under f. It follows from (11.9) that (11.12) h z ( f ) I Chz(f)5

mm.

If the set 2 is f-invariant, we have

By (11.7) we obtain for an invariant set 2 that 1

C h z ( f )= lim lim -log A ( Z , O , U , N ) I W + O N-too N = lim

1

& -log A ( Z , O , U , N ) ,

(UI+O N + w

(11.13)

N

where, in accordance with (11.8), A(Z, O,U, N ) is the smallest number of strings U of length N , for which the sets X ( U ) cover Z . Formula (11.13) reveals the meaning of the quantity C h Z ( f ) :it is the exponential rate of growth in N of the smallest number of strings U of length N , for which the sets X ( U ) cover Z . For a compact invariant set Z we have by Theorem 11.5, that h z ( f ) = Chz(f) = ChZ(f). The topological entropy and lower and upper capacity topological entropies have properties stated in Theorems 11.2 and 11.3 (applied to cp = 0). In particular, they are invariant under a homeomorphism of X which commutes with f . We now proceed with the inequalities (11.12). In examples below we consider the symbolic dynamical system (C,, u),where C, is the space of two-sided infinite sequences on p symbols and u is the (two-sided) shift. We recall that the cylinder set C;,...;, consists of all sequences w = ( j k ) for which j , = i, . . . ,j , = it (see more detailed description in Appendix I1 below).

76

Chapter 4

Example 11.1. There exists a compact non-invariant set Z C C3 for which C h z ( a )< C h z ( a ) .

Proof. Let nk be a strictly increasing sequence of integers. Define the set Z={~=(w,)EC3:w,=lor

2 ifn~e 0 is a constant independent of n. Applying (11.13) with U = Urn (and (UI+ 0 as m + 0) yields

CJzZ(a)= l0g2,

-

C h z ( a )= 10g3.

The desired result follows.

Example 11.2. There is an invariant set 2 c C2 for which hZ(u) < ChZ(a). Proof. Define the sets

It is easy to see that the set Z is invariant and everywhere dense in C2. Therefore, C h z ( a )= Chc, ( u )= log 2. Given m 2 0, consider the cover Urn of C2 by cylinder sets Ci-,...;, . It is easy to see that A(Zk,O,UmLN) 5 C , where C > 0 is a constant independent of N. Therefore, Chz, (a)= ChZ, (a)= 0 and hence hz, (a)= 0 for all m. This implies that h z ( a )= 0.

C-Structures Associated with Dynamical Systems: Thermodynamic Formalism

77

Note that the set Z in this example is the invariant hull of the set 20 and

Chzo(0)= Chz, (u)= 0 while Chz(u)= log 2. Example 11.3.

(1) There is an invariant (non-compact) set Z C CZ f o r which hZ( u) = log 2 while h Z ( 0 - l ) = 0. (2) There is a compact (non-invariant) set Z C Cz for which

Proof. Define the sets ~k = { w = (w,)

E

CZ :

w, = 1 for all n

5 k}, z =

UZ~. k€Z

Obviously, the set Z is invariant (but not compact) and the set 20is compact (but not invariant). We leave it as an exercise to the reader to show that Z fulfills requirements in Statement 1 and so does 20 in Statement 2. Remark. Let U be a finite open cover of X . Given a set Z

c X , the quantities -

def -

h Z ( f , U ) Ef p z ( O , U ) , C J L z ( f , U )gfC p z ( O , U ) , C h z ( f , U ) = C P z ( 0 , U ) are called the topological entropy and lower and upper capacity topological entropies o f f on Z with respect to U. By Theorem 11.5, if Z is invariant, then a z ( f , U ) = C h z ( f , U ) EfC h Z ( f , U ) and if, in addition, Z is compact then h Z ( f , U ) = C h z ( f , U ) . Let V be a finite open cover of X whose diameter does not exceed the Lebesgue number of U.Applying (11.5) with 'p = 0 we obtain that

hx(f,W 5 hx(f,V). In [BGH], Blanchard, Glasner, and Host obtained a significantly stronger statement. Namely, let [ be a finite Borel partition of X such that each element of [ is contained in an element of the cover U. Then there exists an f -invariant ergodic measure p on X for which h x ( f , U ) 5 h p ( f , E ) . Measure-Theoretic Entropy Let p be a Borel probability measure on X (not necessarily invariant under f ) . Consider a finite open cover U of X . According to Section 10, the C-structure 7 = ( S , F , [ , v , $ )on X,introduced by ( l l . l ) , (11.2), and (11.3), generates the Carathkodory dimension of p and lower and upper Carathkodory

78

Chapter 4

capacities of p specified by the cover U and the map f. We denote them by P,(cp,U), Cp..(cp,U), and cp,(cp,U) respectively. We have that

P,(cp,U) = inf{Pz((p,U) : p ( Z ) = I},

CP,(cp,U) -

= 6+0 lim inf {Cpz(cp,U) : p ( Z ) 1 1 -a},

CP,((P,U)=

(11.14)

inf{CPz(cp,U) : p ( z ) 2 1 - 6 ) .

It follows from Theorem 11.1 that there exist the limits

(11.15)

Given a point x E X , we set in accordance with (10.1)

where the infimum and supremum are taken over all strings U with x E X ( U ) and m(U)= N .

Proposition 11.1. If p is a Bore1 probability measure on X invariant under the map f and ergodic, then for every a E R and p-almost every x E X

where h,(f) is the measure-theoretic entropy o f f .

Proof. We need the following statement known as the Brin-Katok local entropy formula (see [BK]). Lemma. For p-almost every x E X we have

where Bn(x, 6 ) is the ( n ,6)-ball at x (see (11.10)).

Proof of the lemma. For the sake of reader's convenience we present a simplified version of the proof in [BK] which exploits the fact that the measure

C-Structures 'Associated with Dynamical Systems: Thermodynamic Formalism


0 and consider a finite measurable partition with max{diamCc : Cc E 0 and a set A2 c Y such that p(A2) 2 1 - S and for any y E A2 and n 2 N z ,

5 exp(-(h,(f, E ) - @n), (A2.3) where C,,, (y) denotes the element of the partition En containing y. At last, using the Birkhoff ergodic theorem one can find N3 > 0 and a set A3 C Y such that p(A3) 2 1 - 6 and for any y E A3 and n 2 N3, P(C€?% (Y))

(A2.4) Set N = max(N1, N2, N3) and A = A1 n A2

n As. We have that

p ( A ) 2 1 - 36.

(A2.5)

Choose any n 2 N and any

A h , ( f , J ) + J cpdp-22~-2~1ogm-y(U)-A>O. Y

By (A2.6) we have that M ( Y ,A, cp,U, N ) 2 1 - 46 2 1/2 if sufficiently small. Therefore, Py(cp,U) > X and hence

Wcp,U)

z h,(f,E)

+

/

Y

E

(and hence 6) is

cpdp - r(U).

Letting E + 0 yields y(U)+ 0 and diamJ -+ 0. The latter also implies that h,(f, J) approaches h,(f) and (A2.1) follows. Consider a Borel f-invariant subset 2 c X . Given a measure p E m(Z) denote by Z, = {z E Z : V(z) = { p } } . It is easy to see that p(Z,) = 1 and that Z, c L(2).Therefore, by (A2.1), PL(Z,(cp) 2

PZ,(cp)

z h,(f)

+

/

Z

PdP.

(A2.11)

We now prove that for any Borel f-invariant (not necessarily compact) subset 0 for any z E Y we have

Y c X with the property that V(z) n m(Y)#

Let E be a finite set and

= (ao, . . . ,ak-1) E

po(e) = :(the

k

Ek.Define the measure pa on E by

number of those j for which aj = e).

Variational Principle for Topological Pressure

91

The following statement describes the asymptotic growth in k of the number of elements in the set R(k, h, E ) . The proof is based on rather standard combinatorial arguments and is omitted.

Lemma 3. (See Lemma 2.16 in [Boll). We have -1 lim -logIR(k,h,E)I 5 h. k

k-tm

Let U = { U l , . . .,U,.} be an open cover of X and

E

> 0.

Lemma 4. Given x E Y and p E V ( x )n 3n(Y), there exists a number m > 0 such that for any n > 0 one can find N > n and a string U E S(U)with m(U) = N satisfying: (1)

2

EX(V;

(3) the string U contains a substring U' of length m(U') = km 2 N - m which, being written as g = (ao,. . . ,ak-l), satisfies the inequality 1

m H ( a ) I h,(f)

+

E.

(A2.13)

ci

Proof. There exists a Bore1 partition = {Cl, . . .,C,.} of X such that c Ui, i = 1 , . . . ,r . By the definition of measure-theoretic entropy there exists a number m > 0 such that

Assume that px,n, converges to the measure p for some subsequence n,. For n' > n we write This shows that if we replace the number nj by the closest integer which is a .factor of m then the new subsequence of measures will still converge to p. Thus, we can assume that nj = mkj. Let D1,. . . ,Dt be the non-empty elements of the partition [ = C V . . . V Fix ,8 > 0. For each D, one can find a compact set Ki C Di such that p(Di \ K ; ) 5 p. Each element Di is contained in an element of the cover

f-("+l)s.

92

Appendix I1

V = U V ... V f-(m+l)U which we denote by Bi. One can find disjoint open subsets & such that Ki c V, c Bi. Moreover, there exist Borel subsets comprising a Borel partition of X such that Vi c V : c Bi. Given nj = mkj, we denote by M y ) the number of those s E [ O , n j ) , for which fS(z)E and by MI,:) the number of those s E M y ) , for which s = q (mod m ) . Define

v

v,

Since P~,,,~converges to the measure p we obtain that

limp?) 2 p

j+oo

( ~ i2) p ( ~ i-) p,

If j is sufficiently large and p is sufficiently small we find that

Since the function g(z) = -z logz is convex we obtain that

and hence

Therefore, the inequality

should hold for some q E [0,m). This implies that

+

Let N = nj q. For some sufficiently large j we choose a string U E S ( U ) with m ( U ) = N in the following way. For s < q we choose Us E U which contains f”(z).Further, for every . such that we choose a string Ui = U O , .~.Um-l,i

K* c u0,,n f - l ( ~ ~. . ., n~f-m+l )

(um-1,i).

Variational Principle for Topological Pressure

93

T h e n , f o r s > q w e w r i t e s = q + m p + e w i t h p > O a n d m > e > O a n d s e t U, = Ue,i, where i is chosen such that fq+"P(x) E Set a, = UO,iUl,i... Urn-l,i and consider the string UO. . . U q - l a o a l . . .a k , - 1 . For a = (ao, . . . ,a k , - 1 ) , the measure p& is given by probabilities p t i , i = 1 . . . t and it satisfies

v.

This proves the first and the third statements of the lemma. Since pz,njconverges to the measure p we obtain for sufficiently large N that

This implies the second statement and completes the proof of the lemma. Given a number m > 0, denote by Y, the set of points y E Y for which Lemma 4 holds for this m and some measure p E V(z) n 3n(Y). We have that Y= Y,. Denote also by Y,,, the set of points y E Y, for which Lemma 3 holds for some measure p E V(z) n M ( Y ) satisfying 'p d p E [u - E , ' 1 ~ E ] . Set c = SUP

PEI)JI(Y)

(h,(l)

+

s,

sy

PdP)

+

.

Note that if z E Y,,, then the corresponding measure p satisfies h,(f) 5 c - ~ + E . Let G , be the collection of all strings U described in Lemma 4 that correspond to all 2 E Y,,, and all N exceeding some number NO. It follows from (A2.13) that for any z E Y,,, the substring constructed in Lemma 3 is contained in R ( k , m(h E),U"), where h = c - u E . Therefore, the total number of the strings constructed in Lemma 4 does not exceed b(N) = IUlmlR(k,m(h+E),U")I. By Lemma 3 we obtain that

+

+

(A2.14)

If NO is sufficiently large, we have that b ( N ) 5 exp(N(h

M(Ym,u,

CP,U ,NO)5

->

PNO

1-0

+ 2.5)). Hence, (A2.15)

94

Appendix I1

where

+ h + cpdp + r(U)+ 3.5) . It follows from (A2.15) that if X > c + r(U)+ 4~ then mC(Yrn,,,,A) = 0. Hence, B = exp (-X

X 2

ern,, (cp,U).

I-IIcpIL IIcpII1.

Assume that points

Then

u1,.

m

. . ,tir

form an &-net of the interval

r

We have that X 2 Py,,,;(cp, U )for any m and i. Therefore,

z SUP PY,,,* m,i

(cpl

U )= f% (cp, U).

This implies that c + r(U)+ 4~ 2 Py(cp,U). Since E can be chosen arbitrarily small it follows that c "y(U) 2 Py(cp,U). Taking the limit as 1241 + 0 yields c 2 Py(cp)and the desired result follows.

+

We state some most interesting corollaries of Theorem A2.1: (1) inequality for topological pressure: for any invariant set Z tinuous function cp on X, and any measure p € Im(Z),

h,(f)

+

1 z

c X, any con-

cpdp I Pz(cp);

(2) classical variational principle for topological pressure on compact sets: for any continuous function cp on X,

(3) classical variational principle for topological entropy on compact sets:

h x ( f ) = SUP h,(f); PE%V

(4) for any set Z P E mw7

c X , any continuous function hp(f

+

s,

cp on

cp dp = PG,(p)i

X, and any measure (A2.16)

where G, is the set of all forward generic points of the measure p (i.e., the points for which the Birkhoff ergodic theorem holds for any continuous function on X); applying this equality to cp = 0 yields Bowen's formula for the measure-theoretic entropy of f hp(f)

(5) let Z

=hG,(f);

c X be an invariant set; if V(z) n m(2)# 0 for every z E 2, then

Variational Principle for Topological Pressure

95

In [PP], Pesin and Pitskel’ described an example that shows that the assumption “V(z) nm(Z)# 0 for every 2 € 2” is crucial and that the variational principle for the topological pressure on non-compact sets may fail otherwise. It also reveals some new and interesting phenomena associated with the topological entropy in the case of non-compact sets.

Proposition A2.1. Let (&,a) be the full shijl on two-sided sequences of two symbols (the classical Bernoulli scheme) and

z = {w

E

CZ

:w

G, for any measure p E

~ ( c Z ) )

= CZ

\

U

G,,

PEW~2)

where G,, is the set of points which are both forward and backward generic with respect to the measure p . Then h z ( a )= hcl(a)= log2.

Remark. Note that p(Z) = 0 for any p E !lX(Cz). Hence,

and the variational principle for the topological pressure fails (note that in this case C ( Z ) = 0 ) . The set 2 is an example of so-called metrically irregular sets for dynamical systems which we describe in detail in Appendix IV.

Proof of the proposition. Consider a Bernoulli measure p on Cz such that p(C0) = p and p(C1) = 1 - p = q, where CO = { w = (in) : io = 0) and C1 = { w = (in) : io = 1) are cylinders and p # q. Given 6, 0 < 6 5 1, we can choose p such that log2 - h,(a) 5 6. Let { n k } be a sequence of positive integers satisfying n k < nk+l and n k -+ 00 as k + 00. We decompose the set of all integers into two disjoint subsets Q1 and Q2 in the following way: i E Q1 if n Z k 5 lil < nZk+l and i E Q2 otherwise. Consider the map I+!J: Cz + C2 given as follows

(I+!J(w))n

=

{ fl+1 (mod 2)

if n E Q1, if n E Qz

where w = (. . . i - l i ~ i l . . .). Note that I+!J is a homeomorphism (even bi-Lipschitz) but it does not commute with the shift. Set Y = I+!J(G,).

Lemma 1. If the sequence

{nk}

grows suficiently fast then Y

c 2.

Proof of the lemma. Let x be the indicator of the set CO.If the sequence { n k } increases sufficiently rapidly then by the Birkhoff ergodic theorem for any w E Y we obtain that lim

k+a,

Since p

nZk+l 1 2nzk+l+ 1.

n2k

r=- - n Z k +l

n-1

# q it follows that the Birkhoff sum 1 C x ( d ( w ) ) does not converge and

hence w E 2.

i=o

Appendix I1

96

Let 6 be the partition of C2 by the sets Co and Cl. Given m > 0 and n m = . $ &qm. denote by qm = . V aj< and 0,

j=-n

Lemma 2. For every w C Gp

+

+

Proof of the lemma. Set Q(i, m, n) = Qi n [-m - n 1,m n - 11, where i = 1 , 2 , m > 0, and n > 0. By the law of large numbers for every w E G, there exists the limit

where IAl denotes the number of elements in the set A . Since the involution 0 e 1, 1 H 0 transfers the measure p into a Bernoulli measure with the same entropy we obtain that

For any n

> 0 we write

The desired result follows. We proceed with the proof of the proposition. Fix m > 0 and consider the partition qm which is also a finite cover of Cz. It follows from Lemma 2 that for any y > 0 there exists a set D C Gp and a number N > 0 such that p ( D ) 2 1-y and for any w E D and n 2 N , P(C(,,)*(@(w)))

I exp (-n(h,(ff) - 7 ) ) .

Fix n 2 N and choose a collection of strings C7 c S(vm) which cover Y (i.e., the sets { & ( U ) : U E 8) cover Y ) and satisfy: m(U) 2 n and

Variational Principle for Topological Pressure

97

Let Ge = { U E C j : m(U) = a} and Ke be the number of elements in Ge. Set

Since the sets &(U') and Cz(U") are disjoint for any distinct U', U" E Ge we obtain that

Therefore,

if n is sufficiently big (such that p(+-'(En) n D ) 2 (1- 27)). For a < h,(a) - y this implies that 1 M(Y,a,O,177n,n) 2 1 - 7 z -. 2 Taking the limit as n -+

00

we obtain that mc(Z,a)= 03 and hence

PY (0, qm) L h,(u)

-

y L log 2 - 6 - y.

Since diamq, -+ 0 as m + co this implies that h y ( u ) = Py(0) 2 log 2 - 6 - y. Taking into account that the numbers 6 and y can be chosen arbitrarily small we conclude that h z ( u ) 2 h y ( a ) 1 log2. This completes the proof of the proposition.

Equilibrium Measures Let cp be a continuous function on X . Given an f-invariant set 2 C X we call a measure p = p,+, an equilibrium measure on 2 corresponding to the function cp if pv E EX(2) and

The following statement establishes the existence of an equilibrium measure for a continuous function cp. Note that, in general, this measure may not be unique. We recall that a homeomorphism f of X is called expansive if there exists E > 0 such that for any two points x , y E X if p(f'(x), f'(y)) 5 E for all k E Z then x = y.

98

Appendix I1

Theorem A2.2. [PP] Assume that the following conditions hold: (1) f is a homeomorphism of X ; ( 2 ) f is expansive; ( 3 ) the set m(2) is closed in m(X)(in the weak*-topology). Then for any continuous function cp there exists an equilibrium measure p,+, on 2. Proof. Let f, be a partition of X with diamf, 5

E.

Set

En

= .

j=-n

fjt. Since f

is expansive we obtain that diamf,, + 0. Therefore, h,(f) = lim h,(f,Cn). On n+w the other hand, it is easy to see that hb(f,f,,) = h,(f,[). Therefore, h , ( f ) = h, (f7 E l . We now show that the map p H h,(f) is upper semi-continuous on m(2). Then the map p H h , ( f ) cpdp is also upper semi-continuous on fUt(2)and the desired result follows from Theorem A2.1 and the fact that an upper semicontinuous function on a compact set attains its supremum. Fix p E m(Z),a > 0, and a partition [ = {C, . . .Cn}of X with diamf, 5 E . For a sufficiently large m we have that

+

Let us fix such an m. Given 0,choose compact sets . . k=O

This implies that

Since Li are disjoint and compact one can find a partition with diamf,’ 5 E such that Li c intCi. We have that

(’ = {C; . . .CA} of X

n f-k(c:,).

m- 1

K~,,...~,,,-~ c int

k=O

If a measure v E m(2)is close to p in the weak*-topology then

k=O

and

99

Variational Principle for Topological Pressure

If p is sufEciently small it follows that 1

L ( f )= L ( ft’) , I ;H,(E’

V

. . . V f -m+l[’)

1

5 -H,(E v . . .v f m

-m+l

0 + I h , ( f ) + 2ff. ff

This completes the proof of the theorem. As a direct consequence of Theorem A2.2 we obtain the following statement.

Theorem A2.3. Assume that a map f satisfies Conditions 1 and 2 of Theorem A2.2. Then for any compact f -invariant set Z c X and any continuous function cp on Z there exists an equilibrium measure plP on Z for which

In particular, if a map f satisfies Conditions 1 and 2 of Theorem A2.2 then for any compact f-invariant set Z C X there exists a measure of maximal entropy, i.e., an equilibrium measure corresponding to the function cp = 0 for which

Let us notice that, in view of (11.14), (11.15), and Theorems 11.6 and A2.1 for any compact invariant set Z we have that

(provided conditions of Theorem 11.6 hold). Therefore, the variational principle for topological pressure can be viewed as a variational principle for CarathCodory dimension (where the dimension is generated by the C-structure defined by (11.1)-(11.3); see Sections 1 and 5). Moreover, any equilibrium measure is a measure of full Carathbodory dimension (with respect to the space M of invariant measures; see (5.4)). In particular, any measure of maximal entropy is a measure of full CarathCodory dimension.

Non-additive Variational Principle We now state a non-additive version of the variational principle for topological pressure established by Barreira in [BarZ]. Let cp = {pn} be a sequence of continuous functions.

Theorem A2.4. Let Z c X be an f -invariant set. Assume that there exists a continuous function $: X + R such that

100

Appendix I1

uniformly on Z

QS

n + 00. Then

As an immediate consequence of the above statement we have that i f Z c X

is an f -invariant compact set, then

where the function @ satisfies (A2.17). Falconer [F3]established another version of the variational principle for topological pressure assuming that the sequence of functions Q is sub-additive (it also should satisfy some other additional requirements) - the so-called “subadditive” variational principle for topological pressure.

Symbolic Dynamical Systems We briefly describe some basic concepts of symbolic dynamics which are used in the second part of the book. For each p E N we denote the space of right-sided infinite sequences of p symbols by

~1.+= { w = (ioil.. . ) } = {I,. . . ,p}’. We call the number ij the j-coordinate of the point w (we also use another notation w j ) . We write w+ for points in C; to stress that we are dealing with a right-sided infinite sequence. A cylinder (or a cylinder set) is defined as

Cio...i, = { ( j o j , . . .) E Cp’ : j k = ik, k = 0,. . . ,n } . We also use more explicit notation C:,.,*,. Given /3 > 1, we endow the space CP+ with the metric (A2.18) where w = (ioil.. .) and w’ = (ibii . . .). It induces the topology on C: such that the space is compact and cylinders are disjoint open (as well as closed) subsets. The (one-sided) shift on C: is defined by a ( w ) k = wk+l

(we also use more explicit notation u+). It is easily seen to be continuous. A subset Q c C$ is said to be u-invariant if a(&) = Q. When the set Q c C: is compact and a-invariant, the map alQ is called a (one-sided) subshift.

Variational Principle for Topological Pressure Let A be a p x p matrix whose entries and u-invariant subset

101 aij

are either 0 or 1. The compact

C z = { (ioil . . .) E Cp+ : ~ i , , i , , + = ~ 1 for all n E N} is called a topological Markov chain with the transfer matrix A . The map ulCf; is called a (one-sided) subshift of finite type. It is topologically transitive (i.e., for any two open subsets U, V c Cf; there exists n > 0 such that u"(U)n V # 0 ) if the matrix A is irreducible, i.e., for each entry aij there exists a positive integer k such that a$ > 0, where ufj is the (i,j)-entry of the matrix A k . The map uICf; is topologically mixing (i.e., for any two open subsets U, V c Cf; there exists N > 0 such that u"(U)n V # 0 for any n > N ) if the transfer matrix A is transitive, i.e., Ak > 0 for some positive integer k. We call (Q, u)a sofic system if Q c C: is a finite factor of some topological + Q such Markov chain C;, i.e., there exists a continuous surjective map C: that uIQoC = Cou. An example is the even system, i.e., the set Q of sequences of 1's and 2's, where the 2's are separated by an even number of 1's. Similarly to the above, we consider the space of left-sided in6nite sequences Cp = { w = (. . . i & l ) } .

We also write w- for points in Cp. A cylinder in Cp is denoted by Ci-,...j, (or more explicitly CtLn,,,io). The (one-sided) shift is defined by

(we also use more explicit notation u-). It is continuous. Further, given a transfer matrix A = ( a i j ) ,we set

The map ulCA is a (one-sided) subshift of finite type. We also consider the space of two-sided infinite sequences of p symbols C,={w=(

...i - ~ i O i l . . . ) } = { l , . . .p}". ,

A cylinder (or a cylinder set) is defined as ~i,...i,,

= { w = (. . . j - l j o j l

. . . ) E C,

: j k = ik, k = m, . . . , n } ,

where m 5 n. Given /3 > 1, we endow the space C , with the metric (A2.18')

102

Appendix I1

where w = (. . .i - l i o i l . . .) and w' = (. . .iLlibii . . . ). It induces the compact topology on C, with cylinders to be disjoint open (and at the same time closed) subsets. The (two-sided)shift o:C, -+ C, is defined by o(w)k = w k + l . It is an expansive homeomorphism. Given a compact o-invariant set Q c C,, we call the map ulQ a (two-sided) subshift. Let A be a p x p transfer matrix with entries 0 and 1. Consider the compact a-invariant subset C A = {w E

C, : aw,w,+I = 1 for all n E

z}.

The map is called a (two-sided) subshift of finite type. Let us notice that given a point w E C A , the set of points w' E C A having the same past as w (i.e., w, = w: for i 5 0) can be identified with the cylinder C; c C:. Similarly, the set of points w' E C A having the same future as w (i.e., w, = w: for i 2 0) can be identified with the cylinder Cz: c C,. Thus, the cylinder Ci, c C A can be identified with the direct product Cif; x Ct;. For symbolic dynamical systems the definitions of topological pressure and lower and upper capacity topological pressure can be simplified based on the following observation (which we have already used in the proof of Proposition A2.1). Let U,,be the open cover of Cg by cylinder sets Cj,...in . Notice that lU,l -+ 0 as n -+ 00 and for any U E S(U,) the set X ( U ) is a cylinder set. N ) can be rewritten according to (11.4) Therefore, the function M ( Z ,a , cp, U,,, as

M ( Z ,a , ~un, , N )=

and the infimum is taken over all finite or countable collections of cylinder with m 2 N > n which cover Z. Furthermore, the function sets Ci,,...i, R(Z,a , cp,U,,, N ) ( N > n ) can be rewritten according to (11.4') as

a ,cp>un, N)

and the sum is taken over the collection of all cylinder sets Cio...iNintersecting 2. Let (Q, o)be a symbolic dynamical system, where Q is a compact o-invariant subset of C; and cp a continuous function on Q. A Bore1 probability measure p = pv on Q is called a Gibbs measure (corresponding to 9) if there exist constants D1 > 0 and D2 > 0 such that for any n > 0, any cylinder set C ~ o . . . ~ , r and any w E Cio..,i,we have that

Variational Principle for Topological Pressure

103

where P = P~(cp).Note that if Condition (A2.20) holds for some number P then P = PQ. Indeed, in this case for every E > 0 by (A2.19) we obtain that

M ( Z ,P

+

E,

cp,U,, N ) I D;'inf

B

p(Cio . . . ~ m )exp(-e

c;o....m EB

I D;lexp(-e

+ r(Un))

+ r(Un)).

+

Letting n + 03 yields that PQ 5 P E and hence Pq 5 P since E is arbitrary. The opposite inequality can be proved in a similar fashion (I thank S. Ferleger for pointing out this argument to me). Any Gibbs measure is an equilibrium measure but not otherwise. It is known that the specification property (see [KH] for definition) of a topologically mixing symbolic dynamical system (Q, u) ensures that any equilibrium measure corresponding to a Holder continuous function is Gibbs. It is known that any subshift of finite type (C2,u)satisfies the specification property. Therefore, an equilibrium measure p,,,, corresponding to a Holder continuous function cp, is a Gibbs measure provided the transfer matrix A is transitive. In this case it is also a Bernoulli measure. For an arbitrary transfer matrix A , by the Perron-F'robenius theorem one can decompose the set C 2 into two shift-invariant subsets: the wandering set Q1 (corresponding to the nonrecurrent states) and the non-wandering set Qz (corresponding to the recurrent states). The latter can be further partitioned into finitely many shift-invariant subsets of the form X i i , where each matrix A, is irreducible and corresponds to a class of equivalent recurrent states (see [KH] for details). Moreover, for each i there exists a number ni such that the map CPS is topologically mixing. Note also that any sofic system satisfies the specification property. We define the notion of Gibbs measures for two-sided subshifts. Let Q be a compact a-invariant subset of Cp and cp a continuous function on Q. A Bore1 probability measure 1.1= p,+, on Q is called a Gibbs measure (corresponding to cp) if there exist constants D1 > 0 and Dz > 0 such that for any rn < 0, n > 0, any cylinder set Ci,...i,, and any w E C,, ...i, we have that (A2.20')

where P = P ~ ( c p )Again, . any Gibbs measure is an equilibrium measure and the specification property ensures otherwise. In the case of subshifts of finite type there is a deep connection between Gibbs measures for one-sided and two-sided subshifts. In order to describe this connection consider a two-sided subshift of finite type ( C A , O ) and a Holder = (u!)) E C A such that continuous function cp on C A . Choose p points di) w!) = i for i = 1,.. . , p and set R = (&), . . . ,w ( P ) ) . We now define the function rn on C A by m ( w ) = r ~...( w-zw-liwlwz

...) = (... ,(i)- 2 w -( a1) z .w 1 w ~ . . . )

104

Appendix I1

provided wo = i. Further, we define the function

@(U)(w)= cp(ra(w))+

c

dU)= 0):

on C A by

W

[ P ( d + ’ ( T o ( w ) ) )- cp(aj(7l2(0w)))]

j=O

Given a Holder continuous function 4 on C A and a positive integer n, we define the n-variation of the function 4 by varn4 = sup{l4(w) - 4(w’)l : w, = w i , 0 5 i

< n}.

If w and w’ are two points whose i-coordinates coincide for all i between -n and co then var,4 5 Cpn, where C > 0 is a constant and /3 is the coefficient in the dp-metric on C A (see (A2.18’).

Lemma A2.1. The function QcU) is Holder continuous, it is cohomologous t o cp and hence has the same topological pressure.

Proof. We have that

c 00

d”)(W)

-k

=Cp(W)

c

[(p(U’+l(TO(w)))- C p ( U ’ + l ( W ) ) ]

j=-1

W

+

[Cp(aj+’((w)) - C p ( 4 T s 2 ( ~ ( ~ ) ) ) = ) 1 Cp(w) -

j=O

+ 4+J)),

In order to complete the proof of the lemma we need only to show that the function u ( w ) is Holder continuous. Since the i-coordinates of the points & ( w ) and U ~ ( T ~ ( Ucoincide ) ) for all i between - j and co we obtain that Icp(+J))

- c p ( 4 m ( w ) ) ) lI varj’p 5 CPj.

If w = ( w i ) and w‘ = (w:) are two points satisfying wi = wi for lil 5 n then for any j E P,nI Icp(aj(w)) - cp(uj(w’))l

5 cpn+

and

Icp(aj(rn(w)))- cp(gj(m(4))l I CP”+. Therefore,

I+)

c

b/21

- 441I

Icp(d(w))

-

cp(C+bJ’))l

j=O

+Icp(aj(ra(w’))) - Cp(4-n(w)))l+

2

c

j>[nl21

This completes the proof of the lemma.

CPj

105

Variational Principle for Topological Pressure

It follows from the lemma that the measure p is the Gibbs measure corresponding to e("). Let x+:C A --t C i and n-: C A --t C, be the projections

x+(.. .i-lioil. . . ) = (ioil . . . ),

x-(. . .L l i 0 i l . . . ) = (. . .i - l i o ) .

One can easily check that O(")(. . .i - l i ~ i l . .) . = 8(")(. . . iLlibii. . . ) whenever ij = i; for every j 2 0. This means that there is a function p(") on C i such that e(") = cp(") o x+ on c A . In a similar fashion, we define the function O(') = 0:) on C A , and find a function cp(') on C, such that e(') = cp(') o x-. The functions cp(") and cp(') can be shown to be Holder continuous. We call them stable and unstable parts of the function cp. Let p(") be the Gibbs measure for p(") on C i and p(') the Gibbs measure for cp(') on C,. We have that p(") = p 0 (x+)-l since both measures are Gibbs measures for the function cp("), and that p(') = p o (x-)-' since both measures are Gibbs measures for the function cp('). This implies that

P(")(C;,,,~,,) = ,4Cio...i,,) and

P ( ' ) ( C ; . . . ~=~p(Ci ) o...i,,),

(A2.21)

where Cz...j,,= x+(Ci0...in) and CL,,... io = x-(Ci _,,... io). We want now to normalize the functions cp(") and cp('), i.e., to switch to the functions +(") on C$ and +(') on C, defined by log

+'"'

= cp(") -

Clearly, Pc=(log+(")) =

Pet; (cp'"'),

log +(') = cp(s) - Pc, (cp'").

log+(^)) = 0.

Lemma A2.2. W e have that

f o r each w+ = (ioil . . .) E C i (with uniform convergence), and

for each w- = (. . .i - l i o ) E C, (with uniform convergence).

Proof. The statement is an immediate corollary of the following property of Gibbs measures (see Proposition 3.2 in [PaPo]): let p be the Gibbs measure corresponding to a Holder continuous function 4 on X i ; then

Here a is the Holder exponent of 4 and lQla = sup{a-nvarn4 : n 2 0)).

141a

is the Holder norm of

4 (i.e.,

Appendix I1

106

The lemma implies that the functions $(") and $(') do not depend on the choice of the point 52. The following statement shows that the measure p is the "direct product" of measures p(u) and p @ ) .

Proposition A2.2. The following properties hold:

(d"')

(1) PEA(V) = PE: = p.1(cp'"'); (2) there exist positive constants A1 and A2 such that for every integers n, m 2 0, and any (. . . i-lid1 . . . ) E C A ,

Proof. Since the functions cp(8) and cp(") are cohomologous the first property follows from Lemma A2.1. The second property is an immediate consequence of identities (A2.21). Bowen's Equation Let ( X ,p) be a compact metric space with metric p, f: X + X a continuous map, and cp:X -+ W a continuous function. Consider the pressure function $(t)= Px(tcp) for t E R By the continuity property of the topological pressure (see Section 11) this function is continuous. If cp is negative the function $(t) can be shown to be strictly decreasing (see below). Since $(O) = h x ( f ) 2 0 the equation Px(tcp) = 0 has a unique root. In [BO~], Bowen discovered this equation while studying the Hausdorff dimension of quasi-circles. This equation is known now in dimension theory as Bowen's equation. If 2 c X is an arbitrary set (not necessarily invariant or compact) one can consider three pressure functions $(t) = Pz(tcp),$(t) = =,(tcp), and $ ( t )= m z (tcp).

Theorem A2.5. Assume that the function cp is negative. Then (1) the functions $ ( t ) ,$ ( t ) , and q(t)are Lipschitz continuous, convex, and strictly decreasing; (2) there exist unique roots s, 2, and 3 of the equations

(3) 0 5 s 5 2 5 3 and S < co if hx (f)< co; (4) s = 0 if and onZy if h ~ ( f=) 0; 2 = 0 if and only i f s = O if and only i f C h Z ( f ) = 0.

G,(f)= 0; and

Variational Principle for Topological Pressure

107

Proof. Lipschitz continuity of the functions $ ( t ) , $(t),and $(t) follows from Theorem 11.4. If t’ 2 t , we obtain that

-(t’ - t)Cl 6 $(t’)- $(t)5 -(t’ - t)Cz, where C1 and Cz are positive constants. This implies that the function $(t) is strictly decreasing. Similar arguments show that the other two functions are also strictly decreasing. The proof of convexity of these functions is straightforward. Since $(O) 2 0, $(O) 2 0, and $(O) 2 0 the second statement follows. The last two statements are consequences of Theorems 11.2 and 11.3. We notice that by Theorem 11.5, if the set 2 is invariant then $ ( t )= $(t) for $(t)= $(t) all t and hence s = 8; if, in addition, the set 2 is compact then $ ( t ) = for all t and hence s = 3 = 8. Following Barreira [Bars], we consider a non-additive version of Bowen’s equation. Let cp = {pn:X + W} be a sequence of continuous functions satisfying (12.1). We assume, in addition, that the following condition holds: there exist negative constants B1 and Bz such that for any suficiently large n (A2.22) This condition is satisfied if cpn is additive. The following statement is an extension of Theorem A2.5 to the non-additive case.

Theorem A2.6. [Bar21 If a sequence of functions cp satisfies Condition (A2.22)

then the pressure functions $(t) = Pz(tcp), $ ( t ) = CP,(tcp), and $(t) = mz(tcp)satisfy Statements (1)-(4) of Theorem A2.5. In particular, there exist unique roots s, g, and 3 of the equations $ ( t ) = 0 , $ ( t )= 0 , and $(t) = 0 respectively, which satisfy 0 6 s 6 g 6 8.

We notice that by Theorem 12.1, if the sequence of functions cp is sub-additive (see (12.4)) and the set 2 is invariant then $ ( t ) = $(t) for all t and hence s = 8; if, in addition, the set 2 is compact and thesequence of functions {cpn} satisfies (12.5) then $(t)= $(t) = $(t) for all t and hence s = 5 = 8. One can use the construction in Example 15.1 below to show that there exists a sequence of functions cp = {cpn} defined on a compact invariant set 2 such that: 1) it satisfies conditions (12.1) and (A2.22) (and hence there exist unique roots g and 8 of the equations $(t)= 0 and $(t) = 0); 2) it is not sub-additive; 3) 3 < 8 (see detailed description in [Bars]). We apply the above results to a symbolic dynamical system ( Q , u ) ,where Q is an invariant compact subset of C;. Assume that for any ( n 1)-tuple (io . . .in) we have a positive number aio..,i,,. Consider a sequence of functions cp = {cpn}, where cpn(w) = 1oga,,,,,in for w = (ioiz . . .). One can verify that the sequence of functions cp satisfies (12.1). The non-additive topological pressure and non-additive lower and upper capacity topological pressures corresponding to cp admit the following explicit description established in [BarZ].

+

108

Theorem A2.7. For every t E W we have: (1) if the sequence of functions cp is sub-additive then 1 t CP,(tcp)= mQ(t9) = lim -log ai,. ..in n-mc n

c

(&...an)

Q-admissible

and the limit exists as n -+ m. (2) if the sequence of functions cp is sub-additive and satisfies (12.5) then

PQ(tV)= =(tP)

=m Q ( t P ) .

Assume that the numbers ai,.,.in satisfy the following condition: (A2.23) ePan 5 ai,...in 5 e-@, where a and ,B are positive constants. Clearly, (A2.23) implies (A2.22), and hence the equations

-

m(t9 = )0, CPQ(t9)= 0 have unique roots s, g, and 3 which satisfy 0 5 s 5 5 5 3 < co. PQ(tcp)= 0,

We consider an important particular case. Let 0 < a l , . . . ,ap < 1 be numbers. Define the function cp on C: by p ( w ) = logai,, where w = ( i o i l . . .).

Theorem A2.8. If Q C C: as a compact a-invariant set, then Bowen's equation PQ(tcp)= 0 has a unique root. If (Q,a ) is the full shift, i.e., Q = C': then Bowen's equation PQ(@)= 0 is equivalent to the equation P

Gait = 1. i=l

If (&,a) i s a subshift of finite type, i.e., Q = E i with a transfer ma= 0 i s equivalent to the equation trix A, then Bowen's equation P~(tcp) p(AMt(a))= 1, where p ( B ) denotes the spectral radius of the matrix B and Mt(a) = diag (alt,.. . ,u p t ) .

Proof. Let f : E i def

(L$f)b) =

-+R be a continuous function.

c

w'Eu-'(w)

exp(cp(w'))f(w') =

Consider the transfer operator

Ck exp(cp(k))f(kioi1 . . .)A@,io),

where w = (ioil. . . ) and the function cp(w) = log ai, = cp(i0) depends on the first coordinate only. The eigenvalue equation (corresponding to an eigenvalue 17) is

According to [Rl],the largest eigenvalue of L , is exp(P(cp)). Hence, exp(P(cp)) is the spectral radius of the matrix A*@,where iP denotes the ( p x p ) diagonal matrix diag(e+'(l), ep(2),. . . ,ev(P)). This implies the third statement. The second one is an immediate consequence of the third statement. The first statement follows from Theorem A2.5.

Variational Principle for Topological Pressure

109

We conclude the appendix with formulae for the topological entropy of a subshift of finite type and the Hausdorff dimension of a topological Markov chain.

Theorem A2.9. Let A be a transfer matrix. Then

w,

(1) hz: ( 0 ) = h q (a)= h ~(0) , = P(A); (2) d i m H z 2 = d i m H z 1 = where p is the coefficient i n the dpmetric (see (A2.18)); (3) dimHCA = 2 w , where p is the coefficient in the do-metric (see (A2.18 I ) ) .

Proof. The first statement follows from the third statement of Theorem A2.8. The second and the third statements can be proved by straightforward calculation which we leave to the reader. Note that the second statement is essentially a corollary of Theorem 13.2 (see Statement 2; see also self-similar constructions in Section 13) while the third statement is a corollary of Theorem 22.2 (see also linear horseshoes in Section 23).

Appendix I11

An Example of Carathkodory Structure Generated by Dynamical Systems

In this Appendix we briefly discuss an example of C-structure generated by dynamical systems. This C-structure was introduced by Barreira and Schmeling [BS] in their study of metrically irregular sets for dynamical systems. See Appendix IV. Let f:X -+ X be a continuous map of a compact metric space X and 'p: X + R a continuous strictly positive function. Consider a finite open cover U of X and define the collection of subsets 3 = 3 ( U ) by (11.2) and three functions E , 7,$: S ( U ) + R as follows E(U)= 1,

7 ( U )= exP(-

SUP Srn(U)'p(4), $(U)= m(U)-l,

XEX(U)

where

c

m(U)-l

Sm(U)'p=

'pOfk.

k=O

One can directly verify that the collection of subsets 3, and the functions 7,[, and II, satisfy Conditions A l , A2, A3, and A3' in Section 10 and hence determine a C-structure 7 = .(U) = ( S , 3 , [ , 7 , $ ) on X . The corresponding Caratheodory function mc(2,a) (where 2 c X and a E W; see Section 10) depends on the cover U (and the function 'p) and is given by m c ( Z , a )= lim M(Z,a,'p,U,N), N+w

and the infimum is taken over all finite or countable collections of strings 0

S ( U ) such that m ( U ) 2 N for all U E B and B covers 2.

c

Furthermore, the Caratheodory functions ~ ~ a( ) 2and, f c ( 2 ,a ) (where 2 C X and a E W; see Section 10) depend on the cover U and are given by

rc(2,a)= Nlim R(Z,cY,'p,U,N), f c ( 2 , a )= N+CO lim R ( Z , a , ' p , U , N ) , -tw 110

An Example of Carathhodory Structure

111

and the infimum is taken over all finite or countable collections of strings 6 c E 6 and 6 covers 2. According to Section 10, given a set Z c X , the C-structure 7 generates the CarathCodory dimension of Z and lower and upper CarathCodory capacities of Z specified by the cover U and the map f. We denote them by BS,,p(Z), &,,(Z), and =,,p(Z) respectively. Repeating arguments in the proof of Theorem 11.1 one can show that for any set Z c X the following limits exist:

S ( U ) such that m ( U ) = N for all U

&(Z)

def

= lim BS.,,,(Z), IW+O

We call the quantities BS,(Z), B S , ( Z ) , and m , ( Z ) , respectively the BSdimension of a set and lower and upper BS-box dimensions of the set Z (specified by the function cp and the map f ; after Barreira and Schmeling). We emphasize that the set Z can be arbitrary and need not be compact or invariant under the map f. The main properties of BS-dimension and lower and upper BS-box dimensions are described below. They are immediate corollaries of the definitions and Theorems 1.1 and 2.1.

Theorem A3.1. BS,0 = 0; BS,(Z) 2 O for any non-empty set Z c X . BS,(Zl) I BS,(ZZ) ifZ1 c ZZ c BS,(Z) = supi2l BS,(Zi), where Z = UiZlZi and 2, c X , i = 1 , 2 , . . . . -0 = = 0; &(Z) 2 0 and m,(Z)2 0 for any non-empty set Z c X . &(ZI) I W ( Z 2 ) and m,(Z1) I %,(ZZ) if ZI c ZZc X. BS.,(Z) 2 supi2l sS,(Zi) and m,(Z)2 sup,,l- BS,(Zi), where Z = U,21Zi and Zi c X , i = 1 , 2 , . . . . If h: X + X is a homeomorphism which commutes with f (i.e., f o h = h o f) then

x.

m,0

BS,(Z) = B ~ , o h - l ( h ( z ) ) ,

BS.,(Z) = &oh-i(h(Z)),

BS,(Z) = BSv0h-i(h(Z)).

For any two continuous functions cp and $ on X IBS,(Z) - B%J(Z)lI IIcp - $11, I&(Z) - Ss.,.(Z)I I IIP - $11, IBS,(Z) - Ss,(Z)I

I IIcp -

$117

Appendix I11

112

where 11.11 denotes the supremum norm in the space of continuousfunctions on X . Remarks. (1) In the case ‘p = 1, the BS-dimension of a set 2 c X and lower and upper BS-box dimensions of 2 coincide with the topological entropy o f f on Z and lower and upper capacity topological entropies o f f on 2 respectively. (2) Compare the definitions of topological pressure and BS-dimension one can obtain that for any set Z c X the BS-dimension of 2 is a unique root of Bowen’s equation Pz(-scp) = 0, i.e., s = BS,(Z). Let p be a Borel probability measure on X which is invariant under f . According to Section 10, the C-structure 7 = (S,F ,[, q,$) on X generates the CarathCodory dimension of p and lower and upper CarathCodory capacities of p specified by the cover U and the map f . We denote them by BS,,u(p), &,u(p), and m , , ~ ( p )respectively. It follows from what was said above that there exist the limits BS,(~L) lim BS,+,,~(~),

ef

IUI-rO

def

B S , ( p ) = lim BS,,u(p). IUI+J

We call these quantities the BS-dimension of a measure and lower and upper BS-box dimensions of the measure p (specified by the function ‘p and the map f ) . Given a point z E X , we set in accordance with (10.1)

where the infimum and supremum are taken over all strings U with z E X ( U ) and m(U)= N . We leave the proof of the following statement to the reader (compare to Proposition 11.1).

Theorem A3.2. If p is a Borel probability measure on X invariant under the map f and ergodic, then (1) for every CY E R and p-almost every z E X ,

where h,(f) is the measure-theoretic entropy o f f . (2) BS,(p) = sS.,(p) = m , ( p ) = d .

113

An Example of Carathkodory Structure

As a consequence of Theorem A3.2 we establish the variational princzple for the BS-dimension: where G =

u

G, and G, is the set of all forward generic points of the

PEWX)

measure p (see (A2.16)). Indeed, by Theorem A3.2 and (A2.16), 0 = hp(f)

-

/

BSv(p)

This implies that BS,(p) = BS,(G,)

X

'f'dp = PG(-BSp(p)P).

and hence

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Part I1

Applications to Dimension Theory and Dynamical Systems

This Page Intentionally Left Blank

Chapter 5

Dimension of Cantor-like Sets and Symbolic Dynamics

It is now the prevailing opinion among experts in dimension theory that the coincidence of the Hausdorff dimension and lower and upper box dimensions of a set is a “rare” phenomenon and can occur only if a set has a “rigid” geometric structure. Nevertheless, there exists a broad class of subsets in Rm, known as Cantor-like sets, for which the coincidence usually takes place. These sets were a traditional object of study in dimension theory for a long time and also served as a touchstone for different techniques. Computing their Hausdorff dimension and box dimension was a great challenge to experts in the field who had to create a number of highly non-trivial methods of study. These sets often were the source for exciting examples that demonstrated different phenomena in dimension theory. The Cantor-like sets are defined by geometric constructions of different types. We begin with the most basic geometric construction. Starting from p arbitrary closed subsets A,, . . . ,Ap of Rm we define a Cantor-like set by M

where the basic sets on the nth step of the geometric construction, A,,,,,,,,, i k = 1,.. . , p (n 2 0) are closed and satisfy the following conditions:

(CG1) A,,, ...i,j C A,,,...,, for .i= 1,.. .,P; (CG2) diamAi,...in + 0 as n + co; (CG3) (separation condition): A,,,,..i, n Aj,,.,.j, n F = 0 for any (io . . . i n ) # (jo..

..in).

The set F is perfect, nowhere dense, and totally disconnected. See Figure 1. The description of a geometric construction includes the description of its symbolic representation and its geometry.

Symbolic representation. Let E; be the set of all one-sided infinite sequences (i& . . . ) of p symbols (p is the number of basic sets on the first step of the construction) endowed with the dp metric given by (A2.18) (see Appendix 11). Given w = (ioil . . . ) E Cp’ the intersection A, ,,...in is non-empty and by Condition (CG2), consists of only one point 2. Thus, the formula x ( w ) = 2 defines correctly a continuous map x: E$ t F which we call the coding map. By the separation condition this map is one-to-one and hence (since Cp’ is compact) it is a homeomorphism.

nr=o 117

118

Chapter 5

We consider a geometric construction modeled by a general symbolic dy: is a compact set invariant under the shift namical system (Q, a),where Q c C map a, i.e., a(&) = Q. Namely, we allow only basic sets that correspond to admissible n-tuples (io . . .in) with respect to Q, i.e., there exists ( j o j l . . .) E Q such that j o = io, j1 = il, . . . ,jn = in. Such constructions are called symbolic geometric constructions. The limit set F is defined by cx)

n=O

(io...in) Q-admissible

One can classify geometric constructions according to their symbolic representation. A construction is called a simple geometric construction if it is : (see Figure 1). Another important class modeled by the full shift, i.e., Q = C of geometric constructions are constructions modeled by subshifts of finite type. Namely, the geometric construction is called a Markov geometric construction if Q = C i (we remind the reader that C i consists of all sequences ( i o i l . . . ) admissible with respect to the transfer matrix A with entries A ( i ,j ) = 0 or 1, i.e., A(ij, = 1 for j = 0 , l . . . ; see Appendix 11; see also Figure 2 in Section 13). Finally, a geometric construction is called a sofic geometric construction if (Q, a) is a sofic system (i.e., a finite factor of a subshift of finite type).

Figure 1. A SIMPLEGEOMETRICCONSTRUCTION. Geometry of the construction. This includes the information on the placement of the basic sets, their geometric shapes, and "sizes". If one has strong control over the shape and sizes of basic sets, then the placement can be fairly

Dimension of Cantor-like Sets and Symbolic Dynamics

119

arbitrary, and vice-versa. To illustrate this we first consider constructions whose geometry seems to be most simple where one has complete control over the shape and sizes of basic sets which are balls. Denote by rio,,.i,the radius and by zio,,.i,E Rm the center of the basic set (ball) A;o,,,;, on the nth step of the construction. The collection of numbers rio...in and coordinates of points zio...i, provide complete information on the geometry of the construction and, in particular, is sufficient to compute the Hausdorff dimension and box dimension of the limit set. Surprisingly, one can often use only the numbers rio...i, to obtain refined estimates for the Hausdorff dimension (from below) and the upper box dimension (from above) of the limit set (see Section 15). The main tool to study this geometric construction is the thermodynamic formalism, developed in Chapter 4 and Appendix I1 (in particular its non-additive version), applied to the underlying symbolic dynamical system. Namely, consider the sequence of functions cpn(w) = logrio...i, on Q, where w = (ioil.. .). Clearly it satisfies Condition (12.1). According to Appendix I1 (see Theorem A2.6) the equations pQ(s{Pn}) = 0, m Q ( S { ‘ P n } ) = 0 (where PQ and CPQ denote the non-additive topological pressure and nonadditive upper capacity topological pressure specified by the sequence of functions { c p n } ) have unique roots provided the sequence of radii admits the asymptotic estimate (A2.23). We denote these roots by 3 and 3 respectively. Clearly, s 5 3. In Section 15 we will show that, under some mild additional assumptions, the number s provides a lower bound for the Hausdorff dimension and the number s provides an upper bound for the upper box dimension of the limit set (see Theorem 15.1). We emphasize that the above approach works well for an arbitrary collection of numbers rio...i, which may depend on the whole “past” and may not admit any asymptotic behavior as n + 03. Yet, the Hausdorff dimension and lower and upper box dimensions may not coincide and their exact values may depend on the placement of basic sets (see Example 15.1). However, if, for instance, the sequence of functions vn is sub-additive (see Condition (12.4)), then the Hausdorff dimension and lower and upper box dimensions coincide and are completely determined by the sizes of basic sets only, i.e., the numbers rio...i, . In this case s =3 s and regardless of the placement of basic sets, the Hausdorff dimension and lower and upper box dimensions of the limit set coincide and are equal to s. Moreover, s is the unique root of the equation

sf

An equation of this type was first discovered by Bowen in his study of the Hausdorff dimension of quasi-circles (see [ B o ~ ]and ) now bears his name. Bowen’s equation is one of the main manifestations of the general CarathCodory construction developed in Chapter 1: it expresses intimate relations between two CarathCodory dimension characteristics: the Hausdorff dimension of the limit set of a geometric construction and non-additive topological pressure specified by the sequence of functions {cpn} for the underlying symbolic dynamical system

120

Chapter 5

( Q ,c). Bowen’s equation contains all necessary information on the geometric construction that matters in computing the Hausdorff dimension of the limit set: the information on the underlying symbolic dynamical system and its realization into the Euclidean space by basic sets. Bowen’s equation seems to be “universal” in dimension theory: we will demonstrate that the Hausdorff dimension of limit sets for various classes of geometric constructions as well as invariant sets for various classes of dynamical systems can be computed as roots of Bowen equations written with respect to the underlying symbolic dynamical systems. In 1946 Moran, in his seminal paper [Mo], introduced and studied geometric constructions with basic sets satisfying the following conditions: (CM1) each basic set A,o...i, is the closure of its interior; (CM2) 4 0 ...i n j C Ai0...in for J’ = 1,.. . ,P; (CM3) each basic set Aio,..injis geometrically similar to the basic set Aio,,,infor every j ; (CM4) for any ( 2 0 . . . i n ) # ( j o . . .jn) the basic sets Aio...in and Ajo,,,jndo not overlap, i.e., their interiors are disjoint; (CM5) diamAiO,,,inj= XjdiamAi,,,,.i,, where 0 < X j < 1 for j = 1,. . . , p are const ants. Such constructions are called Moran geometric constructions (the name coined by Cawley and Mauldin [CM]). The numbers X i are known as ratio coefficients of the construction since they determine the rate of decreasing the sizes of basic sets. We stress that basic sets of a Moran geometric construction at the same step may not be disjoint (they may intersect along boundaries; compare to Condition (CG3)). Therefore, the limit set F may not be a Cantor-like set and the coding map x may not be injective (but it is still surjective). Moran considered only geometric constructions modeled by the full shift. His remarkable observation was that an “optimal” cover, which can be used to obtain the exact values of the Hausdorff dimension of the limit set (we call it a Moran cover) is completely determined by the symbolic representation of the geometric construction. In particular, regardless of the placement of basic sets, the Hausdorff dimension of the limit set is completely determined by the ratio coefficients. Namely, Moran discovered that it is a unique root of the equation P p i

t =1

i=l

which is a particular case of Bowen’s equation corresponding to the full shift. Moran constructions modeled by subshifts of finite type or sofic systems were studied in [MW]. In [PWl], Pesin and Weiss demonstrated that Moran’s approach can be greatly extended to a much more general class of Moran-like geometric constructions w i t h stationary (constant) ratio coefficients. They discovered that the only property of basic sets that matters in constructing Moran covers, is

Dimension of Cantor-like Sets and Symbolic Dynamics the following: there exist closed balls &,...in of radii is a constant) with disjoint interiors such that

121 rio,,,in= Cn;=,

X i J (C > 0

The topology and geometry of basic sets of these constructions may be quite complicated (for example, they may not be connected and their boundary may be fractal). Moreover, the basic sets at step n of the construction need not be geometrically similar to the basic sets at step n - 1 (see Condition (CM3) in the definition of Moran geometric constructions). Another important feature is that basic sets Ai,,.,,in at step n of the construction may intersect each other (although the interiors of the balls Bi,,,,inmust be disjoint). In [PWl], Pesin and Weiss also showed that a slight modification of Moran's approach allows one to deal with geometric constructions modeled by an arbitrary symbolic dynamical system (Q, a).The main tool of study, as explained above, is the thermodynamic formalism described in Chapter 4 and Appendix 11. In particular, regardless of the placement of basic sets the Hausdorff dimension and lower and upper box dimensions of the limit set F coincide and the common value is a unique root of Bowen's equation

where cp(w) = logxi, for any w = (ioil .. . ) E Q. Moreover, the push forward to F by the coding map of an equilibrium measure p on Q corresponding to the function scp is an invariant measure m of full dimension, i.e., dimH F = dimH m (see discussion in Section 5). It is worth emphasizing that if prpis a Gibbs measure (this happens, for example, if Q = Z .: for some transitive transfer matrix A ) the Hausdorff measure of F is equivalent to m,; in particular, it is positive and finite. In [St], Stella considered a particular case of geometric constructions modeled by subshifts of finite type assuming that basic sets do not overlap and satisfy Conditions (CM1) and (CM4). In [Bars], Barreira pointed out that using a non-additive version of the thermodynamic formalism, one can generalize results by Pesin and Weiss to Moran-like geometric constructions with non-stationary ratio coefficients where the radii of balls Bi,,,,inmay depend on the whole past (io . . .in) (see Section 15). Moran-like geometric constructions with stationary or non-stationary ratio coefficients can serve as models to approximate geometric constructions with to be inarbitrary geometry of basic sets. For example, one can choose Bi,...,,, scribed balls and use their radii to estimate the Hausdorff dimension (from below) of the limit set. We note that Moran-like geometric constructions are isotropic, i.e., the ratio coefficients do not depend on the directions in Rm. There are geometric constructions which do not satisfy this assumption and have different rates of contraction in different directions in Rm. Examples include geometric constructions in R2 with basic sets to be rectangles or ellipsis (see Sections 14 and 16). Nevertheless, the idea of approximating geometric constructions (CG1-CG2) with complicated geometry of basic sets (which may not even satisfy separation

122

Chapter 5

condition (CG3)) by Moran-like geometric constructions can be fruitfully used to some extent. We call a geometric construction regular if it admits such an approximation. We stress that basic sets of a regular geometric construction need not resemble balls in any geometrical sense. Roughly speaking the regularity of a geometric construction means that its basic sets can be effectively replaced by balls such that the new geometric construction admits a Moran cover. In Section 14 we give a formal mathematical definition of the notion of regularity and study some properties of regular constructions. In particular, we obtain a general lower bound for the Hausdorff dimension of the limit set of the construction (see Theorem 14.1). One can further use the approach, based on an effective replacement of the basic set of a geometric construction (CG1-CG2) by disjoint balls, to obtain a general upper bound for the upper box dimension of its limit set (see Theorem 14.5). In general, it may be extremely difficult to find an effective approximation of a given geometric construction; for instance, inscribed balls often fail to produce a reasonable approximation (see Example 14.1). A more sophisticated approach to the description of geometric constructions with complicated geometry of basic sets is based upon the study of the induced map G on the limit set F generated by the symbolic dynamical system. We notice that if basic sets of a geometric construction are disjoint the coding map is injective. Therefore, the shift u induces a map G on the limit set F given by G = x o (T o x-'. The dynamics of G bears the complete information on the symbolic representation of the construction (via the map ulQ) as well as on its geometry (via the coding map x,i.e., the embedding of the symbolic dynamics into R" by x). It seems quite plausible that those characteristics of the dynamics of G that are connected to the instability of its trajectories are relevant for computing the Hausdorff dimension and box dimension of F . This can be clearly seen when G is an expanding map. In Section 15 we introduce appropriate characteristics of instability and demonstrate how to use them in estimating the Hausdorff dimension and box dimension of the limit set. In dimension theory there is a popular class of geometric constructions, known as self-similar geometric constructions, which are effected by a finite collection of similarity maps. This means that the basic sets Ai,,,,i,, are given bY Aio,,.in= hi, o hi, o . . . o hin(D), where h l , . . . ,h,: D + D are conformal affine maps, i.e., they satisfy the p r o p erty: dist(hi(x), hi(y)) = X i dist(x, y) for any x,y E D (where D is the unit ball in R") with fixed 0 < X i < 1 (for detailed description of self-similar constructions and related results see [Fl] where further references can also be found). These constructions are a very special type of Moran geometric constructions (CM1-CM5) where not only the sizes of basic sets but also the gaps between them are strongly controlled. A more general class of geometric constructions is formed by geometric constructions with contraction maps where the maps hi are bi-Lipschitz

Dimension of Cantor-like Sets and Symbolic Dynamics

123

contraction maps. This means that for any x, y E D,

Aidist(x, y) 5 dist(hi(x), hi(y)) 5 xi dist(x, y), where 0 < Ai 5 xi

< 1. These constructions are regular (see Section 14).

13. Moran-like Geometric Constructions with Stationary (Constant) Ratio Coefficients We begin with a geometric construction with the simplest geometry of basic sets modeled by a symbolic dynamical system (Q, a),where Q c CP+ is a compact shift invariant set. We assume that basic sets are closed and satisfy: (CPW1) Aio...injC AiO...i,,for j = 1,.. . ,P; (CPW2) C Ai,,...i, C Bio...i,, where balls of radii E ~ ~ ,and , . ~i0,,,,,,; ~ ~ (CPW3)

Bio...i,and BiO . . . i ,

n

are closed

n

(13.1) where 0 < X j < 1 for j = 1,.. . , p , K1 > 0, and Kz (CPW4) intl?io...i, n intEjo,..jm = 0 for any m

2 n.

> 0 are constants; ( 2 0 . . .in) # ( j o , . . . ,jm) and

This class of geometric constructions was introduced by Pesin and Weiss in [PWl]. Basic sets of these constructions are essentially balls, although their topology and geometry may be quite complicated. Furthermore, basic sets Aio.,.inat step n of the construction may intersect each other. The numbers X i are called ratio coefficients. They are fixed and do not depend on the basic sets. Self-similar constructions (see below) or more general Moran geometric constructions (CM1-CM5) (see the introduction to this chapter) are particular examples of geometric constructions (CPW1-CPW4). The geometric simplicity of the geometric constructions (CPW1-CPW4) will allow us to illustrate better the role of symbolic dynamics. Given a ptuple X = (XI,. . . ,A,) such that 0 < X i < 1, there exists a uniquely defined nonnegative number sx such that PQ(sxlOgXi,) = 0 (we remind the reader that PQdenotes the topological pressure on Q with respect to the shift o;see Section 11 and Appendix 11).Denote by x:Q + F the coding map (see definition in the introduction to this chapter). Note that it is Holder continuous. To see this let w1 = (ioil . . . i n j . . .) and w2 = (ioil .. . i n k . . .) be two points in Q with j # k. We have "

124

Chapter 5

where C > 0, 0 < a < 1 are constants and dp is the metric in C: (see (A2.18) in Appendix 11). This implies that any Holder continuous function on F pulls back by x to a Holder continuous function on Q. Notice that, in general, the coding map x is not invertible (since basic sets at the same level of the construction may intersect each other) and even if it is invertible it is not necessarily Holder continuous (this depends on the placement of the basic sets, i.e., the gaps between them). Let px be an equilibrium measure for the function ( i o i l . . . ) ++ sx log Xi, on Q, and mx the push forward measure to F under x (i.e., m x ( 2 ) = px(x-'(Z)) for any Bore1 set 2 c F ) . We describe a special cover of the limit set F for a symbolic geometric construction (CPW1-CPW4) which allows one to build an optimal cover to be used to compute the Hausdorff dimension and box dimension of F . We call this cover a Moran cover. Let X = ( X I , . . . ,Ap) be a vector of numbers with 0 < X i < 1, i = 1,.. . , p . Fix 0 < r < 1. Given a point w = ( i o i l .. .) E Q, let n ( w ) = n ( w , r , A) denote the unique positive integer such that Xi,Xi,

. . . Xin(,, > r , &,Xi, . . .Ain(w)+lI T.

(13.2)

It is easy to see that n ( w ) -+ 00 as r + 0 uniformly in w . Fix w E Q and consider the cylinder set C,,,,,i+,) c Q. We have that w E Ci,...i Furthermore, if w' E C,o,,.in(u) and n(d)5 n ( w ) then

Ci0...in(,) c ciO...in(",). Let C ( w ) be the largest cylinder set containing w with the property that C ( w ) = Cio,,.in(,,,) for some w" E C ( w ) and C~o...~n(w,) c C ( w ) for any w' E C ( w ) . The sets C ( w ) corresponding to different w E Q either coincide or are disjoint. We denote these sets by C(j),j = 1,.. . ,Np. There exist points w j E Q such that C(j) = C;o...in(wj). These sets form a disjoint cover of Q which we denote by U, = U , , Q ( X ) . The sets A(j) = x ( C ( j ) ) j, = 1,. . . ,N,. are not necessarily disjoint and comprise a cover of F (which we will denote by the same symbol U, if it does not cause any confusion). We have that A(j) = Aio...in(zj) for some x ~ jE F . Given a subset R c Q (not necessarily invariant) one can repeat the above arguments to construct a Moran cover of R which we denote by U,,R(X). It where w j E R and the intersection C(j) n consists of sets C(j) = Cio,,,,n(uj),

C@)n R is empty for any j # k (while the intersection C(j)n C @ )may not be

empty1. The crucial role that Moran covers play in studying the Hausdorff dimension and box dimension of F can be understood in view of the following observation. Given a point x E F and a number r > 0, there exists a number M > 0 which does not depend on x, r , and X and satisfies the following property: the number of basic sets A(j) in a Moran coverU,,Q(X) that have non-empty intersection with the ball B ( x ,r ) is bounded f r o m above by M . We call M a Moran multiplicity factor. Moreover, given a subset R c Q, a point x E F , and a number r > 0, the number of basic sets A(j) in a Moran cover U,,R(X) that have non-empty intersection with the ball B ( x ,r ) is bounded from above by M .

Dimension of Cantor-like Sets and Symbolic Dynamics

125

Theorem 13.1. [PWl] Let F be the limit set for a geometric construction (CPWI-CPW4) modeled b y a symbolic dynamical system ( & , a ) .Then (1) dimH F = h B F = d i m ~ F= sx; (2) d'imH mx = sx; (3)

Proof. Set s = sx and d = dimH F . We first show that s 5 d. F i x E > 0. By the definition of Hausdorff dimension there exists a number r > 0 and a cover of F by balls Be, l = 1 , 2 , . . . of radius re 5 r such that (13.3)

For every l > 0 consider a Moran cover qf of F and choose those basic sets Note that from the cover that intersect Be. Denote them by A?', . . . ,AZ""". A?) = Aio,,.in(f,j) for some (io.. . i n ( e , j ) ) . By (13.1) and (13.2) it follows that

where C1 > 0 is a constant independent of l and j . The property of the Moran cover implies that m(l)5 M , where M > 0 is a Moran multiplicity factor (which is independent of l ). The sets {A?), j = 1 , . . .,m ( l ) , l = 1 , 2 , . . . } comprise a cover B of F , and the corresponding cylinder sets Cf' = Cio,,,in(t,j) comprise a cover of Q. By (13.3) and (13.4)

Given a number N > 0 , choose r so small that n ( e , j ) 2 N for all l and j . We now have that for any n > 0 and N > n ,

126

Chapter 5

where M ( Q ,0, cp,Un,N ) is defined by (A2.19) (see Appendix 11) with p ( w ) = (d

+

E)

log xi,

(and a = 0). This implies that PQ((d+E)logXi,) 5 0. Hence, by Theorem A2.5 (see Appendix IT), s 5 d + ~ .Since this inequality holds for all E we conclude that s 5 d. Denote d = d i m ~ F We . now show that d 5 s. Fix E > 0. By the definition of the upper box dimension (see Section 6) there exists a number r = r(&)> 0 such that N ( F , r ) 2 rE-' (recall that N ( F , r ) is the least number of balls of radius r needed to cover the set F ) . Consider a Moran cover U, of Q by basic j = 1,...,N,.. Let A(j) = x ( C ( j ) )= Ai,...i,(,,,, where sets C(j) = Cio,..in(w.), xj = ~ ( w j ) Note . that this cover need not be optimal, i.e., N,. 2 N ( F , r ) . By (13.2) there exists A > 1 such that for j = 1,.. . ,N,.,

and hence

1 A Czlog- - 15 n ( w j ) 5 C310g- + l , r r where Cz > 0 and C3 > 0 are constants. This implies that n ( w j ) can take on at def most B = C3 log - Cz log 2 possible values. We now think of having N,. balls and B baskets. Then there exists a basket containing at least balls. This implies that there exists a positive integer 11 such that N E [C2log - 1,C3 log

4

4+

9

4+

N,. N ( F , r ) card { j : n ( w j ) = N } 2 B -> 2~

rE-'

c,log p '

where card denotes the cardinality of the corresponding set. If r is sufficiently small we obtain that card { j : n ( w j ) = N } 2 rZc-'. Consider an arbitrary cover

of Q by cylinder sets Cio...iN . It follows that

Dimension of Cantor-like Sets and Symbolic Dynamics where

C4

127

> 0 is a constant. We now have that for any n > 0 and N > n,

N

where R(Q,0, p,U,, N ) is defined by (A2.19') (see Appendix 11) with a = 0 and p(w) = ( d - 2&)logXio.

By Theorem 11.5 this implies that

and hence d - 2~ 5 s (see Theorem A2.5 in Appendix 11). Since this inequality holds for all E we conclude that d 5 s. This completes the proof of the first statement. In order to prove the second statement we need only to establish that s I dimH mx. Assume first that the measure p~ is a Gibbs measure corresponding to the function slog Xi,. By (A2.20) (see Appendix 11) there exist positive constants D1 and Dz such that for j = 1,. . .,Nr (13.5) Consider the open Euclidean ball B ( x ,r ) of radius r centered at a point x . Let N ( x , r )denote the number of sets A ( j ) that have non-empty intersection with B ( a , r ) . It follows from the property of the Moran cover that N ( z , r ) 5 M , where M is a Moran multiplicity factor. By (13.5) and (13.2) we obtain that for every x and every r > 0,

j=1

j=1

k=O

where C5 > 0 is a constant. It follows that the measure mx satisfies the uniform mass distribution principle (see Section 7) and hence dimH mx 2 s. We turn to the general case when px is just an equilibrium measure. By definition h,, (4Q) + s log X i o h = 0, (13.7)

Chapter 5

128 def

where h,,(alQ) = h is the measure-theoretic entropy. Let us first assume that is ergodic. Fix E > 0. It follows from the Shannon-McMillan-Breiman theorem that for PA-almost every w E Q one can find N l ( w ) > 0 such that for any n 2 Nl(W), PA(Go..&(w)) I exp(-(h - E b ) , (13.8) where Cio.,,,,,(w) is the cylinder set containing w. If the measure is ergodic it follows from the Birkhoff ergodic theorem, applied to the function slogXi,, that for PA-almost every w E Q there exists N z ( w ) such that for any n 2 N z ( w ) , (13.9)

Combining (13.7), (13.8), and (13.9) we obtain that for PA-almost every w E Q and 71 2 max{Nl(w),Ww)}, n

n

j=O

j=O

where a = 2 ~ minj / log(l/Xj) > 0. This implies that for PA-almost every w E Q and any n 2 max{Nl(w),Nz(w)), (13.10) j=O

If is not ergodic, then (13.10) is still valid and can be shown by decomposing ,LLA into its ergodic components. Given C > 0, denote by Qe = {w E Q : N l ( w ) 5 C and N z ( w ) I C}. It is easy to see that Qe c Qe+l and Q = Qe (mod 0 ) . Thus, there exists CO > 0 such that ,u~(Qe) > 0 if C 2 t o . Let us choose C 2 CO. Given 0 < r < 1, consider a Moran cover &,Q( of the set Qe. It consists of sets Cf), j = 1,. . . ,Nr,e for which there exist points w j E Q such that Cf' = c~~...~,,(~,). Set A?) = x ( ~ f ) ) . Consider the open Euclidean ball B ( x ,T ) of radius r centered at a point 2. Let N = N ( x , r , C ) denote the number of sets A?' that have non-empty intersection with B ( x ,r ) . By the property of the Moran cover we have that N M , where A 4 is a Moran multiplicity factor. It now follows from (13.10) and (13.2) that

u,"=,


0. We now prove that mA(.) 5 const x ~ H ( . , s A ) .Given 6 > 0 and a Bore1 subset 2 c F, there exists E > 0 and a cover of 2 by balls Bk of radius r k 5 E satisfying (Tk)"' 5 mH(F, S x ) + 6.

c k

It follows from (13.6) that

where C1 > 0 is a constant. mx (2)5 ClMmH(F, .A).

Since 6 is chosen arbitrarily this implies that

132

Chapter 5

We now show that m ~ ( SX) . , 5 const x mx(.). Let Z c F be a closed subset. Given 6 > 0, there exists E > 0 such that for any cover U of Z by open sets whose diameter 5 E we have

+

(diamU)SX 6.

~ H ( Zsx) , 5

(13.11)

UEU

Note that one can choose a cover U of Z by basic sets A(k) = diamA(k) 5 E and mx(A(')) 5 mx(2) 6.

C

satisfying

+

A(k)EU

We can apply (13.11) to this cover U and obtain using (13.1) and (13.5) that

C

5K ~ D ; ~

+

+

m x ( ~ ( k ) )6 5 K ~ D ; ~ ( z )( K ~ D ; '

+ i)6.

A(k)EU

Since 6 is chosen arbitrarily this implies the second statement. Fix 0 < T < 1. For each w E Q choose n(w) according to (13.2). It follows from (13.1) that A;o...in(,)+l c B ( x , K z r ) ,where x = ~ ( w ) .By virtue of (13.5) for all w E Q,

where Cz > 0 is a constant. It follows that for all x E F,

This implies the third statement. We now prove the last statement. It follows from the second statement that mH(F n A, , . . . ~ , , , S A ) > 0. Thus, dimH(F n A; o . . . i n ) 2 sx. Now, let U be any open set with F n U # 0 . If x = x(i0il . . .) E F n U and n > 0 is sufficiently large then Ai,..,;,, c U . Therefore, SA

L dimH(F n U ) 2 dimH(F n Aio...in) 2 sA.

This completes the proof of the theorem.

Dimension of Cantor-like Sets and Symbolic Dynamics

133

Self-similar Constructions There is a special class of geometric constructions of type (CG1-CG2) which are most studied in the literature (see for example, [Fl])- self-similar geometric constructions. They are geometric constructions with basic sets Aio,,,in given as follows: Ai,,,,i, = hi, o hi, o . . . 0 hi, (D), where h l , . . . , hp:D + D are conformal affine maps (i.e., maps that satisfy dist(hi(z),hi(y)) = Aidist(z,y) for any z , y E D, where D is the unit ball in Rm). Here 0 < Xi < 1 are ratio coefficients. These geometric constructions can be modeled by an arbitrary symbolic dynamical system (Q, g). Clearly, self-similar constructions are a particular case of Moran geometric constructions with stationary ratio coefficients (CPW1-CPW4). Therefore, by Theorem 13.1, the Hausdorff dimension and lower and upper box dimensions of the limit set of a self-similar construction coincide. The common value is the unique root of Bowen’s equation P ~ ( s ’ p= ) 0, where ~ ( w = ) logxi, for w = (i& . . .) (if a self-similar construction is modeled by a subshift of finite type or the full shift the number s can be computed as stated in Theorem 13.3). If we further assume that basic sets at the same level of a self-similar construction do not overlap (i.e., their interiors are disjoint) then the geometric construction is a Moran geometric construction (CM1-CM5). Moreover, if basic sets at the same level are disjoint the coding map is a homeomorphism. Thus, the induced map G on the limit set of the self-similar construction (which in this case is a smooth map) provides a smooth realization of the subshift ( Q ,a). We also remark that if the ratio coefficients of the maps hi are equal (say, to a number A) then the coding map is an isometry between the limit set of the geometric construction and Q (endowed with the metric dA-l; see (A2.18) in Appendix 11). Thus, it preserves the Hausdorff dimension and box dimension which can be computed by Statement 2 of Theorem 13.2 (compare to Theorem A2.9 in Appendix 11).

14. Regular Geometric Constructions In this section we follow Pesin and Weiss [PWl] and introduce a class of geometric constructions that admit approximations by Moran-like geometric constructions with stationary ratio coefficients. This allows us to obtain effective lower bounds for the Hausdorff dimension of the limit sets. We will control the geometry of basic sets by numbers 71,. . . ,yP such that one can replace the basic sets Ai,,,,i,, by balls of radius yij. In some cases these balls coincide with the largest balls that can be inscribed in the basic sets. However, this is not always the case and below we present an example where the “optimal” numbers 7 1 , . . . ,yP are completely independent of the radii of the largest inscribed balls (see Example 14.1 below). Consider a geometric construction (CG1-CG2) modeled by a symbolic dynamical system (Q, 0) (see the introduction to this chapter; it is worth emphasizing that we do not require the separation condition (CG3)). Given 0 < T < 1 and a vector of numbers y = ( 7 1 , . . . ,y,), 0 < -yi < 1, i = 1,.. . , p , consider a

ny=,

134

Chapter 5

Moran cover U, = 4 ( y ) = {A(j)} of the limit set F constructed in Section 13. Given an open Euclidean ball B ( z ,T ) of radius T centered at z, denote by R(z,r ) the number of sets A(j) that have non-empty intersection with B ( z ,r ) . We call a vector y estimating if R ( z ,r ) 5 constant (14.1) uniformly in z and T . We call a symbolic geometric construction (CG1-CG2) regular if it admits an estimating vector. If y = (71,.. . ,y p ) is an estimating vector for a regular geometric construction, then any vector y = (71,. . .,yP) for which yi 2 Ti,i = 1,. . . , p is also estimating. We provide an example of a regular geometric construction on the plane that illustrates how the choice of the estimating vector can be made.

1

r, Figure 3. A REGULAR GEOMETRIC CONSTRUCTION. Example 14.1 [PWl] Let yl,y2,y3, and A be any numbers in (0,l). Given a number i = 1,2,3 consider a simple geometric construction (CG1-CG2) on the interval [0,1] x {i - 2) with 2" basic sets of size yi" at step n. We denote this construction by CG(yi). Since the 2" intervals at step n in each of these constructions are clearly ordered we may refer to the ith subinterval at step n, 1 5 i 5 2" of these constructions. Consider the 2" polygons in [0,1] x [-1,1] having six vertices which consist of the two endpoints of the ith subinterval at step n for all three constructions. We define the 2" basic sets at step n by intersecting these 2" polygons with the rectangle [0,1] x [-An, A"]. This produces a simple geometric construction (CG1-CG2) on the plane. See Figure 3.

Dimension of Cantor-like Sets and Symbolic Dynamics

135

It is easy to see that the limit set F of this geometric construction coincides with the limit set of the construction CG(y2). Hence, by Theorem 13.3, dimH F = and does not depend on y1,y3, or A. Let us choose numbers y1,y2, 7 3 , and X such that 7 2 < y1 = 7 3 < X and 7 2 < Xyl or 7 2 < Xy3. One can see that the inscribed and circumscribed balls of the basic sets at step n have radii which are bounded from below and above by ClyF and C2Xn, respectively, where C1 and C2 are positive constants which are independent of n. Thus, these balls cannot be used to determine the Hausdorff dimension of the limit set. W Consider the positive number s, such that P~(s,logyi,) = 0, where PQ denotes the topological pressure with respect to the shift u on Q (see Section 11). Let p, denote an equilibrium measure for the function (ioil . . . ) ++ s, log yi, on Q, and let m, be the push forward measure on F under the coding map x (i.e., m7(Z) = p 7 ( x P 1 ( Z ) for ) any Bore1 set Z c F). The following result provides a lower bound for the Hausdorff dimension of the limit set. Its proof is quite similar to the proof of Statement 2 of Theorem 13.1.

Theorem 14.1. [PWl] Let F be the limit set for a regular symbolic geometric construction. Then dimH F 2 s7 for any estimating vector y. Hence, dimH F 2 sup s7, where the supremum i s taken over all estimating vectors y. In the case when the measure p, is Gibbs one can strengthen Theorem 14.1 and prove a statement that is similar to Theorem 13.4.

Theorem 14.2. [PWl] Let F be the limit set of a regular symbolic geometric construction and y an estimating vector. Assume that the measure p7 i s a Gibbs measure. Then (1) the measure m, satisfies the uniform mass distribution principle; (2) 0 < mH(F, s,); moreover, m 7 ( Z ) 5 CmH(Z, s7) for any measurable set Z c F, where C > 0 is a constant; (3) s7 5 &Jx) f o r every x E F; (4) dimH(F fl U ) 2 s, > 0 for any open set U which has non-empty intersection with F. The second statement of Theorem 14.2 is non-trivial only when s, = dimH F. Otherwise, mH(F, s7) = 00. If s7 < s = dimH F, then the s-Hausdorff measure may be zero or infinite. Theorem 14.2 holds for simple geometric constructions or Markov geometric constructions with transitive transfer matrix. In [Barl], Barreira gave sufficient conditions for a geometric construction to be regular. Roughly speaking, it requires that the basic sets contain sufficiently large open balls. We begin with geometric constructions on the line.

Theorem 14.3. Assume that each basic set Aio,,,i,, of a symbolic geometric construction (CG1-CG2) on the line contains an interval Ii,...i,, of length 0 < X i,...in < 1 such that Ii,...in nIj o...jn= 0 for any ( i o . . .in) # ( j o . . .jn). Assume also that there exists 0 < y < 1 such that

lirn min -1 log Xio...in 2 logy,

n--fw

n

136

Chapter 5

where the minimum is taken over all Q-admissible n-tuples (io . . . i n ) . Then the geometric construction is regular with the estimating vector (Ye-€, . . . ,Y e p E )for any E > 0 .

Proof. Given E > 0, we have A i o . . . i n > ( y e - € ) " for every (ioil .. . ) E Q and any sufficiently large n. Given r > 0, one can find a unique number n = n ( r ) > 0 such that 5 r < (re-')". For any interval I of length r there exist at most two basic sets of length 2 (re-')" intersecting I . Therefore, for every point x in the limit set the number R ( x ,r ) in the definition of regular geometric constructions (see (14.1)) we obtain that R ( x , r ) 5 2 for all sufficiently small r . Hence, the construction is regular with the estimating vector (Ye-€, . . . ,Ye-"). H We now formulate a criterion of regularity for a geometric construction in Rm with m > 1. Fix a point x E F and a number A > 0. Given n > 0, consider two basic sets Aio...inand Ajo...jnintersecting the ball B(x,An). Denote by a(Aio...i,,, A, o...jn) the minimum angle of spherical sectors centered at x which ,. Let a,(x, A) be the minimum of all the angles contain both Aio,,.inand Ajo...j a(&o . . . i , , A jO . . . j , , ) .

Theorem 14.4. Assume that each basic set of a symbolic geometric construction (CGI-CG2) in Rm with m > 1 contains a ball Bi,...i, c Aio...i,,of radius A" with 0 < A < 1. Assume also that there exists 6 > 0 such that an(x,A) 2 6A" for all x E F and n 2 1 and that Bi o . . . i ,nBjo...jn = 0 f o r any ( i o . . .in) # ( j o . . .jn). Then the geometric construction is regular with the estimating vector (A,. . . , A).

Proof. By elementary geometry there exists a universal constant C = C(6) such that the maximum number of sets Aio,,,inintersecting a ball B(A") does not exceed C. This proves the result. Consider a simple regular geometric construction in Rm with the limit set F. It follows from Theorem 14.2 that the Hausdorff dimension of any open set U which has non-empty intersection with F satisfies dimH(F n U ) 2 s for some s > 0. The following example shows that the converse statement may not be true.

Example 14.2. [Barl] For each s E (0, l), there exists a geometric construction (CGI-CG2) on the line modeled by the full shift ( C t , u) such that (1) dimH(F n U ) = s for any open set U with F n U # 0 ; ( 2 ) the construction is non-regular.

Proof. Define the function m:Q m(w)= m(ioi1... ) =

+ N U {+m} by

{

w=o

+m,

least j E

N with i j = 1,

w

# 0.

Dimension of Cantor-like Sets and Symbolic Dynamics

137

0

1

Figure 4. A NON-REGULARGEOMETRIC CONSTRUCTION. Define also the numbers Xio . . . i , by

Xo(j)"

+ Xl(j)" = 1

(14.2)

for each j > 0. We consider basic sets spaced as shown on Figure 4. They have ~ b, E Aio,,.i,,l. the following property: if Ai o...in = [a,, b,] then a, E A i o , , , i nand Define intervals A(j) = Aio,,,ij,where (20.. . ij) = (0. . .Ol). Inside each A(j) we have a sub-construction modeled by (Ct, g ) with rates Xo...oli,...i,/X~...~l = Xi, ( j 2). Therefore, the Hausdorff dimension of F n A(j) is equal to s, where s is the unique root of equation (14.2), with j 2 instead of j (see Theorem 13.3). Hence, dimH(F n A(j)) = s. Since F = (0) U Uj>o(Fn A(j)) it follows that dimH F = dimH(F n A(j)) = s. Now, if F n U # 0 , there exists x = x(ioi1.. .) E ( F n U ) \ {0}, and n > 0 such that Aio,,,inc U . Hence, s = dimH F 2 dimH(F n U ) dimH(F n Aio,,,in)= s. This proves the first stat ement .

nz=o +

+

>

Consider a vector (y1,yz).For each n > 0, set T , = 2/(n - I)!. Select now yi, > T, and yij 5 T, for the smallest positive integer k such that some ( i o i l . . .) E E$. Set y = min{yl,ya} and observe that yk+' 5 T,. Hence, k 2 logr,/logy-l. Therefore, fork, = logr,/logy-l weget R(O,T,) 2 2',-" (where R(O,r,) is defined in (14.1)). We also have

n,"=,

k,-n>

log 2 - log(n - I)!

1% 7

-1-n

x

n+m

log n! -- n -logy

n,":

x

,+m

log n! -logy'

-

138

Chapter 5

Therefore, there exists D obtain

> 0 such that k ,

- n 2 Dlogn! for all n

> 0, and we

As the sequence r, decreases monotonically to 0 and (y1,yo) is arbitrary we proved that the construction is non-regular. If a geometric construction is regular one can effectively replace its basic sets by balls to obtain a lower bound for the Hausdorff dimension of its limit set (see Theorem 14.1). We further exploit this approach and show that, under some mild assumption, a geometric construction (CG1-CG2) (whose basic sets are, in general, arbitrary close subsets and are possibly intersecting) can be effectively compared with a geometric construction whose basic sets are disjoint balls.

Theorem 14.5. Let F be the limit set of a geometric construction (CGI-CG2) modeled by a symbolic dynamical system (&,a).Assume that there exist numbers XI,. . .,A,, 0 < X i < 1 such that for any admissible n-tuple ( i o . . .in), ij = 1 , . . . , p , we have

n n

diam Ai ,...in 5 C

Xij,

(14.3)

j=O

where C > 0 is a constant. Then (1) there exists a self-similar geometric construction modeled by a symbolic dynamical system ( Q ,a ) satisfying: a) its basic sets Bio...i, are disjoint X i J , and c) there is a Lipschitz continuballs, b) diam Bi,...i,, = 2C ous map from its limit set F onto F ; (2) d i m ~ F5 S X .

+

nj”=,

Proof. Consider a self-similar geometric construction with ratio coefficients X I , . . . , A, modeled by a symbolic dynamical system (Q, a) whose basic sets Bi,...i, are disjoint balls of radii Cn;=, Ai,. Denote its basic set by F . Let x:Q -+ F and 2:Q -+ F be the coding maps. Consider the map II,= X O X - ~ F: -+ F . We shall show that $ is a (locally) Lipschitz continuous map. Choose x,y E F with p ( x ,y ) 5 E . We have that k-’(x) = (ioil.. .), %-‘(y) = ( j o j l . . .), and i o = . . j o , . . . ,in = jn,z+,1 # jn+l for some n > 0. Therefore, 112 - yII 2 C1 Xij, where C1 > 0 is a constant. One can also see that $ ( x ) , $ ( y ) E Aio...i”. Hence, by (14.3), 11$(z) - $(y)II 5 Cnj”’, Xij. Since the map $ is onto this proves the

nj”=,

first statement. The second statement follows immediately from the first one. W

Geometric Constructions with Ellipsis We describe a special class of regular geometric constructions. We say that a geometric construction (CGlLCG3) in R2 is a construction with ellipsis (see Figure 5) if each basic set Aio.,.inis an ellipse with axes An/2 and r / 2 , for some 0 < < < 1 (we stress that we require the separation condition (CG3)). Such constructions were studied by Barreira in [Barl].

139

Dimension of Cantor-like Sets and Symbolic Dynamics

Theorem 14.6. Let F be the limit set of a geometric construction with ellipsis. Then the construction is regular with the estimating vector (A, A), where X is any number in the interval (0,A2/x). Proof. For each X E ( 0 , l ) define the function g x : ( 0 , l ) x ( 0 , l ) -+ W by

g x ( X , 1)=

{

[(X/X2)"]

(X/X)t,

1'2,

x x

where t = logA/logX. Consider an ellipse En with axes X a n / 2 and p n / 2 , where a = logX/logx. We assume that it is located outside the ball B(O,An),is tangent to this ball at a point, and that the major axis of the ellipse points towards 0. Denote by 2 P(n) the smallest angular sector centered at 0 that contains En. The desired result follows immediately from Theorem 14.4 and the following lemma.

-

Lemma. For each X E (O,x),there exists a number C > 0 such that tanP(n) Cgx(X,X)-" as n -+ m. I n particular, P(n) decreases exponentially with the rate g x ( X , X ) - l when A2/x < X < and is uniformly bounded away from 0 i f 0 < x 5 a"/.

x

Figure 5. A CONSTRUCTION WITH ELLIPSIS. Proof of the lemma. Consider an orthogonal coordinate system centered at 0 with the z-axis directed along the major axis of En. If m(n) = tanP(n) is the slope of a line starting at 0 and tangent to En, the points of tangency are solutions of the equation

(

x - - ~ n - - i ~ ~ / 2 ) (' r+n ( n ) x ) ' = 1, xan/2 xan/2

140

Chapter 5

provided that the discriminant of this equation is zero:

Set b, = A / r . One can see that 4m(n)' = blan/[bE(bE + l)].We consider the following cases:

x;

-

(a) X > then b, > 1 and 2m(n) ( b 1 2 a / b , 2 ) n / 2 = = gx(X,l))-n; (b) X = 1;then b, = 1 and 2 f i m ( n ) (bla)n = g ~ ( X , x ) ; ) - ~ ; (c) X < 1 ;then b, < 1 and 2m(n) (bIza/ba) 4 2 - gx(X,x)-". It is easy to check that if A'/ 1 < X < 1,we have gx(A, 1)> 1 and if 0 < X 5 A'/ we have gx(A, 1)5 1. Since t a n o o as x + 0 the desired result follows. W N

N

-

x,

15. Moran-like Geometric Constructions with Non-stationary Ratio Coefficients

In this section we study Moran-like geometric constructions with ratio coefficients at step n depending on all the previous steps. This class of geometric constructions was introduced by Barreira in [Bar2]. Consider a geometric construction modeled by a symbolic dynamical system (&, .). We assume that alQ is topologically mixing and the following conditions hold: (CB1) Aio ...i n j C Aio ...in for j = 1,.. . ,P; (CB2) &,...in c Aio ...in c Bio ...i n , where &o,,,in and Bio,,.in are closed balls with radii Klr,,,..inand K ~ r i o . . . , , ' and 0 < K1 5 K z ; (CB3) int 13io,,,in n int i3jo,,,jm = 0 for any ( 2 0 . . . i n )# ( j o . . .jn)and m 2 n. The class of geometric constructions (CB1-CB3) is quite broad and includes geometric constructions (CPW1-CPW4) (see Section 13), geometric constructions with contraction maps (see below), and more general geometric constructions with quasi-conformal expanding induced maps (see Theorem 15.5 below). The study of geometric constructions (CB1-CB3) is based upon the non-additive version of the thermodynamic formalism (see Section 12 and Appendix 11). We define the sequence of functions Q = {pn:Q -+ JR} by

for each w = (i0il. . . ) E Q. We wish to consider Bowen's equations

and to show that they have unique roots. We shall use these equations to produce sharp estimates and sometimes obtain exact values of the Hausdorff dimension

Dimension of Cantor-like Sets and Symbolic Dynamics

141

and box dimension of the limit set of a geometric construction (CBl-CB3). We need the following two assumptions on the radii of the basic sets. The first assumption is the following: there exist L1 > 1 and L2 > 0 such that for any (ioil . . .) E Q and n 2 0 , Ti0 ...in

I L;”,

Ti0 ...i,i,+l

2 L2ri o . . . i , .

(15.3)

The role of Condition (15.3) is the following. First of all, one can easily derive from (15.3) that for any (ioil .. . ) E Q and any n 2 1, (15.4) epan I rio...in I e-Dn, where a and ,B are positive constants, i.e., the radii of basic sets decay exponentially. Therefore, the sequence of functions {pn} (see (15.1)) satisfies Condition (A2.22) (see Appendix 11). By Theorem A2.6 Bowen’s equations (15.2) have respectively unique roots 3 and s that satisfy 0 3 < 00. Furthermore, Condition (15.3) implies that for every n 2 0,

<
r and rio...in(~,)+l I r; (b) the corresponding cylinder sets C(j) = C,o...in(zJ) c Q are disjoint. Moran covers have the following crucial property: given a point x E F and a number r > 0 the number of basic sets Aio...i,(,j) - ~ ( C i ~ , , , i , ,in ( =a ~Moran )) cover U, that have non-empty intersection with the ball B ( z ,r ) is bounded from above by a number M , which is independent of z and r (a Moran multiplicity factor). Repeating arguments in the proof of Statement 1 of Theorem 13.1 one can prove the following result.

Theorem 15.1. [Bar21 Let F be the limit set of a geometric construction (CBICB3) modeled by a symbolic dynamical system (&,a).Assume that the numbers rio...i, satisfy Condition (15.3). Then

_s -< dimH F < dim,F I d i m ~ FI 3. The second assumption we need is the following: the sequence cp is sub-additive, i.e., ( P ~ 5 + vn+vm ~ oun on Q (see (12.3)). By virtue of (15.1) the sub-additivity of the sequence cp is equivalent to the following condition: Ti0 ...in+,

I rio ...in x ri,+l ...in+,.

(15.5)

We now establish the remarkable result that under assumptions (15.3) and (15.5) the Hausdorff dimension and box dimension of the limit set coincide regardless of the placement of the basic sets. The common value is the unique root ) 0. The proof uses Theorem A2.7 (see Appenof Bowen’s equation P ~ ( s ’ p= dix 11) and is a modification of the proof of Theorem 13.1.

142

Chapter 5

Theorem 15.2. [Bar21 Let F be the limit set of a geometric construction (CBlCB3) modeled b y a symbolic dynamical system (Q,u). Assume that the numbers ~ i ~ . . . isatisfy ,, Conditions (15.3) and (15.5). Then

-

dimHF = d&lBF= dimBF = 8 = 3 = s, where s is the unique root of Bowen's equations

The assumption that the sequence cp is sub-additive is crucial. The following example demonstrates that the Hausdorff dimension of the limit set of a geometric construction (CB1-CB3) may not agree with the upper box dimension if the sequence cp fails to be sub-additive.

Example 15.1. [Bar21 There exists a geometric construction (CBl-CB3) on the line modeled b y the full shift (E$,u)such that (1) each basic set A;,,...in is a closed interval of length depending only on n; (2) the sequence cp is not sub-additive; (3) 5 = dimHF = d&lBF < dimBF = 3.

Proof. Fix numbers b > a > 0 and choose 6 > 0 such that 6 < i ( b - a). Let mj be a sequence of positive integers. Denote

We define inductively the sequence mj. Set ml = 1 and choose mk such that Ink(a, b ) / n k - bJ < 6 / k , if k is odd, and Ink(a, b ) / n k - a1 < 6 / k , if k is even. w e introduce now the sequence of functions p,: E$ + R by %(U)

= -nrc(a, b) -

{ b(n -

nk)

a(n- n k )

where k is the largest positive integer such that def

nk

cp,(w) = const = A,

if k is odd if k is even

< n. Note that (15.6)

and pnk> nkpo. This implies that cp = (9,) is not sub-additive. One can build a geometric construction (CB1-CB3) on the line modeled by E$ with basic sets A,o..,;,, to be closed intervals of length expX,. Let F be the limit set. Clearly, N ( F ,ex-) 5 2". On the other hand, given an interval of length expX, there exist at most two basic sets which intersect it and at least one, if

Dimension of Cantor-like Sets and Symbolic Dynamics

143

the interval intersects F. Therefore, 2N(F, e'n) 2 2". Since An+l - A, 2 -b by Theorem 6.4 we have

h B F = -

lim n+m

log N ( F ,ex,

-An - log N(F, ex,)

dimBF = lim

n+m

n

log2

) = log2 x lim -= -,

-An

b

n+w -An

- n log2 = log2 x lim -= -. n+w-A, a

By Theorem 15.1 we obtain that dimH F 2 2,where 8 is the unique root of the first equation in (15.2). Since A, 2 -bn for all n > 0 we conclude that log 2 b

-= s = dimH F = h B F

-

__

log 2 a

< dimBF = -.

We need only to show that B = log2/a. Observe that for every s E R, n

log(2"exp(s~,)) = log2 (io...in)

where Cio,.,i, are cylinder sets. By definition A,,

+ -,SAn n

= -nk(a, b ) . Therefore,

Moreover, we have that -bn 5 A, 5 -an and thus, cP,;(scp) This implies that 3 = log 2/a.

= log2 - sa.

Pointwise Dimension of Measures on Limit Sets of Moran-like Geometric Constructions Let v be a Bore1 probability measure on the limit set F of a geometric construction (CBl-CB3) modeled by a symbolic dynamical system (Q, 0).We formulate a criterion that allows one to estimate the lower and upper pointwise dimensions of v. Given x E F, set

-

- log v(Ai0 ...i n )

d(x) = inf lim

n+m

log lAio...i,I '

where IAio...inI denotes the diameter of the basic set Aio,,,inand the infimum is taken over all w = (i&. . .) E Q such that x = ~ ( w ) .

Theorem 15.3. Assume that a geometric construction (CBl-CB3) satisfies Condition (15.3). Then (1) &(x) 5 d(x) for all x E F ; (2) d(x) 5 &(x) for u-almost all x E F ; def (3) if d(x) = a(.) = d(x) for u-almost every x E F , then &(x) = &(x) = d(x) for u-almost every x E F .

Chapter 5

144

Proof. Fix x E F and choose w = (ioil .. .) E Q such that x ( w ) = x. Given r > 0, choose n = n(r,w) such that IAio...i, I < r and lA~o,,,;n-l I 2 r. Since x E Aio.,,;, we have that A,o,..;, c B ( x ,r). This implies logv(B(x, r)) logv(Aio . . . ; , ) log r I logr ' We also have r I &~A~o...;,, 1, where Lz is the constant in (15.3). It follows that

Since this is true for every w = ( i ~ i..l .) E Q with x ( w ) = x the first statement follows. We now prove the second statement. Given a > 0 and C > 0, define

Q,,c = { w = (ioii . . .) E Q : v(Aio,,,i,) 5 CIAio...in, 1 for all n 2 0 ) . Set F,,c = x(Q,,c).If w = ( i o i l .. .) E Qa,cand x ( w ) = x then

Moreover, the set F, = Uc>oF,,c coincides with the set of points for which This will imply Statement 2. Indeed, if d(x) > &(x) on a set of positive measure then there exists a such that d ( x ) > a > &(x) on a set of positive measure. Fix x E F,,c and r > 0. Consider a Moran cover I&. of the set F,,c and choose those basic sets in the cover that have non-empty intersection with the ball B ( z ,r). By the property of the Moran cover there are points xj E F,,c, j = 1 , . . . ,M (where M is a Moran multiplicity factor which is independent of x and r) and basic sets A(j) such that xj E A(j) = Aio,,,,n(zj), diamA(j) I r, and

d(x) 2 a. We will show that &(x) 2 a for almost every x E F,.

B ( z ,r ) n

c

u M

(A(j) n F,,c).

j=1

It follows that V(B(Z,

r) n F , , ~ )I

M

C v ( ~ ( jn) F ) . j=1

Assume that v(F,,c)> 0. By the Bore1 Density Lemma (see Appendix V) for v-almost every x E F,,c there exists a number r o = ro(z) such that for every 0 < r 5 ro we have r) n F a d . ~ ( B ( x4, n F ) I 2 4 % This implies that for v-almost every x E F,,c,

where C1 > 0 is a constant. The last statement is a direct consequence of the preceding statements.

Dimension of Cantor-like Sets and Symbolic Dynamics

145

Let v be a Bore1 probability measure on the limit set F of a geometric construction (CBl-CB3). Even if the pointwise dimension d,(x) of v exists almost everywhere it may not be constant and may essentially depend on x. As Example 25.2 shows, the pointwise dimension may not be constant even if v is a Gibbs measure. This phenomenon is caused by the non-stationarity of geometric constructions (CB1-CB3). The situation is different for geometric constructions (CPW1-CPW4) where ratio coefficients do not depend on the step of the construction.

Theorem 15.4. Let F be the limit set of a geometric construction (CPWlCPWd) modeled by a symbolic dynamical system ( Q ,u) and p an ergodic measure on Q . Let also m be the push forward measure of p to F . Then m is exact dimensional (see Section 7) and f o r p-almost every w = (ioil . . .) E Q we have

where x = ~ ( w ) .

Proof. Since p is ergodic by the Birkhoff ergodic theorem applied to the function w H log A,, (where w = (ioil . . .) E Q) we have that for p-almost every w the following limit exists:

C

-L

l n lim log xi, -

n-tm n k=O

log xi, dp.

Exploiting again the fact that p is an ergodic measure, by the ShannonMcMillan-Breiman theorem we obtain that for p-almost every w = (ioil . . . ) E Q,

It follows from Condition (CPW3) that for p-almost every w = (i0il...) E Q, 1 1 " lim - log diam Ai,,.,in = lim log Xi, = n n-tm n

n-too

k=O

L

log Xi, d p .

The desired result follows now from Theorem 15.3.

Geometric Constructions with Quasi-conformal Induced Map Let F be the limit set of a geometric construction (CG1-CG3) in E P modeled by a subshift ( Q ,a).Since we require the separation condition (CG3) the coding map x:Q + F is a homeomorphism and the induced map G: F + F is well defined by G = x o u o x-'. We have the following commutative diagram

QLQ

FG'F

146

Chapter 5

It is easy to see that G is a continuous endomorphism onto F . By the result of Parry [Pa]it is a local homeomorphism if and only if the subshift is a subshift of ,,: where A is a transfer matrix. From now on we consider finite type, i.e. Q = E this case and assume that A is transitive, i.e., the shift is topologically mixing (see Appendix 11). The induced map encodes information about the sizes, shapes, and placement of the basic sets of the geometric construction and hence can be used to control the geometry of the construction. In order to illustrate this let us fix a number k > 0. For each w = ( i ~ i l .. .) E E , : and n 2 0, we define numbers

where the infimum and the supremum are taken over all distinct x,y E F n Aio...i,+& . It may happen that X ( w , n ) = 0 or X ( w , n ) = 03 for all sufficiently large n. In the case o < ~ ( wn,) 5 X(w,n) < co, consider the limits 1 1 ~ ( x=) lim - logX(w, n), ~ ( x=) lim - logX(w,n), n+wn "+a, n

-

where x = ~ ( w ) .Notice that by the multiplicative ergodic theorem (see for example [KH]), the limits exist for almost all x with respect to any Bore1 Ginvariant measure p on F provided that log- X(x, 1) d p < co,

log+X(x, 1) d p < 00.

(15.8)

If the map G is smooth the numbers X(x) and x ( x ) coincide with the largest and smallest Lyapunov exponents of G at x (see definition of Lyapunov exponents in Section 26). When G is continuous these numbers can serve as a substitution for the Lyapunov exponents (see IKi]). We consider the case when the trajectories of the induced map are strongly unstable. More precisely, we call the induced map expanding if there exist constants b 2 a > 1 and ro > 0, such that for each x E F and 0 < r < ro we have B(G(x), ar) c G(B(x,r)) c B ( G ( x )br). , (15.9) Note that if the induced map G is expanding then it is (locally) bi-Lipschitz. Furthermore, if the induced map G is expanding then the placement of basic sets of the geometric construction cannot be arbitrary (see Theorem 15.5 below). We now specify the choice of the number k in (15.7). Namely, we assume that k is so large that (15.10) diam Ai o...i,+, 5 To.

Dimension of Cantor-like Sets and Symbolic Dynamics

147

We say that an expanding induced map G is quasi-conformalif there exist numbers C > 0 and k > 0 (satisfying (15.10)) such that for each w E C: and n 2 0, (15.11) A(w, n ) 5 C M w , n). As the following example shows geometry of constructions (CGl-CG3) with expanding quasi-conformal induced maps (i.e., the placement of basic sets and their “sizes”) is sufficiently “rigid”.

Theorem 15.5. Let F be the limit set of a geometric construction (CG1-CG3) in Rm modeled by a subshift of finite type ( E f , u ) . Assume that the induced map G is quasi-conformal and that the basic sets on the first step of the construction have non-empty interiors. Then (1) the construction satisfies Conditions (CBl-CB3),i.e., it is a Moran-like geometric construction with non-stationary ratio coeficients; moreover, it also satisfies Conditions (15.3) and (15.5); (2) dimH F = h B F = d i m ~ F= s, where s is a unique number satisfying 1 lim -log (diam(F n Ai0...in))’= 0. n-+m n (io.. .in) Cf;-admissible

Proof. We outline the proof of the theorem. Since the basic sets on the first step of the construction have non-empty interiors we observe that each basic set Aio...inof the geometric construction satisfies

B.io...in C &...in

C

-

Bio...i n ,

and ~ ~ o . . are . i nclosed balls with radii rio,,,,n and where there exist numbers C1 > 0 and C2 > 0 such that

. Moreover,

o...in

Since the induced map G is quasi-conformal this implies that the geometric construction satisfies Condition (CB2). Clearly, Conditions (CB1) and (CB3) hold and thus, the construction is a Moran-like geometric construction with nonstationary ratio coefficients. By straightforward calculations one can show that given w = (ioil . . . ) E C: and n , m 2 1, X(w, n m) 2 X ( w , n) x X(.”(w), 4,

+

and similarly,

~ ( wn, m) 5 I(w,n ) x X(un(w),m).

+

Since the induced map G is quasi-conformal the above inequalities imply Condition (15.5). It follows from (15.9) and (15.10) that Thus, the first inequality in (15.3) holds. Similar arguments show that the second inequality also holds. This implies the first statement. The second statement follows from the first one and Theorem 15.2.

148

Chapter 5

One can build a geometric construction (CPW1-CPW4) for which the induced map on the limit set is not expanding: whether it is expanding depend on the placement of basic sets on each step of the construction (see below). We present now more sophisticated examples which illustrate properties of induced maps.

Example 15.2. [Bar21 There exists a geometric construction (CGI-CG3) on the line modeled by the full shift (C;', u) such that: (1) each basic set Aio,,,i,is a closed interval; (2) there exists a point w E C$ such that X(w, n) = 0 and x(u,n) = 00 for each n 2 1; hence, the induced map G is not expanding.

Proof. Consider a geometric construction on the line modeled by the full shift on 3 symbols for which Aio...i, = [ajn,bj,] for each ( i o . . .in) = ( 0 . . .O) and j = 0, 1, 2. One can choose the basic sets such that: (a) for each n 2 0 the points bo,, al,, bin, and b2, lie in the limit set F ; (b) the difference a l , - bOn is e-a(n+l) for n even and e-b(n+l) for n odd; (c) the difference u2, - bl, is e-a(n+l) for [ n / 2 ] even and e-b(n+l) for [ n / 2 ] odd. Here a and b are positive distinct constants (see Figure 6a where a = log5, b = log6, and the intervals Aio,,,in are of length 5-"). This implies that X ( w , n ) = 0 and x ( w , n ) = 00 for the point (i& . . .) = (00.. .) and all n 2 1. The following example shows that there are geometric constructions (CG1CG3) for which the induced map may be expanding but not quasi-conformal. It also illustrates that in this case Theorem 15.5 may fail.

Example 15.3. [Bar21 There exists a geometric construction (CGl-CG3) o n CT) such that the line modeled by the full shift (Ct, (1) each basic set A,o,,,i,is a closed interval of length depending only o n n ; (2) the induced map G is expanding but not quasi-conformal; (3) dimH F = h B F < d i m ~ F .

Proof. (See Figure 6b.) Let A, be numbers defined by (15.6). Consider a geometric construction (CG1-CG3) on the line modeled by the full shift on 2 symbols with the basic sets Aio...in to be closed intervals of length exp A,. Given a basic set we require that the sets AiO...i,,o and Aio...i,l be attached respectively to the left and right end-points of the interval Aio,,,i,. This guarantees that for each 2,y E F n Aio...in such that x # y , the ratio llGnx - Gnyll/llx - y(I is of the form

eXm

+ Cj"=,kjexm+j

'

for some m 2 0, and k j taking on one of the values { - 2 , -1,O, 1,2} for each j 2 0. Notice that not all sequences k, are admissible. Since

Dimension of Cantor-like Sets and Symbolic Dynamics

149

we obtain

for all b 2 a > log3. If a is sufficiently large we have

It follows that (Y 5 X(w,l) 5 x(w,l) 5 ,f3, where (Y and ,f? are some positive constants. Therefore, for a sficiently small T O and any o , y E F we have that a(Iz- y ( ( 5 llG(z) - G(y)(( 5 PIIo - yII provided 112 - y ( ( 5 T O . Since G is a local homeomorphism one can easily derive from here that G is expanding. Notice that by the construction of the numbers A,, we have that sup (A,

m>O

- Am-,)

= -an,

inf (A,

m>0

- Am-n)

= --bn

(15.12)

for any n 2 0. Hence,

This proves that the induced map G is not quasi-conformal. One can see that 2”-l 5 N(F, ex,) 5 2” (recall that N(F, r ) is the smallest number of balls of radius T needed to cover F ) . Therefore, (15.12) implies that b B F=

-,logb 2

log 2 dimBF = -. a

-

Notice that our construction is a geometric construction (CBl-CB3) with basic sets satisfying (15.3). By Theorem 15.1 we have that dimH F 2 s, where s is the unique root of the equation P +(scp) = 0. Since A, 2 -bn for all n 2 0 we C? conclude that s 2 log 2/b. Hence, dlmH F = b B F = log 2/b.

150

Chapter 5 0

I

d

\ 5

A00 A01 A02 HH H 1 1 36 25

I

1

5

a

b Figure 6. GEOMETRIC CONSTRUCTIONS WITH INDUCED MAPS: a) Non-expanding, b) Non-conformal. Geometric Constructions with Contraction Maps There is a special class of geometric constructions (CG1-CG2) that are wellknown in the literature (see for example, [Fl]) - constructions with contraction maps. Their basic sets Ai,...in are given as follows Ai,...in = hi, o hi, o . . . o hi,, ( D ) Here D is the unit ball in Rm and h l , . . . , hp:D maps, i.e., for any x , y E D ,

+ D are bi-Lipschitz contraction

Aidist(z, y) 5 dist(h,(z), hi(y)) 5 xi dist(x, y), where 0 < Ai 5 xi < 1. They are modeled by a subshift (Q, a). It is easy to see that these geometric constructions are Moran-like geometric constructions with non-stationary ratio coefficients of type (CB1-CB3) and hence can be treated accordingly.

Dimension of Cantor-like Sets and Symbolic Dynamics

151

If we require the separation condition (CG3) then the coding map is a homeomorphism and we can consider the induced map G on the limit set F of the for each z E F n Ai and construction which acts as follows: G(z) = i = 1 , . . . , p . Hence, G is expanding, i.e., it satisfies (15.9). As we mentioned above G is a local homeomorphism if and only if Q is a topological Markov chain, i.e., Q = C i for some transfer matrix A (which we assume to be transitive). One can also verify that for each w = (ioil . . .) E C i and n > 0, n

n

By Theorem 15.2 the Hausdorff dimension and lower and upper box dimensions of the limit set of a construction with contraction maps admit the following estimates: _d < _ dimH F < d i m R F 5 d i m ~ F5 2, where

d and 2 are unique roots of Bowen’s equations Pci(dcp) -

= 0 and

Pct;(dip)= 0 with ~ ( w =) -log&,, and ip(w) = -log&,,. Notice that, in general, d 5 and 2 2 3, where 3 and 3 are roots of Bowen’s equations (15.2) (see Theorem 15.1). The inequalities can be strict (see [BarZ]). This illustrates that one can obtain more refined estimates of the Hausdorff dimension and upper box dimension using the numbers X(w, n) and x(w,n ) than using the Lipschitz constants of the contraction maps (i.e., the numbers Ag and -

Xi).

If Xi = 1;for any i = 1 , . . ., p then the the contraction maps are affine and conformal. The induced map G is quasi-conformal (and in fact, is smooth and conformal; see Section 20) and the Hausdorff dimension and lower and upper box dimensions of the limit set coincide. The common value is given by Theorem 13.3. In particular, if the geometric construction is simple (i.e., is modeled by the full shift) then s is the unique root of the equation XIS

+ . . . + XPS = 1.

Geometric Constructions Associated with Schottky Groups

A Kleinian group is a discrete subgroup of th_egroup of all linear fractional of the complex plane C with the determinant ad transformations z ++ bc = 1 or -1. A linear fractional transformation g is said to be hyperbolic if tr2g = ( a + d)’ > 4 and loxodromic if tr2g E \ [0,4]. A (classical) Schottky group r is a Kleinian group with finitely many generators 91,. . .gp, p 2 1 which act in the following way: there exist 2p disjoint circles 71,r;,. . . ,yp,7;bounding a 2pconnected region D for which gj (D)nD = 0 and gj(7j) = 7; for j = 1 , . . . , p . The group r is known to be free and purely loxodromic, i.e., all non-trivial elements of r have either hyperbolic or loxodromic type (see [Mas], [Krl).

2

152

Chapter 5

A3 Figure 7. A GEOMETRIC CONSTRUCTION CORRESPONDING TO A REFLECTION GROUPWITH THREE GENERATORS. h

The group is said to act discontinuously at a point e E C! if the stabilizer ra = {g E r : g(z) = z } of z in r is finite, and e has a neighborhood U, such that g(Uz) nu, = 0 for all g E r/r, and g(Uz) = U, for g E rZ.The set O(r) of points z E at which I? acts discontinuously is called the region of discontinuity of the group r. It is known that if r is a classical Schottky group then the factor n(I')/r is a closed surface of genus p. The limit set for a classical Schottky group can be viewed as the limit set for a geometric construction with maps which are conformal but, in general, are neither affine nor contracting. This construction is modeled by a (one-sided) subshift of finite type. In the special case rj = 7; Schottky group is called the reflection group (see Figure 7 where p = 3). One can show that the maps, generating the geometric construction associated with a reflection group, are contracting and the transfer matrix has 0s along main diagonal while other entries are equal to 1. Note that the induced map G on the limit set of the geometric construction is smooth conformal and expanding.

Geometric Constructions (CPW1-CPWI) with Disjoint Basic Sets We consider a geometric construction (CPW1-CPW4) (see Section 13) modeled by a subshift of finite type ( E f , g ) . We assume that the basic sets of the construction are disjoint. One can verify that for each w = (ioil.. .) E E a , n > 0, n

n

j=O

j=O

One can construct a geometric construction (CPW1-CPW4) for which the induced map G is not expanding (we leave this as an easy exercise for the reader). However, if it is expanding it is quasi-conformal (this follows from the above inequalities and Condition (13.1)). Thus, 3 = S = sx, where sx is the number in Theorem 13.1(i.e., the Hausdorff dimension and lower and upper box dimensions of the limit set coincide and are equal to SX).

Dimension of Cantor-like Sets and Symbolic Dynamics

153

16. Geometric Constructions with Rectangles; Non-coincidence of Box Dimension and Hausdorff Dimension of Sets

A crucial feature of Moran-like geometric constructions with stationary or non-stationary ratio coefficients is that they are, so to speak, isotropic, i.e., the ratio coefficients do not depend on the directions in the space. This is a very strong requirement and that is why the placement of the basic sets may be fairly arbitrary. A simple example of anisotropic geometric constructions is provided by constructions with rectangles. As in the case of Moran-like geometric constructions one still has a complete control over the sizes and shapes of the basic sets but needs two collections of ratio coefficients to control length and width of rectangles on the nth step. In this section we consider the simplest case when the ratio coefficients are constant and do not depend on the step of the construction. Even so, we will see that the Hausdorff and box dimensions of the limit set may depend on the placement of the basic sets and may not agree. This can happen even if a geometric construction is “most close” to a self-similar construction, i.e., it is given by finitely many affine maps (the so-called general Sierpinski carpets; discussed later in Section 16). In this section we present examples which illustrate how the equality between the Hausdorff dimension and box dimension can be destroyed. These examples are also a source for understanding some “pathological” properties of the pointwise dimension (see Section 25). A surprising phenomenon is that the Hausdorff dimension of the limit set for constructions with rectangles may also depend on some delicate number-theoretic properties of ratio coefficients corresponding to different directions.

Geometric Constructions with Rectangles We call a symbolic geometric construction (CG1-CG3) on the plane modeled by a symbolic dynamical system (Q, CT) a construction with rectangles if there exist 2p numbers Xi and xi, i = 1,.. . ,p , 0 < A, 5 xi < 1 such that the basic set Aio...inis a rectangle with the largest side not exceeding K1 x i j and the smallest side not less than K2 Xij, where K1 and K2 are positive constants. See Figure 8. We stress that we require the separation condition (CG3). Let sx- and sx be the unique roots of Bowen’s equations pQ(S&log&o)= 0 and ~ Q ( s s ; l o g ~=~0orespectively ) and m l and mS; be the push forward measures by the coding map of equilibrium measures p~ and px corresponding to the functions (i&l .. . ) t-)sllog&o and ( i ~ i..l .) +-+ sxlogxi,.

nj”=,

n:=,

Theorem 16.1. Let F be the limit set for a symbolic geometric construction with rectangles. Then (1) sx 5 dimH F < dim,F 5 dimeF 5 SX; (2) If the measures p~ and px on Q are Gibbs, then sx 5 dmA(x)and dmr(z)

I ~1.

154

Chapter 5

Proof. Notice that the geometric construction with rectangle is regular with The result follows from Theorems 14.1, 14.2, the estimating vector (Al,. . . w and 14.5.

,a).

In [Barl], Barreira gave a description of geometric constructions with rectangles. We follow his approach and assume for simplicity that A, = and x i = for i = I,.. . , p for some 0 < A < 1< 1. Given w = (i& . . .) E Q and n > 0 denote by yn(w) E S1 3 [0,2a]/{O, 2 ~ ) the direction of the longest sides of A,,,,.,i,, i.e., the sides of length We first show that yn(w) converges uniformly when n -+ 03 with an exponential rate.

x

r.

Proposition 16.1. (1) For each w E Q, there exists the limit y ( w ) '%iflim y,,(w). (2) There exists C

> 0 such that for all w E Q ,

n+w

(16.1)

I\

Figure 8. A GEOMETRIC CONSTRUCTION WITH RECTANGLES. Proof. Fix w = ( i ~ i l. . . ) E Q. We first consider the "worst" possible placement of rectangles on the nth step of the construction (see Figure 9): a pair of touches the two longest diametrically opposed vertices of a rectangle Ai0 sides of a rectangle Let On be the angle between the longest sides of Aio,,,i,,and A~o...,,+l(see Figure 9). We notice that On does not depend on w and satisfies the equation

x

-+I

sin 0,

+

cos en =

Dimension of Cantor-like Sets and Symbolic Dynamics

155

Setting a = x/A we obtain that sine, satisfies the following equation: (an+l sine,

- 1 / ~ 2) = 1 - sin2 en.

The formal roots of this equation are

Since cose, > 0 we have to choose the root with minus in front of the square root. The discriminant of this equation is non-negative if and only if lAi,,,,.i,, I = [-X2("+') + < An. This takes place if n is sufficiently large, i.e., IA~,,.,,~,,+,I < An. One can also see that sin& is asymptotically equivalent to ( l / A - 1) a-("+') as n + co. Since A < 1 there exist C1 > 0 and Cz > 0 such that Cla-n < sin 0, < C ~ U -for " all sufficiently large n. One can now choose C1 and C2 to obtain in addition that Cla-" < 0, < Cza-,. Therefore, if no > 0 is sufficiently large and n > m 2 n o one has

x2z(n+1)]Y2

We now-consider a general placement Of the basic sets. Given w E Q, there exist angles &(w) (for each k > 0) with le&)l 5 & such that m ( w ) = C;=lek. Therefore, if n > m 2 no, using (16.2) we have

This shows that the sequence ( ~ , ( w ) ) , converges ~ ~ and satisfies (16.1).

A i,... in

-

Figure 9. THE"WORST"DISPOSITION OF RECTANGLES.

156

Chapter 5

Proposition 16.2. The vector field y o x-': F + S1 is uniformly continuous.

Proof. Let w = ( i o i l . . .) E Q. Consider a sequence w, + w as n + 00. There exists a sequence of numbers m, + 00 as n + 00 such that x ( w ) E A.io...i,for each n > 0 and k 2 m,. Therefore, y,(wk) = m ( w ) for n E N and k 2 m,. Using (16.1) we conclude that IY(Wk)

- r c w ) l q r ( w k ) - Yn(wk)J+IYn(wk) - r n ( w ) (

+ Iyn(w) -Y(w)I
0 such that for each (ioil . . .) E Q and each n E N we have that

Dimension of Cantor-like Sets and Symbolic Dynamics

157

dist(Aio . . . i , j , A jo . . . i n k ) 2 SA" whenever j # k and ( i o . . .i,j) and ( 2 0 . . .ink) are &-admissible. With this extra hypothesis we can study the Hausdorff dimension of a geometric construction with rectangles by comparing it with an appropriate geometric construction with aligned rectangles.

Theorem 16.3. Given a geometric constmction with rectangles and exponentially large gaps, modeled b y a symbolic dynamical system (&,a),there exists a geometric construction with aligned rectangles and exponentially large gaps, modeled by (&, a),such that the map f :F + F is Lipschitz with Lipschitz inverse and dimH F = dimH F, where F and F are the limit sets. Non-coincidence of Hausdorff Dimension and Box Dimension We provide several special examples of geometric constructions with rectangles that illustrate some interesting phenomena and reveal the non-trivial structure of these constructions and the crucial difference between them and Moran-like geometric constructions with stationary ratio coefficients. We consider a geometric construction with rectangles in Rz which is generated by finitely many affine maps f i , . . . ,f,: S + s, where S = [0,1] x [0,1] is the unit square. Let F be the limit set. Each map fi can be written as f i ( z , y ) = T i ( z , y ) b,, where Ti is a linear contraction and bi is a two-vector. In [F2] (see also [F4]), Falconer proved that for almost all (bl . . . b,) E RzP (in the sense of 2pdimensional Lebesgue measure), the Hausdorff dimension and the lower and upper box dimensions of F coincide and the common value is completely determined by T I , .. . ,Tp provided that IlTill < On the other hand, as we mentioned above, the Hausdorff and box dimensions of the limit set for geometric constructions with rectangles may depend on the placement of the basic sets. They may not agree even in the case when the construction is generated by finitely many affine maps. The corresponding example is described by McMullen (see [Mu]; in [LG], Lalley and Gatzouras considered a more general version of his construction). Given integers e 2 m 2 2 choose a set A consisting of pairs of integers ( i ,j) wtth 0 5 i < e and 0 5 j < m. Denote by a the cardinality of A (clearly a 5 mn). Let f k , k = 1 , . . . ,a be a f h e maps given in the following way: if k enumerates the element (i,j) E A then fk(s)= Sij, where S = [0,1] x [0,1] and sij = x The affine maps f k generate a simple geometric construction with basic sets A,o.,,i, = fi, o ... o fi,(S) being (L-" x m+)rectangles. We allow some of the basic sets A,, . . .,A, on the first step of the construction to intersect each other by either a common vertex or a common edge (see Figure 11 where e = m = 4 and a = 6 ) . The geometric description of this construction is the following: starting from the ( 1 x m)-grid of the unit square S choose rectangles corresponding to ( i , j ) E A , then repeat this procedure in each chosen rectangle and so on. The limit set F for this construction is

+

i.

[i,y ] [k,G ] .

and is known as a general Sierpiliski carpet (see [Mu]).

158

Chapter 5

Example 16.1.

ty

(1) dimH F = log, (C:&' ") $fs, where t j is the number of those i for which ( i , j ) E A . (2) b B F = d i m ~ F= log,r loge(f), where r is the number of those j for which ( i , j ) E A for some i.

+

Remarks. (1) For the geometric construction shown on Figure 11 we have t o = 1, tl = 2, t z = 2, t 3 = 1, and r = 4. (2) The Hausdorff and box dimensions of the limit set F agree if: a) 1 = m; b) the constants t j take on only one value other then zero. In the first case the construction is conformal self-similar and the common value for dimensions is log, a. Proof. Let x : C z + F be the coding map given as follows:

where k , = (in,jn). This map provides a symbolic representation of the limit set F by the (one-sided) shift on a symbols. It is surjective but may fail to be injective because some points in F may have more than one representation. Define numbers b k , k = 0,. . . , a - 1 in the following way. If k = ( i , j ) E A then bk is the number of 'i such that ( i l , j ) E A . Note that by the definition of the number s, k=O

We define the Bernoulli measure on C: by assigning probabilities Pk

=

b? m - l ms

to each symbol k = 0,. . . , a - 1. In other words, if C ko . . . k , is a cylinder set then p ( C k o . . . k , ) = n : = o p k t . Note that x i = 1 p k = 1. Let X be the measure on F which is the push forward of the measure p (i.e., X(A) = p ( x - ' ( A ) ) for any Bore1 subset A c F ) . We will show that dimH X 2 s. Set q = [nlogt m] 5 n. Given (n 1)-tuple (ko . . . k,) with kt = ( i t , j t ) E A , consider the set R k o . . . k , C F of all points (x,y) for which

+

where 2: = it for t = 0,. . . ,q and ji = j t for t = 0 , . . . ,n. The set R k o . . . k , is "almost" a ball of radius m-". More precisely, there are constants Cl > 0 and CZ> 0 independent of (ko . . . k,) and a point (x,y) E R k o . . . k , such that

B((X,Y), Cim-")

n F c R ko . . . k ,

C B ( ( X , Y),

Czm-") n F.

(16.4)

Dimension of Cantor-like Sets and Symbolic Dynamics

159

We will show that the sets R k o . . . k , comprise an “optimal” cover of J which we use to compute the Hausdorff dimension of the memure A. Note that each set R k o . . . k , can be decomposed into finitely many basic sets A k o . . . k , k ; + , . . . k : , n F , where kl = (zi, j i ) and j l = j , for t = q 1,.. . ,n. Moreover, the number of such sets is equal to bkq+, bk,+= . . .bk,. on Define now the function by

+

+,,

It is easy to see that this function is constant on the cylinder the common value by 4 ko . . . k , .

cko,..kn. w e

denote

Lemma 1. For any (ko . . .k,) we have

+

Proof of the lemma. Let us pick an (n 1)-tuple (ko, k l , . . . ,kn) and set ( S O , s1,..

.,sn) = ( b k o , b k l , . . .

I

bk,).

Note that this sequence is independent of the choice of ( k o , k l , . . . , k,). implies that

As we have mentioned the number of basic sets bk,+, bkq+D . . . bk, . Therefore,

Ako...k,k;+,...k:,

This

is equal to

This completes the proof of the lemma. set

w e now describe the “gaps” between the sets A ~ o , , , k q k ; + l , . , kn : , F inside the in terms of the behavior of the functions & over n.

Rko.,.k,

Lemma 2.

z;.

(1) limn+m+n(W) 2 1 for all w E (2) 4, + 1 as n + co p-almost everywhere.

Proof of the lemma. Define the functions gn and h, on C; by

Chapter 5

160

1

2

3

4

Then & ( w ) = h,(w) x g,(w)logc m. We claim that the functions g, and h, satisfy the following properties which immediately imply the desired result: 1) h,(w) -+ 1 as n -+ co for all w . Indeed, 1 5 bk 5 C for all k and hence

The expression in the right-hand side goes to 1 as n -+ 00 since

since this holds for any positive sequence st bounded away from zero.

Dimension of Cantor-like Sets and Symbolic Dynamics

161

3) gn(u) + 1 p-almost everywhere as n -+ 00. From the definition of p it is clear that the functions ( k o k l . . .) r-) p k , , n = 0,1,2,. . . are independent, identically distributed random variables with respect to p. Hence, the sequence ( P k o P k l . . . p k , ) l / n , n = 0, I, 2,. . . converges for almost every ( k o k l . . . ) E c,+by Kolmogorov's strong law of large numbers. Note that by the definition of p k

and hence gn + 1p-almost everywhere.

It immediately follows from Lemma 1, the second statement of Lemma 2, and 2 s for A-almost every point (x,y) E F. Thus, s 5 dimH F . We now show that dimH F 5 s by constructing an efficient cover of F . Fix E > 0 and consider the collection R,,of those of sets R k o . . . k , for which #Jko...k, 2 m-€. These sets are disjoint and by Lemma 1satisfy (16.4) that &((z, y))

Therefore, the number of such sets is bounded by m(s+E)nas A(F) = 1. Note that any -point (z,y) E F is covered by sets R k o . . . k , E R,,for infinitely many n since limn-tooQn((z,y))2 1 > rn-€ (see Statement 1 of Lemma 2). Therefore, R ( N ) = U , , ~ N R , is a cover of F for any choice of N. Let us choose N so large that E m - € " < E. n>N

It follows that

(here card denotes the cardinality of the corresponding set). This implies that dimH F 5 s + 2~ and the desired result follows since E can be chosen arbitrarily small. In order to compute the box dimension of the limit set F let us choose n > 0 and set r,, = m-". Consider the finite cover of F by sets R k o . . . k , and let N,, be the number of elements in this cover. It is easy to see that

where C1 > 0 and C2 > 0 are constants independent of n (recall that N ( F , r ) is the least number of balls of radius r needed to cover the set F). Note that N,, is precisely the number of ways to choose sequences (it),t = 1,. . . ,q and ( j t ) ,t = 1, . . . , n (recall that q = [nlog, m]) such that a) ( i J , j t ) E A for t = 1,. . . ,q; b) ( i t , j t ) E A for t = q + 1,. . . , n and some choice of &.

162

Chapter 5

It follows that Nn = u'Jrn-q = (f)'Jrn (recall that u is the cardinality of A and r is the number of j such that (i,j)E A for some i). Therefore,

a 4 lim - =log,r+log,-. r n-m n This completes the proof of the statement. = lim

n-m

log Nn

~

- logr,

=log,r+log,-

U

r W

Following Pesin and Weiss [PWll we construct a more sophisticated example than in the previous section, which illustrates that all three characteristics - the Hausdorff dimension, the lower and upper box dimensions - may be distinct.

Example 16.2. There exists a geometric construction with rectangles in the unite square S c R2, modeled by the full shift on two symbols ( C $ , u ) , for which Al = = A, 1 1 = 1 2 = 1,0 < A < < and the limit set F satisfies

x t,

log 2 dimH F = - l0gX'

m B F = 7-

-

log 2

- log1'

log 2 dimBF = - 10gX'

where y E ( 1 , a ) is an arbitrary number and a = logX/logx. Moreover, the induced map G on F is Holder continuous.

Proof. Let no = 0 and for k = 0 , 1 , 2 , . . . , let nk+1 = [an4 and ,& = 2(7-cr)n3k+l. In order to describe the nth step of the construction we use the basic types of spacings: vertical stacking (A) and horizontal staking (B). See Figure 12. (1) We start with two horizontally stacked rectangles. During steps n 3 k < n 5 n3k+1 we use (B). (2) We begin with Y 3 k + l rectangles. Choose ,f?k percent of these rectangles arbitrarily and paint them blue; paint the others green. During steps n3k+1 < n 5 n3k+2, we use (B) in all blue rectangles and use (A) in all green rectangles. (3) During steps n3k+2 < n 5 n3k+3, we use (A) in all blue rectangles and use (B) in all green rectangles. (4) Repaint all 2"3k+3 rectangles white. Repeat steps 1 through 4. The collection of rectangles at the nth step of the construction contains 2" rectangles each with vertical and horizontal sides; the size in the vertical direction is 1" and the size in the horizontal direction is Any two subrectangles at step n 1 that are contained in the same rectangle at step n are stacked either horizontally or vertically and the distance between them is chosen to be at least ;An. The projections of any two distinct rectangles at step n onto the two coordinate axes either coincide or are disjoint. Each point x E X can be coded by a one-sided infinite sequence of two symbols. The induced map G on the limit set F is easily seen to be Holder continuous with Holder exponent log log 1.

r.

+

x/

163

Dimension of Cantor-like Sets and Symbolic Dynamics

-n

h

h

a

b

Figure 12. a) Vertical Stacking, b) Horizontal Stacking. We now compute the Hausdorff dimension and the lower and upper box dimensions of the limit set F.

a) Calculation of Hausdorff Dimension. Given E > 0, choose k > 0 such that AnBk+I 5 E . Consider the cover of F consisting of green rectangles for n = n 3 k + 1 and blue rectangles for n = n 3 k + 2 . Consider a green rectangle Ai,,,.in3k+l.By construction the intersection A = A ~ o . . . ~ n 3nkF + l is contained in 2n3k+2-n3k+1small green rectangles corresponding to n = n 3 k + 2 . These rectangles are vertically aligned and have size Xn3k+2 r 3 k + 2 since p k + 2 = X [ a n 3 k + l l + 2 5 c o n ~ t A ~ the ~ ~ +(1' - Pk)2"3k+' green rectangles in the construction of F are each contained in a green square of size An"+'. Now consider a blue rectangle A ~ o . . . ~ , .3 By k + ,our construction the intersecnF is contained in 2n3k+3-n3k+2small blue rectangles corretion B = Aio,,.in3k+2 "3k+3 sponding to n = n 3 k + 3 . They are vertically aligned and have size n 3 k + 3 x X . Since pk+3 5 ~ons t X " ~the ~ +Pk2"3k+12"3k+2-"3k+1 ~ = Pk2"3k+2 blue rectangles in the construction of F are each contained in a blue square of size A"3k+2. The collection of green and blue squares comprises a cover B = {U;} of F for which

+ Pk2"3k+2 ( f i ~ n 3 k. + 2 ) s )

(diam Ui)" 5 const ((1 - Ic,)2"3k+1 Ui €G

The right-hand side of this inequality tends to 0 as k + 00 if s > *A. This implies that dimH F 5 On the other hand, by Theorem 16.1, we know that dimH F 2

s.

s.

Therefore, we conclude that dimHF = -.

log 2

- log1

We now proceed with the box dimension. b) Calculation of Lower and Upper Box Dimensions.

164

Chapter 5

Choose E > 0. We wish to compute explicitly the number N ( F ,E ) (the least number of ball of radius E needed to cover F ) . There exists a unique integer -+I 0 such that X

r.

A , = 1% " F ,

4

-1ogr

.

We consider the following three cases: Case 1: n 3 k 5 n < n 3 k + 1 . One can easily see that N ( F , E )= 2" and hence

A,

,

log 2

=-

- logX'

+

Case 2: n 3 k + l 5 n < n 3 k + 2 . w e have N(F,E) = Nbiue(F,E) Ngreen(F,E), where Nblue(F, E ) and Ngreen(F, E ) are the numbers of &-ballsin the optimal cover that have non-empty intersection with respectively blue and green rectangles at step n. It follows N ( F ,&) = P k 2 , f (1 - P k ) 2 n 3 k + i .

One can see that for all sufficiently large k (for which

and

provided

n3k+1

< n < n 3 k + 2 . This implies that

Moreover, if n = n 3 k + 1 then log 2 lim A , = n+w - log X'

Pk

5

i),

Dimension of Cantor-like Sets and Symbolic Dynamics

It is easy to see that for sufficiently large k (for which p k 5

165

f),

log 2

lim A, 2 - log A'

n+m

provided

< n < n3k+3. Moreover, if n = n3k+2 then

n3k+2

log 2 lim A, = 00 - log5;'

*. s. n+

It follows that d i m ~ F2 that d i m ~ F=

above, lim A,,,,, k+cc

Combining this with Theorem 16.1, we conclude

It also follows that h B F 2 -y*.

= -y*.

As we have seen

Thus, b B F = -y*.

We consider another example of a simple geometric construction with rectangles in the plane generated by two afKne maps which was introduced by Pollicott and Weiss (see [Pow]). It illustrates that the Hausdorff dimension of the limit set may depend on delicate number-theoretic properties of ratio coefficients while the box dimension is much more robust. Example 16.3 We begin with two disjoint rectangles A,, A2 c I in the unit square I = [0,1] x [0,1] given by Ai = [ai,ai A,] x J i , i = 1,2, where 0 < a1 5 u2< 1 and 51, J2 are disjoint intervals in the vertical axis of the same length A1. We assume that 0 < A 1 5 A2 < 1 and A1 < Consider the two affine maps hi: I + Ai, i = 1 , 2 that contract the unit square by A1 in the vertical direction and by A2 in the horizontal direction. See Figure 13. These maps generate a simple self-similar geometric construction with rectangles in the plane with basic sets at step n Aio,,,in = hi, o hi, 0... o hi,,(I).

+

i.

Let Xk, k = 1 , 2 denote the projections of the unit square I onto the vertical side, for k = 1, and onto the horizontal side, for k = 2. Obviously, 7r1 o h i ( x , y ) = ai A2x for any (2,y) E I and i = 1,2. Hence, for any basic set Aio...inthe left endpoint of the interval ~ 1 ( A i ~ . . . ; is , , )given by

+

Chapter 5

166 Taking the limit when n

{

+ ca yields

03

7r2(F) = c ( a l + ik+l(a2 - al)) A5 : (iO,il,. . .) E {O,l}' k=O

(16.5)

where d = a2 - al. Note that if A 1 = A2 then the construction is a Moran simple geometric construction (CM1-CM5). It then follows from Theorem 13.1 that (16.6) There is a very special - degenerate - case when 7rl(A,) = 7rl(A2) (i.e., the rectangle A1 lies directly above the rectangle A2), It is easy to check that in this case (16.6) still holds. We now compute the Hausdorff dimension and lower and upper box dimensions of the limit set F of the construction assuming that A1 < A2 and d = a2 - a1 > 0. a) Calculation of Box Dimension.

Lemma 1. def (1) h B F=dim~F = dimBF. (2)

< A2 5 f , i f $ 5 x 2 < 1. if0

-log%/logA1

Proof. We first consider the case 0 < A2 5 f . By virtue of (16.5) the set is affinely equivalent to the standard Cantor set

k=O

after scaling by d = disjoint rectangles

a2

- a1

7r2(F)

I > 0 and translating by &.

Consider two new

Dimension of Cantor-like Sets and Symbolic Dynamics

167

hl

Figure 13. A

SELF-SIMILAR CONSTRUCTION WITH

RECTANGLES.

It is easy to see that these rectangles have disjoint projections into the horizontal line and A: c A*, i = 1,2. We consider now the simple self-similar geometric sub-construction using the rectangles A; and A; and the affine maps h l and hz. Obviously, the basic sets of this new construction at step n, i.e., Aio...in, satisfy A:o,,,in c Aio...i,,. Hence, the limit set F' c F. This implies that b B F > dim#* = On the other hand, it follows from Theorem 16.1 that dim~F 5 We turn now to the case f 5 A2 < 1. Given n, consider the cover of the limit set F by squares with sides of length AT such that each rectangle Aio..,i,, is covered by +1 squares aligned in a row. One can see from (16.5) that the projection

-%. -%.

[s]

2.

i , , ) ~= r l ( F n Aio,,,,,,) contains an interval of length ~ J ~ ~ r ~ ( A i ~ .=. . IJlAF Hence, the proportion of the squares in the cover of Aio,..i,, required to cover Aio...i,, n F is at least lJllw(Ai o . . . i n ) l-- d 1- A 2 ' This implies that

A" A; 1 - A 2 2n2 > N(F,A;) 2 2n-A; A? d

(recall that N(F,r ) is the least number of balls of radius r needed to cover F ) . This implies the desired result. w b) Calculation of Hausdorff Dimension.

Lemma 2. If 0

< A2 < !j then dimH F = dimB F = -%.

168

Chapter 5

Proof. First note that d i r n ~ ( ~ ~ (5F d) )h H F 5 dimBF. As we saw in the proof of Lemma 1 the set T ~ ( F is ) the limit set for a simple Moran geometric construction (CM1-CM5) on [0,1] with 2" basic sets at step n of equal length A;&. Hence, dimH(wl(F)) = -*. The result now follows from Theorem 16.1 and Lemma 1. w We turn to the case $ 5 A2 < 1. Following Pollicott and Weiss [Pow], we call a real number /3 E [0, I] a GE-number (after Garsia-ErdBs) if there exists a constant C > 0 such that for all z E [0, +co)

(io, . . . ,in-l) E ( 0 , l}n:

n- 1

irp' E [z, z r=O

+ 0")

The following properties of GEnumbers are known (see [Pow]): (1) no number 0 < < is a GE-number; (2) there exists a non-GE-number that is bigger than $ (for example, the Golden mean, i.e., the positive root of the equation = &); moreover, the reciprocal of any Pisot-Vijayarghavan number (a root of an algebraic equation whose all conjugates have moduli less than one) is a non-GE number [S]; (3) if-B is the reciprocal of a root of 2 then it is a GE-number; (4) almost all numbers on the interval 1) are GE-numbers [So]; (5) if $ < < 1, then ,f3 is a GE-number if and only if for all sequences p E 1) and any d > 0 there exists K > 0 such that Nn(p) 5 K(2p)n-m for any 0 < m < n, where

3

l+i

(3,

nF{O,

(&+I..

Lemma 3. Iff < A2

.in) E {O,l}n-m

:d

< 1 is a GE-number then

dimB F = dimH F = - log

(x)/log xl. 2x2

[ I+

1 and consider Proof. Given r > 0 choose n 2 0 such that n = a Moran cover U, = &(A) constructed in Section 13 with A = (X,,X2). It is easy to see that this cover consists of all rectangles at step n. Given 2 E F , we first compute the number N ( z ,r ) of those rectangles at step n that intersect the ball B ( z , r ) . Choose m = [nlogX2/logX1]. Clearly A? x A;. Assume that z E Ai,...in and consider an asymptotic square S(z) of dimensions AT x A; that prolongates the rectangle A ~ o . . . ~Let n . Nn(z) be the number of those rectangles at step n that intersect the square S(z). Obviously, we have N ( z ,r ) 5 CINn(z), where C1 > 0 is a constant. We now establish an upper estimate for the number Nn(z). First we observe that if a rectangle Ajo...jnintersects the

Dimension of Cantor-like Sets and Symbolic Dynamics

169

square S ( x ) then io = j o , . . . ,im = j,. We note now that the left endpoint of . m Aio...i, is C7=o(al+dit)Xi and the left endpoint of Aio,..i,j,+l...jn is '&o(al+ die)Xi +Cr=m+l(al + d j t ) X i . Hence, for these two rectangles to lie in the same asymptotic square S ( x ) , we should have that I

n

I

(16.8)

It follows fromproperty (5) ofGE-numbers (see (16.7)) that Nn(z) 5 K ( ~ X Z ) ~ - ~ . Set sx = -1og(%)/logX1. Since the function sxlog&o = sxlogXz is constant the corresponding equilibrium measure mx is the measure of maximal entropy for the full shift and hence for any basic set A, o . . . i , we have that mA(A,o,,,,,,) = 2-". Using (13.5) and following (13.6) we obtain by direct calculation that

where K2 > 0 and K3 > 0 are constants. The desired result follows now from the uniform mass distribution principle.

c) Non-coincidence of the Hausdorff Dimension and Box Dimension. We illustrate that the number-theoretic property (16.7) that we have used is not just an artifact of the proof. Consider A 1 = f. In this very special case the limit set F reduces to the graph of a Weierstrass-like function (modulo a countable set). The dimension of such graphs were studied by several authors. In particular, in [PU], Przytycki and Urbaliski showed that if A2 is the reciprocal of a PV number then there exist certain configurations (i.e., a choice of numbers a1 and a2) such that dimH F < dimB F .

Chapter 6

Multifractal Formalism

Let f be a dynamical system acting in a domain U c Rm and 2 an invariant set. We saw in Chapter 2 that the Hausdorff dimension and box dimension of 2 yield information about the geometric (and somehow topological) structure of 2. This information, in fact, may not capture any dynamics. For example, if 2 is a periodic orbit then the Hausdorff and box dimensions of 2 are zero regardless to whether the orbit is stable, unstable, or neutral. In order to obtain relevant information about dynamics one should consider not only the geometry of the set 2 but also the distribution of points on 2 under f . In other words one should be interested in how often a given point x E 2 visits a fixed subset Y c 2 under f. If p is an f -invariant Bore1 ergodic measure for which p ( Y ) > 0, then for a typical point x E 2 the average number of visits is equal to p ( Y ) . Thus, the orbit distribution is completely determined by the measure p. On the other hand, the measure p is completely specified by the distribution of a typical orbit. This fact is widely used in the numerical study of dynamical systems where orbit distributions can easily be generated by a computer. These distributions are, in general, non-uniform and have a clearly visible fine-scaled interwoven structure of hot and cold spots, i.e., regions where the frequency of visitations is either much greater than average or much less than average respectively (see [GOY]for more details; see Figure 14). For dynamical systems possessing strange attractors the computer picture of hot and cold spots reflects the distribution of typical orbits associated with special invariant measures. The latter are naturally interconnected with the geometry of 2 and can be used to describe relations between the dynamics on 2 and the geometric structure of 2.These measures are often called natural measures. If the strange attractor is hyperbolic, the corresponding natural measure is the well-known Sinai-Ruelle-Bowen measure. (Definition and properties of these measures can be found in [KH].) The distribution of hot and cold spots varies with the scale: if a small piece of the invariant set is magnified another picture of hot and cold spots can be seen. In order to obtain a quantitative description of the behavior of hot and cold spots with the scale let us consider a dynamical system f acting on a hypercube K in Itm and a cover of K by a uniform grid of mesh size r . Let pi be an average number of visits of a “typical” orbit to a given box Bi of a grid, i.e., pi = p(B,), where p is a natural measure. The collection of numbers { p i } determine the distribution of hot and cold spots corresponding to the scale level r . Define scaling exponents ai by pi = Tax. In the seminal paper [HJKPS], the authors suggested using the limit distribution of numbers ai when r -+ 0 as a quantitative

170

171

Multifractal Formalism

characteristic of the distribution of hot and cold spots. It is intimately related to the multifractal structure of X (see [M2]).

cold spot Figure 14. AN ORBITDISTRIBUTION. The general concept of multifractality can be formulated as follows (see a more detailed description in Appendix IV). One can say that the set X has the multifractal structure if it admits a decomposition - called multifractal decomposition- into subsets that are homogeneous in a sense. Of course, this is not a rigorous mathematical definition and it depends on how the homogeneity is interpreted. For example, if h: X 4 R is a function then the decomposition of X into level sets X, = {z E X : h ( z ) = a } can be viewed as multifractal with the sets X, being homogeneous. There is a multifractal decomposition of X associated with hot and cold spots of orbit distributions (i.e., the decomposition that is induced by the invariant measure p ) :

Here X, is the set of points where the pointwise dimension takes on the value a and X - the irregular part - is the set of points with no pointwise dimension. This decomposition can be characterized by the dimension spectrum for pointwise dimensions of the measure p or f,(a)-spectrum (for dimensions), where f,(a) = dimH X,. The f,,(a)-spectrum provides a description of the fine-scale geometry of the set X (more precisely, the part of it where the measure p is concentrated) whose constituent components are the sets X, (see Section 18). The straightforward calculation of the f,(a)-spectrum is difficult and one should relate it to observable properties of the invariant measure p. For example, one can use various dimension spectra as most accessible characteristics of orbit

172

Chapter 6

distribution. Among them let us mention the R6nyi spectrum for dimensions introduced by TCl [Tell. It is defined as follows:

where N = N ( r ) is the total number of boxes Bi of the grid with ,u(B,) > 0 (provided the limit exists; see Section 17 and discussion in [Vl]). In [GHP], Grassberger, Hentschel, and Procaccia suggested computing correlations between q-tuples of points in the orbit distribution for q = 2,3, . . . (see also [G, GP, HP]). This led to characteristics known as correlation dimensions of order q (see Section 17). These characteristics proved to be experimentally the most accessible and offered a substantial advantage over the other characteristics of dimension type that were used in numerical study of dynamical systems with chaotic behavior. In Section 17 we give a rigorous mathematical substantiation of the fact that the correlation dimension of order q is completely determined by the function cpq(X,r ) = p ( B ( z ,r))q-’dp(z), r > 0 and thus, depends only on the metric on X and the invariant ergodic measure p, but not on the dynamical system itself - the unexpected phenomenon noticed first by Hentschel and Procaccia in [HP]. They also introduced a family of characteristics depending on a real parameter q 2 0 (except q = 1) of which the correlation dimensions of order q are special cases for integers q 2 2. It is now known as the HP-spectrum for dimensions (after Hentschel and Procaccia). The formal definition and a detailed discussion is given in Section 18. For natural measures the HP-spectrum for dimensions can be defined in the following way:

,s

1 H P q ( p ) = -1im

loginf

{

c

B(si,r)EGP(%..)).) 7

log r

q - 1r-o

where Q is a finite or countable covering of the support of p by balls of radius E (provided the limit exists). A group of physicists in the paper [HJKPS] presented a heuristic argument based on the analogy with statistical mechanics to show that the HP-spectrum for dimensions and the RCnyi spectrum for dimensions coincide and that the latter (multiplied by the factor 1 - q ) and the f,(a)-spectrum for dimensions form a Legendre transform pair (see Sections 18, 19, and 21; see also Figures 17a and 17b in Chapter 7). Roughly speaking their idea goes as follows (see arguments in [CLP]). Given a grid of size r , consider the partition function N

N

i=l

i=l

where q is the “inverse temperature” and Ei is the “energy” of the element B, of the grid (the sum is taken over those elements of the grid B, for which p(&) is positive). The “free energy” of p is defined (when it exists) by F ( q ) = -1im r-0

1%

all

logr

.

Multifractal Formalism

173

The analogy with statistical mechanics is then used to relate the Legendre transform of F (i.e., the function t H inf,[qt-F(q)]) to the distribution of the numbers @i), i.e., to f,(a). Once the Legendre transform relation between the two dimension spectra is established, one can compute the delicate and seemingly intractable f,(a)spectrum through the HP-spectrum, since the latter is completely determined by the statistics of a typical trajectory. In order to convert the above argument into a rigorous mathematical proof one must first establish that the HP-spectrum and the f,(a)-spectrum for dimensions are smooth and strictly convex on some intervals (in q and in a respectively). This seems amazing since a pn'ori one expects the functions f,(a) and HP,(q) to be only measurable. Furthermore, it is not at all clear whether, even if the measure /I is exact dimensional (i.e., d,(z) exists and is constant, say d, almost everywhere) the pointwise dimension attains any values besides d. In Section 18 we will show that the Hentschel-Procaccia spectrum and the RBnyi spectrum for dimensions coincide for any Bore1 finite measure on Rm. This allows us to set up the concept of a (complete) multifractal analysis of dynamical systems as a collection of results which establish smoothness and convexity of these spectra as well as the f,(a)-spectrum for dimensions and the Legendre transform relation between them. Although the multifractal analysis was developed by physicists and applied mathematicians as a tool in numerical study of dynamical systems its significance has not been quite understood. In Section 18 we will demonstrate that dimension spectra - whose study constitutes the multifractal analysis - are CarathCodory dimension characteristics (i.e., they can be introduced within the general CarathCodory construction described in Chapter 1). This alone justifies the study of dimension spectra as part of dimension theory. This also opens new perspectives for the multifractal analysis: one can classify dynamical systems up to an isomorphism that preserves dimension spectra. Such an isomorphism keeps track on stochastic properties of the dynamical system (specified by the invariant measure) as well as dimension properties (of invariant measures or sets) and thus, may be of great importance in describing chaotic dynamical systems (see Appendix IV for more details). In Chapter 7 we effect a complete multifractal analysis of Gibbs measures for smooth conformal expanding maps and axiom A-diffeomorphisms on surfaces. As a part of this analysis we show that any equilibrium measure on a conformal repeller corresponding to a Holder continuous function is diametrically regular (see Proposition 21.4). The same result holds for equilibrium measures on conformal locally maximal hyperbolic sets (see Proposition 24.1). In Section 19 of this chapter we consider another class of examples which includes Gibbs measures on limit sets for a large class of geometric constructions of type (CGl-CG3). Our analysis is based on the dynamical properties of the induced map on the limit set generated by the shift map on the symbolic space. Our main assumption is that this map, being continuous, is expanding and conformal in a weak sense. Whether the induced map satisfies these properties strongly

174

Chapter 6

depends on the symbolic dynamics and its embedding into Euclidean space, i.e., the gaps between the basic sets. Examples include self-similar geometric constructions and geometric constructions associated with some (classical) Schottky groups (see Section 13),as well as geometric constructions effected by a sequence of similarity maps whose ratio coefficients admit some asymptotic estimates. The multifractal analysis of these “expanding and conformal” geometric constructions is intimately related to the analysis for smooth conformal expanding maps (see Section 21).

17. Correlation Dimension One can obtain information about a dynamical system that is related to invariant ergodic measures by observing individual trajectories. In [GHP], Grassberger, Hentschel, and Procaccia introduced the notion of correlation dimension in an attempt to produce a characteristic of dynamics that captures information on the global behavior of “typical” trajectories by observing a single one (see also [GI). This trajectory should be typical with respect to an invariant measure. The formal definition of the correlation dimension is as follows. Let ( X ,p) be a complete separable metric space with metric p and f : X + X a continuous map. Given x E X , n > 0 , and r > 0, we define the correlation sum as 1 C ( z , n , r ) = - card {(i,j): p ( f ” ( x ) , f j ( x ) ) 5 r and 0 5 i , j 5 n } , n2

(17.1)

where card A denotes the cardinality of the set A . Given a point x E X , we call the quantities

the lower and upper correlation dimensions at the point x respectively. We shall first discuss the existence of the limit as n + co. Let p be an f-invariant Bore1 probability measure on X . Consider the function

where D ( x , r ) is the closed ball of radius r centered at 2. It is clearly nondecreasing in r and hence may have only a finite or countable set of discontinuity points. Thus, it is right-continuous; it is continuous at r if and only if p(S(x, r ) ) = 0 for p-almost every z E X ( S ( x ,r ) is the sphere of radius r centered at x).

Theorem 17.1. [P3, PT] Assume that p is ergodic. Then there exists a set Z c X of full measure such that for any E > 0, R > 0 , and any x E Z one can find a positive integer N = N ( x ,E , R) for which the inequality

Multifractal Formalism

175

holds for every n 2 N and 0 < r 5 R. In other words, C ( x ,n , T ) tends to ( P + ( T ) when n 00 ifor p-almost every x E X uniformly over r for 0 < r 5 R.

Proof. For simplicity we provide the proof under the additional assumptions that the function @ ( r ) is continuous and the map f is invertible. The general case is considered in [PT]. We first show that given a number T > 0, there exists a set X, c X of full measure such that for any x E X,, lim C ( x ,n , r ) = (p+(r).

n+cc

(17.3)

Given a point x E X, a measurable set A c X, and integers n 2 0, m 2 0 denote by N ( x , A , n , m ) the number of points fZ(z),-m 5 i 5 n for which T ( x ) E A. Let us fix a countable collection of closed balls Dk = D(yk,rk), k 2 0 forming a basis of the topology in X.We can assume that p(dDk) = 0 for all k 2 0. Since p is ergodic there exists a set Y c X of full measure such that for any x E Y and k 2 0, lim N ( x ,Dk, n , m) = p(Dk). n-m nSm It follows that given z E Y ,E 2 0, and k 2 0, there exists n ( x ,E , k) such that for any n 2 0, m 2 0 , and n m 2 n ( x ,E , k),

+

(17.4)

176

Chapter 6

Thus, for any x E Y and E > 0 there exists n 2 0, m 2 0 with n m 2 n1,

+

n1

= nl(x,E) 2 0 such that for any

(17.7) Since p is ergodic and p ( D ( z , r ) )is a bounded Bore1 function over x E X we have by virtue of the Birkhoff ergodic theorem that for p-almost every x E X ,

This implies that for any E > 0 there exist a measurable set Xi,:' c Y with p(Xi,:') 2 1 - E and a number n2 = n 2 ( E ) such that for any x E Xi,? and n Z 722, (17.8) Moreover, for p-almost every z E X$,

Therefore, there exists a set X i 2 C Xi,:' with p ( X $ ) 2 p ( X $ , y ) - E 2 1 - 2.5 and a number n3 = n 3 ( ~such ) that for any x E X $ )and n 2 723, I ; N ( x , X $ ) , n , O ) - 11 5 2.5. Let us fix z E Xi:? and n 2 0 5 ZE < n ( E ) . Denote

n(E)

(17.9)

+

= max{n1,n2,n3}. Write n = tn(E) Z, with

Ni = N ( f a ( x )D , ( f * ( z )T,) , n - i, i), i 2 0. One can rewrite the expression for C ( x ,n , r ) (see (17.1)) in the following form:

where the first sum is taken over all i, 0 5 i 5 t n ( E ) with f i ( x ) E X $ , the second sum over all i such that tn(E) 1 I i I n, and the third sum over all other i (i.e., those i, 0 5 i 5 tn(E) for which f i ( x ) $ X $ ) . Since Ni 5 n 1 this imr>lies that

+

+

(17.10)

Multifractal Formalism

177

if n is sufficiently large. It follows from (17.9) that (17.11) Let us now estimate the first sum. It follows from (17.8) that

(17.12)

where the first, second, and third sums are taken over i as above. Therefore, by (17.7) and (17.9)-(17.11), we find

This inequality along with (17.10), (17.11), and (17.12), imply (17.3) for all x E Xi?, and hence for any x E X , = u,>oX$. Obviously, X,. is a set of full measure. We now show the uniform convergence of C ( x , n , ~to) ( P + ( T ) over T on an interval 0 < T 5 R. Let us fix E > 0. Since ' p + ( ~ )is continuous there exists 6 > 0 such that I'p+(rl) - 'p+(rz)I 5 E for any T 1 , T 2 E (O,R]with IT1 - T Z 5 ~ 6. Let us fix a countable everywhere dense set T c R and put Y = nrETXr,where X , is the set defined in (17.3). Obviously, p ( Y ) = 1. Given T E R, there exist T ~ , T Z E T satisfying T I < T < T Z and T Z - T I 5 6. It is easy to see that for any n > 0, C ( x ,n, TI) 5 C ( x ,n, T ) 5 C ( x ,n,T Z ) .

If

~bis

sufficiently large we have also that

C(x:,n, TI) 2

- 2 v+(T)

'p+(~l) E

+

- 2.5,

5 (P+(T) + 2 ~ . These inequalities imply that lC(z,n,T ) - ( P + ( T ) ~5 2~ for all 0 < T 5 R if n is C ( x ,n, T Z ) 5 ' p + ( ~ z )

E

sufficiently large.

We now extend the notion of the correlation dimension by considering correlations of higher order. Namely, given x E X , n > 0, T > 0, and an integer q 2 2, we define the correlation sum of order q (specified by the points { f i ( x ) } , i = O , l , . . . ,n ) by 1 Cq(x,n , r ) = - card { (il . . .i 4 ) E {0,1,. . .,n } q : nQ p ( f i 3 ( x ) , f"(x)) 5 T for any

o 5 j ,k 5 q}.

178

Chapter 6 The quantities 1 n, ). q ( x )= lim lim log C,(x, q - l r y 0 n+m log(l/r) 1 logC,(x,n,r) oq(x) = -lim lim q - 1r+O n+m log( 1/r)

’

are called respectively the lower and upper correlation dimensions of order q at the point x or lower and upper q-correlation dimensions at x. The existence of the limit as n + 00 is guaranteed by a more general version of Theorem 17.1 which we shall state now. For any T > 0 and any integer q 2 2 consider the function

where p is an f-invariant Borel probability measure on X (recall that D ( x ,r ) is the closed ball of radius r centered at x). The function cp$(r) is non-decreasing and hence may have only a finite or countable set of discontinuity points. It is clearly right-continuous. The following statement is proved by Pesin and Tempelman in [PT] and is an extension of Theorem 17.1 to correlation dimensions of higher order.

Theorem 17.2. If p is an ergodic measure then for any integer q 2 2 there exists a set 2 C X of full measure such that for any E > 0, R > 0 , and any x E 2 one can find N = N(x, E , R ) for which

with arbitrary n 2 N and 0 < r 5 R.

Remarks. (1) We stress that the lower and upper q-correlation dimensions do not depend either on the dynamical system f or on the point x for p-almost every x (provided p is ergodic). Instead, they are completely specified by the measure p . This allows us to introduce the notion of q-correlation dimension for any finite Borel measure p on a complete separable metric space X (see [PT]). Let 2 c X be a Borel subset of positive measure. Define the lower and upper q-correlation dimensions of the measure p on 2 for q = 2 , 3 , . . . by 1

Corq(p,2 ) = -dim q-1-

1 q-1

2, G q ( p ,2 ) = -dim$.

For q = 2 these formulae define the lower and upper correlation dimensions of the measure p on 2. We also define the (q, H)-correlationdimension of the measure p on 2 for q = 2 , 3 , . . . by 1 Cor,,H(p, 2 ) = -dim, 2. q-1

Multifractal Formalism

179

Consider the function cpq(r)= cpq(X,r)defined by (8.7). We stress that in (8.7) we use open balls of radius T centered at points in X . A simple argument (which we leave to the reader) shows that

If p is an invariant ergodic measure for a dynamical system f acting on X it follows that for p-almost every x E X ,

( X can be also replaced by any set of full measure). These relations expose the “dimensional” nature of the notions of lower and upper q-correlation dimensions: they coincide respectively (up to the constant &) with the upper and lowerq-box dimensions of the space X for q = 2,3, . . . (or any other set of full measure). (2) Example 8.1 shows that the lower and upper q-correlation dimensions of p may not coincide: namely, for any integer q 2 2 there exists a Borel finite measure p on I = [0,1] for which

(and p is absolutely continuous with respect to the Lebesgue measure on I ) . One can check that p is an invariant measure for a continuous map f on I , where f(x) = p([O,x)). Thus, with respect to this map

Gq(x) = Cor_(p,I ) < COT&, I ) = cyq(x)

(17.13)

for p-almost every x E I . Furthermore, using methods in [K] one can construct a diffeomorphism of a compact surface preserving an absolutely continuous Bernoulli measure with non-zero Lyapunov exponents for which (17.13) holds. (3) We describe a more general setup for introducing the lower and upper q-correlation dimensions. Namely, given x,y E X , n > 0, r > 0, and an integer q 2 2, we define the correlation sum of order q (specified by the points {f’(x)} and {p(y)}, i = 0 , 1 , . . ., n ) by 1

C q ( x ,y, n, r ) = - card n Q

{(il

. . .i4)

E {0,1,. . . ,n}Q:

p ( f i i ( x ) ,f i k ( ( y ) )

5 T for any o 5 j , k 5 q}.

Consider the direct product space Y = X x X . We call the quantities

the lower and upper q-correlation dimensions at the point (x,y) E Y . One can prove the following statement: let p be a n f-invariant ergodic Borel

180

Chapter 6

probability measure and u = p x p; then for any integer q 2 2 and u-almost every (x,y ) E Y the limit lim C q ( x ,y , n, r ) = cp:(X, r ) . n+w

exists. (4) The functions cpq(X,r ) , q = 2 , 3 , . . . admit the following interpretation. Consider the direct product space (Yq, pq, uq),where

q times

q times

and p q ( P , g ) = C”,=, P ( X k , Y k ) for P = ( 2 1 , . . . ,xq),jj = ( y l , . . . , yq). Let A = { (x,. . . ,z) E Yq: x E X } be the diagonal. Then for any r > 0, cpq(X,r ) = V q ( U ( A ,T ) ) , where U ( A ,r ) is the r-neighborhood of A. (5) Let p be a Bore1 finite measure on Rm with bounded support. Following Sauer and Yorke [SY] we describe another approach to the notion of correlation dimension of p based on the potential theoretic method (see Section 7). Define the quantity (17.14) D(P) = SUP{S : W )< m}, where &(/I) is the s-energy of p (see (7.5)). The “correlation dimension” D ( p ) is interpreted as the supremum of those s for which the measure p has finite s-energy. Given a subset Z c Rm, one can compute the Hausdorff dimension of Z via the quantity D(p). Namely, by the potential principle (see (7.6)), dimH Z = sup{D(p) : p ( 2 ) = 1).

(17.15)

P

A slight modification of the argument by Sauer and Yorke in [SY] shows that -

Corz(p, X ) = -D(P),

where X is the support of p. Indeed, set D = -,(p,X). By the definition of the lower correlation dimension for any E > 0 there exists ro > 0 such that for any 0 < r 5 ro, (pz(X,r) 5 rD-‘. Given 0 5 s < D , set E = (D - s)/2 > 0. We have that rs

5 D-TOE + constant < 00. E

rs

Multifractal Formalism

181

This implies that I s ( p ) is finite. On the other hand, let 0 < D < s. Set = (s - D)/2 > 0. By the definition of the lower correlation dimension there exists a sequence of numbers r, > 0 such that E

Given n > 0, one can find a number m = m(n) (pz(X,r,)/2. We have that

1 2

1

>

n such that

(pz(X,r,)


1, where

It follows from (8.10) that 1 q-1-

H P , ( p ) = -dim

X,

=,(p)

1 = -dim,X. q-1

(18.1)

Thus, the HP-spectrum for dimensions is a one-parameter family of characteristics of dimension type: f o r every q > 1 the quantities H P , ( p ) and =,(p) coincide (up to a normalizing factor with the lower and upper q-box dimensions of the set X respectively. Equalities (18.1) allow us to rewrite the definition of the HP-spectrum for dimensions in the following way: using (8.5) we obtain that

5)

Multifractal Formalism

183

where y > 1 is an arbitrary number and A,,r(X, r) is defined by (8.6). As Remark 1 in Section 17 shows for g = 2 , 3 , . . . the values H P , ( p ) and H P q ( p ) coincide with the lower and upper correlation dimensions of X , i.e., w ( p ,X ) and =,(p, X). Following Pesin and Tempelman [PT] we introduce the modified HPspectrum for dimensions specified by the measure p as a one-parameter family of pairs of quantities ( H P M , ( p ) , H P M q ( p ) ) ,g > 1, where

and the supremum is taken over all sets 2 c X with p ( 2 ) 2 1 - 6. One can derive from Theorem 8.4 that for any g > 1,

Thus, the modified HP-spectrum for dimensions is a one-parameter family of CarathCodory dimension characteristics specified by the measure p. It follows from Theorem 9.2 that for any q > 1, -essinf;i,(x) XEX

5 H P M , ( ~ )5 H P M , ( ~ )5 -essinf XEX

ci,(z).

In particular, if the measure p satisfies ci,(x) = $(x) = d,(x) then

HPM,(p) = H P M , ( p ) = -essinf d,(z) XEX

.

(compare to (9.4)). The modified HP-spectrum for dimensions is completely specified by the equivalence class of p. This was shown in [PT]. We present the corresponding result omitting the proof.

Theorem 18.1. Let p1 and p2 be Borel measures on X . If these measures are equivalent then HPM,(Pl) = H P M q ( p 2 ) , HPMq(p1) = H P M q ( p 2 ) .

It follows from the definitions that for g > I HPM,(P) 1 H P , ( P ) ,

HPM,(p) 1 HPq(p).

As Example 8.1 shows there exists a Borel finite measure p on [O,p] for some > 0 such that

p

HP,(p) < HP,(p)

< H P M , ( p ) = HPM,(p)

= -1

184

Chapter 6

for all 1 < q 5 Q, where Q > 0 is a given number (the reader can easily check that this holds provided p > a?).

R h y i Spectrum for Dimensions Let U c E P be an open domain. A finite or countable partition E = {Ci, i 1 1) of U is called a (p,r)-grid for some 0 < p < 1, r > 0 if for any i 2 0 one can find a point xi E U such that

B ( z ; , p r )c C;

c B ( xi,r) .

The Euclidean space Rm admits (p,r)-grids for every 0 < p < 1 and r > 0. Let p be a Borel finite measure on Rm and X the support of p. We assume that X is contained in an open bounded domain U c Rm. Given numbers q 2 0 and p > 0, we set

where

and the infimum is taken over all (P,r)-grids in U (compare to (8.6)). We wish to compare the quantities Dim,X,=,X with the quantities dim,X,dim,X. The following result obtained by Guysinsky and Yaskolko [GY] establishes the coincidence of these quantities for q > 1 and demonstrates that one can use grids instead of covers in the definition of q-dimension.

Theorem 18.2. If p is a Borel finite measure on Rm with a compact support X then for any q > 1,

Dim,X

= dim,X,


X

is commutative. Under the coding map the cylinder sets C,,...i, to the basic sets in X generated by the Markov partition

c Ca correspond

The map x is Holder continuous and injective on the set of points whose trajectories never hit the boundary of any element of the Markov partition. The pullback by x of any Holder continuous function on X is a Holder continuous function on Xi. Let f be a continuous expanding map of a compact set X c R". We obtain effective estimates for the Hausdorff dimension and box dimension of X following the approach suggested by Barreira in [BarZ]. Let R = { R l , .. . ,R p }be a Markov partition of X of a small diameter E (which should be less than the number T O in (19.1)) and (Xi, u) the symbolic representation of X by a subshift of finite type generated by R. The Markov partition allows one to view X as the basic set for a geometric construction (modeled by the subshift of finite type) whose basic sets are defined by (19.3) and whose induced map is f. We will extend the approach, developed in Section 15 for expanding induced maps, to arbitrary continuous expanding maps, and we will apply the non-additive version of thermodynamic formalism to compute the Hausdorff dimension and box dimension of X . Fix a number k > 0. For each w = (ioil.. .) E Xi and n 2 1 define numbers

where the infimum and the supremum are taken over all distinct x , y E X n Rio,..,,,+k(compare to (15.7)). Note that since f is expanding Condition (15.8) holds. Define two sequences of functions on Cz Consider Bowen's equations (19.6)

One can show that for any sufficiently large k, any w E C i , and n 2 1, an

5 & ( w , n) 5 & ( w , n) 5 bn

(see the proof of Theorem 15.5). Thus, Condition (A2.23) (see Appendix 11) holds. By Theorem A2.6 this ensures the existence of unique roots of Bowen's equations (19.6). We denote these roots by @) and dk)respectively. Clearly, dk)5 dk). The following result establishes dimension estimates for X .

191

Multifractal Formalism

Proposition 19.1. [Bar21 Assume that f is topologically mixing. Then

SUP^(^) I dimH X < dim,X 5 d i m ~ X5 infd'). k>O

kz0

We say that a continuous expanding map f is quasi-conformal if there exist numbers C > 0 and k > 0 such that for each w E Eft and n 2 0 Condition (15.11) holds. The following result is similar to Theorem 15.5.

Proposition 19.2. [Bar21 Assume that f is topologically mixing. Then f o r any open set U c X and all sufficiently large k , dimH(U n X) = d i m , ( U n X ) = dim~(U n X) = g(') = d k )gfs

and s is a unique number satisfying 1 lim -log

n+wn

(io...in) Ci-admissible

( d i a m ( X n Rio,,,i,))S = 0.

Weakly-conformal Maps We say that a map f of a compact set X is weakly-conformal if there exist a Holder continuous function a ) . ( withI ) . ( . [ > 1 on X and positive constants C1, C2, and ro such that for any two points x , y E X and any integer n 2 0 we have: if p ( f k ( ( z ) ,f k ( y ) ) I ro for all k = 0 , 1 , . . .n then

n n

ClP(X,Y)

k=O

Ia(fk(.))l-'

I dfn(.),fn(Y)) (19.7)

Obviously, a weakly-conformal map is continuous, expanding, and quasi-conformal and Proposition 19.2 applies. We now discuss the diametrically regular property of equilibrium measures corresponding to Holder continuous functions for weakly-conformal maps (see Condition (8.15)). If f were a smooth map this property would follow from Proposition 21.4. The proof in the general case is a modification of arguments in the proof of this proposition and is omitted.

Proposition 19.3. Let f be a weakly-conformal map of a compact set X and cp a Holder continuous function on X. Then any equilibrium measure corresponding to cp is diametrically regular. Te diametrically regular property of an equilibrium measure demonstrates a close connection between the structure of X induced by this measure and metric structure of X induced by the metric p. This property plays an important role in effecting a multifractal analysis of the measure.

192

Chapter 6

Multifractal Analysis of Equilibrium Measures for Weakly-conformal Maps From now on we assume that the map f is topologically mixing. Let cp be a Holder continuous function on X and v = vv an equilibrium measure corresponding to cp. Note that since f is topologically mixing the measure vv is unique and is ergodic (in fact it is a Bernoulli measure). Define the function $ such that log$ = cp - Px(cp). Clearly $ is a Holder continuous function on X such that Px(log$) = 0 and v is a unique equilibrium measure for log$. We denote by m a unique equilibrium measure corresponding to the function x H -slog la(.) I on X , where s is the unique root of Bowen's equation

Px(-slogla(x)l) = 0. Define the one-parameter family of functions cpq, q E (-a, a)on X by

where T ( q ) is chosen such that Px(cp,) = 0. One can show that for every q E R there exists only one number T ( q ) with the above property. It is obvious that the functions cpq are Holder continuous on X . The following statement effects the complete multifractal analysis of equilibrium measures corresponding to Holder continuous functions for weaklyconformal maps. Its proof uses the diametrically regular property of these measures (see Proposition 19.3) and is a slight modification of arguments in the proof of Theorem 21.1 where we consider the case of smooth expanding conformal maps.

Theorem 19.1. [PW2] Let f be a topologically mixing weakly-conformal map o n X . Then f o r any Holder continuous function cp on X we have (1) the pointwise dimension d,(x) exists for v-almost every x E X and

(2) the function T ( q ) is real analytic f o r all q E R; T ( 0 ) = dimH F and T(1) = 0; T'(q) 5 0 and T"(q)2 0 (see Figure 17a in Chapter 7); (3) the function a(q) = -T'(q) takes o n values in an interval [al,az],where 0 5 a1 = a ( a )5 a2 = a(-..) < a;moreover, f,,(a(q))= T ( q ) q a ( q ) (see Figure 17b in Chapter 7); (4) f o r any q E R there exists a unique equilibrium measure vq supported o n the set X a ( q ) ,i e . , vq(Xacq))= 1 (the sets X , are defined by (18.5) with respect to the measure v) and dVq( x ) = Jug( x ) = T ( q ) qa(q) f o r v,-almost every x E X a ( q ) ; (5) if v # m then the functions f,(a) and T ( q ) are strictly convex and f o r m a Legendre transform pair (see Appendix V);

+

+

Multifractal Formalism

193

( 6 ) the u-measure of any open ball centered at points in X is positive and f o r any q E R we have

where the infimum is taken over all finite couers B,. o f X by open balls of radius r . I n particular, for q > 1,

The following statement is an immediate corollary of Theorem 19.1 (see Statement 1).

Proposition 19.4. Any equilibrium measure corresponding to a Holder continuous function f o r a topologically mixing weakly-conformal map f on X is exact dimensional. In fact, this result holds for an arbitrary (not necessarily equilibrium) ergodic measure for a weakly-conformal map. The proof is quite similar to the proof of Theorem 21.3 (which deals with the smooth case).

Induced Maps for Geometric Constructions (CG1-CG3): Multifractal Analysis of Equilibrium Measures Consider a geometric construction (CGl-CG3) in R" (see Section 13) and assume that it is modeled by a transitive subshift of finite type @$,a). Let F be the limit set. Since we require the separation condition (CG3) the coding map x is a homeomorphism and we can consider the induced map G = x o u o x-' on the limit set F . It is a local homeomorphism since the geometric constructions we consider are modeled by a subshift of finite type ( X i , a).Moreover, if one builds a geometric construction modeled by an arbitrary symbolic system (Q, u)with the expanding induced map on the limit set then uIQ must be topologically conjugate to a subshift of finite type. This follows from a result of Parry [Pa]. Therefore, the induced map G is expanding if and only if it satisfies Condition (15.9). Note that the placement of basic sets of a geometric construction, whose induced map is expanding, cannot be arbitrary and must satisfy the following special property: there are constants CI > 0 and CZ > 0 such that for every x E F and any r > 0 there exists n > 0 and basic sets A('), . . . ,A(") (with m = m(z, r ) ) for which

B ( z ,Clr) n F

c

u A(k) F c 3

B ( x ,C z r ) n F.

lsksrn

We assume that the induced map is weakly-conformal. Note that in this case it is also quasi-conformal (see Condition (15.11)). W e conjecture that one can

194

Chapter 6

build a geometric construction modeled by the full shij? with disjoint basic sets such that the induced map is quasi-conformal but not weakly-conformal. Theorem 19.1 can be used to effect the complete multifractal analysis of equilibrium measures, corresponding to Holder continuous functions, supported on the limit sets of geometric constructions (CG1-CG3) with weakly-conformal induced map. We apply this result to a special class of self-similar geometric constructions (see Section 13). Recall that this means that the basic sets A, o . . . i , are given by Aio,,,i,,= hi0 0 . . ' 0hin( D ) where h l , . . . ,h,: D -+ D are conformal affine maps, i.e., Ilhi(z) - hi(y)ll = Xillz-yll with 0 < X i < 1and x , y E D (the unit ball in Rm). Assuming that the basic sets Ai, i = 1,.. . , p are disjoint, one can easily see that the induced map G on the limit set F is weakly-conformal (with a(.) = A;,' where x ( x ) = (Zoil . . .)). Thus, Theorem 19.1 applies. For Bernoulli measures this result was obtained by several authors who used various methods (see, for example, [CM], [EM], [Fl], [01,[Ril). We describe a more general class of geometric constructions to which Theorem 19.1 applies. Namely, consider geometric constructions build up by p sequences of bi-Lipschitz contraction maps h p ) :D + D such that

and for any x , y E D , A?' dist(z, y )

5 dist(hp)(z),h $ ) ( y ) ) 5 1:"'dist(z, y ) ,

5 1:"'< 1 (see [PW2]). We assume that the following asymptotic where 0 < estimates hold: there exist 0 < X i < 1 such that (19.8) One can check that the induced map G is weakly-conformal (with a(.) = A;,' where x ( x ) = (ioil . . .)) and Theorem 19.1 applies. Example 25.2 shows that there exists a geometric construction produced by three sequences of bi-Lipschitz contraction maps which do not satisfy the a s y m p totic estimates (19.8). Although the basic sets at each step of the construction are disjoint it does not admit the multifractal analysis described by Theorem 19.1 (otherwise, by Proposition 19.3 any equilibrium measure corresponding to a Holder continuous function on the limit set F of this construction would be exact dimensional which is false for the geometric construction described in this example). It is still an open problem in dimension theory whether one can effect the complete multifractal analysis of Gibbs measures supported on the limit set of a Moran geometric construction (CM1-CM5) modeled by a transitive subshift

195

Multifractal Formalism

of finite type (for some results in this direction see [PW2, LN]). Notice that by Theorem 15.4, the pointwise dimension of such a measure exists (and is constant) almost everywhere.

Remarks. (1) One can show that if v = r n x then T ( q ) = (1 - 4)s (thus, T ( q ) is a linear function) and fv(a) is the &function (i.e., fv(s) = s and fv(a) = 0 for all a # s; see Remark 1 in Section 21; this case was studied by Lopes [Lo]). (2) The graphs of functions T ( q )and f(a)are shown on Figures 17a and 17b in Chapter 7. Note that the function f(a)is defined on the interval [ a l , a 2 ] , where a1 = - lim T’(q), a2 = - lim T’(q). q++m

q-k--00

It attains its maximal value s at a = a(0). Furthermore, f(a(1)) = a(1) is the common value of the lower and upper information dimensions of Y (see Section 21) and f’(a(1))= 1. We also note that the f(a)-spectrum is complete, i.e., for any a outside the interval [ a 1 , a 2 ] the corresponding set X , is empty (see Section 21).

(3) Consider again a self-similar geometric construction modeled by the full shift D and assume that v is the Bernoulli measure defined by the vector ( P I , . . . ,p,.), where 0 < p k < 1 and x ; = 1 p k = 1. It is easily seen from Theorem A2.8 (see Appendix 11) that

is equivalent to k=l

Chapter 7

Dimension of Sets and Measures Invariant under Hyperbolic Systems

In this Chapter we study the Hausdorff dimension and box dimension of sets invariant under smooth dynamical systems of hyperbolic type. This includes repellers for expanding maps and basic sets for Axiom A diffeomorphisms. The reader who is not quite familiar with these notions, can find all the necessary definitions and brief description of basic results relevant to our study in this chapter. For more complete information we refer the reader to [KH]. We recover two major results in the area: Ruelle’s formula for the Hausdorff dimension of conformal repellers (see Theorem 20.1) and Manning and McCluskey’s formula for the Hausdorff dimension of two-dimensional locally maximal hyperbolic sets (see Theorem 22.2). Our approach differs from the original ones and is a manifestation of our general Caratheodory construction (see Chapter 1) and the dimension interpretation of the thermodynamic formalism: it systematically exploits the “dimension” definition of the topological pressure described in Chapter 4 and Appendix 11. This unifies and simplifies proofs and reveals a non-trivial relation between the topological pressure of some special functions on the invariant set and the Hausdorff dimension of this set. Furthermore, we use Markov partitions to lay down a deep analogy between conformal repellers (as well as two-dimensional locally maximal hyperbolic sets) and the limit sets for Moran-like geometric constructions. We then apply methods developed in Chapter 5 (see Theorem 13.1) to study dimension of these invariant sets. In particular, this allows us to strengthen the results of Ruelle and of Manning and McCluskey by including the box dimension of repellers and hyperbolic sets into consideration. We also cover the case of multidimensional conformal hyperbolic sets. The crucial feature of the dynamics, to which our methods can be applied, is its conformality. In the non-conformal case (multidimensional non-conformal repellers and hyperbolic sets) the approach, based upon the non-additive version of the thermodynamic formalism, allows us to obtain sharp dimension estimates. We stress that in this case the Hausdorff dimension and box dimension may not agree. We describe the most famous examples of repellers (including hyperbolic Julia sets, repellers for one-dimensional Markov maps, and limit sets for reflection groups; see Section 20) and hyperbolic sets (including Smale horseshoes and Smale-Williams solenoids; see Section 23). We also provide a brief exposition of 196

Dimension of Sets and Measures Invariant under Hyperbolic Systems

197

recent results on the Hausdorff dimension of a class of three-dimensional solenoids by Bothe and on the Hausdorff dimension and box dimension of attractors for generalized baker's transformations by Alexander and Yorke, by Falconer, and by Simon (see Section 23). These results illustrate some new methods of study that have been recently developed, as well as reveal some obstructions in studying the Hausdorff dimension in non-conformal and multidimensional cases. A significant part of the chapter is devoted to the recent innovation in the dimension theory of dynamical systems - the multifractal analysis of equilibrium measures (corresponding to Holder continuous functions) supported on conformal repellers (see Theorem 21.1) or two-dimensional locally maximal hyperbolic sets (see Theorem 24.1; in fact, we cover more general multidimensional conformal hyperbolic sets). The first rigorous multifractal analysis of measures invariant under smooth dynamical systems with hyperbolic behavior was carried out by Collet, Lebowitz, and Porzio in [CLP] for a special class of measures invariant under some one-dimensional Markov maps. Lopes [Lo] studied the measure of maximal entropy for a hyperbolic Julia set. Pesin and Weiss [PW2] effected a complete multifractal analysis of equilibrium measures for conformal repellers and conformal Axiom A diffeomorphisms. In this chapter we follow their approach. Simpelaere [Si] used another approach, which is based on large deviation theory, to effect a multifractal analysis of equilibrium measures for Axiom A surface diffeomorphisms. There are two main by-products of our multifractal analysis of equilibrium measures on conformal repellers and hyperbolic sets. The first one is that these measures are exact dimensional (see Theorems 21.3 and 24.2; in Chapter 8 we extend these results to arbitrary hyperbolic measures). The second one is the complete description of the dimension spectrum for Lyapunov exponents for expanding maps on conformal repellers (see Theorem 21.4) and diffeomorphisms on locally maximal conformal hyperbolic sets (see Theorem 24.3 and also Appendix IV). This spectrum provides important additional information on the deviation of Lyapunov exponents from the mean value given by the Multiplicative Ergodic Theorem.

20. Hausdorff Dimension and Box Dimension of Conformal Repellers for Smooth Expanding Maps Repellers for Smooth Expanding Maps Let M be a smooth Riemannian manifold and f : M + M a C1+u-map. Let J be a compact subset of M such that f(J) = J . We say that f is expanding on J and J is a repeller if (a) there exist C > 0 and X > 1 such that IldfEwll 2 CX"llwll for all x E J , v E T,M, and n 2 1 (with respect to a Riemannian metric on M ) ; (b) there exists an open neighborhood V of J (called a basin) such that J = {z E V : fn(x) E V for all n 2 0) Obviously, f is a local homeomorphism, i.e., there exists T O > 0 such that for every x E J the map f l B ( x ,T O ) is a homeomorphism onto its image. Thus, f is expanding as a continuous map (see Section 19 and Condition (19.1)).

198

Chapter 7

We recall some facts about expanding maps. A point x E M is called non-wandering if for each neighborhood U of x there exists n 2 1 such that f"(U) n U # 0. We denote by O(f) the set of all non-wandering points o f f . It is a closed f-invariant set. The Spectral Decomposition Theorem claims that the set O(f) can be decomposed into finitely many disjoint closed f-invariant subsets, n(f) = J l U . . . U Jm, such that f I Ji is topologically transitive. Moreover, for each i there exist a number ni and a set Ai c Ji such that the sets fk(A,) are disjoint for 0 5 k < n,, their union is the set Ji, fn.(Ai) = Ai, and the map f n a 1 A, is topologically mixing (see [KH]for more details). From now on we will assume that the map f is topologically mixing. This is just a technical assumption that will allow us to simplify proofs. In view of the Spectral Decomposition Theorem our results can be easily extended to the general case (with some obvious modifications). An expanding map f has Markov partitions of arbitrarily small diameter (see definition of the Markov partition in Section 19; see also [R2]). Let R = {R1,.. . ,Rp} be a Markov partition for f. It generates a symbolic model of the repeller by a subshift of finite type ( E f , a ) . Namely, consider the basic sets &,,,,in defined by (19.3) and the coding map x:Cf + J defined by (19.2). Then the following diagram

q

X i

D x;

Ix

J f > J is commutative. We remind the reader that the map x is Holder continuous and injective on the set of points whose trajectories never hit the boundary of any element of the Markov partition. We need the following well-known estimates of the Jacobian of C1+aexpanding maps. Proposition 20.1. Let $ be a Holder continuous function on J such that $(x) 2 c > 0. There exist positive constants L1 = L1($) and Lz = L z ( $ ) such that for any (n l)-tupple (io . . . i n ) and any x,y E Rio...in,

+

Let $ be a Holder continuous function on J such that $(x) 1 c > 0. There exist positive constants L1 = L1($) and Lz = Lz($) such that for any n > 0 , any branch h of f-", and any points x E J , y E h(B(x,ro)) Statement 1 holds. There exist positive constants L1 and Lz such that for any ( n 1)-tupple (io.. .in) and any x,y E Ri,...in,

+

where Jac f" denotes the Jacobian off".

Dimension of Sets and Measures Invariant under Hyperbolic Systems

Proof. Let p

199

> 0 be the Holder exponent and C1 > 0 the Holder constant of $.

By the expanding property we find that where Cz > 0 is a constant. Therefore,

This implies the upper bound. The lower bound follows by interchanging x and y. This completes the proof of the first statement. The second statement can be proved in a similar fashion. The third statement follows by applying the first one to the function $(x) = Jac f (x) which is Holder continuous since f is of class

e+=.

A Markov partition R = (R1,.. . ,Rp}allows one to set up a complete analogy between limit sets of geometric constructions (CG1-CG2) and repellers of expanding maps by considering the sets Rio,,,inas basic sets. Namely, n>O (io ...i n )

+

where the union is taken over all admissible (n 1)-tuples ( 2 0 . . .in). By the = h(Ri,) n Ri, for some branch h of Markov property every basic set Rio...in f-n+l.

A smooth map f : M -+ M is called conformal if for each x E X we have dfz = a(.) Isom=, where Isom, denotes an isometry of T,M and a(.) is a scalar. A smooth conformal map f is expanding if Ia(x)I > 1 for every point x E M . The repeller J for a conformal expanding map is called a conformal repeller. Note that a smooth conformal expanding map is weakly conformal (as a continuous expanding map; see Condition 19.7). The converse is not true in general: there exists a C"-map which is not conformal in the above sense but is weakly-conformal (see [BarP]). Conformal repellers can be viewed as limit sets for Moran-like geometric constructions with non-stationary ratio coefficients since the basic sets &o...,n satisfy Condition (B2). Proposition 20.2. (1) Each basic set Rio.,.incontains a ball of radius I - ~ ~ . , and , ~ ~ is contained in a ball of radius Tio...;,. (2) There exist positive constants K1 and Kz such that f o r every basic set RiO,..i,, and every x E Rio . . . i , , n ~1l-J j=O

~zn n

~ a ( f j ( x ) ) ~I - ' ~i~...i, I pi0 ...in I

la(fj(x~))~-'.

j=O

(20.1)

200

Chapter 7

Proof. Since f is conformal and expanding on J we have for every x E J ,

n n

Ildf,"ll =

= IJacfn(x)l.

la(fj(z))l

j=O

This fact and the third statement of Proposition 20.1 imply that for every z E Rio . . . i n , diamRi,...i,

I diamRi,

x max Ildh,II = diamRi, x max lJach(y)I YE&,

YE&,

where h is some branch of f-" and C1 > 0 is a constant. Since each Rj is the closure of its interior we have diamRi,...,, (x)2 diamRi, x min Ildh,II = diamRi, x min lJach(y)I YE&,

where

C2

YE%,

> 0 is a constant. This completes the proof of Proposition 20.2.

W

We use the analogy with geometric constructions to define a Moran cover of the conformal repeller. It allows us to build up an "optimal" cover for computing the Hausdorff dimension and lower and upper box dimensions of the repeller. Given r > 0 and a point w E C;, let n ( w ) denote the unique positive integer such that .(W)

J-J

k=O

la(x(g"4))l-l

> r,

+)+I

J-J

1 4 x ( g k ( 4 ) ) l - l L T.

(20.2)

k=O

It is easy to see that n ( w ) -+00 as r -+ 0 uniformly in w . Fix w E C: and consider the cylinder set Ci,,.,i,,(,) c C;. We have that w E Ci,,,,i,,(,). Furthermore, if w' E Cjo...i,(,tand n(w') 5 n(u) then

Ci,...i,(,) c Ci,...i,(,,). Let C ( w ) be the largest cylinder set containing w with the property that C ( w ) = Cio...in(u,,) for some w" E C ( w ) and C,,...i,(,,) c C ( w ) for any u' E C ( w ) . The sets C ( w ) corresponding to different w E C: either coincide or are disjoint. We denote these sets by C(j),j = 1 , . . . ,N,.. There exist points w j E C; such that C(j) = Cio,,,~n-cuj,. These sets comprise a disjoint cover of C: which we denote by U, and call a Moran cover. The sets R(j) = x(C(j)),j = 1 , . . . ,N,. may overlap along their boundaries. They comprise a cover of J (which we will

Dimension of Sets and Measures Invariant under Hyperbolic Systems

201

denote by the same symbol & if it does not cause any confusion). We have that R(j) = Ri, ...in(, j ) for some xj E J . Let Q c C i be a (not necessarily invariant) subset. One can repeat the above arguments to construct a Moran cover of the set Q. It consists of cylinder sets C(j),j = 1 , . . . , N , for which there exist points w j E Q such that C(j) = C~o...~n~u,) and the intersection C ( j )n di) n Q is empty if j # i (while the intersection C(j)n C(*)may not be empty). We denote this cover by &,Q. Moran covers have a property that plays a crucial role in studying the Hausdorff dimension and box dimension of the conformal repeller J . Namely, given a point z E J and a sufficiently small r > 0, the number of basic sets R(j) in a Moran cover U, that have non-empty intersection with the ball B ( z ,r ) is bounded from above by a number M , which is independent of x and r . Analogously to Moran-like geometric constructions (CPWl-CPW4) we call this number a Moran multiplicity factor. In order to explain this property of a Moran cover let 5; = max{diamRi : i = 1 , . . . , p } . Since the sets Ri are the closure of their interiors there exists a number 0 < r1 5 i such that each Ri contains a ball of radius r1. By Proposition 20.2 each basic set R(j) in the Moran cover contains a ball of radius C r , where C > 0 is a constant independent of T and j. This implies the desired property of the Moran cover. Examples of Conformal Expanding Maps

e

(1) Rational Maps[CG]. Let R:? + be a rational map of degree 2 2, where ? denotes the Riemann sphere. The map R, being holomorphic, is clearly conformal. The Julia set J of R is the closure of the set of repelling periodic points of R (recall that a periodic point p of period m is repelling if I(Rm)’(p)l > 1). One says that R is a hyperbolic rational map (and that J is a hyperbolic Julia set) if the map R is expanding on J (i.e., it satisfies Conditions (1)-(2) in the definition of smooth expanding map with respect to the spherical metric on It is known that the map z t+ z2 c is hyperbolic provided IcI < (see Figure 15). It is conjectured that there is a dense set of hyperbolic quadratic maps. (2) One-Dimensional Markov Maps [Ra]. Assume that there exists a finite family of disjoint closed intervals I l , I z , . . . Ip c I and a map f : U j Ij + I such that

e).

+

a

(a) for every j , there is a subset P = P ( j ) of indices with f(Ij) = u k E p I k (mod 0); (b) for every z E UjintIj, the derivative of f exists and satisfies I f’(z)I 2 (Y for some fixed (Y > 0; (c) there exists X > 1 and no > 0 such that if f m ( x ) E UjintIj, for all 0 5 m 5 no - 1 then l(fno)’(z)l 2 A. Let J = {z E I : fn(z)E u;=, Ij for all n E N}. The set J is a repeller for the map f . It is conformal because the domain of f is one-dimensional (see Figure 16).

202

Chapter 7

a

b

Figure 15. THEBOUNDARIES OF THESE “BLACKSPOTS” ARE JULIASETS FOR THE POLYNOMIAL z2 + c WITH

0

4 1 4 2

I, = A,

A21A22

1

U

I, = A 2

Figure 16. A ONE-DIMENSIONAL MARKOV MAP.

(3) Conformal Toral Endomorphisms.This is a map of multidimensional torus defined by a diagonal matrix (k, . . .,k), where k is an integer and Jkl > 1.

Dimension of Sets and Measures Invariant under Hyperbolic Systems

203

(4) Induced Maps. Let h l , . . . , hp:D + D be conformal affine maps of the unit ball D in Rm. Assume that the sets hi(D) are disjoint. Consider the self-similar construction generated by these maps and modeled by the full shift (or a subshift of finite type; see Section 13). Define the map G on Uiinthi(D) by G ( x ) = hT1(x)if x E inthi(D). Clearly, G is a smooth conformal expanding map and the repeller for G is the limit set of the geometric construction (i.e., G is a smooth extension of the induced map). Similar result holds for the induced map on the limit set of the geometric construction generated by reflection groups (see Section 15).

Hausdorff Dimension and Box Dimension of Conformal Repellers We compute the Hausdorff dimension and box dimension of a conformal repeller. Let f : M -+ M be a C1+"-conformal expanding map with a conformal repeller J . Let rn be a unique equilibrium measure corresponding to the Holder continuous function -slog la(.) I on M , where s is the unique root of Bowen's equation PJ(-slog 1). =0 (see Appendix 11).

Theorem 20.1. (1) dimH J = d&J,

=d

im~= J s; moreover

S=

hdf)

s, 1% la(.)

I dmk).

(2) The s-Hausdorff measure of J is positive and finite; moreover, it is equivalent to the measure rn. (3) s = dimHrn, in other words, the measure m is an invariant measure of full Carathe'odory dimension (see Section 5).

In [R2], Ruelle proved that the Hausdorff dimension of the conformal repelier J of a C1+a-map is given by the root of Bowen's equation PJ(-slog 1). = 0. He also showed that the s-Hausdorff measure of J is positive and finite. In [F4], Falconer showed that the Hausdorff and box dimensions of J coincide. One can derive Theorem 20.1 from a more general result for continuous weaklyconformal expanding maps (see [Bar21 and Section 19). In particular, this shows that Theorem 20.1 holds for C1-conformal expanding maps. This result was also established by Gatzouras and Peres [Gap]; see also Takens [T2] for a particular case. We provide here an independent and straightforward proof of Theorem 20.1 which is similar to the proof of Theorem 13.1 (where we dealt with limit sets of geometric constructions (CPW1-CPW4)).

Proof of the theorem. Set d = dimH J . We first show that s 5 d. Fix E > 0. By the definition of the Hausdorff dimension there exists a number r > 0 and a cover of J by balls Be, !. = 1,2,. . . of radius re 5 r such that (20.3)

204

Chapter 7

Let R = {Rl,. . . ,R p } be a Markov partition for f . For every L > 0 consider a Moran cover U,.!of J and choose those basic sets from the cover that intersect Be. Denote them by Ril),. . . ,RLm(e)).Note that RF’ = R,o..,;n(t,j) for some (io . . .in(e,j)). By Proposition 20.2 and (20.2) it follows that for every x E RF),

n

n(e,j) ~1

(20.4)

5 ~ i ~ . . . i ~ (I ~ , ClT-e, ~)

ta(fk(x))I-’I ~,~...i,,(,,~)

k=O

where Cl > 0 is a constant independent of e and j . By the property of the Moran cover we conclude that m(e)5 M , where M > 0 is a Moran multiplicity factor (which is independent of l ) . The sets { R y ) ,j = 1,.. . ,m(e),l = 1 , 2 , . . . } comprise a cover of J , and the corresponding cylinder sets Cp) = C;o...~n(,,j) comprise a cover of X i . By (20.3) and (20.4) we have

Given a number N > 0, choose r so small that n ( e , j ) 2 N for all now have that for any n > 0 and N > n,

e and j .

where M(C$,O,p,Un,N) is defined by (A2.19) (see Appendix 11) with and v ( w ) = -(d + E ) 1% la(x(w))l. This implies that

Pci (-(d

+ &) log la XI) 0

(Y

We

=0

I 0,

where Pc: is the topological pressures on X i . Hence, by Theorem A2.5 (see Appendix 11),s 5 d E . Since this inequality holds for all E we conclude that s 5 d. Denote d = d i m ~ J .We now proceed with the upper estimate and show that dI s. Fix E > 0. By the definition of the upper box dimension (see Section 6) there exists a number r = T ( E ) > 0 such that N ( J ,r ) 2 rE-’ (recall that N ( J ,T - ) is the least number of balls of radius r needed to cover the set J ) .

+

Dimension of Sets and Measures Invariant under Hyperbolic Systems

205

Let R = { R l , . . . ,R , } be a Markov partition for f and U, a Moran cover of J by basic sets R ( j ) = Ri,...i,,(=, ) , j = 1,.. . ,N,. The sets C(j) = x - l ( R ( j ) ) = C~o,..,~ (where ~wj~ w j = x-'(zj)) comprise a Moran cover of (recall that x is the coding map generated by the Markov partition). Repeating arguments in the proof of Theorem 13.1 one can show that there exist A > 0 and a positive integer N such that for any sufficiently small r , card { j : n ( w j ) = N } 2 r2€-' Consider an arbitrary cover 4 of C i by cylinder sets Cio,,.;N. It follows that

where Cz

> 0 is a constant. We now have that for any n > 0 and N > n,

=

c

C;,...,,

n N

sup

la(fk(z))l-d+2E2

cz,

EB ZER?) k=O

where R(Cf;,O,p,U,,,N)is defined by (A2.19') (see Appendix 11) with a = 0 and P(W) = 4-2E) logla(x(w))l. By Theorem 11.5 this implies that

CPJ(- (a- 2E) log)1.1

= PJ

(-(a- 2 4 log)1.1

= Pc;

(- (d - 2 4 log la 0 X I ) 2 0

and hence d - 2~ 5 s. Since this inequality holds for all E we conclude that d 5 s. The last equality in Statement 1follows from the variational principle (see Theorem A2.1 in Appendix 11). This completes the proof of the first statement. We now prove the other two statements. Note that

where hrn(f) is the measure-theoretic entropy of m. One can use formulae (21.20) and (21.21) below to conclude that drn(z) = s for m-almost every z E J . The third statement follows now from Theorem 7.1. However, we present here a simple and straightforward proof of the third statement. Having in mind that the measure m is an equilibrium measure and

206

Chapter 7

P ~ ( - s l o g)1.1 = 0, there exists a constant C3 > 0 such that for any x E any basic set &o...i, (x)

M

and

n

m(Ri, ...i"(2)) I c3

Ia(fkx)I-s

(20.5)

k=O

(see Condition (A2.20) in Appendix 11). Given r > 0, consider a Moran cover a Markov partition R for f. It follows from the property of the Moran cover and (20.5) that

U, = { R ( j ) }of J constructed from rn(~(x,r)) 5

C

m(Rp))5 MC3rS.

R ( i )ELL,

Thus, m satisfies the uniform mass distribution principle (see Section 7) and hence dimH m 2 s. The above arguments also imply the second statement.

Remark. By slight modification of arguments in the proof of Theorem 20.1 one can strengthen the first statement of this theorem and prove the following result (see [BarZ]; compare to Statement 4 of Theorem 13.4): given an open set U c M such that U n J # 0 we have

(recall that s is the unique root of Bowen's equation PJ(-sloglal) = 0). Analysing the proof of Theorem 20.1 one can also obtain a lower bound for the Hausdorff dimension as well as an upper bound of the upper box dimension of any set Z C J (which need not be invariant or compact). Namely,

where - s and 3 are unique roots of Bowen's equations Pz(-sloglal) = 0 and CPz(-slog lal) = 0 (existence and uniqueness of these roots are guaranteed by Theorem A2.5 in Appendix 11). One can apply Theorem 20.1 to conformal expanding maps described in Examples 1-4 above. We leave it as an exercise to the reader to show that: 1) if f is a Einear one-dimensional Markov map with the repeller J then dimH J = d&BJ

=d

i m ~ J= S,

where s is the unique root of the equation cl-s

+ . . . + cp-s

=1

(here cj > 1 is the slope of f on the interval I j , j = 1,. . . , p ) .

Dimension of Sets and Measures Invariant under Hyperbolic Systems

207

2) iff is the induced map generated by conformal affine maps h l , . ..,h,: D -+ D (Dis the unit ball in EP), h j ( x ) = Xjx uj,then the number s (that is the common value of the Hausdorff dimension and lower and upper box dimensions) is the unique root of the equation

+

XIS

+ . . . + XPS = 1.

One can also use Theorem 20.1 to compute dimension of hyperbolic Julia sets of rational maps. We present two additional results (without proofs) that provide more detailed information on the Hausdorff dimension of Julia sets. The first result is due to Ruelle [R2] and deals with the two-parameter family of rational maps z e zq - p . Let Jq,p be the corresponding Julia set.

Proposition 20.3. If Jq,p is hyperbolic then dimH Jq,p = 1+ 'I2 + (terms of order 41og Q

> 2 in p ) .

Consider a family {Rx : X E A} of rational maps. Given A, let J x be the corresponding Julia set. The family Rx is said to be J-stable at XO E A if there exists a continuous map h: A' x Jxo + such that A' is a neighborhood of XO in A and h(X, .) is a conjugacy from (Jx,, Rxo) to (Jx, Rx) satisfying h(Xo, .) = idlJx,,. The second result was obtained by Shishikura [Shi]. Consider the oneparameter family of complex quadratic polynomials Rx(z) = z2 A. The set aM = {A E CC : Rxis not J-stable} is known to be the boundary of a set M c C called the Mandelbrot set.

+

Proposition 20.4. There exists a residual subset A C aM such that dimH J x = 2 for any X E A. Estimates of Hausdorff Dimension and Box Dimension of Repellers: N?n-conformal Case Let J be a repeller for an expanding C1+a-map f. If f is not conformal the Hausdorff dimension and the lower and upper box dimension of J may not coincide. An example is given by the induced map on the limit set of the selfsimilar geometric construction, described in Section 16.1 (or Section 16.3), which is smooth expanding but is not conformal. Nevertheless, a slight modification of the above approach, which is relied upon the thermodynamic formalism (and its non-additive version), can be still used to establish effective dimension estimates (i.e., estimates that can not be improved). We follow Barreira [Bars]. Consider two Holder continuous functions 'p and ?j? on J 'p(X)

= - 1% Ildzfll,

a x ) = 1% ll(dzfr1Il.

Let & and 3 be the unique roots of Bowen's equations PJ(tcp)= 0 ,

PJ(t?j?) = 0.

(20.6)

208

Chapter 7

Let R = {RI,. . . ,R p } be a Markov partition of J of a small diameter and ( C f , u)the symbolic representation of J by a subshift of finite type. Since f is a

continuous expanding map we can consider two sequences of functions y ( k and ) on C i defined by (19.5). One can verify that given E > 0, there exists k 2 0 and n 2 1 we have such that for every w E $k)

n

n

where x = x ( w ) E J . This implies that t I Q ( ~ 5 ) idk)5 t (we remind the reader that g@) and dk)are unique roots of Bowen's equations (19.6)). By Proposition 19.3 we obtain the following dimension estimates of the repeller J :

tI dimH J < dim,J 5 dim~J5 t.

(20.7)

We describe another method which allows one to obtain even sharper dimension estimates. Consider two sequences of functions on J defined as follows:

CP == {'P -n

(XI

F = {Fn(x) = log II(dzf")-lII}.

= -log IIdzfnII}i

Note that there exists a constant C

(20.8)

> 1 such that

C" I ll(dzfn)-lll-l

5

Il&fnll

I Kn

for any x E J and n 2 1, where K = max{Ildz f 11 : x E J } . We wish to apply the non-additive version of thermodynamic formalism and find roots of Bowen's equations c p J ( S'p)

= 0,

PJ(SCp) = 0.

(20.9)

In order to do this we ought to establish Property (12.1). Given 6 E (0,1], we call the map f 6-bunched if for every x E J we have Il(dzf)-lII1+alldzf

II < 1.

(20.10)

Proposition 20.5. [Bar21 Iff is a C1+a-expanding &bunched map (for some satisfy Condition (12.1). More6 > 0) then the sequences of functions cp and over, there exists E 2 0 such that for every w E C f , --E

+ -ncp

(XI I p ( w ) I Fik'(w) I cpn(x)

+

-E

for all suficiently large n and some k 2 1 (where x = ~ ( w ) ) .

In view of Proposition 19.1 this implies dimension estimates for the repeller J ,

-s -< dimH J < dim, J 5 d i m ~ J5 3,

(20.11)

Dimension of Sets and Measures Invariant under Hyperbolic Systems

209

where 2 and B are unique roots of Bowen's equations (20.9) (one can show that under the assumptions of Proposition 20.5 Bowen's equations (20.9) have unique roots). The lower and upper estimates in (20.7) and (20.11) cannot be improved. Note that n n

C(p(fj(4) I: pn(4I Pn@) I CP(fj(4) j=O

j=O

for every x = x ( w ) E J and n 1 1. These inequalities imply that if f is an &bunched C1+a-expanding map then t I: 3 I: 3 5 Z . If f is conformal it is easy-to see that t = 3 = 3 = z. If f is not conformal the numbers 3 and 3 may provide sharper estimates then the numbers t and f. Indeed, in [BarZ], Barreira constructed an example of a 1-bunched Cm-expanding map of a compact manifold for which t < s = 3 < 5. This map is not conformal but can be shown to be weakly-conformal (as a continuous expanding map; see (19.7)).

21. Multifractal Analysis of Gibbs Measures for Smooth Conformal Expanding Maps

We undertake the complete multifractal analysis of Gibbs measures for smooth conformal expanding maps. Let J be a conformal repeller for a C1+aconformal expanding map f : M + M of a compact smooth Riemannian manifold M . We assume that f is topologically mixing. The general case can be reduced to this one (with obvious modifications) using the Spectral Decomposition Theorem.

Thermodynamic Description of the Dimension Spectrum Let R = (R1,. . . ,R p } be a Markov partition for f and x the corresponding coding map from C i to J (see Section 20). Consider a Holder continuous function 'p on J . The pull back by x of 'p is a Holder continuous function (p on C i , i.e., @(w)= ' p ( x ( w ) ) for w E X i . Since f is topologically mixing (and so is the shift c) an equilibrium measure corresponding to this function, p = p+, is the unique Gibbs measure for o (see Appendix 11).Its push forward is a measure on J which is a unique equilibrium measure corresponding to 'p. We denote it by v = vv. Define the function 11, on J such that log$ = 'p - P~(cp).Clearly 11, is a Holder continuous function such that PJ(l0g 11,) = 0 and v is a unique equilibrium measure for log$. By the variational principle (see Theorem A2.1 in Appendix 11) we obtain that

Define the one parameter family of functions 'pq(4

=

'pq, q

E

(--00,

m) on J by

-w1% 1441+ 4log11,bL

where T ( q ) is chosen in such a way that P J ( ( P = ~ )0. It is obvious that the functions 'pq are Holder continuous. Clearly, T ( 0 )= dimH J = s.

210

Chapter 7

The function T(q)can also be described in terms of symbolic representation of the repeller by a subshift of finite type ( E f , a). Namely, let 6, and Qq be the pull back by the coding map x of the functions a , $J,and pq respectively. Note that log4 = Q - PE;(@).Therefore, the function T ( q ) satisfies

4,

@q@)

=

-w1% I6(w)l+ q l o g 4 4 4

with P + ( Q q ) = 0. This simple observation allows us to work with the funcE.4 tion T ( q ) using either the dynamical system ( J ,f ) or the underlying symbolic dynamical system ( E f , a). We first study some basic properties of the function T ( q ) .

Proposition 21.1. The function T ( q ) is real analytic for all q E R. Proof. Consider the function c: It2 + C"(Ef,R) defined by c(r,q ) = -r log I&[+ qlogq. This function is clearly real analytic. It is known that the pressure P = P + is a real analytic function on the space of Holder continuous functions E* (see for example, [Rl]), i.e. the function P(r,q ) = Pc+ (c(r,q ) ) is real analytic with respect to r and q. The desired result follows immediately from the Implicit Function Theorem once we verify the non-degeneracy hypothesis. The latter is

In order to compute the partial derivative we use the well-known formula for the derivative of the pressure (see [Rl]): given two Holder continuous functions hl and hz on E f , we have (21.1)

where p h l denotes the Gibbs measure for the function hl. Applying this formula with hl = -ro log 161 qo logq, hz = log 161, and E = ro - r we obtain that

+

(21.2)

where

is the Gibbs measure for c(r0,qo).

We show that the function T(q) is strictly decreasing by computing its first derivative. Fix q E R and let pq denote the Gibbs measure corresponding to the function (Pq. We also denote the push forward of pq to J by vq. The measure vq is a unique equilibrium measure corresponding to the function pq. Given q E R, let us set

The following statement establishes monotonicity of the function T ( q ) .

Dimension of Sets and Measures Invariant under Hyperbolic Systems

211

Proposition 21.2. For all q me have a(q) = -T’(q). In particular, T‘(q) < 0 for all q. Proof. Since P(cpq)= 0 for all q we have

Hence,

In order to compute the partial derivative we apply (21.1) with hi = -ro log 161 + qo log&, hz = log&, and E = qo - q. This results in

Using (21.2) and assuming that ro = T(q0) we conclude that

The lemma follows.

m

We show now that the function T ( q ) is convex by computing its second derivative. We recall that by m we denote the measure of full dimension (see Theorem 20.1), i.e., a unique equilibrium measure corresponding to the Holder continuous function -slogla(z)l on J , where s is the unique root of Bowen’s equation PJ(-slog )1.1 = 0.

Proposition 21.3. The function T(q) is convex, i.e., T”(q) >_ 0. It is strictly convex if and only zf u # m (see Figure 1‘7b). Proof. Differentiating twice the equation P(cpq)= 0 we obtain that

where the partial derivatives are evaluated at (9, r ) = (q, T ( q ) ) . We use the explicit formula for the second derivative of pressure for the shift map on C: obtained by Ruelle in [Rl]. Namely,

212

Chapter 7

where Qh is the bilinear form defined for hl, h2 E Cu(Ci,R) by

and /& is the Gibbs measure for the potential h. Ruelle also showed that Qh(g,g) 2 0 for all functions g E ca(ci,R) and that Qh(g,g) > 0 if and only if g is not cohomologous to a constant function (see definition of cohomologous functions in Appendix V). Applying the second derivative formula we obtain that d2 -P(-rlogIiiI

dr2

a2

+qlog4) = T P ( - ( T +~2)logI?il ~ ++qlog4), E~ dr = Q h b g 161, log 161).

Arguing similarly we find that

This implies that

It follows that T”(q) 2 0 for any q. Moreover, T”(q)> 0 for some q if the function logq(w) -T’(q) log l6(w)l is not cohomologous to a constant function. The latter can be assured provided that the functions logq(w) and -T’(q) log l6(w)l are not cohomologous. On the other hand, if they are cohomologous for some q then Pcip+(q)log161) = Pc+(l0g$J)= 0. Hence, -T’(q) = s. This implies that v = m.

Diametrical Regularity of Equilibrium Measures An important ingredient of the multifractal analysis of equilibrium measures is the remarkable fact that these measures are diametrically regular (see Condition (8.5)) as the following statement shows.

Proposition 21.4. Let p be a Holder continuous function on a conformal repeller J . Then any equilibrium measure for p with respect t o f is diametrically regular. Proof. Let v = v,+,be an equilibrium measure for p. Choose a Markov partition

R of J . Given a number r > 0, consider a Moran cover U, of J . Fix a point z E J

Dimension of Sets and Measures Invariant under Hyperbolic System

213

and choose those elements . . .,R(m) from the Moran cover that intersect the ball B ( x , 2 r ) . We recall the following properties of the Moran cover: for e v e r y j = l , ..., m, (1) R ( j )= Ri*...in(=.), where xj E J is a point; (2) diam R(j)5 r; (3) m 5 K, where K is a constant independent of x and r. There is an element of the Moran cover that contains x. We have that

R@)c ~ ( xr), c ~ ( x27') , c

m

U

~ ( j ) .

j=1

Define the function 11, such that log$ = cp - P(cp). Clearly, $ is a Holder continuous function on X such that P(log$) = 0 and v is an equilibrium measure for log$. By (A2.20) (see Appendix 11) we also have that for any j = 1,.. . ,m,

In view of the second statement of Proposition 20.1, if the diameter of the Moran cover does not exceed ro, then for any j = 1,.. . ,m,

This completes the proof.

Multifractal Analysis of Equilibrium Measures We state the result that establishes the multifractal analysis for equilibrium measures (corresponding to Holder continuous functions) supported on repellers of smooth conformal expanding maps. One can apply this result to conformal expanding maps listed in the previous section (i.e., hyperbolic rational maps, one-dimensional Markov maps, etc.). For a 2 0 consider the sets

J,

= {x E J : d&) = a }

(compare to (18.5)) and the f,(a)-spectrum for dimensions f u ( a )= dimH J,.

214

Chapter 7

Theorem 21.1. [PW2] (1) The pointwise dimension d u ( x ) exists for v-almost every x E J and

(2) The function f u ( a )is defined on the interval [al,a21 which is the range of the function a(q) (i.e., 0 5 a1 5 a2 < -00, a1 = a(-00)and a2 = a(--00)); this function is real analytic and fu(a(q)) = T ( q ) + q a ( q ) (see Figure 17b). (3) For any q E W we have that vq(JU(,)) = 1 and & ( x ) = d,(x) = T ( q ) qa(q) f o r v,-almost every

x

E

+

Ja(q).

(4) If v # m (i.e., v is not the measure of full dimension) then the functions fu(a)and T ( q ) are strictly convex and form a Legendre transform pair (see Appendix V). (5) The v-measure of any open ball centered at points in J is positive and f o r any q E R we have

where the infimum is taken over all finite covers 8, of J by open balls of radius r. I n particular, f o r every q > 1,

Proof. We begin with the following lemma. Lemma 1. There exist constants C1 > 0 and and f o r all basic sets Rio,,.in,

C2

> 0 such that f o r every q 6 W (21.4)

Proof of the lemma. Note that p and pq are Gibbs measures corresponding to the Holder continuous functions log4 and @, = - T ( q ) log 161 qlogq whose topological pressure is zero. Note also that m is a unique equilibrium measure corresponding to the Holder continuous function -slog la1 whose topological pressure is zero. It now follows from (A2.20) (see Appendix 11) that the ratios

+

v(Ri0...in)

vq(Ri0...in )

m(Rio..in)

r I L 0 + ( f k ( x )’) r I L 0 l a ( f k ( x ) I-T‘q’+(fk(x))q’ ) r I L 0 14fk(x))l-s (where x E Rio...in)are bounded from below and above by constants independent of n. The desired result follows.

215

Dimension of Sets and Measures Invariant under Hyperbolic Systems Given 0

RP) =

< r < 1, consider a Moran cover 4 of

Ri,,.,in(+

the repeller J by basic sets with radius approximately equal to r (see Section 20). Let

N(z,r) denote the number of sets RP) that have non-empty intersection with a given ball B ( z , r ) centered at z of radius r . By the property of the Moran cover N ( z ,r ) 5 M uniformly in z and r , where M is a Moran multiplicity factor (recall that M does not depend on z and r ) . It follows from (20.1) and (20.2) that there exist positive numbers C3 and C4 such that for every RP) E &, c3r5

I ~ ( R P )5) c4rS.

(21.5)

Since 4 is a disjoint cover of J we have

vq(RP))= 1. R?)ELL,

Hence, summing (21.4) over the elements of the cover 4, we obtain that there exist positive constants C5and c6 such that

c55 rT(q)

C

I

v ( ~ P ) ) qc6.

R2)ELL,

Taking logarithm and dividing by log r yields for all q E R,

(21.6) Given 0 < r < 1 and a point w E C i , consider the unique positive integer n ( w ) = n ( w , r ) defined by (20.2). For a number a 2 0 denote by j, the “symbolic” level set, i.e., the set of points w E C i for which the limit

(21.7)

exists and is equal to a. Define the “symbolic” spectrum for dimensions by

-

-

fu ( a )= dimH J ,

.

(21.8)

Chapter 7

216

Proof of the lemma. Consider the functions w r j log lii(w) I and w Since pq is ergodic the Birkhoff ergodic theorem yields that

5log4(ak(w))

lim

k=O

n+w

c log l+k(w))l-l

rj

log 4 ( w ) .

= 4Q)

k=O

for pq-almost every w E C i . This implies the first statement. It follows that for any E > 0 and every w E ja(q) there exists r ( w ) such that r(w), for any r I

c log4(ak(w)) I c log la(.k(w))l-l n(w,r)-

a((?)- E

1

n(wo):l,

I 4Q) + E.

(21.9)

k=O

i}.

Given -! > 0, denote by Qe = { w E ja(*) : T(W) 5 It is easy to see that Qe C Qe+i and ja(q) = Qe. Thus, there exists l o > 0 such that vq(Qe) > 0 if l 1 lo.Let us choose l Z 4,. Fix r , 0 < r < 1 and consider a Moran cover & , Q ~ of the set Qe (see Section 20). It consists of cylinder sets Cf), j = 1 , . . . ,Nr,e for which there exist points w j E Qe such that Cf) = C~. o . .. . z n ~ ~ jIf~ .r is sufficiently small we have n ( w j ) 2 e for all j. Since pq is a Gibbs measure we obtain by (A2.20) (see Appendix 11) that for every w = (i& . . . ) E and n > 0,

u,"=,

(21.10)

where C, and Cs are positive constants. It follows from (21.9), (21.10), and (20.2) that for all n 2 -! and any x E x(Qe), M ~q(B(T 5 ), n x(Qe))

5

pq(Cf)) j=1

n

M n(wj)-1

< CSC

j=1

j=1

-T(q)

l~(flk(w))l

- k !@

k=O

k=O

where C9 > 0 is a constant and M is a Moran multiplicity factor (we stress that both Cg and M do not depend on 1 and j ) . Since vq(x(Qe)) > 0, by the Bore1

Dimension of Sets and Measures Invariant under Hyperbolic Systems

217

Density Lemma (see Appendix V), for vq-almost every x E x ( Q ! ) there exists a number ro = ro(x) such that for every 0 < r 5 T O we have

This implies that for any l > l o and almost every x 6 x ( Q t ) ,

+

Since sets Qe are nested and exhaust the set Q we obtain that &(x) 2 T ( q ) q(a(q)- E ) for vq-almost every x E J,(q). Since E is arbitrary this implies that &(x) 2 T ( q ) qa(q) for vq-almost every x E x ( J , ( ~ ) ) By . Theorem 7.1 we obtain that dimH x(S,(,)) T ( q ) q a ( q ) . Fix 0 < T < 1. For each w = ( i ~ i l .. . ) E Qe choose n ( w ) = n ( w , r ) according to (20.2). It follows that R;o.,,in(w) c B ( x , r ) ,where x = ~ ( w ) By . virtue of (21.9) and (21.10) for all w E Qe,

+

>

+

-

d v q ( x )= --ogvq(B(xc, lim 7-0 logr

5 T ( q )+ q ( a ( q )+ E ) .

Since E is arbitrary this implies that J , (x)5 T(q)+qa(q)for every x E ~ ( j , ( ~ ) ) . Therefore, &(x) = au,(x) = T ( q ) g a ( q ) for vq-almost every x E x(S,(,)) and the second statement of the lemma follows. Moreover, by Theorem 7.2, we obtain that dimH x ( S , ( ~ )5) T(q)+qa(q).This implies the last statement and completes the proof of the lemma.

+

It immediately follows from (21.6) that T(1) = 0 and thus, p = p1. The first statement of the theorem now follows from Lemma 2 and the following lemma.

Lemma 3. J , = x(S,). Proof of the lemma. Fix 0 < r < 1. For each w = n = n ( w , r ) according to (20.2). It follows that

(2021

...) E Qe choose

218

Chapter 7

where x = ~ ( w ) y, E Ri,...in is a point, and (710 > 0 is a constant. Since the measure v is diametrically regular (see Proposition 21.4) we obtain v(%...i") I Zmz,

I @(Y, 2r)) I Ciiv(B(y,GOT)) I 4Ri0...in)7

where C11 > 0 is a constant. Note that p is a Gibbs measure corresponding to the Holder continuous function log4 whose topological pressure is zero. Note also that m is a unique equilibrium measure corresponding to the Holder continuous function --slog la[ whose topological pressure is zero. It now follows from (A2.20) (see Appendix 11) that the ratios

(where x E Rio,,,i,,)are bounded from below and above by constants independent of n. This implies that the two limits nfw.rl-I

exist simultaneously, and the proof of the lemma follows.

+

Applying Lemma 3 we conclude that dimH Ja(q)= T ( q ) qa(q), and Statements 1-5 follow. We now prove the final statement of the theorem. Given r > 0, consider a Moran cover U,. = {RP)} of J . There are positive constants Cl2 and c13 independent of r such that for every j one can find a point xj E RLj) satisfying

B(xj,C12T) C RP)

C

(21.11)

B(xj7 C13~).

Since the measure v is diametrically regular (see Proposition 21.4) it follows from (21.11) that for every q E R,

c

v(B(Xj, C13r))'

v(B(xj, c12r))' 5

5 c14

c 1 4

j

j

c

v(RP))',

(21.12)

j

where C14 > 0 is a constant independent of j and r . Let Gr be a cover of the repeller J by balls B(yi,r). For each j 2 0 there exists B(yij, r ) E 8,. such that B(yij ,r ) n RP) # 0. Consider the new cover of J by the balls Bj = B(yi,, 2C13r). By (21.11) each basic set RLj) is contained in at least one element of the new cover. Since the measure v is diametrically regular we obtain by (21.11) that for any q E R,

c

Rktk

v(RY))' I cl5v(Bk)',

(21.13)

Dimension of Sets and Measures Invariant under Hyperbolic Systems

219

where CIS > 0 is a constant. Exploiting again the fact that the measure diametrically regular we conclude using (21.13) that for any q E W,

Y

v(R;)' 5

c:~

V(Bk)'

5 c16

k

R!j'€U,

is

v(B)'i B€P,

> 0 is a constant. Statement 5 of the theorem follows immediately where from (21.6), (21.12), and (21.13). Remarks. (1) Assume that v = m is the measure of full dimension. This implies that the functions -slog ( ao X I and log+ are cohomologous to each other (see qlog+) = 0 it follows that T ( q ) = Appendix V). Since PE; (-T(q ) log la o X I (1 - q)s (thus, T ( q ) is a linear function). Since the pointwise dimension of m is equal to s everywhere in J we have that f v ( s ) = s and f v ( a )= 0 for all a # s.

+

slope = -a(--)

\

s= dim,, J

\ \

Figure 17a. "TYPICAL" GRAPHOF

THE

FUNCTION T(q).

(2) Assume that Y # m. There is an interesting manifestation of the multifractal analysis which immediately follows from Statement 4 of Theorem 21.1: for any E > 0 there exists a distinct (singular) equilibrium measure p on J such that I dimH p - dimH 5 E . In fact, using analyticity of the topological pressure and Statement 1 of Theorem 21.1 (see also (21.20) and (21.22) below) one can show that (we leave detail arguments to the reader): the Hausdorff dimension

YI

220

Chapter 7 tangent to the curve at a(l ) , with slope = 1

\/

f(a)A f(a(1))

_---------------

s = dimH(support p)

I I I I I I

I I I I I I I

I I

I

I

I

I

I

I

I

I

0

a(m)

Nl)

I

I

:

a(-)

a(0)

q>o

; I

a

b

I

q

(4+0O)),

={

f u ( 4 + 4 ) , fu(a(-0O))}.

(5) Assume that vq is the measure of maximal entropy for some q # 0. This implies that the function pq is cohomologous to 0, and hence the functions T ( q )log la1 and qlog $ are cohomologous. Therefore, the measure v must be a unique equilibrium measure corresponding to the Holder continuous function t log lal, where t = ( 6 ) In [Si], Simpelaere obtained a variational description of the function

y.

(21.16) where the infimum is taken over all Borel ergodic measures p on J . It follows from Statements 1 and 3 of Theorem 21.1 and (21.3) that the infimum is attained at p = vq.

Completeness of the f,,(a)-Spectrum Schmeling [Sch] showed that the f,(a)-spectrum is complete, i.e., for any ] set J , is empty. More precisely, the number a outside the interval [ c q , a ~the following result holds.

Theorem 21.2. We have that

and hence f,,(a) = 0 zf and only ifa $ [al,a2].

Proof. Using (21.16) we find that for every q 2 0,

where the infimum is taken over all Borel ergodic measures p on J . Dividing by q and letting q -+ 00 yields that (21.17)

Chapter 7

222

Recall that a1 = limq+ma(q). Therefore, in view of Theorem 21.1, a1 2 inf,,Jd,(x). Assume that this inequality is strict. One can find E > 0, x E X , and a sequence of numbers n k such that

log$(.) j=O

(Io ) Clogla(x)I

-1

La1 --.

Let p be an accumulation measure of the sequence of measures pk =

nk-I

7=0

sf,(.)

(here 6, denotes the unit mass concentrated at the point y). We havethat

Passing to the limit as k

+ co yields that

which contradicts (21.17). This implies the first equality. The second one can be proved in a similar fashion. Schmeling also proved that for any 71, 7 2 E [0, s] there exists an equilibrium measure v on J corresponding to a Holder continuous function for which f,(al) = 71 and fv(a2) = 72. However, “generically” fv(al)= fv(az) = 0; more precisely, this holds for equilibrium measures whose potential functions belong to an open and dense set in the space of Holder continuous functions on J - the phenomenon noticed previously in physical literature (most of the graphs of the function f,(a) that one can find there end up on the x-axis). Note that in the latter case the measures va(-..) and have zero measure-theoretic entropy. We have the following multifractal decomposition of a conformal repeller J associated with the pointwise dimension of an equilibrium measure supported on J and corresponding to a Holder continuous function: (21.18)

Here J , is the set of points where the pointwise dimension takes on the value a. Each set J , is everywhere dense in J and supports a measure va (i.e., va(J,) = l),which is a unique equilibrium measure corresponding to the Holder continuous qlog$(x), where q is chosen such that a = function (pq(z)= -T(q)logla(x)I a(q). The set j - the irregular part of the multifractal decomposition consists of points with no pointwise dimension. We will see in Appendix IV that j # 0 ; moreover, it is everywhere dense in J and carries full Hausdorff dimension (i.e., dimH = dimH J ) and full topological entropy (i.e., hj(f)h~(f)).

+

Dimension of Sets and Measures Invariant under Hyperbolic Systems

223

Pointwise Dimension of Measures on Conformal Repellers Given a point x E J , we define the Lyapunov exponent at x by (21.19)

(provided the limit exists; see Section 26 for more details). If v is an f-invariant measure then by the Birkhoff ergodic theorem, the above limit exists v-almost everywhere, and if v is ergodic then it is constant almost everywhere. We denote the corresponding value by A, > 0. Let v be an equilibrium measure corresponding to a Holder continuous function on J . It follows from Statement 1 of Theorem 21.1 that for v-almost every x E J, (21.20)

where h,(f) is the measure-theoretic entropy of f and

Furthermore, let G, be the set of all forward generic points of v (i.e., points for which the Birkhoff ergodic theorem holds for any continuous function on X ; see Appendix 11). It follows from Lemmas 2 and 3 in the proof of Theorem 21.1 (applied to the measure v1 = v) that for every x E G,,

In view of Theorems 7.1 and 7.2, this implies that h'(f) A,

- dimH v = dimH G,.

(21.22)

We extend (21.20) and (21.22) to any Borel ergodic measure on J of positive measure-theoretic entropy which is not necessarily an equilibrium measure.

Theorem 21.3. For any Borel ergodic measure v of positive measure-theoretic entropy supported on a conformal repeller J we have (1) the equality (21.20) holds for almost every x E J ; = dimH v = dimH G, = h B G , = dim~G,. (2)

y.

Proof. Set d, = We first show that d,(x) 2 d,. The proof is a slight modification of the Goof of Statement 2 of Theorem 13.1. Consider a Markov partition R = { R l , .. .,R p } for f and the corresponding symbolic model (Cf;,0) (see Section 20). Let p be the pullback to of the measure v by the coding map x.

224

Chapter 7

Fix E > 0. It follows from the Shannon-McMillan-Breiman theorem that for p-almost every o E C i one can find N l ( w ) > 0 such that for any n 2 N l ( w ) ,

where Cia,,,in(w) is the cylinder set containing w and h = h,(a) is the measuretheoretic entropy of the shift 0 . It follows from the Birkhoff ergodic theorem, applied to the function log la(z)l, that for u-almost every z E M there exists N z ( z ) such that for any n 2 N z ( z ) , (21.24) In order to prove the desired lower bound for d,(z) it remains only to use (21.23) and (21.24) and to repeat readily the argument in the proof of Statement 2 of Theorem 13.1. We now prove the opposite inequality. Fix 0 < r < 1. By (20.1) it follows that R~a...;n~z)+l c B ( z ,C l r ) , where C1 > 0 is a constant. Therefore,

4 B ( z 1 C r ) )2 where Cz

4 % . . . i n ~ z ) + l2)

CZexp(-(h

+&)n(z)),

> 0 is a constant. By virtue of (20.1) we obtain for all z E J ,

Since E can be arbitrarily small this proves that &(z) 5 d,. In order to prove the second statement we note that d , = dimH u 5 dimH G,. On the other hand, by (A2.16) (see Appendix 11), we conclude that 0 = h,(f) d,X,du = PG,(-~,X,), i.e., d , is the root of Bowen's equation. Repeating arguments in the proof of Theorem 20.1 (applied to the set G, instead of J ) we obtain that d i m ~ G ,5 d,. This completes the proof of the theorem.

s,

Some results, similar to Theorem 21.3 for measures supported on conformal repellers of holomorphic maps, were obtained in [Ma2, PUZ]. Since the measure-theoretic entropy is a semi-continuous function we obtain as an immediate corollary of Theorem 21.3 and (21.21) that: the Hausdorff dimension of a Borel ergodic measure u on a conformal repeller is a semicontinuous function of u.

Information Dimension We compute the information dimension of a Gibbs measure u on a conformal repeller J . Applying Theorem 21.1 and taking into account that the function T ( q )is differentiable we obtain that the limit T(q) lim 1- q

q+l

Dimension of Sets and Measures Invariant under Hyperbolic Systems

225

exists and is equal to --T”(1) = ( ~ ( 1 ) As . we know the latter coincides with the Hausdorff dimension of Y (see (21.22) and Remark 3 above). This implies that f”((Y(1)) = (Y(1) = -T‘(l) = I(v) = 7(v), where I ( v ) and 7 ( v )are the lower and upper information dimensions of v (see Section 18). We note that Statement 5 of Theorem 21.1 allows one to extend the Hentschel-Procaccia spectrum and Rhnyi spectrum for dimensions for any q # 1. Moreover, the above argument makes it possible to define these spectra even for q = 1 (as being equal to ( ~ ( 1 ) ) .

Dimension Spectrum for Lyapunov Exponents We consider the multifractal decomposition of the repeller J associated with the Lyapunov exponent X ( x ) (see (21.19) (21.25) where

L ={x E J

: the limit in (21.19) does not exist)

is the irregular part and

Lp = { x E J : the limit in (21.19) exists and X(x) = p}. If v is an ergodic memure for f we obtain that X(x) = A, for v-almost every x E J . Thus, the set LA, # 0. Moreover, if Y is an equilibrium measure corresponding to a Holder continuous function the set LA, is everywhere dense (since in this case the support of v is the set J ) . We note that if the set L p is not empty then it supports an ergodic memure v p for which A, = p (indeed, for every x E L p the sequence of measures 6 f k ( s ) has an accumulation measure whose ergodic components satisfy the above property). There are several fundamental questions related to the above decomposition, for example, (1) Are there points x for which the limit in (21.19) does not exist, i.e., i# 0 7 ( 2 ) How large is the range ofvalues o f X ( x ) ? (3) Is there any number p for which any ergodic measure Y with u(Lp) > 0 is not an equilibrium measure? ~~~~

In order to characterize the above multifractal decomposition quantitatively, we introduce the dimension spectrum for Lyapunov exponents o f f by

f(p) = dimH L p .

Chapter 7

226

This definition is inspired by the work of Eckmann and Procaccia [EP]. In [We], Weiss derived the complete study of the Lyapunov dimension spectrum for conformal expanding maps by establishing its relation to the fuma, (a)-spectrum, where u,, is the measure of maximal entropy. Notice that the measure of maximal entropy is a unique equilibrium measure corresponding to the function cp = 0, and hence 1c, = constant = exp(-hJ(f)), where h J ( f )is the topological entropy of f on J . Therefore, for every x E Lp,

This implies the following result.

Theorem 21.4. [We] Let J be a conformal repeller f o r a smooth expanding map f . Then (1) If vm, # m (m is the measure of full dimension) then the dimension spectrum f o r Lyapunov exponents

is a real analytic strictly convex function o n a n open interval [PI, p z ] containing the point P = h J ( f ) / s . (2) If v,, = m then the dimension spectrum f o r Lyapunov exponents is a delta function, i.e.,

As immediate consequences of Theorem 21.4 we obtain that i f urn, # m then the range of the function X(x) contains an open interval, and hence the Lyapunov exponent attains uncountably many distinct values. On the contrary, if the Lyapunov exponent X(x) attains only countably many values, then urn, = m. There is an interesting application of this result to rational maps. In [Z], Zdunik proved that in the case of rational maps the coincidence v,, = rn implies that the map must be of the form t + zfn. Therefore, we obtain the following rigidity theorem for rational maps. Theorem 21.5. I f the Lyapunov exponent of a rational map with a hyperbolic Julia set attains only countably many values, then the map must be of the f o r m z 3 zfn. We can now answer the above questions. Namely, (1) the set is not empty and has full Hausdorff dimension (see Appendix IV; compare to (21.18)); (2) the range of values of X(x) is an interval [P,,p,] and f o r any P outside this interval the set Lp is empty (i.e., the spectrum is complete); (3) f o r any p E [pl,p,] there exists an equilibrium measure u corresponding to a Holder continuous function f o r which u ( L p ) = 1.

e

Dimension of Sets and Measures Invariant under Hyperbolic Systems

227

22. Hausdorff Dimension and Box Dimension of Basic Sets for Axiom A DifFeomorphisms

Axiom A Diffeomorphisms In this section we study the Hausdorff dimension and box dimension of sets invariant under smooth dynamical systems with strong hyperbolic behavior. Let M be a smooth finite-dimensional Riemannian manifold and f : A4 + M a C1+a-diffeomorphism (i.e., f is a C1+a-invertible map whose inverse is of class C'+"). A compact f-invariant set A c M is said to be hyperbolic if there exist a continuous splitting of the tangent bundle TAM = E(') CBE(") and constants C > 0 and 0 < X < 1 such that for every x E A (1) dfE(")(z)= E ( S ) ( f ( x ) )dfE(")(x) , = E(")(f(z)); (2) for all n 2 0

~ldj"vll5 ~ ~ " l l v l l if v E E ( ' ) ( z ) , ~ldf-~vll5 ~ ~ " l l v l l if v E ~ ( " ) ( x ) . The subspaces E(')(x)and E ( " ) ( z )are called stable and unstable subspaces at z respectively and they depend Holder continuously on z. It is well-known (see, for example, [KH]) that for every x E A one can construct stable and unstable local manifolds, W,b?(z)and W,b"(x).They have the following properties: (3)

2

E W,b",'(z), 2 E W k ) ( x ) ;

(4) T,W,;;(z) = E ( S ) ( z ) T,W,bu((X) , = E(")(x); ( 5 ) f(W,b",'(.,) c W,;",'fc.)), f-'CW,b",'(.,) c W,:2(f-'(4); (6) there exist K > 0 and 0 < ,u < 1 such that for every n 2 0,

PV(Y), ~ ( z )5) ~,u"p(y,x) for all Y E ~, b" , ' x)

(22.1)

5 K,u"P(Y,Z) for all Y E W,b"(4,

(22.2)

and p(f-"(y),f-"(4)

where p is the distance in M induced by the Riemannian metric; (7) there exist E > 0 such that the intersection W,',",'(z)nB(z,E)(respectively, W,b",'(x)n B ( z ,E ) ) consists of all points in B ( z ,E ) that satisfy (22.1) (respect ively, (22.2)).

A hyperbolic set A is called locally maximal if there exists a neighborhood U of A such that for any closed f -invariant subset A' c U we have A' c A. In this case A= fyq.

n

-w 0. For almost every y E R ( z ) the conditional measure v(')(y) generated by Y on W[zJ(y) n R ( z ) (respectively, the conditional measure v(")(y) generated by v on W,b?(y) nR(z))coincide with the push forward of the measure p(') (respectively, p ( " ) ) . By Proposition A2.2 (see Appendix 11) we obtain the following result. Proposition 22.2. There are positive constants A1 and A2 such that for a n y n ~ ( z and ) F E ~ , b " ( zn ) ~(z), Borel sets E E ~,b",'(z)

A l ( v ( ' ) ( E )x v ( " ) ( F ) )5 v ( E x F ) 5 A z ( d ' ) ( E ) x d " ) ( F ) ) . In other words, the measure v on R ( z )is equivalent to the direct product of measures v(')(z)and v(")(z). Conformal Axiom A Diffeomorphisms We say that a diffeomorphism f of a locally maximal hyperbolic set A is u-conformal (respectively, s-conformal) if there exists a continuous function u(")(z) (respectively, a(")(.)) on A such that df I E ( " ) ( z )= u(")(z) Isom, for every z E A (respectively, df I E ( ' ) ( z )= u ( ' ) ( z ) Isom,; recall that Isom, denotes an isometry of E ( " ) ( z )or E(')(z)). Since the subspaces E(")(z)and E(')(z) depend Holder continuously on z the functions u ( " ) ( z )and a(")(.) are also Holder continuous. Note that Ia(")(z)I > 1 and I U ( ~ ) ' ( Z ) I < 1 for every z E A. A diffeomorphism f on A is called conformal if it is u-conformal and s-conformal as well. Consider a Markov partition R = { E l , .. . ,R p } of A of a small diameter (which is less than the number E in Property (7)). Given a point z E A, define the sets (22.7) A(")(z) = w,b",' (z) n ~ ( z ) , A(') (z) = w/;J(z) n ~ ( z )

Dimension of Sets and Measures Invariant under Hyperbolic Systems

231

(we assume that x E intR(x)). Note that by (22.5) the rectangle R ( x ) has the direct product structure (defined up to a homeomorphism 0): R ( x ) = A(")(x)x

A(a)(x).

From now on we assume that f is u-conformal. Note that one can view A ( " ) ( x )as the limit set for a geometric construction (CG1-CG2) with Ri:,),i, to be the basic sets. Namely,

A(")(%)=

n

U R::,),~~n ~ , b " ( ~ ) ,

n20 (i,...in)

where the union is taken over all admissible n-tuples ( 2 0 . . . in) with Ri, = R(x). Moreover, this geometric construction is a Moran-like geometric construction with non-stationary ratio coefficients since its basic sets satisfy Condition (B2) (see Section 15) as the following statement shows. Proposition 22.3. Let A be a locally maximal hyperbolic set f o r a u-conformal C'+a-diffeomorphism f . W e have that (1) each basic set Ri:,).i,nW/:)(x) contains a ball in W[b",'(x) of radius~io,,,i, and is contained in a ball in W,b"(x) of radius Fio,,,in; (2) There exist positive constants K1 and Kz such that f o r every basic set Ri:,!,in and every x E R!:,),i,,

Proof. See the proof of Proposition 20.2. Using the analogy with geometric constructions described above we will construct a Moran cover of the set A(")(x).It is comprised from basic sets and it allcws us to build up an optimal cover for computing the Hausdorff dimension and lower and upper dimensions of A. .. ^ box ^ Let D = (. . .i-lioil.. .) E CA is chosen such that x(D) = x. We identify the set of points in CA having the same past as D with the cylinder set C$ c X i . Given r > 0 and a point w E C?, let n ( w ) denote the unique positive integer 10 such that n(w)

J-J I,(")

k=O

I-'

(x(ak(w))) > T ,

n

n(w)+l

I,(")

(x(ak(w))) 1-l I r.

(22.8)

k=O

It is easy to see that n ( w ) + co as T + 0 uniformly in w. Fix w = (ioil . . .) E C$ 10 and consider the cylinder set C;o,,.~n~w) c C;. We have w E Cio...,n(W) and if w' E Cio,..in(W) and n(w') 2 n ( w ) then

232

Chapter 7

Let C ( w ) be the largest cylinder set containing w with the property that C ( w ) = C,o,,,jn(w,,) for some w" E C ( w ) and C~o...jn(w) c C ( w ) for any w' E C ( w ) . The sets C ( w ) corresponding to different w E C? either coincide or are disjoint. We 10

denote these sets by C,'"),j = 1,.. . ,N,. There exist points w j E C? such that 10

Cy' = C j o , , . j , ( w jThese ). sets form a disjoint cover of C? which we denote by 20

&I. The sets Ry' = x(C,'"'),j = 1 , . . . ,N , may overlap along their boundaries

and comprise a cover of A ( " ) ( z )(which we will denote by the same symbol &) if it does not cause any confusion). We have that RY) = Rio...in(yj) for some yj E A ( " ) ( x ) . Let Q c C? be a subset (not necessarily invariant). One can repeat the 10 above arguments to construct a Moran cover of the set Q. It consists of cylinder sets C y ) , j = 1,.. . , N , for which there exist points w j E Q such that C,'") =

Cio..,in(,j) and the intersection Cj'"'

n C,!")n Q is empty as soon as i # j .

We

denote this cover by .;:U Moran covers have the following crucial property. Given a point z E A(")(z) and a sufficiently small r > 0, the number of basic sets RP) in a Moran cover &) that have non-empty intersection with the ball B ( % ) ( z , ris ) bounded from above by a number M which is independent of z and r . We call this number a

Moran multiplicity factor.

In order to establish this property let i = max {diamRi : i = 1,.. .,p}. Since 5i the sets Rj are the closure of their interiors there exists a number 0 < such that for every x E A,

B(")(z,r1) c A ( % ) ( z ) . The desired property of Moran covers follows from Proposition 22.3.

Hausdorff Dimension and Box Dimension of Locally Maximal Hyperbolic Sets for u-Conformal and s-Conformal Diffeomorphisms Let A be a locally maximal hyperbolic set for a C'+a-diffeomorphism f . Assume that f is topologically mixing. We denote by d")a unique equilibrium measure corresponding to the Holder continuous function -t(")logla(")(z)l on A, where t(") is the unique root of Bowen's equation PA(-tlogla(")l) = 0. For every y E R(z) we also denote by m ( " ) ( y )the conditional measure on W ( " ) ( y ) n R ( o )generated by du) as explained above (see Proposition 22.2; recall that m ( " ) ( y ) is the push forward by the coding map of the unstable part of the measure d")). We now state our main result assuming that f is u-conformal.

Theorem 22.1. Let f be a u-conformal C1+a-diffeomorphism of a locally maximal hyperbolic set A. T h e n (1) f o r any z E A and f o r any open set U c W ( " ) ( z such ) that U n A # 0, dimH(U n A) = b B ( U n A) = dimB(Un A) = t(");

Dimension of Sets and Measures Invariant under Hyperbolic Systems (2)

hnw (

233

f1

s, log Ia(")(x)I d d " ) ( x ) '

'("I =

where h,(,)(f) is the measure-theoretic entropy o f f with respect to the measure d"); (3) the t(")-Hausdorff measure of U n A is positive and finite; moreover, it is equivalent to the measure m(")(x)lUfor every x E A; (4) d") = dimHm(")(X) f o r every x E A, i.e., the measure m(")(x) is the measure of full dimension (see Section 5).

Proof. It is sufficient to prove the theorem assuming that U = A(") (y) for some point y E W ( " ) ( x ) . The arguments are very similar to the arguments in the proof of Theorem 20.1 and exploit the crucial property of Moran covers observed above. ^ ^ Choose a point & = (. . . L ~ i ~ i.)l .E. C A such that ~ ( 2=) y and set d = dimH(A(")(y) nA). We show first that t(")5 d. Fix E > 0. By the definition of the Hausdorff dimension there exists a number r > 0 and a cover of A(")(y)nA by balls Be, l = 1,2,. . . of radius re 5 r such that A

For every l > 0 consider a Moran cover $) of A(")(y) n A and choose those basic sets from the cover that intersect Be. Denote them by RP),. . .,Rim(e)). Note that R y ) = R$'!,in(l,j)for some ( 2 0 . . . i n ( e , j ) ) . Using Proposition 22.1, the property of the Moran cover, and repeating the argument in the proof of Statement 1 of Theorem 20.1 one can show that

where C1 > 0 is a constant. Given a number N > 0, choose r so small that n ( l , j ) _> N for all l and j . We now have that for any n > 0 and N > n,

where M ( C $ , O , p , U , , , N ) is defined by (A2.19) (see Appendix 11) with and p(w) = -(d E ) log la(")(x(w))l.

+

(Y

=0

234

Chapter 7

This implies that

Pc; (-(d

+ E) log )"(,I

0 XI)

5 0.

Notice that for any continuous function cp: A + R pA(CP)

= PEA(X'cp) = suPP,,~(c$)(x*'P) = pc$ (x*cp). n>O

Hence, PA

+ E ) log

(-(d

'0

'0

lU(")l)

5 0.

Therefore, by Theorem A2.5 (see Appendix II), d") 5 d + ~ .Since this inequality holds for all E we conclude that d") 5 d. Denote d = dim~(A(")(y) fl A). We now proceed with the upper estimate and show that d 5 d"). Fix E > 0. Repeating arguments in the proof of Theorem 20.1 one can show that there exists a positive integer N such that for an arbitrary cover 6 of C? by cylinder sets Cio..,iN, ao

c

n N

sup

C;o...,N€G ZERP) k=O

where

C2

la(")(fk(z))l-z+ze 2 cz,

> 0 is a constant. We now have that for any n > 0 and N > n,

=

C

Ci,,... ,EG

I-IIa'"'(fk(z))I-6+2E 2 1"

sup E€RP)

c2,

k=O

where R(C?, 0, c p , U ~N) , is defined by (A2.19') (see Appendix 11) with a = 0 20 and qJ(w) = - ( d - 2E) log lu(")(x(w))l. This implies that

CP,; ( - ( d -

2 4 log Iu(U) 0 XI) 2 0.

Notice that for any continuous function p: A pA(cp)

+R

= pEA(X*(P) =cpEA(x*cp) 2 s u P C P ~ ~ ( C ~ ) ( x *= ( PCPC$ ) (x*cp)' n20

It follows that PA

(-(a-

'0

'0

2E) log I U ( " ) l ) 2 0

and hence d - 2~ 5 d"). Since this inequality holds for all d 5 t("). This completes the proof of the first statement.

E

we conclude that

Dimension of Sets and Measures Invariant under Hyperbolic Systems

235

Since the measure dU) is an equilibrium measure corresponding to the function -t(") log Ia(")(x)I on A, whose topological pressure is zero, we obtain in view of the variational principle (see Appendix 11) that

This implies the second statement. Since the measure p(") is the stable part of the measure d")(i.e., the projection of &(") to Z f ) for every x E A and any basic set Rir,),,incontaining x we have that n

m(u)(x)(~!r,!,in n w ( u ) (5~ )const ) nla(")(fk(x))l-'. k=O

Fix z E A, r > 0 and consider a Moran cover of the set A ( " ) ( z )by basic sets R(j).It follows from (22.8) that n

Therefore, by the property of the Moran cover we find that

m(")(~)(B(~)(x,r)) 5 E m ( " ) ( x ) ( R ( j5) )Mrt (=), j

where M is a Moran multiplicity factor (which does not depend on T ) . Thus, m(")(x)satisfies the uniform mass distribution principle (see Section 7) and hence dimH m ( " ) ( x ) 2 t("). The opposite inequality follows from the first statement. This also shows that the t(")-Hausdorff measure of U n A is positive and finite and is equivalent to the measure m(")(x)lA(")(x). Let A be a locally maximal hyperbolic set for a C1+a-diffeomorphism f . Assume that f is s-conformal. Denote by dS)a unique equilibrium measure corresponding to the Holder continuous function t(')log Ia(")(x)Ion A, where t(') is the unique root of Bowen's equation pA(t log Id')I) = 0. Consider the system of conditional measures m(')(y) on W(')(y) n R ( x ) ,y E R ( x ) generated by (see Proposition 22.2; recall that m(')(y) is the push forward by the coding map of the unstable part of the measure d');see Appendix 11). Similarly to Theorem 22.1, one can prove that for any x E A and any open set u c ~ ( ' ) ( x ) , dimH(U n A) = h B ( U n A) = dim~(U n A) = t('). Moreover, t(S)

hd.)

=JA

(f)

log la(s)(x)ld &( S ) ( Z ) '

236

Chapter 7

where h,(.) (f)is the measure-theoretic entropy of f with respect to the measure d'). In addition, the t(')-Hausdorff measure of U n A is positive and finite. We assume now that A is a locally maximal hyperbolic set for a diffeomorphism f which is both s- and u-conformal. By results of Hasselblatt [Ha] in this case the stable and unstable distributions on A are smooth. Hence, given a point z 6 A and a rectangle R containing x, the homeomorphism 0 in (22.5) is Lipschitz continuous. Therefore, using Theorems 6.5 and 22.1 we compute the Hausdorff dimension and box dimension of A.

Theorem 22.2. We have

+ d'),

dimH A = b B A = d i m ~ A= t(")

where t(") and t(")are unique roots of Bowen's equations pA(-tbg

I,(")[

= 0,

PA(tlOg la(')l) = 0

respectively and can be computed by the formulae

This result applies and produces a formula for the Hausdorff dimension and box dimension of a basic set of an Axiom A C1+a-surface diffeomorphism, which is clearly seen to be both s- and u-conformal. For this case, McCluskey and Manning [MM] proved that dimH A = t(") d'). For diffeomorphisms of class C2 Takens [T2] showed that the Hausdorff dimension of A coincides with its lower and upper box dimensions. Palis and Viana [PV] extended this result to diffeomorphisms of class C1. Barreira [Bar21 obtained the same result using another approach based on the non-additive version of the thermodynamic formalism. This approach allowed him to extend Theorem 22.2 to "conformal" homeomorphisms possessing locally maximal "topologically hyperbolic" sets. Consider the measures m(")(x)and m(')(x)for z E A. By (22.9), Proposition 26.1, and Theorem 7.1,

+

dimH m(")(z)= t(")+ d'), dimH m(')(z)= t(') + d"), where d(') 5 t(') and d(") 5

d"). Moreover, the equalities hold if and only if &(") = /€(')

%f /€.

(22.10)

In this case, K is the measure of full dimension. Condition (22.10) is a "rigidity" type condition. It holds if and only if the functions -t(") log Iu(")(x)Iand t(')logla(')(z)l are cohomologous (see [KH] and Appendix V). One can show that this is the case if and only if for any periodic point z E A of period p ,

n

P- 1

k=O

a(u)(fk(x))t(U) a(s)(fk(z))t(') = 1.

Dimension of Sets and Measures Invariant under Hyperbolic Systems

237

Estimates of Hausdorff Dimension and Box Dimension of Locally Maximal Hyperbolic Sets: Non-conformal Case Let A be a locally maximal hyperbolic set for a topologically transitive C1+adiffeomorphism f of a compact manifold M . In the multidimensional case the map f may not be either u- or s-conformal and the Hausdorff dimension of the intersection U n A (where U c W(')(x)is an open set) may not coincide with its lower and upper box dimensions (see example of a multidimensional horseshoe in Section 23). Nevertheless, the above approach, which relied upon the thermodynamic formalism (or its non-additive version), can be still used to establish effective dimension estimates (i.e., estimates that can not be improved). We follow Barreira [Bars]. Define functions p(") and p(U) on A by

One can obtain even sharper dimension estimates using another approach. Define two sequences of functions 9;) and p?) on A by p(U)(z) n =

-log Ild,f-nlE("'II,p,"'(x) = log l~(d5f")-11Eq

and two other sequences of functions --n p(')(z) =

-log Ild&"IE(')II,

(22.11)

9:) and i&) on A by &)(x)

= log lJ(dzf-")-lIE(')II (22.12)

238

Chapter 7

If the derivatives dflE(") and df-lIE(') are &bunched (see Condition (20.9)), one can show that all four sequences of functions, (22.11) and (22.12), satisfy Condition (12.1). Thus, by Theorem A2.6 (see Appendix 11),Bowen's equations

have unique roots ~(")(x) and d")(z),and Bowen's equations

have unique roots r(')(x) and T(')(x). We can now state dimension estimates.

Proposition 22.5. If the derivatives df IE(u) and df-lIE(') are 6-bunched for some 6 > 0 then for any x E A,

~(")(x) 5 dimH(W/b"(x) n A) < dim,(W/:)(x)

n A)

5 dimB(w/~)(X) n A) 5 +)(x), r(')(x) 5 dimH(W,bs,)(x)n A)

< dimB(W/:,!(x) n A) 5 dimB(W/:2(~) n A) 5 T(')(z).

23. Hausdorff Dimension of Horseshoes and Solenoids Multidimensional Horseshoes Let B1 c Rk and Bz c Rt be balls and A = B1 x Bz C Rk @ R' = R". We refer to Rk as "horizontal" and R' as "vertical" directions in R". Let also U c R" be an open set and f : U -+ R" a C"-diffeomorphism, r 2 1. The set A = B1 x Bz c U is called a horseshoe for f if the intersection A' = A n f(A) can be decomposed into simply connected closed disjoint sets A' = A1 U. . .U Ap such that ) Bz,where T I : R" -+ Rk and T Z : R" + Rk (1) q ( A , ) = B1, ~ ~ ( f - ' ( A i ) = are the canonical projections; (2) wzlf-'(Ai n (B1 x ~ ( z ) )is) a bijection onto Bz for any z E Ai; (3) the map df preserves and expands a vertical cone family C1(z,a) and the map df-lIA' preserves and expands a horizontal cone family Cz(z, a). The latter means that Cl(z,a) and C ~ ( z , aare ) cones in R" at z of angle a around R' and Rk respectively such that

df(Cl(z,a)) c G ( f ( z ) , a )

df-l(Cz(z, a)) c Cz(f-l(z), a) and

ifz E A if z E A'

Dimension of Sets and Measures Invariant under Hyperbolic Systems

239

where X > 1 is a constant. One can show that the set

is a locally maximal hyperbolic set with “almost” vertical unstable and “almost” horizontal stable subspaces (see [KH]). In the two-dimensional case the above construction was first introduced by Smale [Sm] and the set A is known as a “Smale horseshoe.” We describe the topological structure of A. Note that for every n 2 0 the set f” (A’) consists of pn simply connected close disjoint “almost” vertical components which we denote by Ajr),,,,,. Similarly, for every n 2 0 the set f-”(A’) consists of pn simply connected close disjoint “almost” horizontal components which we denote by A:;!,,in (see Figure 18). Given a point z E A, denote by A!:.),i, ( z ) and A!;!,,,, ( z ) the vertical and, respectively, horizontal component that contains a (clearly, it is uniquely defined). One can show that

n,,,

unstable - Air,),,, ( z ) is a smooth .!-dimensional A,!;),,,,( z ) submanifold that is isomorphic to Bz;similarly, the set is a smooth k-dimensional stable submanifold that is isomorphic to B1; (2) every point z E A can be coded by a two-sided infinite sequence of integers (. . . iLli0il . . . ), ij = 1,.. . , p such that (1) for every z E A the set

n,

moreover, the coding map x : A + C, defined by x(z) = (. . .i-lioil.. .) is bijective and onto. The map x establishes the symbolic representation of the horseshoe by the full shfft on p symbols. There is another description of the horseshoe which is more suitable for studying its dimension. Notice first that the sets

define a geometric construction in A (modeled by the full shift on 2p symbols) whose limit set is A. Furthermore, consider the sets

They define geometric constructions in B1 and B2 respectively (modeled by the full shift on p symbols). We denote these constructions by CG(”)and CG(”). If F ( s ) and F(”) are their limit sets then A = F(”)x F(”).

Chapter 7

240

Figure 18. A SMALEHORSESHOE. A horseshoe A is called linear if every component A; is a vertical strip and every component f-'(Ai) is a horizontal strip and the maps hi = flf-'(Ai) are linear. In this case, the constructions CG(") and CG(") are generated by affine maps g!"' and g!"' respectively given as follows: gis)(Y) = nl(j-l(n;l(Y)

n A,)),

n nil). (23.1)

g,!")(x) = .2(f-1(.;1(.)

On the other hand, let CG(") and CG(") be two geometric constructions in B1 and B2 modeled by the full shift on p symbols and generated by affine maps 9:") and g,!"' respectively. Let also 3"")and F(") be their limit sets. One can build a linear horseshoe A for a map f such that A = F(")x F(") and relations (23.1) hold. If the maps g!"' and g,!"' are conformal then by Theorem 22.2, we obtain that dimH A = h B A = d i m ~ A= dimH F ( " ) dimH F(") .

+

On the other hand, using Example 16.1 we conclude that there exist a linear horseshoe in R4 for which dimH A

< dimah = d i m ~ h.

We also remark that if the ratio coefficients of the affine maps gj"' and g,!"' are equal then the coding map is an isometry between the the horseshoe A and C,. Thus, it preserves the Hausdorff dimension and box dimension (compare to Theorem A2.9 in Appendix 11).

Three-Dimensional Solenoids We follow Bothe [Bot]. Let S' be the unit circle and D2 the unit disk in R2. Then V = S' x D2 is a solid torus. The projections K : V -+ S1 and p1, p2 : V -+ S' x [-1,1] are defined by

+,

z, Y) = t ,

Pl(t, 2,Y) = 0 7 ).

7

Y = ( 4 Y) .

P 2 @ , 2,

Dimension of Sets and Measures Invariant under Hyperbolic Systems

241

We also denote by D ( t ) = { t }x V 2= E - ' ( t ) (where t E S'). Let 5 be the space of all C1-embeddings f : V -+ V of the form

f ( t , X , Y ) = (cp(t),A l @ b + Z l ( t ) ,

A2(t)Y

+ z2(t)),

(23.2)

where cp :

s'

-+ s2,

A1, A 2

:

s' -+

(0,l),

d', 2 2 :

s'

-+ ( - 1 , l )

are C1-maps, and cp is expanding, i.e.

The last condition implies that the degree 6' of cp is at least 2 and that f stretches the solid torus V in the direction of S'. Since 0 < X,I A 2 < 1 the disks D ( t ) are contracted. This implies that the image f(V) is thinner then V and it is wrapped around inside V exactly 6' times. The set

np(v)

A=

n10

is called a solenoid. One can show that A is a hyperbolic attractor for f (see definition in Section 26). The local topological structure of A is described as follows. For each t E S' the set A(t) = A n D ( t ) is a Cantor-like set; it is the limit set for a geometric construction in D ( t ) modeled by the full shift on 6' symbols; its basic sets on the step n are mutually disjoint ellipses Di,,,,i,, (where 23' = 1 , . . . ,e) which comprise the intersection f"(V) n D ( t ) (see Figure 19). Furthermore, for any arc B c S' containing t there is a homeomorphism

h: B x A(t) -+ A n 7r-'(B) which can be chosen such that for q E B and x 7r h(q,x) = q

,

E

(23.3)

A(t),

h(t,x) = 2.

For each x E A(t) the embedding h, = h ( . , x ) : B depends continuously on x in the C1-topology.

+ V is of class C',

and h,

A map f E 5 is called intrinsically transverse (with respect to the projections and p2) if for any arc B c S' and any two components B1, B2 c A n 7r-'(B) the arcs pi(&) and pi(&) are transverse in S' x [-I, I] at each point of pi(B1)n pi(&). We denote by 5' the set of all intrinsically transverse maps f E 5. For i = 1 , 2 we also denote by p1

f E 5 : sup A;

< inf

- 4 log inf A;

/ log sup A;

where the infimum and supremum are taken over 0 5 t 5 1. Obviously, & is open in 5. In [Bot], Bothe proved the following statement.

Chapter 7

242

Figure 19. THECROSS-SECTION FOR t = 0. Proposition 23.1. T h e set

OF A SMALE-WILLIAMS SOLENOID

5’ n 5i is open

and dense an 5,.

We now compute the Hausdorff dimension of the cross-sections h(t).Since the map cp of the circle S1is smooth and expanding and the functions X i are of class C’ for = 1 , 2 there exists a unique root of Bowen’s equation

PSI

(Si

log Xi) = 0

(where PSI is the topological pressure with respect to the map cp). One can show that si is the unique number of which the functional equation

has a positive continuous solution

t : S1 -+ R (see [Bot]).

Proposition 23.2. (1) For every f E 5 and every t E S1, dimH h(t)5 max(s1,

s2)

.

(2) If sio = max(s1, s2) then for every f E $i0 n 5’ and every t E S’, dimHA(t) = si0 .

Dimension of Sets and Measures Invariant under Hyperbolic Systems

243

Note that the stable foliation of the solid torus by disks D ( t ) is obviously smooth. However, the one-dimensional unstable distribution on A is, in general, only Holder continuous. So is the homeomorphism h defined by (23.3). We set

x = Omin min(xl(t), ~ 2 ( t ) } , the map T = Tp possesses the Sinai-Bowen-Ruelle measure pp, that is the limit evolution of the Lebesgue measure mes on S, i.e.,

4,

.n-1

pp = lim -x(Tpk),mes. 1 n+w

n

k=O

The measure pp has a characteristic property that its conditional measures on vertical lines (where Tp expands) are equivalent to the linear Lebesgue measure. Let v p be the factor-measure induced by ,up. It has the following “measurearihmetic” interpretation and plays an important role in the old and not yet completely solved problem by Erdos. Let E,, n = 0,1,2, . . . be a sequence of independent random variables, each with the values +1 and -1 with equal probabilities. The measure v p can be shown to be the measure with distribution function of the random variable W

n=O

In other words, for any interval ( a , b ) and any integer N let v p , ~ ( a , bbe ) the proportion of points of the form

N-1

C

n=O

en(l - p)pn that lie in (a,b), i.e., N-1

v p , ~ ( ab), = 2-Ncard

en(l - p)p”,

a

0 is a constant independent of y and r . By virtue of (24.7) and (24.8) for all w E Qe, vq(B(y,K r ) ) 2 vq(Ri-n-...i n + )

k=-n-+l

k=-n-+I

k=O

It follows that for all y E ~ ( Q L ) ,

253

Dimension of Sets and Measures Invariant under Hyperbolic Systems

Since E is arbitrary this implies that Zvq (y) 5 T(q)+qa(q)for every y E x(&,))n R( z ) . The third statement of the lemma follows. Moreover, by Theorem 7.2 we obtain that dirnH(X(i,(,)) n R ( z ) )5 T ( q ) qa(q). This completes the proof of the lemma.

+

We also need the following statement.

Lemma 2. A, = ~ ( i , ) . and v(") are diametrically Proof of the lemma. Notice that the measures regular (see Proposition 24.1) and that the measure v is locally equivalent to their direct product (see Proposition 22.2). One can now apply arguments in the proof of lemma 3 of Theorem 21.1 and conclude that the limit lim log4B(zc, r ) ) logr

r-m

exists simultaneously with the limit (24.4). This completes the proof of the lemma. W Since T(1) = 0 we have that ,uIR(z)= 771. The first statement of the theorem now follows from Lemmas 1 and 2 and the following obvious observation:

+

Note that by Lemma 2, A,(,) = ~ ( h , ( ~ ) )Hence, . dimHA,(,) = T ( q ) qa(q). This implies Statements 2, 3, and 4. Since the measure v is diametrically regular and has local product structure one can use (24.2) to prove 'Statement 5 by W repeating arguments in the proof of the last statement of Theorem 21.1.

Remarks. (1) Assume that v is the measure of full dimension. One can show (see Remark 1 in Section 21) that T(q) = (1 - q)dimHA (thus, T ( q ) is a h e a r fupction) and that fv(dimH A) = dimH A and fv(a)= 0 for all a # dimH A. (2) (see Remark 2 in Section 21). Assume that v # rn. Then for any E > 0 there. exists a distinct (singular) equilibrium measure ,u on A such that I d i m ~ p- dimHV/ 5 E . One can also show that: the Hausdorff dimension of an equilibrium measure on a A (corresponding to a Holder continuous potential) depends continuously on the potential. (3) The function fv(a)has properties described in Remarks 3 and 4 in Section 21. (4) We have the following multifractal decomposition of the hyperbolic set A associated with the pointwise dimension of an equilibrium measure on A corresponding to a Holder continuous function (compare to the case of conformal repellers of smooth expanding maps; see (21.18)): (24.9)

254

Chapter 7

where A, is the set of points for which the pointwise dimension takes on the value cy and the irregular part is the set of points with no pointwise dimension. It follows from results in [BS] that A # 0 ; moreover, it is everywhere dense in A and dimH A = dimH A. We also have that each set A, is everywhere dense in A.

A

Pointwise Dimension of Measures on Hyperbolic Sets Let v be an equilibrium measure corresponding to a Holder continuous function on A. According to Statement 1 of Theorem 24.1 for v-almost every z E A, (24.10) It follows from a result by Young [Y2] that this formula holds for an arbitrary ergodic measure v on A which is not necessarily an equilibrium measure (see Proposition 26.3; it is a particular case of Theorem 26.1). We present a straightforward proof of this result based upon the approach which was used in the proof of Theorem 21.3. Theorem 24.2. Let v be a Bore1 ergodic measure on A. Then for u-almost every x E A the equality (24.10) holds.

($ *).

Proof. Set d, = h,(f) Let R = { R l , .. . ,R p }be a Markov partition for f, ( C A ,a) the corresponding symbolic model, and x the corresponding coding map from C A to A (see Section 20). Also, let p be the pullback of v by

x.

Fix E > 0. It follows from the Shannon-McMillan-Breiman theorem that for p-almost every w E C A one can find Nl(w) > 0 such that for any n, m 2 Nl(w),

where Ci-,...i, ( w ) is the cylinder set containing w and h = h,(a) is the measurekheoretic entropy. It follows from the Birkhoff ergodic theorem, applied to the functions logla(u)(z)l and logla(')(x)1, that for v-almost every z E M there exists N z ( z ) such that for any n, m 1 N z ( z ) , l n log )a(")(x)ldv5 -log Ia(")(fj(x))l j=o

+ E.

For w E C A we set Nz(w) = Nz(x(w)). Given l > 0, denote by Qe = { w E C A : Nl(w) 5 l and N z ( w ) 5 l } . It is easy to see that the sets Qe are nested and exhaust CA. Thus, there exists l o > 0 such that p(Qe) > 0 if! 2 l o . Let us choose l 1. l o .

Dimension of Sets and Measures Invariant under Hyperbolic Systems

255

Given 0 < r < 1, consider a Moran cover & , Q ~ of the set x(Qe). It consists of sets R p ) ,j = 1,. . . ,Nr,e for which there exist points xj E A such that Rp) = %+ ),

...in(*j).

Consider the open Euclidean ball B ( x ,r ) of radius r centered at a point x. Let N ( x , r, .t)denote the number of sets R f ) that have non-empty intersection with B ( x ,r ) . We have that N ( x ,r , l ) 5 M , where M is a Moran multiplicity factor (which is independent of r and 1 ) . It now follows that

Since v(x(Qe))> 0 by the Borel Density Lemma (see Appendix V) for v-almost every x E x ( Q l ) there exists a number ro = ro(x) such that for every 0 < r 5 ro we have v ( B ( x ,r)) I 2 v ( B ( x ,r ) n x(Qe)).

By virtue of (22.8) this implies that for any e > .to and v-almost every x E x(Qe),

Since sets Qe are nested and exhaust the set Q we obtain that &(x) 2 d, - 2~ for v-almost every x E A. Since E can be arbitrarily small this proves that d , ( x ) 2 du. We now prove the opposite inequality. Fix 0 < r < 1. By (22.8) it follows + 1 C B ( x ,C l r ) , where C1 > 0 is a constant. This implies that Ri-(,,+)+ ,).,, i that v(B(x1, CU-1) 2 u ( ~ ~ - ( ~ ( = ) + l ) . . . i "2( =c 2 ) +exP(-(h ~)

+ E)(+)

+m ( x ) ) ) ,

where C2 > 0 is a constant. By virtue of (22.8) we obtain for all x E A,

Since E can be arbitrarily small this proves that & ( x ) 5 d,.

w

Since the measure-theoretic entropy is a semi-continuous function we obtain as an immediate corollary of Theorem 22.2 and (24.10) that: the Hausdorff dimension of a Borel ergodic measure Y on A is a semi-continuous function of v. Let v be again an equilibrium measure corresponding to a Holder continuous function on A and G, the set of all forward generic points of v (i.e., points for which the Birkhoff ergodic theorem holds for any continuous function on X ) . It follows from Lemma 1 in the proof of Theorem 24.1 (applied to the measure vl = v) that for every x E G,,

256

Chapter 7

In view of Theorems 7.1 and 7.2 this implies that

We extended this result to any Borel ergodic measure on A which is not necessarily an equilibrium measure.

Theorem 24.3. Let every z E A,

u be a

Borel ergodic measure on A. Then for u-almost

Proof readily repeats arguments in the proof of Theorem 21.3. Manning [Mall proved the part of Theorem 24.2 involving the Hausdorff dimension of the sets G, n W/:?(z)and G, n W,b"(z). Information Dimension We study the information dimension of the measure u. As in the case of conformal repellers one can show that fv(a(l))= a(1) = -T'(l) = L(v) = f ( u ) = dimH u, where L(u) and ?(u) are the lower and upper information dimensions of u (see Section 18). We note that Statement 6 of Theorem 24.1 allows us to extend the notion of the Hentschel-Procaccia spectrum and RBnyi spectrum for dimensions for any q # 1 and the above argument defines these spectra for q = 1.

Dimension Spectrum for Lyapunov Exponents Consider the following multifractal decomposition of the set A associated with positive values of the Lyapunov exponent A+(x) at points z E A (see Section 26): (24.11) where

L+ = {z E A : the limit in (26.1) does not exist for any v E E ( " ) ( z ) ) is the irregular part and

L$ = {X E A : A+(x) = p}.

257

Dimension of Sets and Measures Invariant under Hyperbolic Systems

If v is an ergodic measure for f we obtain that A+(x) = A?) for v-almost every x E A. Thus, the set ~5:~) # 0. Moreover, if v is an equilibrium measure corresponding to a Holder continuous function this set is everywhere dense (since in this case the support of v is the set A). We note that if a set Lp is not empty then it supports an ergodic measure v p for which A,, = /3 (indeed, for every x E Lo the sequence of measures C;:,' d f k ( = ) has an accumulation measure whose ergodic components satisfy the above property). As in the case of conformal repellers there are several fundamental questions related to the above multifractal decomposition, for example: (1) Are there points x for which the limit in (26.1) does not exist for any

+

v E E ( ~ ) ( X ) ,i.e., 2+ a? (2) How large is the range of values of X+(x)? (3) Is there any number p for which any ergodic measure v with v(Lp) > 0 is not an equilibrium measure? We introduce the dimension spectrum for (positive) Lyapunov exponents o f f by !+(p) = dimH L;. In [PW3], Pesin and Weiss established the relation between the Lyapunov dimension spectrum and the fvmax(a)-spectrum,where vmax is the measure of maximal entropy. Notice that the measure of maximal entropy is a unique equilibrium measure corresponding to the function cp = 0 and hence $ = constant = exp(-hA(f)), where h A ( f ) is the topological entropy o f f on A. Therefore, for every x E L;

vk&

(recall that denotes the conditional measure induced by ,,v stable manifolds). This implies the following result.

on local un-

Theorem 24.4. [PW3] (1) Ifvk&lR(x)is not equivalent to the measure rncU)(x) for some x E A then

the Lyapunov spectrum

is a real analytic strictly convex function on an interval [pi,pz] containing the point p = hA(f)/dimH(A n W/b",'(x)). (2) Ifvg&lR(x) is equivalent to m(")IR(x)for some x E A then the Lyapunov spectrum is a delta function, i.e.,

!+(PI

=

{

dimHh, f o r @ = hA(f)/dirnH(ArlW/:?(x)) 0,

f o r p # hA(f)/dimH(AnW~~'(x)).

258

Chapter 7

As immediate consequences of this result we obtain that i f the measure for somex E A then the range uk&lR(x) is not equivalent to the measure mcU)(x) of the function X+(x) contains an open interval, and hence, the Lyapunov exponent attains uncountably many distinct values. On the contrary, i f the Lyapunov exponent X+(x) attains only countably many values then uk&lR(x) is equivalent to m(")IR(x)for some x E A. We can now answer the above questions. Namely, (1) The set i+ is not empty and has full Hausdorff dimension (see Remark (3) in this Section). (2) The range of values of X+(x) is an interval [PI, p2] and for any outside this interval the set L i is empty (i.e., the spectrum is complete). (3) For any p E [pl,pz] there exists an equilibrium measure u corresponding to a Holder continuous function for which u(Lp) = 1. Similar statements hold true for dimension spectrum for (negative) Lyapunov exponents of f corresponding to negative values of the Lyapunov exponent A-(z) at points x E A.

Appendix IV

A General Concept of Multifractal Spectra; Multifractal Rigidity

Multifractal Spectra In Sections 19, 21, and 24 we observed two dimension spectra - the dimension spectrum for pointwise dimensions and the dimension spectrum for Lyapunov exponents. The first one captures information about various dimensions associated with the dynamics (including the Hausdorff dimension, correlation dimension, and information dimension of invariant measures) while the second one yields integrated information on the instability of trajectories. These spectra are examples of so-called multifractal spectra which were introduced by Barreira, Pesin, and Schmeling in [BPS21 in an attempt to obtain a refined quantitative description of various multifractal structures generated by dynamical systems. The formal description follows. Let X be a set, Y c X a subset, and g: Y -+ [-m, +co] a function. The level sets KZ = {Z E X : g(Z) = a } , -CO 5 (Y 5 +CO are disjoint and produce a multifractal decomposition of X ,

where the set X = X \ Y is called the irregular part. If g is a smooth function on a Euclidean space then the non-empty sets KZ are smooth hypersurfaces except for some critical values a. We will be interested in the case when g is not even a continuous (but Borel) function on a metric space so that KZ may have a very complicated topological structure. Now, let G be a set function, i.e., a real function that is defined on subsets of X. Assume that G(&) 5 G(Z2) if Z1 c 2 2 . We introduce the multifractal spectrum specified by the pair of functions (9,G) (or simply the ( 9 ,G)multifractal spectrum) as the function F:[-m, +m] -+ R defined by

F ( a )= G(KZ). The function g generates a special structure on X , called the multifractal structure, and the function F captures important information about this structure. 259

Appendix IV

260

Given a, let u, be a probability measure on KE. If

F(a) = inf(G(2) : 2 c Kz, v,(Z) = 1) then we call v, a (9,G)-full measure. Constructing a one-parameter family of (g,G)-full probability measures v, seems an effective way of studying multifractal decompositions (see examples of v,-measures below). We consider the case when X is a complete separable metric space. Let f : X + X be a continuous map. There are two natural set functions on X . The first one is generated by the metric structure on X : GD(2)= dimH 2, and the second one is generated by the dynamics on X :

Multifractal spectrum generated by the function GD is called the dimension spectrum while multifractal spectrum generated by the function GE is called the entropy spectrum. There are also three natural ways to choose the function g. (1) Let p be a Bore1 finite measure on X . Consider the subset Y consisting of all points x E X for which the limit

cX

exists (i.e., the lower and upper pointwise dimensions coincide). We set g ( 0 ) = d,(x)

Sfg o ( x ) ,

2

E Y.

This leads to two multifractal spectra DD = Dg) and DE = V$" specified respectively by the pairs of functions (90,Go) and (gD,GE). w e call them the multifractal spectra for (pointwise) dimensions (note that V D is just the f,(a)-spectrum studied before). We stress that these spectra do not depend on the map f . (2) Assume that the measure ,u is invariant with respect to f . Consider a finite measurable partition of X . For every n > 0, we write f,, = $, V f - l < v + . V f-" 0 such that

Cia-" 5 diam 0.

(a AT

Lemma 3. For all m 2 2 and i = 1 , . . . ,2m-' lf"(Km+l)I

we have

< If"Lm)I 5 PIGi,ml.

Proof of the lemma. Note that by linearity of f-'lf(Km) the above inequalities are equivalent to

278

Chapter 8

or The first inequality holds since A inequality

> 1. The second inequality follows from the

which is clearly true since l/C 5 Qm/Orn+l. The attractor A for the map f is defined as

n uf 00 2n-1

A=

w ~ .

i=l By Lemmas 1 and 2 A is a Cantor-like set of zero Lebesgue measure. We define a probability measure /I on A by assigning to every set f"(Krn),i = 1 , . . . ,2"+' the measure p ( f i ( K m ) )= 21-rn. The measure /I is clearly non-atomic and invariant. Each point x E A admits a symbolic coding x H w = ( ~ ~ ( xwhere ) ) , E ~ ( z )= 0 or 1. Namely, for x E fi(Kn+l),1 5 i 5 2" we set e,(x) = 0 if i 5 2n-1 and cn(x) = 1 if i > 2"-l. Using this coding map one can show that the measure /I is ergodic and has zero measure-theoretic entropy (see [CE]). n=l

Lemma 4. For all x E A

Proof of the lemma. For x E fg(Kn+l),i = 1,. . . ,2* we set n

7*(4 = Ifi(Kn+l)l= l - J C k ( E k ( 4 ) k+l (see Lemma 2). Fix no such that p < C-"o. Clearly, the interval [x- qn(x), x qn(x)] contains fi(Kn+l). For each k < n there exists i ( k ) such that 1 5 i(k) 5 2k and x E fi@)(Kn+l).By Lemma 2 we have

+

c-(n-k) Ifi(Kn+l)I

Jfi(k)(Kk+l)l 2

and hence if k 5 n - no then

Ifi("(~k+l)I

2 Pqn(x).

By Lemma 3 we obtain that if k 5 n - no then IGi(k),kl

2 hn(x).

Therefore, since the sets G i ( k ) , k are gaps and hence are disjoint from A the set [z - 71,(z),x 77,(z)]n A is contained in fi(n-no)(Kn-no+l).It follows that

+

2-" 5 p ( [ x - 7 3 , ( x ) x ,

+ vn(x)]) 5 2-("-*0).

The desired result follows immediately from this relation and the inequalities

The lemma is proved.

Relations between Dimension, Entropy, and Lyapunov Exponents

279

Note that logC’((0) = log% 0, Since have

E ~ ( z ) are

- logA,

logCj(1) = log-ej+l .

4

independent stationary sequences for p-almost every z E A we

Let us choose the sequence 8, such that B 5 1 lim -log& = B n+mn

,

% 5 C and

-1 lim -log& = C. n

n-im

From Lemma 4 it follows that for p-almost every z E A,

This completes the construction of the example.

1

26. Dimension of Measures with Non-zero Lyapu& Exponents; The Eckmaut-Ruelle Conjecture

We assume that M is a compact smooth Riemannianpdimensional manifold and f :M + M is a C1+a-diffeomorphism. We recall some basic notions of the theory of dynamical systems with non-zero Lyapunov exponents (see [KH], [MI for more details). Given 2 E M and w E T,M, define the Lyapunov exponent of u at x by the formula A(z, w ) = lim 1% Ildf3ll (26.1) n+m n

If z is fixed then the function X(x,.) can take on only finitely many distinct values A(l)(z) > ... > A(Q)(z),where q = q(z) and 1 5 q 5 p . Let ki(z) be the multiplicity of the value A(*)(%), i = 1,..., q. The functions q(z), X(g)(z), and ki(z),i = 1, . . . ,q are measurable and invariant under f . Let p be a Bore1 f-invariant measure on M . We will always assume that p is ergodic. Then the function q(z) = q is constant p-almost everywhere and so are the functions X(a)(z) and k i ( z ) , i = 1 , . . . ,q. We denote the corresponding values by A$’ and k!). A measure p is said to be hyperbolic if

280

Chapter 8

for some k, 1 I k < q. If p is such a measure then for p-almost every point x E M there exist stable and unstable subspaces E ( S ) ( x )E(")(x) , c T,M such

that

(1) E(S)(z)@ E(")(x) = T,M, df,E(S)(x) = E ( " ( f ( x ) ) , df,E(")(x) =

E(")(f(4);

(2) for any n 2 0 ,

< y < 1 is a constant and C,(z) > 0 is a measurable function; (3) L(E(')(z),E ( " ) ( x ) 2 ) Cz(z) > 0, where Cz(x)is a measurable function

where 0

and L denotes the angle between subspaces E ( S ) ( xand ) E(")(x); 6

(4)Cl(f"(x)) 5 C1(x)en6, Cz(f"(x)) 2 Cz(z)e-"6for any > 0 is a constant which is sufficiently small compare to 1 - y.

n

2 0, where

Denote by At, t? 2 1 the set of points z for which Cl(x) 5 t?, CZ(Z)2 $. The sets Ae are closed, he c &+1, and the set A = Ue,l A! is f-invariant and coincides with M up to a set of p-measure zero. For any x E A one can construct stable and unstable smooth local manifolds which we denote by W,b"(x)and W,"(x) respectively. They have the following properties: (5) x E W{$(x) and x E W{b",)(x);T , W { ~ ~ (=x )E ( S ) ( x and ) T,W/b",)(x)= E(")(x); (6) for any n 2 0,

p(f"(z),f"(!/)) i C3(x)-Y"p(x,y) if Y E W,b",'4, p(f-"(.),

f-"(Y))

I C 3 ( Z ) Y " P ( ~Y,)

if Y E W,b",'(.),

where p is the distance in M generated by the Riemannian metric and C3(x) > 0 is a measurable function satisfying C3(f"(x)) 5 C 3 ( ~ ) e ~ ' for " ~ every n 2 0;

(7) W,b",'(x)and W,b",'(x)depend continuously on x E he in the C'-topology and have size at least > 0; this means that they contain respectively balls B(')(x,re) and B(")(x,re) centered at x of radius re in the intrinsic topology generated by the Riemannian metric; (8) there exists a function = $ ( l ) < 00, such that for any x , y E At, if the intersection B(')(x,re) n B(")(y, re) # 0 , then it consists of a single point E he1;

(9) there exists K

> 0 such that for any x E he,

Relations between Dimension, Entropy, and Lyapunov Exponents

281

where and p(") and p(') are the intrinsic distances in the local unstable and stable manifolds induced by the Riemannian metric. A closed set ll c A is called a rectangle if : a) there exists l > 1 such that IInAt is an open subset of At l (with respect to the induced topology of At); moreover, for any x E II there exists y E II n A! such that z E B(')(y, re) or x E B(")(y, re); b) for any x,y E II the intersection B ( S ) ( xre) , n B(")(y, re) is not empty and consists of a single point z E II (one can see that II c At1). Given a rectangle II and points x,y E II, define the u-holonomy map H$'d: B(')(x,re) n II + B(')(y,re) n II by H$')(z) = B(")(z,re) n B(')(y, re). One can similarly define the s-holonomy map Hi:;. For each rectangle II such that p ( I I ) > 0 let 5'") and (('1 be the measurable partitions of ll : [("'(x) = II n B(")(x,rt) and $')(x) = II n B(')(x,re) (see [KH]). Let { p p ) } z E n and { p p ) } z E nbe the canonical families of conditional measures associated with the partitions @") and ((') respectively. We say that a measure p has the local semi-product structure if for any rectangle II with p ( I I ) > 0 and p-almost every x,y E II the associated holonomy map Hi$ is absolutely continuous with respect to the conditional measures p p ) and p p ) , i.e., the measure ( H i : i ) * p p ) is absolutely continuous with respect to the measure p p ) (or if the same is true for the holonomy maps H$'J and the conditional measures ,up),p g ) for p-almost every x,y E II). We say that a measure p has the local product structure if for any rectangle II with p ( I I ) > 0 and p-almost every z, y E II the associated holonomy maps H g J ,H$i both are absolutely continuous with respect to the conditional measures p p ) , p c ) and p p ) , p p ) , respectively. There are two special cases of measures which have local product structure. A hyperbolic measure p is called a Sinai-Ruelle-Bowen measure (or SRB-measure) if for every rectangle II with p ( I I ) > 0 and p-almost every x E II the, conditional measure is absolutely continuous with respect to the Riemannian volume on W[b",'(x).SRB-measures have semi-local product structure. Let A be a locally maximal hyperbolic set for a C1+a-diffeomorphism on a compact smooth manifold M (see definition in Section 22) and p = plPa unique equilibrium measure corresponding to a Holder continuous function 'p on A. By Proposition 22.2 p has local product structure. Given x E A and a sufficiently small r > 0, define

(assuming that the limits exist). One can show that the functions dp)(x) and d p ) ( x ) are measurable and invariant under f. Since the measure p is ergodic these functions are almost everywhere constant. We denote the corresponding values by d(") and d(').

282

Chapter 8

We first consider the cases k = 1 and k = q - 1. The following result is a slightly more general multidimensional version of the result obtained by Young in the two-dimensional case (see [Y2]).

Proposition 26.1. For any ergodic hyperbolic measure p invariant under a C1+a-diffeomorphism the limits (26.2) exist almost everywhere and (1) i f k = 1 then d(") = $$!$;

(2) i f k = q - 1 then d(') = -*. The existence of the values d f ) ( z ) and d p ) ( x ) in the general case was established by Ledrappier and Young in [LY]. We recall that since p is ergodic & ( x ) = const = d ( p ) and &(z) = const = a ( p ) almost everywhere (see Section 7).

Proposition 26.2. For any ergodic hyperbolic measure p invariant under a C1+Ol-diffeomorphism the limits (26.2) exist almost everywhere and

+

(1) a ( p ) 5 d(") d('); and d ( s ) 2 (2) d(") 2

*

-h

(f).

In [ER], Eckmann and Ruelle discussed the existence of pointwise dimension for hyperbolic invariant measures. We summarize this discussion in the following statement, which was proved by Barreira, Pesin, and Schmeling in [BPSl].

Theorem 26.1. Let f : M + M be a C1+a-diffeomorphism of a compact smooth Riemannian manifold M and p a hyperbolic ergodic measure.. Then for p-almost any x E M , d ( p ) = a ( p ) = d(") d(').

+

Remark. In [Y2], Young proved this theorem in the two-dimensional case. Proposition 26.3. Let f be a C'+a-diffeomorphism of a smooth compact sur>0 > face M and p a hyperbolic ergodic measure with Lyapunov exponents 'A: A!?. Then

Proof of the theorem. For the sake of reader's convenience we first consider the special and simpler case of measures with local semi-product structure. The proof in the general case will be given later. Since it is technically more complicated it can be omitted in the first reading. We follow [PY]. Without loss of generality we may assume that the holonomy maps Hi$ are absolutely continuous with respect to the measures p e l , p p ) for any rectangle II with p ( I I ) > 0 and yalmost every x , y E II. We fix such a rectangle II. According to Proposition 26.2 it is sufficient to show that d ( p ) 2 d(') d(").

+

Relations between Dimension, Entropy, and Lyapunov Exponents

283

Proposition 26.2 (that states the existence of the limits in (26.2)) implies that there exist a closed set A1 c n with p(A1) > 0 and a number r1 > 0 satisfying the following condition: (10) for any 0 < r 5 rl and any z E A1, p p ) ( ~ ( u ) ( z ,n r n) ) 5 r d ( " ) - ~&, ) ( B ( s ) ( ~ ,7 ) n n) I #')--E.

(26.3)

It follows from the Bore1 Density Lemma (see Appendix V) that one can find a closed set Az c A1 with ~ ( A z>) 0 and rz > 0 such that for any 0 < r 5 rz and any x E Az, P(+, 5 2 P ( B b , ). n A d . (26.4) Fix a point z o E Az for which p&)(W/;J(zo)n Az)

> 0 and

We first study the factor measure p induced by the measurable partition $"). Denote by du) the projection of B(z0,r ) n A1 into W/:J ( 2 0 )given by d ' ) (y) = W,b",'(zo) n W/:,!(y). For any x E B(z0,r ) n A1 we also have that

dU)Iw/:J(z) n

=

~is,!,Iw/;)(x) nA ~ .

Without loss of generality we may assume that the holonomy map H& is absolutely continuous with respect to the conditional measures &) and pp). Therefore, the factor measure ji = A P )is~absolutely continuous with respect to &): d j i ( z ) = c p ( ~ ) d p & ) ( ~ )z, E B(")(zo,Q )

n Al,

where p(z) 2 0 is an L1-function. Since &)(W/;)(z0)n Az) > 0, without loss of generality, we may assume that the Radon-Nikodym derivative dj?/d&) is bounded. Therefore, we have for sufficiently small r > 0,

ii;(B(")(zo, I C4&)(B(")(zo,r ) ) ,

(26.5)

where C, > 0 is a constant. We now apply the Fhbini theorem to the measurable partition @") and write

/

P(B(zo,r))=

Cll"'(~/:h

nw,6",'a.(

($0 ,r)

n B(zo,r))djZ(+

(26.6)

)

If the intersection W,b",'(z)n B(z0,r ) n A1 contains a point z then by Condition (9)7

n~ ( z ~ c , rB )( s ) ( 2~ ,~ r )

w/:,!(Z)

284

Chapter 8

(26.7)

We also have that

Conditions (26.5) - (26.8) imply that for sufficiently small T ,

where C5 > 0 is a constant. Consider a decreasing sequence of positive numbers pk + 0 (k + 00) such that p ( B ( z 0 , p k ) ) 2 pkd(P)-€ for all k. We can also assume that p1 < min(r1, ~ 2 r g, } . It follows now from (26.4) that pkd(P)+E

< - p(B(z0,Pk)) I 2CL(B(zo, P k ) n

A)

I~ ~ ~ ~ d ( ' ) + d ( ' ' ) - - 2 ~ If k

-+ 00

this yields d(p)

+ & 2 dS)+ dU)- 2E.

Since E is arbitrary this implies the desired result. We now proceed with measuws which do not have local (semi-) product stmcture. We will first establish a crucial property of hyperbolic measures: they have nearly local product structure. This enables us to apply a slight modification of the above approach to obtain the desired result. In order to highlight the main idea and avoid some complicated technical constructions in the theory of dynamical systems with non-zero Lyapunov exponents we assume that the map f possesses a locally maximal hyperbolic set A which supports the measure p. Although the set A has direct product structure hyperbolic measures supported on it, in general, do not have local (semi-) product structure. For the general case see [BPSl]. Consider a Markov partition R = { R l , .. . ,%}. In order to simplify no) element of the partition V;J% that tations we set RL(z) = R ; k , , , i , ( z(the contains z). We point out the following properties of the Markov partition. Given 0 < E < 1, there exists a set r c M of measure p ( r ) > 1- ~ / 2 an , integer

Relations between Dimension, Entropy, and Lyapunov Exponents

2 1, and a number C > 1 such that for every x E the following properties hold: (a) for all integers k, 1 2 1 we have no

285

r and any integer n 2 no

where h = h,(f) (the measure-theoretic entropy o f f with respect to p); (26.12)


0 one can choose a sequence of points { z i E F' n A(')(y)}E1 and a sequence of integers { t i } z l , where ti E { m j ( z ; ) } g l and ti > L for each i, such that the collection of balls C = { B ( z i , e-") : i = 1 , 2 , . .. } comprises a cover of F' n A(')(y) whose multiplicity does not exceed p. We write Q ( i ) = Q t i ( z i ) . The Hausdorff sum corresponding to this cover is 00

By (26.27) we obtain

i=l

i=l

00

54c

C q=1

e--aqh--laq~

):fi(")(~,z i , Q(i)). i:t,=q

Since the multiplicity of the subcover C is at most p each set Q(i)appears in the sum Ci:t,=q fi(')(y, zi,&(i)) at most p times. Hence, i:t;=q

It follows from Lemma 4 that 00

BEC

q=1

Since the number L can be chosen arbitrarily large (and so are the numbers ti) it follows that dimH(F' n A(')(y)) 5 d') - E < d"). This contradicts (26.28). Hence, p ( F ) = 0. This yields the first inequality. The proof of the second inequality is similar.

292

Chapter 8

Lemmas 3, 4, and 5 show that the deviation of the distribution of rectangles in Z from direct product structure is subexponentially small. The rest of the proof simulates the case of measures which have local (semi-) product structure considered above. By Lemmas 2 and 3 for p-almost every y E R(x)flf and n 2 n2(y) we obtain

~(B(Y e-*-')) ,

5 fi("(n7 Y, Q n b ) ) x fi(")(n, Y,Qn(Y)) 4~3~4a(h+~),-Zanh+6ana

By Lemma 5 for p-almost every y E R(x)nf there exists an integer n3(y) 2 nz(y) such that for all n 1 n3(y), fic5)(n,Y, Qn(y)) < N ( ' ) ( ~ , YQ , n(~))e~~~', *(")(n, Y, Qn(Y))

< N(")(n, Y,Q n ( ~ ) ) e ~ ~ ~ " .

This implies that

P(B(Y,e-n-2)) 5 N("(n7 Y,Qn(y)) x N(")(w Y,Qn(y)) 4 ~ 3 ~ 4 a ( he --2anh+20an~ +~) Applying Lemma 1 we obtain p ( ~ ( ye-"-')) ,

5 p!)(~(")(y, 4e-"))

x p P ) ( ~ ( " ) ( y4e-n)) ,

4~5~4a(h+~)~22anr This implies that

for p-almost every y E

f . Since p ( f ) > 1 - E and E > 0 is arbitrary we conclude

for p-almost every y E M . ,This completes the proof.

Let A be a hyperbolic set for a C'+a-diffeomorphism f on a compact smooth manifold M . Assume that flA is topologically transitive. The set A is called a hyperbolic attractor if there exists an open set U such that A C U and f(U) c U . Clearly, A = nn,,fn(U). One can show that if A is a hyperbolic attractor then W/',",'(x) c A for every x E A (see, for example, [KH]). Consider the unique equilibrium measure p = pv on A corresponding to the function cp = J(")(x). It is known that pv is the SRB-measure (see [Bo~]).Obviously, d(") = dimW/bU,)(x) m for every x E A. By Proposition 26.2 and the entropy formula (see [LY]) we have that

zf

This implies that

Appendix V

Some Useful Information

1. Outer Measures [Fe].

Let ( X ,p ) be a complete metric space and m a a-sub-additive outer measure on X , i.e., a set function which satisfies the following properties: (1) m ( 0 )= 0; (2) m(21)5 7 4 2 2 ) if 21 c 2 2 c X ; m( u Zi)5 Em(&),where Zi c X,i= 0 , 1 , 2 , .. .. (3) i20 i20 A set E c X is called measurable (with respect to m or simply m-measurable) if for any A c X , m(A) = m ( A n E ) m( A \ E ) .

+

The collection M of all m-measurable sets can be shown to be a 0-field and the restriction of m to B to be a 0-additive measure (which we will denote by the same symbol m). An outer measure m is called (1) Borel if all Borel sets are m-measurable; (2) metric if m(Eu F ) = m(E) m ( F ) for any positively separated sets E and F (i.e., p(E,F ) = inf{p(x, y) : x E E , Y E F ) > 0); (3) regular if for any A c X there exists an m-measurable set E containing A for which m(A) = m(E). One can prove that any metric outer measure is Borel.

+

2. Borel Density Lemma [Gu].

We state the result known in the general measure theory as the Borel Density Lemma. We present it in the form which best suits to our purposes. Borel Density Lemma. Let A Then f o r p-almost every x E A,

cX

be a measurable set of positive measure.

Furthermore, if p ( A ) > 0 then for each 6 > 0 there is a set A c A with p ( A ) > p ( A ) - 6 and a number ro > 0 such that for all x E A and 0 < r < ro, 1 CL(B(X, ). n A ) 2 p ( B h r ) ) . 293

294

Appendix V

3. Covering Results [F4], [Fe].

In the general measure theory there is a number of “covering statements” which describe how to obtain an “optimal” cover from a given one. We describe two of them which we use in the book. Consider the Euclidean space Rm endowed with a metric p which is equivalent to the standard metric.

Vitali Covering Lemma. Let A be a collection of balls contained in some bounded region of Rm. Then there is a (finite or countable) disjoint subcollection B = {Bi}c A such that IJ B C UBi, B€d

i

where Bi is the closed ball concentric with Bi and of four times the radius.

Besicovitch Covering Lemma. Let Z c Rm be a set and r : Z -+ R+ a bounded function. Then the cover C7 = { B ( z ,r ( 2 ) ) : 2 E 2 ) contains a finite or countable subcover of finite multiplicity which depends only on m. As an immediate consequence of the Besicovitch Covering Lemma we obtain that for any set Z c Rm and E > 0 there exists a cover of Z by balls of radius E of finite multiplicity which depends only on m. 4. Cohomologous Functions [Rl].

Two functions cp1 and cpz on a compact metric space X are called cohomologous if there exist a Holder continuous function 8: X + R and a constant C such that cp1-9z=17-17Of+C.

In this case we write (PI N cpz. If the above equality holds with C = 0 the functions are called strictly cohomologous. We recall some well-known properties of cohomologous functions: (1) if 91 cp2 then for every z E X ,

-

-

(2) (pl cpz if and only if equilibrium measures of cp1 and cp2 on X coincide; (3) if the functions cp1 and 9 2 are strictly cohomologous then Px(cp1) =

PX(9Z). 5. Legendre Transform [Ar].

We remind the reader of the notion of a Legendre transform pair of functions. Let h be a strictly convex C2-function on an interval I , i.e., h”(z) > 0 for all x E I . The Legendre transform of h is the differentiable function g of a new variable p defined by (A5.1) d P ) = F$P” - h(z)). One can show that: 1) g is strictly convex; 2) the Legendre transform is involutive; 3) strictly convex functions h and g form a Legendre transform pair if and only if g(a)= h(q) qa, where a(q)= -h‘(q) and q = g’(a).

+

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Index

Axiom A diffeomorphism, 228 u (s)-conformal, 230

coding map, 117, 229, 239 conformal map, 199 repeller, 199 toral endomorphism, 202 j-coordinate, 100 correlation sum, 174, 177, 181 cylinder (cylinder set), 100, 101

baker’s transformation classical, 244 fat, 245 generalized, 244 skinny, 245 slanting, 246 basic set, 117, 190 Besicovitch cover, 31 Bowen’s equation, 106 box dimension of a measure, 41, 61 of a set, 36, 61 BS-box dimension of a measure, 112 of a set, 111 q-box dimension of a measure, 58 of a set, 52, 62 (q, 7)-box dimension of a measure, 57, 63 of a set, 49, 62

@-density,45 dimension box of a measure, 61 of a set, 61 Caratheodory of a measure, 21, 67 of a set, 14, 66 pointwise, 24, 67 correlation at a point, 174, 178, 179 of a measure, 178 specified by the data, 181 Hausdorff of a measure, 41, 61 of a set, 36, 61 information, 186, 224, 256 pointwise, 42, 61, 143, 223, 254 BS-dimension of a measure, 112 of a set, 111 q-dimension of a measure, 58 of a set, 52, 62 (4, 7)-dimension of a measure, 57, 63 of a set, 49, 62 pointwise, 58

Carathkodory capacity of a measure, 22, 67 of a set, 16, 67 dimension of a measure, 21, 67 of a set, 14, 66 dimension spectrum, 32 dimension structure, 12, 66 measure, 13 outer measure, 13 pointwise dimension, 24, 67 chain topological Markov, 101 301

302 entropy capacity topological, 75 with respect to a cover, 77 measure-theoretic, 77 of a partition, 186 topological, 75 with respect to a cover, 77 equilibrium measure, 97 estimating vector, 134 expanding map, 146, 189, 197 expansive homeomorphism, 97 function stable, unstable, 105 geometric construction associated with Schottky group, 151 Markov, 118 Moran, 120 Moran-like asymptotic, 275 with non-stationary ratio coefficients, 121 with stationary (constant) ratio coefficients, 120, 152 regular, 122, 134 self-similar, 122, 133 simple, 118 sofic, 118 symbolic, 118 with contraction maps, 122, 150 with ellipsis, 138 with exponentially large gaps, 156 with quasi-conformal induced map, 145 with rectangles, 153 geometry of a construction, 118 Gibbs measure, 102, 103 grid, 184 group Kleinian, 151 reflection, 152 Schottky, 151 Hausdorff dimension of a measure, 41, 61 of a set, 36, 61

Index

measure, 36, 61 outer measure, 35, 61 holonomy map, 281 horseshoe, 238 linear, 240 hyperbolic attractor, 292 measure, 279 set, 227 locally maximal, 227 induced map, 122, 145, 203 irregular part, 171, 188, 222, 225, 254, 256, 259, 264 Julia set, 201 Kleinian group, 151 length of string, 68 limit set, 118 local entropy, 260 Lyapunov exponent, 223, 279 Mandelbrot set, 207 manifold stable, unstable global, 228 local, 227 map coding, 117, 229, 239 conformal, 199 conformal affine, 133 expanding, 146, 189, 197 induced, 122, 145, 203 one-dimensional Markov, 201 quasi-conformal, 147, 191 weakly-conformal, 191 Markov partition, 189, 228 mass distribution principle non-uniform, 43 uniform, 43 matrix irreducible, 101 transfer, 101 transitive, 101 measure (9,G)-full, 260 a-Carathbodory outer, 13 a-Hausdorff, 36, 61

Index outer, 35, 61 CarathCodory, 13 diametrically regular, 55, 62, 212 doubling, 55 equilibrium, 97 exact dimensional, 44 Federer, 55 Gibbs, 102, 103 hyperbolic, 279 of full Carathkodory dimension, 33 of full dimension, 129 of maximal entropy, 99 Sinai-Ruelle-Bowen (SRB), 281 (q, y)-measure, 49 Moran cover, 120, 124, 141, 200, 231 multiplicity factor, 124, 141, 201, 232 multifractal analysis, 173, 213, 249 classification, 267 decomposition, 171, 222, 225, 253, 256, 259 rigidity, 266 spectrum, 259 structure, 171, 259 point irregular, 47 regular, 47 potential principle, 44 potential theoretic method, 44 pressure capacity topological, 70 non-additive, 84 topological, 70 non-addit ive, 84 pressure function, 106 ratio coefficients, 120, 123, 133 rectangle, 189, 228, 281 reflection group, 152 repeller, 197 conformal, 199 scaling exponents, 170 Schottky group, 151 separation condition, 117

303 sequence of functions additive, 85 sub-additive, 85 set basic, 117, 190 hyperbolic, 227 locally maximal, 227 invariant, 100 metrically irregular, 95, 264, 265 a-set, 47 shift one-sided, 100, 101 two-sided, 102 Sierpiliski carpet, 157 similarity maps, 122 solenoid, 241 space isotropic, 62 metric Besicovitch, 62 of finite multiplicity, 61 spectrum complete, 189, 195, 221, 226, 258 dimension, 260 CarathCodory, 32 for Lyapunov exponents, 225,257, 258 for pointwise dimensions, 171, 188 entropy, 260 multifractal, 259 for local entropies, 261 for Lyapunov exponents, 261 for pointwise dimensions, 260 Rknyi for dimensions, 172, 185 f,(a)-spectrum, 171, 188 HP-spectrum for dimensions, 172, 182 modified, 183 string, 68 structure local product, 281 local semi-product, 281 multifractal, 171, 259 subordinated, 15 C-structure, 12, 66 subshift one-sided, 100

304

of finite type, 101 topologically mixing, 101 topologically transitive, 101 two-sided, 102 of finite type, 102 subspace stable, unstable, 227 symbolic dynamical system, 100 representation, 117 system even, 101 SOfiC, 101

Index topological entropy, 75 with respect to a cover, 77 Markov chain, 101 pressure, 70 non-additive, 84 variational principle, 33 for topological entropy, 94 for topological entropy inverse, 83 for topological pressure, 88 non-additive, 99 for topological pressure inverse, 83

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