Non-Invertible Dynamical Systems. Volume 2: Finer Thermodynamic Formalism – Distance Expanding Maps and Countable State Subshifts of Finite Type, Conformal GDMSs, Lasota-Yorke Maps and Fractal Geometry 9783110700619, 9783110702699

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Mariusz Urbański, Mario Roy, Sara Munday Non-Invertible Dynamical Systems

De Gruyter Expositions in Mathematics

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Edited by Lev Birbrair, Fortaleza, Brazil Victor P. Maslov, Moscow, Russia Walter D. Neumann, New York City, New York, USA Markus J. Pflaum, Boulder, Colorado, USA Dierk Schleicher, Bremen, Germany Katrin Wendland, Freiburg, Germany

Volume 69/2

Mariusz Urbański, Mario Roy, Sara Munday

Non-Invertible Dynamical Systems |

Volume 2: Finer Thermodynamic Formalism – Distance Expanding Maps and Countable State Subshifts of Finite Type, Conformal GDMSs, Lasota-Yorke Maps and Fractal Geometry

Mathematics Subject Classification 2010 37A05, 37A25, 37A30, 37A35, 37A40, 37B10, 37B25, 37B40, 37B65, 37C05, 37C20, 37C40, 37D20, 37D35, 37E05, 37E10 Authors Prof. Dr. Mariusz Urbański University of North Texas Department of Mathematics 1155 Union Circle #311430 Denton, TX 76203-5017 USA [email protected]

Dr. Sara Munday John Cabot University Via della Lungara 233 00165 Rome Italy [email protected]

Prof. Dr. Mario Roy York University Glendon College 2275 Bayview Avenue Toronto, M4N 3M6 Canada [email protected]

ISBN 978-3-11-070061-9 e-ISBN (PDF) 978-3-11-070269-9 e-ISBN (EPUB) 978-3-11-070273-6 ISSN 0938-6572 Library of Congress Control Number: 2021951113 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2022 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

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Mariusz Urbański dedicates this book to his wife, Irena. À mes parents Thérèse et Jean-Guy, à ma famille et à mes amis, sans qui ce livre n’aurait pu voir la vie... du fond du coeur, merci! Mario

Preface Dynamical systems and ergodic theory is a rapidly evolving field of mathematics with a large variety of subfields, which use advanced methods from virtually all areas of mathematics. These subfields comprise but are by no means limited to: abstract ergodic theory, topological dynamical systems, symbolic dynamical systems, smooth dynamical systems, holomorphic/complex dynamical systems, conformal dynamical systems, one-dimensional dynamical systems, hyperbolic dynamical systems, expanding dynamical systems, thermodynamic formalism, geodesic flows, Hamiltonian systems, KAM theory, billiards, algebraic dynamical systems, iterated function systems, group actions, and random dynamical systems. All of these branches of dynamical systems are mutually intertwined in many involved ways. Each of these branches nonetheless also has its own unique methods and techniques, in particular embracing methods which arise from the fields of mathematics the branch is closely related to. For example, complex dynamics borrows advanced methods from complex analysis, both of one and several variables; geodesic flows utilize methods from differential geometry; and abstract ergodic theory and thermodynamic formalism rely heavily on measure theory and functional analysis. Indeed, it is truly fascinating how large the field of dynamical systems is and how many branches of mathematics it overlaps with. In this book, we focus on some selected subfields of dynamical systems, primarily noninvertible ones. In the first volume, we give introductory accounts of topological dynamical systems acting on compact metrizable spaces, of finite-state symbolic dynamical systems, and of abstract ergodic theory of measure-theoretic dynamical systems acting on probability measure spaces, the latter including the metric entropy theory of Kolmogorov and Sinai. More advanced topics include infinite ergodic theory, general thermodynamic formalism, and topological entropy and pressure. This volume also includes a treatment of several classes of dynamical systems, which are interesting on their own and will be studied at greater length in the second volume: we provide a fairly detailed account of distance expanding maps and discuss Shub expanding endomorphisms, expansive maps, and homeomorphisms and diffeomorphisms of the circle. The second volume is somewhat more advanced and specialized. It opens with a systematic account of thermodynamic formalism of Hölder continuous potentials for open transitive distance expanding systems. One chapter comprises no dynamics but rather is a concise account of fractal geometry, treated from the point of view of dynamical systems. Both of these accounts are later used to study conformal expanding repellers. Another topic exposed at length is that of thermodynamic formalism of countable state subshifts of finite type. Relying on this latter, the theory of conformal graph directed Markov systems, with their special subclass of conformal iterated function systems, is described. Here, in a similar way to the treatment of conformal expanding repellers, the main focus is on Bowen’s formula for the Hausdorff dimension https://doi.org/10.1515/9783110702699-201

VIII | Preface of the limit set and multifractal analysis. A rather short examination of Lasota–Yorke maps of an interval is also included in this second volume. The third volume is entirely devoted to the study of the dynamics, ergodic theory, thermodynamic formalism, and fractal geometry of rational functions of the Riemann sphere. We present a fairly complete account of classical as well as more advanced topological theory of Fatou and Julia sets. Nevertheless, primary emphasis is placed on measurable dynamics generated by rational functions and fractal geometry of their Julia sets. These include the thermodynamic formalism of Hölder continuous potentials with pressure gaps, the theory of Sullivan’s conformal measures, invariant measures and their dimensions, entropy, and Lyapunov exponents. We further examine in detail the classes of expanding, subexpanding, and parabolic rational functions. We also provide, with proofs, several of the fundamental tools from complex analysis that are used in complex dynamics. These comprise Montel’s Theorem, Koebe’s Distortion Theorems and Riemann–Hurwitz formulas, with their ramifications. In virtually each chapter of this book, we describe a large number of concrete selected examples illustrating the theory and serving as examples in other chapters. Also, each chapter of the book is supplied with a number of exercises. These vary in difficulty, from very easy ones asking to verify fairly straightforward logical steps to more advanced ones enhancing largely the theory developed in the chapter. This book originated from the graduate lectures Mariusz Urbański delivered at the University of North Texas in the years 2005–2010 and that Sara Munday took notes of. With the involvement of Mario Roy, the book evolved and grew over many years. The last 2 years (2020 and 2021) of its writing were most dramatic and challenging because of the COVID-19 pandemic. Our book borrows widely from many sources including the books [63, 81, 98]. We nevertheless tried to keep it as self-contained as possible, avoiding to refer the reader too often to specific results from special papers or books. Toward this end, an appendix comprising classical results, mostly from measure theory, functional analysis and complex analysis, is included. The book covers quite a many topics treated with various degrees of completeness, none of which are fully exhausted because of their sheer largeness and their continuous dynamical growth.

List of Figures Figure 15.1 Figure 16.1 Figure 16.2

As the parameter α crosses the Hausdorff dimension HD(A), the α-dimensional Hausdorff measure Hα (A) collapses from ∞ to 0. | 577 Graph of T . | 654 Graph of Fμφ . | 654

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Introduction to Volume 2 In the first volume, we developed the general theory of thermodynamic formalism, i. e., for general topological dynamical systems (i. e., continuous self-maps of a compact metrizable space) subject to general potentials (i. e., continuous potentials). In this second volume, we carry on with the study of thermodynamic formalism and its dynamical and geometric applications, but this time for specific classes of systems subject to specific classes of potentials. We now describe in more detail the content of each chapter in this second volume, including their mutual dependence and interrelations. Chapter 13 – Gibbs states and transfer operators for open, distance expanding systems In this chapter, we introduce the fundamental concept of Gibbs states. Originating from statistical mechanics, Gibbs states play a prominent role in dynamical systems theory as (1) they describe, from a measure-theoretic viewpoint, the long-term behavior of typical orbits; (2) they are part of the core of thermodynamic formalism; and (3) they are indispensable for studying the fractal geometry of many dynamicallydefined invariant sets. We also introduce the transfer operator (sometimes called Perron–Frobenius operator), one of the most powerful tools in thermodynamic formalism. This operator is designed to construct Gibbs states. All of this is done for open, distance expanding maps (cf. Chapter 4) under continuous potentials. The first five sections are devoted to preparatory material. Section 13.1 concerns potentials. Section 13.2 introduces Gibbs states, specifies their basic properties and reveals that invariant Gibbs measures are equilibrium measures (cf. Chapter 12). Section 13.3 discusses Jacobians and changes of variables in purely measure-theoretic terms. Section 13.4 outlines a general way of constructing invariant measures from quasi-invariant ones. Section 13.5 presents general transfer operators and their fundamental properties. The existence of Gibbs states generally requires two additional assumptions: (1) that the map be topologically transitive; and (2) that the potential be Hölder continuous. Under these conditions, we establish in Section 13.6 the existence of an ergodic, though not necessarily invariant, Gibbs state. Under these same conditions, we show in Section 13.7 that every system admits exactly one invariant Gibbs state. This measure is also ergodic and is the unique equilibrium measure. In Section 13.8, we study more advanced properties of transfer operators, especially the behavior of the iterates of these operators. Among others, we show that the transfer operator is quasi-compact, i. e., it exhibits a spectral gap. We do this by proving the IonescuTulcea and Marinescu inequality and using the Ionescu-Tulcea and Marinescu theorem (see Appendix B). We further establish the continuity (1) of invariant Gibbs https://doi.org/10.1515/9783110702699-203

XII | Introduction to Volume 2 states; (2) of eigenmeasures of dual operators; and (3) of the Radon–Nikodym derivative of the former with respect to the latter; in their dependence on the underlying potential. Section 13.9 demonstrates that measure-theoretic dynamical systems generated by invariant Gibbs states truly deserve to be called random or chaotic, as they obey stochastic laws such as the exponential decay of correlations, the central limit theorem, the law of the iterated logarithm, K-mixing and the Bernoulli property. They also have asymptotic variances. Finally, in Section 13.10 we show that if a potential depends real analytically on a parameter, then the corresponding topological pressure also depends real analytically on that parameter. This is proved by complex-analytic methods and by means of the perturbation theory for linear operators applied in our context to transfer operators. This topic will have notable sequels in Chapters 16 and 20. Chapter 14 – Lasota–Yorke maps Roughly speaking, Lasota–Yorke maps are piecewise monotone, piecewise differentiable expanding maps of a compact interval. They have finitely many pieces and the reciprocal of the absolute value of their derivative is of bounded variation. After defining them in Section 14.1, we describe the form of their Perron–Frobenius operator in Section 14.2 and use this operator in Section 14.3 to establish the existence of an invariant measure which is absolutely continuous with respect to the Lebesgue measure. In Section 14.4, we demonstrate the exponential decay of correlations with respect to that invariant measure under two additional assumptions on the spectrum of the transfer operator. Chapter 15 – Fractal measures and dimensions In this “nondynamical” chapter, we discuss different ways of evaluating the size of sets, especially fractal sets. Classical geometrical objects such as lines, squares or cubes, have integer Euclidean and topological dimensions. In contrast, because of their irregularities at infinitesimal scale, fractal sets cannot be accurately described and distinguished from one another by their Euclidean or topological dimensions. In order to evaluate their size, we need geometric measures (Hausdorff and packing) and the dimensions they induce, to examine the local and global structures of these sets at arbitrarily small scales. We also introduce the box-counting dimensions. In Sections 15.2 and 15.3, we establish the fundamental properties of these measures and dimensions. In Section 15.5, we consider volume lemmas and their geometric consequences. Their main purpose is to tell us when a Hausdorff measure or a packing measure is positive, finite, zero or infinite. We refer to them as Frostman converse theorems. At the end of that section, we formulate Frostman’s (direct) lemma and compare it with the Frostman converse theorems. The advantage of the latter is that they provide tools to calculate, or at least estimate, both Hausdorff and packing

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measures and dimensions. Finally, in Section 15.6, we deal with dimensions of measures. Chapter 16 – Conformal expanding repellers In this chapter, we return to the class of expanding repellers that were introduced in Chapter 4. As these maps are open and distance expanding, the theory of Gibbs states developed in Chapter 13 applies. However, this class is too large to achieve many interesting results about the fractal structure of the limit set the repellers act on. So in this chapter, we restrict our attention to the subclass of conformal expanding repellers. In Section 16.1, we briefly review the notion of conformality. In Section 16.2, we define the class of conformal expanding repellers, provide some examples, and derive some consequences of results presented in Chapters 4 and 13, in particular bounded distortion properties. Beginning with Section 16.3, we engage with the fractal geometry of the repeller’s limit set. In Section 16.3, we prove the celebrated Bowen’s formula, according to which the unique zero of the pressure function for a particular geometric potential is equal to the Hausdorff dimension of the limit set of the repeller. Bowen’s formula is a generalization of Hutchinson’s formula, a well-known result in fractal geometry. In Section 16.4, we examine the behavior of the Hausdorff dimension of the limit set of complex repellers under perturbation. This technically-involved study relies on transfer operators and on powerful complex- and functional-analytic methods, including the perturbation theory of bounded linear operators. It necessitates the use of quasiconformal mappings, holomorphic motions and the celebrated λ-lemma, whose complete proof is provided in Theorem 16.4.4. Ultimately, we show in Theorem 16.4.11 that the Hausdorff dimension of the repellers’ limit set varies real analytically under analytic perturbations of the repellers. In Section 16.5, we develop a formula expressing the Hausdorff dimension of an ergodic invariant Borel probability measure as the ratio of its measure-theoretic entropy and its Lyapunov exponent. For historical reasons, this formula is frequently referred to as a volume lemma. Finally, in Section 16.6, we perform a multifractal analysis of the Hausdorff dimension of the level sets of Gibbs states for any Hölder continuous potential. This is a fertile field of applications of the theory of conformal expanding repellers and more largely of this book. In addition to its intrinsic interest and importance, the multifractal analysis conducted in this chapter bears witness to the strength and sophistication of the tools and methods developed in this book. Chapters 17–18 – Countable state thermodynamic formalism In these two chapters, we present thermodynamic formalism for countable-alphabet subshifts of finite type EA∞ generated by transition matrices A : E ×E → {0, 1} on countable alphabets E. If the alphabet E is finite, then the symbolic space EA∞ is compact

XIV | Introduction to Volume 2 and all the knowledge acquired in Chapters 11–13 about pressure, variational principle, equilibrium states, Gibbs states and transfer operators, applies to the shift map σ : EA∞ → EA∞ . If the alphabet E is infinite and the incidence matrix A is finitely irreducible, then the phase space EA∞ is not compact and essentially none of the results proved in the compact case holds directly. The primary goal of Chapter 17 is to introduce the appropriate concept of topological pressure and to prove the existence and uniqueness of equilibrium states and invariant Gibbs states for potentials that are summable and Hölder continuous on cylinders. The definitions of pressure, equilibrium state and Gibbs state reduce to the usual ones in the case of a finite alphabet. In Section 17.1, we give the basic definitions of finite irreducibility and finite primitivity. Unless otherwise stated, the results mentioned in this paragraph assume that the subshift is finitely irreducible. In Section 17.2, we define the pressure. In order to speak of equilibrium states, one needs a variational principle for the pressure and we prove in Section 17.3 three such principles. In Section 17.4, we study Gibbs states and prove the existence of a unique invariant Gibbs state. This state is ergodic, and even totally ergodic when the subshift is finitely primitive. In Section 17.5, we identify a simple condition under which the unique invariant Gibbs state is an equilibrium state. Moreover, we show that equilibrium states are unique. Consequently, we deduce that invariant Gibbs states and equilibrium states exist, are unique, and coincide. In Section 17.6, we examine the transfer operator and we demonstrate that its dual admits at most one eigenmeasure, which turns out to be a (nonnecessarily invariant) Gibbs state. In Section 17.7, the existence of such an eigenmeasure is established. So the dual of the transfer operator has a unique eigenmeasure and its eigenvalue is the exponential of the pressure. The aforementioned facts are summarized in Corollary 17.7.5. Finally, in Section 17.8 we confirm the presence of a pressure gap and, consequently, the positivity of the measure-theoretic entropy of the subshift with respect to the unique equilibrium state. Chapter 18 treats of refined properties of σ-invariant Gibbs states μf for potentials f : EA∞ → ℝ that are summable and Hölder continuous on cylinders. We show that the measure-theoretic dynamical systems (σ : EA∞ → EA∞ , μf ) deserve to be called random or chaotic by proving in Section 18.3 that they obey classical stochastic laws. This is achieved via a refined study of transfer operators acting on the Banach space of Hölder continuous functions and especially by establishing their spectral properties, such as quasicompactness (spectral gap) and simplicity of the leading eigenvalue eP(f ) . To set the table, we prove in Section 18.1 the famous Ionescu-Tulcea and Marinescu inequality. In Section 18.2, we establish the continuity (1) of the pressure function f 󳨃→ P(f ); (2) of the unique σ-invariant Gibbs states f 󳨃→ μf ; (3) of the unique eigenmeasures of the dual operators f 󳨃→ mf ; and (4) of the Radon–Nikodym derivative of μf with respect to mf , i. e. the continuity of f 󳨃→ dμf /dmf . Finally, in Section 18.4 we characterize, primarily by means of cohomological equations, those potentials that have the same invariant Gibbs state.

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Chapter 19 – Conformal graph directed Markov systems (CGDMSs) Chapter 20 – Real analyticity of topological pressure and Hausdorff dimension for CGDMSs Chapter 21 – Multifractal analysis for CGDMSs In these three chapters, we apply the thermodynamic formalism developed in Chapters 17–18 to CGDMSs. In Chapter 19, we describe a powerful method for constructing and studying the geometric and dynamical properties of fractal sets, including the middle-third Cantor set, the Sierpiński triangle, the Sierpiński carpet, the Koch curve, various kinds of Menger sets and the limit set of real and complex continued fractions. This method arises from the theory of countable-alphabet conformal graph directed Markov systems, which are a generalization of iterated function systems (IFSs). These systems are built from a graph comprising vertices and directed edges and from an edge-transition matrix. In Section 19.1, we start with the introduction of basic notions and notation for graph directed Markov systems (GDMSs): in particular, the underlying symbolic space and the shift map on it, the limit set, and the projection (coding map) from the symbolic space onto the limit set. Section 19.2 consists of properties of conformal maps in Euclidean spaces that will be crucial in deriving in Section 19.3 the core properties of conformal IFSs (CIFSs) and their generalization, conformal GDMSs (CGDMSs). In Section 19.4, we define an appropriate version of the topological pressure, the finiteness parameter, and the Bowen parameter. Based on the behavior of the pressure function, we then classify in Section 19.5 CGDMSs as irregular or regular, and in this latter case a finer splitting into strongly regular and critically regular systems is described. In Section 19.6, a variation of Bowen’s formula is derived. Among its many consequences, we get an almost cost-free, effective and non-trivial lower estimate for the Hausdorff dimension of the limit sets of finitely irreducible CGDMSs. We further deduce from it the classical version of Hutchinson’s formula. A particularly important tool to study the fractal geometry of limit sets of CGDMSs is the concept of conformal measure. We introduce and study these measures at length in Section 19.8, and we give a formula, called volume lemma, for the Hausdorff dimension of projections of ergodic invariant measures from the underlying symbolic space. This formula is expressed in the familiar form of the ratio of the measure-theoretic entropy and the Lyapunov exponent (cf. Section 16.5). Beforehand, though, we examine in Section 19.7 different separation and geometric conditions: the Strong Open Set Condition (SOSC), the Boundary Separation Condition (BSC), the Strong Separation Condition (SSC) and the Cone Condition (CC). Any one of those, when satisfied, has a profound impact on conformal measures, invariant measures, and the whole dynamics and geometry of CGDMSs. Finally, we provide a variety of examples in Section 19.9. The first three sections of Chapter 20 are related to our considerations in Section 13.10. They are written in the framework of Chapters 17 and 18. Under various hypotheses, we show that the pressure function P(φ + g) depends real-analytically

XVI | Introduction to Volume 2 on g (in an appropriately defined way). The potential φ is assumed to be strongly summable and Hölder continuous on cylinders. The methods of proof are of complexand functional-analytic character. Our strategy consists in complexifying potentials first, and thus Perron–Frobenius operators, too. Then we prove that these complexified operators depend holomorphically on the perturbation g. This is where infinitedimensional complex analysis (on Banach spaces) is used. Next, enters functional analysis. Indeed, we know that the exponential of the pressure function is a simple isolated eigenvalue of the transfer operator. So the Kato–Rellich perturbation theorem yields holomorphic dependence of appropriate eigenvalues of the complexified transfer operators. Restricting these operators to the real realm gives real-analyticity of the exponential of the pressure function, and hence of the pressure itself. We further identify the pressure’s derivative. In Section 20.4, we consider families of CGDMSs in the complex plane ℂ. We assume that these families depend in an appropriate way on a complex parameter λ and we show the real-analytic dependence of the Bowen parameter on λ. We deduce from this the real-analyticity of the Hausdorff dimension of the limit set. Holomorphic motions are once again encountered in this section (cf. Section 16.4). In Chapter 21, we perform a multifractal analysis of the Hausdorff dimension of the level sets of Gibbs states for any Hölder family of functions of the form ge + u log |φ′e |, where {ge }e∈E is a bounded Hölder family of functions and Φ = {φe }e∈E is a CGDMS. In [46, 81], the multifractal analysis used, at each point of the limit set J, the natural filtration generated by the initial cylinders associated to the word that encodes the given point. Aiming to give the multifractal analysis a transparent geometrical meaning, we derive in this chapter an analysis using as the filtration a base of balls centred at the point. We will conduct the analysis of the conformal measure (or, equivalently, of the invariant measure) associated to a family of weights imposed upon a (finite or infinite) CGDMS. We also conduct this analysis over the full set of parameters on which it can be expected to hold. However, we restrict our attention to cofinitely regular, finitely irreducible, conformal-like CGDMSs that satisfy the SOSC and to a large and dynamically significant subset J0 of the limit set J, as opposed to the entire limit set, though the sets J0 and J coincide under a mild boundary separation condition. An additional geometric flavor of this analysis results from the fact that we concentrate on a geometrically meaningful family of Hölder weights. The results apply to large classes of CGDMSs, one- and multi-dimensional alike, including real and complex continued fractions. In Section 21.1, we describe the Hölder families of weights F and Fq,t we shall work with and we study the properties of the pressure function P(q, t) and of the temperature function T(q) they determine. (The general properties of Hölder families of functions and the amalgamated function they induce are studied in Section 19.8.) In Section 21.2, we carry out the multifractal analysis of the conformal measure mF (or, equivalently, of the invariant measure μF ) associated to the family of weights F. This analysis is done over a set J0 ⊆ J of full μF -measure and conducted by means of balls. In particular, we show that the Legendre transform of the temperature function T(q)

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describes the local structure of the set J0 . This is accomplished via the introduction of auxiliary measures that capture subsets of J0 where the pointwise dimension is constant. Part of the analysis is restricted to a subclass called almost-CIFSs. In Section 21.3, we derive the multifractal analysis of mF under additional conditions on the almost-CIFSs. Finally, in Section 21.4 we conduct a similar analysis on smaller subsets of points of the limit set J (that have a symbolic coding with at least one infinitely-seen letter from a finite subalphabet) but for all CGDMSs that are cofinitely regular, finitely irreducible, conformal-like and satisfy the SOSC.

Contents Volume 1 1 1.1 1.2 1.3 1.4 1.5 1.5.1 1.5.2 1.5.3 1.6 1.6.1 1.6.2 1.7

Dynamical systems Basic definitions Topological conjugacy and structural stability Factors Subsystems Mixing and irreducibility Minimality Transitivity and topological mixing Topological exactness Examples Rotations of compact topological groups Maps of the interval Exercises

2 2.1 2.2 2.2.1 2.3 2.3.1 2.3.2 2.4

Homeomorphisms of the circle Lifts of circle maps Orientation-preserving homeomorphisms of the circle Rotation numbers Minimality for homeomorphisms and diffeomorphisms of the circle Denjoy’s theorem Denjoy’s counterexample Exercises

3 3.1 3.2 3.2.1 3.2.2 3.2.3 3.3 3.4

Symbolic dynamics Full shifts Subshifts of finite type Topological transitivity Topological exactness Asymptotic behavior of periodic points General subshifts of finite type Exercises

4 4.1 4.1.1 4.1.2 4.2 4.3

Distance expanding maps Definition and examples Expanding repellers Hyperbolic Cantor sets Inverse branches Shadowing

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XX | Contents 4.4 4.5 4.6

Markov partitions Symbolic representation generated by a Markov partition Exercises

5 5.1 5.2 5.3 5.4 5.5

(Positively) expansive maps Expansiveness Uniform expansiveness Expansive maps are expanding with respect to an equivalent metric Parabolic Cantor sets Exercises

6 6.1 6.2 6.3 6.3.1 6.3.2 6.4 6.4.1 6.4.2 6.4.3 6.5

Shub expanding endomorphisms Shub expanding endomorphisms of the circle Definition, characterization, and properties of general Shub expanding endomorphisms A digression into algebraic topology Deck transformations Lifts Dynamical properties Expanding property Topological exactness and density of periodic points Topological conjugacy and structural stability Exercises

7 7.1 7.1.1 7.2 7.2.1 7.2.2 7.2.3 7.3 7.4 7.5 7.6

Topological entropy Covers of a set Dynamical covers Definition of topological entropy via open covers First stage: entropy of an open cover Second stage: entropy of a system relative to an open cover Third and final stage: entropy of a system Bowen’s definition of topological entropy Topological degree Misiurewicz–Przytycki theorem Exercises

8 8.1 8.1.1 8.1.2 8.1.3 8.2

Ergodic theory Measure-preserving transformations Examples of invariant measures Poincaré’s recurrence theorem Existence of invariant measures Ergodic transformations

Contents | XXI

8.2.1 8.2.2 8.2.3 8.2.4 8.3 8.3.1 8.3.2 8.3.3 8.4 8.5

Birkhoff’s ergodic theorem Existence of ergodic measures Examples of ergodic measures Uniquely ergodic transformations Mixing transformations Weak mixing Mixing K-mixing Rokhlin’s natural extension Exercises

9 9.1 9.2 9.3 9.4 9.4.1 9.4.2 9.4.3 9.5 9.6 9.7

Measure-theoretic entropy An excursion into the origins of entropy Partitions of a measurable space Information and conditional information functions Definition of measure-theoretic entropy First stage: entropy and conditional entropy for partitions Second stage: entropy of a system relative to a partition Third and final stage: entropy of a system Shannon–McMillan–Breiman theorem Brin–Katok local entropy formula Exercises

10 10.1 10.2 10.3 10.4 10.5

Infinite invariant measures Quasi-invariant measures, ergodicity and conservativity Invariant measures and inducing Ergodic theorems Absolutely continuous σ-finite invariant measures Exercises

11 Topological pressure 11.1 Definition of topological pressure via open covers 11.1.1 First stage: pressure of a potential relative to an open cover 11.1.2 Second stage: the pressure of a potential 11.2 Bowen’s definition of topological pressure 11.3 Basic properties of topological pressure 11.4 Examples 11.5 Exercises 12 The variational principle and equilibrium states 12.1 The variational principle 12.1.1 Consequences of the variational principle

XXII | Contents 12.2 12.3 12.4

Equilibrium states Examples of equilibrium states Exercises

Volume 2 Preface | VII List of Figures | IX Introduction to Volume 2 | XI 13 13.1 13.2 13.2.1 13.2.2 13.3 13.4 13.5 13.6 13.6.1 13.6.2 13.6.3 13.6.4 13.6.5 13.7 13.7.1 13.7.2 13.7.3 13.7.4 13.7.5 13.7.6 13.8 13.8.1 13.8.2

Gibbs states and transfer operators for open, distance expanding systems | 431 Hölder continuous potentials | 433 Gibbs measures | 438 Definition and properties | 438 Invariant Gibbs states are equilibrium states | 446 Jacobians and changes of variables | 447 Construction of invariant measures from quasi-invariant ones | 449 Transfer operators | 452 Nonnecessarily-invariant Gibbs states | 455 Existence of eigenmeasures for the dual of the transfer operator | 456 Eigenmeasures are conformal measures | 463 Eigenmeasures are Gibbs states for transitive systems | 466 Ergodicity of the eigenmeasures for transitive systems | 467 Metric exactness of the eigenmeasures for topologically exact systems | 470 Invariant Gibbs states | 471 Almost periodicity of normalized transfer operators | 471 Existence, uniqueness and ergodicity of invariant Gibbs states for transitive systems | 474 Invariant Gibbs states and equilibrium states coincide and are unique | 479 Hölder continuous potentials with the same Gibbs states | 488 Invariant Gibbs states have positive entropy; pressure gap | 489 Absolutely continuous invariant measures for Shub expanding maps | 491 Finer properties of transfer operators and Gibbs states | 492 Iterates of transfer operators | 492 Ionescu-Tulcea and Marinescu inequality and spectral gap | 502

Contents | XXIII

13.8.3 13.9 13.9.1 13.9.2 13.9.3 13.9.4 13.9.5 13.10 13.11 14 14.1 14.2 14.3 14.4 14.5 15 15.1 15.2

Continuity of Gibbs states | 507 Stochastic laws | 509 Exponential decay of correlations | 511 Asymptotic variance | 512 Central limit theorem | 526 Law of the iterated logarithm | 529 Metric exactness, K-mixing and weak Bernoulli property | 529 Real analyticity of topological pressure | 534 Exercises | 544 Lasota–Yorke maps | 549 Definition | 549 Transfer operator | 550 Existence of absolutely continuous invariant probability measures | 555 Exponential decay of correlations | 555 Exercises | 557

15.4 15.5 15.6 15.7

Fractal measures and dimensions | 563 Outer measures | 564 Geometric (Hausdorff and packing) outer measures and dimensions | 567 Gauge functions | 567 Hausdorff measures | 567 Packing measures | 571 Packing versus Hausdorff measures | 575 Dimensions of sets | 576 Hausdorff dimension | 576 Packing dimensions | 578 Packing versus Hausdorff dimensions | 579 Box-counting dimensions | 579 Alternate definitions of box dimensions | 580 Hausdorff versus packing versus box dimensions | 581 Hausdorff, packing and box dimensions under (bi-)Lipschitz mappings | 584 A digression into geometric measure theory | 587 Volume lemmas—Frostman converse theorems | 588 Dimensions of measures | 595 Exercises | 600

16 16.1

Conformal expanding repellers | 603 Conformal maps | 604

15.2.1 15.2.2 15.2.3 15.2.4 15.3 15.3.1 15.3.2 15.3.3 15.3.4 15.3.5 15.3.6 15.3.7

XXIV | Contents 16.2 16.3 16.3.1 16.4 16.5 16.6 16.7

Conformal expanding repellers | 605 Bowen’s formula | 614 Special case of Hutchinson’s formula | 617 Real-analytic dependence of Hausdorff dimension of repellers in ℂ | 619 Dimensions of measures, Lyapunov exponents and measure-theoretic entropy | 640 Multifractal analysis of Gibbs states | 648 Exercises | 663

17 Countable state thermodynamic formalism | 667 17.1 Finitely irreducible subshifts | 668 17.1.1 Finitely primitive subshifts | 670 17.2 Topological pressure | 671 17.2.1 Potentials: acceptability and Hölder continuity on cylinders | 671 17.2.2 Partition functions | 673 17.2.3 The pressure function | 677 17.3 Variational principles and equilibrium states | 681 17.4 Gibbs states | 683 17.5 Gibbs states versus equilibrium states | 691 17.6 Transfer operator | 693 17.7 Existence and uniqueness of eigenmeasures of the dual transfer operator, of Gibbs states and of equilibrium states | 697 17.8 The invariant Gibbs state has positive entropy; pressure gap | 702 17.9 Exercises | 703 18 Countable state thermodynamic formalism: finer properties | 707 18.1 Ionescu-Tulcea and Marinescu inequality | 707 18.2 Continuity of Gibbs states | 719 18.3 Stochastic laws | 725 18.3.1 Exponential decay of correlations | 725 18.3.2 Asymptotic variance | 725 18.3.3 Central limit theorem | 726 18.3.4 Law of the iterated logarithm | 727 18.4 Potentials with the same Gibbs states | 727 18.5 Exercises | 730 19 Conformal graph directed Markov systems | 733 19.1 Graph directed Markov systems | 735 19.1.1 The underlying multigraph 𝒢 | 735 19.1.2 The underlying matrix A | 736 19.1.3 The system itself | 737

Contents | XXV

19.2 19.3 19.4 19.5 19.6 19.6.1 19.6.2 19.6.3 19.7 19.7.1 19.7.2 19.7.3 19.7.4 19.7.5 19.8 19.8.1 19.8.2 19.8.3 19.8.4 19.9 19.10 20 20.1 20.2 20.3 20.4 20.5

Properties of conformal maps in ℝd , d ≥ 2 | 739 Conformal graph directed Markov systems | 742 Topological pressure, finiteness parameter, and Bowen parameter for CGDMSs | 749 Classification of CGDMSs | 753 Bowen’s formula for CGDMSs | 754 The finite case | 754 The general case | 758 Hutchinson’s formula | 759 Other separation conditions and cone condition | 760 The strong open set condition | 760 The boundary separation condition | 763 The strong separation condition | 763 The cone condition | 764 Conformal-likeness | 771 Hölder families of functions and conformal measures | 773 Basic definitions and properties | 773 Conformal measures for summable Hölder families of functions | 776 Conformal measures for CGDMSs | 784 Dimensions of measures for CGDMSs | 788 Examples | 793 Exercises | 797 Real analyticity of topological pressure and Hausdorff dimension | 799 Real analyticity of pressure: part I | 800 Real analyticity of pressure: part II | 809 Real analyticity of pressure: part III | 812 Real analyticity of Hausdorff dimension for CGDMSs in ℂ | 815 Exercises | 826

21 21.1 21.2

Multifractal analysis for conformal graph directed Markov systems | 827 Pressure and temperature | 828 Multifractal analysis of the conformal measure mF over a subset of J | 835 21.3 Multifractal analysis over J | 845 21.3.1 Under the boundary separation condition | 845 21.3.2 Under other conditions | 846 21.4 Multifractal analysis over another subset of J | 850 21.5 Exercises | 853 A A.1

A selection of classical results | 855 Measure theory | 855

XXVI | Contents A.1.1 A.1.2 A.1.3 A.1.4 A.1.5 A.1.6 A.1.7

Collections of sets and measurable spaces | 855 Measurable transformations | 859 Measure spaces | 861 Extension of set functions to measures | 864 Integration | 866 Convergence theorems | 868 Mutual singularity, absolute continuity and equivalence of measures | 874 The space C(X ), its dual C(X )∗ and the subspace M(X ) | 875 Expected values and conditional expectation functions | 879 Characteristic functions and distributions | 887 Functional analysis | 888 Complex analysis in one variable | 890 Complex analysis in several variables | 893

A.1.8 A.1.9 A.1.10 A.2 A.3 A.4 B

The Ionescu-Tulcea and Marinescu theorem | 895

Bibliography | 913 Index | 919

Volume 3 22 22.1 22.2 22.3 22.4 22.5 22.6

The Riemann–Hurwitz formula Proper analytic maps and their degree The Euler characteristic of plane bordered surfaces ̂ The Riemann–Hurwitz formula for bordered surfaces in ℂ ̂ Euler characteristic: the general ℂ case ̂ case Riemann–Hurwitz formula: the general ℂ Exercises

23 23.1 23.2

Selected tools from complex analysis Koebe’s distortion theorems Normal families and Montel’s theorem

24 Dynamics and topology of rational functions: their Fatou and Julia sets 24.1 Fatou set, Julia set and periodic points 24.1.1 Attracting periodic points 24.1.2 Non-attracting periodic points 24.2 Rationally indifferent periodic points

Contents | XXVII

24.2.1 24.2.2 24.2.3

24.3 24.4 24.4.1 24.4.2 24.5 24.6 24.7 24.8

Local and asymptotic behavior of holomorphic functions around rationally indifferent periodic points; part I Leau–Fatou flower petals Local and asymptotic behavior of rational functions around rationally indifferent periodic points; part II: Fatou’s flower theorem and fundamental domains Non-attracting periodic points revisited: total number and denseness of repelling periodic points The structure of the Fatou set Forward invariant components of the Fatou set Periodic components of the Fatou set Cremer points, boundary of Siegel disks and Herman rings Continuity of Julia sets Polynomials Exercises

25 Selected technical properties of rational functions 25.1 Passing near critical points; results and applications 25.2 Two rules for critical points 25.3 Expanding subsets of Julia sets 25.4 Lyubich’s geometric lemma 25.5 Two auxiliary partitions 25.5.1 Boundary partition 25.5.2 Exponentially large partition 25.5.3 Mañé’s partition 25.6 Miscellaneous facts 25.6.1 Miscellaneous general facts 25.6.2 Miscellaneous facts about rational functions 25.7 Exercises 26 26.1 26.2 26.3 26.4 27 27.1 27.2

Expanding (or hyperbolic), subexpanding, and parabolic rational functions: topological outlook Expanding rational functions Expansive and parabolic rational functions Subexpanding rational functions Exercises Equilibrium states for rational functions and Hölder continuous potentials with pressure gap Bad and good inverse branches The transfer operator ℒφ : C(𝒥 (T )) → C(𝒥 (T )); its lower and upper bounds

XXVIII | Contents 27.2.1 27.2.2 27.2.3 27.3 27.4 27.5 27.5.1 27.6 27.7 27.8 27.9 27.10

First lower bounds for ℒφ Auxiliary operators Lβ Final bounds Equicontinuity of iterates of ℒφ ; “Gibbs” states mφ and μφ ̂φ and their dynamical Spectral properties of the transfer operator ℒ consequences Equilibrium states for φ : 𝒥 (T ) → ℝ The “Gibbs” state μφ is an equilibrium state for φ : 𝒥 (T ) → ℝ Continuous dependence on φ of the “Gibbs” states mφ , μφ and the density ρφ Differentiability of topological pressure Uniqueness of equilibrium states for Hölder continuous potentials Assorted remarks Exercises

28 28.1 28.2 28.3 28.4

Invariant measures: fractal and dynamical properties Lyapunov exponents are non-negative Ruelle’s inequality Pesin’s theory in a conformal setting Volume lemmas, Hausdorff and packing dimensions of invariant measures 28.5 HD(𝒥 (T )) > 0 and radial Julia sets 𝒥r (T ), 𝒥re (T ), 𝒥ure (T ) 28.6 Conformal Katok’s theory of expanding sets and topological pressure 28.6.1 Pressure-like definition of the functional hμ + ∫ φ dμ 28.6.2 Conformal Katok’s theory 28.7 Exercises 29 Sullivan’s conformal measures for rational functions 29.1 General concept of conformal measures 29.1.1 Motivation for and definition of general conformal measures 29.1.2 Selected properties of general conformal measures 29.1.3 The limit construction and PS limit measures 29.1.4 Conformality properties of PS limit measures 29.2 Sullivan’s conformal measures 29.3 Pesin’s formula 29.4 Exercises 30 30.1 30.2 30.3

Conformal measures, invariant measures, and fractal geometry of expanding rational functions Fundamental fractal geometry Geometric rigidity Exercises

Contents | XXIX

31

Conformal measures, invariant measures, and fractal geometry of parabolic rational functions 31.1 General conformal measures for expansive topological dynamical systems 31.2 Geometric topological pressure and generalized geometric conformal measures for parabolic rational functions 31.3 Sullivan’s conformal measures for parabolic rational functions 31.3.1 Technical preparations 31.3.2 The atomless hT -conformal measure mT : existence, uniqueness, ergodicity and conservativity 31.3.3 The complete structure of Sullivan’s conformal measures for parabolic rational functions: the atomless measure mT and purely atomic measures; Bowen’s formula 31.4 Invariant measures equivalent to mT : existence, uniqueness, ergodicity and the finite-infinite dichotomy 31.5 Hausdorff and packing measures 31.6 Exercises 32

Conformal measures, invariant measures, and fractal geometry of subexpanding rational functions 32.1 Sullivan’s conformal measures for subexpanding rational functions 32.1.1 Technical preparations 32.1.2 The atomless hT -conformal measure mT : existence, uniqueness, ergodicity and conservativity 32.1.3 The complete structure of Sullivan’s conformal measures for subexpanding rational functions: the atomless measure mT and purely atomic measures; Bowen’s formula 32.2 Invariant probability measure equivalent to mT : existence, uniqueness, and ergodicity 32.3 Hausdorff and packing measures 32.4 Exercises

13 Gibbs states and transfer operators for open, distance expanding systems In the first volume, we developed the general theory of thermodynamic formalism, i. e., for general topological dynamical systems (i. e., continuous self-maps of a compact metrizable space) subject to general potentials (i. e., continuous potentials). In this second volume, we carry on with the study of thermodynamic formalism and its dynamical and geometric applications, but this time for specific classes of systems subject to specific classes of potentials. In this chapter, we introduce the fundamental concept of Gibbs states. Motivated by statistical mechanics, this concept was introduced to dynamical systems theory (primarily to subshifts of finite type with finite alphabets) by David Ruelle, Rufus Bowen and Yakov Sinai, in their foundational works [11, 111–113] and [117], respectively. Gibbs states play a prominent role in dynamical systems theory as (1) they describe from a measure-theoretic viewpoint the long-term unstable behavior of typical orbits; (2) they are part of the core of thermodynamic formalism; and (3) thanks to the seminal paper [12], they are indispensable for studying the fractal geometry of many dynamically defined invariant sets. We also introduce the transfer operator (sometimes called Perron–Frobenius operator), one of the most powerful tools in thermodynamic formalism. This operator is designed to construct Gibbs states. All of this is done for open, distance expanding maps (cf. Chapter 4) under continuous potentials. Our exposition stems and expands the account given in [98]. We also borrowed from [91]. First of all, we prove the existence and uniqueness of (invariant) Gibbs states, along with the existence and uniqueness of equilibrium states. We further show that equilibrium states and invariant Gibbs states coincide. We also establish the basic properties of Gibbs states as well as some refined ones. We also show that the transfer operator is quasi-compact (i. e., it exhibits a spectral gap). This operator, and especially its spectral properties, are essential for establishing fine and transparent stochastic properties of invariant Gibbs states, such as the central limit theorem, the law of the iterated logarithm and the exponential decay of correlations. Moving on to more detailed content, (X, T) will stand for a topological dynamical system, i. e., a continuous self-map T : X → X of a compact metrizable space X. We further assume that T is an open, distance expanding map with respect to some metric d which is compatible with the topology on X. Per Definition 4.1.1, the map T is distance expanding if there exist two constants δ > 0 and λ > 1 such that d(x, y) < 2δ

󳨐⇒

d(T(x), T(y)) ≥ λ d(x, y).

(13.1)

For such a system, we proved in Chapter 4 that there exists a constant 0 < ξ ≤ δ such that T and all its iterates have local inverse branches on all balls of radius ξ . More https://doi.org/10.1515/9783110702699-013

432 | 13 Gibbs states and transfer operators for open, distance expanding systems precisely, for any x ∈ X and n ∈ ℕ the local inverse branch of T n that maps T n (x) to x is defined to be (cf. (4.29), (4.30) and (4.31)) −1 Tx−n := Tx−1 ∘ TT(x) ∘ ⋅ ⋅ ⋅ ∘ TT−1n−1 (x) : B(T n (x), ξ ) → B(x, λ−n ξ ).

(13.2)

Obviously, these inverse branches of T n are contractions with ratio λ−n . Initially, the system T : X → X will be submitted to continuous potentials φ : X → ℝ, though we will eventually restrict our attention to Hölder ones. The first five sections are devoted to preparatory material. Section 13.1 concerns potentials. Section 13.2 introduces Gibbs states, specifies their basic properties and establishes that invariant Gibbs measures are equilibrium measures. Section 13.3 discusses Jacobians and changes of variables in purely measure-theoretic terms. Section 13.4 describes a general way of constructing invariant measures from quasiinvariant ones. Section 13.5 presents general transfer operators and their fundamental properties. The existence of Gibbs states generally requires two additional assumptions: (1) An assumption on the map T: we impose that T be topologically transitive; and (2) An assumption on the potential φ: we demand that φ be Hölder continuous. As we will observe, both assumptions arise naturally. Under these conditions, we establish in Section 13.6 the existence of an ergodic, though not necessarily invariant, Gibbs state. Under these same conditions, we show in Section 13.7 that every system admits exactly one invariant Gibbs state. This measure is also ergodic and is the unique equilibrium measure. This section is quite long, with plenty of interesting results. In particular, we prove that Hölder continuous potentials entail a pressure gap, and consequently, the positivity of entropy. We also fully characterize, primarily by means of cohomological equations, those Hölder continuous potentials that have the same invariant Gibbs state. In Section 13.8, we study more advanced properties of transfer operators, especially the behavior of the iterates of these operators. Among others, the transfer operator is quasi-compact, i. e., it exhibits a spectral gap. We achieve this by proving the Ionescu-Tulcea and Marinescu inequality and using the Ionescu–Tulcea and Marinescu theorem (see Appendix B). We also establish the continuity of invariant Gibbs states, of eigenmeasures of dual operators, and of the Radon–Nikodym derivative of the former with respect to the latter, in their dependence on the potential φ. Section 13.9 demonstrates that dynamical systems generated by invariant Gibbs states truly deserve to be called random or chaotic. We treat, by means of spectral properties of transfer operators, stochastic laws such as the exponential decay of correlations, the central limit theorem, the law of the iterated logarithm, K-mixing, and the Bernoulli property. Another feature essential for randomness, which we also prove by means of spectral properties of transfer operators, is the existence of asymptotic

13.1 Hölder continuous potentials | 433

variances of Hölder continuous functions. We establish two very useful formulas for them. Finally, in Section 13.10, we prove that if a potential depends real analytically on a parameter (in particular, if it is of the form t 󳨃→ tφ) then the corresponding topological pressure also depends real analytically on that parameter. This is proved by complex analytic methods and by means of the perturbation theory for linear operators applied in our context to transfer operators. This topic will have notable sequels: in Chapter 16, where will be shown the real-analytic dependence of the Hausdorff dimension of limit sets of conformal expanding repellers, and in Chapter 20, where the real-analytic dependence of the topological pressure for countable-alphabet symbolic dynamical systems will be substantiated, which will lead to the real-analytic dependence of the Hausdorff dimension of limit sets of conformal graph directed Markov systems. Once it has been ascertained that the topological pressure is real analytic, it is natural to ask what its first and second derivatives are. We give a full answer to this question at the end of Section 13.10. Namely, the first derivative is the integral of the “perturbation” against the appropriate Gibbs state while the second derivative is the asymptotic variance of the “perturbation” with respect to that same state. To our knowledge, the first result on the real-analytic dependence of the topological pressure on a parameter can be found in [111] and [113]. A number of similar results ensued since then; we mention here very few of them, not covered in this book, such as [2, 84, 121, 127, 135, 137].

13.1 Hölder continuous potentials Let (X, d) be a compact metric space. Recall that a function φ : X → ℝ is said to be Hölder continuous with exponent α > 0 if there exists some constant c ≥ 0 such that α 󵄨󵄨 󵄨 󵄨󵄨φ(x) − φ(y)󵄨󵄨󵄨 ≤ c[d(x, y)] ,

∀x, y ∈ X.

The least constant c with this property will be denoted by υα (φ). To simplify notation, we will at times write dα (x, y) for [d(x, y)]α . Clearly, every Hölder continuous function is continuous. Also, observe that a Hölder continuous function with exponent α = 1 is simply a Lipschitz continuous function. (For more, see Exercise 13.11.1.) For now, we will consider only real-valued functions. Denote by C(X) the Banach space of all continuous functions defined on X. For a given α > 0, the space of all Hölder continuous functions defined on X with exponent α will be denoted by Hα (X). For every φ ∈ Hα (X), set ‖φ‖α := υα (φ) + ‖φ‖∞ . Then (Hα (X), ‖ ⋅ ‖α ) is a Banach space (see Exercise 13.11.2).

(13.3)

434 | 13 Gibbs states and transfer operators for open, distance expanding systems The following result affirms that the nth ergodic sum of any Hölder potential φ along each of the inverse branches of the nth iterate of T is Hölder continuous with Hölder constant independent of the branch and of n. Lemma 13.1.1 (Local bounded variation principle for ergodic sums). If T : X → X is an open, distance expanding system and φ ∈ Hα (X), then for all w ∈ X, all n ∈ ℕ and all x, y ∈ B(T n (w), ξ ) we have 󵄨󵄨 󵄨 −n −n α 󵄨󵄨Sn φ(Tw (x)) − Sn φ(Tw (y))󵄨󵄨󵄨 ≤ Cα (φ) d (x, y), where Cα (φ) :=

υα (φ) λα −1

(13.4)

while ξ and λ come from (13.1)–(13.2). Thus

exp(Sn φ(Tw−n (x))) ≤ Dα (φ) exp(Sn φ(Tw−n (y))),

(13.5)

where Dα (φ) := exp((2ξ )α Cα (φ)). Furthermore, if d(x, y) < ξ then for all n ∈ ℕ, ∑ x∈T −n (x)

eSn φ(x) ≤ Dα (φ)

∑ y∈T −n (y)

eSn φ(y) .

(13.6)

j Proof. Recall that Sn φ := ∑n−1 j=0 φ ∘ T . Then n−1

󵄨󵄨 󵄨 󵄨 󵄨 −n −n j −n j −n 󵄨󵄨Sn φ(Tw (x)) − Sn φ(Tw (y))󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨φ ∘ T ∘ Tw (x) − φ ∘ T ∘ Tw (y)󵄨󵄨󵄨 j=0

n−1

󵄨 󵄨 −(n−j) −(n−j) = ∑ 󵄨󵄨󵄨φ ∘ TT j (w) (x) − φ ∘ TT j (w) (y)󵄨󵄨󵄨 j=0

n−1

≤ ∑ υα (φ) dα (TT j (w) (x), TT j (w) (y)) −(n−j)

−(n−j)

j=0

n−1

≤ υα (φ) ∑ λ−α(n−j) dα (x, y) j=0

∞

≤ υα (φ) dα (x, y) ∑ λ−αk k=1

1 = α υ (φ) dα (x, y). λ −1 α The second part of the statement follows from the first one by summing over all branches of T n . Indeed, by Lemma 4.2.8 we know that Tu−n (B(z, ξ )) ∩ Tυ−n (B(z, ξ )) = 0 for all z ∈ X, n ∈ ℕ and u, υ ∈ T −n (z) with u ≠ υ. Moreover, by Lemma 4.2.9 we know that T −n (A) = ⋃u∈T −n (z) Tu−n (A) for all z ∈ X, all n ∈ ℕ and all A ⊆ B(z, ξ ). As d(x, y) < ξ ,

13.1 Hölder continuous potentials |

435

there is then a bijection between the sets T −n (x) and T −n (y) defined by b : T −n (x) 󳨀→ T −n (y) x 󳨃󳨀→ y = b(x) := Tx−n (y). Therefore, using the exponential form of the first part on each branch, we get ∑ x∈T −n (x)

eSn φ(x) = ≤

eSn φ(Tx

(x))

eSn φ(Tx

(y))

−n

∑ x∈T −n (x)

−n

∑ x∈T −n (x)

≤ Dα (φ)

∑ y∈T −n (y)

⋅ exp(Cα (φ) dα (x, y))

eSn φ(y) .

Under further assumptions, this local version has a global analogue. Corollary 13.1.2 (Global bounded ratio of exponentials of ergodic sums). If, additionally, T : X → X is topologically transitive or X is connected, then there exists a constant Dα (φ) ≥ 1 such that ∑ x∈T −n (x)

eSn φ(x) ≤ Dα (φ)

∑ y∈T −n (y)

eSn φ(y) ,

∀x, y ∈ X, ∀n ∈ ℕ.

Proof. When X is connected, the result follows from (13.6) in Lemma 13.1.1 by constructing a finite chain of balls of radius ξ /2 covering X. If T is topologically transitive, then due to the compactness of X there exists M ∈ ℕ such that for all x, y ∈ X there is 0 ≤ m ≤ M such that T m (B(x, ξ )) ∩ B(y, ξ ) ≠ 0. Indeed, select a point z ∈ X such that ω(z) = X. Then there are ξ -dense suborbits ′ ′ T k (z), . . . , T k (z) and T l (z), . . . , T l (z) with l > k ′ . Set M = l′ − k. There then exists k ≤ ′′ ′′ k ′′ ≤ k ′ and l ≤ l′′ ≤ l′ such that T k (z) ∈ B(x, ξ ) while T l (z) ∈ B(y, ξ ). With m = l′′ −k ′′ , ′′ ′′ we get T m (T k (z)) = T l (z) ∈ T m (B(x, ξ )) ∩ B(y, ξ ). Thus, there is a point y′ ∈ B(x, ξ ) ∩ T −m (B(y, ξ )). Using (13.6) in Lemma 13.1.1 twice (at the beginning with the pair (x, y′ ) and at the end with the pair (y′ , y)), we obtain ∑ x∈T −n (x)

eSn φ(x) ≤ Dα (φ)

eSn φ(y ) ′

∑ y′ ∈T −n (y′ )

= Dα (φ)

eSm+n φ(y ) e−Sm φ(T ′

∑

n

(y′ ))

y′ ∈T −n (y′ )

≤ Dα (φ)e−m inf φ

eSm+n φ(y ) ′

∑ y′ ∈T −n (y′ )

≤ Dα (φ)e−m inf φ

∑ y′ ∈T −(m+n) (T m (y′ ))

eSm φ(y ) eSn φ(T ′

m

(y′ ))

436 | 13 Gibbs states and transfer operators for open, distance expanding systems ≤ Dα (φ)em(sup φ−inf φ)

eSn φ(T

∑

m

(y′ ))

y′ ∈T −(m+n) (T m (y′ )) m

≤ Dα (φ)e2m‖φ‖∞ ⋅ (#T −1 ) M

2

eSn φ(y ) ′

∑ y′ ∈T −n (T m (y′ ))

≤ (#T −1 ) (Dα (φ)) e2M‖φ‖∞

∑ y∈T −n (y)

eSn φ(y) ,

where #T −1 := maxz∈X #T −1 (z) < ∞ by the expanding property of T and the compactness of X. Finally, note that M depends only on z, and thus is independent of x, y and n. The pressure of an open, distance expanding system can be expressed as a pointwise pressure. Proposition 13.1.3. Let T : X → X be an open, distance expanding system. For any potential φ ∈ C(X), we have P(T, φ) = max Px (T, φ) and x∈X

P(T, φ) = Px0 (T, φ) for some x0 ∈ X,

where Px (T, φ) := lim sup n→∞

1 log ∑ eSn φ(x) . n x∈T −n (x)

If φ is Hölder continuous and, moreover, T is topologically transitive or X is connected, then P(T, φ) = lim

n→∞

1 log ∑ eSn φ(x) = Px (T, φ), n x∈T −n (x)

∀x ∈ X.

Proof. Let x ∈ X and n ∈ ℕ. Given ζ > 0, recall that a set E ⊆ X is ζ -separated if d(e1 , e2 ) ≥ ζ for all e1 ≠ e2 ∈ E; it is (n, ζ )-separated if max0≤j 0 and P ∈ ℝ such that for every 0 < ξ ′ ≤ ξ there is a

13.2 Gibbs measures | 439

constant C(ξ ′ ) ≥ 1 such that C(ξ ′ )

−1

≤

μ(Tx−n (B(T n (x), ξ ′ ))) exp(Sn φ(x) − nP)

≤ C(ξ ′ ),

∀n ∈ ℕ, ∀x ∈ X.

Lemma 13.2.3. Let T : X → X be an open, distance expanding system. Suppose that μ is a Gibbs state for a potential φ and that a Borel probability measure ν is equivalent to μ, i. e., μ ≺≺ ν and ν ≺≺ μ. Suppose further that the Radon–Nikodym derivative of μ with respect to ν is essentially uniformly bounded away from 0 and ∞, i. e., 0 < ess inf(

dμ dμ ) ≤ ess sup( ) < ∞, dν dν

where ess inf(

dμ dμ ) := sup inf (x) dν ν(N)=0 x∈X\N dν

and ess sup(

dμ dμ ) := inf sup (x). ν(N)=0 x∈X\N dν dν

Then the measure ν is also a Gibbs state for the same potential φ with the same constant P. dμ −1

dν Proof. Since dμ = ( dν ) a. e., the Radon–Nikodym derivative of ν with respect to μ is also essentially uniformly bounded away from 0 and ∞. Therefore, there exists a constant D ≥ 1 such that

D−1 μ(B) ≤ ν(B) ≤ Dμ(B),

∀B ∈ ℬ(X),

where ℬ(X) is the Borel σ-algebra on X. It is then easy to see that ν, like μ, is a Gibbs state with the same constant P: the constant C for μ can be replaced by CD for ν. The following proposition states that the converse of Lemma 13.2.3 is also true, and identifies the constant P as P(T, φ), the topological pressure of φ. Proposition 13.2.4. Let T : X → X be an open, distance expanding system. Suppose that μ and ν are both Gibbs states for a potential φ ∈ C(X) and that (C, P) and (D, Q) are the couples of constants arising from Definition 13.2.1 for these respective measures. Then (a) μ and ν are equivalent, (b) P = Q = P(T, φ), and (c) the mutual Radon–Nikodym derivatives of μ and ν are essentially uniformly bounded away from 0 and ∞. Proof. Let ξ come from (13.2). Let E be a finite ξ -spanning set for X, i. e., a finite set such that X = ⋃e∈E B(e, ξ ). Observe that for every n ∈ ℕ, X = T −n (X) = ⋃ T −n (B(e, ξ )) = ⋃ e∈E

e∈E

⋃ x∈T −n (e)

Tx−n (B(e, ξ )).

(13.14)

440 | 13 Gibbs states and transfer operators for open, distance expanding systems We first claim that for all compact sets A ⊆ X, for all γ > 0 and for all n ≥ N(A, γ) large enough, μ(A) ≤ CD(#E)(ν(A) + γ)en(Q−P) .

(13.15)

To prove this claim, note that by the compactness of A and the outer regularity of μ and ν (see Definition A.1.23 and Theorem A.1.24), there exists ε > 0 such that μ(B(A, ε)) ≤ μ(A) + γ and ν(B(A, ε)) ≤ ν(A) + γ. Fix n ∈ ℕ so large that 2λ−n ξ < ε. For each e ∈ E, define the set Xn (e) by 󵄨 Xn (e) := {x ∈ T −n (e) 󵄨󵄨󵄨 A ∩ Tx−n (B(e, ξ )) ≠ 0}. By (13.14) and (13.2) (or, alternatively, (4.31)), we have A⊆ ⋃

⋃ Tx−n (B(e, ξ )) ⊆ B(A, 2λ−n ξ ) ⊆ B(A, ε).

e∈E x∈Xn (e)

(13.16)

For each e ∈ E the family {Tx−n (B(e, ξ ))}x∈T −n (e) consists of preimages under different inverse branches of T n at e. According to Lemma 4.2.8, these preimages are mutually disjoint. Therefore, μ(A) ≤ μ(⋃

⋃ Tx−n (B(e, ξ )))

e∈E x∈Xn (e)

∑ μ(Tx−n (B(T n (x), ξ )))

≤∑

e∈E x∈Xn (e)

≤C∑

∑ eSn φ(x)−nP

e∈E x∈Xn (e)

= Cen(Q−P) ∑

(13.17)

∑ eSn φ(x)−nQ

e∈E x∈Xn (e)

≤ CDe

n(Q−P)

∑

∑ ν(Tx−n (B(T n (x), ξ )))

e∈E x∈Xn (e)

= CDen(Q−P) ∑ ν( ⋃ Tx−n (B(e, ξ ))) e∈E

x∈Xn (e)

≤ CDen(Q−P) ∑ ν(B(A, ε)) e∈E

≤ CDe

n(Q−P)

(#E)(ν(A) + γ).

This proves the claim. If Q < P, letting A = X in the above inequality and letting n → ∞ would imply that μ(X) ≤ 0. This means that Q ≥ P. Reversing the roles of μ and ν and correspondingly

13.2 Gibbs measures | 441

those of P and Q gives P ≥ Q. So P = Q. This is part of (b). Thus, (13.15) reduces to μ(A) ≤ CD(#E)(ν(A) + γ). Letting γ → 0 results in μ(A) ≤ CD(#E) ⋅ ν(A). By the inner regularity of the measures μ and ν, it follows that μ(B) = sup μ(A) ≤ CD(#E) ⋅ sup ν(A) = CD(#E) ⋅ ν(B) A⊆B, A compact

A⊆B, A compact

for every Borel set B ⊆ X. Hence, μ ≺≺ ν. Exchanging the roles of μ and ν yields ν(B) ≤ CD(#E) ⋅ μ(B) for every Borel set B ⊆ X. Hence, ν ≺≺ μ. So μ and ν are equivalent. This proves (a). Moreover, since ∫ B

dμ dν = μ(B) ≤ CD(#E) ⋅ ν(B) = ∫ CD(#E) dν, dν

∀B ∈ ℬ(X),

B

dμ

dμ

we deduce that dν ≤ CD(#E) ν-a. e. on X. By a similar argument, dν ≥ (CD(#E))−1 dν ν-a. e. on X. Interchanging μ and ν yields the same lower and upper bounds on dμ dμ

dμ

dν dν (alternatively, recall that dμ = ( dν )−1 a. e.). So dν and dμ are essentially uniformly bounded away from 0 and ∞. This demonstrates (c). We now establish that P ≤ P(T, φ). Using (13.17) with A = X, we get

1 = μ(X) ≤ C ∑

e∈E

∑ x∈T −n (e)

≤ C(#E)

eSn φ(x)−nP

∑ x∈T −n (emax )

eSn φ(x)−nP

≤ C(#E) ∑ eSn φ(x)−nP , x∈En (δ)

where emax ∈ E is chosen so that the sum ∑x∈T −n (e) exp(Sn φ(x) − nP) is maximal among all e ∈ E and where En (δ) is a maximal (n, δ)-separated set containing T −n (emax ). Taking logarithms, dividing by n and rearranging, we obtain P≤

1 1 log(C(#E)) + log ∑ eSn φ(x) . n n x∈E (δ) n

Upon letting n → ∞ and δ → 0, we deduce from Theorem 11.2.1 that P ≤ P(T, φ).

442 | 13 Gibbs states and transfer operators for open, distance expanding systems It remains to show that P ≥ P(T, φ). In virtue of Proposition 13.1.3, let x0 ∈ X be such that Px0 (T, φ) = P(T, φ). It follows from the definition of a Gibbs state that ∑ x∈T −n (x0 )

eSn φ(x)−nP ≤ C

μ(Tx−n (B(x0 , ξ )))

∑ x∈T −n (x

0)

= Cμ(

Tx−n (B(x0 , ξ )))

⋃ x∈T −n (x

0)

= Cμ(T (B(x0 , ξ ))) −n

≤ C.

Therefore, enP ≥ C −1

∑ x∈T −n (x0 )

eSn φ(x) ,

and hence 1 1 P ≥ − log C + log ∑ eSn φ(x) . n n x∈T −n (x ) 0

Letting n → ∞, we conclude that P ≥ Px0 (T, φ) = P(T, φ). The following is a generalization of Definition 8.2.39. Definition 13.2.5. Let ℝX be the set of all real-valued functions defined on X, and let F ⊆ ℝX . A function φ ∈ ℝX is said to be cohomologous in F to a function ψ ∈ ℝX provided that there exists a function u ∈ F such that φ − ψ = u ∘ T − u. Any such function u is called a coboundary. Note that cohomology of functions in a subset F of ℝX is an equivalence relation on ℝX if F is a group with respect to pointwise addition. We now show that continuous functions that are cohomologous, up to a constant, in the group of bounded functions have (1) pressures that differ by that constant, and (2) the same Gibbs states. Lemma 13.2.6. Let T : X → X be an open, distance expanding system. If two potentials φ, ψ ∈ C(X) are such that φ − ψ is cohomologous to a constant c in the additive group of bounded functions on X, then P(T, φ) = P(T, ψ) + c and 𝒢φ = 𝒢ψ . Proof. Temporarily assume that φ is cohomologous to ψ in the additive group of bounded functions on X, i. e., assume that c = 0. Let u be a bounded coboundary

13.2 Gibbs measures | 443

such that φ − ψ = u ∘ T − u. Then for every n ∈ ℕ, Sn φ − Sn ψ = u ∘ T n − u. Given any family {En (ε)}n∈ℕ, ε>0 of maximal (n, ε)-separated sets, by Theorem 11.2.1 we then have P(T, φ) = lim lim sup ε→0

n→∞

1 log ∑ eSn φ(x) n x∈E (ε) n

n 1 = lim lim sup log ∑ eSn ψ(x) eu∘T (x)−u(x) ε→0 n→∞ n x∈E (ε) n

1 ≤ lim lim sup log ∑ eSn ψ(x) e2‖u‖∞ ε→0 n→∞ n x∈E (ε)

= P(T, ψ).

n

By symmetry (and replacing u by −u), we deduce that P(T, ψ) = P(T, φ). Suppose now that φ − ψ is cohomologous to a constant c ∈ ℝ in the additive group of bounded functions on X, and let u be a bounded coboundary so that φ − ψ − c = u ∘ T − u. This is equivalent to saying that φ is cohomologous to ψ + c via u. By the first part and Proposition 11.3.1, we get P(T, φ) = P(T, ψ + c) = P(T, ψ) + c. Moreover, Sn φ = Sn (ψ + c + u ∘ T − u)

= Sn ψ + Sn c + Sn u ∘ T − Sn u

= Sn ψ + nc + u ∘ T n − u. Hence,

Sn φ − nP(T, φ) = Sn ψ − nP(T, ψ) + u ∘ T n − u. Let ξ come from (13.2). Let μ ∈ 𝒢φ and C ≥ 1 be a Gibbs constant for μ, i. e. such that C −1 ≤

μ(Tx−n (B(T n (x), ξ )))

exp(Sn φ(x) − nP(T, φ))

≤ C,

∀n ∈ ℕ, ∀x ∈ X.

Then C −1 ≤

μ(Tx−n (B(T n (x), ξ )))

exp(Sn ψ(x) − nP(T, ψ) + u ∘ T n (x) − u(x))

≤ C,

∀n ∈ ℕ, ∀x ∈ X.

444 | 13 Gibbs states and transfer operators for open, distance expanding systems Consequently, (Ce2‖u‖∞ )

−1

≤

μ(Tx−n (B(T n (x), ξ )))

exp(Sn ψ(x) − nP(T, ψ))

≤ Ce2‖u‖∞ ,

∀n ∈ ℕ, ∀x ∈ X.

So μ ∈ 𝒢ψ , and hence 𝒢φ ⊆ 𝒢ψ . By symmetry, 𝒢φ ⊇ 𝒢ψ . We now compare classes of Gibbs states for various Hölder continuous potentials. Theorem 13.2.7. If T : X → X is a topologically transitive, open, distance expanding system and φ, ψ : X → ℝ are Hölder continuous potentials such that 𝒢φ ≠ 0 and 𝒢ψ ≠ 0, then the following statements are equivalent: (a) There exists a constant R1 ∈ ℝ such that Sn φ(x) − Sn ψ(x) = R1 n whenever n ∈ ℕ and x ∈ X is a periodic point of T of period n. (b) There exist constants R2 ∈ ℝ and C ≥ 0 such that 󵄨󵄨 󵄨 󵄨󵄨Sn φ(x) − Sn ψ(x) − R2 n󵄨󵄨󵄨 ≤ C,

∀n ∈ ℕ, ∀x ∈ X.

(c) φ − ψ is cohomologous to a constant R3 ∈ ℝ in the additive group of bounded functions on X. (d) φ − ψ is cohomologous to a constant R4 ∈ ℝ in the additive group of continuous functions on X. (e) φ − ψ is cohomologous to a constant R5 ∈ ℝ in the additive group of Hölder continuous functions on X. (f) 𝒢φ = 𝒢ψ . (g) 𝒢φ ∩ 𝒢ψ ≠ 0. In addition, if any of the above statements holds, then all the constants R1 –R5 are the same and equal to P(T, φ) − P(T, ψ). Proof. Of course (e)⇒(d)⇒(c). It is straightforward to see that (c)⇒(b). To see that (b)⇒(a), take a periodic point x of period n (by Corollary 4.3.10, every transitive, open, distance expanding system has a dense set of periodic points). Write the left-hand side of the relation in (b) with n replaced by kn for an arbitrary k ∈ ℕ. Let k → ∞ and conclude that (a) holds. It also then follows that R5 = R4 = R3 = R2 = R1 . The implication (c)⇒(f) follows immediately from Lemma 13.2.6. Let us now show that (f)⇒(a). Suppose that μ ∈ 𝒢φ = 𝒢ψ . Let ζ = [φ − P(T, φ)] − [ψ − P(T, ψ)] = [φ − ψ] − [P(T, φ) − P(T, ψ)]. As μ is a Gibbs state for both φ and ψ (cf. Definition 13.2.1), there exists a constant D ≥ 1 such that D−1 ≤ exp([Sn φ(x) − nP(T, φ)] − [Sn ψ(x) − nP(T, ψ)]) = eSn ζ (x) ≤ D

(13.18)

13.2 Gibbs measures | 445

for all n ∈ ℕ and all x ∈ X. In particular, if x is a periodic point of period n for T, then Sn ζ (x) = 0. Otherwise, we would have Skn ζ (x) = kSn ζ (x) for all k ∈ ℕ and hence (13.18) would not hold. Thus, item (a) holds with R1 = P(T, φ) − P(T, ψ). With this value for R1 , let us move on to showing that (a) implies (e). Keep ζ as above. Since T is transitive, let x ∈ X be such that ω(x) = X. Define u(T n (x)) := Sn ζ (x) for every n ≥ 0, where we define S0 ζ ≡ 0. This function u is well-defined on 𝒪+ (x). Indeed, if x is periodic then u is well-defined since SN ζ (x) = 0, where N is the prime period of x. If x is not periodic, then u is well-defined simply because T m (x) ≠ T n (x) whenever m ≠ n. Observe further that ζ (T n (x)) = Sn+1 ζ (x) − Sn ζ (x) = u(T n+1 (x)) − u(T n (x))

= u ∘ T(T n (x)) − u(T n (x)) = (u ∘ T − u)(T n (x))

for all n ≥ 0, i. e., ζ is cohomologous to 0 on 𝒪+ (x) with u as a coboundary. Equivalently, φ − ψ is cohomologous to P(T, φ) − P(T, ψ) on 𝒪+ (x) with u as a coboundary. If x is periodic, then 𝒪+ (x) = X and u is the sought coboundary. If x is not periodic, then 𝒪+ (x) ≠ X and we need to extend u continuously to 𝒪+ (x) = X. To do this (and more), given that ω(x) = X let m < n be such that d(T m (x), T n (x))
0 such that ℒφ (g)(x) = C

∑ y∈T −1 (x)

g(y)eφ(y) ,

∀x ∈ X, ∀g ∈ L1 (m).

(13.23)

13.5 Transfer operators | 453

The dual operator of ℒφ shall be denoted by ℒ∗φ : L1 (m)∗ → L1 (m)∗ and is defined in the usual manner by ℒφ (F)(g) := F(ℒφ (g))

(13.24)

∗

for every bounded linear functional F : L1 (m) → ℝ and g ∈ L1 (m). In particular, we have ℒ∗φ (m)(g) = m(ℒφ (g)), where m denotes the positive bounded linear functional m(g) = ∫ g dm.

(13.25)

X

The following proposition reveals that any positive fixed point of the dual ℒ∗φ of a transfer-type operator ℒφ is a quasi-invariant measure. Proposition 13.5.2. Let T : X → X be an open, distance expanding system. If a transfertype operator ℒφ : L1 (m) → L1 (m) is such that ℒ∗φ (m) = m, then m is quasi-T-invariant. Moreover, m ∘ T ≺≺ m. Furthermore, the Jacobian of m is φ(z) J−1 m (z) = Ce

and

Jm (z) = C −1 e−φ(z)

for m-a. e. z ∈ X,

when ℒφ (g)(x) = C ∑y∈T −1 (x) g(y)eφ(y) . Note: We slightly abuse notation when writing m ∘ T ≺≺ m. By this, we mean that m(T(A)) = 0 whenever A is a Borel subset of X such that m(A) = 0. (Observe that T(A) is a Borel set for all Borel sets A since the map T is open.) This is equivalent to 󵄨 󵄨 m ∘ Tx 󵄨󵄨󵄨T −1 (B(T(x),ξ )) ≺≺ m󵄨󵄨󵄨T −1 (B(T(x),ξ )) for all x ∈ X. x

x

Proof. Let w ∈ X and A a Borel subset of B(w, ξ ). Then m(T −1 (A)) = m(1T −1 (A) ) = ℒ∗φ (m)(1T −1 (A) )

= m(ℒφ (1T −1 (A) )) = ∫ ℒφ (1T −1 (A) )(z) dm(z) X

= ∫C

∑

X

y∈T −1 (z)

= ∫C

∑

X

y∈T −1 (z)

= ∫ 1A (z) ⋅ C X

φ(y)

dm(z)

φ(y)

dm(z)

1T −1 (A) (y)e 1A (T(y))e

∑ y∈T −1 (z)

eφ(y) dm(z)

= ∫ ℒφ (1) dm. A

So, if m(A) = 0 then m ∘ T −1 (A) = 0, and it follows that m ∘ T −1 ≺≺ m.

454 | 13 Gibbs states and transfer operators for open, distance expanding systems Furthermore, for any Borel set A we have m(A) = m(1A ) = ℒ∗φ (m)(1A ) = ∫ ℒφ (1A )(z) dm(z) X

= ∫C X

∑ y∈T −1 (z)

≥ ∫ C T(A)

≥

1A (y)e

∑ y∈T −1 (z)

φ(y)

1A (y)e

dm(z)

φ(y)

dm(z)

1 ∫ C ∑ eφ(y) dm(z) (#T −1 )e2‖φ‖∞ y∈T −1 (z) T(A)

=

1 ∫ ℒφ (1) dm, (#T −1 )e2‖φ‖∞ T(A)

where #T −1 := maxx∈X #T −1 (x) < ∞ by the expanding property of T and the compactness of X. Since ℒφ is a positive operator, we deduce that m(T(A)) = 0 whenever m(A) = 0. Take x ∈ X and a Borel set A ⊆ B(T(x), ξ ). We identify the Jacobians J−1 m (z) := dm∘Tx−1 |B(T(x),ξ ) (T(z)) dm|B(T(x),ξ )

and Jm (z) :=

dm∘Tx |T −1 (B(T(x),ξ )) x

dm|T −1 (B(T(x),ξ )) x

−1 (z) = (J−1 m (z)) that are induced by the

homeomorphism Tx : Tx−1 (B(T(x), ξ )) → B(T(x), ξ ). Observe that m ∘ Tx−1 (A) = m(1Tx−1 (A) ) = ℒ∗φ (m)(1Tx−1 (A) ) = m(ℒφ (1Tx−1 (A) )) = ∫C X

∑ y∈T −1 (z)

= ∫ Ceφ(Tx

−1

1Tx−1 (A) (y)e

(z))

φ(y)

dm(z)

dm(z).

A

So

dm∘Tx−1 (z) dm

= Ceφ(Tx

−1

(z))

for m-a. e. z ∈ B(T(x), ξ ). Thus, for m-a. e. w ∈ Tx−1 (B(T(x), ξ )), φ(Tx J−1 m (w) = Ce

−1

(T(w)))

= Ceφ(w) .

As {Tx−1 (B(T(x), ξ ))}x∈X is an open cover of the compact space X, one can extract a finite φ(w) subcover and conclude that J−1 m-a. e. on X. According to Lemma 13.3.3, m (w) = Ce −1 −φ(w) Jm (w) = C e m-a. e. on X.

13.6 Nonnecessarily-invariant Gibbs states | 455

13.6 Nonnecessarily-invariant Gibbs states In this section, we will establish the existence of at least one Gibbs state for every topologically transitive, open, distance expanding system under a Hölder continuous potential. This Gibbs state will turn out to be ergodic, though perhaps not invariant. In general, Gibbs states are not unique. In the next section, we will pursue the question of existence and uniqueness of an invariant Gibbs state. We now outline the basic structure of the argument to come in this section. (a) Start with a restriction of a transfer-type operator ℒφ : C(X) → C(X), where ℒφ (g)(x) :=

∑ y∈T −1 (x)

g(y)eφ(y)

(13.26)

for all g ∈ C(X) and x ∈ X. Here, φ is a continuous potential and there is not yet any involvement of a measure. (b) Define the dual operator ℒ∗φ : C(X)∗ → C(X)∗ by setting ℒφ (F)(g) := F(ℒφ (g)) ∗

(13.27)

for every F ∈ C(X)∗ and g ∈ C(X). (c) Under the additional assumption that T is surjective, find an eigenfunctional mφ ∈ C(X)∗ of the dual ℒ∗φ : C(X)∗ → C(X)∗ such that mφ is positive and mφ (1) = 1. Let Λφ ∈ ℝ be the associated eigenvalue, i. e., ℒφ (mφ ) = Λφ mφ . ∗

(13.28)

(d) In view of Riesz representation theorem, mφ ∈ M(X). Thus, it makes sense to examine the extension ℒφ : L1 (mφ ) → L1 (mφ ) and its dual operator ℒ∗φ : L1 (mφ )∗ → L1 (mφ )∗ . Also, mφ ∈ L1 (mφ )∗ . Since mφ is regular and X is compact, the subspace C(X) is dense in L1 (mφ ). Demonstrate that ℒ∗φ (mφ ) = Λφ mφ for the extension, too. ̂φ := Λ−1 ℒφ , and hence ℒ ̂∗ = (e) Next, normalize the operators ℒφ and ℒ∗φ by setting ℒ φ φ −1 ∗ Λφ ℒφ . ̂∗ (mφ ) = mφ , i. e., mφ is a fixed point of ℒ ̂∗ . In light of Proposition 13.5.2, (f) Then ℒ φ φ −1 deduce that mφ ∘ T ≺≺ mφ and mφ ∘ T ≺≺ mφ , and that the Jacobian of mφ is −1 φ(z) ̂φ = ℒm . J−1 . By Proposition 13.4.2, it further follows that ℒ mφ (z) = Λφ e φ ̂φ , and their duals. Infer formulas for the mea(g) Investigate the iterates of ℒφ and ℒ

sure of backward and forward images of small sets, and establish that mφ is a (Λφ e−φ )-conformal measure. See Proposition 13.6.9, Corollaries 13.6.10–13.6.11 and Definition 13.6.12. (h) Show that mφ is a Gibbs measure. It is at that point that the supplementary assumptions that φ is Hölder continuous and that T is transitive are needed. It further turns out that mφ is ergodic. (i) Finally, if T is topologically exact then the Gibbs state mφ is metrically exact.

456 | 13 Gibbs states and transfer operators for open, distance expanding systems Recall that throughout this chapter, T : X → X is an open, distance expanding system on a compact metric space (X, d) and δ, λ and ξ are constants arising from Definition 4.1.1 and relations (4.29), (4.30) and (4.31); alternatively, see (13.1) and (13.2).

13.6.1 Existence of eigenmeasures for the dual of the transfer operator Let us begin with the restriction of the transfer-type operator ℒφ : C(X) → C(X) as defined in (13.26). Lemma 13.6.1. For every open, distance expanding system T : X → X and every potential φ ∈ C(X), the positive linear operator ℒφ : C(X) → C(X) is well defined and bounded. Proof. First, let us show that ℒφ is well defined, i. e., that it maps continuous functions to continuous functions. Let g ∈ C(X) and ε > 0. By the expanding property of T and the compactness of X, we know that #T −1 := maxx∈X #T −1 (x) < ∞. As the functions g and eφ are uniformly continuous on X, there exists ζ > 0 such that if d(z, w) < ζ then 󵄨󵄨 󵄨 󵄨󵄨g(z) − g(w)󵄨󵄨󵄨
0 such that ℒ∗φ (mφ ) = Λφ mφ . Proof. For each m ∈ M(X), set H(m) :=

ℒ∗φ (m) ℒ∗φ (m)(1)

.

Because T is surjective, ℒ∗φ (m)(1) := m(ℒφ (1)) ≥ einf φ > 0. Then H : M(X) → C(X)∗ is a positive linear operator with H(m)(1) = 1. So, by the Riesz representation theorem (Theorem A.1.54), H(m) ∈ M(X). We claim that H : M(X) → M(X) is a continuous operator in the weak∗ topology on M(X). As M(X) is metrizable in the weak∗ topol∗ ogy, it is enough to show sequential continuity, i. e., that if (mn )∞ n=1 weak converges ∗ ∞ to m, then (H(mn ))n=1 weak converges to H(m). This amounts to establishing that ℒ∗φ (mn )(g) → ℒ∗φ (m)(g) for all g ∈ C(X), which holds as ℒ∗φ (mn )(g) := mn (ℒφ g) → m(ℒφ g) =: ℒ∗φ (m)(g). Since M(X) is a compact convex subset of the locally convex topological vector space C(X)∗ , by virtue of the Schauder–Tychonov fixed-point theorem (cf. Theorem V.10.5 in Dunford and Schwartz [26]), the continuous map H : M(X) → M(X) has a fixed point mφ ∈ M(X). Thus, ℒ∗φ (mφ ) ℒ∗φ (mφ )(1)

= mφ .

Setting Λφ = ℒ∗φ (mφ )(1) > 0, we see that ℒ∗φ (mφ ) = Λφ mφ . As pointed out earlier, since mφ ∈ M(X) it makes sense to examine the extension

ℒφ : L1 (mφ ) → L1 (mφ ) and its dual operator ℒ∗φ : L1 (mφ )∗ → L1 (mφ )∗ . First, let us

make sure that this extension exists.

Theorem 13.6.3. For every surjective, open, distance expanding system T : X → X and every potential φ ∈ C(X), the positive linear operator ℒφ : C(X) → C(X) is uniformly continuous and extends continuously to a positive linear operator ℒφ : L1 (mφ ) → L1 (mφ )

458 | 13 Gibbs states and transfer operators for open, distance expanding systems defined mφ -a. e. by formula (13.26) for every g ∈ L1 (mφ ), where mφ is an eigenmeasure for the dual operator ℒ∗φ : C(X)∗ → C(X)∗ . Proof. According to Theorem 13.6.2, the positive linear operator ℒφ : C(X) → C(X) satisfies mφ (ℒφ (f )) = Λφ mφ (f ) for all f ∈ C(X). Let g ∈ C(X). Using (13.26), 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩ℒφ (g)󵄩󵄩󵄩1 ≤ 󵄩󵄩󵄩ℒφ (|g|)󵄩󵄩󵄩1 = mφ (ℒφ (|g|)) = Λφ mφ (|g|) = Λφ ‖g‖1 . It immediately follows that ℒφ : C(X) → C(X) is uniformly continuous. Since C(X) is dense in L1 (mφ ), the operator ℒφ : C(X) → C(X) extends continuously to a positive linear operator ℒφ : L1 (mφ ) → L1 (mφ ). Let

󵄨󵄨 󵄨 Z := {f ∈ L1 (mφ ) 󵄨󵄨󵄨 ℒφ (f )(x) = ∑ f (y)eφ(y) for mφ -a. e. x ∈ X}. 󵄨󵄨 y∈T −1 (x) We will show that Z = L1 (mφ ) through a series of claims. Recall that ℝX denotes the set of all real-valued functions defined on X. Let β be an ordinal number such that β > |ℝX |. Let ℬ0 := C(X). Let γ be an ordinal such that γ < β and suppose that ℬα has been defined for all ordinals α < γ. Set 󵄨 󵄨

X 󵄨 ℬγ := {f ∈ ℝ 󵄨󵄨󵄨 f = lim fn , where ∀n ∈ ℕ, fn ∈ ℬαn for some αn < γ}. n→∞

It is obvious that ℬα ⊆ ℬα′ whenever α ≤ α′ . Claim 1. 󵄨 ⋃ ℬγ = {f ∈ ℝX 󵄨󵄨󵄨 f Borel function}. γ 0. Then ∀q ≥ 0 ∃cq > 0 ∀ε > 0 ∃kε ∈ ℕ ∀k ≥ kε ∃zk ∈ E such that mφ (T q (B(zk , ξ ) \ T k (A))) ≤ cq

ε . mφ (A)

Proof. Let q ≥ 0 and ε > 0. Recall that every Borel probability measure on a compact metrizable space X is regular. By the inner regularity of mφ , it suffices to prove the lemma for all compact sets A. As in the proof of Proposition 13.2.4, for each k ∈ ℕ put Xk := ⋃ Xk (e), e∈E

where E is a finite ξ -spanning set and 󵄨 Xk (e) := {x ∈ T −k (e) 󵄨󵄨󵄨 A ∩ Tx−k (B(T k (x), ξ )) ≠ 0}. Like in (13.16), A ⊆ ⋃ Tx−k (B(T k (x), ξ )) ⊆ B(A, 2λ−k ξ ). x∈Xk

(13.38)

On one hand, it ensues that ∑ mφ (Tx−k (B(T k (x), ξ ))) ≥ mφ (A).

x∈Xk

(13.39)

On the other hand, by the outer regularity of mφ and the compactness of A, there exists kε ∈ ℕ so large that for all k ≥ kε , mφ (B(A, 2λ−k ξ ) \ A) ≤ ε.

(13.40)

For each e ∈ E, the family {Tx−k (B(e, ξ ))}x∈T −k (e) consists of preimages under different inverse branches of T k at e. According to Lemma 4.2.8, these preimages are mutually disjoint. This and relations (13.38) and (13.40) give ∑ mφ (Tx−k (B(T k (x), ξ )) \ A) = ∑

x∈Xk

∑ mφ (Tx−k (B(T k (x), ξ )) \ A)

e∈E x∈Xk (e)

= ∑ mφ ( ⋃ Tx−k (B(T k (x), ξ )) \ A) e∈E

x∈Xk (e)

≤ ∑ mφ (B(A, 2λ−k ξ ) \ A) e∈E

≤ #E ⋅ ε.

(13.41)

13.6 Nonnecessarily-invariant Gibbs states | 469

From (13.39) and (13.41), it follows that ∑x∈Xk mφ (Tx−k (B(T k (x), ξ )) \ A)

≤

∑x∈Xk mφ (Tx−k (B(T k (x), ξ )))

#E ⋅ ε . mφ (A)

(13.42)

We claim that there exists y ∈ Xk such that mφ (Ty−k (B(T k (y), ξ )) \ A) mφ (Ty−k (B(T k (y), ξ )))

Otherwise, i. e., if this ratio was greater than k

≤

#E⋅ε mφ (A)

#E ⋅ ε . mφ (A)

(13.43)

for every y ∈ Xk , summing over

all y would contradict (13.42). Set zk := T (y) ∈ E. Using Corollary 13.6.10, Proposition 13.6.16 and Lemma 13.1.1, we get mφ (B(zk , ξ ) \ T k (A)) mφ (B(zk , ξ ))

≤

mφ (T k (Ty−k (B(zk , ξ )) \ A))

mφ (B(zk , ξ )) 1 = ∫ mφ (B(zk , ξ ))

eP(T,φ)k−Sk φ dmφ

Ty−k (B(zk ,ξ ))\A

≤

≤

Dα (φ)eP(T,φ)k−Sk φ(y) mφ (Ty−k (B(zk , ξ )) \ A) mφ (T k (Ty−k (B(zk , ξ ))))

Dα (φ)eP(T,φ)k−Sk φ(y) mφ (Ty−k (B(zk , ξ )) \ A) Dα (φ)−1 eP(T,φ)k−Sk φ(y) mφ (Ty−k (B(zk , ξ ))) 2

≤

(Dα (φ)) ⋅ #E ⋅ ε . mφ (A)

(13.44)

Let 0 < η ≤ ξ be a Lebesgue number for the open cover {Tx−q (B(T q (x), ξ ))}x∈X of the compact set X. Let 𝒜 = {A1 , A2 , . . . , An } be a finite Borel partition of X with diam(𝒜) < η. Using Corollary 13.6.10 and Proposition 13.6.16 once again, for any Borel subset Y of X we obtain that n

mφ (T q (Y)) = mφ (⋃ T q (Y ∩ Ai )) n

i=1

≤ ∑ mφ (T q (Y ∩ Ai )) i=1 n

= ∑ ∫ eP(T,φ)q−Sq φ dmφ i=1 Y∩A

i

n

≤ eP(T,φ)q+‖φ‖∞ q ∑ mφ (Y ∩ Ai ) = eq(P(T,φ)+‖φ‖∞ ) mφ (Y). i=1

(13.45)

470 | 13 Gibbs states and transfer operators for open, distance expanding systems From (13.44) and (13.45), we conclude that mφ (T q (B(zk , ξ ) \ T k (A))) ≤ eq(P(T,φ)+‖φ‖∞ ) mφ (B(zk , ξ ) \ T k (A)) ε 2 ≤ (Dα (φ)) ⋅ #E ⋅ eq(P(T,φ)+‖φ‖∞ ) . mφ (A) Setting cq = (Dα (φ))2 ⋅ #E ⋅ eq(P(T,φ)+‖φ‖∞ ) , we are done. Theorem 13.6.18. If T : X → X is a transitive, open, distance expanding system and φ : X → ℝ is a Hölder continuous potential, then mφ is ergodic. Proof. Let A ∈ ℬ(X) be such that T −1 (A) = A. Then T n (A) ⊆ A for all n ∈ ℕ. Suppose further that mφ (A) > 0. Let E be a finite ξ -spanning set. By Proposition 13.6.14, there exists Q ≥ 0 such that Q

⋃ T q (B(z, ξ )) = X,

q=0

∀z ∈ E.

Let ε > 0. Take c0 , . . . , cQ , take k big enough, and zk ∈ E satisfying Lemma 13.6.17. Then Q

mφ (X \ A) ≤ mφ (X \ ⋃ T q+k (A)) q=0

Q

Q

q=0

q=0

= mφ ( ⋃ T q (B(zk , ξ )) \ ⋃ T q+k (A)) Q

≤ mφ ( ⋃ T q (B(zk , ξ )) \ T q+k (A)) Q

q=0

≤ ∑ mφ (T q (B(zk , ξ ) \ T k (A))) q=0

≤

Q ε ∑ cq . mφ (A) q=0

Since this is true for an arbitrary ε > 0, we deduce that mφ (X \ A) = 0. Hence, mφ is ergodic. 13.6.5 Metric exactness of the eigenmeasures for topologically exact systems Theorem 13.6.19. If T : X → X is a topologically exact, open, distance expanding system and φ : X → ℝ is a Hölder continuous potential, then mφ is metrically exact. Proof. Let A ∈ ℬ(X) be such that mφ (A) > 0. Let E be a finite ξ -spanning set. The topological exactness of T guarantees that there exists some q ≥ 0 such that T q (B(z, ξ )) = X

13.7 Invariant Gibbs states | 471

for all z ∈ E. Let ε > 0. Take cq , kε , and for all k ≥ kε pick zk ∈ E so that Lemma 13.6.17 is satisfied. Then mφ (X \ T q+k (A)) = mφ (T q (B(zk , ξ )) \ T q+k (A)) ≤ mφ (T q (B(zk , ξ ) \ T k (A))) ≤ cq

ε . mφ (A)

Thus, for all k ≥ kε , mφ (T q+k (A)) ≥ 1 − cq

ε . mφ (A)

So lim inf mφ (T n (A)) ≥ 1 − cq n→∞

ε . mφ (A)

Since this holds for an arbitrary ε > 0, we conclude that lim mφ (T n (A)) = 1,

n→∞

i. e., mφ is metrically exact (cf. Definition 8.4.4 and Proposition 8.4.5).

13.7 Invariant Gibbs states In this section, we will show that under the assumptions that T is transitive and φ is Hölder continuous, there exists a unique invariant Gibbs state and that this state is ergodic. Recall that invariant Gibbs states are equilibrium states according to Proposition 13.2.9. We will prove conversely that every equilibrium state is an invariant Gibbs state. More precisely, we will: ̂φ : L1 (mφ ) → L1 (mφ ) such that ϱφ ≥ 0 and ∫ ϱφ dmφ = 1. (a) Find a fixed point ϱφ of ℒ X ̂ Thus, ℒmφ (ϱφ ) = ℒφ (ϱφ ) = ϱφ and Theorem 13.4.1 affirms that the measure μφ := ϱφ mφ is T-invariant.

(b) Further show that ϱφ , which is the Radon–Nikodym derivative

dμφ , dmφ

is a Hölder

continuous function uniformly bounded away from 0 and ∞. It follows from Lemma 13.2.3 that μφ is a Gibbs measure, like mφ . It is also ergodic, like mφ . (c) Prove the uniqueness of an invariant Gibbs state. (d) Demonstrate that a measure is an invariant Gibbs state if and only if it is an equilibrium state. 13.7.1 Almost periodicity of normalized transfer operators We first examine a few more properties of the iterates of the transfer operator ℒφ and ̂φ . its normalized counterpart ℒ

472 | 13 Gibbs states and transfer operators for open, distance expanding systems Lemma 13.7.1. If T : X → X is a topologically transitive, open, distance expanding system and φ ∈ Hα (X), then for all n ∈ ℕ and all x, y ∈ X we have −1

(Dα (φ))

≤

ℒnφ (1)(y)

ℒnφ (1)(x)

=

̂n (1)(y) ℒ φ

̂n (1)(x) ℒ φ

≤ Dα (φ).

Proof. This follows from Proposition 13.6.7 and Corollary 13.1.2. ̂n (1), n ∈ ℕ, are uniformly bounded It ensues from this lemma that the functions ℒ φ away from 0 and ∞. Proposition 13.7.2. If T : X → X is a transitive, open, distance expanding system and φ ∈ Hα (X), then −1

(Dα (φ))

̂n (1) ≤ sup ℒ ̂n (1) ≤ Dα (φ), ≤ inf ℒ φ φ

∀n ∈ ℕ.

Proof. Fix y ∈ X. Using Lemma 13.7.1, Corollary 13.6.6 as well as (13.21) with m = mφ , we obtain that n

n

n

̂ (1)(y) ≤ Dα (φ) ∫ ℒ ̂ (1)(x) dmφ (x) = Dα (φ) ∫ ℒ (1)(x) dmφ (x) ℒ φ φ mφ X

X

= Dα (φ) ∫ 1(x) dmφ (x) = Dα (φ). X

The lower bound results from a similar argument. In particular, this means that the supremum norms of the iterates of the normal̂φ : C(X) → C(X) are uniformly bounded from above (by ized restricted operator ℒ Dα (φ)). Let us recall the concept of almost periodicity of an operator. Definition 13.7.3. A bounded linear operator L : E → E on a Banach space E is said to be almost periodic if the forward orbit of every point x ∈ E under L, i. e., the set {Ln (x) : n ≥ 0}, is relatively compact in E. Theorem 13.7.4. If T : X → X is a transitive, open, distance expanding system and ̂n (g)}∞ φ : X → ℝ is a Hölder continuous potential, then for every g ∈ C(X) the family {ℒ φ n=0 ̂φ : C(X) → C(X) is equicontinuous and pointwise bounded in C(X). Hence the operator ℒ is almost periodic. ̃ = φ − P(T, φ). Proof. Let g ∈ C(X) and n ≥ 0. Fix x, y ∈ X with d(x, y) < ξ . Let φ ̃ = Sn φ − nP(T, φ) and one can set Cα (φ ̃) = Cα (φ) and Dα (φ ̃) = Dα (φ) in Then Sn φ Lemma 13.1.1. Observe also that ̂φ = e ℒ

−P(T,φ)

ℒφ = ℒφ̃ ,

̂n = e−nP(T,φ) ℒn = ℒñ . and hence ℒ φ φ φ

13.7 Invariant Gibbs states | 473

Using Proposition 13.6.7, the second part of the proof of Lemma 13.1.1, that lemma itself, the mean value theorem, and Proposition 13.7.2, we deduce that 󵄨󵄨 ̂n ̂n (g)(x)󵄨󵄨󵄨 󵄨󵄨ℒφ (g)(y) − ℒ φ 󵄨 󵄨 󵄨󵄨 n n = 󵄨󵄨ℒφ̃ (g)(y) − ℒφ̃ (g)(x)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ̃ −n ̃ −n 󵄨 󵄨 = 󵄨󵄨󵄨 ∑ eSn φ(Tz (y)) g(Tz−n (y)) − ∑ eSn φ(Tw (x)) g(Tw−n (x))󵄨󵄨󵄨 󵄨󵄨 −n 󵄨󵄨 −n z∈T (y) w∈T (x) 󵄨󵄨 󵄨󵄨 ̃ −n ̃ −n 󵄨 󵄨 = 󵄨󵄨󵄨 ∑ [eSn φ(Tw (y)) g(Tw−n (y)) − eSn φ(Tw (x)) g(Tw−n (x))]󵄨󵄨󵄨 󵄨󵄨 −n 󵄨󵄨 w∈T (x) ≤

w∈T −n (x)

+

eSn φ(Tw ̃

∑

−n

(y)) 󵄨󵄨

󵄨 −n −n 󵄨󵄨g(Tw (y)) − g(Tw (x))󵄨󵄨󵄨

󵄨󵄨 ̃ −n 󵄨󵄨 󵄨󵄨󵄨 S φ̃ (T −n (y)) −n − eSn φ(Tw (x)) 󵄨󵄨󵄨 󵄨󵄨g(Tw (x))󵄨󵄨󵄨󵄨󵄨󵄨e n w 󵄨 󵄨

∑ w∈T −n (x)

≤

w∈T −n (y)

+

eSn φ(Tw ̃

∑

−n

(y))

󵄨 󵄨 max{󵄨󵄨󵄨g(Tz−n (y)) − g(Tz−n (x))󵄨󵄨󵄨 : z ∈ T −n (x)}

̃ (T −n (x)) 󵄨󵄨 󵄨󵄨 󵄨 󵄨 −n S φ −n −n 󵄨󵄨g(Tw (x))󵄨󵄨󵄨Dα (φ)e n w 󵄨󵄨Sn φ(Tw (y)) − Sn φ(Tw (x))󵄨󵄨󵄨

∑ w∈T −n (x)

̂n (1)(y) max{󵄨󵄨󵄨g(y′ ) − g(x′ )󵄨󵄨󵄨 : d(y′ , x′ ) ≤ λ−n d(y, x)} ≤ℒ φ 󵄨 󵄨 ̂n (1)(x)Cα (φ) dα (x, y) + ‖g‖∞ Dα (φ)ℒ φ ≤ Dα (φ) = Dα (φ)

max

󵄨󵄨 ′ α ′ 󵄨 󵄨󵄨g(y ) − g(x )󵄨󵄨󵄨 + ‖g‖∞ Dα (φ)Dα (φ)Cα (φ) d (x, y)

max

󵄨󵄨 ′ ′ 󵄨 Dα (φ)‖g‖∞ dα (x, y), 󵄨󵄨g(y ) − g(x )󵄨󵄨󵄨 + ̃

d(y′ ,x ′ )≤λ−n d(y,x)

(13.46)

d(y′ ,x ′ )≤λ−n d(y,x)

where (13.47)

̃ Dα (φ) := Cα (φ)Dα (φ)Dα (φ).

̂n (g)}∞ is equicontinuAs g is uniformly continuous, we conclude that the family {ℒ φ n=0 ous. By Proposition 13.7.2, this family is also uniformly bounded since 󵄩󵄩 ̂n 󵄩󵄩 󵄩 ̂n 󵄩󵄩 󵄩󵄩 ̂n 󵄩󵄩 󵄩󵄩ℒφ (g)󵄩󵄩∞ ≤ 󵄩󵄩󵄩ℒ φ (‖g‖∞ ⋅ 1)󵄩 󵄩∞ = ‖g‖∞ 󵄩󵄩ℒφ (1)󵄩󵄩∞ ≤ Dα (φ)‖g‖∞ ,

∀n ≥ 0.

The assumptions of the Arzelà–Ascoli theorem (Theorem A.2.3) being satisfied, we ̂n (g)}∞ is relatively compact. Consequently, the operator conclude that the family {ℒ φ n=0 ̂φ : C(X) → C(X) is almost periodic. ℒ

474 | 13 Gibbs states and transfer operators for open, distance expanding systems 13.7.2 Existence, uniqueness and ergodicity of invariant Gibbs states for transitive systems Lemma 13.7.5. If T : X → X is a topologically transitive, open, distance expanding system and φ ∈ Hα (X), then there exists ϱφ ∈ Hα (X) with the following properties: ̂φ (ϱφ ) = ϱφ . (a) ℒ (b) (Dα (φ)) ≤ ϱφ ≤ Dα (φ). (c) ∫X ϱφ dmφ = 1. (d) υα (ϱφ ) ≤ max{̃ Dα (φ), 2ξ −α Dα (φ)}, −1

where ̃ Dα (φ) is defined in (13.47) and Dα (φ) in Corollary 13.1.2. Proof. It is a general fact, proved via the triangle inequality, that if a family {gn }∞ n=0 ∞ is equicontinuous, then so are its Cesàro averages { n1 ∑n−1 j=0 gj }n=1 . Per Theorem 13.7.4, ̂n (g)}∞ is equicontinuous for every g ∈ C(X). In particular, {ℒ ̂n (1)}∞ is the family {ℒ φ n=0 φ n=0 ̂j (1)}∞ is equicontinuous. equicontinuous. Consequently, the Cesàro family { 1 ∑n−1 ℒ n

j=0

φ

n=1

By Proposition 13.7.2, we further know that this Cesàro family is uniformly bounded from below by (Dα (φ))−1 and from above by Dα (φ). By the Arzelà–Ascoli theorem (Theorem A.2.3), there is then a strictly increasing sequence (nk )∞ k=1 such that the se∞ nk −1 ̂j ℒφ (1))k=1 converges uniformly to a continuous function, denoted by quence ( n1 ∑j=0 k

ϱφ , which is bounded below by (Dα (φ))−1 and above by Dα (φ). This establishes (b). ̂φ : C(X) → C(X) is a continuous linear To see that ϱφ satisfies part (a), since ℒ operator we obtain that n −1

n −1

1 k ̂j 1 k ̂j+1 ∑ ℒφ (1)) = lim ∑ ℒφ (1) k→∞ nk k→∞ nk j=0 j=0

̂φ (ϱφ ) = ℒ ̂φ ( lim ℒ = lim [ k→∞

n −1

1 ̂nk 1 k ̂j (1) − 1)] ∑ ℒ (1) + (ℒ nk j=0 φ nk φ n −1

1 k ̂j ∑ ℒφ (1) k→∞ nk j=0

= lim

using Proposition 13.7.2

= ϱφ .

̂j )∗ (mφ ) = mφ for each We now show that ϱφ fulfills (c). By Corollary 13.6.5, we have (ℒ φ ̂j (1)) = mφ (1) = 1. By Lebesgue’s dominated convergence j ∈ ℕ and in particular mφ (ℒ φ

theorem (Theorem A.1.38), we get

X

X

n −1

n −1

1 k ̂j 1 k ̂j (1) dmφ ∑ ℒφ (1) dmφ = lim ∑ ∫ℒ φ k→∞ nk k→∞ nk j=0 j=0

∫ ϱφ dmφ = ∫ lim

= lim

k→∞

nk −1

X

1 ̂j (1)) = 1. ∑ m (ℒ nk j=0 φ φ

13.7 Invariant Gibbs states | 475

Finally, we establish that ϱφ ∈ Hα (X) whenever φ ∈ Hα (X). Fix x, y ∈ X. Suppose first that d(x, y) < ξ . From (13.46) in the proof of Theorem 13.7.4, we have 󵄨󵄨 ̂n α ̂n (1)(y)󵄨󵄨󵄨 ≤ ̃ 󵄨󵄨ℒφ (1)(x) − ℒ φ 󵄨 Dα (φ) d (x, y), This also holds for the Cesàro averages

1 nk

∀n ∈ ℕ.

nk −1 ̂j ℒφ (1), k ∈ ℕ: ∑j=0

󵄨󵄨 󵄨󵄨 nk −1 n −1 󵄨󵄨 󵄨󵄨 1 1 k ̂j α 󵄨 ̃ 󵄨󵄨 ̂j 󵄨󵄨 n ∑ ℒφ (1)(x) − n ∑ ℒφ (1)(y)󵄨󵄨󵄨 ≤ Dα (φ) d (x, y), 󵄨󵄨 󵄨󵄨 k j=0 k j=0

∀k ∈ ℕ.

Passing to the limit k → ∞, we deduce that if d(x, y) < ξ then 󵄨󵄨 󵄨 Dα (φ) dα (x, y). 󵄨󵄨ϱφ (x) − ϱφ (y)󵄨󵄨󵄨 ≤ ̃

(13.48)

2‖ϱφ ‖∞ α 󵄨󵄨 󵄨 d (x, y). 󵄨󵄨ϱφ (x) − ϱφ (y)󵄨󵄨󵄨 ≤ 2‖ϱφ ‖∞ ≤ ξα

(13.49)

If d(x, y) ≥ ξ , then

By (13.48) and (13.49), we conclude that ϱφ ∈ Hα (X). Note that part (d) is a consequence of (13.48), (13.49) and part (b). Remark 13.7.6. Carefully tracing the constants Dα (φ) and ̃ Dα (φ), we observe that ̃α : [0, ∞) → (0, ∞) such that there exist two monotone increasing functions V α , V ̃α (‖φ‖α ). Dα (φ) ≤ V α (‖φ‖α ) and ̃ Dα (φ) ≤ V

(13.50)

By parts (b) and (d) of Lemma 13.7.5, we conclude that ̃α (‖φ‖α ), 2ξ −α V α (‖φ‖α )} ‖ϱφ ‖α ≤ Vα (‖φ‖α ) := V α (‖φ‖α ) + max{V

(13.51)

and Vα : [0, ∞) → (0, ∞) is a monotone increasing function. We are now in a position to demonstrate the existence of an invariant Gibbs state. We further prove its uniqueness as well as its ergodicity. Theorem 13.7.7. If T : X → X is a topologically transitive, open, distance expanding system and φ : X → ℝ is a Hölder continuous potential, then the measure μφ := ϱφ mφ , i. e., the measure determined by the requirement that dμφ

dmφ

= ϱφ ,

is the unique T-invariant Gibbs state for φ, is ergodic, and is equivalent to mφ with Radon–Nikodym derivative ϱφ bounded away from 0 and ∞.

476 | 13 Gibbs states and transfer operators for open, distance expanding systems Proof. By virtue of its definition and Lemma 13.7.5(c), we know that μφ is a Borel probability measure. By virtue of its definition and Lemma 13.7.5(b), it is clear that μφ is equivalent to mφ with Radon–Nikodym derivative ϱφ bounded away from 0 and ∞. In particular, μφ is absolutely continuous with respect to mφ , and hence the ergodicity of mφ is transmitted to μφ . From Corollaries 13.6.5–13.6.6, Lemma 13.7.5 and Theorem 13.4.1, we deduce that μφ is T-invariant. Since mφ is a Gibbs state and since μφ is equivalent to mφ with Radon–Nikodym derivative ϱφ bounded away from 0 and ∞, we know by Lemma 13.2.3 that μφ is a Gibbs state, too. Finally, to show uniqueness, suppose that μ is a T-invariant Gibbs state for φ. By Proposition 13.2.4, we know that μ is equivalent to μφ . So, like μφ , the measure μ is ergodic and T-invariant. But Theorem 8.2.21 then tells us that either μ = μφ or μ ⊥ μφ . This latter case is impossible given that μ is equivalent to μφ . Thus, μ = μφ . This result has several fundamental repercussions. The following three corollaries are rather straightforward. Corollary 13.7.8. Let T : X → X be a topologically transitive, open, distance expanding system and φ : X → ℝ a Hölder continuous potential. Any nonnegative Borel measurable function satisfying Lemma 13.7.5(a,c) must be mφ -almost everywhere equal to ϱφ . Furthermore, any nonnegative continuous function satisfying Lemma 13.7.5(a,c) must be equal to ϱφ . Proof. Let ϱ be a nonnegative Borel measurable function satisfying Lemma 13.7.5(a,c), ̂φ (ϱ) = ϱ and ∫ ϱ dmφ = 1. The latter property makes ϱmφ a Borel i. e. such that ℒ X probability measure. The former property guarantees that ϱmφ is T-invariant. Obviously, ϱmφ is absolutely continuous with respect to mφ . Therefore the ergodicity of mφ is passed on to ϱmφ . This signifies that the measures ϱmφ and ϱφ mφ are two ergodic T-invariant Borel probability measures. Theorem 8.2.21 then affirms that either ϱmφ = ϱφ mφ or ϱmφ ⊥ ϱφ mφ . This latter case being impossible since ϱφ > 0 and supp(mφ ) = X, the former case holds. This means that ϱ = ϱφ mφ -almost everywhere. Since the only invariant Borel probability measure absolutely continuous with respect to an ergodic invariant Borel probability measure is this ergodic measure itself, another simple consequence of Theorem 13.7.7 is the following. Corollary 13.7.9. Let T : X → X be a topologically transitive, open, distance expanding system. If φ : X → ℝ is a Hölder continuous potential, then μφ is the only T-invariant Borel probability measure which is absolutely continuous with respect to mφ .

13.7 Invariant Gibbs states | 477

After Corollary 13.6.8, we observed that we may set mT n ,Sn φ := mφ since all that is ̂∗n . Adopting this, we obtain yet required of mT n ,Sn φ is that it be a fixed point of ℒ T ,Sn φ another direct ramification of Theorem 13.7.7. Corollary 13.7.10. Let T : X → X be a topologically transitive, open, distance expanding system and φ : X → ℝ be a Hölder potential. Then μT n ,Sn φ = μφ and ϱT n ,Sn φ = ϱφ for every n ∈ ℕ. Proof. We give three different proofs of this fact. In all three cases, we need to observe that T n : X → X is a topologically transitive, open, distance expanding system and that Sn φ : X → ℝ is a Hölder continuous potential. First, Corollary 13.7.9 (with the pair (T, φ) replaced with (T n , Sn φ)) tells us that μT n ,Sn φ is the only T n -invariant Borel probability measure which is absolutely continuous with respect to mT n ,Sn φ . But given that mT n ,Sn φ := mφ and that μφ is a (in fact, the only) T-invariant Borel probability measure which is absolutely continuous with respect to mφ , we know that μφ is a T n -invariant Borel probability measure which is absolutely continuous with respect to mT n ,Sn φ . We deduce that μT n ,Sn φ = μφ . An alternative argument consists in using Exercise 13.11.15(a) and the uniqueness of invariant Gibbs states from Theorem 13.7.7. We leave it to the reader to fill in the blanks. Finally, we will demonstrate in Proposition 13.7.12 that μφ is the unique equilibrium state for φ. A third argument relies on that and on Exercise 13.11.15(b). The details are left to the reader. In all three cases, ϱφ =

dμφ

dmφ

=

dμT n ,Sn φ

dmT n ,Sn φ

= ϱT n ,Sn φ .

We now demonstrate that from a measure-theoretic viewpoint, the dynamics of T are essentially the same as those of a subshift of finite type. Theorem 13.7.11. Let T : X → X be a topologically transitive, open, distance expanding system and let φ : X → ℝ be a Hölder continuous potential. For any Markov partition ℛ of sufficiently small diameter, the coding map π : (EA∞ , σ, μφ∘π ) → (X, T, μφ∘π ∘ π −1 ) induced by ℛ is a measure-preserving isomorphism, where μφ∘π is the unique σ-invariant −n Gibbs state for the potential φ∘π : EA∞ → ℝ. Moreover, the set π −1 (X\⋃∞ n=0 T (𝜕ℛ)) is of full μφ∘π -measure and the restriction of π to that set is injective. Furthermore, P(T, φ) = P(σ, φ ∘ π). Proof. As before, the letters δ, λ and ξ are the constants arising from Definition 4.1.1 and relations (4.29), (4.30) and (4.31); alternatively, see (13.1) and (13.2).

478 | 13 Gibbs states and transfer operators for open, distance expanding systems Since T is open and distance expanding, Theorem 4.4.6 asserts that there exists a Markov partition ℛ for T with diam(ℛ) < ξ . Write ℛ = {R1 , R2 , . . . , Rp }. Let E = {1, 2, . . . , p}, let Aij := {

1

if T(Int(Ri )) ∩ Int(Rj ) ≠ 0

0

otherwise.

(cf. (4.46)) and let σ : EA∞ → EA∞ be the symbolic representation of T induced by ℛ. To shorten notation, write 𝜕ℛ := ⋃ 𝜕Re . e∈E

Let π : EA∞ → X be the coding map induced by ℛ per (4.48), i. e., π : (EA∞ , σ, μφ∘π ) 󳨀→ ω

(X, T, μφ∘π ∘ π −1 )

−n 󳨃󳨀→ π(ω) := ⋂∞ n=0 T (Rωn ).

According to Theorem 4.5.2, the map π is a Hölder continuous surjection which is in−n jective on π −1 (X \ ⋃∞ n=0 T (𝜕ℛ)). Furthermore, every point of the forward T-invariant, −n dense Gδ -set X \⋃∞ n=0 T (𝜕ℛ) has a unique preimage under π. Finally, the coding map π is a factor map between σ : EA∞ → EA∞ and T : X → X. As both φ : X → ℝ and π : EA∞ → X are Hölder continuous, so is their composition φ ∘ π : EA∞ → ℝ. Moreover, since T is topologically transitive, so is σ according to Lemma 4.5.4. We have also observed on several occasions that the map σ is open and distance expanding. By virtue of Theorem 13.7.7, the potential φ ∘ π then admits a unique T-invariant Gibbs state μφ∘π and this state is ergodic. Given that diam(ℛ) < ξ , the map T is a local homeomorphism on a neighborhood of any element of the Markov partition ℛ. According to Lemma 4.5.4, the set π −1 (X \ 𝜕ℛ) is then backward σ-invariant. By Exercise 8.5.36 and the ergodicity of μφ∘π , we deduce that μφ∘π (π −1 (X \ 𝜕ℛ)) ∈ {0, 1}. However, since the set 𝜕ℛ is closed in X and π is continuous, the set π −1 (X \ 𝜕ℛ) is open in EA∞ . As μφ∘π is a Gibbs measure, it has full support and hence μφ∘π (π −1 (X \ 𝜕ℛ)) > 0. We conclude that μφ∘π (π −1 (X \ 𝜕ℛ)) = 1. By Exercise 8.5.36 again, it follows that ∞

1 = μφ∘π (π −1 (X \ 𝜕ℛ)) = μφ∘π ( ⋂ σ −n (π −1 (X \ 𝜕ℛ))) n=0

∞

∞

= μφ∘π ( ⋂ (π ∘ σ ) (X \ 𝜕ℛ)) = μφ∘π ( ⋂ (T n ∘ π) (X \ 𝜕ℛ)) n=0

n −1

n=0

−1

13.7 Invariant Gibbs states | 479

∞

∞

= μφ∘π ( ⋂ π −1 (T −n (X \ 𝜕ℛ))) = μφ∘π (π −1 ( ⋂ T −n (X \ 𝜕ℛ))) n=0

∞

n=0

∞

= μφ∘π (π ( ⋂ X \ T (𝜕ℛ))) = μφ∘π (π (X \ ⋃ T −n (𝜕ℛ))). −1

−n

−1

n=0

n=0

−n ∞ So π −1 (X\⋃∞ n=0 T (𝜕ℛ)) is of full μφ∘π -measure, and hence the map π : (EA , σ, μφ∘π ) → −1 (X, T, μφ∘π ∘ π ) is a measure-preserving isomorphism. Turning our attention to the relationship between the pressures P(T, φ) and P(σ, φ ∘ π), one inequality is obvious: since π is a factor map, Proposition 11.1.20 asserts that

P(T, φ) ≤ P(σ, φ ∘ π).

(13.52)

Regarding the opposite inequality, since the map π : (EA∞ , σ, μφ∘π ) → (X, T, μφ∘π ∘π −1 ) is a measure-preserving isomorphism, Exercise 9.7.15 affirms that hμφ∘π (σ) = hμφ∘π ∘π −1 (T). By Proposition 13.2.9, we also know that μφ∘π is an equilibrium state for φ ∘ π and the variational principle (Theorem 12.1.1) then yields that P(σ, φ∘π) = hμφ∘π (σ)+ ∫ φ∘π dμφ∘π = hμφ∘π ∘π −1 (T)+∫ φ dμφ∘π ∘π −1 ≤ P(T, φ). EA∞

(13.53)

X

By (13.52)–(13.53) the equality follows. 13.7.3 Invariant Gibbs states and equilibrium states coincide and are unique According to Proposition 13.2.9, every T-invariant Gibbs state for a potential φ on an open, distance expanding system T is an equilibrium state for φ. If, in addition, T is topologically transitive and φ is Hölder continuous, then Theorem 13.7.7 affirms that φ admits a unique T-invariant Gibbs state and that this state is ergodic. This measure is denoted by μφ . In this section, we show the converse. Proposition 13.7.12. Let T : X → X be a topologically transitive, open, distance expanding system and let φ : X → ℝ be a Hölder continuous potential. Then φ has a unique equilibrium state and this state is μφ , the unique T-invariant Gibbs state for φ. This state is ergodic. Rather than proving directly this result in its full generality, we will first establish it in a symbolic setting. Proposition 13.7.13. Let E be a finite alphabet and A : E → E an irreducible matrix. Let σ : EA∞ → EA∞ be the subshift of finite type generated by A and φ : EA∞ → ℝ be a Hölder continuous potential. Then φ has a unique equilibrium state and this state is μφ , the unique σ-invariant Gibbs state for φ. This state is ergodic.

480 | 13 Gibbs states and transfer operators for open, distance expanding systems Let us first convince ourselves that this is really a special case of Proposition 13.7.12. We have observed before that EA∞ is a compact space when E is finite and that the shift map σ is open and distance expanding with respect to any one of the usual metrics on EA∞ (the distance between two infinite words depends on the length of their initial common word). Per Theorem 3.2.14, the irreducibility of A is equivalent to the transitivity of σ. So Proposition 13.7.13 is indeed a special case of Proposition 13.7.12. For the sake of completeness, we prove an auxiliary lemma which is a weak version of a well known density lemma from probability theory (see Theorem 35.8 in Billingsley [8]). In its full generality, this density lemma is proved as a consequence of the martingale convergence theorem (Theorem A.1.65) (see also Williams [138]). Recall that there is a bijection between the set of finite measurable partitions of a measurable space (X, ℱ ) and the set of finite sub-σ-algebras of ℱ . Indeed, given a finite measurable partition α = {A1 , . . . , An } of (X, ℱ ) and letting A(0) := Ak while A(1) := 0 k k for all 1 ≤ k ≤ n, the family 𝒜(α) = {⋃nk=1 Ak k : ω ∈ {0, 1}n } of all sets that can be expressed as unions of atoms of α, is a finite sub-σ-algebra of ℱ . Conversely, given a finite sub-σ-algebra 𝒜 = {B1 , . . . , Bm } of ℱ and letting B(0) := Bk while B(1) := X\Bk k k (ω )

k : ω ∈ {0, 1}m } is a finite measurable for all 1 ≤ k ≤ m, the family α(𝒜) = {⋂m k=1 Bk partition of (X, ℱ ). It is easy to see that α(𝒜(α)) = α and 𝒜(α(𝒜)) = 𝒜. Moreover, given two finite measurable partitions α, β, it is obvious that β is a refinement of α (α ≺ β) if and only if 𝒜(α) ⊆ 𝒜(β). Finally, given x ∈ X recall that α(x) denotes the atom of α that contains the point x.

(ω )

Lemma 13.7.14. Let (X, ℱ ) be a measurable space and let (αn )∞ n=1 be a sequence of increasingly finer countable measurable partitions generating ℱ , i. e., σ(∪∞ n=1 αn ) = ℱ . If μ and ν are mutually singular probability measures on (X, ℱ ), then for every t > 0 we have lim ν({x ∈ X : ν(αn (x)) ≤ tμ(αn (x))}) = 0.

n→∞

Proof. Suppose for a contradiction that there exists t > 0 such that Δ := lim sup ν(Xn ) > 0 n→∞

where Xn := {x ∈ X : ν(αn (x)) ≤ tμ(αn (x))}.

Let ∞

Sn := ⋃ Xk k=n

and

∞

S := ⋂ Sn . n=1

Clearly (Sn )∞ n=1 forms a descending family of sets. Moreover, ν(Sn ) ≥ Δ for all n ∈ ℕ and thus ν(S) ≥ Δ. Since the measures ν and μ are mutually singular, there exists a set Y ∈ ℱ such that ν(Y) = 1 and μ(Y) = 0. Setting F := S ∩ Y,

13.7 Invariant Gibbs states | 481

we then have that ν(F) = ν(S) ≥ Δ

and μ(F) = 0.

(13.54)

Since (αn )∞ n=1 is a sequence of increasingly finer countable measurable partitions generating ℱ , Exercise 13.11.12 states that there exists a countable family ℐ ⊆ ⋃∞ n=1 αn of mutually disjoint sets such that F ⊆ ⋃ I and μ(⋃ I) ≤ I∈ℐ

I∈ℐ

Δ . 2t

(13.55)

Then for every x ∈ F there exists a unique I(x) ∈ ℐ such that x ∈ I(x). Given that for every I ∈ ℐ there is n(I) ∈ ℕ such that I ∈ αn(I) , we deduce that I(x) = αn(I(x)) (x). Let 𝒥 := {αn(I(x)) (x) : x ∈ F} = {I(x) : x ∈ F} ⊆ ℐ .

As ℐ is countable, so is 𝒥 . Similarly, as the elements of ℐ are mutually disjoint sets, so are the elements of 𝒥 . Furthermore, observe that F ⊆ ⋃ J = ⋃ αn(I(x)) (x) = ⋃ I(x) ⊆ ⋃ I. J∈𝒥

x∈F

x∈F

I∈ℐ

(13.56)

In addition, every x in F is in S and, therefore, lies in Sn(I(x)) . Consequently, x ∈ Xn(I(x)) . By definition of the sets Xn , it ensues that ν(αn(I(x)) (x)) ≤ tμ(αn(I(x)) (x)). So ν(J) ≤ tμ(J) for every J ∈ 𝒥 . Using this and (13.54)–(13.56), we deduce that Δ Δ ≥ μ(⋃ I) ≥ μ( ⋃ J) = ∑ μ(J) ≥ t −1 ∑ ν(J) = t −1 ν( ⋃ J) ≥ t −1 ν(F) ≥ . 2t t I∈ℐ J∈𝒥 J∈𝒥 J∈𝒥 J∈𝒥 This is the contradiction that was sought. Using this result, we prove Proposition 13.7.13. Proof of Proposition 13.7.13. Exercise 12.4.4 asserts that for any topological dynamical system T : X → X and continuous potential φ : X → ℝ, the set of all equilibrium states for φ is a convex subset of the set M(T) of all T-invariant measures. Furthermore, Theorem 8.2.23 affirms that the ergodic measures E(T) constitute the extreme points of M(T). It ensues that if φ has a unique ergodic equilibrium state, then it admits a unique equilibrium state. So it suffices to demonstrate that any ergodic equilibrium state ν coincides with μφ . Let T = σ : EA∞ → EA∞ be a subshift of finite type generated by an irreducible matrix A over a finite alphabet E, and let φ : EA∞ → ℝ be a Hölder continuous potential. Pursuant to Theorem 13.7.7, let μφ be the unique σ-invariant Gibbs state for φ. Recall that this state is ergodic. Let ν be an ergodic equilibrium state for φ. In particular, ν is σ-invariant. Our goal is to show that ν = μφ .

482 | 13 Gibbs states and transfer operators for open, distance expanding systems Let α := {[e]}e∈E be the partition of EA∞ into its one-cylinders. It is easy to see that −i n ∞ α := ⋁n−1 i=0 σ (α) = {[ω]}ω∈EAn , i. e., α is the partition of EA into its n-cylinders. Recall that α is a generator for σ (see Definition 9.4.19 and Example 9.4.23). By Theorem 9.4.20 and Definition 9.4.10, we know that n

hν (σ) = hν (σ, α) = inf

n∈ℕ

1 H (αn ). n ν

For every n ∈ ℕ, the variational principle (Theorem 12.1.1) affirms that 0 = n[hν (σ) + ∫ (φ − P(σ, φ)) dν] EA∞

≤ Hν (αn ) + ∫ (Sn φ − nP(σ, φ)) dν EA∞

= − ∑ ν(A)[log ν(A) − A∈αn

1 ∫(Sn φ − nP(σ, φ)) dν] ν(A) A

≤ − ∑ ν(A)[log ν(A) − (Sn φ(ωA ) − nP(σ, φ))] A∈αn

for some ωA ∈ A for each A ∈ αn . Using the fact that μφ is a Gibbs state for φ, we can continue this inequality as follows: 0 ≤ − ∑ ν(A) log[ν(A) exp(−(Sn φ(ωA ) − nP(σ, φ)))] A∈αn

≤ − ∑ ν(A) log[ν(A)C −1 (μφ (A)) ] −1

A∈αn

= log C − ∑ ν(A) log A∈αn

ν(A) , μφ (A)

(13.57)

where the constant C comes from the Gibbs property of μφ . Assume, for a contradiction, that the measure ν is an ergodic equilibrium state distinct from μφ . Then, in light of Theorem 8.2.21, the measures ν and μφ must be mutually singular. In search of a contradiction it suffices, in view of (13.57), to show that lim ∑ ν(A) log

n→∞

A∈αn

ν(A) = ∞. μφ (A)

To do this, for every j ∈ ℤ and every n ∈ ℕ set Fn,j := {ω ∈ EA∞ : ej
ek }) − e ∑ je−j . j=1

As the measures ν and μφ are mutually singular, applying Lemma 13.7.14 with αn = αn and t = ek , we infer that lim ν({ω ∈ EA∞ :

n→∞

ν(αn (ω))

μφ (αn (ω))

> ek }) = 1.

Therefore, lim inf ∑ ν(A) log n→∞

A∈αn

∞ ν(A) ≥ k − e ∑ je−j . μφ (A) j=1

−j Since the series ∑∞ j=1 je is convergent, letting k → ∞ yields (13.58). This contradicts (13.57). Consequently, the only ergodic equilibrium state for φ is the unique σ-invariant Gibbs state μφ . This ends the proof in the special case of a topologically transitive subshift of finite type.

484 | 13 Gibbs states and transfer operators for open, distance expanding systems In order to prove the result in its full generality, i. e., Proposition 13.7.12, we will need a measure-lifting lemma. It is kind of, but not quite, a Hahn–Banach theorem, and does not really follow from this latter. We will formulate and prove it in a slightly more general case than we need. In a sense, a more general result can be found in [85], where, with the notation of Theorem 13.7.16 below, neither X nor Y were assumed to be compact but merely Polish spaces; however, X was therein a subshift of finite type over a countable alphabet and all the fibers π −1 (y), y ∈ Y, were assumed to be compact. The strongest theorem, with the weakest hypotheses, can be found in [20]; however, compactness of all fibers was still assumed. This will not be assumed here. We start with the following well-known fact from measure theory. We provide its proof since it is easy, short and this fact is not so commonly formulated and proved in textbooks. We do it also for the sake of completeness and convenience for the reader. Lemma 13.7.15. Let W and Z be Polish spaces. Let μ be a Borel probability measure on Z, ̂ be its completion, and denote by ℬ̂μ the complete σ-algebra of all μ ̂ -measurable let μ subsets of Z. Let f : W → Z be a Borel measurable surjection and let g : W → ℝ =: [−∞, +∞] be a Borel measurable function. Define the functions g∗ , g ∗ : Z → ℝ respectively by 󵄨 g∗ (z) := inf{g(w) 󵄨󵄨󵄨 w ∈ f −1 (z)} and

󵄨 g ∗ (z) := sup{g(w) 󵄨󵄨󵄨 w ∈ f −1 (z)}.

Then g∗ , g ∗ are measurable with respect to the σ-algebra ℬ̂μ . If, in addition, the map f is locally one-to-one, then g∗ , g ∗ are Borel measurable. Proof. Replacing g by −g, it suffices to prove the lemma for the function g ∗ : Z → ℝ. Fix t ∈ ℝ. For any z ∈ Z, we have that g ∗ (z) ∈ (t, ∞] if and only if g(w) ∈ (t, ∞] for some w ∈ f −1 (z). Thus, (g ∗ )−1 ((t, ∞]) = f (g −1 ((t, ∞])). Hence, (g ∗ )−1 ((t, ∞]) is an analytic set since g −1 ((t, ∞]) is a Borel set, f : W → Z is a Borel map, and both spaces W and Z are Polish. The first assertion now follows from the fact that all analytic subsets of Z belong to ℬ̂μ . If, in addition, the map f : W → Z is locally 1-to-1, then the f -images of all Borel subsets of W are Borel in Z, so f (g −1 ((t, ∞])) ⊆ Z is Borel. We now return to measure-theoretic dynamical systems. Theorem 13.7.16. Let X and Y be compact metrizable spaces, and let T : X → X and S : Y → Y be continuous maps. Assume that there is a continuous surjection π : X → Y making S a topological factor of T, i. e., π ∘ T = S ∘ π,

13.7 Invariant Gibbs states | 485

or, equivalently, that the following diagram commutes: T

X 󳨀󳨀󳨀󳨀󳨀󳨀→ ↑ ↑ π↑ ↑ ↓ Y 󳨀󳨀󳨀󳨀󳨀󳨀→ S

X ↑ ↑ ↑ ↑π ↓ Y

Then the map M(T) ∋ ν 󳨃󳨀→ ν ∘ π −1 ∈ M(S) is surjective. Recall that M(T) (resp., M(S)) denotes the set of all T-invariant (resp., S-invariant) Borel probability measures on X (resp., Y). Proof. Fix μ ∈ M(S). Let ℬb (X) and ℬb (Y) be the vector spaces of all bounded Borel measurable real-valued functions defined on X and Y, respectively. Let ℬb+ (X) and ℬb+ (Y) be the corresponding convex cones consisting of all nonnegative functions. Let 󵄨

̂b (X) := {g ∘ π 󵄨󵄨 g ∈ ℬb (Y)}. ℬ 󵄨 Clearly ℬ̂b (X) is a vector subspace of ℬb (X) and, as π : X → Y is a surjection, for each f ∈ ℬ̂b (X) there exists a unique g ∈ ℬb (Y) such that f = g ∘ π. Thus, treating (via integration) μ as a linear functional from ℬb (Y) to ℝ, the formula ̂b (X) ∋ g ∘ π 󳨃󳨀→ μ ̂ (g ∘ π) := μ(g) ∈ ℝ ℬ ̂ from ℬ̂b (X) to ℝ. By Lemma 13.7.15 applied to defines a positive linear functional μ π : X → Y, for every h ∈ ℬb (X) the function h∗ ∘ π : X → ℝ belongs to ℬ̂b (X). Also, h − h∗ ∘ π ≥ 0. Hence, h − h∗ ∘ π ∈ ℬb+ (X). Riesz’s Theorem (Theorem A.2.6) then produces a positive linear functional μ∗ : ℬb (X) → ℝ such that ̂ (f ), μ∗ (f ) = μ

∀h ∈ ℬ̂b (X).

And the restriction of μ∗ to the space C(X) of continuous functions on X, is obviously linear and positive. Claim 1. If (gn )∞ n=1 is a monotone decreasing sequence of nonnegative functions in C(X) converging pointwise to the constant function 0, then lim μ∗ (gn ) = 0. n→∞

Proof of Claim 1. Clearly, is a monotone decreasing sequence of nonnegative bounded functions that, by Lemma 13.7.15, belong to ℬ(Y), and thus to ℬb+ (Y). Fix (gn∗ )∞ n=1

486 | 13 Gibbs states and transfer operators for open, distance expanding systems y ∈ Y. As the set π −1 (y) ⊆ X is compact, Dini’s theorem (Theorem A.1.43) confirms that the sequence (gn |π −1 (y) )∞ n=1 converges uniformly to zero. Since all these functions are ∗ ∞ nonnegative, the sequence (gn∗ )∞ n=1 converges pointwise to zero. In summary, (gn )n=1 is a monotone decreasing sequence of functions in ℬb+ (Y) converging pointwise to zero. Therefore, using the fact that gn ≤ gn∗ ∘ π and the monotone convergence theorem (Theorem A.1.35), we get ̂ (gn∗ ∘ π) = lim μ(gn∗ ) = 0. 0 ≤ lim μ∗ (gn ) ≤ lim μ∗ (gn∗ ∘ π) = lim μ n→∞

n→∞

n→∞

n→∞

So limn→∞ μ∗ (gn ) = 0.

◼

Having Claim 1, the Daniell–Stone representation theorem (a generalization of the Riesz representation theorem (Theorem A.1.54)) immediately entails the following. Claim 2. The positive linear functional μ∗ : ℬb (X) → ℝ extends uniquely from C(X) to an element of M(X). We denote this extension by μ∗ as well. Now, for every n ∈ ℕ set μ∗n :=

1 n−1 ∗ ∑ μ ∘ T −j . n j=0

Since the set M(X) of all Borel probability measures on X is compact with respect to the weak∗ topology on C(X)∗ , there exists an increasing sequence (nk )∞ k=1 of positive ∗ ∗ ∞ integers such that the sequence (μnk )k=1 weak converges. Denote its limit by ν ∈ M(X). A standard argument, employed for instance in the proofs of the Krylov–Bogolyubov theorem (Theorem 8.1.22) and the variational principle (Theorem 12.1.1), shows that ̂ and μ∗ , we obtain for every g ∈ C(X) and every n ∈ ℕ ν ∈ M(T). By the definitions of μ that μ∗n ∘ π −1 (g) = μ∗n (g ∘ π) =

1 n−1 1 n−1 ∗ ∑ μ ∘ T −j (g ∘ π) = ∑ μ∗ (g ∘ π ∘ T j ) n j=0 n j=0

=

1 n−1 ∗ 1 n−1 ̂ ((g ∘ Sj ) ∘ π) ∑ μ (g ∘ Sj ∘ π) = ∑ μ n j=0 n j=0

=

1 n−1 1 n−1 ∑ μ(g ∘ Sj ) = ∑ μ(g) = μ(g). n j=0 n j=0

So μ∗n ∘ π −1 = μ for all n ∈ ℕ. Consequently, ν ∘ π −1 = lim μ∗nk ∘ π −1 = lim μ = μ. k→∞

k→∞

Finally, we can prove the main result of this section, Proposition 13.7.12.

13.7 Invariant Gibbs states | 487

Proof of Proposition 13.7.12. Let T : X → X be a topologically transitive, open, distance expanding system subject to a Hölder continuous potential φ : X → ℝ. By Theorem 13.7.11, given a Markov partition ℛ = {R1 , . . . , Rp } of sufficiently small diameter, we know that the coding map π : (EA∞ , σ, μφ∘π )

→

(X, T, μφ∘π ∘ π −1 )

{π(ω)}

=

⋂ T −n (Rωn )

∞

n=0

is a measure-preserving isomorphism, where μφ∘π is the unique σ-invariant Gibbs state for the Hölder potential φ ∘ π : EA∞ → ℝ. Furthermore, P(T, φ) = P(σ, φ ∘ π). By Proposition 13.7.13 (the symbolic case already proved), we know that the only equilibrium state for φ ∘ π is μφ∘π . Let ν be any equilibrium state for φ on X. We will show that ν = μφ∘π ∘ π −1 = μφ . Per Theorem 4.5.2, the coding map π : EA∞ → X is a factor map between (EA∞ , σ) and (X, T). According to Theorem 13.7.16, there then exists a σ-invariant Borel probability measure ν̃ on EA∞ such that ν̃ ∘ π −1 = ν. Since π is a factor map, Exercise 9.7.15(j) states that h̃ν (σ) ≥ hν (T). Using the variational principle (Theorem 12.1.1), we get h̃ν (σ) + ∫ φ ∘ π dν̃ = h̃ν (σ) + ∫ φ dν̃ ∘ π −1 = h̃ν (σ) + ∫ φ dν EA∞

X

X

≥ hν (T) + ∫ φ dν X

= P(T, φ) = P(σ, φ ∘ π). Thus, ν̃ is an equilibrium state for φ ∘ π. From Proposition 13.7.13, we infer that ν̃ = μφ∘π . Hence, ν = ν̃ ∘ π −1 = μφ∘π ∘ π −1 . Therefore, φ has a unique equilibrium state, namely μφ∘π ∘ π −1 . Since μφ , the unique T-invariant Gibbs state for φ, is an equilibrium state for φ by Theorem 13.7.7 and Proposition 13.2.9, we conclude that μφ∘π ∘ π −1 = μφ .

488 | 13 Gibbs states and transfer operators for open, distance expanding systems 13.7.4 Hölder continuous potentials with the same Gibbs states We now provide a complete characterization of when two Hölder continuous potentials share the same invariant Gibbs/equilibrium state. As an immediate consequence of Theorem 13.2.7 and Proposition 13.7.12, we have the following. Theorem 13.7.17. If T : X → X is a topologically transitive, open, distance expanding system and φ, ψ : X → ℝ are Hölder continuous potentials, then the following statements are equivalent: (a) The invariant Gibbs/equilibrium states of φ and ψ coincide, i. e., μφ = μψ . (b) There exists a constant R1 ∈ ℝ such that Sn φ(x) − Sn ψ(x) = R1 n whenever n ∈ ℕ and x ∈ X is a periodic point of T of period n. (c) There exist constants R2 ∈ ℝ and C ≥ 0 such that 󵄨󵄨 󵄨 󵄨󵄨Sn φ(x) − Sn ψ(x) − R2 n󵄨󵄨󵄨 ≤ C,

∀n ∈ ℕ, ∀x ∈ X.

(d) φ − ψ is cohomologous to a constant R3 ∈ ℝ in the additive group of bounded functions on X. (e) φ − ψ is cohomologous to a constant R4 ∈ ℝ in the additive group of continuous functions on X. (f) φ − ψ is cohomologous to a constant R5 ∈ ℝ in the additive group of Hölder continuous functions on X. (g) 𝒢φ = 𝒢ψ . (h) 𝒢φ ∩ 𝒢ψ ≠ 0. In addition, if any of the above statements holds, then all the constants R1 –R5 are the same and equal to P(T, φ) − P(T, ψ). We would like to stress that this theorem shows that a transitive, open, distance expanding system admits an abundance of different Gibbs states. For instance, take a Hölder continuous potential φ. Fix two different periodic orbits. While preserving the Hölder continuity, perturb slightly φ on a neighborhood of one point of the first orbit, a neighborhood so small that it does not intersect the second orbit, and call the resulting potential ψ. Then from the second orbit we would deduce that R1 = 0 in condition (b), whereas from the first orbit we would infer that R1 ≠ 0. From this contradiction, we conclude that the perturbed potential must have a different Gibbs state from the one of the original potential. As the reader can foresee, condition (b) is a very powerful tool to show that Gibbs states are different.

13.7 Invariant Gibbs states | 489

13.7.5 Invariant Gibbs states have positive entropy; pressure gap Definition 13.7.18. Let T : X → X be a topological dynamical system. A continuous potential φ : X → ℝ is said to have a pressure gap with respect to T if there exists k ∈ ℕ such that P(T k , Sk φ) > sup(Sk φ). Our first goal in this section is to show that if T is a transitive, open, distance expanding map and the space X is infinite, then each Hölder continuous potential φ : X → ℝ has a pressure gap with respect to T. We will deduce from this that hμφ (T) > 0. We start with the following well-known fact from measure theory. Proposition 13.7.19. Let (X, d) be a compact metric space and μ a Borel probability measure on X. For every r > 0, define 󵄨 Sμ (r) := sup{μ(B(x, r)) 󵄨󵄨󵄨 x ∈ X}. If the measure μ is atomless, then lim Sμ (r) = 0.

r→0

Proof. Suppose that limr→0 Sμ (r) > 0. As the function (0, ∞) ∋ r 󳨃→ Sμ (r) is monotone increasing, this assumption implies that there exists ε > 0 such that Sμ (r) > ε,

∀r > 0.

Therefore, for every r > 0 there is x(r) ∈ X such that μ(B(x(r), r)) > ε. Given that X is compact, there is a strictly increasing sequence (nk )∞ k=1 of positive integers such that the sequence (x(1/nk ))∞ converges. Denote its limit by x. For every k=1 s > 0, there exists k ∈ ℕ so large that d(x, x(1/nk ))
ε.

490 | 13 Gibbs states and transfer operators for open, distance expanding systems Hence, μ({x}) = lim μ(B(x, s)) ≥ ε > 0. s→0

This contradicts the atomless nature of μ. Now we can establish the announced existence of pressure gaps. Theorem 13.7.20. If T : X → X is a transitive, open, distance expanding system on an infinite space X, then each Hölder potential φ : X → ℝ has a pressure gap with respect to T. Proof. Since the Gibbs state μφ is ergodic, supp(μφ ) = X and X is infinite, we infer that μφ is atomless. Let Cφ ≥ 1 be the constant C from Definition 13.2.1 ascribed to the potential φ and P = P(T, φ). As diam(Tx−n (B(T n (x), ξ ))) ≤ ξλ−n for all x ∈ X and all n ≥ 0, we deduce from Proposition 13.7.19 that there exists k ∈ ℕ such that μφ (Tx−k (B(T k (x), ξ ))) ≤ (eCφ )−1 ,

∀x ∈ X.

Given any x ∈ X, it follows from this and (13.13) that Sk φ(x) − kP(T, φ) ≤ log Cφ + log μφ (Tx−k (B(T k (x), ξ ))) ≤ log Cφ − log e − log Cφ = −1. Using this and Theorem 11.1.22 (P(T k , Sk φ) = P(T, φ)), we conclude that Sk φ(x) ≤ kP(T, φ) − 1 = P(T k , Sk φ) − 1 for any x ∈ X. There is a pressure gap. A pressure gap entails strictly positive entropy with respect to the unique equilibrium state μφ . Theorem 13.7.21. If T : X → X is a transitive, open, distance expanding map of an infinite space X and φ : X → ℝ is a Hölder continuous potential, then hμφ (T) > 0. Proof. We will provide two proofs. By Theorem 13.7.20, there exists k ∈ ℕ such that P(T k , Sk φ) > sup(Sk φ). Using successively Theorem 9.4.13 (hμφ (T k ) = khμφ (T)), Corollary 13.7.10 and Theorem 12.2.2 (applied to the pair (T k , Sk φ) rather than (T, φ)), we infer that hμφ (T) =

1 1 h (T k ) = hμ k (T k ) > 0. k μφ k T ,Sk φ

13.7 Invariant Gibbs states | 491

An alternative argument goes as follows. Per Theorem 11.1.22, we know that P(T k , Sk φ) = kP(T, φ). Since μφ is an equilibrium state, it ensues from the variational principle (Theorem 12.1.1) and the T-invariance of μ that hμφ (T) = P(T, φ) − ∫ φ dμφ X

1 = [kP(T, φ) − k ∫ φ dμφ ] k X

1 = [P(T k , Sk φ) − ∫ Sk φ dμφ ] k X

>

1 [sup(Sk φ) − ∫ Sk φ dμφ ] k X

≥ 0. So hμφ (T) > 0. 13.7.6 Absolutely continuous invariant measures for Shub expanding maps If M is a compact connected smooth manifold, then each Riemannian metric ρ on M induces a unique volume (Lebesgue) measure λρ on M and all such measures are mutually equivalent. Call the corresponding measure class the Lebesgue measure class on M. As announced at the end of Chapter 6, using the theory of Gibbs states we prove that every C 1+ε (ε > 0) Shub expanding endomorphism admits a unique invariant Borel probability measure absolutely continuous with respect to the Lebesgue measure class. This fact was first proved for C 2 maps in [64]. It is also in that paper that the appropriate transfer operator was for the first time explicitly used in dynamical systems. As it is based on the theory of Gibbs states developed in this chapter, our proof is different but nevertheless presents significant similarities with that of Krzyżewski and Szlenk. Theorem 13.7.22. If T : M → M is a C 1+ε Shub expanding endomorphism on a compact connected smooth manifold M, then there exists a unique T-invariant Borel probability measure μ on M which is absolutely continuous with respect to the Lebesgue measure class on M. In fact, μ is equivalent to the Lebesgue measure class on M and μ = μφ , where φ(x) = − log det(Dx T). Proof. Let ρ be a Riemannian metric on M. For every x ∈ M, let Dx T : Tx M → TT(x) M

492 | 13 Gibbs states and transfer operators for open, distance expanding systems be the corresponding derivative map and λx be the Lebesgue measure on Tx M induced by the Euclidean structure on Tx M which is engendered by the inner product ρx . Let det(Dx T) be the determinant of the map Dx T. It is positive since M is orientable and T is orientation preserving. As T is a C 1+ε endomorphism, the potential M ∋ x 󳨃󳨀→ φ(x) = − log det(Dx T) ∈ ℝ is Hölder continuous with exponent ε. Let ℒφ : Hε (M) → Hε (M) be the associated transfer operator defined by the formula ℒφ (g)(x) =

∑ (det(Dy T)) g(y). −1

y∈T −1 (x)

Let ξ be the number coming from the distance expandingness of T. Then for every x ∈ M and every n ≥ 0, we have λρ (Tx−n (Bρ (T n (x), ξ ))) =

∫

det(Dz Tx−n ) dλρ (z) ≍ (det(Dx T n ))

−1

= exp(Sn φ(x)).

Bρ (T n (x),ξ )

Thus, λρ is a Gibbs measure for the potential φ, and P(T, φ) = 0 according to Proposition 13.2.4. By that same proposition, the ergodic T-invariant Gibbs state μφ produced in Theorem 13.7.7 is equivalent to λρ , and the proof of the existence part is complete. Since the only T-invariant Borel probability measure absolutely continuous with respect to an ergodic T-invariant Borel probability measure, is this ergodic measure itself, the uniqueness part is an immediate consequence of the ergodicity of μφ .

13.8 Finer properties of transfer operators and Gibbs states 13.8.1 Iterates of transfer operators Let T : X → X be a topologically transitive, open and distance expanding system and φ ∈ Hα (X). Define ̂ := φ + log ϱφ − log ϱφ ∘ T, φ ̂ ∈ C(X) may not be Hölder continuwhere ϱφ comes from Lemma 13.7.5. Note that φ ous. (Nevertheless, Lemma 13.7.5 affirms that ϱφ ∈ Hα (X) and that ϱφ is bounded away from 0 and ∞. As log is Lipschitz on any compact subset of (0, ∞), we then have that ̂ is log ϱφ ∈ Hα (X). If T is additionally Hölder, then log ϱφ ∘ T is Hölder, and thus φ Hölder. Though this special case is worth mentioning, we will stick to the general case ̂ may not be Hölder continuous.) In any case, since φ ̂ is cohomologous to φ where φ in the additive group of bounded (even Hölder continuous) functions, it follows from

13.8 Finer properties of transfer operators and Gibbs states | 493

̂) = P(T, φ) and 𝒢φ̂ = 𝒢φ . In particular, this implies that the Lemma 13.2.6 that P(T, φ ̂, has a unique invariant Gibbs state, namely μφ the system T, under the potential φ unique invariant Gibbs state under the potential φ (see Theorem 13.7.7). A straightforward calculation shows that the operator ℒφ̂ : C(X) → C(X) satisfies ℒφ (ϱφ g)

ℒφ̂ (g) =

ϱφ

.

Yet another calculation yields that its dual ℒ∗φ̂ : C(X)∗ → C(X)∗ is such that ℒ∗φ̂ (μφ ) =

eP(T,φ) μφ . Therefore, one can set Λφ̂ = eP(T,φ) and mφ̂ = μφ in Theorem 13.6.2 and ̂. Among others, extending and norsubsequent applicable results for the potential φ ̂φ̂ : L1 (μφ ) → L1 (μφ ) and its malizing leads to the positive bounded linear operator ℒ ̂∗̂ : L1 (μφ )∗ → L1 (μφ )∗ determined by the formula dual ℒ φ ̂φ̂ (g)(x) = ℒ

̂φ (ϱφ g)(x) ℒ ϱφ (x)

=

e−P(T,φ) ϱφ (x)

∑ y∈T −1 (x)

ϱφ (y)g(y)eφ(y)

for μφ -a. e. x ∈ X,

̂∗̂ (μφ ) = μφ . Then with equality holding everywhere if g ∈ C(X). Moreover, ℒ φ ̂φ̂ = R ∘ ℒ ̂φ ∘ Rφ , ℒ φ −1

where Rφ : L1 (μφ ) → L1 (μφ ) is given by Rφ (g) := ϱφ ⋅ g. Consequently, for every n ∈ ℕ, n

n

̂̂ = R ∘ ℒ ̂ ∘ Rφ ℒ φ φ φ −1

̂n = Rφ ∘ ℒ ̂n̂ ∘ R−1 . and ℒ φ φ φ

So n

̂̂ (g)(x) = ℒ φ

̂n (ϱφ g)(x) ℒ φ ϱφ (x)

=

e−nP(T,φ) ϱφ (x)

∑ y∈T −n (x)

ϱφ (y)g(y)eSn φ(y)

for μφ -a. e. x ∈ X, (13.61)

with equality holding everywhere if g ∈ C(X). ̂φ have counterparts for ℒ ̂φ̂ . The latter Lemma 13.7.1 and Proposition 13.7.2 for ℒ directly follow from the former through relation (13.61) and Lemma 13.7.5. Lemma 13.8.1. If T : X → X is a topologically transitive, open, distance expanding system and φ ∈ Hα (X), then for all n ∈ ℕ and all x, y ∈ X we have −3

(Dα (φ))

≤

̂n̂ (1)(y) ℒ φ

̂n (1)(x) ℒ ̂ φ

3

≤ (Dα (φ)) .

494 | 13 Gibbs states and transfer operators for open, distance expanding systems Proposition 13.8.2. If T : X → X is a transitive, open, distance expanding system and φ ∈ Hα (X), then −3

(Dα (φ))

̂n̂ (1) ≤ sup ℒ ̂n̂ (1) ≤ (Dα (φ))3 , ≤ inf ℒ φ φ

∀n ∈ ℕ.

Theorem 13.7.4 also has a counterpart. The latter follows from the former, relation (13.61) and Lemma 13.7.5. Theorem 13.8.3. If T : X → X is a transitive, open, distance expanding system and ̂n̂ (g)}∞ is equiconφ : X → ℝ is a Hölder potential, then for every g ∈ C(X) the family {ℒ n=0 φ ̂φ̂ : C(X) → C(X) is almost tinuous and pointwise bounded in C(X). Hence the operator ℒ periodic.

Now we present an analogue of Lemma 13.7.5 and a little more. While ϱφ is a fixed ̂φ , the constant function 1 is a fixed point of ℒ ̂φ̂ . Moreover, whereas the meapoint of ℒ ∗ ̂ ̂∗̂ . sure mφ is a fixed point for ℒφ , the measure μφ is a fixed point for ℒ φ Proposition 13.8.4. If T : X → X is a transitive, open, distance expanding system and ̂φ̂ possesses the following properties: φ : X → ℝ is a Hölder potential, then the operator ℒ ̂φ̂ (1) = 1. (a) ℒ 󵄩̂ 󵄩󵄩 ̂φ̂ : C(X) → C(X) Moreover, 󵄩󵄩󵄩ℒ 󵄩∞ ≤ ‖f ‖∞ for every f ∈ C(X), i. e. the operator ℒ ̂ (f )󵄩 φ 󵄩󵄩 ̂ 󵄩󵄩 has for norm 󵄩󵄩ℒφ̂ 󵄩󵄩∞ = 1. ̂̂ (g) dμφ = ∫ g dμφ for all g ∈ L1 (μφ ). ̂∗̂ (μφ ) = μφ , i. e. ∫ ℒ (b) ℒ φ X X φ 󵄩̂ 󵄩󵄩 1 ̂φ̂ : L1 (μφ ) → L1 (μφ ) 1 (μ ) , ∀g ∈ L (μφ ), i. e. the operator ℒ Also, 󵄩󵄩󵄩ℒ ≤ ‖g‖ 1 ̂ (g)󵄩 φ L 󵄩L (μφ ) φ 󵄩 ̂ 󵄩󵄩 has for norm 󵄩󵄩󵄩ℒ ̂󵄩 φ 󵄩L1 (μφ ) = 1. ̂n̂ (g)))∞ is increasing while the sequence (c) For all g ∈ C(X), the sequence (inf(ℒ φ

̂n̂ (g))) (sup(ℒ φ n=0 is decreasing. In particular, ∞

n=0

̂n̂ (g)) ≤ sup(ℒ ̂n̂ (g)) ≤ sup(g), inf(g) ≤ inf(ℒ φ φ

∀n ∈ ℕ.

Proof. Using Lemma 13.7.5(a), we get ̂φ̂ (1) = R ∘ ℒ ̂φ ∘ Rφ (1) = ℒ φ −1

1 ̂ ℒ (ϱ ) = 1. ϱφ φ φ

󵄩󵄩 ̂φ̂ yield 󵄩󵄩󵄩ℒ ̂ The positivity and linearity of ℒ 󵄩 φ̂ (f )󵄩󵄩∞ ≤ ‖f ‖∞ for any f ∈ C(X), so (a) holds. 1 For part (b), let g ∈ L (μφ ). Using Corollary 13.6.6 and (13.21), observe that ̂φ ∘ Rφ (g) ⋅ ϱφ dmφ = ∫ ℒ ̂φ (Rφ (g)) dmφ ̂φ̂ (g) dμφ = ∫ 1 ⋅ ℒ ∫ℒ ϱφ

X

X

X

= ∫ ℒmφ (Rφ (g)) dmφ = ∫ Rφ (g) dmφ = ∫ g ⋅ ϱφ dmφ = ∫ g dμφ . X

X

X

X

13.8 Finer properties of transfer operators and Gibbs states |

495

Furthermore, using (13.22) we have 1 󵄨󵄨 ̂ 󵄨󵄨 󵄨̂ 󵄨󵄨 󵄨 󵄨̂ 󵄩 󵄩󵄩 ̂ ⋅ 󵄨ℒ ∘ R (g)󵄨󵄨 ⋅ ϱ dmφ = ∫󵄨󵄨󵄨ℒ 󵄩󵄩ℒφ̂ (g)󵄩󵄩󵄩L1 (μφ ) = ∫󵄨󵄨󵄨ℒ ̂ (g)󵄨󵄨 dμφ = ∫ φ (Rφ (g))󵄨󵄨 dmφ φ ϱ 󵄨 φ φ 󵄨 φ φ

X

X

X

󵄨 󵄨 󵄨 󵄨 = ∫󵄨󵄨󵄨ℒmφ (Rφ (g))󵄨󵄨󵄨 dmφ ≤ ∫󵄨󵄨󵄨Rφ (g)󵄨󵄨󵄨 dmφ = ∫ |g|ϱφ dmφ = ‖g‖L1 (μφ ) . X

X

X

̂φ̂ is positive, it is monotone and thus for all g ∈ C(X), Concerning part (c), as ℒ ̂φ̂ (g)(x) ≤ ℒ ̂φ̂ (sup(g) ⋅ 1)(x) = sup(g) ⋅ ℒ ̂φ̂ (1)(x) = sup(g) ⋅ 1(x) = sup(g) ℒ ̂φ̂ (g)) ≤ sup(g). Similarly, inf(g) ≤ inf(ℒ ̂φ̂ (g)). Replacing g by for all x ∈ X. So sup(ℒ n ̂ ℒ ̂ (g) yields (c). φ

In light of Proposition 13.8.4(c), one may ask when the extreme inequalities turn out to be strict. Under the stronger assumption of topological exactness of T, the next lemma provides such conditions on g. Lemma 13.8.5. Let T : X → X be a topologically exact, open, distance expanding system and φ : X → ℝ a Hölder continuous potential. If g ∈ C(X) is not identically equal to 0 but ∫X g dμ = 0 for some μ ∈ M(X) with supp(μ) = X, then there exists n ∈ ℕ with ̂n̂ (g)) < sup(g). ̂n̂ (g)) ≤ sup(ℒ inf(g) < inf(ℒ φ φ Proof. Since g is not identically 0, the open set U0 := {x ∈ X | g(x) ≠ 0} is not empty. Because supp(μ) = X, we deduce that μ(U0 ) > 0. As ∫ g dμ = 0, there exist z1 , z2 ∈ X such that g(z1 ) < 0 and g(z2 ) > 0. Since g is a continuous function, there are neighborhoods U1 and U2 of z1 and z2 , respectively, on which g(y) ≤ g(z1 )/2 < 0 for all y ∈ U1 and g(y) ≥ g(z2 )/2 > 0 for all y ∈ U2 . The system T being topologically exact, there exists n ∈ ℕ such that T n (U1 ) = X = T n (U2 ). Therefore, for all x ∈ X, n

̂̂ (g)(x) = e ℒ φ

−nP(T,̂ φ)

[

g(y)eφ(y) + ̂

∑ y∈T −n (x)\U

1

≤ e−nP(T,̂φ) [sup(g) = e−nP(T,̂φ) [sup(g)

y∈T −n (x)\U1

∑ y∈T −n (x)

̂n̂ (1)(x) + ( = sup(g)ℒ φ ≤ sup(g) + (

eφ(y) + ̂

∑

eφ(y) + ( ̂

g(y)eφ(y) ] ̂

∑ y∈T −n (x)∩U

1

g(z1 ) ̂ eφ(y) ] ∑ 2 y∈T −n (x)∩U 1

g(z1 ) ̂ − sup(g)) eφ(y) ] ∑ 2 −n y∈T (x)∩U 1

g(z1 ) ̂ − sup(g))e−nP(T,̂φ) eφ(y) ∑ 2 −n y∈T (x)∩U

g(z1 ) − sup(g))e−(‖̂φ‖∞ +nP(T,̂φ)) . 2

1

496 | 13 Gibbs states and transfer operators for open, distance expanding systems ̂n̂ (g)) < sup(g). The Taking the supremum over all x on the left-hand side yields sup(ℒ φ n ̂̂ (g)) > inf(g) proceeds in an analogous fashion using the set U2 to proof that inf(ℒ split T −n (x).

φ

From this point on, we consider C(X) and Hα (X) as complex Banach spaces. Nevertheless, the potentials, usually denoted by φ, will remain real valued, which we will indicate by adding ℝ to the notation of the space; for example, φ ∈ Hα (X, ℝ). ̂φ , ℒ ̂φ̂ : C(X) → C(X) naturally extend to complexThe bounded linear operators ℒφ , ℒ valued functions g ∈ C(X) and it is easy to see that ℒφ (g) = ℒφ (Re(g)) + iℒφ (Im(g)).

̂φ and ℒ ̂φ̂ hold. We have already seen in Consequently, corresponding relations for ℒ ̂ ̂φ̂ : C(X, ℝ) → C(X, ℝ) are alTheorems 13.7.4 and 13.8.3 that the real operators ℒφ , ℒ most periodic and their complex counterparts inherit this property, and many others. We observed in Proposition 13.8.4 that for every real-valued g ∈ C(X), the sên̂ (g)))∞ is increasing while the sequence (sup(ℒ ̂n̂ (g)))∞ is decreasquence (inf(ℒ φ φ n=0 n=0 ̂n̂ (g))∞ converges uniformly and we will identify ing. It turns out that the sequence (ℒ φ

n=0

its limit. This holds for any real-valued g, and thus for complex-valued ones, too, by linearity. To accomplish this, we will need the following auxiliary result, which is also interesting on its own. Proposition 13.8.6. If T : X → X is a topologically exact, open, distance expanding system then lim sup dist(x, T −n (y)) = 0.

n→∞ x,y∈X

Proof. Fix ε > 0. As X is compact, there exists a finite (ε/2)-spanning set F ⊆ X, i. e. ⋃ B(z, ε/2) = X.

z∈F

Since the map T : X → X is topologically exact, for every z ∈ F there exists Nz ≥ 0 such that T Nz (B(z, ε/2)) = X. Setting Nε = max{Nz : z ∈ F} yields T n (B(z, ε/2)) = X,

∀z ∈ F, ∀n ≥ Nε .

Take x, y ∈ X. Then x ∈ B(z, ε/2) for some z ∈ F. For every n ≥ Nε , there exists yn ∈ B(z, ε/2) such that T n (yn ) = y. Thus, dist(x, T −n (y)) ≤ d(x, yn ) ≤ d(x, z) + d(z, yn ) < ε/2 + ε/2 = ε,

∀n ≥ Nε .

13.8 Finer properties of transfer operators and Gibbs states |

497

Lemma 13.8.7. If T : X → X is a topologically exact, open, distance expanding system and φ : X → ℝ is a Hölder continuous potential, then for every function g ∈ C(X), the ̂n̂ (g))∞ converges uniformly in the complex Banach space C(X) and sequence (ℒ n=0 φ ̂n̂ (g) = μφ (g)1. lim ℒ φ

n→∞

Proof. Splitting g into its real and imaginary parts, we may assume without loss of gen̂n̂ (g))∞ . To lighten erality that g is a real-valued function. Take any subsequence of (ℒ n=0 φ notation, we will identify this subsequence in the same way as the sequence. By Theorem 13.8.3, there exists a strictly increasing sequence (nk )∞ k=1 such that the sequence ̂nk (g))∞ converges uniformly. Denote its limit by g ∗ ∈ C(X). By Proposition 13.8.4(c), (ℒ k=1 ̂ φ ̂n̂ (g ∗ )))∞ is decreasing and the sequence (sup(ℒ φ

n=0

̂n̂ (g ∗ )) ≤ sup(g ∗ ), sup(ℒ φ

∀n ≥ 0.

(13.62)

̂nk (g)). The deBut the uniform convergence imposes that sup(g ∗ ) = limk→∞ sup(ℒ ̂ φ ̂n̂ (g)))∞ implies that creasing nature of the sequence (sup(ℒ φ

n=0

n

̂n̂ (g)) = lim sup(ℒ ̂n̂ (g)) = lim sup(ℒ ̂ k (g)) = sup(g ∗ ). inf sup(ℒ φ φ ̂ φ

n≥0

n→∞

k→∞

Hence, ̂n̂ (g ∗ )) ≥ sup(g ∗ ), sup(ℒ φ

∀n ≥ 0.

(13.63)

Inequalities (13.62)–(13.63) give ̂n̂ (g ∗ )) = sup(g ∗ ), sup(ℒ φ

∀n ≥ 0.

Since X is compact, for every n ≥ 0 there exists a point xn ∈ X such that ̂n̂ (g ∗ )) = ℒ ̂n̂ (g ∗ )(xn ). sup(ℒ φ φ Therefore, −P(T,φ)n ̂n̂ (g ∗ )(xn ) = e sup(g ∗ ) = ℒ φ ϱφ (xn )

∑ y∈T −n (xn )

ϱφ (y)g ∗ (y)eSn φ(y) .

By Proposition 13.8.4(a), the right-hand side of this equation is a convex combination of real numbers smaller than or equal to sup(g ∗ ). Consequently, g ∗ (y) = sup(g ∗ ),

∀y ∈ T −n (xn ), ∀n ≥ 0.

498 | 13 Gibbs states and transfer operators for open, distance expanding systems As the function g ∗ : X → ℝ is continuous, it follows from Proposition 13.8.6 that g ∗ is constant. From Proposition 13.8.4(b), we then get that ̂nk (g)) = lim μφ (ℒ ̂nk (g)) = lim μφ (g) = μφ (g). g ∗ = μφ (g ∗ ) = μφ ( lim ℒ ̂ ̂ φ φ k→∞

k→∞

k→∞

∞ So we have proved that every subsequence (nj )∞ j=1 contains a sub-subsequence (nji )i=1 nji ̂ (g))∞ converges to μφ (g)1 in C(X). This means that the sequence such that (ℒ ̂ φ

i=1

̂n̂ (g))∞ converges uniformly. (ℒ n=0 φ Let

󵄨 ℂ⊥ := {g ∈ C(X) 󵄨󵄨󵄨 μφ (g) = 0}. As a straightforward consequence of Lemma 13.8.7 and Proposition 13.8.4, the space ̂φ̂ -invariant subspaces and the C(X) is the direct sum of two simply expressed, ℒ ̂φ̂ -orbits of its points converge as follows. ℒ Theorem 13.8.8. Let T : X → X be a topologically exact, open, distance expanding system and let φ : X → ℝ be a Hölder continuous potential. Then: (a) ℂ1 and ℂ⊥ are both closed vector subspaces (so are Banach subspaces) of C(X). ̂φ̂ (ℂ1) = ℂ1 while ℒ ̂φ̂ (ℂ⊥ ) ⊆ ℂ⊥ . (b) ℒ ⊥ (c) C(X) = ℂ1 ⨁ ℂ . ̂n̂ (g))∞ converges uniformly, i. e., in the Banach (d) If g ∈ C(X) then the sequence (ℒ n=0 φ space C(X), and ̂n̂ (g) = μφ (g)1. lim ℒ φ

n→∞

(e) In particular, if g ∈ ℂ⊥ then ̂n̂ (g) = 0. lim ℒ φ

n→∞

Let 󵄨 (ℂϱφ )⊥ := {g ∈ C(X) 󵄨󵄨󵄨 mφ (g) = 0}. As a direct repercussion of Theorem 13.8.8 and (13.61), the space C(X) is the direct ̂φ -invariant subspaces and the ℒ ̂φ -orbits of its points sum of two simply expressed, ℒ converge as follows. Theorem 13.8.9. Let T : X → X be a topologically exact, open, distance expanding system and let φ : X → ℝ be a Hölder continuous potential. Then: (a) ℂϱφ and (ℂϱφ )⊥ are closed vector subspaces (so are Banach subspaces) of C(X). ̂φ (ℂϱφ ) = ℂϱφ while ℒ ̂φ ((ℂϱφ )⊥ ) ⊆ (ℂϱφ )⊥ . (b) ℒ ⊥ (c) C(X) = (ℂϱφ ) ⨁(ℂϱφ ) .

13.8 Finer properties of transfer operators and Gibbs states | 499

̂n (g))∞ converges uniformly and (d) If g ∈ C(X) then the sequence (ℒ φ n=0 ̂n (g) = mφ (g)ϱφ . lim ℒ φ

n→∞

(e) In particular, if g ∈ (ℂϱφ )⊥ then ̂n (g) = 0. lim ℒ φ

n→∞

̂φ Finally, we establish the uniqueness of the eigenfunction ϱφ for the operator ℒ and the uniqueness of the eigenpair (eigenvalue, eigenmeasure) = (Λφ , mφ ) for the dual operator ℒ∗φ . Proposition 13.8.10. Let T : X → X be a topologically exact, open, distance expanding system and let φ : X → ℝ be a Hölder continuous potential. Then: (a) There exists a unique pair (Λ, m) with Λ > 0 and m ∈ M(X) such that ℒ∗φ (m) = Λm; namely, Λ = Λφ = eP(T,φ) and m = mφ . ̂φ (ϱ) = ϱ with ϱ ≥ 0 and ∫ ϱ dmφ = 1; (b) There exists a unique ϱ ∈ C(X) such that ℒ X namely ϱ = ϱφ .

Proof. Starting with Theorem 13.6.2, all that was assumed about mφ and Λφ was that mφ is an eigenmeasure for the dual operator ℒ∗φ : C(X)∗ → C(X)∗ with eigenvalue Λφ > 0, i. e., ℒ∗φ (mφ ) = Λφ mφ . Thus, all the results from that point on apply to any pair (Λ, m), with Λ ∈ (0, ∞) and m ∈ M(X) such that ℒ∗φ (m) = Λm. Eventually, we proved in Proposition 13.6.9 that such a measure is conformal and this ultimately helped us establish in Proposition 13.6.16 that such a measure is a Gibbs state for φ. The uniqueness of Λ as Λφ = eP(T,φ) ensues from Proposition 13.2.4. For this, only transitivity of T was needed. Likewise, under the stronger assumption that T is topologically exact but assuming only the equality ℒ∗φ (m) = Λm, we derived Theorem 13.8.9. Part (d) of that theorem (with g = 1) reveals that ̂n (1), ϱ = lim ℒ φ n→∞

which is completely independent of m while Λ has already been shown to be unique. So ϱ is unique, i. e., ϱ = ϱφ , and the above is a formula for it. Per Corollary 13.7.8, the ̂φ : C(X) → C(X) such that ϱφ ≥ 0 function ϱφ is the unique fixed point of the operator ℒ and ∫X ϱφ dmφ = 1. In fact, ϱφ is bounded away from 0 and ∞. Replacing the above expression for ϱφ in part (d) of Theorem 13.8.9 yields m(g) = lim

n→∞

̂n (g)(x) ℒ φ ̂n (1)(x) ℒ φ

,

∀x ∈ X.

This formula establishes the uniqueness of m as mφ .

500 | 13 Gibbs states and transfer operators for open, distance expanding systems Corollary 13.8.11. Let T : X → X be a topologically exact, open, distance expanding, Hölder continuous system and let φ : X → ℝ be a Hölder continuous potential. Then: (a) There exists a unique pair (Λ, m) with Λ > 0 and m ∈ M(X) such that ℒ∗φ̂ (m) = Λm;

namely, Λ = Λφ = eP(T,φ) and m = mφ̂ = μφ . ̂φ̂ (ϱ) = ϱ with ϱ ≥ 0 and ∫ ϱ dmφ̂ = 1; (b) There exists a unique ϱ ∈ C(X) such that ℒ X namely ϱ = ϱφ̂ = 1. (c) μφ̂ = μφ .

̂ at the start of Subsection 13.8.1, we observed that φ ̂ Proof. When we introduced φ is Hölder continuous when T is. So we can use Proposition 13.8.10 with φ replaced ̂. We also noted at the beginning of Subsection 13.8.1 that P(T, φ ̂) = P(T, φ) and with φ ∗ P(T,φ) ℒφ̂ (μφ ) = e μφ . Thus, (a) holds. Furthermore, Proposition 13.8.4 shows that ϱφ̂ = 1. Hence, (b) holds. Finally, we noticed at the outset of Subsection 13.8.1 that 𝒢φ̂ = 𝒢φ . The uniqueness of the T-invariant Gibbs states μφ and μφ̂ implies that μφ̂ = μφ . In Theorem 13.8.8, we described the uniform convergence of the iterates of a con̂φ̂ . We now establish the L1 (μφ )-convergence tinuous function g under the operator ℒ 1 of the iterates of a L (μφ )-function g. Theorem 13.8.12. Let T : X → X be a topologically exact, open, distance expanding system and let φ : X → ℝ be a Hölder continuous potential. For every g ∈ L1 (μφ ), 󵄩 ̂n lim 󵄩󵄩ℒ ̂ (g) n→∞󵄩 φ

󵄩 − μφ (g)1󵄩󵄩󵄩L1 (μ ) = 0. φ

Proof. Fix ε > 0. As C(X) is a dense subset of L1 (μφ ), there exists f ∈ C(X) such that ‖g − f ‖L1 (μφ ) < ε/2. Then 󵄨󵄨 󵄨 󵄨󵄨μφ (g) − μφ (f )󵄨󵄨󵄨 < ε/2. It follows from Proposition 13.8.4(b) and Lemma 13.8.7 that 󵄩 ̂n 󵄩󵄩 lim sup󵄩󵄩󵄩ℒ ̂ (g) − μφ (g)1󵄩 φ 󵄩L1 (μ n→∞

φ)

󵄩 ̂n ̂n̂ (f ) − μφ (f )1) + (μφ (f ) − μφ (g))1󵄩󵄩󵄩 1 = lim sup󵄩󵄩󵄩ℒ ̂ (g − f ) + (ℒφ φ 󵄩L (μφ ) n→∞ 󵄩 ̂n 󵄩 󵄩 ̂n 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ lim sup󵄩󵄩ℒφ̂ (g − f )󵄩󵄩L1 (μ ) + lim sup󵄩󵄩ℒφ̂ (f ) − μφ (f )1󵄩󵄩L1 (μ ) φ φ n→∞ n→∞ 󵄩 󵄩 󵄩 󵄩 + 󵄩󵄩(μφ (f ) − μφ (g))1󵄩󵄩L1 (μ ) φ 󵄩 ̂n 󵄩󵄩 󵄨󵄨 󵄨󵄨 ≤ ‖g − f ‖L1 (μφ ) + lim sup󵄩󵄩󵄩ℒ ̂ (f ) − μφ (f )1󵄩 φ 󵄩∞ + 󵄨󵄨μφ (f ) − μφ (g)󵄨󵄨 n→∞

< ε/2 + 0 + ε/2 = ε.

󵄩󵄩 󵄩 ̂n The arbitrariness of ε means that lim sup󵄩󵄩󵄩ℒ ̂ (g) − μφ (g)1󵄩 φ 󵄩L1 (μ ) = 0. n→∞

φ

13.8 Finer properties of transfer operators and Gibbs states | 501

As a fairly immediate consequence of this theorem, we obtain the following remarkable result. Theorem 13.8.13. Let T : X → X be a topologically exact, open, distance expanding system and let φ : X → ℝ be a Hölder continuous potential. Then the measure-preserving dynamical system (T, μφ ) is totally ergodic. That is, all iterates T q : X → X, q ∈ ℕ, are ergodic with respect to the measure μφ , including T itself. Proof. Fix q ∈ ℕ. Suppose that A ⊆ X is a completely T q -invariant Borel set, i. e., T −q (A) = A. Then T −qn (A) = A for all n ≥ 0. Equivalently, 1A ∘ T qn = 1A for all n ≥ 0. Therefore, qn

qn

qn

qn

̂ (1) = 1A . ̂ (1A ∘ T ) = 1A ⋅ ℒ ̂ (1A ) = ℒ ℒ ̂ ̂ ̂ φ φ φ It follows from Theorem 13.8.12 that 1A = μφ (A)1 μφ -almost everywhere. Thus, μφ (A) ∈ {0, 1}, yielding the ergodicity of T q . In fact, we can easily prove more. Theorem 13.8.14. Let T : X → X be a topologically exact, open, distance expanding system and let φ : X → ℝ be a Hölder continuous potential. Then the measure-preserving dynamical system (T, μφ ) is metrically exact, and hence its Rokhlin natural extension is a K-system. Proof. Let ℬ(X) be the Borel σ-algebra of X. Recall that the measure-preserving dynamical system (X, T, μφ ) is metrically exact if and only if the tail σ-algebra ∞

⋂ T −n (ℬ(X))

n=0

consists solely of sets of μφ -measure 0 or 1 (cf. Definition 8.4.4 and Proposition 8.4.5). −n −n Let A ∈ ⋂∞ n=0 T (ℬ (X)). For every n ≥ 0 there is a set An ∈ ℬ (X) such that A = T (An ). Equivalently, 1A = 1An ∘ T n . Consequently, n

n

n

n

̂̂ (1A ) = ℒ ̂̂ (1A ∘ T ) = 1A ⋅ ℒ ̂̂ (1) = 1A . ℒ φ φ φ n n n It therefore follows from Theorem 13.8.12 that 1An converges in L1 (μφ ) to the constant function μφ (A)1. But since 1An takes on only the two values 0 and 1, we conclude that μφ (A) ∈ {0, 1}. The metric exactness of (X, T, μφ ) is thus proved. That its Rokhlin natural extension is a K-system follows directly from Theorem 8.4.6.

502 | 13 Gibbs states and transfer operators for open, distance expanding systems 13.8.2 Ionescu-Tulcea and Marinescu inequality and spectral gap The complex Banach space (Hα (X), ‖ ⋅ ‖α ) is a vector subspace of the complex Banach space (C(X), ‖ ⋅ ‖∞ ). We shall now derive a Ionescu-Tulcea and Marinescu type of inequality (see condition (c1) at the beginning of Appendix B). We first get such an inequality with respect to υα (⋅); see Section 13.1. Theorem 13.8.15. Let T : X → X be a topologically transitive, open, distance expanding ̂n (g) ∈ Hα (X) and system and φ ∈ Hα (X, ℝ). If g ∈ Hα (X), then for all n ∈ ℕ we have ℒ φ ̂n (g)) ≤ Dα (φ)λ−αn υα (g) + D′ (φ)‖g‖∞ , υα (ℒ α φ ̃α (φ), 2Dα (φ)/ξ α } and the constants Dα (φ) and D ̃α (φ) respecwhere D′α (φ) := max{D tively arise from Corollary 13.1.2 and (13.47) in the proof of Theorem 13.7.4. Proof. Fix x, y ∈ X. Suppose first that d(x, y) < ξ . From a slight alteration of (13.46) in the proof of Theorem 13.7.4, we obtain 󵄨󵄨 ̂n ̂n (g)(x)󵄨󵄨󵄨 ≤ Dα (φ) max{󵄨󵄨󵄨g(T −n (y)) − g(T −n (x))󵄨󵄨󵄨 : z ∈ T −n (x)} 󵄨󵄨ℒφ (g)(y) − ℒ φ z 󵄨 󵄨 z 󵄨 α ̃ + Dα (φ)‖g‖∞ d (x, y)

≤ Dα (φ)υα (g) max{dα (Tz−n (y), Tz−n (x)) : z ∈ T −n (x)} +̃ Dα (φ)‖g‖∞ dα (x, y)

≤ Dα (φ)υα (g)λ−αn dα (x, y) + ̃ Dα (φ)‖g‖∞ dα (x, y) = [Dα (φ)λ−αn υα (g) + ̃ Dα (φ)‖g‖∞ ]dα (x, y). (13.64) If d(x, y) ≥ ξ , then using Proposition 13.7.2 we get dα (x, y) 󵄨󵄨 ̂n 󵄩󵄩 ̂n 󵄩󵄩 ̂n (g)(x)󵄨󵄨󵄨 ≤ 2󵄩󵄩󵄩ℒ ̂n 󵄩󵄩 󵄨󵄨ℒφ (g)(y)− ℒ φ 󵄨 󵄩 φ (g)󵄩󵄩∞ ≤ 2‖g‖∞ 󵄩󵄩ℒφ (1)󵄩󵄩∞ ≤ 2‖g‖∞ Dα (φ) ξ α . (13.65) Combining (13.64) and (13.65), we conclude that the expression Dα (φ)λ−αn υα (g) + ̂n (g). max{̃ Dα (φ), 2Dα (φ)/ξ α }‖g‖∞ is a Hölder constant for ℒ φ We now obtain the sought Ionescu-Tulcea and Marinescu inequality. Corollary 13.8.16. Let T : X → X be a topologically transitive, open, distance expanding system and φ ∈ Hα (X, ℝ). There exists N ∈ ℕ such that for all n ≥ N and all g ∈ Hα (X), we have 1 󵄩󵄩 ̂n 󵄩󵄩 ′′ 󵄩󵄩ℒφ (g)󵄩󵄩α ≤ ‖g‖α + Dα (φ)‖g‖∞ , 2 ′ where D′′ α (φ) := Dα (φ) + Dα (φ).

13.8 Finer properties of transfer operators and Gibbs states |

503

Proof. Using Theorem 13.8.15 and Proposition 13.7.2, we obtain for all n ∈ ℕ that 󵄩 ̂n 󵄩󵄩 󵄩󵄩 ̂n 󵄩󵄩 −αn ̂n 󵄩󵄩 ̂n (g)) + 󵄩󵄩󵄩ℒ υα (g) + D′α (φ)‖g‖∞ + 󵄩󵄩󵄩ℒ 󵄩󵄩ℒφ (g)󵄩󵄩α := υα (ℒ φ φ (1)󵄩 󵄩 φ (g)󵄩󵄩∞ ≤ Dα (φ)λ 󵄩∞ ‖g‖∞ ≤ Dα (φ)λ−αn ‖g‖α + (D′α (φ) + Dα (φ))‖g‖∞ .

Take N so large that Dα (φ)λ−αN ≤ 1/2. As a consequence of the extended Ionescu-Tulcea and Marinescu Theorem (see Appendix B), our main result affirms that the normalized transfer operator is quasicompact. We first recall the definition of this concept. Definition 13.8.17. Let B be a Banach space. A bounded linear operator L : B → B is called quasi-compact if it exhibits a spectral gap, i. e., if its spectral radius r(L) is larger than its essential spectral radius ress (L). The spectral radius r(L) is the largest modulus of the eigenvalues of the operator L (i. e., the supremum of the modulus of all elements in the spectrum of L). In terms of the operator norm, 󵄩 󵄩1/n r(L) = lim 󵄩󵄩󵄩Ln 󵄩󵄩󵄩 . n→∞

The essential spectral radius is 󵄩 󵄩1/n ress (L) := lim [inf 󵄩󵄩󵄩Ln − K 󵄩󵄩󵄩 ], n→∞

K

where the infimum is taken over all compact operators K : B → B. Theorem 13.8.18. If T : X → X is a topologically exact, open, distance expanding system and φ ∈ Hα (X, ℝ), then: ̂φ : Hα (X) → Hα (X). (a) The number 1 is a simple isolated eigenvalue of ℒ (b) The eigenspace of the eigenvalue 1 is generated by the function ϱφ ∈ Hα (X, ℝ) arising from Lemma 13.7.5 (see also Theorem 13.7.7), i. e., the eigenspace of 1 is ℂϱφ . (c) For all n ∈ ℕ, n

n

̂ = L1 + S , ℒ φ where (d) L1 : Hα (X) → Hα (X) is the projector onto the eigenspace ℂϱφ given by L1 (ψ) = [∫ ψ dmφ ]ϱφ ; X

̂φ − L1 ; (e) S : Hα (X) → Hα (X) is simply S = ℒ (f) L1 ∘ S = S ∘ L1 = 0;

(13.66)

504 | 13 Gibbs states and transfer operators for open, distance expanding systems (g) r(S) < 1, where r(S) denotes the spectral radius of S; and ̂φ ) \ {1} = σ(S) ⊆ B(0, r(S)), where σ(R) denotes the spectrum of R. (h) σ(ℒ Proof. We want to apply the Ionescu-Tulcea and Marinescu theorem (Theorem B.1.1) ̂φ and with N with (B, ‖ ⋅ ‖) = (Hα (X), ‖ ⋅ ‖α ), with (E, | ⋅ |) = (C(X), ‖ ⋅ ‖∞ ), with T = ℒ coming from Corollary 13.8.16. We must check that conditions (a1)–(d1) are satisfied. Condition (b1) follows from Proposition 13.7.2 while (c1) is precisely Corollary 13.8.16 with n = N, r = 1/2 and R = D′′ α (φ). By Arzelà–Ascoli’s theorem (Theorem A.2.3), every bounded subset of (Hα (X), ‖ ⋅ ‖α ) is relatively compact in (C(X), ‖ ⋅ ‖∞ ). Since ̂φ : Hα (X) → Hα (X) is a bounded operator, this establishes (d1) per Remark B.1.2. ℒ Finally, for (a1), suppose that (gn )∞ n=1 ⊆ Hα (X) with ‖gn ‖α ≤ K and gn → g uniformly in C(X). We need to verify that g ∈ Hα (X) and ‖g‖α ≤ K. Let x, y ∈ X. For every n ∈ ℕ, we have 󵄨󵄨 󵄨 α 󵄨󵄨gn (x) − gn (y)󵄨󵄨󵄨 ≤ υα (gn ) d (x, y), where 0 ≤ υα (gn ) ≤ ‖gn ‖α ≤ K. Letting n → ∞, we get 󵄨󵄨 󵄨 inf υα (gn ) dα (x, y). 󵄨󵄨g(x) − g(y)󵄨󵄨󵄨 ≤ lim n→∞ Hence, υα (g) ≤ lim infn→∞ υα (gn ) ≤ K. Moreover, since gn → g uniformly, we have ‖gn ‖∞ → ‖g‖∞ as n → ∞. Therefore, ‖g‖α = υα (g) + ‖g‖∞ ≤ lim inf υα (gn ) + lim ‖gn ‖∞ = lim inf ‖gn ‖α ≤ K. n→∞

n→∞

n→∞

So (a1) is satisfied. We can thus apply the Ionescu-Tulcea and Marinescu theorem. ̂φ ) \ {λ1 , . . . , λp } = σ(S) ⊆ B(0, r(S)) ⊆ ℂ, By (f2) in that theorem, we know that σ(ℒ 1 where λ1 , . . . , λp ∈ 𝕊 and 0 < r(S) < 1. ̂φ : Hα (X) → Hα (X) and its Claim 1. The number 1 is a simple isolated eigenvalue for ℒ corresponding eigenspace is ℂϱφ . Proof of Claim 1. We already know from Lemma 13.7.5 that ϱφ ∈ Hα (X, ℝ) is an eigen̂φ with eigenvalue 1 and it is clear from (f2) above that the number 1 is an vector for ℒ ̂φ . isolated eigenvalue of ℒ ̂ Suppose that ℒφ (g) = 1 ⋅ g = g for some nonzero function g ∈ Hα (X). Since g is nonzero, at least one of its real and imaginary part functions Re(g) and Im(g) does not identically vanish. Denote by ψ whichever is nonvanishing. Then ψ ∈ Hα (X) and ̂φ (ψ) = ψ. It remains to show that ψ is a real multiple of ϱφ . By Lemma 13.7.5(b), ℒ there exists M > 0 so large that c−1 ≤ ψ + Mϱφ ≤ c for some constant c ≥ 1. But ̂φ (ψ + Mϱφ ) = ψ + Mϱφ and, therefore, the function ℒ ψ + Mϱφ

∫X (ψ + Mϱφ ) dmφ

13.8 Finer properties of transfer operators and Gibbs states | 505

satisfies properties (a–c) in Lemma 13.7.5. According to Corollary 13.7.8, this function is nothing else than ϱφ . Consequently, ψ + Mϱφ = [∫(ψ + Mϱφ ) dmφ ]ϱφ . X

Equivalently, ψ = [∫(ψ + Mϱφ ) dmφ − M]ϱφ =: Cϱφ . X

Thus, the number 1 is a simple eigenvalue and its corresponding eigenspace is ℂϱφ . An alternative argument for this latter property is the following. ̂φ (h) = h and Subclaim 1*. If h : X → [0, ∞) is a continuous function such that ℒ h(x) = 0 for some x ∈ X, then h ≡ 0. Proof of Subclaim 1*. Since h is continuous, relation (13.30) in Proposition 13.6.7 holds everywhere. Then for every n ∈ ℕ we have that ̂n (h)(x) = e−nP(T,φ) ℒn (h)(x) = e−nP(T,φ) 0 = h(x) = ℒ φ φ

∑ y∈T −n (x)

h(y)eSn φ(y) .

−n Therefore, h(y) = 0 for every y ∈ ⋃∞ n=0 T (x). Since h is continuous, it ensues that ∞ −n h(y) = 0 for every y ∈ ⋃n=0 T (x). As T is transitive, open and distance expanding, recall from Proposition 13.6.14 that T is very strongly transitive (alternatively, see Lemma 4.2.10). This implies that T is strongly transitive, i. e. the backward T-orbit of any point is dense in X (see Definition 1.5.14). So h is the constant function 0 and Subclaim 1* follows. ⧫

Now assume that g ∈ C(X) is an eigenfunction associated with the eigenvalue 1 ̂φ : C(X) → C(X). Given that ℒ ̂φ preserves the vector subspace of of the operator ℒ C(X) consisting of real-valued functions, both the real and imaginary parts of g are also eigenfunctions associated with the eigenvalue 1. So we may assume without loss of generality that g is real valued. By Lemma 13.7.5, the function g/ϱφ : X → ℝ is continuous. Since X is compact, this function attains its minimum value; denote this latter by s. Then g−sϱφ ≥ 0 and there exists x ∈ X such that (g−sϱφ )(x) = 0. Since g−sϱφ is an eigenfunction associated with the eigenvalue 1, too, it follows from Subclaim 1* that g − sϱφ ≡ 0; equivalently, g = sϱφ . So, again, the number 1 is a simple eigenvalue and its corresponding eigenspace is ℂϱφ . Note that this alternative argument considers ̂φ : C(X) → C(X), as opposed to ℒ ̂φ : Hα (X) → Hα (X), and does not the operator ℒ require the use of Corollary 13.7.8. In any case, Claim 1 is proved. ◼

506 | 13 Gibbs states and transfer operators for open, distance expanding systems Note that until this point, topological transitivity of T is sufficient. We will need topological exactness to establish the next claim. ̂φ : Hα (X) → Hα (X) having moduClaim 2. The number 1 is the only eigenvalue of ℒ lus 1. ̂φ . Then there exists a Proof of Claim 2. Suppose that λ ≠ 1 is a unitary eigenvalue of ℒ ̂φ (g) = λg. Therefore, nonzero function g ∈ C(X) such that ℒ ̂φ (g) dmφ = ∫ ℒm (g) dmφ = ∫ g dmφ . λ ∫ g dmφ = ∫ ℒ φ X

X

X

X

̂n (g) → 0 Since λ ≠ 1, we deduce that ∫ g dmφ = 0. By Theorem 13.8.9(d), λn g = ℒ φ uniformly in C(X). So g is identically equal to 0. This contradiction finishes the proof of Claim 2. ◼ ̂φ ) \ {1} = σ(S) ⊆ B(0, r(S)). In particular, this shows that σ(ℒ All that is left to show (the rest follows directly from the Ionescu-Tulcea and Marinescu theorem) is (13.66). From (b), it follows that L1 (ψ) = aϱφ for some a ∈ ℂ. By (c), n

n

n

̂ (ψ) = L1 (ψ) + S (ψ) = aϱφ + S (ψ). ℒ φ Using Lemma 13.7.5(c), we deduce that ̂n (ψ) dmφ − ∫ Sn (ψ) dmφ a = ∫ aϱφ dmφ = ∫ ℒ φ X

X

=

X

∫ ℒnmφ (ψ) dmφ

− ∫ Sn (ψ) dmφ

X

X n

= ∫ ψ dmφ − ∫ S (ψ) dmφ X

→ ∫ ψ dmφ

X

when n → ∞ by (g).

X

Hence a = ∫X ψ dmφ and (13.66) holds. Corollary 13.8.19. If T : X → X is a topologically exact, open, distance expanding syŝφ is quasi-compact. tem and φ ∈ Hα (X, ℝ), then the normalized transfer operator ℒ ̂φ is r(ℒ ̂φ ) = 1. Note that each Proof. Per Theorem 13.8.18, the spectral radius of ℒ finite-rank operator (i. e., whose range is finite-dimensional) is compact. So the onedimensional projection operator L1 onto the eigenspace ℂρφ is compact. By Theôn − L1 ‖1/n = limn→∞ ‖Sn ‖1/n = r(S) < 1 and rem 13.8.18 still, we know that limn→∞ ‖ℒ φ ̂ ̂ ̂ ̂φ has a spectral gap; in other words, it thus ress (ℒφ ) < 1. So r(ℒφ ) − ress (ℒφ ) > 0, i. e., ℒ is quasi-compact.

13.8 Finer properties of transfer operators and Gibbs states | 507

Note: For some dynamicists, a spectral gap means quasi-compactness of the operator plus simplicity of its leading eigenvalue as well as the absence of any other eigenvalue of maximum modulus (i. e., equal to the operator’s spectral radius). Theorem 13.8.18 asserts that the normalized transfer operator has all of these features. Distinguishing those parts of the proofs of Theorem 13.8.18 and Corollary 13.8.19 that require topological exactness from those that do not, we get a similar result assuming only topological transitivity. Theorem 13.8.20. If T : X → X is a transitive, open, distance expanding dynamical system and φ ∈ Hα (X, ℝ), then: ̂φ : Hα (X) → Hα (X). (a) The number 1 is a simple isolated eigenvalue of ℒ (b) The eigenspace of the eigenvalue 1 is generated by the function ϱφ ∈ Hα (X, ℝ) arising from Lemma 13.7.5 (see also Theorem 13.7.7), i. e., the eigenspace of 1 is ℂϱφ . (c) For all n ∈ ℕ, n

p

n

n

̂ = ∑ λ Lj + S , ℒ φ j j=1

where (d) L1 : Hα (X) → Hα (X) is the projector onto the eigenspace ℂϱφ given by L1 (ψ) = [∫ ψ dmφ ]ϱφ ;

(13.67)

X

(d′ ) For every j ≥ 2, Lj is a finite-rank projector, meaning that L2j = Lj and ̂φ with modulus 1 and dim(Range(Lj )) < ∞; moreover, λj is an eigenvalue of ℒ ̂φ ) = Range(Lj ); Ker(λj Id − ℒ ̂φ − ∑p λj Lj ; (e) S : Hα (X) → Hα (X) is simply S = ℒ j=1 (f) Lj ∘ S = S ∘ Lj = 0 for all j = 1, 2, . . . , p and Li ∘ Lj = 0 whenever i ≠ j; (g) r(S) < 1, where r(S) denotes the spectral radius of S; ̂φ ) \ {λj : 1 ≤ j ≤ p} = σ(S) ⊆ B(0, r(S)), where σ(S) is the spectrum of S; and (h) σ(ℒ ̂ (i) ℒφ is quasi-compact.

13.8.3 Continuity of Gibbs states Theorem 13.8.21. If T : X → X is a topologically exact, open, distance expanding system, then for every α > 0 and every R > 0 the function BHα (X) (0, R) ∋ φ 󳨃󳨀→ ϱφ ∈ C(X)

508 | 13 Gibbs states and transfer operators for open, distance expanding systems is continuous with respect to the topologies of uniform convergence on both the domain and codomain, and the functions BHα (X) (0, R) ∋ φ 󳨃󳨀→ mφ , μφ ∈ M(X) are continuous with respect to the topology of uniform convergence on the domain and the weak∗ topology on the codomain. Proof. We begin with the assertion about the eigenmeasures mφ . Let (ψn )∞ n=1 be a sequence of potentials in BHα (X) (0, R) converging uniformly to some potential ψ : X → ℝ. This latter belongs to BHα (X) (0, R) as that set is compact by Arzelà–Ascoli’s theorem (Theorem A.2.3 in Appendix A). Per Theorem A.1.59 the set M(X) is compact, so let ∞ (mψn )∞ k=1 be any convergent subsequence of the sequence (mψn )n=1 . Denote its limit k by m. We obviously have the following. Claim 1. The function C(X) ∋ h 󳨃󳨀→ ℒh ∈ L(C(X))

(13.68)

is continuous. As an immediate consequence, we get the following. Claim 2. The function C(X) ∋ h 󳨃󳨀→ ℒ∗h ∈ L(C(X)∗ )

(13.69)

is continuous. According to Exercise 11.5.7, the topological pressure function P : C(X) → ℝ is Lipschitz with constant 1, and thus lim P(T, ψn ) = P(T, ψ). We then deduce that n→∞

m = lim mψn = lim e−P(T,ψnk ) ℒ∗ψn (mψn ) = e−P(T,ψ) ℒ∗ψ (m). k→∞

k

k→∞

k

k

So ℒ∗ψ (m) = eP(T,ψ) m, whence m = mψ by Proposition 13.8.10(a). The proof of the continuity of φ 󳨃→ mφ is complete. Moving on to the assertion about the eigenfunctions ϱφ , suppose for a contradiction that its continuity claim fails at some potential ψ ∈ BHα (X) (0, R). This means that there exist δ > 0 and a sequence (ψk )∞ k=1 of potentials in BHα (X) (0, R) which converge uniformly to ψ but are such that ‖ϱψk − ϱψ ‖∞ > δ,

∀k ∈ ℕ.

(13.70)

By Remark 13.7.6, ‖ϱψk ‖α ≤ Vα (R) for all k ∈ ℕ. By virtue of Arzelà–Ascoli’s theorem (Theorem A.2.3) and by passing to a subsequence if necessary, we may assume that

13.9 Stochastic laws | 509

the sequence (ϱψk )∞ k=1 converges uniformly to some function ϱ ∈ C(X). By (13.70), ϱ ≠ ϱψ .

(13.71)

On the other hand, using Claim 1 and the continuity of the pressure function, we get ℒψ (ϱ) = e

P(T,ψ)

ϱ.

(13.72)

Furthermore, using the known continuity of φ 󳨃→ mφ and Lemma 13.7.5, we have ∫ ϱ dmψ = lim ∫ ϱψk dmψk = lim 1 = 1.

X

k→∞

(13.73)

k→∞

X

Finally, the nonnegativity of the functions (ϱψk )∞ k=1 ensures that ϱ ≥ 0.

(13.74)

According to Proposition 13.8.10, it ensues from relations (13.72)–(13.74) that ϱ = ϱψ , contrary to (13.71). The continuity of the invariant Gibbs states μφ is a consequence of the continuity of the eigenmeasures mφ and of the eigenfunctions ϱφ . Indeed, if ψ ∈ BHα (X) (0, R) and (ψn )∞ n=1 is a sequence of functions in BHα (X) (0, R) converging uniformly to ψ, then for every function g ∈ C(X) we obtain that lim ∫ g dμψn = lim ∫ gϱψn dmψn = ∫ gϱψ dmψ = ∫ g dμψ .

n→∞

n→∞

X

X

X

X

∗ This means that the sequence (μψn )∞ n=1 converges to μψ in the weak topology on M(X). The proof of Theorem 13.8.21 is thus complete.

13.9 Stochastic laws Let (X, ℱ , μ) be a probability space and (Yn )∞ n=0 a sequence of identically distributed random variables in L2 (μ). Define the (m, n)-correlation of this sequence to be 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 Cm,n := 󵄨󵄨󵄨∫ Ym ⋅ Yn dμ − ∫ Ym dμ ⋅ ∫ Yn dμ󵄨󵄨󵄨. 󵄨󵄨 󵄨󵄨 󵄨X 󵄨 X X If the variables (Yn )∞ n=0 are independent, then Cm,n = 0 whenever m ≠ n. The random variables (Yn )∞ are mixing if n=0 Cm,n → 0 as |m − n| → ∞. Intuitively, the faster the correlations approach 0, the bigger the mixing.

510 | 13 Gibbs states and transfer operators for open, distance expanding systems Now suppose that T : (X, ℱ , μ) → (X, ℱ , μ) is a measure-preserving transformation. Suppose also that f , g ∈ L2 (μ) and set Yn = f ∘ T n . The sequence (f ∘ T n )∞ n=0 is identically distributed because of the T-invariance of μ. Let m ≤ n. Then 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 Cm,n (f ) := 󵄨󵄨󵄨∫(f ∘ T m )(f ∘ T n ) dμ − ∫ f ∘ T m dμ ⋅ ∫ f ∘ T n dμ󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨X X X 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 = 󵄨󵄨󵄨∫(f ⋅ (f ∘ T n−m )) ∘ T m dμ − ∫ f dμ ⋅ ∫ f dμ󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨X X X 󵄨󵄨 2 󵄨󵄨 󵄨󵄨 󵄨󵄨 = 󵄨󵄨󵄨∫ f ⋅ (f ∘ T n−m ) dμ − (∫ f dμ) 󵄨󵄨󵄨 = Cn−m (f , f ), 󵄨󵄨 󵄨󵄨 󵄨X 󵄨 X where (with Eμ (g) := μ(g) := ∫X g dμ) 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 n 󵄨 Cn (f , g) := 󵄨󵄨∫ g ⋅ (f ∘ T ) dμ − ∫ g dμ ⋅ ∫ f dμ󵄨󵄨󵄨 = 󵄨󵄨󵄨Eμ (g ⋅ (f ∘ T n )) − Eμ (g) ⋅ Eμ (f )󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨X 󵄨 X X

(13.75)

are the correlations of f and g. Observe that ∫(g − Eμ (g)) ⋅ [(f − Eμ (f )) ∘ T n ]dμ X

= ∫ g ⋅ (f ∘ T n ) dμ − ∫ g ⋅ Eμ (f ) dμ − ∫ Eμ (g) ⋅ (f ∘ T n ) dμ + ∫ Eμ (g) ⋅ Eμ (f ) dμ X

X

X

X

n

= ∫ g ⋅ (f ∘ T ) dμ − Eμ (f )Eμ (g) − Eμ (g)Eμ (f ) + Eμ (g)Eμ (f ) X

= Eμ (g ⋅ (f ∘ T n )) − Eμ (g)Eμ (f ). Thus, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 Cn (f , g) = 󵄨󵄨󵄨∫(g − Eμ (g)) ⋅ [(f − Eμ (f )) ∘ T n ]dμ󵄨󵄨󵄨. 󵄨󵄨 󵄨󵄨 󵄨X 󵄨

(13.76)

Cn (f , g) = Cn (f − Eμ (f ), g − Eμ (g)).

(13.77)

Note further that

So, when calculating correlations, we may replace f by f − Eμ (f ) and g by g − Eμ (g). And, theoretically, we may assume without loss of generality that Eμ (f ) = 0 = Eμ (g). We shall abbreviate Cn (f ) := Cn (f , f ).

(13.78)

13.9 Stochastic laws |

511

Obviously, the correlations Cn (f , g) and Cn (f ) depend on the underlying measure μ. When pertinent, we will use the respective notations Cμ,n (f , g) and Cμ,n (f ).

13.9.1 Exponential decay of correlations Let us go back to dynamical systems and potentials. We will demonstrate that the correlations Cn (f , g) decay exponentially under the additional condition that T is Hölder ̂ is Hölder continuous, too. continuous. This entails that φ Theorem 13.9.1. Let T : X → X be a topologically exact, open, distance expanding, Hölder continuous system and φ ∈ Hα (X, ℝ) a potential. Per Proposition 13.7.12, let μφ be the unique T-invariant Gibbs state and equilibrium state for φ. There exist constants c ≥ 0 and 0 < ζ < 1 such that for all f ∈ L1 (μφ ) and all g ∈ Hα (X) we have 󵄩 󵄩 󵄩 󵄩 Cμφ ,n (f , g) ≤ cζ n 󵄩󵄩󵄩f − Eμφ (f )󵄩󵄩󵄩L1 (μ ) 󵄩󵄩󵄩g − Eμφ (g)󵄩󵄩󵄩α , φ

∀n ∈ ℕ.

(13.79)

Proof. Replacing f by f − Eμφ (f ) and g by g − Eμφ (g), we may assume without loss of generality that Eμφ (f ) = Eμφ (g) = 0. Using (13.76) and Proposition 13.8.4(b), we obtain 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 ̂n ̂n̂ (g) dμφ 󵄨󵄨󵄨. Cμφ ,n (f , g) = 󵄨󵄨󵄨∫ g ⋅ (f ∘ T n ) dμφ 󵄨󵄨󵄨 = 󵄨󵄨󵄨∫ ℒ (g ⋅ (f ∘ T n )) dμφ 󵄨󵄨󵄨 = 󵄨󵄨󵄨∫ f ⋅ ℒ ̂ φ φ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨X 󵄨 󵄨X 󵄨 󵄨X

(13.80)

̂ and using Corollary 13.8.11 as well as Applying Theorem 13.8.18 with φ replaced by φ ̂n̂ (g) = Sn (g). Therefore, the fact that Eμφ (g) = 0, we infer that L1 (g) = 0, and thus ℒ φ 󵄩󵄩 ̂n 󵄩󵄩 󵄩 n 󵄩 󵄩 n 󵄩 n 󵄩󵄩ℒφ̂ (g)󵄩󵄩∞ = 󵄩󵄩󵄩S (g)󵄩󵄩󵄩∞ ≤ 󵄩󵄩󵄩S (g)󵄩󵄩󵄩α ≤ cζ ‖g‖α for some c ≥ 0 and r(S) < ζ < 1. Inserting this into (13.80), we conclude that 󵄨 ̂n 󵄨󵄨 󵄩󵄩 ̂n 󵄩󵄩 n Cμφ ,n (f , g) ≤ ∫󵄨󵄨󵄨f ⋅ ℒ ̂ (g)󵄨󵄨 dμφ ≤ 󵄩 φ 󵄩ℒφ̂ (g)󵄩󵄩∞ ∫ |f | dμφ ≤ cζ ‖g‖α ‖f ‖L1 (μφ ) . X

X

Example 13.9.2. Let σ : EA∞ → EA∞ be a subshift of finite type generated by a primitive matrix A on a finite alphabet E. According to Theorems 3.2.16 and 3.2.12 and Example 4.1.3, such a subshift is a topologically exact, open, distance expanding, Hölder system. Let φ ∈ Hα (EA∞ , ℝ). Per Proposition 13.7.12, let μφ be the unique σ-invariant Gibbs state and equilibrium state for φ. Let f ∈ L1 (μφ ) and g be constant on cylinders of length k for some k ∈ ℕ. Using the metric d(ω, τ) = e−|ω∧τ| , it can be observed that 󵄩󵄩 󵄩 󵄩󵄩g − Eμφ (g)󵄩󵄩󵄩∞ ≤ 2‖g‖∞

and υα (g − Eμφ (g)) = υα (g) ≤ 2‖g‖∞ e−αk .

512 | 13 Gibbs states and transfer operators for open, distance expanding systems By Theorem 13.9.1, there exist constants c ≥ 0 and 0 < ζ < 1 such that for all such f and g we have 󵄨 󵄨 Cμφ ,n (f , g) ≤ cζ n (1 + e−αk )‖g‖∞ ∫ 󵄨󵄨󵄨f − Eμφ (f )󵄨󵄨󵄨 dμφ ,

∀n ∈ ℕ.

EA∞

13.9.2 Asymptotic variance Lemma 13.9.3. Let (X, ℱ , μ) be a probability space. Let T : (X, ℱ , μ) → (X, ℱ , μ) be a measure-preserving transformation and g ∈ L2 (μ). If ∞ 󵄨󵄨 󵄨󵄨 󵄨 󵄨 ∑ 󵄨󵄨󵄨∫ g ⋅ g ∘ T n dμ󵄨󵄨󵄨 < ∞, 󵄨 󵄨󵄨 n=1 󵄨 X

then 2

n−1 ∞ 1 ∫( ∑ g ∘ T k ) dμ = ∫ g 2 dμ + 2 ∑ ∫ g ⋅ g ∘ T n dμ. n→∞ n n=1 k=0

lim

X

X

X

Proof. Almost all of this proof is based on sheer calculations. For every n ∈ ℕ, using the T-invariance of μ we obtain n−1

∫(Sn g)2 dμ = ∑ ∫ g ∘ T i ⋅ g ∘ T j dμ i, j=0 X

X

n−1

= ∑ ∫ g ∘ T i ⋅ g ∘ T i dμ + 2 i=0 X

n−2 n−1

∫ g ∘ T i ⋅ g ∘ T j dμ

∑

0≤i 0. By hypothesis, the series ∑∞ j=1 󵄨󵄨∫X g ⋅ g ∘ T dμ󵄨󵄨 converges, so there exists N1 ∈ ℕ such that ∞ 󵄨󵄨 󵄨󵄨 ε 󵄨 󵄨 ∑ 󵄨󵄨󵄨∫ g ⋅ g ∘ T j dμ󵄨󵄨󵄨 < . 󵄨󵄨 󵄨󵄨 3 j=N1

(13.83)

X

Therefore, n−1

∑ k=N1

󵄨󵄨 n−1 󵄨󵄨 󵄨󵄨 ε k 󵄨󵄨󵄨󵄨 󵄨 󵄨 󵄨 k k 󵄨󵄨∫ g ⋅ g ∘ T dμ󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨∫ g ⋅ g ∘ T dμ󵄨󵄨󵄨 < . 󵄨󵄨 󵄨󵄨 󵄨󵄨 3 n 󵄨󵄨 k=N X

1

(13.84)

X

Finally, observe that there exists N2 ≥ N1 such that N1 −1

∑ k=1

󵄨󵄨 ε k 󵄨󵄨󵄨󵄨 󵄨 k 󵄨󵄨∫ g ⋅ g ∘ T dμ󵄨󵄨󵄨 < , 󵄨󵄨 3 n 󵄨󵄨

∀n ≥ N2 .

X

Inserting this, (13.84) and (13.83) into (13.82), we get for every n ≥ N2 that N1 −1 󵄨 󵄨󵄨 n−1 k 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 ∞ 󵄨󵄨󵄨 Δn k 󵄨󵄨 ε 󵄨 󵄨 󵄨 ≤ ∑ 󵄨󵄨󵄨∫ g ⋅ g ∘ T k dμ󵄨󵄨󵄨 + ∑ 󵄨󵄨󵄨∫ g ⋅ g ∘ T k dμ󵄨󵄨󵄨 + ∑󵄨󵄨󵄨∫ g ⋅ g ∘ T j dμ󵄨󵄨󵄨 < 3 = ε. 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 2 n n 3 j=n k=1 k=N1 X

X

X

Hence lim Δn = 0. The proof of Lemma 13.9.3 is complete. n→∞

As a direct consequence of this lemma, we have the following. Recall that Cμ,n (g) is defined through (13.75)–(13.78). Proposition 13.9.4. Let T : (X, ℱ , μ) → (X, ℱ , μ) be a measure-preserving transformation of a probability space (X, ℱ , μ) and g ∈ L2 (μ). If ∞ ∞ 󵄨󵄨 󵄨󵄨 󵄨 󵄨 ∑ Cμ,n (g) = ∑ 󵄨󵄨󵄨∫(g − μ(g)1) ⋅ (g − μ(g)1) ∘ T n dμ󵄨󵄨󵄨 < ∞, 󵄨 󵄨󵄨 n=1 n=1󵄨 X

(13.85)

514 | 13 Gibbs states and transfer operators for open, distance expanding systems then the limit 2

n−1 1 ∫( ∑ (g − μ(g)1) ∘ T k ) dμ n→∞ n k=0

2 σμ,∞ (g) := lim

(13.86)

X

exists and ∞

2

2 σμ,∞ (g) = ∫(g − μ(g)1) dμ + 2 ∑ ∫(g − μ(g)1) ⋅ (g − μ(g)1) ∘ T n dμ. X

n=1

(13.87)

X

2 The number σμ,∞ (g) is called the asymptotic variance of g (with respect to T and μ). It is in general different than the common variance 2

σμ2 (g) := ∫(g − μ(g)1) dμ. X

However, if the random variables g ∘ T n , n ≥ 0, are independent with respect to μ, then 2 σμ,∞ (g) = σμ2 (g). Our goal in this subsection is to provide several characterizations of the positivity 2 of the asymptotic variance σμ,∞ (g). Essentially all of those equivalencies will be ex-

2 pressed by means of cohomology of functions. Knowing that σμ,∞ (g) > 0 is important because it is needed for the central limit theorem and the law of the iterated logarithm, which are the subject of the next two subsections. We start with the following result, which has as a by-product the addition of one item to Theorem 13.7.17 characterizing potentials that have the same Gibbs/equilibrium states.

Theorem 13.9.5. Let (X, ℱ , μ) be a probability space. Let T : (X, ℱ , μ) → (X, ℱ , μ) be a measure-preserving transformation and g ∈ L2 (μ). If ∞

∑ nCμ,n (g) < ∞,

n=0

(13.88)

2 then σμ,∞ (g) exists and the following three conditions are equivalent: 2 (a) σμ,∞ (g) = 0. 󵄩 󵄩 (b) sup{󵄩󵄩󵄩Sn (g − μ(g)1)󵄩󵄩󵄩L2 (μ) : n ∈ ℕ} < ∞.

(c) g is cohomologous to a constant in the Hilbert space L2 (μ). 2 Proof. By Proposition 13.9.4, we know that σμ,∞ (g) exists. The implication (c)⇒(a) follows immediately from (13.87) after substituting g = h ∘ T − h, where h ∈ L2 (μ). For all other implications established later in this proof, assume without loss of generality that μ(g) = 0.

13.9 Stochastic laws |

515

Let us prove that (a)⇒(b). Set Ck∗ (g) := ∫ g ⋅ g ∘ T k dμ,

∀k ≥ 0.

X

Using (13.81), we get n−1

∫(Sn g)2 dμ = nC0∗ (g) + 2 ∑ (n − k)Ck∗ (g) k=1

X

∞

∞

n−1

k=n

k=1

= n(C0∗ (g) + 2 ∑ Ck∗ (g)) − 2n ∑ Ck∗ (g) − 2 ∑ kCk∗ (g) k=1

n−1

∞

2 = nσμ,∞ (g) − 2n ∑ Ck∗ (g) − 2 ∑ kCk∗ (g) k=n

∞

n−1

k=n

k=1

k=1

= −2n ∑ Ck∗ (g) − 2 ∑ kCk∗ (g). By (13.88), the first summand in this formula converges to 0 as n → ∞ and the second summand is uniformly bounded. We conclude that (b) holds. Finally, we show that (b)⇒(c). By our hypothesis there exists M ≥ 1 such that ‖Sn g‖L2 (μ) ≤ M for all n ∈ ℕ. Consequently, the same holds for the Cesàro averages: 󵄩󵄩 n 󵄩󵄩󵄩 󵄩󵄩 1 󵄩󵄩 ∑ Sj g 󵄩󵄩󵄩 ≤ M, 󵄩󵄩 󵄩󵄩 n 󵄩󵄩L2 (μ) 󵄩󵄩 j=1

∀n ∈ ℕ.

As a Hilbert space, L2 (μ) is reflexive and hence its closed ball BL2 (μ) (0, M) is weakly compact in L2 (μ). Thus there is u ∈ BL2 (μ) (0, M) and an increasing sequence (nk )∞ k=1 of nk 1 ∞ positive integers such that the sequence ( n ∑j=1 Sj g)k=1 converges weakly to u. Then k

n

1 k ∑ Sj (g ∘ T) = u ∘ T k→∞ nk j=1 lim

weakly in L2 (μ)

(13.89)

and n

n

n

k 1 k 1 1 k 1 (∑ Sj+1 g − nk g) = ∑ Sj (g ∘ T) = ∑ S g + (Snk +1 g − g) − g. nk j=1 nk j=1 nk j=1 j nk

Taking the weak limits of both sides of this equation and invoking (13.89) we get u∘T = u − g or, equivalently, g = u − u ∘ T. The proof of the implication (b)⇒(c) and of the whole theorem is complete. As an immediate consequence of Theorems 13.9.5 and 13.9.1, we get the following.

516 | 13 Gibbs states and transfer operators for open, distance expanding systems Corollary 13.9.6. Let T : X → X be a topologically exact, open, distance expanding, Hölder continuous system and let φ : X → ℝ be a Hölder potential. If g : X → ℝ is a Hölder function, then σμ2 φ ,∞ (g) exists and the following three conditions are equivalent:

(a) σμ2 φ ,∞ (g) = 0. 󵄩 󵄩 (b) sup{󵄩󵄩󵄩Sn (g − μφ (g)1)󵄩󵄩󵄩L2 (μ ) : n ∈ ℕ} < ∞. φ (c) g is cohomologous to a constant in the Hilbert space L2 (μφ ).

Our next goal is to promote the L2 (μφ )-cohomology in item (c) of Corollary 13.9.6 to cohomology in the class of Hölder continuous functions. Of course, we need a much more special setting and much stronger hypotheses. Though we ultimately aim at carrying out this promotion for transitive, open, distance expanding, Hölder continuous systems, in a first stage we will accomplish this promotion for irreducible/transitive subshifts of finite type and then use the Markov partitions and the symbolic representation from Sections 4.4–4.5, to propagate it to all transitive, open, distance expanding, Hölder continuous maps. 2 As we will also encounter the issue of characterization of the equation σμ,∞ (g) = 0 in Section 18.4 in the context of transitive subshifts of finite type over a countable (possibly infinite) alphabet, we will work in that setting for a while. The reader who is not interested in countable state shifts may assume without losing anything that the countable set E below is finite. The reader familiar with countable state shifts can smoothly read what follows and the reader interested in countable state subshifts but not familiar with them yet, is advised to first read Chapter 17. Let E be a countable set and A : E × E → {0, 1} be a finitely irreducible incidence matrix. Let a potential f : EA∞ → ℝ be summable and Hölder continuous on cylinders. In addition to the usual one-sided subshift EA∞ = EAℕ = EA{1,2,...} , let 󵄨 EAℤ := {(ωn )n∈ℤ 󵄨󵄨󵄨 Aωn ωn+1 = 1, ∀n ∈ ℤ} be the two-sided subshift generated by the transition matrix A and let 󵄨 EA−∞ := {(ωn )n≤0 󵄨󵄨󵄨 Aωn ωn+1 = 1, ∀n ≤ −1}. Given −∞ ≤ m ≤ n ≤ ∞ and ω ∈ EAℤ , let ω|nm = ωm ωm+1 . . . ωn and let 󵄨 [ω]nm := {τ ∈ EAℤ 󵄨󵄨󵄨 τk = ωk , ∀m ≤ k ≤ n}. Given e ∈ E, let 󵄨 EA∞,e := {ρ ∈ EA∞ 󵄨󵄨󵄨 Aeρ1 = 1} and

󵄨 EA∗,e := {τ ∈ EA∗ 󵄨󵄨󵄨 Aeτ1 = 1}.

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517

Let α− := {[χ]}χ∈EA−∞ be the partition of EAℤ whose atoms are [χ] = {χτ : τ ∈ EA

∞,χ0

} = {γ ∈ EAℤ : γ|0−∞ = χ} = [χ]0−∞ ,

where χ ∈ EA−∞ . For every k ∈ ℤ, the map πk : EAℤ → E is the projection onto the kth coordinate given by the formula πk (ω) := ωk . ∞ Also, for every set B ⊆ EAℤ , the set B|∞ 1 is the projection of B onto EA . Finally, let μf be the unique one-sided shift-invariant Gibbs state for the poteñ f be its Rokhlin’s natural extension to the two-sided subshift EAℤ . Let tial f , and let μ ̃ f is us recall that, given finitely many sets C1 , C2 , . . . , Ck contained in E, the measure μ defined on a cylinder C = πn−11 (C1 ) ∩ πn−12 (C2 ) ∩ ⋅ ⋅ ⋅ ∩ πn−1k (Ck ) with −∞ < n1 ≤ n2 ≤ ⋅ ⋅ ⋅ ≤ nk < ∞, by the formula ̃ f (C) = μf (σ̃ n1 −1 (C)|∞ μ 1 ), ̃ f is two-sided shiftwhere σ̃ : EAℤ → EAℤ is the left-shift map on EAℤ . Let us recall that μ invariant. We shall prove the following. ̃ f ,α− (ω) on Lemma 13.9.7. For every ω ∈ EAℤ , there exists a Borel probability measure μ 0 the atom α− (ω) = [ω]−∞ with the following properties: (a) For every Borel set B ⊆ EAℤ the function ̃ f ,α− (τ) (B ∩ α− (τ)) ∈ [0, 1] EAℤ ∋ τ 󳨃󳨀→ μ is measurable. (b) For every Borel set B ⊆ EAℤ , ̃ f (B) = ∫ μ ̃ f ,α− (τ) (B ∩ α− (τ)) dμ ̃ f (τ). μ EAℤ

̃ f ,α− (ω) on EA 0 ⊆ EA∞ is equiva(c) The measure μf ,α− (ω) defined as the projection of μ ∞,ω0 lent to μf restricted to EA . dμf ,α− (ω) is uniformly bounded away from 0 and ∞. (d) The Radon–Nikodym derivative dμf ∞,ω

Proof. Let lB : ℓ∞ → ℝ be a Banach limit. Given ω ∈ EAℤ and a Borel set B ⊆ EAℤ , define ̃ f ,α− (ω) (B) := lB (( μ

̃ f (B ∩ [ω]0−n ) μ ̃ f ([ω]0−n ) μ

∞

) ). 0

(13.90)

518 | 13 Gibbs states and transfer operators for open, distance expanding systems Since the functional lB is linear, as an immediate consequence of this definition, we get the following. ̃ f ,α− (ω) (B ∩ α− (ω)) ∈ [0, 1], defined on Claim 1. For every ω ∈ EAℤ , the function B 󳨃󳨀→ μ the Borel σ-algebra of EAℤ , is finitely additive. Now we shall show that for every τ ∈ EA

∗,ω0

Q−3 ≤

,

̃ f ,α− (ω) ([ω|0−∞ τ]) μ μf ([τ])

≤ Q3 ,

(13.91)

where Q ≥ 1 is a Gibbs constant for μf . (Note: To lighten notation, we will use P(f ) in lieu of P(σ, f ) in the following calculations.) Let n ≥ 0. In view of the Gibbs property, we get ̃ f ([ω]0−n ) = μf (σ̃ −(n+1) ([ω]0−n )|∞ μ 1 ) ≤ Q exp(inf(Sn+1 f |σ̃ −(n+1) ([ω]0−n )|∞ ) − P(f )(n + 1)) 1

= Q exp(inf(Sn+1 f |[ω−n ...ω0 ] ) − P(f )(n + 1)).

(13.92)

Similarly, ̃ f ([ω]0−n ) ≥ Q−1 exp(sup(Sn+1 f |[ω−n ...ω0 ] ) − P(f )(n + 1)). μ

(13.93)

Setting k = |τ|, by the same Gibbs property we obtain that ̃ f ([ω]0−n ∩ [τ]k1 ) = μ

= μf (σ̃ −(n+1) ([ω]0−n ∩ [τ]k1 )|∞ 1 ) ≥ Q−1 exp(inf(Sn+1+k f |σ̃ −(n+1) ([ω]0

k ∞ −n ∩[τ]1 )|1

) − P(f )(n + 1 + k))

= Q−1 exp(inf(Sn+1+k f |[ω−n ...ω0 τ1 ...τk ] ) − P(f )(n + 1 + k)) ≥ Q−1 exp(inf(Sn+1 f |[ω−n ...ω0 ] ) + inf(Sk f |[τ1 ...τk ] ) − P(f )(n + 1 + k)) = Q−1 exp(inf(Sn+1 f |[ω−n ...ω0 ] ) − P(f )(n + 1)) exp(inf(Sk f |[τ] ) − P(f )k).

(13.94)

Similarly, ̃ f ([ω]0−n ∩[τ]k1 ) ≤ Q exp(sup(Sn+1 f |[ω−n ...ω0 ] )−P(f )(n+1)+sup(Sk f |[τ] )−P(f )k). (13.95) μ Using (13.92) and (13.94) and applying the Gibbs property once more, we deduce that ̃ f ([ω]0−n ∩ [τ]k1 ) μ ̃ f ([ω]0−n ) μ

≥ Q−2 exp(inf(Sk f |[τ] ) − P(f )k) ≥ Q−3 μf ([τ]).

13.9 Stochastic laws |

519

On the other hand, using (13.93) and (13.95) and applying the Gibbs property once more, we arrive at ̃ f ([ω]0−n ∩ [τ]k1 ) μ ̃ f ([ω]0−n ) μ

≤ Q2 exp(sup(Sk f |[τ] ) − P(f )k) ≤ Q3 μf ([τ]).

Inserting these two inequalities into (13.90) and using the monotonicity of Banach limits, we conclude that (13.91) holds. ̂ f be the measure on α− (ω) = [ω]0−∞ given by Let μ ̂ f (ω|0−∞ F) := μf (F) μ for every Borel set F ⊆ EA 0 , where ω|0−∞ F := {ω|0−∞ ρ : ρ ∈ F}. Let ℬω∗ be the algebra of ∗,ω sets on [ω]0−∞ generated by the sets of the form [ω|0−∞ τ], τ ∈ EA 0 . Since each element of this algebra can be represented as a disjoint union of sets of the form [ω|0−∞ τ], τ ∈ ∗,ω EA 0 , a consequence of (13.91) is that ∞,ω

̂ f (G) ≤ μ ̃ f ,α− (ω) (G) ≤ Q3 μ ̂ f (G), Q−3 μ

∀G ∈ ℬω∗ .

(13.96)

̃ f ,α− (ω) is absolutely continuous with respect to μ ̂ f on ℬω∗ . In conjunction In particular, μ ̃ f ,α− (ω) is a (σ-additive) Borel probabilwith Claim 1 and Lemma A.1.31, this gives that μ ity measure on α− (ω). Furthermore, items (c,d) of our lemma follow from (13.96). It immediately ensues from Theorem A.1.68 (martingale convergence theorem for conditional expectations) and Example A.1.66 applied with the function φ = 1B and n,ω 𝒜n = {[ω|0−∞ τ] : τ ∈ EA 0 }, that ̃ f ,α− (ω) (B) := lim μ

n→∞

̃ f (B ∩ [ω]0−n ) μ ̃ f ([ω]0−n ) μ

= Eμ̃f (1B |α− )(ω)

(13.97)

̃ f -a. e. ω ∈ EAℤ (depending on B). Consequently, item (a) holds. Item (b) directly for μ follows from the definition of conditional expected value. The proof of Lemma 13.9.7 is complete. An immediate ramification of this lemma is the following. Corollary 13.9.8. For every ω ∈ EAℤ , ̃ f ,α− (ω) ) = α− (ω). supp(μ We now make a first step toward establishing cohomology in the class of Hölder continuous functions for symbolic systems. Lemma 13.9.9. Let E be a countable set and A : E × E → {0, 1} be a finitely irreducible incidence matrix. Let a potential f : EA∞ → ℝ be summable and Hölder continuous on cylinders. Suppose that g ∈ Hβ (EA∞ , ℝ) has the following properties:

520 | 13 Gibbs states and transfer operators for open, distance expanding systems (a) g ∈ L2 (μf ); ∞

(b) ∑ nCμf ,n (g) < ∞ (in particular, σμ2 f ,∞ (g) exists by Proposition 13.9.4); and n=0

(c) σμ2 f ,∞ (g) = 0. Then there exists u ∈ Hβb (EA∞ , ℝ) such that g = μf (g) + u − u ∘ σ. In particular, the function g turns out to be bounded, so g ∈ Hβb (EA∞ , ℝ).

Proof. Assume without loss of generality that μf (g) = 0. It follows from Theorem 13.9.5 that there exists u ∈ L2 (μf ) such that g =u−u∘σ

μf -a. e. on EA∞ .

(13.98)

Our goal is to show that u can be extended to a version which is Hölder continuous of order β on cylinders. We first extend g and u to the two-sided shift space EAℤ by declaring g̃ (ω) := g(ω|∞ 1 ) and

̃ (ω) := u(ω|∞ u 1 )

̃ : EAℤ → ℝ is wherever u(ω|∞ 1 ) is defined and fulfills (13.98). Like g, the function g ̃ is measurable (and even belongs to Hölder continuous of order β on cylinders, while u ̃ f ), though we will not need this fact). The cohomological equation (13.98) carries L2 (μ ̃ on a μ ̃ f -a. e. basis since over to g̃ and u ∞ ∞ ∞ ̃ (ω)|∞ ̃ (ω) (13.99) ̃ (ω) − u ̃ ∘ σ̃ (ω) = u(ω|∞ u 1 ) − u(σ 1 ) = u(ω|1 ) − u(σ(ω|1 )) = g(ω|1 ) = g

wherever u(ω|∞ 1 ) is defined and satisfies (13.98). By virtue of Luzin’s theorem (Theorem A.1.49), there exists a compact set D ⊆ EAℤ ̃ f (D) > 1/2 and the restricted function u ̃ |D is continuous. Corollary 8.2.14 such that μ (Ergodic case of Birkhoff’s ergodic theorem), or Corollary 8.2.15, then affirms that there ̃ f (B) = 1; (2) for every γ ∈ B, exists a Borel set B ⊆ EAℤ such that (1) μ lim

n→∞

1 ̃ f (D) > 1/2; #{0 ≤ j < n : σ̃ −n (γ) ∈ D} = μ n

(13.100)

̃ is well-defined on the union ⋃n∈ℤ σ̃ −n (B); and (4) relation (13.99) (3) the function u holds on the same set ⋃n∈ℤ σ̃ −n (B). By Lemma 13.9.7(b), there is a Borel set F ⊆ EAℤ ̃ f (F) = 1 and such that μ ̃ α− (ω) (B ∩ α− (ω)) = 1, μ

∀ω ∈ F.

It then follows from Corollary 13.9.8 that the set B ∩ α− (ω) is dense in α− (ω) for every ̃ f ([e]) > 0, there exists ω ∈ F such ω ∈ F. Now fix an arbitrary element e ∈ E. Since μ

13.9 Stochastic laws |

521

that ω1 = e. Let Be := {χ ∈ EA∞ : χ1 = e and ω|0−∞ χ ∈ B}. Now consider two arbitrary elements ρ, τ ∈ Be . Because of (13.100), there exists a strictly increasing sequence (nj )∞ j=1 of positive integers such that σ̃ −nj (ω|0−∞ ρ), σ̃ −nj (ω|0−∞ τ) ∈ D,

∀j ∈ ℕ.

Write ω := ω|0−∞ . Using (13.99), we get 󵄨󵄨 󵄨 󵄨̃ ̃ (ωτ)󵄨󵄨󵄨󵄨 (ωρ) − u 󵄨󵄨u(ρ) − u(τ)󵄨󵄨󵄨 = 󵄨󵄨󵄨u 󵄨󵄨 󵄨󵄨 nj nj 󵄨󵄨 󵄨󵄨󵄨 ̃ ̃ −nj −k −nj −k ̃ ̃ ̃ ̃ ̃ ̃ = 󵄨󵄨(u(σ (ωρ)) − ∑ g (σ (ωρ))) − (u(σ (ωτ)) − ∑ g (σ (ωτ)))󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 k=1 k=1 󵄨 󵄨 nj 󵄨 ̃ ̃ −nj ̃ (σ̃ −nj (ωτ))󵄨󵄨󵄨󵄨 + ∑ 󵄨󵄨󵄨󵄨g̃ (σ̃ −k (ωρ)) − g̃ (σ̃ −k (ωτ))󵄨󵄨󵄨󵄨. (13.101) ≤ 󵄨󵄨󵄨u (σ (ωρ)) − u k=1

Since limj→∞ d(σ̃ −nj (ωρ), σ̃ −nj (ωτ)) = 0, since both σ̃ −nj (ωρ) and σ̃ −nj (ωτ) belong to D, and since u|D is uniformly continuous (as D is compact), we deduce that 󵄨 ̃ ̃ −nj ̃ (σ̃ −nj (ωτ))󵄨󵄨󵄨󵄨 = 0. lim 󵄨󵄨󵄨u (σ (ωρ)) − u

(13.102)

j→∞

As g̃ is Hölder continuous of order β on cylinders, we obtain that nj

nj

k=1

k=1 nj

β 󵄨 󵄨 ∑ 󵄨󵄨󵄨g̃ (σ̃ −k (ωρ)) − g̃ (σ̃ −k (ωτ))󵄨󵄨󵄨 ≤ ∑ υβ (g̃ )[d(σ̃ −k (ωρ), σ̃ −k (ωτ))]

= ∑ υβ (g̃ )e−βk [d(ωρ, ωτ)] k=1 nj

≤ ∑ υβ (g̃ )e−βk [d(ρ, τ)]

β

β

k=1

≤

υβ (g̃ )

1 − e−β

β

[d(ρ, τ)] .

(13.103)

It follows from (13.101)–(13.103) that υβ (g̃ ) β 󵄨󵄨 󵄨 [d(ρ, τ)] . 󵄨󵄨u(ρ) − u(τ)󵄨󵄨󵄨 ≤ 1 − e−β As Be is dense in [e], it ensues that u has a Hölder extension from Be to [e]. Since e ∈ E is arbitrary, we have thus defined an extension u of u to EA∞ which is Hölder continuous of order β on cylinders. Given that (13.98) holds on a dense subset of EA∞ , we infer that g =u−u∘σ

(13.104)

522 | 13 Gibbs states and transfer operators for open, distance expanding systems holds on this dense set, and consequently on the whole of EA∞ . We are left to prove that the function u : EA∞ → ℝ is bounded. Let Λ ⊆ EA∗ be a finite set of words witnessing the finite irreducibility of the matrix A. Let |Λ| := max{|α| : α ∈ Λ}

and Λ1 := {τ1 : τ ∈ Λ}.

As u and g are Hölder continuous on cylinders, we have that sup(|g||[e] ) and sup(|u||[e] ) are finite for each e ∈ E. So 󵄨 Mg := max{sup(|g|󵄨󵄨󵄨[e] ) : e ∈ Λ1 } < ∞

󵄨 and Mu := max{sup(|u|󵄨󵄨󵄨[e] ) : e ∈ Λ1 } < ∞.

For every ρ ∈ EA∞ , choose any ρ̂ ∈ Λ such that ρ̂ρ ∈ EA∞ . Let k := |ρ̂| ≤ |Λ|. It follows from (13.104) iterated k times that Sk g(ρ̂ρ) = u(ρ̂ρ) − u(σ k (ρ̂ρ)) = u(ρ̂ρ) − u(ρ). Therefore, 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨u(ρ)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨Sk g(ρ̂ρ)󵄨󵄨󵄨 + 󵄨󵄨󵄨u(ρ̂ρ)󵄨󵄨󵄨 ≤ Sk |g|(ρ̂ρ) + sup(|u|󵄨󵄨󵄨[̂ρ1 ] ) ≤ kMg + Mu ≤ |Λ|Mg + Mu . Hence ‖u‖∞ ≤ |Λ|Mg + Mu < ∞ and the proof of Lemma 13.9.9 is complete. When the alphabet E is finite, Lemma 13.9.9’s statement reduces to the following. Lemma 13.9.10. Let E be a finite set and A : E × E → {0, 1} be an irreducible matrix. Let also f : EA∞ → ℝ be a Hölder continuous potential. If g ∈ Hβ (EA∞ , ℝ) is such that ∞

∑ nCμf ,n (g) < ∞,

n=0

(13.105)

then σμ2 f ,∞ (g) exists according to Proposition 13.9.4. If furthermore σμ2 f ,∞ (g) = 0, then there exists u ∈ Hβ (EA∞ , ℝ) such that g = μf (g) + u − u ∘ σ. A straightforward repercussion of Lemma 13.9.10 and Theorem 13.9.1 is next. Lemma 13.9.11. Let E be a finite set and A : E × E → {0, 1} be a primitive incidence matrix. Let also f : EA∞ → ℝ be a Hölder continuous potential. If g ∈ Hβ (EA∞ , ℝ), then σμ2 f ,∞ (g) exists. Furthermore, if σμ2 f ,∞ (g) = 0 then there exists u ∈ Hβ (EA∞ , ℝ) such that g = μf (g) + u − u ∘ σ. The cohomology in the class of Hölder continuous functions established in a finite-state symbolic setting in Lemma 13.9.10, will now be transferred to the more

13.9 Stochastic laws |

523

general class of transitive, open, distance expanding, Hölder continuous systems. To accomplish this, we will use the existence of Markov partitions and of a symbolic representation derived in Sections 4.4–4.5. Lemma 13.9.12. Let T : X → X be a topologically transitive, open, distance expanding, Hölder continuous map and let φ : X → ℝ be a Hölder continuous potential. If g : X → ℝ is a Hölder continuous function such that ∞

∑ nCμφ ,n (g) < ∞,

n=0

(13.106)

then σμ2 φ ,∞ (g) exists according to Proposition 13.9.4. If furthermore σμ2 φ ,∞ (g) = 0, then there exists a Hölder continuous function u : EA∞ → ℝ such that g = μφ (g) + u − u ∘ σ. Proof. By Theorem 4.4.6, let ℛ = {Re }e∈E be a Markov partition with diameter smaller than an expansive constant for the map T : X → X. Let A be the corresponding irreducible matrix defined in (4.46), and let π : EA∞ → X be the coding map in Theorem 4.5.2. By Theorem 13.7.11, this map π is a metric isomorphism of the measurepreserving dynamical systems (σ : EA∞ → EA∞ , μφ∘π ) and (T : X → X, μφ ). Since π is Hölder continuous per Theorem 4.5.2, the function g ∘ π : EA∞ → ℝ is also Hölder continuous and μφ∘π (g ∘ π) = μφ (g) and Cμφ∘π ,n (g ∘ π) = Cμφ ,n (g) for every n ∈ ℕ. Therefore, Lemma 13.9.10 applies and asserts that σμ2 φ∘π ,∞ (g ∘π) exists. As σμ2 φ∘π ,∞ (g ∘π) = σμ2 φ ,∞ (g), we know that this latter quantity exists. If it equals 0, then Lemma 13.9.10 produces a ̂ : EA∞ → ℝ such that Hölder continuous function u ̂−u ̂ ∘ σ = μφ (g) + u ̂−u ̂ ∘ σ. g ∘ π = μφ∘π (g ∘ π) + u Assume without loss of generality that μφ (g) = 0. The above equation reduces to ̂−u ̂ ∘ σ. g∘π =u

(13.107)

Let ∞

ℛ∞ := X \ ⋃ T n=0

−n

(⋃ 𝜕Re ). e∈E

By Theorem 4.5.2, each point x ∈ ℛ∞ has a unique coding π −1 (x) ∈ EA∞ . So there is a well-defined function u : ℛ∞ → ℝ given by the formula ̂ (π −1 (x)). u(x) := u Fix any e ∈ E. Take x, y ∈ ℛ∞ ∩ Re . Let ω = π −1 (x) and τ = π −1 (y). Then ω1 = τ1 = e, x = π(ω) and y = π(τ). Pick any word ρ ∈ EA∗ such that Aρk e = 1, where k = |ρ|. Then

524 | 13 Gibbs states and transfer operators for open, distance expanding systems ρω, ρτ ∈ EA∗ . Using (13.107) and Lemma 13.1.1, we get 󵄨 󵄨̂ 󵄨󵄨 ̂ (ω)󵄨󵄨󵄨󵄨 (τ) − u 󵄨󵄨u(y) − u(x)󵄨󵄨󵄨 = 󵄨󵄨󵄨u 󵄨󵄨 ̂ ̂ (ρω) − Skσ (g ∘ π)(ρω))󵄨󵄨󵄨󵄨 = 󵄨󵄨(u(ρτ) − Skσ (g ∘ π)(ρτ)) − (u 󵄨̂ ̂ (ρω)󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨Skσ (g ∘ π)(ρω) − Skσ (g ∘ π)(ρτ)󵄨󵄨󵄨󵄨 (ρτ) − u ≤ 󵄨󵄨󵄨u 󵄨̂ ̂ (ρω)󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨SkT g(π(ρω)) − SkT g(π(ρτ))󵄨󵄨󵄨󵄨 (ρτ) − u = 󵄨󵄨󵄨u 󵄨̂ ̂ (ρω)󵄨󵄨󵄨󵄨 + Cβ (g)dβ (π(ω), π(τ)) (ρτ) − u ≤ 󵄨󵄨󵄨u 󵄨̂ ̂ (ρω)󵄨󵄨󵄨󵄨 + Cβ (g)dβ (x, y). = 󵄨󵄨󵄨u (ρτ) − u ̂ : EA∞ → ℝ is Taking words ρ with length k = |ρ| increasing to ∞ and noting that u uniformly continuous, we deduce that 󵄨󵄨 󵄨 β 󵄨󵄨u(y) − u(x)󵄨󵄨󵄨 ≤ Cβ (g)d (y, x).

(13.108)

This means that u is Hölder continuous on ℛ∞ ∩ Re = ℛ∞ ∩ Int(Re ). We would like to show that u is locally Hölder continuous and uniformly continuous on ℛ∞ to later extend it to X. Take any s ∈ Int(Re ). Let η > 0 be so small that B(s, 2η) ⊆ Int(Re ).

(13.109)

By Proposition 13.6.14, the map T is very strongly transitive, i. e., there is N ≥ 0 such that N

⋃ T k (B(s, η)) = X.

k=0

(13.110)

Let λ > 1 and ξ > 0 come from the definition of T : X → X being open and distance expanding ((13.1)–(13.2)). Let ξ ′ := min{ξ , η}. Fix any two points w, z ∈ ℛ∞ such that d(w, z) < ξ ′ . By (13.110), there exist z ′ ∈ B(s, η) and 0 ≤ k ≤ N such that T k (z ′ ) = z. Let w′ := Tz−k ′ (w).

(13.111)

d(w′ , z ′ ) ≤ λ−k d(w, z) < ξ ′ ≤ η

(13.112)

Since

and since z ′ ∈ B(s, η), we deduce that w′ ∈ B(s, 2η) and it ensues from (13.109) that z ′ , w′ ∈ Int(Re ).

(13.113)

As T is a local homeomorphism on a neighbourhood of each element of ℛ, Lemma 4.5.4 states that T(⋃e′ ∈E 𝜕Re′ ) ⊆ ⋃e′ ∈E 𝜕Re′ . As w, z ∈ ℛ∞ , we infer that T n (z ′ ), T n (w′ ) ∈ ℛ∞ ,

∀n ≥ 0.

(13.114)

13.9 Stochastic laws |

525

In consequence, using (13.107) we obtain that ̂ (π −1 (z ′ )) − u ̂ (σ k (π −1 (z ′ ))) = u(z ′ ) − u(z). SkT g(z ′ ) = Skσ (g ∘ π)(π −1 (z ′ )) = u

(13.115)

Likewise, SkT g(w′ ) = u(w′ ) − u(w). Moreover, using (13.108) in conjunction with (13.114) (when n = 0), we infer that 󵄨󵄨 β ′ ′ ′ ′ 󵄨 󵄨󵄨u(w ) − u(z )󵄨󵄨󵄨 ≤ Cβ (g)d (w , z ).

(13.116)

Furthermore, using d(w, z) < ξ and the definitions of z ′ and w′ (cf. (13.111)), Lemma 13.1.1 yields 󵄨󵄨 T ′ T ′ 󵄨 β 󵄨󵄨Sk g(z ) − Sk g(w )󵄨󵄨󵄨 ≤ Cβ (g)d (w, z).

(13.117)

From (13.115)–(13.117) and (13.112), it ensues that 󵄨󵄨 󵄨 󵄨 ′ ′ T ′ T ′ 󵄨 󵄨󵄨u(w) − u(z)󵄨󵄨󵄨 = 󵄨󵄨󵄨(u(w ) − u(z )) + (Sk g(z ) − Sk g(w ))󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨u(w′ ) − u(z ′ )󵄨󵄨󵄨 + 󵄨󵄨󵄨SkT g(z ′ ) − SkT g(w′ )󵄨󵄨󵄨

≤ Cβ (g)dβ (w′ , z ′ ) + Cβ (g)dβ (w, z) ≤ 2Cβ (g)dβ (w, z).

(13.118)

This holds for any z, w ∈ ℛ∞ such that d(z, w) < ξ . This inequality shows that the function u : ℛ∞ → ℝ is locally Hölder continuous and also uniformly continuous. The uniform continuity implies that u has a unique continuous extension to ℛ∞ = X. Call it u : X → ℝ. Inequality (13.118) clearly extends to u. So u is locally Hölder continuous, and hence Hölder continuous since X is compact. Finally, (13.107) and the definition u imply that g = u−u∘T on ℛ∞ . As the functions g, u and u ∘ T are continuous, we conclude that g = u − u ∘ T on ℛ∞ = X. As an immediate consequence of Lemma 13.9.12 and Theorem 13.9.1, we get the following. Lemma 13.9.13. Let T : X → X be a topologically exact, open, distance expanding, Hölder system and let φ : X → ℝ be a Hölder potential. If g : X → ℝ is a Hölder function, 2 then σμ,∞ (g) exists. Furthermore, if σμ2 φ ,∞ (g) = 0 then there exists a Hölder function ∞ u : EA → ℝ such that g = μφ (g) + u − u ∘ σ. Thanks to Lemma 13.9.13 and Corollary 13.9.6, we can add two more equivalences to Theorem 13.7.17 when the system T is exact and Hölder continuous. Theorem 13.9.14. If T : X → X is a topologically exact, open, distance expanding, Hölder continuous system and φ, ψ : X → ℝ are Hölder continuous potentials, then the following statements are equivalent:

526 | 13 Gibbs states and transfer operators for open, distance expanding systems (a) The T-invariant Gibbs/equilibrium states of φ and ψ coincide, i. e., μφ = μψ . (b) There exists a constant R1 ∈ ℝ such that Sn φ(x) − Sn ψ(x) = R1 n whenever n ∈ ℕ and x ∈ X is a periodic point of T of period n. (c) There exist constants R2 ∈ ℝ and C ≥ 0 such that 󵄨󵄨 󵄨 󵄨󵄨Sn φ(x) − Sn ψ(x) − R2 n󵄨󵄨󵄨 ≤ C,

∀n ∈ ℕ, ∀x ∈ X.

(d) φ − ψ is cohomologous to a constant R3 ∈ ℝ in the additive group of bounded functions on X. (e) φ − ψ is cohomologous to a constant R4 ∈ ℝ in the additive group of continuous functions on X. (f) φ − ψ is cohomologous to a constant R5 ∈ ℝ in the additive group of Hölder continuous functions on X. (g) 𝒢φ = 𝒢ψ . (h) 𝒢φ ∩ 𝒢ψ ≠ 0. (i) σμ2 f ,∞ (φ − ψ) = 0 for some Hölder continuous potential f : X → ℝ. (j) σμ2 f ,∞ (φ − ψ) = 0 for all Hölder continuous potentials f : X → ℝ.

In addition, if any of the above conditions holds, then all the constants R1 –R5 are the same and equal to P(T, φ) − P(T, ψ). 13.9.3 Central limit theorem We now have all the tools to provide a short demonstration of a dynamical variation of the famous central limit theorem, which generally states that the distribution of the difference between (1) the sample average of a sequence of independent and identically distributed random variables with mean μ and variance σ 2 and (2) their expected value is, after adjusting for the size of the sample, approaching the normal (also called bell curve or Gaussian) distribution with the same mean μ and the same variance σ 2 as that of the variables. Our approach, stemming from the one in [91], is based on the concept of characteristic functions (Fourier transforms). For a short recap on this topic, please see Subsection A.1.10 near the end of Appendix A. Recall that the normal (Gaussian) distribution 𝒩0 (σ 2 ) with mean zero and variance σ 2 > 0 is the Borel probability measure on the real line ℝ whose Radon–Nikodym derivative with respect to the Lebesgue measure is d −t 2 2 𝒩0 (σ ) = exp( 2 ). dt 2σ

13.9 Stochastic laws |

527

Theorem 13.9.15 (Central limit theorem). Let T : X → X be a topologically exact, open, distance expanding, Hölder continuous system and φ : X → ℝ a Hölder continuous potential. If g : X → ℝ is Hölder continuous but is not cohomological to a constant in the class of Hölder continuous functions on X, then for every Lebesgue measurable set B ⊆ ℝ with boundary of Lebesgue measure zero we have that lim μ ({x n→∞ φ

∈X:

Sn g(x) − μφ (g)n √n

∈ B}) = 𝒩0 (σμ2 φ ,∞ (g))(B) =

1 √2πσμ2 φ ,∞ (g)

∫ exp( B

−t 2

2σμ2 φ ,∞ (g)

) dt,

where σμ2 φ ,∞ (g) is the asymptotic variance of g, whose existence and positivity are guaranteed by Lemma 13.9.13. Proof. Let φ, g ∈ Hα (X, ℝ). Replacing φ by φ − P(T, φ), we may assume without loss ̂φ = ℒφ . Replacing g by of generality that P(T, φ) = 0. In particular, this implies that ℒ g − μφ (g)1, we may assume without loss of generality that μφ (g) = 0. For every n ∈ ℕ, consider the Hölder continuous function Yn : X → ℝ defined by Yn (x) =

Sn g(x) . √n

As μφ = ϱφ mφ , ℒ∗φ (mφ ) = eP(T,φ) mφ = mφ and ℒnφ (eitYn ⋅ h) = ℒnφ+ ∫ eitYn dμφ = ∫ eitYn ϱφ dmφ = ∫ ℒnφ (eitYn ϱφ ) dmφ = ∫ ℒnφ+

X

X

X

X

it g √n

it g √n

(h), we get

(ϱφ ) dmφ .

(13.119)

For every s ∈ ℂ, let φs := φ + sg. Thanks to Theorem 13.8.18, we may apply Theorem A.2.5 (Kato–Rellich perturbation theorem) with L0 = ℒφ0 = ℒφ . Set Qs = Qℒφ , s Ss = Sℒφ , and λs = α(ℒφs ) for all s ∈ ℂ in some small enough neighborhood of zero, s say s ∈ Bℂ (0, R), determined by that theorem. This application of Kato–Rellich perturbation theorem, in conjunction with (13.119), yields ∫ eitYn dμφ = ∫ ℒnφ+

X

X

it g √n

(ϱφ ) dmφ = ∫(λnit Q it (ϱφ ) + Snit (ϱφ )) dmφ X

√n

√n

√n

(13.120)

for all n ∈ ℕ large enough. Given that φ0 = φ and μφ0 (g) = 0, this same application of Kato–Rellich perturbation theorem, this time in combination with Taylor’s theorem and Theorems 13.10.6–13.10.7, further gives λs = exp(log λs ) = exp(

σμ2 φ,∞ (g) 2

s2 + O(|s|3 )),

∀s ∈ Bℂ (0, R).

(13.121)

528 | 13 Gibbs states and transfer operators for open, distance expanding systems Therefore, for all n ∈ ℕ large enough, λnit = exp(−

σμ2 φ,∞ (g)t 2 2

√n

+ |t|3 O(n−1/2 )).

(13.122)

Also by virtue of Theorem 13.8.18 and Theorem A.2.5, we have 󵄩 󵄩 󵄩 󵄩 0 ≤ lim sup󵄩󵄩󵄩Snit (ϱφ )󵄩󵄩󵄩∞ ≤ lim sup󵄩󵄩󵄩Snit (ϱφ )󵄩󵄩󵄩α = 0, n→∞

n→∞

√n

√n

which means that 󵄩 󵄩 lim 󵄩󵄩Snit (ϱφ )󵄩󵄩󵄩∞ = 0. √n

(13.123)

n→∞󵄩

Again by means of these same two theorems and (13.122), λnit Q it (ϱφ ) √n

√n

n→∞

󳨀→

uniformly on X

exp(−

σμ2 φ,∞ (g)t 2 2

)Q0 (ϱφ ) = exp(−

σμ2 φ,∞ (g)t 2 2

)ϱφ .

Using this uniform convergence and (13.120), we conclude that lim ∫ eitYn dμφ = exp(−

n→∞

σμ2 φ,∞ (g)t 2 2

X σμ2

)mφ (ϱφ ) = exp(−

σμ2 φ,∞ (g)t 2 2

).

(g)t 2

Since the function ℝ ∋ t 󳨃→ exp(− φ,∞2 ) ∈ ℝ is the characteristic function (Fourier transform) of the Gaussian (normal) distribution 𝒩0 (σμ2 φ ,∞ (g)), the proof is completed by invoking Theorem A.1.78. Remark 13.9.16. The central limit theorem (Theorem 13.9.15) also follows from a result of Gordin [40] once we know that the series ∞

󵄩 ̂n 󵄩󵄩2 ∑ 󵄩󵄩󵄩ℒ ̂ (g)󵄩 φ 󵄩L2 (μ

φ)

n=0

converges. And indeed, assuming without loss of generality that μφ (g) = 0, we get ∞

∞

󵄩 ̂n 󵄩󵄩2 󵄨󵄨 ̂n 󵄨󵄨2 ∑ 󵄩󵄩󵄩ℒ ̂ (g)󵄩 φ 󵄩L2 (μ ) = ∑ ∫󵄨󵄨ℒφ̂ (g)󵄨󵄨 dμφ

n=0

φ

n=0

X

∞

∞

2 󵄨 󵄨2 = ∑ ∫󵄨󵄨󵄨Sn (g)󵄨󵄨󵄨 dμφ ≤ ∑ ∫(cζ n ‖g‖α ) dμφ n=0

n=0

X 2

∞

= (c‖g‖α ) ∑ ζ n=0

2n

< ∞,

X

13.9 Stochastic laws |

529

where, as in the proof of Theorem 13.9.1, the second equality sign and the following ̂φ replaced by ℒ ̂φ̂ . inequality sign both result from Theorem 13.8.18 applied with ℒ 13.9.4 Law of the iterated logarithm For the sake of completeness, we state a dynamical version of the celebrated law of the iterated logarithm. Its full proof relies on deep results from probability theory and is too involved to be presented in this book. We instead refer the reader to [98], where a complete proof is provided. Given the spectral results derived in this chapter, the law of the iterated logarithm also easily ensues from the seminal paper [41]. Theorem 13.9.17 (Law of the iterated logarithm). Let T : X → X be a topologically exact, open, distance expanding, Hölder continuous system and φ : X → ℝ a Hölder continuous potential. If g : X → ℝ is Hölder continuous but is not cohomological to a constant in the class of Hölder continuous functions on X, then lim sup n→∞

Sn g(x) − n ∫X g dμφ √n log log n

= √2σμ2 φ ,∞ (g) for μφ -a. e. x ∈ X.

13.9.5 Metric exactness, K -mixing and weak Bernoulli property An immediate consequence of Theorems 8.4.6, 13.6.19 and 13.7.7 is the following. Theorem 13.9.18. If T : X → X is a topologically exact, open, distance expanding map and φ : X → ℝ is a Hölder continuous potential, then the T-invariant Gibbs state μφ is ̃→X ̃ is K-mixing. metrically exact and the Rokhlin’s natural extension T̃ : X In fact, a much stronger result holds: the Rokhlin’s natural extension T̃ is metrically isomorphic to a two-sided Bernoulli shift. In order to establish this, we need to introduce and discuss an important concept commonly referred to as weak Bernoulli. Definition 13.9.19. Given two finite measurable partitions 𝒜 and ℬ of a probability space (X, ℱ , μ) and ε ≥ 0, we say that the partition ℬ is ε-independent of 𝒜 if there exists a subfamily 𝒜′ ⊆ 𝒜 such that μ( ⋃ A) ≥ 1 − ε A∈𝒜′

and

󵄨󵄨 μ(A ∩ B) 󵄨󵄨 󵄨 󵄨 − μ(B)󵄨󵄨󵄨 ≤ ε, ∑ 󵄨󵄨󵄨 󵄨󵄨 μ(A) 󵄨󵄨 B∈ℬ

∀A ∈ 𝒜′ .

Definition 13.9.20. Given an ergodic measure-preserving endomorphism T : X → X of a Lebesgue space (X, ℱ , μ), a finite measurable partition 𝒜 is called weakly Bernoulli

530 | 13 Gibbs states and transfer operators for open, distance expanding systems if for every ε > 0 there exists an integer k = k(ε) ≥ 0 such that the partition n+1

n

𝒜m+k := ⋁ T (𝒜) −j

j=m+k

is ε-independent of the partition m+1

𝒜0

m

:= ⋁ T −j (𝒜) j=0

for all integers m ≥ 0 and n ≥ m + k. The endomorphism (T : X → X, μ) is said to be weakly Bernoulli if it has a weakly Bernoulli one-sided generating partition 𝒜. “Onen sided generating” means that σ(⋃∞ n=1 𝒜 ) = ℱ . If ε = 0 and k = 1, then all partitions T −j (𝒜), j ≥ 0, are mutually independent, which means that μ(A ∩ B) = μ(A)μ(B) for all A ∈ T −i (𝒜), all B ∈ T −j (𝒜) and all i ≠ j. We then say that the partition 𝒜 is Bernoulli. If 𝒜 is a one-sided generating partition (resp., a two-sided generating partition), then obviously the measure-preserving endomorphism T : X → X is isomorphic to a one-sided (resp., a two-sided) Bernoulli shift with #𝒜 symbols. The following famous theorem of Friedman and Ornstein [33] shows the high importance of the “weak Bernoulli” concept. Theorem 13.9.21. If an ergodic measure-preserving automorphism T : X → X of a Lebesgue space (X, ℱ , μ) is weakly Bernoulli, then T is isomorphic to a two-sided Bernoulli shift. In order to formulate an appropriate “Bernoulli theorem” for noninvertible systems, we bring up the following simple observation. Observation 13.9.22. If an ergodic measure-preserving endomorphism T : X → X of a Lebesgue space (X, ℱ , μ) has a weakly Bernoulli (one-sided) generating partition 𝒜, then π0−1 (𝒜) = {π0−1 (A) : A ∈ 𝒜} is a weakly Bernoulli generating partition for the ̃ → X. ̃ Rokhlin’s natural extension T̃ : X As a direct consequence of this observation, Theorem 8.4.7 and Theorem 13.9.21, we get the following. Theorem 13.9.23. If an ergodic measure-preserving endomorphism T : X → X of a Lebesgue space (X, ℱ , μ) is weakly Bernoulli, then its Rokhlin’s natural extension is isomorphic to a two-sided Bernoulli shift.

13.9 Stochastic laws |

531

However, remark that being weakly Bernoulli is usually far from sufficient to conclude that the endomorphism is isomorphic to a one-sided Bernoulli shift. The reason/obstacle is that the endomorphism’s Jacobian is invariant under an isomorphism and the Jacobian of each one-sided Bernoulli shift is locally constant. In order to have a more complete picture, we now formulate another famous theorem of Ornstein [90]. Theorem 13.9.24. Two Bernoulli shifts (with a finite alphabet) are metrically isomorphic if and only if they have the same entropy. Heading toward proving that natural extensions of Gibbs measures of distance expanding maps are isomorphic to two-sided Bernoulli shifts, we shall first show an intermediate (and interesting in itself) version for subshifts of finite type. We start with the following improvement of Theorem 13.9.1 on the exponential decay of correlations in the symbolic case. Theorem 13.9.25. Let E be a finite set having at least two elements and A : E×E → {0, 1} be a primitive incidence matrix. Let 𝒜 := {[e] : e ∈ E} be the partition of EA∞ into cylinders of length one and σ(𝒜) be the σ-algebra generated by 𝒜. Let φ : EA∞ → ℝ be a Hölder continuous potential. Then there exists a constant D > 0 such that if f ∈ L1 (μφ ) and g : EA∞ → ℝ is measurable with respect to the σ-algebra σ(𝒜k ) for some k ∈ ℕ, then 󵄩 󵄩 󵄩 󵄩 Cμφ ,n (f , g) ≤ Dζ n−k 󵄩󵄩󵄩f − Eμφ (f )󵄩󵄩󵄩L1 (μ ) 󵄩󵄩󵄩g − Eμφ (g)󵄩󵄩󵄩L1 (μ ) , φ φ

∀n ≥ k,

where ζ ∈ (0, 1) comes from Theorem 13.9.1. Proof. Let φ ∈ Hα (EA∞ , ℝ) and n ≥ k. Replacing f by f − Eμφ (f ) and g by g − Eμφ (g), we ̂k̂ (g) ⋅ (f ∘ σ n−k ). ̂k̂ (g ⋅ (f ∘ σ n )) = ℒ may assume that Eμ (f ) = 0 = Eμ (g). Observe that ℒ φ

φ

φ

φ

̂∗̂ (μφ ) = μφ ̂k̂ (g)) since μφ is σ-invariant and since ℒ Moreover, Eμφ (f ∘ σ n−k ) = 0 = Eμφ (ℒ φ φ by Proposition 13.8.4, respectively. Using all of the above and Theorem 13.9.1, we get 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 n k n 󵄨 󵄨 󵄨 ̂ Cμφ ,n (f , g) = 󵄨󵄨 ∫ g ⋅ (f ∘ σ ) dμφ 󵄨󵄨 = 󵄨󵄨 ∫ ℒφ̂ (g ⋅ (f ∘ σ )) dμφ 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨E ∞ 󵄨 󵄨E ∞ 󵄨 A

A

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 ̂k̂ (g) ⋅ (f ∘ σ n−k ) dμφ 󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∫ ℒ φ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨E ∞ A

̂k̂ (g)) = Cμφ ,n−k (f , ℒ φ 󵄩 ̂k 󵄩󵄩 n−k ≤ cζ ‖f ‖L1 (μφ ) 󵄩󵄩󵄩ℒ ̂ (g)󵄩 φ 󵄩α .

(13.124)

̂k̂ (g) ∈ Hα (E ∞ ), This last inequality arises from Theorem 13.9.1 and presupposes that ℒ A φ ∞ though g may not be in Hα (EA ). It thus remains to find an appropriate upper bound 󵄩 ̂k 󵄩󵄩 for 󵄩󵄩󵄩ℒ ̂ (g)󵄩 󵄩α . φ

532 | 13 Gibbs states and transfer operators for open, distance expanding systems Decomposing g = g+ − g− into its positive and negative parts, we may assume that g ≥ 0. Since the function g : EA∞ → ℝ depends only on the first k coordinates, using Lemma 13.1.1 we obtain for all ω, τ ∈ EA∞ with ω1 = τ1 that k

ℒφ (g)(ω) =

∑ γ∈EAk : Aγk ω1 =1

=

∑ γ∈EAk : Aγk τ1 =1

g(γω) exp(Sk φ(γω)) g(γτ) exp(Sk φ(γω))

≤ exp(Cα (φ)dα (ω, τ))

∑ γ∈EAk : Aγk τ1 =1

g(γτ) exp(Sk φ(γτ))

= exp(Cα (φ)dα (ω, τ))ℒkφ (g)(τ) ≤ eCα (φ) ℒkφ (g)(τ).

(13.125)

Therefore, sup(ℒkφ (g)|[e] ) ≤ eCα (φ) inf(ℒkφ (g)|[e] ),

∀e ∈ E.

̂∗ (mφ ) = mφ ), the fact that μφ := ϱφ mφ and Lemma 13.7.5, Using this, Corollary 13.6.5 (ℒ φ it ensues that ̂k (g)|[e] ) ≤ eCα (φ) inf(ℒ ̂k (g)|[e] ) ≤ eCα (φ) sup(ℒ φ φ

1 ̂k (g) dmφ ∫ℒ φ mφ ([e]) [e]

≤e

Cα (φ)

1 1 ̂k (g) dmφ = eCα (φ) ∫ℒ ∫ g dmφ mφ ([e]) ∞ φ mφ ([e]) ∞ EA

= eCα (φ)

EA

1 1 ‖g‖ 1 ≤ eCα (φ) mφ ([e]) L (mφ ) mφ ([e]) inf(ϱφ )

‖g‖L1 (μφ )

≤ C1 (φ)‖g‖L1 (μφ ) , where

C1 (φ) := Dα (φ)eCα (φ) max a∈E

1 . mφ ([a])

Hence, 󵄩󵄩 ̂k 󵄩󵄩 ̂k (g)) ≤ C1 (φ)‖g‖L1 (μ ) . 󵄩󵄩ℒφ (g)󵄩󵄩∞ = sup(ℒ φ φ

(13.126)

Using (13.125)– (13.126), we have for all ω, τ ∈ EA∞ with ω1 = τ1 that 󵄨󵄨 ̂k ̂k (g)(τ)󵄨󵄨󵄨 ≤ [exp(Cα (φ)dα (ω, τ)) − 1] max{ℒ ̂k (g)(ω), ℒ ̂k (g)(τ)} 󵄨󵄨ℒφ (g)(ω) − ℒ φ φ φ 󵄨 ≤ [exp(Cα (φ)dα (ω, τ)) − 1]C1 (φ)‖g‖L1 (μφ ) ≤ C2 (φ)Cα (φ)dα (ω, τ) ⋅ C1 (φ)‖g‖L1 (μφ ) ,

13.9 Stochastic laws |

533

where C2 (φ) > 0 is a constant such that ex ≤ 1 + C2 (φ)x for all x ∈ [0, Cα (φ)]. Consequently, ̂k (g)) ≤ C2 (φ)Cα (φ) ⋅ C1 (φ)‖g‖L1 (μ ) . υα (ℒ φ φ Along with (13.126), this gives 󵄩󵄩 ̂k 󵄩󵄩 󵄩󵄩ℒφ (g)󵄩󵄩α ≤ (1 + C2 (φ)Cα (φ)) ⋅ C1 (φ) ⋅ ‖g‖L1 (μφ ) . ̂ (this is possible because φ ̂ ∈ Hα (EA∞ , ℝ), as the shift map σ is Now replace φ by φ Lipschitz; see the start of Subsection 13.8.1). Since μφ̂ = μφ (see Corollary 13.8.11), we deduce that 󵄩󵄩 ̂k 󵄩󵄩 ̂)Cα (φ ̂)) ⋅ C1 (φ ̂) ⋅ ‖g‖L1 (μφ ) . 󵄩󵄩ℒφ̂ (g)󵄩󵄩α ≤ (1 + C2 (φ Inserting this into (13.124), we conclude that ̂)Cα (φ ̂)) ⋅ C1 (φ ̂)] ⋅ ζ n−k ‖f ‖L1 (μφ ) ‖g‖L1 (μφ ) . Cμφ ,n (f , g) ≤ [c(1 + C2 (φ Theorem 13.9.26. Let E be a finite set with #E ≥ 2 and let A : E × E → {0, 1} be a primitive incidence matrix. Then the partition 𝒜 := {[e] : e ∈ E} of EA∞ into cylinders of length one is a weak Bernoulli one-sided generator for the Gibbs state μφ for every Hölder continuous potential φ : EA∞ → ℝ. In consequence, the Rokhlin’s natural extension of the dynamical system (σ : EA∞ → EA∞ , μφ ), which is in fact the double-sided shift ̃ φ ), is isomorphic to a two-sided Bernoulli shift. (σ̃ : EAℤ → EAℤ , μ Proof. Fix integers k ∈ ℕ, ℓ ≥ 0 and N ≥ k + ℓ. Fix also ω ∈ EAk and τ ∈ EAN−(k+ℓ) . Recall that μφ is σ-invariant. Applying Theorem 13.9.25 with n = k + ℓ, k = k, f = 1[τ] and g = 1[ω] and using (13.75), we obtain that 󵄨󵄨 󵄨 󵄨󵄨μφ (σ −(k+ℓ) ([τ]) ∩ [ω]) − μφ (σ −(k+ℓ) ([τ]))μφ ([ω])󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 = 󵄨󵄨󵄨μφ (1[ω] ⋅ (1[τ] ∘ σ k+ℓ )) − μφ (1[ω] )μφ (1[τ] )󵄨󵄨󵄨 󵄨 󵄨 = Cμφ ,k+ℓ (1[τ] , 1[ω] ) 󵄩 󵄩 󵄩 󵄩 ≤ Dζ ℓ 󵄩󵄩󵄩1[τ] − μφ (1[τ] )󵄩󵄩󵄩L1 (μ ) 󵄩󵄩󵄩1[ω] − μφ (1[ω] )󵄩󵄩󵄩L1 (μ ) φ φ ≤ 4Dζ ℓ μφ ([τ])μφ ([ω]). Therefore, for every ω ∈ EAk we get 󵄨󵄨 󵄨󵄨 μφ (σ −(k+ℓ) ([τ]) ∩ [ω]) 󵄨 󵄨󵄨 − μφ (σ −(k+ℓ) ([τ]))󵄨󵄨󵄨 ≤ 4Dζ ℓ ∑ μφ ([τ]) = 4Dζ ℓ . 󵄨󵄨 󵄨󵄨 󵄨 μ ([ω]) 󵄨 φ τ∈E N−(k+ℓ) τ∈E N−(k+ℓ) ∑ A

A

534 | 13 Gibbs states and transfer operators for open, distance expanding systems As ζ ∈ (0, 1), given ε > 0 there is ℓ = ℓ(ε) ∈ ℕ so large that 4Dζ ℓ < ε. Hence, the k+1 partition 𝒜N+1 k+ℓ is ε-independent of the partition 𝒜0 . Since, moreover, 𝒜 is a topo∞ ∞ logical generator for the shift map σ : EA → EA , the measure-preserving dynamical system (σ : EA∞ → EA∞ , μφ ) is weakly Bernoulli. Its Rokhlin’s natural extension ̃ φ ) is isomorphic to a two-sided Bernoulli shift by Theorem 13.9.23. (σ̃ : EAℤ → EAℤ , μ A straightforward consequence of this theorem and Theorem 13.7.11 is next. Theorem 13.9.27. If T : X → X is a topologically exact, open, distance expanding system and φ : X → ℝ is a Hölder continuous potential, then the measure-preserving dynamical system (T : X → X, μφ ) is weakly Bernoulli and its Rokhlin’s natural extension is isomorphic to a two-sided Bernoulli shift.

13.10 Real analyticity of topological pressure Let T : X → X be an open, distance expanding system on a compact metric space (X, d). Let ε > 0 be so small that all the inverse branches of T are well defined on any ball B(x, ε). For instance, take any ε ≤ ξ , where ξ is the constant arising from relation (13.2) (cf. (4.29)). Fix α > 0. Recall that C(X) and Hα (X) are now considered complex Banach spaces. The supremum norm ‖ ⋅ ‖∞ , the Hölder variation υα and the Hölder norm ‖ ⋅ ‖α are defined exactly as in Section 13.1. For every point x ∈ X, let Hα,x,ε (X) be the Banach space of all complex-valued Hölder continuous functions with exponent α whose domain is the ball B(x, ε). The supremum norm ‖ ⋅ ‖∞,x,ε , the Hölder variation υα,x,ε and the Hölder norm ‖ ⋅ ‖α,x,ε are defined in a similar way as in Section 13.1 by simply replacing X with B(x, ε). Let F, G be complex Banach spaces. Let L(F) and L(F, G) respectively denote the Banach spaces of bounded ℂ-linear operators from F to itself and from F to G. For any operator L ∈ L(F, G), recall that the operator norm of L is defined by ‖L‖ := inf{c ≥ 0 : ‖L(υ)‖G ≤ c‖υ‖F , ∀υ ∈ F} = sup{

‖L(υ)‖G : υ ∈ F \ {0}}. ‖υ‖F

In particular, if L ∈ L(Hα (X), Hα,x,ε (X)) then ‖L‖α := inf{c ≥ 0 : ‖L(g)‖α,x,ε ≤ c‖g‖α , ∀g ∈ Hα (X)}. Given an arbitrary set Λ and a function Φ : Λ → L(Hα (X)), for every point x ∈ X define the function Φx : Λ → L(Hα (X), Hα,x,ε (X))

13.10 Real analyticity of topological pressure | 535

by Φx (λ)(g) := Φ(λ)(g)|B(x,ε) . Obviously, if Λ ⊆ ℂ and the function Φ is holomorphic then so is the function Φx for all x ∈ X. A natural endeavor then consists in identifying conditions under which the converse holds. This is the motivation behind the following statement. Lemma 13.10.1. Let Λ be an open subset of ℂ. If a function Φ : Λ → L(Hα (X)) is such that the function Φx : Λ → L(Hα (X), Hα,x,ε (X)) is holomorphic for every x ∈ X and 󵄩 󵄩 sup{󵄩󵄩󵄩Φx (λ)󵄩󵄩󵄩α : x ∈ X, λ ∈ Λ} < ∞, then Φ : Λ → L(Hα (X)) is holomorphic. Proof. Fix λ0 ∈ Λ and take r > 0 so small that the disc D(λ0 , 2r) centered at λ0 and of radius 2r is contained in Λ. By hypothesis, for each x ∈ X the Taylor series expansion of Φx around λ0 , i. e., ∞

n

Φx (λ) := ∑ ax,n (λ − λ0 ) , n=0

λ ∈ D(λ0 , r),

with some ax,n ∈ L(Hα (X), Hα,x,ε (X)), converges uniformly in the Banach space L(Hα (X), Hα,x,ε (X)). Set M = sup{‖Φx (λ)‖α : x ∈ X, λ ∈ Λ} < ∞. It follows from Cauchy’s estimates (Theorem A.3.2 in Appendix A) that ‖ax,n ‖α ≤ Mr −n ,

∀x ∈ X, ∀n ≥ 0.

(13.127)

For every n ≥ 0, define the operator an by an (g)(x) := ax,n (g)(x),

∀g ∈ Hα (X), ∀x ∈ X.

Let g ∈ Hα (X). Then 󵄩󵄩 󵄩 󵄩 󵄩 −n 󵄩󵄩an (g)󵄩󵄩󵄩∞ ≤ sup󵄩󵄩󵄩ax,n (g)󵄩󵄩󵄩α ≤ sup ‖ax,n ‖α ‖g‖α ≤ Mr ‖g‖α . x∈X

x∈X

Moreover, if d(x, y) < ε then for every w ∈ B(x, ε) ∩ B(y, ε), we have ∞

n

∑ ax,n (g)(w)(λ − λ0 ) = (Φx (λ)(g))(w) = (Φ(λ)(g))(w) = (Φy (λ)(g))(w)

n=0

∞

n

= ∑ ay,n (g)(w)(λ − λ0 ) , n=0

∀λ ∈ D(λ0 , r).

(13.128)

536 | 13 Gibbs states and transfer operators for open, distance expanding systems The uniqueness of the coefficients of Taylor series expansions imposes that ax,n (g)(w) = ay,n (g)(w),

∀n ≥ 0.

(13.129)

Since x, y ∈ B(x, ε) ∩ B(y, ε), we get from (13.129) and (13.127) that 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨an (g)(y) − an (g)(x)󵄨󵄨󵄨 = 󵄨󵄨󵄨ay,n (g)(y) − ax,n (g)(x)󵄨󵄨󵄨 = 󵄨󵄨󵄨ax,n (g)(y) − ax,n (g)(x)󵄨󵄨󵄨 󵄩 󵄩 ≤ 󵄩󵄩󵄩ax,n (g)󵄩󵄩󵄩α,x,ε dα (x, y) ≤ ‖ax,n ‖α ‖g‖α dα (x, y) ≤ Mr −n ‖g‖α dα (x, y). Consequently, υα,x,ε (an (g)|B(x,ε) ) ≤ Mr −n ‖g‖α for all x ∈ X. Due to the compactness of X, one can extract a finite number of finite chains of open balls of radius ε covering X, chains that are positively separated from one another. There hence exists a constant C ≥ 1 such that υα (an (g)) ≤ CMr −n ‖g‖α . Combining this with (13.128), we obtain 󵄩󵄩 󵄩 −n 󵄩󵄩an (g)󵄩󵄩󵄩α ≤ (C + 1)Mr ‖g‖α . Thus, an ∈ L(Hα (X)) and ‖an ‖α ≤ (C + 1)Mr −n . Therefore, the series ∞

n

∑ an (λ − λ0 ) ,

n=0

λ ∈ D(λ0 , r/2),

converges absolutely uniformly in the Banach space L(Hα (X)) with 󵄩󵄩 ∞ 󵄩 󵄩󵄩 󵄩 0 n󵄩 󵄩󵄩 ∑ an (λ − λ ) 󵄩󵄩󵄩 ≤ 2(C + 1)M, 󵄩󵄩n=0 󵄩󵄩α

∀λ ∈ D(λ0 , r/2).

0 n The function D(λ0 , r/2) ∋ λ 󳨃󳨀→ ∑∞ n=0 an (λ − λ ) ∈ L(Hα (X)) is thereby holomorphic. Finally, for every g ∈ Hα (X) and every x ∈ X, ∞

n

∞

n

∞

n

( ∑ an (λ − λ0 ) )(g)(x) = ∑ an (g)(x)(λ − λ0 ) = ∑ ax,n (g)(x)(λ − λ0 ) n=0

n=0 ∞

n=0

n

= ( ∑ ax,n (λ − λ0 ) )(g)(x) = Φx (λ)(g)(x) n=0

= Φ(λ)(g)(x).

13.10 Real analyticity of topological pressure | 537

So ∞

n

Φ(λ) = ∑ an (λ − λ0 ) , n=0

∀λ ∈ D(λ0 , r/2)

and Φ is thus holomorphic on D(λ0 , r/2). Since holomorphicity is a local property, the result ensues. The main technical result of this section concerns the analytic dependence of the transfer operator ℒφλ on the parameter λ. Theorem 13.10.2. Let T : X → X be an open, distance expanding system. Suppose that Λ is an open subset of ℂk for some k ∈ ℕ. If φλ ∈ Hα (X) for every λ ∈ Λ, if sup{‖φλ ‖α : λ ∈ Λ} < ∞ and if the function Λ ∋ λ 󳨃󳨀→ φλ (x) ∈ ℂ is holomorphic for every x ∈ X, then the map Λ ∋ λ 󳨃󳨀→ ℒφλ ∈ L(Hα (X)), is holomorphic, where ℒφλ is the natural complexification of the real transfer operator ℒφ . Proof. By Hartogs’ theorem (Theorem A.4.1 in Appendix A), we may assume without loss of generality that k = 1, i. e., Λ ⊆ ℂ. Let H := sup{‖φλ ‖α : λ ∈ Λ} < ∞. For all λ ∈ Λ, all x ∈ X and all y ∈ T −1 (x), we know that 󵄩󵄩 −1 󵄩 H 󵄩󵄩exp(φλ ∘ Ty )󵄩󵄩󵄩∞,x,ε ≤ e ,

(13.130)

where Ty−1 is the branch of T −1 on B(x, ε) mapping x to y. Fix λ0 ∈ Λ and take a radius r > 0 so small that D(λ0 , 2r) ⊆ Λ. Let also x ∈ X and y ∈ T −1 (x). By hypothesis, the function D(λ0 , 2r) ∋ λ 󳨃󳨀→ exp(φλ ∘ Ty−1 (w)) is holomorphic for every w ∈ B(x, ε). Consider its Taylor series expansion ∞

n

exp(φλ ∘ Ty−1 (w)) = ∑ ay,n (w)(λ − λ0 ) , n=0

∀λ ∈ D(λ0 , r).

In view of Cauchy’s estimates (Theorem A.3.2) and (13.130), we get 󵄨󵄨 󵄨 H −n 󵄨󵄨ay,n (w)󵄨󵄨󵄨 ≤ e r ,

∀n ≥ 0, ∀w ∈ B(x, ε).

(13.131)

Moreover, as the local inverse branches of T are contractions, we know that d(Ty−1 (w), Ty−1 (υ)) ≤ d(w, υ) for all w, υ ∈ B(x, ε). Cauchy’s estimates and the mean value inequal-

538 | 13 Gibbs states and transfer operators for open, distance expanding systems ity then yield 󵄨 󵄨 󵄨 −n 󵄨󵄨 −1 −1 󵄨󵄨ay,n (w) − ay,n (υ)󵄨󵄨󵄨 ≤ r sup󵄨󵄨󵄨exp(φλ ∘ Ty (w)) − exp(φλ ∘ Ty (υ))󵄨󵄨󵄨 λ∈Λ

≤r

−n

≤r

−n

󵄨 󵄨 sup emax Re(φλ ) 󵄨󵄨󵄨φλ ∘ Ty−1 (w) − φλ ∘ Ty−1 (υ)󵄨󵄨󵄨 λ∈Λ

sup e‖φλ ‖α υα (φλ )dα (Ty−1 (w), Ty−1 (υ)) λ∈Λ

≤ eH Hr −n dα (w, υ).

(13.132)

Take an arbitrary g ∈ Hα (X). In view of relations (13.131)–(13.132), we obtain that 󵄨󵄨 󵄨 −1 −1 󵄨󵄨ay,n (w)g(Ty (w)) − ay,n (υ)g(Ty (υ))󵄨󵄨󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨(ay,n (w) − ay,n (υ))g(Ty−1 (w)) + ay,n (υ)(g(Ty−1 (w)) − g(Ty−1 (υ)))󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨ay,n (w) − ay,n (υ)󵄨󵄨󵄨 ⋅ ‖g‖∞ + 󵄨󵄨󵄨ay,n (υ)󵄨󵄨󵄨 ⋅ υα (g)dα (Ty−1 (w), Ty−1 (υ)) ≤ eH Hr −n dα (w, υ) ⋅ ‖g‖α + eH r −n ⋅ ‖g‖α dα (w, υ)

= (H + 1)eH r −n ‖g‖α dα (w, υ). Combining this and (13.131), we deduce that the formula Ny,n (g)(w) := ay,n (w) ⋅ g(Ty−1 (w)) engenders a bounded linear operator Ny,n : Hα (X) 󳨀→ Hα,x,ε (X) such that υα,x,ε (Ny,n (g)) ≤ (H + 1)eH r −n ‖g‖α and ‖Ny,n (g)‖∞,x,ε ≤ eH r −n ‖g‖α . So ‖Ny,n (g)‖α,x,ε ≤ (H + 2)eH r −n ‖g‖α , and hence ‖Ny,n ‖α ≤ (H + 2)eH r −n . Consequently, the function n

D(λ0 , r/2) ∋ λ 󳨃󳨀→ Ny,n (λ − λ0 ) is holomorphic and 󵄩󵄩 0 n󵄩 −n H 󵄩󵄩Ny,n (λ − λ ) 󵄩󵄩󵄩α ≤ 2 (H + 2)e ,

∀λ ∈ D(λ0 , r/2).

Thus, the series ∞

n

Aλ,y := ∑ Ny,n (λ − λ0 ) , n=0

λ ∈ D(λ0 , r/2),

13.10 Real analyticity of topological pressure | 539

converges absolutely uniformly in the Banach space L(Hα (X), Hα,x,ε (X)), with norm ‖Aλ,y ‖α ≤ 2(H + 2)eH , and the function D(λ0 , r/2) ∋ λ 󳨃󳨀→ Aλ,y ∈ L(Hα (X), Hα,x,ε (X)) is holomorphic. Hence, ℒλ,x :=

∑ y∈T −1 (x)

Aλ,y ∈ L(Hα (X), Hα,x,ε (X)),

(13.133)

with ‖ℒλ,x ‖α ≤ 2#T −1 (H +2)eH , and the function D(λ0 , r/2) ∋ λ 󳨃󳨀→ ℒλ,x is holomorphic. Note that the above estimate on ‖ℒλ,x ‖α is independent of λ and λ0 , and thus holds for all λ ∈ Λ. It is also independent of x. Moreover, analyticity is a local property. A straightforward calculation establishes that ℒλ,x (g)(x) = ℒφλ (g)(x) for all g ∈ Hα (X), all x ∈ X and all λ ∈ Λ. Invoking Lemma 13.10.1 concludes the proof. A function from a complex vector space to a complex Banach space is called holomorphic if its restriction to any complex finite-dimensional affine subspace is holomorphic; see [26, Definition VI.10.5]. Theorem 13.10.2 yields the holomorphicity of Hα (X) ∋ φ 󳨃󳨀→ ℒφ ∈ L(Hα (X))

(13.134)

mentioned in the introduction to this section. It further yields real analyticity after restricting this function to the real Banach space Hα (X, ℝ). From Theorem 13.10.2, we can infer the real analyticity of the pressure function for a “nicely” parametrized family of Hölder continuous potentials. Theorem 13.10.3. Let T : X → X be a topologically transitive, open, distance expanding system. Suppose that Λ is an open subset of ℂk for some k ∈ ℕ. Assume that φλ ∈ Hα (X) for every λ ∈ Λ, that sup{‖φλ ‖α : λ ∈ Λ} < ∞, and that the function Λ ∋ λ 󳨃󳨀→ φλ (x) ∈ ℂ is holomorphic for every x ∈ X. If M is a real-analytic submanifold of Λ such that φλ (X) ⊆ ℝ,

∀λ ∈ M,

then the pressure function M ∋ λ 󳨃󳨀→ P(T, φλ ) ∈ ℝ is real analytic. Proof. Fix λ0 ∈ M. Since Λ ⊆ ℂk is open, there exists R > 0 such that Bℂk (λ0 , R) ⊆ Λ. By Theorem 13.8.20, the number exp(P(T, φλ )) is a simple isolated eigenvalue of the operator ℒφλ ∈ L(Hα (X)) for all λ ∈ M ∩ Bℂk (λ0 , R). It follows from Theorem 13.10.2 and

540 | 13 Gibbs states and transfer operators for open, distance expanding systems the Kato–Rellich theorem for perturbations of linear operators (Theorem A.2.5) that there exist r ∈ (0, R] and a holomorphic function γ : Bℂk (λ0 , r) → ℂ such that γ(λ) is an eigenvalue of the operator ℒφλ for all λ ∈ Bℂk (λ0 , r) and γ(λ) = exp(P(T, φλ )),

∀λ ∈ M ∩ Bℂk (λ0 , r).

Consequently, the function M ∩ Bℂk (λ0 , r) ∋ λ 󳨃󳨀→ P(T, φλ ) ∈ ℝ is real analytic. Since real analyticity is a local property, the result holds. As a special case of this theorem, we get the real analyticity of the pressure function for the parametrized family {tφ + ψ}t∈ℝ for any Hölder functions φ, ψ. Corollary 13.10.4. Let T : X → X be a topologically transitive, open, distance expanding system. If φ, ψ : X → ℝ are Hölder continuous functions, then the pressure function ℝ ∋ t 󳨃󳨀→ P(T, tφ + ψ) ∈ ℝ is real analytic. Proof. Let α be a Hölder exponent common to φ and ψ. Fix t ∈ ℝ. Let Λ = Bℂ (t, 1) and φλ = λφ + ψ for every λ ∈ Λ. Obviously, φλ ∈ Hα (X) for every λ ∈ Λ and sup{‖φλ ‖α : λ ∈ Λ} = sup{‖λφ + ψ‖α : λ ∈ Λ} ≤ sup{|λ| ⋅ ‖φ‖α + ‖ψ‖α : λ ∈ Λ} = sup{|λ| : λ ∈ Bℂ (t, 1)}‖φ‖α + ‖ψ‖α = max{−(t − 1), t + 1}‖φ‖α + ‖ψ‖α < ∞. Furthermore, the function Λ ∋ λ 󳨃󳨀→ φλ (x) ∈ ℂ is clearly holomorphic for all x ∈ X. Moreover, φλ (X) ⊆ ℝ for all λ ∈ (t − 1, t + 1) = Λ ∩ ℝ = M. So Theorem 13.10.3 applies, giving the desired outcome. We will need another real analyticity result similar to Theorem 13.10.3. Theorem 13.10.5. Let T : X → X be a topologically transitive, open, distance expanding dynamical system. Suppose that Λ is an open subset of ℂk for some k ∈ ℕ. Assume also that φλ ∈ Hα (X) for every λ ∈ Λ, that sup{‖φλ ‖α : λ ∈ Λ} < ∞, and that the function Λ ∋ λ 󳨃󳨀→ φλ (x) ∈ ℂ is holomorphic for every x ∈ X. If M is a real-analytic submanifold of Λ such that φλ (X) ⊆ ℝ,

∀λ ∈ M,

13.10 Real analyticity of topological pressure

| 541

then the function M ∋ λ 󳨃󳨀→ ϱφλ (x) ∈ ℝ is real analytic for every x ∈ X, where ϱφλ ∈ Hα (X) arises from Lemma 13.7.5. Proof. Like the proof of Theorem 13.10.3, the current theorem follows immediately from Theorem 13.8.20, Theorem 13.10.2 and Kato–Rellich theorem for perturbations of linear operators (Theorem A.2.5), applied with the vector υ0 = 1. It follows from Corollary 13.10.4 that both the first and second derivatives P′ (tφ+ψ) and P′′ (tφ + ψ) exist. We will now express them in familiar dynamical terms. Theorem 13.10.6. Let T : X → X be a topologically transitive, open, distance expanding dynamical system. If φ, ψ : X → ℝ are Hölder continuous functions, then for every s ∈ ℝ, 󵄨󵄨 d P(T, tφ + ψ)󵄨󵄨󵄨 = ∫ φ dμsφ+ψ , 󵄨t=s dt

(13.135)

X

where μsφ+ψ is the unique T-invariant Gibbs state for sφ + ψ (see Theorem 13.7.7). Proof. With the notation and terminology of Theorems 13.10.2–13.10.3 and Corollary 13.10.4, it follows from these results and the proof of Theorem 13.10.2 (particularly formula (13.133) and the fact that ℒλ,x (g)(x) = ℒφλ (g)(x)) that the derivative of the operator ℒλ = ℒφλ , λ ∈ Λ = Bℂ (s, 1), is given by the formula d d φ (y) φ (y) ℒ (g)(x) = ∑ g(y) e λ = ∑ g(y)φ(y)e λ = ℒλ (gφ)(x). dλ λ dλ −1 −1 y∈T (x) y∈T (x) So d ℒ (g) = ℒλ (gφ). dλ λ

(13.136)

For every t ∈ ℝ, set μt := μtφ+ψ and mt := mtφ+ψ . Then P(T, tφ + ψ) = P(T, t(φ − μs (φ)) + ψ + tμs (φ)) = P(T, t(φ − μs (φ)) + ψ) + tμs (φ). Therefore, 󵄨󵄨 󵄨󵄨 d d P(T, tφ + ψ)󵄨󵄨󵄨 = P(T, t(φ − μs (φ)) + ψ)󵄨󵄨󵄨 + μs (φ). 󵄨t=s dt 󵄨t=s dt

(13.137)

Furthermore, as s(φ − μs (φ)) + ψ − (sφ + ψ) = −sμs (φ), Theorem 13.7.17 asserts that μsφ+ψ = μs(φ−μs (φ))+ψ .

(13.138)

542 | 13 Gibbs states and transfer operators for open, distance expanding systems Replacing φ by φ − μs (φ), it ensues from (13.137)–(13.138) that we may assume that μs (φ) = 0.

(13.139)

Proving (13.135) then reduces to showing that 󵄨󵄨 d P(T, tφ + ψ)󵄨󵄨󵄨 = 0. 󵄨t=s dt

(13.140)

Moreover, since P(T, tφ + ψ − P(T, sφ + ψ)) = P(T, tφ + ψ) − P(T, sφ + ψ), we get 󵄨󵄨 󵄨󵄨 d d P(T, tφ + ψ)󵄨󵄨󵄨 = P(T, tφ + ψ − P(T, sφ + ψ))󵄨󵄨󵄨 . 󵄨 󵄨t=s t=s dt dt

(13.141)

Also, μsφ+ψ = μsφ+ψ−P(T,sφ+ψ) by Theorem 13.7.17, whence μsφ+ψ (φ) = μsφ+ψ−P(T,sφ+ψ) (φ).

(13.142)

By (13.141)–(13.142), we may replace ψ by ψ − P(T, sφ + ψ) and assume without losing generality that P(T, sφ + ψ) = 0.

(13.143)

Set P(t) := P(T, tφ + ψ) and ϱt := ϱtφ+ψ , where ϱtφ+ψ arises from Lemma 13.7.5. Take t ∈ (s − 1, s + 1). By Theorem 13.10.5, the function (s − 1, s + 1) ∋ t 󳨃󳨀→ ϱt (x) ∈ ℝ is real analytic for every x ∈ X. Differentiating with respect to t the equation ℒt (ϱt ) = e

P(t)

ϱt

with the help of (13.136), we obtain that ℒt (ϱt φ) + ℒt (ϱt ) = e ′

P(t)

(P′ (t)ϱt + ϱ′t ).

Taking t = s and using (13.143), we deduce that ℒs (ϱs φ) + ℒs (ϱs ) = P (s)ϱs + ϱs . ′

′

′

Integrating this equality with respect to the measure ms , we have ms (ℒs (ϱs φ)) + ms (ℒs (ϱ′s )) = P′ (s)ms (ϱs ) + ms (ϱ′s ). Using Corollary 13.6.4, this equation boils down to ms (ϱs φ) + ms (ϱ′s ) = P′ (s)ms (ϱs ) + ms (ϱ′s ).

(13.144)

13.10 Real analyticity of topological pressure

| 543

By virtue of Theorem 13.7.7 and Lemma 13.7.5, this reduces to μs (φ) + ms (ϱ′s ) = P′ (s) + ms (ϱ′s ). Invoking (13.139), we conclude that P′ (s) = μs (φ) = 0, meaning that (13.140) holds, as sought. An alternative proof, analogous to that of Theorems 16.4.10 and 20.2.4 and based on the variational principle (Theorem 12.1.1), is left to the reader as an exercise. We now provide a formula for the second derivative of the topological pressure function for those systems that are additionally exact and Hölder continuous. Theorem 13.10.7. Let T : X → X be a topologically exact, open, distance expanding, Hölder continuous system. If φ, ψ : X → ℝ are Hölder continuous functions, then for every s ∈ ℝ, 󵄨󵄨 d2 P(T, tφ + ψ)󵄨󵄨󵄨 = σμ2 sφ+ψ,∞ (φ), 󵄨t=s dt 2 where σμ2 sφ+ψ,∞ (φ) is the asymptotic variance of φ, is defined in (13.86) and satisfies (13.87) in Proposition 13.9.4. Proof. Adapting the proof of Theorem 13.10.6 we may again assume without loss of generality that both (13.139) and (13.143) hold. Equation (13.144) holds, too. Differentiating it we get, with the use of (13.136), that 2

ℒt (ϱt φ ) + 2ℒt (ϱt φ) + ℒt (ϱt ) = e ′

′′

P(t) ′

P (t)(P′ (t)ϱt + ϱ′t ) + eP(t) (P′′ (t)ϱt + P′ (t)ϱ′t + ϱ′′ t ).

Setting t = s and using (13.140), we deduce that 2

ℒs (ϱs φ ) + 2ℒs (ϱs φ) + ℒs (ϱs ) = P (s)ϱs + ϱs . ′

′′

′′

′′

Integrating this equality with respect to the measure ms , we obtain ′′ ′′ μs (φ2 ) + 2ms (ϱ′s φ) + ms (ϱ′′ s ) = P (s) + ms (ϱs ).

Equivalently, P′′ (s) = μs (φ2 ) + 2ms (ϱ′s φ).

(13.145)

Now, fix any n ∈ ℕ. The iterate T n : X → X is also a topologically exact, open, distance expanding, Hölder continuous dynamical system and Sn φ, Sn ψ : X → ℝ are

544 | 13 Gibbs states and transfer operators for open, distance expanding systems Hölder continuous potentials. So is the potential tSn φ + Sn ψ = Sn (tφ + ψ). According to Theorem 11.1.22, P(T n , tSn φ + Sn ψ) = nP(T, tφ + ψ). Moreover, per Corollaries 13.6.8 and 13.7.10, we know that mT n ,sSn φ+Sn ψ = mT,sφ+ψ ,

μT n ,sSn φ+Sn ψ = μT,sφ+ψ

and

ϱT n ,sSn φ+Sn ψ = ϱT,sφ+ψ .

Applying (13.145) to the iterate T n and the potential sSn φ + Sn ψ, we deduce that nP′′ (s) = μs ((Sn φ)2 ) + 2ms (ϱ′s Sn φ). Equivalently, P′′ (s) =

1 1 μs ((Sn φ)2 ) + 2ms (ϱ′s Sn φ). n n

(13.146)

′ The sequence (ϱ′s n1 Sn φ)∞ n=1 is uniformly bounded by ‖ϱs ‖∞ ⋅ ‖φ‖∞ , and by Birkhoff’s Ergodic theorem (Theorem 8.2.11) and (13.139) it converges μs -a. e. to the zero function. But since the measures ms and μs are equivalent, that sequence converges ms -a. e. to the zero function as well. Therefore, it follows from Lebesgue’s dominated convergence theorem (Theorem A.1.38) that

1 lim 2ms (ϱ′s Sn φ) = 0. n

n→∞

Thus, (13.146) yields P′′ (s) = lim

n→∞

1 μ ((S φ)2 ) = σμ2 sφ+ψ,∞ (φ). n s n

This is the sought result. Note that the last equality follows from the definition of σμ2 sφ+ψ,∞ (φ) (cf. (13.86)) and the reduction of the proof to the case where μsφ+ψ (φ) = 0 per (13.139). Proposition 13.9.4 applies according to Theorem 13.9.1.

13.11 Exercises Exercise 13.11.1. Let (X, d) be a metric space. A function φ : X → ℝ is said to be locally Hölder continuous with exponent α > 0 if for every z ∈ X there exist constants cz ≥ 0 and δz > 0 such that α 󵄨󵄨 󵄨 󵄨󵄨φ(x) − φ(y)󵄨󵄨󵄨 ≤ cz [d(x, y)] ,

∀x, y ∈ B(z, δz ).

Show that local Hölder continuity and (global) Hölder continuity are equivalent notions when the space X is compact.

13.11 Exercises | 545

Exercise 13.11.2. Let (X, d) be a compact metric space and α > 0. Show that the space (Hα (X), ‖ ⋅ ‖α ) is a Banach space. Show further that (Hα (X), ‖ ⋅ ‖α ) is a vector subspace of the Banach space (C(X), ‖ ⋅ ‖∞ ). Finally, show that (Hα (X), ‖ ⋅ ‖α ) is not a Banach subspace of (C(X), ‖ ⋅ ‖∞ ). Exercise 13.11.3. Let (X, d) be a compact metric space. If φ ∈ Hα (X) and ψ ∈ Hβ (X), then show that φ, ψ ∈ Hmin{α,β} (X). Exercise 13.11.4. Let μ be a Gibbs state for a potential φ and for some constant P. Suppose that a Borel probability measure ν is equivalent to μ and is such that 0 < dμ dμ ess inf( dν ) ≤ ess sup( dν ) < ∞. Show that ν is also a Gibbs state for the same potential φ and for the same constant P. Exercise 13.11.5. Let T : X → X be a dynamical system with at least one periodic point. Show that if a function g : X → ℝ is cohomologous to a constant c, then c is unique. Exercise 13.11.6. Prove the first change-of-variables formula in Proposition 13.3.1. Exercise 13.11.7. Prove relation (13.22). −1 Exercise 13.11.8. Show that J−1 m (z) is independent of x for z ∈ Tx (B(T(x), ξ )).

Exercise 13.11.9. Consider the transfer-type operator ℒφ : L1 (m) → L1 (m) introduced in Definition 13.5.1 and its dual ℒ∗φ : L1 (m)∗ → L1 (m)∗ . (a) Show that ℒφ is positive, bounded and linear. (b) Prove that ℒφ (C(X)) ⊆ C(X). (c) Show that the dual operator ℒ∗φ is well defined, positive, bounded and linear. (d) Prove that the functional m(g) = ∫X g dm is positive, bounded and linear. Exercise 13.11.10. Prove part (c) of Corollary 13.6.5. Exercise 13.11.11. Prove Corollary 13.6.10. Exercise 13.11.12. Let (X, ℱ ) be a measurable space and let (αn )∞ n=1 be a sequence of increasingly finer countable measurable partitions generating ℱ , i. e., σ(∪∞ n=1 αn ) = ℱ . Let also μ be a probability measure on (X, ℱ ). (a) Let A, B ∈ ⋃n∈ℕ αn . Prove that A ∩ B = A or A ∩ B = B or A ∩ B = 0. (b) Without loss of generality, assume that 0 ∈ α1 . Let α+ be the family of all sets Z ⊆ X that can be expressed as a countable union of sets in ⋃n∈ℕ αn , i. e., for which there exists a countable set ℐ + ⊆ ⋃n∈ℕ αn such that Z = ⋃I∈ℐ + I. Show that any such set Z can be expressed as a countable union of disjoint sets in ⋃n∈ℕ αn . Show also that α+ is closed under countable unions and under finite intersections. (c) Let 󵄨 α− := {X \ Z 󵄨󵄨󵄨 Z ∈ α+ }.

546 | 13 Gibbs states and transfer operators for open, distance expanding systems Deduce that α− is closed under countable intersections and under finite unions. (d) Let 󵄨󵄨 ∀ε > 0 ∃F ∈ α− ∃G ∈ α+ such that ε ε ̂ := {W ∈ ℱ 󵄨󵄨󵄨󵄨 }. α 󵄨󵄨 Fε ⊆ W ⊆ Gε and μ(Gε \ Fε ) < ε Observe that ̂. ⋃ αn ⊆ α+ ∩ α− ⊆ α

n∈ℕ

̂ is a σ-algebra. (e) Demonstrate that α ̂ = ℱ. (f) Infer that α Exercise 13.11.13. Let E be a finite set having at least two elements and P : E → (0, 1) a probability vector. Let μP be the Bernoulli measure on the shift space E ℕ , as defined in Examples 8.1.14 and 8.2.32. Show that μP is the unique shift-invariant Gibbs state and the unique equilibrium state for the potential φP : E ℕ → ℝ given by the formula φP (ω) = log Pω1 . More generally, let A : E × E → (0, 1) be a stochastic matrix and μA be the Markov chain measure on the shift space E ℕ , as defined in Exercise 8.5.48. Show that μA is the unique shift-invariant Gibbs state and the unique equilibrium state for the potential φA : E ℕ → ℝ given by the formula φA (ω) = log Aω1 ω2 . Exercise 13.11.14. Establish that the equilibrium states that you identified in Exercise 12.4.2 are Gibbs states for the potential φ dealt with therein. Compare this exercise with Exercise 13.11.13. Exercise 13.11.15. Let T : X → X be an open, distance expanding system and φ : X → ℝ a continuous potential. Let also μ be a Borel probability measure on X. Fix any k ∈ ℕ. Prove the following statements: (a) If μ is a Gibbs state for the potential φ on the system T, then μ is a Gibbs state for the potential Sk φ on the system T k . (b) If μ is an equilibrium state for the potential φ on the system T, then μ is an equilibrium state for the potential Sk φ on the system T k . Exercise 13.11.16. Inspiring yourself from Lemma 13.9.3, show the following generalization of Proposition 13.9.4. Suppose that T : (X, ℱ , μ) → (X, ℱ , μ) is a measurepreserving transformation of a probability space (X, ℱ , μ) and that f , g ∈ L2 (μ). Show

13.11 Exercises | 547

that if ∞

∑ Cμ,n (f , g) < ∞

n=1

∞

and

∑ Cμ,n (g, f ) < ∞,

n=1

(13.147)

then the limit n−1 n−1 1 ∫[ ∑ (g − μ(g)1) ∘ T k ] ⋅ [ ∑ (f − μ(f )1) ∘ T j ]dμ n→∞ n j=0 k=0

2 σμ,∞ (f , g) := lim

X

exists and ∞

2 σμ,∞ (f , g) = ∫(f − μ(f )1)(g − μ(g)1) dμ + ∑ ∫(f − μ(f )1) ⋅ [(g − μ(g)1) ∘ T n ] dμ n=1

X

X

∞

+ ∑ ∫(g − μ(g)1) ⋅ [(f − μ(f )1) ∘ T n ] dμ. n=1

(13.148)

X

2 The number σμ,∞ (f , g) is called the asymptotic covariance of f and g (with respect to T and μ). It is in general different than the usual covariance

σμ2 (f , g) := ∫(f − μ(f )1)(g − μ(g)1) dμ. X

However, if the random variables f ∘ T n , g ∘ T n , n ≥ 0, are independent with respect 2 2 2 to μ, then σμ,∞ (f , g) = σμ2 (f , g). Moreover, σμ,∞ (g, g) = σμ,∞ (g) is simply the asymptotic variance.

14 Lasota–Yorke maps In this chapter, we study Lasota–Yorke maps. Roughly speaking, Lasota–Yorke maps are piecewise monotone, piecewise differentiable expanding maps of a compact interval. They have finitely many pieces and the reciprocal of the absolute value of their derivative is of bounded variation. These maps appeared for the first time in the literature in Andrzej Lasota and James Yorke’s 1973 paper [65]. After defining them in Section 14.1, we describe the form of their transfer (Perron–Frobenius) operator in Section 14.2 and use this operator in Section 14.3 to establish the existence of an invariant measure absolutely continuous with respect to the Lebesgue measure. Moreover, we demonstrate in Section 14.4 the exponential decay of correlations with respect to that invariant measure, under two additional assumptions: (1) the transfer operator has 1 as its only eigenvalue of modulus one, and (2) the eigenspace of the eigenvalue 1 is one-dimensional. Lasota–Yorke maps have attracted dynamicists’ attention ever since they were introduced. After Lasota and Yorke’s original paper, Gerhard Keller and Franz Hofbauer’s article [54] substantially contributed to their rise in popularity. Marek Rychlik [114] successfully extended the class of Lasota–Yorke maps to the ones having infinitely many pieces of monotonicity. A book exposition of Lasota–Yorke maps is provided in [13]. Finally, it is worthwhile to point out that it is in Lasota and Yorke’s groundbreaking paper [65] that a Ionescu-Tulcea and Marinescu inequality was for the first time used in dynamical systems.

14.1 Definition Let I = [a, b] ⊆ ℝ be a nondegenerate compact interval endowed with the normalized Lebesgue measure λ, i. e., λ(I) = 1. In the sequel, a partition 𝒬 of I shall be a finite set {x0 , x1 , . . . , xn } such that xi−1 < xi for all 1 ≤ i ≤ n, with a = x0 and xn = b. The set of compact subintervals determined by 𝒬 will be denoted by 𝒬, i. e., 𝒬 = {J1 , J2 , . . . , Jn }, where Ji := [xi−1 , xi ]. The restriction of T to a subinterval J ∈ 𝒬 will be denoted by TJ := T|J : J → T(J). Definition 14.1.1. A map T : I → I is called Lasota–Yorke if: (a) T is piecewise expanding, meaning that there exists a partition 𝒫 of I and a constant α > 1 such that TJ is differentiable for every J ∈ 𝒫 and |TJ′ (x)| ≥ α for all x ∈ J and all J ∈ 𝒫 . (The derivative at an endpoint of a subinterval J ∈ 𝒫 is the one-sided derivative defined by approaching that endpoint from within J.) (b) 1/|TJ′ | is a function of bounded variation for every J ∈ 𝒫 . (For more information on functions of bounded variation, see Exercises 14.5.8–14.5.17.) https://doi.org/10.1515/9783110702699-014

550 | 14 Lasota–Yorke maps

14.2 Transfer operator Let (X, ℬ(X)) be a Borel measurable space and let T : X → X be a Borel measurable transformation. Recall from Sections 10.1 and 8.1 that a Borel probability measure m on X is quasi-T-invariant if m ∘ T −1 ≺≺ m and T-invariant if m ∘ T −1 = m. Clearly, any T-invariant measure is quasi-T-invariant. Though the converse is generally not true, we have learned in Section 13.4 a way of constructing an invariant measure from a quasi-invariant one. Suppose that m is a quasi-T-invariant measure. Let g : X → ℝ be any m-integrable nonnegative function. Define the finite Borel measure gm(A) := ∫A g dm for every A ∈ ℬ(X). Observe that gm ≺≺ m and thereby gm ∘ T −1 ≺≺ m ∘ T −1 ≺≺ m. This allows us to define a transfer operator ℒm : L1 (m) → L1 (m) as follows. Using the Radon–Nikodym theorem (Theorem A.1.51), for every nonnegative g ∈ L1 (m) define the function ℒm g (or, more formally, ℒm (g)) by ℒm g :=

d(gm ∘ T −1 ) . dm

The following equality characterizes this transfer operator: ∫ ℒm g dm = ∫ g dm, A

(14.1)

∀A ∈ ℬ(X).

T −1 (A)

For any g ∈ L1 (m), set ℒm g := ℒm (g + )−ℒm (g − ). Defining ℒm : L1 (m) → L1 (m) in this way makes the transfer operator a positive bounded linear operator that conforms to (14.1) for all g ∈ L1 (m) and whose norm satisfies ‖ℒm ‖1 ≤ 1. A counterpart of Theorem 13.4.1 for Lasota-Yorke maps provides a characterization of the nonnegative fixed points of ℒm . Indeed, it affirms that if m is a quasi-T-invariant measure and if g ∈ L1 (m) is such that g ≥ 0 and ∫X g dm = 1, then ℒm g = g if and only if gm ∘ T −1 = gm. Clearly, the Lebesgue measure λ is quasi-T-invariant for any Lasota–Yorke map T since TJ : J → T(J) is invertible and satisfies |(TJ−1 )′ (x)| = |TJ′ (TJ−1 (x))|−1 ≤ α−1 < 1 for all x ∈ T(J). The transfer operator ℒλ associated to a Lasota–Yorke map has the following form. Theorem 14.2.1. If T : I → I is a Lasota–Yorke map, then for all g ∈ L1 (λ) we have ℒλ g(x) = ∑ g(TJ (x))󵄨󵄨󵄨(TJ ) (x)󵄨󵄨󵄨1T(J) (x) = −1

󵄨

−1 ′

󵄨

J∈𝒫

g(y) ′ (y)| |T y∈T −1 (x) ∑

for λ-a. e. x ∈ I.

Proof. Let B ⊆ I be a Borel set. Since λ is atomless and T is bounded-to-one, the change-of-variables formula (13.19) applied to each invertible map TJ yields ∫ ℒλ g dλ = ∫ g dλ = B

T −1 (B)

= ∑

∫ ⋃J∈𝒫

∫

J∈𝒫 T −1 (B)∩J

g dλ

T −1 (B)∩J

g dλ = ∑

∫

J∈𝒫 T −1 (B∩T(J)) J

g dλ

14.2 Transfer operator

| 551

′ 󵄨 󵄨 ∫ g(TJ−1 (x))󵄨󵄨󵄨(TJ−1 ) (x)󵄨󵄨󵄨 dλ(x)

= ∑

J∈𝒫 B∩T(J)

′ 󵄨 󵄨 = ∑ ∫ g(TJ−1 (x))󵄨󵄨󵄨(TJ−1 ) (x)󵄨󵄨󵄨1T(J) (x) dλ(x) J∈𝒫 B

󵄨 󵄨−1 = ∫ ∑ g(TJ−1 (x))󵄨󵄨󵄨TJ′ (TJ−1 (x))󵄨󵄨󵄨 1T(J) (x) dλ(x) B J∈𝒫

=∫

∑

−1 B y∈T (x)

󵄨 󵄨−1 g(y)󵄨󵄨󵄨T ′ (y)󵄨󵄨󵄨 dλ(x).

Since the Borel set B is arbitrary, the result follows immediately. The iterates of the transfer operator have the following form. Corollary 14.2.2. If T : I → I is a Lasota–Yorke map, then for every n ∈ ℕ and all g ∈ L1 (λ) we have n

ℒλ g(x) = ∑ g(TJ (x))󵄨󵄨󵄨(TJ ) (x)󵄨󵄨󵄨1T n (J) (x) = −n

J∈𝒫

n

󵄨

−n ′

󵄨

g(y) 󵄨󵄨 n ′ 󵄨󵄨 y∈T −n (x) 󵄨󵄨(T ) (y)󵄨󵄨 ∑

for λ-a. e. x ∈ I.

Proof. The proof by induction is left to the reader as an exercise. The advantage of the first expression for ℒnλ g(x) is that it is defined for all x ∈ I, whereas the second expression generally is not (as (T n )′ is not defined at the endpoints n of the subintervals in 𝒫 , except at a and b). For this reason, we will henceforth define n ℒλ g(x) as the first expression. We now show that ℒλ g is of bounded variation whenever g is. Lemma 14.2.3. Let T : I → I be a Lasota–Yorke map and δ = minJ∈𝒫 λ(J). Then for every g ∈ BV(I), we have that VI (ℒλ g) ≤ AVI (g) + B‖g‖1 , 2 where A = α3 + maxJ∈𝒫 VJ (|TJ′ |−1 ) and B = αδ + δ1 maxJ∈𝒫 VJ (|TJ′ |−1 ). Recall that α arises from Definition 14.1.1 while BV(I) and VI are defined in Exercise 14.5.8.

Proof. Let 𝒬 be an arbitrary partition of I, generally unrelated to the partition 𝒫 defining T as a Lasota–Yorke map. For each K ∈ 𝒬, write K = [aK , bK ]. Using the triangle inequality, we get 󵄨󵄨 ′ 󵄨 󵄨 󵄨 󵄨 󵄨 ∑ 󵄨󵄨󵄨ℒλ g(bK ) − ℒλ g(aK )󵄨󵄨󵄨 = ∑ 󵄨󵄨󵄨 ∑ g(TJ−1 (bK ))󵄨󵄨󵄨(TJ−1 ) (bK )󵄨󵄨󵄨1T(J) (bK ) 󵄨󵄨 K∈𝒬 K∈𝒬 J∈𝒫 󵄨󵄨 ′ 󵄨 󵄨 󵄨 − ∑ g(TJ−1 (aK ))󵄨󵄨󵄨(TJ−1 ) (aK )󵄨󵄨󵄨1T(J) (aK )󵄨󵄨󵄨 󵄨󵄨 J∈𝒫

552 | 14 Lasota–Yorke maps 󵄨󵄨 ′ 󵄨 󵄨 󵄨 ≤ ∑ ∑ 󵄨󵄨󵄨g(TJ−1 (bK ))󵄨󵄨󵄨(TJ−1 ) (bK )󵄨󵄨󵄨1T(J) (bK ) 󵄨󵄨 K∈𝒬 J∈𝒫 󵄨󵄨 ′ 󵄨 󵄨 󵄨 − g(TJ−1 (aK ))󵄨󵄨󵄨(TJ−1 ) (aK )󵄨󵄨󵄨1T(J) (aK )󵄨󵄨󵄨. 󵄨󵄨 We divide the sum on the right-hand side into three subsums: ∑ = sum over the pairs of intervals for which 1T(J) (bK ) = 1T(J) (aK ) = 1. 1

∑ = sum over the pairs for which 1T(J) (bK ) = 1 and 1T(J) (aK ) = 0. 2

∑ = sum over the pairs for which 1T(J) (bK ) = 0 and 1T(J) (aK ) = 1. 3

Then ∑= 1

∑ (K,J)∈𝒬×𝒫: aK ,bK ∈T(J)

≤

󵄨󵄨 󵄨 󵄨󵄨g(T −1 (bK ))󵄨󵄨󵄨(T −1 )′ (bK )󵄨󵄨󵄨−g(T −1 (aK ))󵄨󵄨󵄨(T −1 )′ (aK )󵄨󵄨󵄨󵄨󵄨󵄨 J J 󵄨󵄨 󵄨 J 󵄨 󵄨 J 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨󵄨 −1 ′ 󵄨 󵄨 −1 ′ 󵄨󵄨󵄨 −1 󵄨󵄨g(TJ (bK ))󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨(TJ ) (bK )󵄨󵄨󵄨−󵄨󵄨󵄨(TJ ) (aK )󵄨󵄨󵄨󵄨󵄨󵄨 󵄨 󵄨

∑

(J,K)∈𝒫×𝒬: aK ,bK ∈T(J)

+

󵄨󵄨 󵄨󵄨 −1 ′ 󵄨󵄨󵄨 −1 −1 󵄨󵄨(TJ ) (aK )󵄨󵄨󵄨󵄨󵄨󵄨g(TJ (bK )) − g(TJ (aK ))󵄨󵄨󵄨 󵄨 󵄨

∑

(J,K)∈𝒫×𝒬: aK ,bK ∈T(J)

= ∑

J∈𝒫

󵄨󵄨 󵄨󵄨󵄨󵄨 ′ −1 󵄨−1 󵄨 ′ −1 󵄨−1 󵄨󵄨 −1 󵄨󵄨g(TJ (bK ))󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨TJ (TJ (bK ))󵄨󵄨󵄨 − 󵄨󵄨󵄨TJ (TJ (aK ))󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨 󵄨

∑

K∈𝒬: aK ,bK ∈T(J)

+ ∑

J∈𝒫

∑

󵄨󵄨 󵄨󵄨 ′ −1 󵄨−1 󵄨󵄨 −1 −1 󵄨󵄨TJ (TJ (aK ))󵄨󵄨󵄨 󵄨󵄨󵄨g(TJ (bK )) − g(TJ (aK ))󵄨󵄨󵄨 󵄨 󵄨

K∈𝒬: aK ,bK ∈T(J)

󵄨 󵄨 󵄨 󵄨−1 󵄨 󵄨−1 ≤ ∑ sup󵄨󵄨󵄨g(x)󵄨󵄨󵄨 ⋅ VJ (󵄨󵄨󵄨TJ′ 󵄨󵄨󵄨 ) + ∑ max󵄨󵄨󵄨TJ′ (x)󵄨󵄨󵄨 ⋅ VJ (g). x∈J J∈𝒫

x∈J

J∈𝒫

Using Exercises 14.5.15 and 14.5.8(d) successively, we obtain 1 󵄨 󵄨−1 󵄨 󵄨−1 ∑ ≤ max VJ (󵄨󵄨󵄨TJ′ 󵄨󵄨󵄨 ) ⋅ ∑ (VJ (|g|) + ∫ |g| dλ) + max max󵄨󵄨󵄨TJ′ (x)󵄨󵄨󵄨 ⋅ ∑ VJ (g) x∈J λ(J) J∈𝒫 J∈𝒫 1 J∈𝒫

J

J∈𝒫

1 1 󵄨 󵄨−1 ≤ max VJ (󵄨󵄨󵄨TJ′ 󵄨󵄨󵄨 )(VI (g) + ∫ |g| dλ) + VI (g). δ α J∈𝒫 I

We consider ∑2 and ∑3 together. Since 1T(J) (bK ) = 1 and 1T(J) (aK ) = 0 if and only if bK ∈ T(J) and aK ∉ T(J), for each J ∈ 𝒫 there is at most one K(J) ∈ 𝒬 for which these two conditions are satisfied. Similarly, there exists at most one K ′ (J) such that

14.2 Transfer operator

| 553

aK ′ ∈ T(J) and bK ′ ∉ T(J). Thus, ′ ′ 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∑ + ∑ ≤ ∑ (󵄨󵄨󵄨g(TJ−1 (bK(J) ))󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨(TJ−1 ) (bK(J) )󵄨󵄨󵄨 + 󵄨󵄨󵄨g(TJ−1 (aK ′ (J) ))󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨(TJ−1 ) (aK ′ (J) )󵄨󵄨󵄨) 3

2

J∈𝒫

≤

1 󵄨 󵄨 󵄨 󵄨 ∑ (󵄨󵄨󵄨g(TJ−1 (bK(J) ))󵄨󵄨󵄨 + 󵄨󵄨󵄨g(TJ−1 (aK ′ (J) ))󵄨󵄨󵄨). α J∈𝒫

For each J ∈ 𝒫 , choose yJ ∈ J such that 1 󵄨󵄨 󵄨 ∫ |g| dλ. 󵄨󵄨g(yJ )󵄨󵄨󵄨 ≤ λ(J) J

Then ∑+∑ ≤ 3

2

≤

1 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∑ (2󵄨󵄨󵄨g(yJ )󵄨󵄨󵄨 + 󵄨󵄨󵄨g(TJ−1 (bK(J) )) − g(yJ )󵄨󵄨󵄨 + 󵄨󵄨󵄨g(TJ−1 (aK ′ (J) )) − g(yJ )󵄨󵄨󵄨) α J∈𝒫

2 1 2 1 ∑( ∫ |g| dλ + VJ (g)) ≤ [ ∫ |g| dλ + VI (g)]. α λ(J) α δ J∈𝒫

J

I

Hence, ∑+∑+∑ ≤ ( 1

2

3

3 1 2 󵄨 󵄨−1 󵄨 󵄨−1 + max VJ (󵄨󵄨󵄨TJ′ 󵄨󵄨󵄨 ))VI (g) + ( + max VJ (󵄨󵄨󵄨TJ′ 󵄨󵄨󵄨 )) ∫ |g| dλ. α J∈𝒫 αδ δ J∈𝒫 I

Since the partition 𝒬 was chosen arbitrarily, we conclude that VI (ℒλ g) ≤ (

2 1 3 󵄨 󵄨−1 󵄨 󵄨−1 + max VJ (󵄨󵄨󵄨TJ′ 󵄨󵄨󵄨 ))VI (g) + ( + max VJ (󵄨󵄨󵄨TJ′ 󵄨󵄨󵄨 ))‖g‖1 . α J∈𝒫 αδ δ J∈𝒫

The set BV(I) of all functions of bounded variation on I is a vector space in itself (see Exercise 14.5.12). Functions of bounded variation are also Lebesgue integrable (see Exercise 14.5.14). By identifying functions that are λ-almost everywhere equal on I, the set BV(I) can be seen as a vector subspace of L1 (λ). Under a norm ‖ ⋅ ‖BV := ‖ ⋅ ‖∗BV + ‖ ⋅ ‖1 , this set becomes a Banach space on its own. The definition and basic properties of that norm are given in Exercise 14.5.17. It is then desirable to derive a form of IonescuTulcea and Marinescu inequality (see condition (c1) at the beginning of Appendix B). We make a first attempt now. Corollary 14.2.4. Let T : I → I be a Lasota–Yorke map. For every g ∈ BV(I), ‖ℒλ g‖BV ≤ A‖g‖BV + (B + 1)‖g‖1 , where A and B are defined in Lemma 14.2.3.

554 | 14 Lasota–Yorke maps Proof. Let g ∈ BV(I). Recall that ‖ℒλ ‖1 ≤ 1. It is also easy to see that if h ∈ [g]1 , then ℒλ h ∈ [ℒλ g]1 . Using Lemma 14.2.3, we obtain that ‖ℒλ g‖BV = ‖ℒλ g‖∗BV + ‖ℒλ g‖1 = ≤ ≤

inf

H∈[ℒλ g]1

VI (H) + ‖ℒλ g‖1

inf

VI (ℒλ h) + ‖ℒλ ‖1 ‖g‖1

inf

(AVI (h) + B‖h‖1 ) + ‖g‖1

h∈[g]1 ∩BV(I) h∈[g]1 ∩BV(I)

= A inf VI (h) + B‖g‖1 + ‖g‖1 h∈[g]1

≤ A‖g‖BV + (B + 1)‖g‖1 . We would like A < 1. For this, we look at the iterates of T and of 𝒫 . Lemma 14.2.5. Let T be a Lasota–Yorke map. For every n ∈ ℕ, we have ′ 󵄨−1 󵄨 Wn := maxn VJ (󵄨󵄨󵄨(TJn ) 󵄨󵄨󵄨 ) ≤ J∈𝒫

n

αn−1

W1 .

Proof. We proceed by induction on n. The case n = 1 is trivial. Now suppose that the n+1 n = 𝒫 ∨ T −n (𝒫 ) and let 𝒬 be a partition inequality holds for some n ∈ ℕ. Let J ∈ 𝒫 of J. Then 󵄨󵄨󵄨 ′ ′ 󵄨−1 󵄨 󵄨−1 󵄨󵄨 ∑ 󵄨󵄨󵄨󵄨󵄨󵄨(TJn+1 ) (bK )󵄨󵄨󵄨 − 󵄨󵄨󵄨(TJn+1 ) (aK )󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨 󵄨

K∈𝒬

󵄨󵄨󵄨 ′ ′ 󵄨−1 󵄨 󵄨−1 󵄨 󵄨−1 󵄨 󵄨−1 󵄨󵄨 = ∑ 󵄨󵄨󵄨󵄨󵄨󵄨(TJn ) (T(bK ))󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨TJ′ (bK )󵄨󵄨󵄨 − 󵄨󵄨󵄨(TJn ) (T(aK ))󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨TJ′ (aK )󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨 󵄨 K∈𝒬

󵄨󵄨󵄨 ′ ′ 󵄨−1 󵄨 󵄨−1 󵄨󵄨 󵄨 󵄨−1 ≤ ∑ 󵄨󵄨󵄨󵄨󵄨󵄨(TJn ) (T(bK ))󵄨󵄨󵄨 − 󵄨󵄨󵄨(TJn ) (T(aK ))󵄨󵄨󵄨 󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨TJ′ (bK )󵄨󵄨󵄨 󵄨 󵄨 K∈𝒬

′ 󵄨 󵄨−1 󵄨󵄨󵄨 󵄨−1 󵄨 󵄨−1 󵄨󵄨 + ∑ 󵄨󵄨󵄨(TJn ) (T(aK ))󵄨󵄨󵄨 󵄨󵄨󵄨󵄨󵄨󵄨TJ′ (bK )󵄨󵄨󵄨 − 󵄨󵄨󵄨TJ′ (aK )󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨 󵄨 K∈𝒬

1 n 1 1 n+1 1 ≤ Wn + n W1 ≤ ⋅ n−1 W1 + n W1 = n W1 . α α α α α α Since the partition 𝒬 of J is arbitrary, we deduce that VJ (|(TJn+1 )′ |−1 ) ≤

any subinterval in 𝒫

n+1

, we conclude that Wn+1 ≤

n+1 W1 . αn

n+1 W1 . As J αn

was

As a consequence of Corollary 14.2.4 and Lemma 14.2.5, we get the following Ionescu-Tulcea and Marinescu inequality. Proposition 14.2.6. There exists N ∈ ℕ and C > 0 such that 1 󵄩󵄩 N 󵄩󵄩 󵄩󵄩ℒλ g 󵄩󵄩BV ≤ ‖g‖BV + C‖g‖1 , 2

∀g ∈ BV(I).

14.3 Existence of absolutely continuous invariant probability measures | 555

14.3 Existence of absolutely continuous invariant probability measures We can now prove the main result about Lasota–Yorke maps. Theorem 14.3.1. Every Lasota–Yorke map T : I → I admits a T-invariant Borel probability measure μ which is absolutely continuous with respect to the Lebesgue measure λ on I. More precisely, μ = ρλ for some ρ ∈ BV(I) such that ρ ≥ 0, ∫I ρ dλ = 1 and ℒλ ρ = ρ.

Proof. From Lemma B.1.6 in the proof of the Ionescu-Tulcea Marinescu theorem (see Appendix B), there exists a constant M ≥ 0 such that ‖ℒnλ 1‖BV ≤ M for all n ≥ 0. Consequently, 󵄩󵄩 n−1 󵄩󵄩 󵄩󵄩 1 󵄩 󵄩󵄩 ∑ ℒj 1󵄩󵄩󵄩 ≤ M, 󵄩󵄩 n λ 󵄩 󵄩󵄩 󵄩󵄩 j=0 󵄩BV

∀n ∈ ℕ. n −1

j

j

1 ∞ ∞ k So the sequence ( n1 ∑n−1 j=0 ℒλ 1)n=1 contains a subsequence ( n ∑j=0 ℒλ 1)k=1 converging k

to, say, ρ in L1 (λ) since all bounded subsets of BV(I) are relatively compact in L1 (λ). In fact, ρ ∈ BV(I) per Exercise 14.5.17(f). Furthermore, ρ ≥ 0 and

I

n −1

n −1

1 k 1 k j ∑ ∫ ℒλ 1 dλ = lim ∑ ∫ 1 dλ = 1. k→∞ nk k→∞ nk j=0 j=0

∫ ρ dλ = lim

I

I

Moreover, 󵄩󵄩 󵄩󵄩 nk −1 󵄩󵄩 󵄩󵄩 nk −1 nk −1 nk −1 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩ℒλ ( 1 ∑ ℒj 1) − 1 ∑ ℒj 1󵄩󵄩󵄩 = 󵄩󵄩󵄩 1 ∑ ℒj+1 1 − 1 ∑ ℒj 1󵄩󵄩󵄩 󵄩 󵄩󵄩 󵄩 λ λ 󵄩 λ λ 󵄩 󵄩󵄩 󵄩 n n n n 󵄩󵄩1 󵄩󵄩 k j=0 󵄩󵄩 k j=0 k j=0 k j=0 󵄩1 󵄩󵄩 1 󵄩󵄩 2 n 󵄩 󵄩 = 󵄩󵄩󵄩 (ℒλ k 1 − 1)󵄩󵄩󵄩 ≤ → 0 as k → ∞. 󵄩󵄩 nk 󵄩󵄩1 nk Therefore, ‖ℒλ ρ − ρ‖1 = 0, that is, ℒλ ρ = ρ in L1 (λ). As ρ satisfies ρ ≥ 0, ∫I ρ dλ = 1 and ℒλ ρ = ρ, and as λ is quasi-T-invariant, Theorem 13.4.1 affirms that ρλ is T-invariant. Obviously, ρλ is absolutely continuous with respect to λ.

14.4 Exponential decay of correlations By the Ionescu-Tulcea and Marinescu theorem (Theorem B.1.1 in Appendix B), there exist a finite number of distinct eigenvalues of ℒλ : BV(I) → BV(I) denoted by λ1 , . . . , λp ∈ 𝕊1 := {z ∈ ℂ : |z| = 1} and operators L1 , . . . , Lp , S ∈ C(BV(I), L1 (λ)) such that n

p

n

ℒλ = ∑ λi Li + S

n

i=1

for all n ∈ ℕ, with L2i = Li , Li ∘ Lj = 0 whenever i ≠ j, Li ∘ S = S ∘ Li = 0 and r(S) < 1.

556 | 14 Lasota–Yorke maps Under two additional assumptions, we shall prove the exponential decay of correlations with respect to the absolutely continuous T-invariant measure μ = ρλ uncovered in Theorem 14.3.1. Theorem 14.4.1. Let T : I → I be a Lasota–Yorke map and μ = ρλ be the T-invariant Borel probability measure from Theorem 14.3.1. Assume further that: (1) {λi }pi=1 = {1}; and (2) dim(L1 (L1 (λ))) = 1. Then there exist constants c ≥ 0 and r(S) < r < 1 such that for all f ∈ L∞ (λ) and all g ∈ L1 (λ) we have Cn (f , g) ≤ cr n ‖f ‖∞ ‖gρ‖BV ,

∀n ∈ ℕ.

Proof. The two additional assumptions ensure that n

ℒλ = L1 + S

n

for all n ∈ ℕ, and lim ℒnλ g = L1 g = C(g)ρ

n→∞

for some C(g) ∈ ℝ and all g ∈ L1 (λ). Therefore, C(g) = ∫ C(g)ρ dλ = lim ∫ ℒnλ g dλ = lim ∫ g dλ = ∫ g dλ. I

n→∞

I

n→∞

I

I

So L1 g = (∫ g dλ)ρ. Now, let f ∈ L∞ (λ) and g ∈ L1 (λ). Thanks to (13.77), we may assume without loss of generality that Eμ (f ) = Eμ (g) = 0 by replacing f with f − Eμ (f ) and g with g − Eμ (g). Then C(gρ) = ∫ gρ dλ = ∫ g dμ = 0. Hence, L1 (gρ) = 0 and ℒnλ (gρ) = Sn (gρ). By (13.76), 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 Cn (f , g) = 󵄨󵄨󵄨∫ g ⋅ (f ∘ T n ) dμ󵄨󵄨󵄨 = 󵄨󵄨󵄨∫(gρ) ⋅ (f ∘ T n ) dλ󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 I

I

󵄨󵄨 󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨∫ ℒnλ ((gρ) ⋅ (f ∘ T n )) dλ󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 I

󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 = 󵄨󵄨󵄨∫ f ⋅ ℒnλ (gρ) dλ󵄨󵄨󵄨 ≤ ‖f ‖∞ ∫󵄨󵄨󵄨ℒnλ (gρ)󵄨󵄨󵄨 dλ 󵄨󵄨 󵄨󵄨 I I 󵄩 󵄩 󵄩 󵄩 ≤ ‖f ‖∞ 󵄩󵄩󵄩ℒnλ (gρ)󵄩󵄩󵄩BV = ‖f ‖∞ 󵄩󵄩󵄩Sn (gρ)󵄩󵄩󵄩BV 󵄩 󵄩 󵄩 󵄩 ≤ ‖f ‖∞ 󵄩󵄩󵄩Sn 󵄩󵄩󵄩BV 󵄩󵄩󵄩(gρ)󵄩󵄩󵄩BV ≤ ‖f ‖∞ ⋅ cr n ‖gρ‖BV

14.5 Exercises | 557

for some constants c ≥ 0 and r(S) < r < 1. That is, the correlations Cn (f , g) decay exponentially (to 0 as n → ∞).

14.5 Exercises Exercise 14.5.1. Show that the Lebesgue measure λ is quasi-T-invariant for any Lasota–Yorke map T, i. e. λ ∘ T −1 ≺≺ λ. Exercise 14.5.2. Is the composition of any two Lasota–Yorke maps a Lasota–Yorke map? Exercise 14.5.3. Establish that under the hypotheses of Theorem 14.3.1 there exists at least one ergodic measure μ claimed therein. Exercise 14.5.4. Prove that under the hypotheses of Theorem 14.3.1 there exists at most finitely many ergodic measures μ claimed therein. Exercise 14.5.5. Show that under the hypotheses of Theorem 14.3.1, if the map T is additionally topologically exact then the hypotheses of Theorem 14.4.1 are satisfied. Exercise 14.5.6. Prove that under the hypotheses of Theorem 14.4.1 there exists exactly one measure μ as claimed in Theorem 14.3.1. Show that this measure is ergodic and even more, that the Rokhlin’s natural extension of the measure-preserving dynamical system (T, μ) is K–mixing. Exercise 14.5.7. Deduce in the same way as in Chapter 13 that under the hypotheses of Theorem 14.4.1 the central limit theorem and the law of the iterated logarithm hold. The remainder of the exercises in this chapter pertain to functions of bounded variation. For all of them, let I = [a, b] be a nondegenerate compact interval in ℝ. Exercise 14.5.8. Let f : I → ℝ. Given a partition 𝒫 = {xi }ni=0 of I, the variation of f over the interval I with respect to the partition 𝒫 is defined as n

󵄨 󵄨 VI (f , 𝒫 ) = ∑󵄨󵄨󵄨f (xi ) − f (xi−1 )󵄨󵄨󵄨. i=1

The variation of f over the interval I is then defined by VI (f ) = sup{VI (f , 𝒫 ) : 𝒫 is a partition of I}. The function f is said to be of bounded variation on I if VI (f ) < ∞. We shall denote the set of functions of bounded variation on I by BV(I). (a) Let 𝒫 , 𝒬 be two partitions of I such that 𝒬 is a refinement of 𝒫 , i. e., 𝒫 ⊆ 𝒬. Prove that VI (f , 𝒫 ) ≤ VI (f , 𝒬).

558 | 14 Lasota–Yorke maps (b) Let 𝒫 be a partition of I and suppose that f , g : I → ℝ are functions that differ only at one point x ∈ I. Show that |VI (f , 𝒫 ) − VI (g, 𝒫 )| ≤ 2|f (x) − g(x)|. (c) Deduce that if f , g : I → ℝ differ at at most finitely many points, then f ∈ BV(I) if and only if g ∈ BV(I). (d) Show that VI (|f |) ≤ VI (f ). Hence, f ∈ BV(I) 󳨐⇒ |f | ∈ BV(I). (e) Find a function f such that |f | ∈ BV(I) but f ∉ BV(I). (f) Prove that if f is continuous on I, then |f | ∈ BV(I) ⇐⇒ f ∈ BV(I). Exercise 14.5.9. Let a < c < b. Show that V[a,b] (f ) = V[a,c] (f ) + V[c,b] (f ). Exercise 14.5.10. (a) Prove that VI (f ) ≥ |f (b) − f (a)|, and that the equality holds if and only if f is monotone on I. In particular, this implies that any monotone function on I is of bounded variation on I. (b) Show that |f (x)| ≤ |f (a)| + VI (f ) for all x ∈ I. In particular, this means that any function of bounded variation on I is bounded on I. (c) Identify bounded functions which are not of bounded variation. Exercise 14.5.11. (a) Prove that any function f ∈ C 1 (I) is of bounded variation on I. That is, C 1 (I) ⊆ BV(I). (b) Show that there exist functions of bounded variation which are not continuously differentiable. Thus, C 1 (I) ⊊ BV(I). (c) Find continuous functions which are not of bounded variation. So C(I) ⊄ BV(I). Exercise 14.5.12. The aim of this exercise is to establish that BV(I) is a vector space and that the variation of functions is a seminorm on that space. (a) Show that VI (f ± g) ≤ VI (f ) + VI (g). Deduce that f , g ∈ BV(I) 󳨐⇒ f ± g ∈ BV(I). (b) Prove that VI (cf ) = |c|VI (f ). Deduce that f ∈ BV(I) 󳨐⇒ cf ∈ BV(I) for all c ∈ ℝ. (c) Show that VI (f ) ≥ 0, and that the equality prevails if and only if f is a constant function. Exercise 14.5.13. The goal of this exercise is to study the operations of multiplication and division on functions of bounded variation. (a) Prove that VI (fg) ≤ M(VI (f ) + VI (g)), where M = max{sup |f |, sup |g|}. Deduce that f , g ∈ BV(I) 󳨐⇒ fg ∈ BV(I). (b) Let f ∈ BV(I) be such that |f | ≥ ε for some ε > 0. Prove that VI (1/f ) ≤ (1/ε2 )VI (f ). Deduce that 1/f ∈ BV(I) for such a function f . (c) Show that f ∈ BV(I) 󳨐⇒ ̸ 1/f ∈ BV(I) in general.

14.5 Exercises | 559

Exercise 14.5.14. Exercise 14.5.10(a) affirms that any nondecreasing (resp., nonincreasing) function on I is of bounded variation on I. Moreover, Exercise 14.5.12(a) states that the sum and difference of two functions of bounded variation on I are also of bounded variation. The aim of this exercise is to prove a converse statement, namely, that any function of bounded variation on I is the difference of two nondecreasing functions on I. This fact was uncovered by Jordan. We further deduce that a function of bounded variation has at most countably many points of discontinuity and thus is Riemann integrable. Let f : I → ℝ. The total variation of f is the function F : I → [0, ∞] defined by F(a) = 0 and F(x) = V[a,x] (f ). (a) (b) (c) (d) (e)

Prove that F is a nondecreasing function. Show that |f (y) − f (x)| ≤ F(y) − F(x) for all a ≤ x ≤ y ≤ b. Deduce that F − f is a nondecreasing function. Deduce further that VI (f ) ≤ VI (F). Deduce from (a) and (c) that f ∈ BV(I) if and only if there exist nondecreasing functions g, h : I → ℝ such that f = g − h. (f) Deduce from (e) that any f ∈ BV(I) has a left-hand limit at each x ∈ (a, b] and a right-hand limit at each x ∈ [a, b). (This can also be proved by contradiction.) (g) Deduce from (f) that any f ∈ BV(I) has at most countably many points of discontinuity. (h) Conclude that any f ∈ BV(I) is Riemann integrable and hence Lebesgue integrable, i. e., BV(I) ⊆ L1 (λ). Exercise 14.5.15. Let f ∈ BV(I). Recall that the average of f over I is defined by AvgI (f ) =

1 ∫ f dλ. λ(I) I

(This average is well defined according to Exercise 14.5.14(h).) Prove that AvgI (f ) − VI (f ) ≤ inf f ≤ sup f ≤ AvgI (f ) + VI (f ). I

I

Exercise 14.5.16. The objective of this exercise is to show that pointwise convergence, and even uniform convergence, are insufficient on their own to ensure the transmission of the bounded variation property from a sequence of functions to its limit. Under an additional condition, though, the bounded variation property is preserved. (a) Construct a sequence (fn )∞ n=1 in BV(I) which converges pointwise to a function f ∉ BV(I).

560 | 14 Lasota–Yorke maps (b) Build a sequence (fn )∞ n=1 in BV(I) which converges uniformly to a function f ∉ BV(I). (c) Let (fn )∞ n=1 be a sequence in BV(I) which converges pointwise to a function f . Show that if supn∈ℕ VI (fn ) < ∞, then f ∈ BV(I). More specifically, prove that VI (f ) = limn→∞ VI (fn ). Exercise 14.5.17. The objective of this exercise is to study the properties of a norm that makes BV(I) a Banach space. Given that any two functions that are λ-almost everywhere equal are identified in L1 (λ), they must be identified in BV(I) as well and we cannot simply use the variation of functions VI (⋅) as a seminorm (cf. Exercise 14.5.12). Rather, we need to take the infimum of the variations over all functions in the same L1 (λ)-class. Let ‖f ‖∗BV = inf VI (g) g∈[f ]1

and ‖f ‖BV = ‖f ‖∗BV + ‖f ‖1 ,

where [f ]1 denotes the equivalence class of f in L1 (λ) and ‖f ‖1 = ∫I |f | dλ is its L1 -norm. (a) Prove that the set BV(I) resulting from the aforementioned identification is a real vector subspace of L1 (λ). (b) Show that the function ‖⋅‖∗BV : BV(I) → ℝ defines a seminorm on that vector space. Demonstrate also that ‖f ‖∗BV = 0 if and only if f is constant λ-almost everywhere. (c) Prove that the function ‖ ⋅ ‖BV : BV(I) → ℝ defines a norm on that vector space. (d) Demonstrate that BV(I) is a Banach space, i. e., a complete normed vector space, when endowed with the norm ‖⋅‖BV . One way to do this is the following. Let (fn )∞ n=1 be a Cauchy sequence in (BV(I), ‖ ⋅ ‖BV ). ∗ 1 1. Deduce that (fn )∞ n=1 is a Cauchy sequence in (BV(I), ‖ ⋅ ‖BV ) and in L (λ). 1 2. Deduce that (fn )∞ n=1 converges to some function f in L (λ). ∞ ∞ 3. Prove that (fn )n=1 admits a subsequence (gm )m=1 that converges λ-almost everywhere to f . ∗ m 4. Show that the subsequence (gm )∞ m=1 can be chosen so that ‖gm+1 −gm ‖BV < 1/2 for all m ∈ ℕ. ∗ ∞ ∗ 5. Establish the existence of a sequence (gm )m=1 such that gm ∈ [gm ]1 and ∗ ∗ m VI (gm+1 − gm ) < 1/2 for all m ∈ ℕ. ∗ ∞ 6. Deduce that the sequence (gm )m=1 converges pointwise to a function f ∗ ∈ [f ]1 . 7. Using Exercise 14.5.16(c), deduce that f ∗ has bounded variation. ∗ 8. Show that VI (gm − f ∗ ) → 0 as m → ∞. 9. Deduce that ‖gm − f ‖∗BV → 0 as m → ∞. 10. Deduce that ‖fn − f ‖∗BV → 0 as n → ∞. 11. Conclude that (fn )∞ n=1 converges to f in (BV(I), ‖ ⋅ ‖BV ). (e) Show that the Banach space (BV(I), ‖⋅‖BV ) is not a Banach subspace of the Banach space (L1 (λ), ‖ ⋅ ‖1 ).

14.5 Exercises | 561

(f) Let (fn )∞ n=1 be a sequence in BV(I) such that lim ‖fn − f ‖1 = 0 for some f ∈ L1 (λ)

n→∞

and

sup ‖fn ‖BV ≤ K for some K ≥ 0. n∈ℕ

Show that f ∈ BV(I)

and ‖f ‖BV ≤ K.

Note: This demonstrates that condition (a1) in the Ionescu-Tulcea and Marinescu theorem (Theorem B.1.1 at the beginning of Appendix B) is fulfilled.

15 Fractal measures and dimensions In a sense, this chapter is exceptional as it does not directly involve any dynamics: only fractal geometry. However, the concepts and results it contains will be frequently used in most of the forthcoming chapters. In this chapter, we shall discuss different ways of evaluating the size of sets, especially fractal sets. Unlike classical geometrical objects such as lines, squares or cubes which have integer Euclidean and topological dimensions, fractal sets cannot, because of their irregularities at all scales, be accurately described and distinguished from one another by these conventional dimensions. We need less common but more befitting geometric measures, such as the Hausdorff and packing measures, as well as the dimensions they induce, to examine the local and global structures of these sets at arbitrarily small scales. It is important to emphasize that unlike the topological dimension, the Hausdorff and packing dimensions are not topological invariants, meaning that they are generally not preserved by homeomorphisms. They are however almost trivially preserved by isometries and, much more significantly, by bi-Lipschitz maps. In particular, these dimensions constitute useful tools for discerning metric spaces which, despite being homeomorphic, are not bi-Lipschitz equivalent. To be more specific, in this chapter we introduce several basic geometric concepts on metric spaces. These are: Hausdorff measures and the Hausdorff dimension; packing measures and packing dimensions; box-counting dimensions; and dimensions of measures. In Sections 15.2 and 15.3, we establish the fundamental properties of these measures and dimensions. While Hausdorff measures and the Hausdorff dimension were introduced in 1919 by Felix Hausdorff [47], it took many more decades for packing measures and packing dimensions to be defined. This was done in stages in [126, 130, 131]. There are now plenty of books on these concepts; we refer the reader to [28–30, 72, 98], among others. Also interesting (not only for historical reasons) is the classical book [101] by C. A. Rogers, which originally appeared in 1970. The 1998 edition is particularly appealing thanks to Kenneth Falconer’s foreword. In Section 15.5, we consider volume lemmas and their geometrical consequences. Their main purpose is to tell when a Hausdorff or a packing measure is null/positive and finite/infinite. We refer to them as Frostman converse theorems. Somewhat strangely, these lemmas/theorems are frequently called mass redistribution principle in the fractal geometry literature, as if to indicate that such measures always appear in the context of an iterative construction. At the end of that section, we formulate Frostman’s (direct) lemma and compare it with the Frostman converse theorems. As already mentioned, the advantage of the latter is that they provide tools to calculate, or at least to estimate, both Hausdorff and packing measures and dimensions. In Section 15.6, we deal specifically with dimensions of measures. We approach them by means of pointwise dimensions and show how to calculate them by proving results frequently referred to as volume lemmas. https://doi.org/10.1515/9783110702699-015

564 | 15 Fractal measures and dimensions The exposition given in this chapter is brief but self-contained, and comprises those results that are relevant for many dynamical applications. We start with outer measures, a classical concept in measure theory. (Basic measure-theoretic concepts and results can be found at the beginning of Appendix A.)

15.1 Outer measures While the family 𝒫 (X) of all subsets of a set X is a σ-algebra on X, some measures are not and cannot be defined on 𝒫 (X). For example, Vitali (see 5.7, The Vitali Monsters on p. 120 of [51]) constructed a subset of ℝ which is not measurable under the Lebesgue measure. That is, the Lebesgue measure cannot be defined on all subsets of ℝ. However, some measures arise from so-called “outer measures”. Outer measures are generally not measures themselves but they are defined on 𝒫 (X) and when restricted to an appropriate sub-σ-algebra of 𝒫 (X), they constitute a measure. Definition 15.1.1. Let X be a set. An outer measure on X is a function μ : 𝒫 (X) → [0, ∞] satisfying the following three conditions: (a) μ(0) = 0. (b) If A ⊆ B ⊆ X, then μ(A) ≤ μ(B). ∞ ∞ (c) μ(⋃∞ i=1 Ai ) ≤ ∑i=1 μ(Ai ) for any sequence (Ai )i=1 in 𝒫 (X). The great Greek mathematician Carathéodory discovered that there is a natural σ-algebra that can be associated to every outer measure. Given an outer measure μ, he defined a μ-measurable set to be a set which splits every other set without losing any measure in the process. Definition 15.1.2. Let μ : 𝒫 (X) → [0, ∞] be an outer measure. A set A ⊆ X is called μ-measurable if μ(B ∩ A) + μ(B \ A) = μ(B),

∀B ⊆ X.

Remark 15.1.3. Due to property (c) of an outer measure (see Definition 15.1.1), the above condition is equivalent to μ(B ∩ A) + μ(B \ A) ≤ μ(B),

∀B ⊆ X.

Theorem 15.1.4. Let μ : 𝒫 (X) → [0, ∞] be an outer measure and let ℱμ denote the family of all μ-measurable sets. Then ℱμ is a σ-algebra and μ|ℱμ is a measure. Proof. The proof is left to the reader as an exercise. When the underlying set X has a metric structure, one can define a concept of metric outer measure as follows.

15.1 Outer measures | 565

Definition 15.1.5. Let (X, d) be a metric space. An outer measure μ on X is said to be a metric outer measure if d(A, B) > 0 󳨐⇒ μ(A ∪ B) = μ(A) + μ(B), where d(A, B) = inf{d(x, y) : x ∈ A, y ∈ B}. In other words, a metric outer measure is an outer measure that behaves like a measure for any pair of sets that lie at a positive distance from each other. Henceforth, let μ be a metric outer measure on a metric space (X, d). Our first goal is to show that the Borel σ-algebra is contained in ℱμ , i. e., to show that all Borel sets are μ-measurable. The following result, which will help us achieve our goal, is well known for any measure and any ascending sequence of measurable sets. Here, we prove it for any metric outer measure and any ascending sequence of (not necessarily measurable) sets whose boundaries stand at a positive distance from each other. Lemma 15.1.6. Let μ be a metric outer measure on a metric space (X, d). Let (An )∞ n=1 be an ascending sequence of subsets of X and set A = ⋃∞ A . If d(A , A \ A ) > 0 for all n n n+1 n=1 n ∈ ℕ, then μ(A) = lim μ(An ). n→∞

Proof. Since the sequence (μ(An ))∞ n=1 is nondecreasing, it converges (perhaps to infinity). Also, An ⊆ A implies that μ(An ) ≤ μ(A) for all n ∈ ℕ. Thus, limn→∞ μ(An ) ≤ μ(A). It remains to show the opposite inequality, namely, that M := lim μ(An ) ≥ μ(A). n→∞

This is trivial if M = ∞. So assume that M < ∞. Define the sequence (Bn )∞ n=1 in the following way. Let B1 := A1 , and Bn := An \ An−1 for each n > 1. The sequence (Bn )∞ n=1 consists of pairwise disjoint sets. If n, m ∈ ℕ are such that n ≥ m + 2, then Bm ⊆ Am and, because the sequence (An )∞ n=1 satisfies An ⊆ An+1 for all n ∈ ℕ, we have that Bn ⊆ A \ An−1 ⊆ A \ Am+1 . As d(Am , A \ Am+1 ) > 0 for all m ∈ ℕ, it follows that for all n ≥ m + 2, d(Bm , Bn ) ≥ d(Am , A \ Am+1 ) > 0. Thus, for each k ∈ ℕ we have k

k

i=1

i=1

∑ μ(B2i−1 ) = μ(⋃ B2i−1 ) ≤ μ(A2k−1 ) ≤ M

and similarly

k

∑ μ(B2i ) ≤ μ(A2k ) ≤ M. i=1

566 | 15 Fractal measures and dimensions ∞

Hence, ∑ μ(Bi ) ≤ 2M < ∞. Then, for all n ∈ ℕ we obtain that i=1

∞

∞

∞

∞

k=n

k=n+1

k=n+1

μ(A) = μ( ⋃ Ak ) = μ(An ∪ ⋃ Bk ) ≤ μ(An ) + ∑ μ(Bk ) ≤ lim μ(Aj ) + ∑ μ(Bk ). j→∞

k=n+1

Since the series ∑∞ k=n+1 μ(Bk ) converges, letting n → ∞ yields μ(A) ≤ limj→∞ μ(Aj ). Using this result, we can now prove that every Borel set is μ-measurable whenever μ is a metric outer measure. Theorem 15.1.7. If μ : 𝒫 (X) → [0, ∞] is a metric outer measure on a metric space (X, d), then ℬ(X) ⊆ ℱμ , where ℬ(X) denotes the Borel σ-algebra on X, and the restriction μ : ℬ(X) → [0, ∞] is a measure. Proof. Theorem 15.1.4 affirms that ℱμ is a σ-algebra and that μ|ℱμ is a measure. Therefore, it is enough to show that all the closed sets are in ℱμ , since these generate the Borel sets. Let A be a closed subset of X and B an arbitrary subset of X. It suffices to show that μ(B ∩ A) + μ(B \ A) ≤ μ(B) (cf. Remark 15.1.3). For each n ∈ ℕ, define Bn := {x ∈ B \ A : d(x, A) ≥ 1/n}. Then d(A, Bn ) ≥ 1/n. Moreover, (Bn )∞ n=1 is an ascending sequence of sets such that B \ ∞ A = ⋃n=1 Bn . Let x ∈ (B \ A) \ Bn+1 and y ∈ Bn . Then d(x, A) < 1/(n + 1). Thus, there exists z ∈ A such that d(x, z) < 1/(n + 1). Therefore, d(y, x) ≥ d(y, z) − d(z, x) ≥

1 1 1 − = , n n + 1 n(n + 1)

and hence d(Bn , (B\A) \ Bn+1 ) ≥ 1/[n(n + 1)] > 0. Consequently, Lemma 15.1.6 asserts that μ(B \ A) = limn→∞ μ(Bn ). It follows that μ(B ∩ A) + μ(B \ A) = μ(B ∩ A) + lim μ(Bn ) n→∞

= lim (μ(B ∩ A) + μ(Bn )) n→∞

= lim μ((B ∩ A) ∪ Bn ) n→∞

≤ lim μ((B ∩ A) ∪ B) n→∞

= lim μ(B) = μ(B). n→∞

The third equality above is a consequence of the fact that μ is a metric outer measure and d(B ∩ A, Bn ) ≥ d(A, Bn ) > 0 for all n.

15.2 Geometric (Hausdorff and packing) outer measures and dimensions | 567

15.2 Geometric (Hausdorff and packing) outer measures and dimensions In this section, we introduce two families of outer measures which are in some sense defined in a geometrical way. The first one is the family of Hausdorff measures while the second is the family of packing measures. We show how they give rise to Borel measures, also referred to as Hausdorff and packing measures, and we prove some of their fundamental properties and relations between them.

15.2.1 Gauge functions Definition 15.2.1. A function φ : [0, ∞) → [0, ∞) is called a gauge function provided that: (a) φ is nondecreasing. (b) φ(0) = 0. (c) φ((0, ∞)) ⊆ (0, ∞). (d) 0 = φ(0) = limt→0 φ(t), i. e., φ is continuous at 0. Conditions (b) and (c) immediately imply that φ−1 (0) = {0}. Example 15.2.2. The foremost family of such functions is φ(t) = t α , α > 0. Later in the chapter, we shall see why these functions are especially important. A family of gauges for which it is somewhat easier to calculate the Hausdorff and packing measures is the following. Definition 15.2.3. A gauge function φ : [0, ∞) → [0, ∞) is called evenly varying if there exists a function cφ : [0, ∞) → [1, ∞) such that for all a > 0 and for all sufficiently small t > 0, −1

(cφ (a)) φ(t) ≤ φ(at) ≤ cφ (a)φ(t). Example 15.2.4. For each α > 0, the gauge function φα (t) = t α is evenly varying. To see this, note that φα (at) = aα t α . So it suffices to let cφ (a) := max{a−α , aα }. 15.2.2 Hausdorff measures Given a set A ⊆ X and δ > 0, we endeavor to confer a “measure” to A by covering it with the “best” δ-covers possible. How well a particular covering covers a given set shall be assessed by the values that the gauge function assigns to the diameter of each set in the covering. The lower the sum of these values, the better the covering is considered to be (since the gauge function is nondecreasing). This is the justification for the name given

568 | 15 Fractal measures and dimensions to the gauge functions—they allow us to gauge the quality of a cover. Note that there may not actually be a best cover, in this sense. This explains the use of the infimum, rather than a minimum, in the following definition. Definition 15.2.5. Let (X, d) be a metric space and φ : [0, ∞) → [0, ∞) be a gauge function. Let also δ > 0 and A ⊆ X. A countable cover {Ci }∞ i=1 of A such that sup{diam(Ci ) : i ∈ ℕ} ≤ δ is said to be a δ-cover of A. Define ∞

Hφδ (A) := inf{∑ φ(diam(Ci )) : {Ci }∞ i=1 is a δ-cover of A}. i=1

Lemma 15.2.6. Hφδ : 𝒫 (X) → [0, ∞] is an outer measure. Proof. We must check that the three conditions in Definition 15.1.1 are satisfied. First, it is clear that Hφδ (0) = 0. For the second condition, let A ⊆ B ⊆ X. Then Hφδ (A) ≤ Hφδ (B) since any δ-cover of B is also a δ-cover of A. ∞ δ ∞ It remains to show that Hφδ (⋃∞ n=1 An ) ≤ ∑n=1 Hφ (An ) for every sequence (An )n=1 in

𝒫 (X). This is certainly true if ∑n=1 Hφδ (An ) = ∞. So assume that ∑n=1 Hφδ (An ) < ∞. Let ∞

ε > 0 and for each n ∈ ℕ, let

∞

{Cn,i }∞ i=1

be a δ-cover of An such that

∞

∑ φ(diam(Cn,i )) ≤ Hφδ (An ) + i=1

ε . 2n

∞ Then the countable family {Cn,i }∞ n,i=1 is a δ-cover of the set ⋃n=1 An . As φ is a nonnegative function, we have ∞

∞ ∞

∞

n=1

n=1 i=1

n=1

Hφδ ( ⋃ An ) ≤ ∑ ∑ φ(diam(Cn,i )) ≤ ∑ [Hφδ (An ) +

∞ ε ] = Hφδ (An ) + ε. ∑ 2n n=1

Since this holds for any ε > 0, letting ε → 0 yields the desired result. In general, Hφδ is not a metric outer measure. However, as δ decreases, the family

of δ-covers shrinks. Thus, Hφδ (A) increases. Passing to the limit as δ tends to zero, we obtain a metric outer measure.

Definition 15.2.7. Let (X, d) be a metric space and φ be a gauge function. For every A ⊆ X, define Hφ (A) := lim Hφδ (A) = sup Hφδ (A). δ→0

δ>0

Lemma 15.2.8. Hφ : 𝒫 (X) → [0, ∞] is a metric outer measure and its restriction Hφ : ℬ(X) → [0, ∞] is a measure.

15.2 Geometric (Hausdorff and packing) outer measures and dimensions | 569

Proof. Clearly, Hφ (0) = 0, and Hφ (A) ≤ Hφ (B) for all A ⊆ B, by the corresponding

properties of Hφδ , δ > 0. To prove countable subadditivity, note first that Hφ (A) ≥ Hφδ (A) for all δ > 0 and all A ⊆ X. Let A = ⋃∞ n=1 An and let δ > 0 be arbitrary. Then ∞

∞

∞

n=1

n=1

n=1

Hφδ ( ⋃ An ) ≤ ∑ Hφδ (An ) ≤ ∑ Hφ (An ). Letting δ → 0 yields ∞

∞

n=1

n=1

Hφ ( ⋃ An ) ≤ ∑ Hφ (An ). So Hφ is an outer measure. It remains to show that Hφ is a metric outer measure, i. e., to prove that Hφ (A ∪ B) = Hφ (A) + Hφ (B) for all A, B ⊆ X such that d(A, B) > 0. That Hφ (A ∪ B) ≤ Hφ (A) + Hφ (B) comes from the subadditivity of Hφ . For the opposite inequality, let δ < d(A, B)/2 and choose a δ-cover {Ci }∞ i=1 of A ∪ B. Without loss of generality, we may assume that each Ci is contained in A ∪ B. Also, by the choice of δ, if Ci ∩ A ≠ 0 then Ci ∩ B = 0, while Ci ∩ B ≠ 0 implies Ci ∩ A = 0. So we can split the cover as follows: CA = {i ∈ ℕ : Ci ∩ A ≠ 0}

CB = {i ∈ ℕ : Ci ∩ B ≠ 0}.

Then the subfamily {Ci }i∈CA covers A whereas the subfamily {Ci }i∈CB covers B. Therefore, ∞

∑ φ(diam(Ci )) = ∑ φ(diam(Ci )) + ∑ φ(diam(Ci )) ≥ Hφδ (A) + Hφδ (B). i=1

i∈CA

i∈CB

Taking the infimum over all δ-covers of A ∪ B on the left-hand side, we deduce that Hφδ (A ∪ B) ≥ Hφδ (A) + Hφδ (B). Letting δ → 0 yields Hφ (A ∪ B) ≥ Hφ (A) + Hφ (B). So Hφ is a metric outer measure. Then Theorem 15.1.7 states that Hφ |ℬ(X) is a measure. Definition 15.2.9. The metric outer measure Hφ is called the Hausdorff outer measure induced by the gauge function φ. Remark 15.2.10. (a) If two gauge functions φ and ψ are such that φ|[0,η) = ψ|[0,η) for some η > 0, then Hφ = Hψ .

(b) Hφ has no atoms, i. e., Hφ ({x}) = 0 for every x ∈ X. This is because Hφδ ({x}) ≤ φ(diam({x})) = φ(0) = 0 for each δ > 0.

570 | 15 Fractal measures and dimensions ∞ (c) If {Ci }∞ i=1 is a δ-cover of a set A ⊆ X, then so is {Ci ∩ A}i=1 . Moreover, ∞

∞

i=1

i=1

∑ φ(diam(Ci ∩ A)) ≤ ∑ φ(diam(Ci )). This means that the structure of the space outside of A is irrelevant, i. e., if d, ρ are two metrics on X with the property that d|A×A = ρ|A×A , then the quantities Hφ (A) with respect to d and ρ are equal. In other terms, the Hausdorff measure Hφ (A) is an intrinsic property of the metric subspace (A, d|A×A ). We shall soon see that the case when φ(t) := φα (t) = t α for some α > 0 is of particular interest and for this reason we abridge the notation Hφα (A) = Ht α (A) to Hα (A). The outer measure Hα also deserves a special name. Definition 15.2.11. The metric outer measure Hα is called the α-dimensional Hausdorff measure. It is sometimes more convenient to consider covers consisting of balls instead of arbitrary sets. Definition 15.2.12. Let (X, d) be a metric space, and let A ⊆ X and r > 0. A countable family of open balls {B(xi , ri )}∞ i=1 is said to be an r-ball cover of A if it is a cover of A such that the centers xi belong to A and sup{ri : i ∈ ℕ} ≤ r. Let ∞

HφB,r (A) := inf{∑ φ(ri ) : {B(xi , ri )}i=1 is an r-ball cover of A}. ∞

i=1

Following the same limit procedure as in the definition of Hφ , define HφB (A) := lim HφB,r (A) = sup HφB,r (A). r→0

r>0

Remark 15.2.13. Ball covers can also be defined using closed balls rather than open ones. The avid reader may check that the most fundamental forthcoming results also hold in that case. In contrast with Hφ , the set function HφB is not necessarily an outer measure. For example, if A ⊆ B, a ball cover of B need not be a ball cover of A, since some centers that are in B may not lie in A. Nevertheless, the set functions HφB and Hφ are closely related to one another when φ is evenly varying.

15.2 Geometric (Hausdorff and packing) outer measures and dimensions |

571

Lemma 15.2.14. Let φ be an evenly varying gauge function. For all A ⊆ X, we have 2

HφB (A) ≤ cφ (2)Hφ (A) ≤ (cφ (2)) HφB (A), where cφ (2) is the constant from Definition 15.2.3. Proof. Let {B(xi , ri )}∞ i=1 be an r-ball cover of A. Since every r-ball cover of A is a (2r)-cover of A, we get that ∞

∞

∞

i=1

i=1

i=1

Hφ2r (A) ≤ ∑ φ(diam(B(xi , ri ))) ≤ ∑ φ(2ri ) ≤ cφ (2) ∑ φ(ri ). As the r-ball cover of A was chosen arbitrarily, we conclude that Hφ2r (A)

≤

cφ (2)HφB,r (A). Letting r → 0 yields Hφ (A) ≤ cφ (2)HφB (A). For the other inequality, let δ > 0 and let {Ci }∞ i=1 be a δ-cover of A. We may assume that Ci ∩ A ≠ 0 for all i. Let xi ∈ Ci ∩ A and let ri be the diameter of Ci . Then {B(xi , 2ri )}∞ i=1 is a (2δ)-ball cover of A since Ci ⊆ B(xi , 2ri ) for each i ∈ ℕ. Thus, ∞

∞

i=1

i=1

HφB,2δ (A) ≤ ∑ φ(2ri ) ≤ cφ (2) ∑ φ(diam(Ci )). B,2δ δ As {Ci }∞ i=1 is an arbitrary δ-cover of A, we deduce that Hφ (A) ≤ cφ (2)Hφ (A). Letting

δ → 0 gives HφB (A) ≤ cφ (2)Hφ (A).

Remark 15.2.15. As noted above, HφB (A) need not be an outer measure. However, as long as we are only interested in whether Hφ (A) is positive and finite (properties which determine the Hausdorff dimension of a set; see Subsection 15.3.1), we can consider either HφB (A) or Hφ (A). 15.2.3 Packing measures We now aim to introduce the packing measure that any gauge function induces. Let us first define the notion of packing. Definition 15.2.16. Let (X, d) be a metric space and φ : [0, ∞) → [0, ∞) be a gauge function. A countable collection {(xi , ri )}∞ i=1 in X × (0, ∞) is said to be a packing of a set A ⊆ X if xi ∈ A,

∀i ∈ ℕ and d(xi , xj ) ≥ ri + rj ,

Moreover, a packing {(xi , ri )}∞ i=1 is called an r-packing if sup{ri : i ∈ ℕ} ≤ r.

∀i, j ∈ ℕ with i ≠ j.

572 | 15 Fractal measures and dimensions ∞ Observe that if {(xi , ri )}∞ i=1 is a packing of A, then the balls {B(xi , ri )}i=1 are mutually disjoint. In this sense, a packing can be thought of as a counterpart to a covering. Whereas the Hausdorff measure of a set is determined by the way the set can be covered, the packing measure is defined by the way the set can be packed. The Hausdorff measure is obtained by covering the set as well as possible at every scale, which means finding “minimal” coverings with respect to the gauge function at every scale. So it is natural to define the packing measure by packing the set the “best” way (i. e., as “much” as) possible at every scale. This justifies the use of the supremum, rather than the infimum, in the following definition.

Definition 15.2.17. Let (X, d) be a metric space and φ : [0, ∞) → [0, ∞) be a gauge function. For every r > 0 and A ⊆ X, define ∞

∞

Pφ∗r (A) := sup{∑ φ(ri ) : {(xi , ri )}i=1 is an r-packing of A}. i=1

In general, Pφ∗r is not an outer measure. Nevertheless, as r decreases, the family of r-packings shrinks. Thus, Pφ∗r (A) decreases. Passing to the limit as r tends to zero, we obtain the following quantity. Definition 15.2.18. Let (X, d) be a metric space and φ be a gauge function. For every A ⊆ X, define Pφ∗ (A) := lim Pφ∗r (A) = inf Pφ∗r (A). r→0

r>0

It is natural to ask whether Pφ∗ is an outer measure. Obviously, Pφ∗ (0) = 0. If A ⊆ B, then each r-packing of A is an r-packing of B. Therefore, Pφ∗r (A) ≤ Pφ∗r (B). Letting r → 0 yields Pφ∗ (A) ≤ Pφ∗ (B). However, countable subadditivity may fail to hold. For example, if X = [0, 1] and Q = ℚ ∩ [0, 1], then Pφ∗ ({x}) = 0 and so ∑x∈Q Pφ∗ ({x}) = 0 for any gauge function φ. On n

1 )}2i=0 is a the other hand, take φ(t) = t. For all n ∈ ℕ, the family {( 2in , 2n+1 of Q. Thus,

Pφ∗2

−(n+1)

2n

(Q) ≥ ∑

i=0

1

2n+1

=

1 -packing 2n+1

2n + 1 . 2n+1

Hence, Pφ∗ (Q) ≥ 1/2 and Pφ∗ is not countably subadditive in this case. Though countable subadditivity does not hold in general, perhaps finite subadditivity does. This latter can be reduced to two sets, say A and B. Pick any r-packing {(zk , rk )}∞ k=1 of A ∪ B. Observe that the couples {(zk , rk ) : zk ∈ A} and {(zk , rk ) : zk ∈ B} form r-packings of A and B, respectively. Consequently, ∞

∑ φ(rk ) ≤ ∑ φ(rk ) + ∑ φ(rk ) ≤ Pφ∗r (A) + Pφ∗r (B).

k=1

zk ∈A

zk ∈B

15.2 Geometric (Hausdorff and packing) outer measures and dimensions |

573

Since the r-packing {(zk , rk )}∞ k=1 of A ∪ B was arbitrarily chosen, we infer that Pφ∗r (A ∪ B) ≤ Pφ∗r (A) + Pφ∗r (B),

∀r > 0.

(15.1)

Letting r → 0 shows that finite subadditivity prevails: Pφ∗ (A ∪ B) ≤ Pφ∗ (A) + Pφ∗ (B). Let us push our investigation one step further. Suppose that d(A, B) > 0. We want to establish that Pφ∗r (A ∪ B) = Pφ∗r (A) + Pφ∗r (B) for every 0 < r < d(A, B)/2. It suffices to show that Pφ∗r (A ∪ B) ≥ Pφ∗r (A) + Pφ∗r (B), as the opposite inequality is simply (15.1). Let ∞ ∞ {(xi , ri )}∞ i=1 be an r-packing of A and let {(yj , sj )}j=1 be an r-packing of B. Then {(xi , ri )}i=1 ∪ ∞ {(yj , sj )}j=1 is an r-packing of A ∪ B since d(xi , yj ) ≥ d(A, B) > 2r ≥ ri + sj for all i, j. Therefore, ∞

∞

i=1

j=1

Pφ∗r (A ∪ B) ≥ ∑ φ(ri ) + ∑ φ(sj ). Upon taking the supremum over all r-packings of A and B, it follows that Pφ∗r (A ∪ B) ≥ Pφ∗r (A) + Pφ∗r (B). In summary, Pφ∗r (A ∪ B) = Pφ∗r (A) + Pφ∗r (B),

∀0 < r < d(A, B)/2.

Letting r → 0 yields Pφ∗ (A ∪ B) = Pφ∗ (A) + Pφ∗ (B) when d(A, B) > 0.

(15.2)

In retrospect, only countable subadditivity is missing in order for Pφ∗ to be a metric outer measure. To obtain one, we have to go through one further step. Definition 15.2.19. For each A ⊆ X, let ∞

Pφ (A) := inf{∑ Pφ∗ (Ci ) : {Ci }∞ i=1 is a countable partition of A} i=1 ∞

= inf{∑ Pφ∗ (Ci ) : {Ci }∞ i=1 is a countable cover of A}. i=1

Remark 15.2.20. Clearly, Pφ (A) ≤ Pφ∗ (A) for all A ⊆ X. The reader also ought to convince themself that countable partitions and countable covers can be used interchangeably in the definition of Pφ (A). We can thus use partitions or covers, whichever are more suitable to the situation at hand.

574 | 15 Fractal measures and dimensions Lemma 15.2.21. Pφ : 𝒫 (X) → [0, ∞] is a metric outer measure and its restriction Pφ : ℬ(X) → [0, ∞] is a measure. Proof. For the first two conditions of an outer measure, notice that Pφ (0) = 0 since Pφ∗ (0) = 0 while the monotonicity of Pφ follows directly from that of Pφ∗ . It remains to establish countable subadditivity. To that end, let A = ⋃∞ n=1 An . Let ε > 0 and for each n ∈ ℕ, choose a countable cover {Cn,i }∞ of A such that n i=1 ∞

∑ Pφ∗ (Cn,i ) ≤ Pφ (An ) + i=1

ε . 2n

Then {Cn,i }∞ n,i=1 is a countable cover of A and thus ∞ ∞

∞

n=1 i=1

n=1

Pφ (A) ≤ ∑ ∑ Pφ∗ (Cn,i ) ≤ ∑ [Pφ (An ) +

∞ ε ] = ∑ Pφ (An ) + ε. n 2 n=1

Letting ε → 0 leads to the inequality Pφ (A) ≤ ∑∞ n=1 Pφ (An ). So Pφ is an outer measure. It remains to show that Pφ is a metric one. Suppose that d(A, B) > 0. We want to show that Pφ (A ∪ B) ≥ Pφ (A) + Pφ (B) (the opposite inequality follows from the ∞ subadditivity of Pφ ). Let {Ci }∞ i=1 be any countable cover of A ∪ B. Then {Ci ∩ A}i=1 covers A and {Ci ∩ B}∞ i=1 covers B. Moreover, d(Ci ∩ A, Ci ∩ B) ≥ d(A, B) > 0 for all i. Hence Pφ∗ ((Ci ∩ A) ∪ (Ci ∩ B)) = Pφ∗ (Ci ∩ A) + Pφ∗ (Ci ∩ B) for all i by (15.2). Thus, ∞

∞

i=1

i=1 ∞

∑ Pφ∗ (Ci ) ≥ ∑ Pφ∗ (Ci ∩ (A ∪ B)) = ∑ Pφ∗ ((Ci ∩ A) ∪ (Ci ∩ B)) i=1 ∞

∞

i=1

i=1

= ∑ Pφ∗ (Ci ∩ A) + ∑ Pφ∗ (Ci ∩ B) ≥ Pφ (A) + Pφ (B).

Taking the infimum over all countable covers of A ∪ B gives Pφ (A ∪ B) ≥ Pφ (A) + Pφ (B). So Pφ is a metric outer measure. Theorem 15.1.7 states that Pφ |ℬ(X) is a measure. Definition 15.2.22. The metric outer measure Pφ is called the packing measure induced by the gauge function φ. We shall soon see that the case φ(t) := φα (t) = t α , with α > 0, is of particular im∗ portance and for this reason we shorten the notations Pt∗α (A) and Pt α (A) to Pα and Pα , respectively. The outer measure Pα accordingly deserves a special name.

15.2 Geometric (Hausdorff and packing) outer measures and dimensions |

575

Definition 15.2.23. The metric outer measure Pα is called the α-dimensional packing measure.

15.2.4 Packing versus Hausdorff measures There is a general relationship between the Hausdorff and packing measures induced by an evenly varying gauge function. Proposition 15.2.24. Let A ⊆ X. If φ is an evenly varying gauge function, then 2

Hφ (A) ≤ (cφ (2)) Pφ (A). Proof. Let r > 0. First, we will show that HφB,2r (A) ≤ cφ (2)Pφ∗r (A). Suppose that there is no finite, maximal (in the sense of cardinality) r-packing of A of the form {(xi , r)}∞ i=1 . k Then for every k ∈ ℕ there is a packing {(xi , r)}i=1 of A. Hence, k

Pφ∗r (A) ≥ ∑ φ(r) = kφ(r) → ∞ as k → ∞. i=1

Thus, Pφ∗r (A) = ∞ and we are done in this case.

Otherwise, there exists a finite maximal r-packing {(xi , r)}ℓi=1 of A. Then {B(xi , 2r)}ℓi=1 is a (2r)-ball cover of A. Therefore, ℓ

ℓ

i=1

i=1

HφB,2r (A) ≤ ∑ φ(2r) ≤ cφ (2) ∑ φ(r) ≤ cφ (2)Pφ∗r (A). Upon taking the limit as r tends to zero, we obtain that HφB (A) ≤ cφ (2)Pφ∗ (A). Now, if A ⊆ ⋃∞ i=1 Ci then using the above and Lemma 15.2.14 we get ∞

∞

i=1

i=1

2

∞

Hφ (A) ≤ ∑ Hφ (Ci ) ≤ cφ (2) ∑ HφB (Ci ) ≤ (cφ (2)) ∑ Pφ∗ (Ci ). i=1

Taking the infimum over all countable covers of A, we finally deduce that 2

Hφ (A) ≤ (cφ (2)) Pφ (A). Though we do not provide a proof of the following fact, this latter will prove quite useful in Chapter 16 (in particular, see Definition 16.2.1 and Theorem 16.3.2; for a definition of conformality, see Section 16.1).

576 | 15 Fractal measures and dimensions Lemma 15.2.25. Let n ∈ ℕ. If U is an open subset of ℝn and f : U → ℝn is a conformal C 1 local diffeomorphism, then for every Borel set B ⊆ U it turns out that 󵄨α 󵄨 Hα (f (B)) = ∫󵄨󵄨󵄨f ′ (x)󵄨󵄨󵄨 dHα (x)

and

B

󵄨α 󵄨 Pα (f (B)) = ∫󵄨󵄨󵄨f ′ (x)󵄨󵄨󵄨 dPα (x). B

15.3 Dimensions of sets In this section, we shall see that Hausdorff and packing measures that are induced by some parameterized families of gauge functions change drastically at a certain parameter. This special parameter, which depends on the set A ⊆ X (and the metric d on X) under consideration, is called the Hausdorff (resp., packing) dimension of the set A. The changing behavior of these measures can be described as follows. At any parameter smaller than the dimension of the set, the measure of the set is infinite (as the gauge function is then too fine to estimate the size of the set), while at any parameter larger than the dimension of the set, the measure of the set is zero (as the gauge function is then too coarse to perceive the structure of the set). At the critical/transition parameter, the Hausdorff (resp., packing) measure of the set can be zero, positive and finite, or infinite; anything is possible. 15.3.1 Hausdorff dimension Let us begin by proving a straightforward lemma that will allow us to formulate the definition of the Hausdorff dimension of a set. As we will shortly see, the Hausdorff dimension has the advantage of being defined for any set. We restrict ourselves to the case φα (t) = t α . δ δ Lemma 15.3.1. Let A ⊆ X. If 0 < α1 ≤ α2 , then Hα (A) ≤ δα2 −α1 Hα (A) for all δ > 0. 2 1

Proof. Let δ > 0 and {Ci }∞ i=1 be a δ-cover of A. Then ∞

α2

∑(diam(Ci )) i=1

∞

α2 −α1

= ∑(diam(Ci )) i=1

α1

(diam(Ci ))

∞

≤ δα2 −α1 ∑(diam(Ci )) 1 . α

i=1

δ δ Taking the infimum over all δ-covers of A gives Hα (A) ≤ δα2 −α1 Hα (A). 2 1

Corollary 15.3.2. Let A ⊆ X. If Hα1 (A) < ∞ for some α1 > 0, then Hα2 (A) = 0 for all α2 > α1 . Proof. Assume that Hα1 (A) < ∞ for some α1 ≥ 0 and let α2 > α1 . By Lemma 15.3.1, δ δ Hα2 (A) = lim Hα (A) ≤ lim δα2 −α1 ⋅ lim Hα (A) = 0 ⋅ Hα1 (A) = 0. 2 1 δ→0

δ→0

δ→0

15.3 Dimensions of sets | 577

Hence, there is a unique parameter α ∈ [0, ∞] such that Hβ (A) = ∞ for all β < α and Hβ (A) = 0 for all β > α. This unique α is called the Hausdorff dimension of the set A. The value of Hα (A) may be 0, positive and finite, or ∞. Figure 15.1 shows a graph of the function α 󳨃→ Hα (A). Hα (A)

∞

0

HD(A)

α

Figure 15.1: As the parameter α crosses the Hausdorff dimension HD(A), the α-dimensional Hausdorff measure Hα (A) collapses from ∞ to 0.

Definition 15.3.3. Let A ⊆ X. The Hausdorff dimension of A is defined to be HD(A) := sup{α > 0 : Hα (A) = ∞} = inf{α > 0 : Hα (A) = 0}. We now examine the basic properties of the Hausdorff dimension. First of all, the Hausdorff dimension is an increasing set function. Theorem 15.3.4. If A ⊆ B, then HD(A) ≤ HD(B). Proof. If HD(B) = ∞, we are done. So suppose that HD(B) < ∞. Let HD(B) < α. Then Hα (B) = 0, and thus Hα (A) = 0 since A ⊆ B. Hence, HD(A) ≤ α. Since this is true for all HD(B) < α, we conclude that HD(A) ≤ HD(B). We shall now show that the Hausdorff dimension function is countably stable (also called σ-stable). Theorem 15.3.5. If {An }∞ n=1 is a countable family of subsets of X, then ∞

HD( ⋃ An ) = sup HD(An ). n=1

n∈ℕ

Proof. By Theorem 15.3.4, it is clear that HD(An ) ≤ HD(∪∞ m=1 Am ) for every n ∈ ℕ. Taking the supremum over all n ∈ ℕ, we obtain the inequality supn∈ℕ HD(An ) ≤ HD(∪∞ n=1 An ).

578 | 15 Fractal measures and dimensions For the opposite inequality, let s := supn∈ℕ HD(An ). If s = ∞, we are done. So assume that s < t < ∞. Then HD(An ) ≤ s < t for all n ∈ ℕ, and hence Ht (An ) = 0 for all n. Recall that Ht is an outer measure, so is countably subadditive. Therefore Ht (∪∞ n=1 An ) ≤ ∞ ∞ ∞ ∑n=1 Ht (An ) = 0. Thus, HD(∪n=1 An ) ≤ t for every t > s. That is, HD(∪n=1 An ) ≤ s. An immediate consequence of this result is that any countable set has Hausdorff dimension zero. Proposition 15.3.6. If A ⊆ X is countable, then HD(A) = 0. Proof. By Theorem 15.3.5, it suffices to show that HD({x}) = 0 for every x ∈ X. We already know from Remark 15.2.10(b) that each Hα is atomless, i. e., Hα ({x}) = 0 for every x ∈ X and every α > 0. Thus, HD({x}) = 0 for all x. 15.3.2 Packing dimensions We can also define packing∗ and packing dimensions in a similar natural way, from the definitions of the quantity Pφ∗ and the metric outer measure Pφ . We again restrict ourselves to the case φα (t) = t α . Lemma 15.3.7. Let A ⊆ X. If 0 < α1 ≤ α2 , then for all r > 0, ∗r ∗r Pα (A) ≤ r α2 −α1 Pα (A). 2 1

Proof. The proof goes along the same lines as that of Lemma 15.3.1 and is left to the reader as an exercise. ∗ ∗ Corollary 15.3.8. Let A ⊆ X. If Pα (A) < ∞ for some α1 , then Pα (A) = 0 for all α2 > α1 . 1 2

Proof. The proof follows the same lines as that of Corollary 15.3.2. ∗ Thus, the graph of the function α 󳨃→ Pα (A) has the same general shape as that of α 󳨃→ Hα (A), as there exists a unique parameter α∗ ∈ [0, ∞] such that Pβ∗ (A) = ∞ for all β < α∗ and Pβ∗ (A) = 0 for all β > α∗ . This unique α∗ is called the packing∗ ∗ dimension of A. The value of Pα (A) may be 0, positive and finite, or ∞.

Definition 15.3.9. Let A ⊆ X. The packing∗ dimension of A is ∗ ∗ PD∗ (A) := sup{α > 0 : Pα (A) = ∞} = inf{α > 0 : Pα (A) = 0}.

By taking the infimum over all countable covers of the set A, we further obtain some information about the packing measures. Corollary 15.3.10. Let A ⊆ X. If Pα1 (A) < ∞ for some α1 , then Pα2 (A) = 0 for all α2 > α1 . Proof. This ensues from Definition 15.2.19 and Corollary 15.3.8.

15.3 Dimensions of sets | 579

∗ Consequently, the graph of the function α 󳨃→ Pα (A) resembles those of α 󳨃→ Pα (A) and α 󳨃→ Hα (A), as there exists a unique α ≥ 0 such that Pβ (A) = ∞ for all β < α and Pβ (A) = 0 for all β > α. This unique α is called the packing dimension of A. The value of Pα (A) may be 0, positive and finite, or ∞.

Definition 15.3.11. Let A ⊆ X. The packing dimension of A is PD(A) := sup{α > 0 : Pα (A) = ∞} = inf{α > 0 : Pα (A) = 0}. Like the Hausdorff dimension, the packing dimension respects the set-inclusion order and is countably stable. Theorem 15.3.12. If A ⊆ B ⊆ X, then PD∗ (A) ≤ PD∗ (B) and PD(A) ≤ PD(B). Moreover, if {An }∞ n=1 is a countable family of subsets of X, then ∞

PD( ⋃ An ) = sup PD(An ). n=1

n∈ℕ

Proof. A proof, analog to those of Theorems 15.3.4–15.3.5, is left as an exercise. The following is a direct consequence of this theorem and is proved like its counterpart for the Hausdorff dimension, Proposition 15.3.6. Proposition 15.3.13. If A ⊆ X is countable, then PD(A) = 0. 15.3.3 Packing versus Hausdorff dimensions We now describe a relationship between the Hausdorff, packing∗ and packing dimensions of any given set. Proposition 15.3.14. For any A ⊆ X, we have that HD(A) ≤ PD(A) ≤ PD∗ (A). Proof. By Proposition 15.2.24 and the fact that φα (t) = t α is evenly varying, we get Hα (A) ≤ 22α Pα (A). Thus, Pα (A) = 0 implies that Hα (A) = 0 and, therefore, HD(A) ≤ PD(A). ∗ On the other hand, since {A} is a cover of A, it is clear that Pα (A) ≤ Pα (A) for ∗ every α, and hence PD(A) ≤ PD (A). 15.3.4 Box-counting dimensions We now examine a slightly different type of dimension, the box-counting dimension. This dimension is not defined via an outer measure. It behaves somewhat awkwardly:

580 | 15 Fractal measures and dimensions it is not σ-stable, and a set and its closure have the same dimension. However, its definition is substantially simpler than those of the Hausdorff and packing dimensions; it is frequently easier to calculate or to estimate; it often agrees with the Hausdorff and packing dimensions; and it is widely used in physics. Definition 15.3.15. Let r > 0, and let A ⊆ X be a bounded set. Define N(A, r) to be the minimum cardinality of an r-ball cover of A. Then the upper and lower box-counting (or, more simply, box) dimensions of A are respectively defined to be BD(A) := lim sup r→0

log N(A, r) − log r

and

BD(A) := lim inf r→0

log N(A, r) . − log r

If these two quantities are equal, their common value is called the box-counting (or simply box) dimension of A and is denoted by BD(A). The box-counting dimensions do not share all the congenial properties of the Hausdorff dimension. In particular, they are not σ-stable. To see this, observe that BD(ℚ ∩ [0, 1]) = 1 ≠ 0 = sup{BD({q}) : q ∈ ℚ ∩ [0, 1]}. 15.3.5 Alternate definitions of box dimensions The terminology “box-counting” comes from the fact that in Euclidean spaces we may use boxes from a lattice, rather than balls, to cover the set under scrutiny. Indeed, let n ∈ ℕ, X = ℝn and let ℒ(r) be any lattice in ℝn consisting of closed n-dimensional cubes (also called boxes) with edges of length r. For any A ⊆ X, define L(A, r) := #{C ∈ ℒ(r) : C ∩ A ≠ 0}. Proposition 15.3.16. If A is a bounded subset of ℝn , then BD(A) = lim sup r→0

log L(A, r) − log r

and

BD(A) = lim inf r→0

log L(A, r) . − log r

Proof. Without loss of generality, let 0 < r < 1. Select points xi ∈ A so that N(A,r)

A ⊆ ⋃ B(xi , r). i=1

Momentarily fix 1 ≤ i ≤ N(A, r). If C ∈ ℒ(r) is such that d(C, xi ) < r, where d denotes the standard Euclidean metric on ℝn , one immediately verifies that C ⊆ B(xi , (1 + √n)r). Thus, for any 1 ≤ i ≤ N(A, r), we have that #{C ∈ ℒ(r) : d(C, xi ) < r} ≤

n

λn (B(xi , (1 + √n)r)) cn [(1 + √n)r] ≤ = cn (1 + √n)n , λn (cube of side r) rn

15.3 Dimensions of sets | 581

where λn denotes the Lebesgue measure on ℝn and cn denotes the volume of the unit ball in ℝn . Since every C ∈ L(A, r) admits at least one 1 ≤ i ≤ N(A, r) such that d(C, xi ) < r, we deduce that L(A, r) ≤ N(A, r)cn (1 + √n)n . Therefore, log L(A, r) ≤ log[cn (1 + √n)n ] + log N(A, r). Hence, log L(A, r) log[cn (1 + √n)n ] log N(A, r) ≤ + . − log r − log r − log r So lim sup r→0

log L(A, r) ≤ BD(A) − log r

and

lim inf r→0

log L(A, r) ≤ BD(A). − log r

For the opposite inequalities, let again 0 < r < 1 and for each C ∈ L(A, r) choose xC ∈ C ∩A. Then C ∩A ⊆ B(xC , 2r√n). Thus, the family of balls {B(xC , 2r√n) : C ∈ L(A, r)} covers A. Therefore, N(A, 2r√n) ≤ L(A, r). It then follows that BD(A) ≤ lim sup r→0

log L(A, r) − log r

and

BD(A) ≤ lim inf r→0

log L(A, r) . − log r

15.3.6 Hausdorff versus packing versus box dimensions We now present general relationships between the Hausdorff, packing and box dimensions. For this, we return to our general setting. Let (X, d) be a metric space, let A ⊆ X and r > 0. Define P(A, r) to be the supremum of the cardinalities of all r-packings of A of the form {(xi , r)}ki=1 , i. e., k

P(A, r) := sup{k : {(xi , r)}i=1 is an r-packing}.

(15.3)

Lemma 15.3.17. If A ⊆ X and r > 0, then N(A, 2r) ≤ P(A, r) ≤ N(A, r). Proof. The first inequality certainly holds if P(A, r) = ∞. So assume this is not the case and let {(xi , r)}ki=1 be an r-packing of A with maximal cardinality k = P(A, r) < ∞. Then {B(xi , 2r)}ki=1 is a (2r)-cover of A. Consequently, N(A, 2r) ≤ k = P(A, r). For the second inequality, there is nothing to prove if N(A, r) = ∞. So, again, let {(xi , r)}ki=1 be a finite r-packing of A. Let also {B(yj , r)}ℓj=1 be a finite r-ball cover of A. Then, for each 1 ≤ i ≤ k there exists 1 ≤ j(i) ≤ ℓ such that xi ∈ B(yj(i) , r). We will show that k ≤ ℓ. It is enough to show that the function i 󳨃→ j(i) is injective. But for each 1 ≤ j ≤ ℓ, the cardinality of the set {xi }ki=1 ∩ B(yj , r) is at most 1; otherwise, {(xi , r)}ki=1 would not be an r-packing. Thus, i 󳨃→ j(i) is injective, as required. Hence, k ≤ l and P(A, r) ≤ N(A, r).

582 | 15 Fractal measures and dimensions These inequalities have the following immediate implications. Corollary 15.3.18. If A ⊆ X, then BD(A) = lim sup r→0

log P(A, r) − log r

and

BD(A) = lim inf r→0

log P(A, r) . − log r

Proof. This follows directly from Lemma 15.3.17. A surprising consequence of this is that the packing∗ and upper box dimensions coincide, as we show in the next lemma. Lemma 15.3.19. For every set A ⊆ X, we have PD∗ (A) = BD(A). Proof. Let t < BD(A). We will show that PD∗ (A) ≥ t for all such t, which will ensure that PD∗ (A) ≥ BD(A). Indeed, Corollary 15.3.18 implies that there is a sequence (rn )∞ n=1 , with 0 < rn < 1, such that limn→∞ rn = 0 and log P(A, rn ) > t. − log rn Then ∞

Pt n (A) = sup{∑ sti : {(xi , si )}i=1 is an rn -packing of A} ≥ rnt P(A, rn ) > 1. ∗r

∞

i=1

Letting n → ∞, we infer that Pt∗ (A) ≥ 1. So PD∗ (A) ≥ t, as sought. For the converse inequality, let s < t < PD∗ (A). We must show that BD(A) ≥ s. Since t < PD∗ (A), we have that Pt∗ (A) = ∞. Thus, for each n ∈ ℕ large enough there is t a (2−n )-packing {(xn,i , rn,i )}k(n) of A such that ∑k(n) i=1 rn,i > 1. i=1

Fix n ∈ ℕ large enough. The idea now is to split the (2−n )-packing into subpackings with nearly the same radius in order to get closer to the definition of the box-counting dimension. For each m ≥ n, let In,m := {1 ≤ i ≤ k(n) : 2−(m+1) < rn,i ≤ 2−m }. Observe that {(xn,i , rn,i )}i∈In,m is a 2−(m+1) -packing of A. We would like to estimate the cardinality of In,m . First, note that k(n)

∞

∞

t t 1 < ∑ rn,i = ∑ ∑ rn,i ≤ ∑ (#In,m )2−mt . i=1

m=n i∈In,m

m=n

Now, if it turned out that #In,m < 2ms (1 − 2s−t ) for all m ≥ n, then we would have ∞

∞

m=n

m=1

1 < ∑ (#In,m )2−mt < (1 − 2s−t ) ∑ 2m(s−t) = 1,

15.3 Dimensions of sets | 583

which is impossible. Thus, for all n ∈ ℕ, there exists m(n) ≥ n with #In,m(n) ≥ 2m(n)s (1 − 2s−t ). Obviously, m(n) → ∞ as n → ∞. Hence, P(A, 2−(m(n)+1) ) ≥ #In,m(n) ≥ 2m(n)s (1 − 2s−t ). Therefore, for all n ∈ ℕ we deduce that log P(A, 2−(m(n)+1) ) m(n)s log 2 + log(1 − 2s−t ) ≥ . (m(n) + 1) log 2 − log 2−(m(n)+1) By Corollary 15.3.18, we conclude that BD(A) = lim sup r→0

log P(A, 2−(m(n)+1) ) log P(A, r) ≥ s. ≥ lim sup − log r n→∞ − log 2−(m(n)+1)

We then have the following general relationships between the various dimensions introduced so far. Theorem 15.3.20. If A ⊆ X, then HD(A) ≤ min{PD(A), BD(A)} ≤ max{PD(A), BD(A)} ≤ BD(A) = PD∗ (A). Proof. That BD(A) = PD∗ (A) is just the statement of Lemma 15.3.19. The second inequality is obvious. The last inequality follows from Proposition 15.3.14 and the trivial fact that BD(A) ≤ BD(A). Concerning the first inequality, since HD(A) ≤ PD(A) by Proposition 15.3.14, all that remains to be shown is that HD(A) ≤ BD(A). If HD(A) = 0 then the inequality readily holds. So suppose that HD(A) > 0 and let 0 < α < HD(A). Then 2δ lim Hα (A) = Hα (A) = ∞.

δ→0

2δ Since A can be covered by N(A, δ) balls of radii at most δ, it ensues that Hα (A) ≤ α N(A, δ)(2δ) . Thus, for all sufficiently small δ > 0, 2δ log N(A, δ) + α log(2δ) ≥ log Hα (A) > 0.

Therefore, α ≤ lim inf δ→0

log N(A, δ) log N(A, δ) = lim inf = BD(A). − log(2δ) − log δ δ→0

Since this holds for all α < HD(A), we deduce that HD(A) ≤ BD(A).

584 | 15 Fractal measures and dimensions 15.3.7 Hausdorff, packing and box dimensions under (bi-)Lipschitz mappings We now investigate the behavior of the Hausdorff and packing measures and dimensions, as well as the box-counting dimensions, under Lipschitz and bi-Lipschitz maps. It is obvious that isometries preserve all these quantities. However, it is fairly easy to see that, unlike the topological dimension of a set, the Hausdorff dimension of a set is not invariant under homeomorphisms. We will show that bi-Lipschitz maps preserve dimensions, and preserve “so much” of Hausdorff and packing measures (namely, their nullness and finiteness/infiniteness) that it would be well justified to call them (iso)morphisms for the category of metric spaces with respect to fractal measures and dimensions. We first look at the behavior of the Hausdorff measures and dimension. Theorem 15.3.21. If (X, dX ) and (Y, dY ) are metric spaces and f : X → Y is a Lipschitz map with Lipschitz constant L > 0, then: (a) Hα (f (X)) ≤ Lα Hα (X) for every α ≥ 0. (a′ ) If Hα (X) < ∞, then Hα (f (X)) < ∞. (b) HD(f (X)) ≤ HD(X). If f is bi-Lipschitz, meaning that f is bijective and both f and f −1 are Lipschitz, then (c) Hα (Y) ≤ Lα Hα (X) ≤ Lα Lα −1 Hα (Y) for every α ≥ 0, where L−1 is a Lipschitz constant for f −1 . (c′ ) Hα (Y) > 0 if and only if Hα (X) > 0 and Hα (Y) < ∞ if and only if Hα (X) < ∞. (d) HD(Y) = HD(X). −1 ∞ Proof. Let α ≥ 0. Fix δ > 0 and let {An }∞ n=1 be a (L δ)-cover of X. Then {f (An )}n=1 is a δ-cover of f (X) and ∞

∞

∑ (diam(f (An ))) ≤ Lα ∑ (diam(An )) .

n=1

α

α

n=1

Thus, δ L δ Hα (f (X)) ≤ Lα Hα (X). −1

Letting δ ↘ 0, part (a) ensues. The other parts are direct consequences of (a) and are left as an exercise. The behavior exhibited by the packing measures and dimension is identical. Theorem 15.3.22. If (X, dX ) and (Y, dY ) are metric spaces and f : X → Y is a Lipschitz map with Lipschitz constant L > 0, then: (a) Pα (f (X)) ≤ Lα Pα (X) for every α ≥ 0. (a′ ) If Pα (X) < ∞, then Pα (f (X)) < ∞. (b) PD(f (X)) ≤ PD(X).

15.3 Dimensions of sets | 585

If f is bi-Lipschitz, then (c) Pα (Y) ≤ Lα Pα (X) ≤ Lα Lα −1 Pα (Y) for every α ≥ 0, where L−1 is a Lipschitz constant −1 for f . (c′ ) Pα (Y) > 0 if and only if Pα (X) > 0 and Pα (Y) < ∞ if and only if Pα (X) < ∞. (d) PD(Y) = PD(X). Proof. Let A ⊆ X. Fix r > 0 and let {(yj , rj )}∞ j=1 be an r-packing of f (A). For every j ∈ ℕ, pick an element xj ∈ f −1 (yj ) ∩ A. If k ≠ ℓ, then dX (xk , xℓ ) ≥ L−1 rk + L−1 rℓ as, otherwise, we would have rk + rℓ ≤ dY (yk , yℓ ) = dY (f (xk ), f (xℓ )) ≤ LdX (xk , xℓ ) < L(L−1 rk + L−1 rℓ ) = rk + rℓ , −1 which is impossible. Hence, {(xj , L−1 rj )}∞ j=1 is an (L r)-packing of A. Thus, ∞

∞

j=1

j=1

∗L r (A). ∑ rjα = Lα ∑(L−1 rj ) ≤ Lα Pα −1

α

Taking the supremum over all r-packings of f (A), it follows that ∗r ∗L r Pα (f (A)) ≤ Lα Pα (A). −1

So ∗ ∗r ∗L r ∗ Pα (f (A)) = inf Pα (f (A)) ≤ Lα inf Pα (A) = Lα Pα (A). −1

r>0

r>0

∞ Therefore, if {Ak }∞ k=1 is a countable cover of X then {f (Ak )}k=1 is a countable cover of f (X), and consequently, ∞

∞

k=1

k=1

∗ ∗ Pα (f (X)) ≤ ∑ Pα (f (Ak )) ≤ Lα ∑ Pα (Ak ).

Taking the infimum over all covers of X, we deduce that Pα (f (X)) ≤ Lα Pα (X). Part (a) is established. The other parts are repercussions of (a) and are left as an exercise. The following is a fairly immediate consequence of Theorems 15.3.21–15.3.22. Theorem 15.3.23. If (X, dX ) is a separable metric space, (Y, dY ) is a metric space and f : X → Y is a locally Lipschitz map on some co-countable subset of X, then HD(f (X)) ≤ HD(X)

and PD(f (X)) ≤ PD(X).

586 | 15 Fractal measures and dimensions Proof. By hypothesis, there exists a countable set F ⊆ X such that for every x ∈ X\F there is r(x) > 0 for which f |B(x,r(x)) is Lipschitz continuous. Since X is separable, so is X\F. Therefore, there is a countable set E ⊆ X\F such that ⋃ B(x, r(x)) ⊇ X\F.

x∈E

Hence, {f (B(x, r(x))) : x ∈ E} is a countable cover of f (X\F). It follows from Proposition 15.3.6 and Theorems 15.3.5 and 15.3.21(b) that HD(f (X)) = max{HD(f (F)), HD(f (X\F))} = max{0, HD(f (X\F))} = HD(f (X\F)) ≤ sup{HD(f (B(x, r(x)))) : x ∈ E} ≤ sup{HD(B(x, r(x))) : x ∈ E} ≤ HD(X).

The inequality PD(f (X)) ≤ PD(X) is proved in the same way. Finally, let us glance at the behavior of the box-counting dimensions. Theorem 15.3.24. If (X, dX ) and (Y, dY ) are metric spaces and f : X → Y is a Lipschitz map with Lipschitz constant L > 0, then: (a) BD(f (X)) ≤ BD(X). (b) BD(f (X)) ≤ BD(X). (c) If BD(X) exists, then BD(f (X)) ≤ BD(f (X)) ≤ BD(X). If f (d) (e) (f)

is bi-Lipschitz, then BD(Y) = BD(X). BD(Y) = BD(X). If BD(X) exists, then so does BD(Y) and BD(Y) = BD(X).

Proof. Fix r > 0. Since f (B(x, L−1 r)) ⊆ B(f (x), r) for all x ∈ X, it is obvious that N(f (X), r) ≤ N(X, L−1 r). Therefore, log N(X, L−1 r) log N(X, s) log N(f (X), r) ≤ lim inf = lim inf r→0 r→0 s→0 − log r − log r − log(Ls) log N(X, s) log N(X, s) log N(X, s) = lim inf = lim inf = lim inf s→0 − log L − log s s→0 − log s(1 + log L ) s→0 − log s

BD(f (X)) = lim inf

= BD(X).

log s

Thus, part (a) holds. Part (b) is proved in the same way. The other parts follow immediately from (a) and (b).

15.4 A digression into geometric measure theory | 587

15.4 A digression into geometric measure theory In order to estimate measures and dimensions of sets, as well as dimensions of measures, we will need covering theorems. In geometric measure theory, there are essentially three fundamental covering theorems: Vitaly, 5r (or 4r), and Besicovitch. We will look at the last two. In what follows, we use the notation tB(x, r) := B(x, tr). Furthermore, for every ball B := B(x, r), we let r(B) := r and c(B) := x. Theorem 15.4.1 (5r-covering theorem). Let (X, d) be a metric space and ℬ a family of balls in X such that sup{r(B) : B ∈ ℬ} < ∞. Then there exists a subfamily ℬ′ ⊆ ℬ consisting of mutually disjoint balls such that for every B ∈ ℬ there is B′ ∈ ℬ′ with B ∩ B′ ≠ 0 and r(B) ≤ 2r(B′ ). In particular, this implies that ⋃ B ⊆ ⋃ 5B′ .

B∈ℬ

B′ ∈ℬ′

In addition, if X is separable and r(B) > 0 for all but countably many B ∈ ℬ, then ℬ′ is countable. For the proof, recall the following version of Zorn’s lemma: If every chain in a nonempty partially ordered set has an upper bound, then the partially ordered set has a maximal element. Proof. Let Ω denote the partially ordered (by inclusion) set consisting of all disjointed subfamilies ω of ℬ with the following property: If a ball B ∈ ℬ intersects some ball from ω, then there is B′ ∈ ω with B ∩ B′ ≠ 0 and r(B) ≤ 2r(B′ ). First, note that Ω ≠ 0 because the one-ball family ω = {B1 } is in Ω whenever B1 ∈ ℬ is such that r(B1 ) ≥ 21 sup{r(B) : B ∈ ℬ}. Second, if 𝒞 is a chain in Ω then it is easy to see that ω𝒞 := ⋃ ω ω∈𝒞

belongs to Ω. Thus every chain in Ω has an upper bound and Ω has a maximal element, denoted by ℬ′ , according to Zorn’s lemma. By construction, ℬ′ is disjointed. Third, if there were a ball B ∈ ℬ that did not meet any ball from ℬ′ , then one could pick a ball B0 ∈ ℬ such that the radius of B0 is at least half of the radius of any other ball that does not meet the balls from ℬ′ . Then, if a ball B ∈ ℬ met a ball from ℬ′′ := ℬ′ ∪ {B0 }, by construction it would meet at least one ball from ℬ′′ whose radius is at least half that of B, showing that ℬ′′ belongs to Ω. But this would contradict the maximality of ℬ′ . Consequently, every ball B ∈ ℬ meets a ball B′ ∈ ℬ′ such that r(B) ≤ 2r(B′ ). The triangle inequality shows that B ⊆ 5B′ . The last part of the theorem follows immediately from the fact that any family of mutually disjoint open subsets of a separable space is countable.

588 | 15 Fractal measures and dimensions Remark 15.4.2. (a) The proof of this theorem comes from Heinonen [48]. It applies to more general coverings than those by balls (see 2.8 in Federer [32]). (b) Zorn’s lemma is equivalent to the axiom of choice; see, e. g., Chapter 1 in [52] or [51]. (c) If there exists a finite Borel measure μ on X such that μ(B) > 0 for all B ∈ ℬ, then ℬ′ is countable. (d) Simple examples show that the assumption of uniformly bounded radii, i. e., sup{r(B) : B ∈ ℬ} < ∞, is necessary. (e) As a matter of fact, 5r in Theorem 15.4.1 can be replaced by 4r. This improvement has virtually no consequences whatsoever, but its proof is much longer and much more involved as it relies on transfinite induction, and thus requires knowledge of ordinal numbers. Such a proof can be found in [63] but stems from the one in [72], where it is assumed that all closed balls are compact and an ordinary induction suffices. In the following section, we shall derive several geometrical consequences of the 5r-covering theorem. Later, we will also need the following theorem, a proof of which can be found in [63]. That reference also contains the proofs of all other basic classical covering theorems used in geometric measure theory. Theorem 15.4.3 (Besicovitch covering theorem). Let n ∈ ℕ. There exists a constant c(n) ∈ ℕ such that the following holds: If A is a bounded subset of ℝn then for any function r : A → (0, ∞) there exists a countable subset {xk }∞ k=1 of A such that the collection ℬ(A, r) := {B(xk , r(xk )) : k ∈ ℕ}

covers A and can be decomposed into c(n) packings of A.

15.5 Volume lemmas—Frostman converse theorems In this section, we identify conditions under which some sets have a positive or finite Hausdorff (resp., packing) measure. In general, this is much harder than showing that this measure is zero or infinite. But this will be instrumental in getting upper and lower bounds for the Hausdorff (resp., packing) dimension of those sets. Moreover, this will help us in estimating the dimension of some measures in Section 15.6. Throughout, 1 := 0 and 01 := ∞. we adopt the convention that ∞ Theorem 15.5.1 (Frostman converse theorem for generalized Hausdorff measures). Let φ : [0, ∞) → [0, ∞) be a continuous, evenly varying gauge function. Let μ be a Borel probability measure on a metric space X, and A ⊆ X.

15.5 Volume lemmas—Frostman converse theorems | 589

(a) If there exists a constant c ∈ (0, ∞] such that for all but countably many x ∈ A, lim sup r→0

μ(B(x, r)) ≥ c, φ(r)

(15.4)

then Hφ (E) ≤ c−1 cφ (10)μ(E) for every Borel set E ⊆ A. In particular, Hφ (E) < ∞, and Hφ (E) = 0 if c = ∞ or μ(E) = 0. (b) If there exists a constant c ∈ [0, ∞) such that for all x ∈ A, lim sup r→0

μ(B(x, r)) ≤ c, φ(r)

(15.5)

then Hφ (E) ≥ (ccφ (2))−1 μ(E) for every Borel set E ⊆ A. In particular, Hφ (E) > 0 whenever μ(E) > 0. Proof. (a) We shall prove this statement when c < ∞. The case c = ∞ is left as an exercise. Let E be a Borel set contained in A. Since Hφ is an outer measure that equals 0 on any countable set, we may assume that E does not contain any point of the exceptional countable set at which (15.4) is not satisfied. Fix ε > 0. By the regularity of μ, there exists an open set G ⊇ E such that μ(G) ≤ μ(E) + ε. Fix R > 0. By (15.4), for each x ∈ E there exists 0 < r(x) < R such that B(x, r(x)) ⊆ G and μ(B(x, r(x))) ≥c−ε φ(r(x))

or, equivalently, φ(r(x)) ≤

μ(B(x, r(x))) . c−ε

(15.6)

In particular, this implies that the family {B(x, r(x))}x∈E consists of balls of positive μ-measure. According to the 5r-covering theorem (Theorem 15.4.1) and Remark 15.4.2(c), there exists a countable subfamily of mutually disjoint balls {B(xk , r(xk ))}∞ k=1 such that ∞

E ⊆ ⋃ B(x, r(x)) ⊆ ⋃ B(xk , 5r(xk )) x∈E

k=1

for appropriately chosen xk ∈ E. For each k ∈ ℕ, we obviously have that diam(B(xk , 5r(xk ))) ≤ 10r(xk ) ≤ 10R. Using this and (15.6), we obtain that ∞

Hφ10R (E) ≤ ∑ φ(diam(B(xk , 5r(xk )))) k=1 ∞

≤ ∑ cφ (10)φ(r(xk )) k=1

590 | 15 Fractal measures and dimensions ∞

1 μ(B(xk , r(xk ))) c − ε k=1

≤ cφ (10) ∑ = ≤ ≤

cφ (10) c−ε

cφ (10)

c−ε cφ (10) c−ε

∞

μ( ⋃ B(xk , r(xk ))) k=1

μ(G) (μ(E) + ε).

Letting successively R → 0 and ε → 0 yields Hφ (E) ≤

cφ (10) c

μ(E).

(b) For every r > 0, the regularity of μ implies that the function from X to ℝ given by x 󳨃→ μ(B(x, r))/φ(r) is lower semicontinuous, and hence Borel measurable. Since the supremum of a countable family of measurable functions is a measurable function, it follows that for every k ∈ ℕ the function x 󳨃󳨀→ ψk (x) := sup{

μ(B(x, r)) 1 : r ∈ ℚ ∩ (0, ]} φ(r) k

is Borel measurable. Observe also that (ψk )∞ k=1 forms a decreasing sequence of func∞ tions. Let s > c. Let A0 = 0. For each k ∈ ℕ, let Ak := ψ−1 k ((0, s]). Since (ψk )k=1 is decreasing, the sequence (Ak )∞ k=1 is ascending. Moreover, since s > c, it follows from (15.5) that ⋃∞ A ⊇ A. Furthermore, since (0, s] is a Borel subset of ℝ, the measurability of k k=1 ψk ensures that Ak is a Borel subset of X. Fix k ∈ ℕ. Choose some arbitrary r ∈ (0, k1 ] and pick a sequence of rational numbers (rj )∞ j=1 such that rj increases to r. Since μ is a measure and φ is continuous, we obtain for all x ∈ Ak that μ(B(x, rj )) μ(B(x, r)) = lim ≤ ψk (x) ≤ s. j→∞ φ(r) φ(rj ) So, if x ∈ Ak then sup{

μ(B(x, r)) 1 : r ∈ (0, ]} ≤ s. φ(r) k

(15.7)

Now, fix any set F ⊆ Ak , any r ∈ (0, k1 ], and let {B(xi , ti )}∞ i=1 be an r-ball cover of F. By (15.7), ∞

∞

∞

i=1

i=1

i=1

∑ φ(ti ) ≥ s−1 ∑ μ(B(xi , ti )) ≥ s−1 μ(⋃ B(xi , ti )) ≥ s−1 μ(F).

15.5 Volume lemmas—Frostman converse theorems | 591

Taking the infimum over all r-ball covers yields HφB,r (F) ≥ s−1 μ(F). As this holds for all 0 < r ≤ 1/k, we deduce that HφB (F) ≥ s−1 μ(F). By Lemma 15.2.14, we obtain that −1

Hφ (F) ≥ (scφ (2)) μ(F),

∀F ⊆ Ak , ∀k ∈ ℕ.

To complete the proof, consider the sequence (Bk )∞ k=1 , where Bk = (Ak \Ak−1 ) ∩ A. ∞ ∞ Given that the sequence (Ak )∞ is ascending and ⋃ k=1 Ak ⊇ A, the sequence (Bk )k=1 k=0 ∞ ∞ consists of mutually disjoint sets such that Bk ⊆ Ak for all k and ⋃k=1 Bk = ⋃k=1 Ak ∩A = A. Hence, if E is a Borel subset of A then ∞

∞

k=1

k=1

−1

Hφ (E) = Hφ (E ∩ ⋃ Bk ) = ∑ Hφ (E ∩ Bk ) ≥ (scφ (2))

∞

−1

∑ μ(E ∩ Bk ) = (scφ (2)) μ(E).

k=1

Letting s ↘ c finishes the proof. Theorem 15.5.2 (Frostman converse theorem for generalized packing measures). Let φ be a continuous, evenly varying gauge function. Let μ be a Borel probability measure on a metric space X, and A ⊆ X. (a) If there exists a constant c ∈ (0, ∞] such that for all but countably many x ∈ A, lim inf r→0

μ(B(x, r)) ≥ c, φ(r)

(15.8)

then Pφ (E) ≤ c−1 μ(E) for each Borel set E ⊆ A. In particular, Pφ (E) < ∞, and Pφ (E) = 0 if c = ∞ or μ(E) = 0. (b) If X is separable and there exists a constant c ∈ [0, ∞) such that for all x ∈ A, lim inf r→0

μ(B(x, r)) ≤ c, φ(r)

(15.9)

then Pφ (E) ≥ (ccφ (10))−1 μ(E) for every Borel set E ⊆ A. In particular, Pφ (E) > 0 whenever μ(E) > 0. Proof. (a) Since Pφ is an outer measure that equals 0 on any countable set, we may assume that A does not contain any point of the exceptional countable set at which (15.8) is not satisfied. The sequence of functions (ψk )∞ k=1 , where x 󳨃󳨀→ ψk (x) := inf{

μ(B(x, r)) 1 : r ∈ ℚ ∩ (0, ]} φ(r) k

forms an increasing sequence of Borel measurable functions. Let 0 < s < c. For each ∞ ∞ k ∈ ℕ, let Ak := ψ−1 k ([s, ∞)). As (ψk )k=1 is increasing, so is the sequence (Ak )k=1 . Moreover, since s < c, it follows from (15.8) that ⋃∞ k=1 Ak ⊇ A. Furthermore, since [s, ∞) is a Borel subset of ℝ, the measurability of ψk ensures that Ak is a Borel subset of X. Fix k ∈ ℕ. Choose some arbitrary r ∈ (0, k1 ] and pick a sequence of rational numbers (rj )∞ j=1

592 | 15 Fractal measures and dimensions such that rj increases to r. Since μ is a measure and φ is continuous, we obtain for all x ∈ Ak that μ(B(x, rj )) μ(B(x, r)) = lim ≥ ψk (x) ≥ s. j→∞ φ(r) φ(rj ) μ(B(x,r))

So, if x ∈ Ak then inf{ φ(r) : r ∈ (0, k1 ]} ≥ s. Now, fix any set F ⊆ Ak , any r ∈ (0, k1 ], ∞ and let {(xi , ti )}∞ i=1 be an r-packing of F. Then the balls {B(xi , ti )}i=1 are mutually disjoint, and hence ∞

∞

∞

i=1

i=1

i=1

∑ φ(ti ) ≤ s−1 ∑ μ(B(xi , ti )) = s−1 μ(⋃ B(xi , ti )) ≤ s−1 μ(B(F, r)), where B(F, r) = {x ∈ X : d(x, F) < r} is the open r-neighborhood of F. Taking the supremum over all r-packings yields Pφ∗r (F) ≤ s−1 μ(B(F, r)). Thus, Pφ (F) ≤ s−1 μ(B(F, r)) for all r ∈ (0, 1/k] and each subset F ⊆ Ak . Consequently, Pφ (F) ≤ s−1 μ(∩n∈ℕ B(F, 1/n)) = s−1 μ(F) for all F ⊆ Ak . In particular, if C is a closed subset of E then Pφ (C ∩ Ak ) ≤ s−1 μ(C ∩ Ak ) ≤ s−1 μ(C) ≤ s−1 μ(E). ∞ As this holds for all k ∈ ℕ and since C ⊆ E ⊆ A ⊆ ∪∞ k=1 Ak and the sequence (Ak )k=1 is −1 ascending, we deduce that Pφ (C) ≤ s μ(E). Upon taking the supremum over all closed sets C contained in E, the regularity of μ confirms that Pφ (E) ≤ s−1 μ(E). Letting s ↗ c completes the proof. (b) Let ε > 0. Choose an arbitrary subset F ⊆ A. Define a descending sequence (Gn )∞ n=0 of open sets containing F, as follows. Let G0 = X. By (15.9), for every x ∈ A there exists 0 < r1 (x) < 1 such that

μ(B(x, r1 (x))) ≤ c + ε. φ(r1 (x)) Take the family of balls {B(x, 51 r1 (x))}x∈F . According to the 5r-covering theorem (Theorem 15.4.1), there is a countable set F1 ⊆ F such that the subfamily {B(x, 51 r1 (x))}x∈F 1 consists of mutually disjoint balls satisfying 1 F ⊆ ⋃ B(x, r1 (x)) ⊆ ⋃ B(x, r1 (x)). 5 x∈F x∈F 1

Set G1 := ⋃x∈F1 B(x, r1 (x)). For the inductive step, suppose that Gn has been defined for 1 some n ∈ ℕ. By (15.9), for every x ∈ A there exists some 0 < rn+1 (x) < n+1 such that B(x, rn+1 (x)) ⊆ Gn and μ(B(x, rn+1 (x))) ≤ c + ε. φ(rn+1 (x))

(15.10)

15.5 Volume lemmas—Frostman converse theorems | 593

Consider the family of balls {B(x, 51 rn+1 (x))}x∈F . According to the 5r-covering theorem (Theorem 15.4.1), there exists a countable set Fn+1 ⊆ F such that the subfamily {B(x, 51 rn+1 (x))}x∈F consists of mutually disjoint balls satisfying n+1

1 F ⊆ ⋃ B(x, rn+1 (x)) ⊆ ⋃ B(x, rn+1 (x)). 5 x∈F x∈F n+1

Let Gn+1 := ⋃x∈Fn+1 B(x, rn+1 (x)). It is clear that Gn+1 is an open set and F ⊆ Gn+1 ⊆ Gn . Moreover, for all pairs x, y ∈ Fn+1 ⊆ F we know that d(x, y) ≥

1 1 max{rn+1 (x), rn+1 (y)} ≥ (r (x) + rn+1 (y)). 5 10 n+1

1 1 Therefore, the collection {(x, 10 rn+1 (x))}x∈F forms a ( n+1 )-packing of F. Using (15.10), n+1 it follows that ∗

1 r (x)) 10 n+1

1

Pφ n+1 (F) ≥ ∑ φ( x∈Fn+1

−1

≥ (cφ (10))

∑ φ(rn+1 (x))

x∈Fn+1

−1

≥ (cφ (10))

∑

x∈Fn+1

μ(B(x, rn+1 (x))) c+ε

−1

=

(cφ (10)) c+ε

μ( ⋃ B(x, rn+1 (x))) x∈Fn+1

−1

=

(cφ (10)) c+ε

μ(Gn+1 ).

Letting n → ∞, it ensues that −1

Pφ∗ (F)

≥

(cφ (10)) c+ε

lim μ(Gn ) =

n→∞

(cφ (10)) c+ε

−1

μ(GF ),

where GF := ⋂∞ n=1 Gn is a Gδ -set, and thereby a Borel set, containing F. Consequently, for every Borel set E ⊆ A we have ∞

Pφ (E) = inf{∑ Pφ∗ (Ci ) : {Ci }∞ i=1 is a partition of E} i=1

−1

≥

(cφ (10)) c+ε

i=1

−1

≥

(cφ (10)) c+ε

∞

inf{∑ μ(GCi ) : {Ci }∞ i=1 is a partition of E} ∞

inf{μ(⋃ GCi ) : {Ci }∞ i=1 is a partition of E} i=1

594 | 15 Fractal measures and dimensions −1

≥

(cφ (10)) c+ε

∞

inf{μ(⋃ Ci ) : {Ci }∞ i=1 is a partition of E} i=1

−1

=

(cφ (10)) c+ε

μ(E). −1

Since this holds for all ε > 0, we deduce that Pφ (E) ≥ (ccφ (10)) μ(E). Replacing φ(r) by r t , Hφ by Ht , and Pφ by Pt in Theorems 15.5.1 and 15.5.2 respectively, we immediately get the following two results. Theorem 15.5.3 (Frostman converse theorem for Hausdorff measures). Fix t > 0. Let μ be a Borel probability measure on a metric space X, and A ⊆ X. (a) If there exists a constant c ∈ (0, ∞] such that μ(B(x, r)) ≥c rt

lim sup r→0

for all but countably many points x ∈ A, then the Hausdorff measure Ht satisfies Ht (E) ≤ c−1 10t μ(E) for every Borel set E ⊆ A. In particular, Ht (E) < ∞ and

(Ht (E) = 0 if c = ∞ or μ(E) = 0).

(b) If there exists a constant c ∈ [0, ∞) such that μ(B(x, r)) ≤c rt

lim sup r→0

for all x ∈ A, then Ht (E) ≥ (c2t ) μ(E) −1

for every Borel set E ⊆ A. In particular, μ(E) > 0 󳨐⇒ Ht (E) > 0. Theorem 15.5.4 (Frostman converse theorem for packing measures). Fix t > 0. Let μ be a Borel probability measure on a metric space X, and A ⊆ X. (a) If there exists a constant c ∈ (0, ∞] such that lim inf r→0

μ(B(x, r)) ≥c rt

15.6 Dimensions of measures | 595

for all but countably many x ∈ A, then the packing measure Pt satisfies Pt (E) ≤ c−1 μ(E) for every Borel set E ⊆ A. In particular, Pt (E) < ∞

and (Pt (E) = 0 if c = ∞ or μ(E) = 0).

(b) If X is separable and there exists a constant c ∈ [0, ∞) such that lim inf r→0

μ(B(x, r)) ≤c rt

for all x ∈ A, then Pt (E) ≥ (c10t ) μ(E) −1

for every Borel set E ⊆ A. In particular, μ(E) > 0 󳨐⇒ Pt (E) > 0. In the literature, Frostman converse theorems are sometimes referred to as mass redistribution principles (especially for Hausdorff measures) or volume lemmas. In the opposite direction to Frostman converse theorems, there is the following well-known fact (for a proof of this result, see Theorems 8.8 and 8.17 in [72]). Theorem 15.5.5 (Frostman direct lemma). If X is a Borel subset of a Euclidean space ℝn or an arbitrary compact metric space and if Ht (X) > 0 for some t > 0, then there exists a Borel probability measure μ on X such that μ(B(x, r)) ≤ r t for every point x ∈ X and all radii r > 0. This is an interesting theorem, although Frostman converse theorems are more convenient to estimate and calculate Hausdorff and packing measures and dimensions.

15.6 Dimensions of measures The dimension of a measure is defined as the infimum of the dimensions of the sets of full measure. Definition 15.6.1. Let μ be a Borel probability measure on a metric space (X, d). (a) The Hausdorff dimension of μ is HD(μ) := inf{HD(Y) : μ(Y) = 1} = inf{HD(Y) : μ(X \ Y) = 0}.

596 | 15 Fractal measures and dimensions (b) The packing dimension of μ is PD(μ) := inf{PD(Y) : μ(Y) = 1} = inf{PD(Y) : μ(X \ Y) = 0}. We now introduce the well-studied and extremely useful concept of pointwise dimensions of a measure. These will be featured in Chapters 16 and 21 where multifractal analysis will be discussed. Definition 15.6.2. Let μ be a Borel probability measure on a metric space X. The lower and upper pointwise dimensions of μ at a point x ∈ X are respectively defined by dμ (x) := lim inf r→0

log μ(B(x, r)) log r

and dμ (x) := lim sup r→0

log μ(B(x, r)) . log r

If dμ (x) = dμ (x), this common value is called the pointwise dimension of μ at x and is denoted by dμ (x). As a direct consequence of Theorem 15.5.3, we obtain an upper (resp., lower) bound for the Hausdorff dimension of sets whose lower pointwise dimensions have common upper (resp., lower) bounds. Theorem 15.6.3. Let μ be a Borel probability measure on a metric space X and A ⊆ X be a Borel set. (a) If μ(A) > 0 and there exists θ1 ≥ 0 such that for all x ∈ A we have dμ (x) ≥ θ1 ,

(15.11)

then HD(A) ≥ θ1 . (b) If there exists θ2 ≥ 0 such that for all x ∈ A we have dμ (x) ≤ θ2 ,

(15.12)

then HD(A) ≤ θ2 . Proof. (a) If θ1 = 0, the statement is trivial. So assume θ1 > 0 and pick an arbitrary 0 ≤ θ < θ1 . By (15.11), for every x ∈ A there exists 0 < R < 1 such that for all 0 < r < R, log μ(B(x, r)) ≥θ log r Therefore, lim supr→0

μ(B(x,r)) rθ

or, equivalently, μ(B(x, r)) ≤ r θ .

≤ 1. Applying Theorem 15.5.3(b), we obtain that Hθ (A) ≥ 2−θ μ(A) > 0.

Thus, HD(A) ≥ θ. Letting θ ↗ θ1 , we conclude that HD(A) ≥ θ1 .

15.6 Dimensions of measures | 597

(b) Fix θ > θ2 . By (15.12), for each x ∈ A there is a sequence (rn )∞ n=1 with 0 < rn < 1 and rn → 0 such that log μ(B(x, rn )) ≤θ log rn Hence, lim supn→∞

μ(B(x,rn )) rnθ

or, equivalently, μ(B(x, rn )) ≥ rnθ .

≥ 1. By Theorem 15.5.3(a), we deduce that Hθ (A) ≤ 10θ μ(A) < ∞.

Thus, HD(A) ≤ θ. Letting θ ↘ θ2 , we conclude that HD(A) ≤ θ2 . As a corollary to this theorem, we can also estimate the Hausdorff dimension of certain measures. Corollary 15.6.4. Let μ be a Borel probability measure on a metric space X. (a) If there exist θ1 ≥ 0 and a Borel set A ⊆ X such that μ(A) > 0 and dμ (x) ≥ θ1 ,

∀x ∈ A,

(15.13)

for μ-a. e. x ∈ X,

(15.14)

then HD(μ) ≥ θ1 . (b) If there exists θ2 ≥ 0 such that dμ (x) ≤ θ2 , then HD(μ) ≤ θ2 . Proof. (a) Let Y ⊆ X be a Borel set such that μ(Y) = 1. Then Y ∩ A is a Borel set such that μ(Y ∩ A) = μ(A) > 0 and (15.13) is satisfied at each point of that set. By Theorem 15.6.3(a), we deduce that HD(Y) ≥ HD(Y ∩ A) ≥ θ1 . The arbitrary choice of Y implies that HD(μ) ≥ θ1 . ̂ be the Borel set of all points x ∈ X at which (15.14) holds. By hypothesis, (b) Let X ̂ = 1. Moreover, HD(X) ̂ ≤ θ2 by Theorem 15.6.3(b). Hence, HD(μ) ≤ HD(X) ̂ ≤ θ2 . μ(X) Our next aim is to obtain a characterization of the Hausdorff dimension of any measure. Before embarking upon this, though, let us recall the definition of the essential infimum and supremum of a measurable function. Definition 15.6.5. Let μ be a Borel probability measure on a metric space X and f : X → ℝ be a Borel measurable function. The μ-essential infimum and supremum of f are defined by ess inf(f ) := sup

inf f (x)

μ(N)=0 x∈X\N

and

ess sup(f ) := inf

sup f (x).

μ(N)=0 x∈X\N

598 | 15 Fractal measures and dimensions The following characterizations of the essential infimum and supremum, which we state here without proof, are also often used in practice. Lemma 15.6.6. Let μ be a Borel probability measure on a metric space X and f : X → ℝ be a Borel measurable function. Then: (a) ess inf(f ) = θ1 , where θ1 is the unique real number such that μ({x ∈ X : f (x) < θ1 }) = 0

and

μ({x ∈ X : f (x) < θ}) > 0,

∀θ > θ1 .

(b) ess sup(f ) = θ2 , where θ2 is the unique real number such that μ({x ∈ X : f (x) > θ2 }) = 0

and

μ({x ∈ X : f (x) > θ}) > 0,

∀θ < θ2 .

We can now formulate a characterization of the Hausdorff dimension of any measure. Corollary 15.6.7. If μ is a Borel probability measure on a metric space X, then HD(μ) = ess sup(dμ ). In particular, if there exists θ ≥ 0 such that dμ (x) = θ for μ-a. e. x ∈ X, then HD(μ) = θ. Proof. Let θ2 := ess sup(dμ ). By Lemma 15.6.6(b), it immediately follows that μ({x ∈ X : dμ (x) > θ2 }) = 0 or, equivalently, μ({x : dμ (x) ≤ θ2 }) = 1. By Corollary 15.6.4(b), we deduce that HD(μ) ≤ θ2 . On the other hand, fix Y ⊆ X with μ(Y) = 1. Let θ < θ2 . By Lemma 15.6.6(b), we have that μ({x : dμ (x) > θ}) > 0, and consequently μ(Y ∩ {x : dμ (x) > θ}) > 0. It follows from Theorem 15.6.3(a) that HD(Y ∩ {x : dμ (x) > θ}) ≥ θ. So HD(Y) ≥ θ for all θ < θ2 . In other words, HD(Y) ≥ θ2 . As this holds for all Y ⊆ X such that μ(Y) = 1, we infer that HD(μ) ≥ θ2 . Let us now present results analogous to Theorem 15.6.3 as well as Corollaries 15.6.4 and 15.6.7 for packing measures and dimensions. Theorem 15.6.8. Let μ be a Borel probability measure on a metric space X and A ⊆ X be a Borel set. (a) If X is separable, μ(A) > 0, and there exists θ1 ≥ 0 such that dμ (x) ≥ θ1 ,

∀x ∈ A,

dμ (x) ≤ θ2 ,

∀x ∈ A,

then PD(A) ≥ θ1 . (b) If there exists θ2 ≥ 0 such that

then PD(A) ≤ θ2 .

15.6 Dimensions of measures | 599

Proof. Similar to that of Theorem 15.6.3 and hence is left as an exercise. Corollary 15.6.9. Let μ be a Borel probability measure on a metric space X. (a) If X is separable and there exist θ1 ≥ 0 and a Borel set A ⊆ X such that μ(A) > 0 and dμ (x) ≥ θ1 ,

∀x ∈ A,

then PD(μ) ≥ θ1 . (b) If there exists θ2 ≥ 0 such that dμ (x) ≤ θ2 ,

for μ-a. e. x ∈ X,

then PD(μ) ≤ θ2 . Proof. Similar to that of Corollary 15.6.4 and hence is left as an exercise. Corollary 15.6.10. If μ is a Borel probability measure on a separable metric space X, then PD(μ) = ess sup(dμ ). In particular, if there exists θ ≥ 0 such that dμ (x) = θ for μ-a. e. x ∈ X, then PD(μ) = θ. Proof. Similar to that of Corollary 15.6.7 and hence is left as an exercise. As an immediate consequence of this corollary and Corollary 15.6.7, we get the following result. Corollary 15.6.11. Let μ be a Borel probability measure on a separable metric space X. If dμ (x) exists for μ-a. e. x ∈ X and takes on the same value d for μ-a. e. x ∈ X, then HD(μ) = PD(μ) = d. Any measure satisfying the hypotheses of this corollary is said to be dimensionally exact. Measures which confer to open balls weights that are comparable to their radii, as long as those radii are sufficiently small, are called geometric measures. Definition 15.6.12. A Borel probability measure μ on a metric space X is said to be geometric or Ahlfors if there exist constants t > 0 and c ≥ 1 such that c−1 ≤

μ(B(x, r)) ≤ c, rt

∀x ∈ X, ∀0 < r ≤ 1.

We call t the exponent of the geometric measure μ. Note that t is unique. The following observation is immediate.

(15.15)

600 | 15 Fractal measures and dimensions Lemma 15.6.13. Any geometric measure μ is dimensionally exact. More precisely, dμ (x) = t for every x ∈ X, where t is the exponent of μ. We end this section and this chapter with fundamental properties of geometric measures. Theorem 15.6.14. Suppose that X is a separable metric space and μ is a geometric measure on X with exponent t. Then the following statements hold: (a) The measures μ, Ht and Pt are mutually equivalent. Furthermore, 0 < inf

(b)

dHt dHt ≤ sup 0, show that the function from X to ℝ given by x 󳨃→ μ(B(x, r)) is lower semicontinuous. Exercise 15.7.7. Prove Theorem 15.5.1(a) when c = ∞. Exercise 15.7.8. Prove Lemma 15.6.6. Exercise 15.7.9. Prove Theorem 15.6.8. Exercise 15.7.10. Prove Corollary 15.6.9. Exercise 15.7.11. Prove Corollary 15.6.10. Exercise 15.7.12. Show that the upper bound 1 on r in Definition 15.6.12 can be replaced by any r ∗ > 0. Exercise 15.7.13. Show that for every t ∈ (0, ∞) there exists n ∈ ℕ and a Borel set Bt ⊆ ℝn such that HD(Bt ) = t. Exercise 15.7.14. Show that for every t ∈ (0, ∞) there exists n ∈ ℕ and a Borel set Bt ⊆ ℝn such that PD(Bt ) = t. Exercise 15.7.15. Does there exist for every t ∈ (0, ∞) some n ∈ ℕ and a Borel set Bt ⊆ ℝn such that HD(Bt ) ⋅ PD(Bt ) = t? Exercise 15.7.16. Given n ∈ ℕ, prove that the Lebesgue measure λn , the Hausdorff measure Hn and the packing measure Pn are nonzero constant multiples of each other. Exercise 15.7.17. Show that for every n ∈ ℕ there is a set E ⊆ ℝn such that HD(E) = n but λn (E) = 0. Exercise 15.7.18. Given n ∈ ℕ, show that every Lebesgue nonmeasurable set F ⊆ ℝn has a Hausdorff dimension equal to n. Exercise 15.7.19. For every p > 0, calculate the box-counting dimension of the set {np : n ∈ ℕ} ⊆ ℝ. Exercise 15.7.20. Give an example of a set G ⊆ ℝ whose dimensions HD(G), BD(G), BD(G) and PD(G), are all distinct. Exercise 15.7.21. Let f : [0, 1] → ℝ be a Hölder continuous function with exponent α ∈ [0, 1]. Show that the upper box-counting dimension of the graph of f is bounded above by 2 − α. Exercise 15.7.22. Show that there exists a continuous function f : [0, 1] → ℝ whose graph has Hausdorff dimension equal to 2. Can it have a positive 2–dimensional Lebesgue measure?

16 Conformal expanding repellers In this chapter, we return to the class of expanding repellers that were introduced in Chapter 4. As these maps are open and distance expanding, the theory of Gibbs states developed in Chapter 13 applies to them. However, this class is a little too big to achieve many interesting results about the fractal structure of the limit set these repellers act on. So in this chapter we will impose the extra condition that the repellers be conformal. In Section 16.1, we briefly review the notion of conformality. In Section 16.2, we define the subclass of conformal expanding repellers, provide some examples and derive some consequences of results presented in Chapters 4 and 13, in particular, bounded distortion properties. Beginning with Section 16.3, we deal with the fractal geometry of the limit set of conformal expanding repellers. We consider all fractal dimensions (Hausdorff, packing and box-counting dimensions) of the limit set, as well as the dimensions of invariant measures on that limit set. To do this, we heavily rely on results and techniques from Chapter 15. In Section 16.3, we prove the celebrated Bowen’s formula, according to which the unique zero of the pressure function for a particular, geometric potential is equal to the Hausdorff dimension of the limit set of the repeller. The history of Bowen’s formula began with Bowen’s breakthrough work on quasi-Fuchsian groups [12]. Since then, it has attracted the attention of many researchers and has been adapted, in various forms, to many contexts of conformal dynamics. Conformal expanding repellers are just one such instance. Conformal graph directed Markov systems are another one (see Chapter 19, and especially Section 19.6). Bowen’s formula is a generalization of Hutchinson’s formula [56], a well-known result in fractal geometry. The fine observer will note that Bowen’s formula is entirely expressed in dynamical terms, more precisely in thermodynamic formalism terms. Using transfer (i. e., Perron–Frobenius) operators, this enables us to study in Section 16.4 the behavior of the Hausdorff dimension of the limit sets of complex repellers under perturbations. This technically-involved study also requires the use of powerful methods from functional analysis, more specifically the perturbation theory of bounded linear operators. It further necessitates the use of quasiconformal mappings, holomorphic motion and the celebrated λ-lemma, whose complete proof is provided in Theorem 16.4.4. Ultimately, we prove in Theorem 16.4.11 that the Hausdorff dimension of the limit set of repellers varies real analytically under analytic perturbations of the repellers. In Section 16.5, we develop a formula expressing the Hausdorff dimension of an ergodic invariant Borel probability measure as the ratio of its measure-theoretic entropy and its Lyapunov exponent. For (partly justified) historical reasons, this formula is frequently referred to as a volume lemma. Its first forms can be traced back to the works of Eggleston [27] and Billingsley [7]. From a dynamical viewpoint, a certain breakthrough was achieved by Lai-Sang Young [139]. Since then, a multitude of papers appeared. https://doi.org/10.1515/9783110702699-016

604 | 16 Conformal expanding repellers Some of the first few are [69, 71, 97]. The early papers [12] and [99] shed some light on the nature of dimensions of measures. Finally, in Section 16.6 we perform a multifractal analysis of the Hausdorff dimension of the level sets of Gibbs states for any Hölder continuous potential. This is a fertile field of applications of the theory of conformal expanding repellers and more largely of this book. In addition to its intrinsic interest and importance, the multifractal analysis conducted in this chapter bears witness to the strength and sophistication of the tools and methods developed in this book. To our knowledge, rigorous mathematical research on multifractal analysis began in 1989 with David Rand’s work on cookie-cutter Cantor sets [100]. In its full generality, nearly as in Section 16.6, a multifractal analysis was carried out by Pesin and Weiss in the paper [96]. Other valuable resources are the books [95] and [98]. Our exposition stems from this latter. By now, there are legions of books and papers on multifractal analysis of Hausdorff dimension of level sets of various measures and other functions, both from a mathematical perspective and from a more informal, but rich in ideas and stimulating, physical perspective. A quite general approach to the subject can be found in Lars Olsen’s papers [87, 88]. We regret that, due to a sheer lack of space, we did not do a multifractal analysis of the Hausdorff dimension of level sets of Birkhoff’s averages of Hölder continuous functions. This is an important, quickly developing research field and we encourage the reader to familiarize themself with it.

16.1 Conformal maps Let d ∈ ℕ. The usual inner product of two vectors υ, w ∈ ℝd will be denoted by ⟨υ, w⟩ := ∑dj=1 υj wj . The norm ‖υ‖ ≥ 0 of a vector υ is defined by ‖υ‖2 := ⟨υ, υ⟩ = ∑dj=1 υ2j . The angle 0 ≤ ∠(υ, w) ≤ π between two vectors υ, w satisfies cos ∠(υ, w) =

⟨υ, w⟩ . ‖υ‖ ⋅ ‖w‖

By definition, a conformal map is an angle-preserving transformation. To be precise, let {ej }dj=1 be the standard basis of ℝd . Let I ⊆ ℝ be an interval. Let γ, Γ : I → ℝd be two differentiable curves that meet at a point P = γ(t0 ) = Γ(t0 ). If the curve γ : I → ℝd , given by γ(t) = (γ1 (t), . . . , γd (t)) = ∑dj=1 γj (t)ej , is such that γ′i (t0 ) ≠ 0 for some 1 ≤

i ≤ d, then γ has a tangent vector at γ(t0 ), namely γ′ (t0 ) = ∑dj=1 γ′j (t0 )ej . Likewise for Γ. By definition, the angle between γ and Γ at P is the angle between their tangent vectors at P, i. e., ∠P (γ, Γ) := ∠(γ′ (t0 ), Γ′ (t0 )).

16.2 Conformal expanding repellers | 605

Let U be a nonempty open subset of ℝd . A transformation T : U → ℝd is said to be conformal at a point P ∈ U if it preserves the angle between any two curves at P, i. e., ∠T(P) (T(γ), T(Γ)) = ∠P (γ, Γ) for all differentiable curves with a tangent vector at P. This map is said to be conformal if it is conformal at each point of its domain. Analytically, the conformality of a differentiable map can be characterized in terms of its derivative. For instance, a C 1 local diffeomorphism T : U → ℝd is conformal if and only if for all vectors υ, w ∈ ℝd , ⟨T ′ (x)υ, T ′ (x)w⟩ = eσ(x) ⟨υ, w⟩ for some real-valued function σ : U → ℝ (see Theorem 3.8 in Blair [10]). In slightly different terms, a C 1 local diffeomorphism T : U → ℝd is conformal if and only if at every point x ∈ U its derivative T ′ (x) : ℝd → ℝd is a similarity without a translation component, i. e., of the form rA, where r > 0 and A is an orthogonal matrix. Recall that the operator norm of T ′ (x) is 󵄨󵄨 ′ 󵄨󵄨 󵄩 ′ 󵄩 󵄨󵄨T (x)󵄨󵄨 := sup{󵄩󵄩󵄩T (x)υ󵄩󵄩󵄩 : ‖υ‖ ≤ 1}. A conformal C 1 local diffeomorphism T : U → ℝd has, among others, the following properties: 󵄩󵄩 ′ 󵄩 󵄨 ′ 󵄨 󵄩󵄩T (x)υ󵄩󵄩󵄩 = 󵄨󵄨󵄨T (x)󵄨󵄨󵄨 ⋅ ‖υ‖,

∀υ ∈ ℝd , ∀x ∈ U

(16.1)

and 󵄨 󵄨2 ⟨T ′ (x)υ, T ′ (x)w⟩ = 󵄨󵄨󵄨T ′ (x)󵄨󵄨󵄨 ⟨υ, w⟩,

∀υ, w ∈ ℝd , ∀x ∈ U.

(16.2)

Moreover, if S : V → ℝd and T : U → ℝd are two conformal C 1 local diffeomorphisms such that T(U) ⊆ V, then 󵄨󵄨 󵄨 󵄨 ′ 󵄨 󵄨 ′ 󵄨 ′ 󵄨󵄨(S ∘ T) (x)󵄨󵄨󵄨 = 󵄨󵄨󵄨S (T(x))󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨T (x)󵄨󵄨󵄨,

∀x ∈ U.

(16.3)

16.2 Conformal expanding repellers Now that we have described conformal maps, let us define the family of conformal expanding repellers. The larger class of expanding repellers was introduced in Definition 4.1.4. Definition 16.2.1. Let U be an nonempty open subset of ℝd and T : U → ℝd a map. Also let X be a nonempty compact subset of U. The triple (X, U, T) is said to be a conformal expanding repeller provided that it satisfies the following conditions:

606 | 16 Conformal expanding repellers (a) T ∈ C 1+ϵ (U), i. e., T ∈ C 1 (U) and the map U ∋ x 󳨃→ T ′ (x) ∈ L(ℝd ) is Hölder continuous with exponent ϵ, where L(ℝd ) is the d2 -dimensional Banach space of linear operators from ℝd to ℝd . (b) T is conformal. (c) T(X) = X. (d) There exists λ > 1 such that |T ′ (x)| ≥ λ for all x ∈ X. −n (e) ⋂∞ n=0 T (U) = X. (f) The restriction T|X : X → X is topologically transitive. The set X is sometimes called the limit set of the repeller T. In other words, T is an expanding repeller satisfying the additional conditions (a), (b) and (f). Condition (b) in Definition 4.1.4 reduces to condition (d) by (16.1). In Section 4.1.2, we constructed conformal expanding repellers called hyperbolic Cantor sets. These sets were built within a compact set X0 , from which a neighborhood U, and thereafter the limit set X, were constructed. We now present an alternative construction method. We begin with an open, bounded, convex set B, construct U and then X. Example 16.2.2. Let 2 ≤ k ∈ ℕ. For each 1 ≤ i ≤ k, let Fi : ℝd → ℝd be a contracting similarity map, i. e., a map of the form Fi (x) = ri Ai (x) + bi , where 0 < ri < 1, Ai is an orthogonal matrix and bi ∈ ℝd . The maps Fi , 1 ≤ i ≤ k, are often called generators. Bundled together, they form a so-called iterated function system (IFS). Let B be an open, bounded, convex subset of ℝd . By choosing the ri ’s small enough and adjusting the bi ’s appropriately, the images Fi (B), 1 ≤ i ≤ k, can be made to be: (1) mutually disjoint, i. e., Fi (B) ∩ Fj (B) = 0 whenever i ≠ j; this is frequently called the Open Set Condition (abbreviated to OSC); and (2) such that Fi (B) ⊆ B. (Note that Fi (B) = Fi (B).) Together, these two conditions are commonly called the Strong Separation Condition (SSC) in the theory of IFSs. Nevertheless, a lot of the theory of IFSs can be developed without condition (2), i. e., by only requiring the OSC. The reader will encounter that situation in much greater generality in Chapter 19: condition (2) will not be assumed, the generators will be conformal (not merely affine), the alphabet will be countable (not necessarily finite) and, lastly, the system will be graph directed and Markov (so some transitions will be forbidden, unlike in IFSs where all transitions are allowed). Let U := ⋃ki=1 Fi (B) ⊆ B and define the map f : U → ℝd by setting f (x) := Fi−1 (x)

if x ∈ Fi (B).

By virtue of property (1), the map f is well defined. Furthermore, f ′ (x) = ri−1 A−1 i (x) for each x ∈ Fi (B). Thus, f satisfies conditions (a) and (b) in the definition of a conformal

16.2 Conformal expanding repellers | 607

−n expanding repeller. Defining X := ⋂∞ n=0 f (U), condition (e) is automatically satisfied.

Setting λ = (max{ri : 1 ≤ i ≤ k}) > 1, condition (d) holds on U ⊇ X. Regarding condition (c), observe that f −1 (U) ⊆ U by definition. Therefore, −(n+1) f (U) ⊆ f −n (U) for all n ≥ 0, and hence f −1 (X) ⊆ X. The opposite inclusion being obvious from the definition of X, we deduce that f −1 (X) = X. It immediately follows that f (X) ⊆ X. To prove that f (X) ⊇ X, let x ∈ X, fix any 1 ≤ i ≤ k, and let y = Fi (x). Since x ∈ X ⊆ U ⊆ B, we have that y ∈ Fi (B) ⊆ U. Moreover, f (y) = f (Fi (x)) = Fi−1 (Fi (x)) = x ∈ X and so f n (y) = f n−1 (x) ∈ X for all n ∈ ℕ. −n Hence, f n (y) ∈ U for all n ≥ 0, i. e., y ∈ ⋂∞ n=0 f (U) = X. Consequently, x = f (y) ∈ f (X). So X ⊆ f (X). In summary, f (X) = X, i. e., condition (c) is fulfilled. We also need to show that X is nonempty and compact. For any set C ⊆ B, observe that f −1 (C) = ⋃ki=1 Fi (C). In particular, f −1 (B) = U. Therefore, −1

f −(n+1) (C) ⊆ f −(n+1) (B) = f −n (U),

∀n ≥ 0.

(16.4)

So ∞

∞

n=0

n=0

⋂ f −n (C) ⊆ ⋂ f −n (U) = X.

(16.5)

Let K := ⋃ki=1 Fi (B). By the definitions of U and K and property (2), U ⊆ K ⊆ B.

(16.6)

f −1 (K) ⊆ f −1 (B) = U ⊆ K.

(16.7)

Hence, by (16.6) and (16.4),

So (f −n (K))∞ n=0 is a descending sequence. Moreover, by (16.6) and (16.5) with C = K, observe that ∞

∞

n=0

n=0

X := ⋂ f −n (U) ⊆ ⋂ f −n (K) ⊆ X. −n That is, X = ⋂∞ n=0 f (K). Since B is bounded and closed, it is compact according to the Heine–Borel theorem. So is each Fi (B), as the image of a compact set under a continuous map. Thus, as a finite union of compact sets, K is compact. Every f −n (K) is compact since it is a bounded preimage of a closed set under a continuous map. Hence, X is the intersection of a descending sequence of nonempty compact sets, and is consequently nonempty and compact. Finally, we prove that f |X : X → X is topologically exact. We will use symbolic n dynamics notation. Let E = {1, . . . , k} be the alphabet. For every ω ∈ E ∗ := ⋃∞ n=0 E , we

608 | 16 Conformal expanding repellers have that f |ω| (x) = Fω−1 (x) for all x ∈ Fω (B), where Fω = Fω1 ∘⋅ ⋅ ⋅∘Fω|ω| . Notice that Fω (B) is

open in ℝd and diam(Fω (B)) ≤ λ−|ω| diam(B) for all ω ∈ E ∗ since B is convex. Moreover, f |ω| (Fω (B)) = B ⊇ X. As f (X) = X = f −1 (X), we deduce that f |ω| (Fω (B) ∩ X) = X. Let V be a nonempty open subset of X. That is, V = W ∩ X for some nonempty open set ∞ −n −|ω| W ⊆ B. Since X = ⋂∞ diam(B) n=0 f (U) = ⋂n=0 ⋃ω∈E n Fω (B) and diam(Fω (B)) ≤ λ ∗ ∗ for all ω ∈ E , there exists τ ∈ E such that Fτ (B) ⊆ W. Then Fτ (B) ∩ X ⊆ V. Therefore, X = f |τ| (Fτ (B) ∩ X) ⊆ f |τ| (V) ⊆ f |τ| (X) = X. Hence, f |τ| (V) = X. So f : X → X is topologically exact and thus transitive. Condition (f) is established.

Remark 16.2.3. (a) Note that any conformal expanding repeller built from an open, bounded, convex set B in Example 16.2.2 results in the same hyperbolic Cantor set as the one constructed by setting X0 = ⋃ki=1 Fi (B) in Section 4.1.2. Conversely, any conformal expanding repeller built from a compact set X0 in Section 4.1.2 leads to the same hyperbolic Cantor set constructed by setting B = B(X0 , ε) in Example 16.2.2. These two construction methods are equivalent. (b) In Example 16.2.2, property (1) ensures that the map f is well defined. Property (2) guarantees that X is nonempty and compact. See Exercises 16.7.1–16.7.3. Example 16.2.4. Fix 0 < δ < 1/8 and let U := (−δ, (1/4) + δ) ∪ ((1/2) − δ, 1 + δ) ⊆ ℝ. Define the map T : U → ℝ by setting T(x) := {

4x 1 x

−1

if x ∈ (−δ, 41 + δ)

if x ∈ ( 21 − δ, 1 + δ).

It is clear that T ∈ C 1+ϵ (U) for any ϵ > 0. Furthermore, T is conformal since it is a differentiable map on U ⊆ ℝ. Thus, T satisfies conditions (a) and (b) in the definition −n of a conformal expanding repeller. As in Example 16.2.2, define X := ⋂∞ n=0 T (U). So ′ condition (e) is satisfied. Note that |T (x)| ≤ 1 for all x ∈ [1, 1 + δ) ⊆ U. However, X ⊆ T −1 (U) ⊆ (− 4δ , 3 1 ). Then condition (d) holds with λ = (11/8)2 . We leave the 2

−δ

verification of the remaining conditions as an exercise to the reader.

According to Theorem 4.1.5, conformal expanding repellers are distance expanding maps on X. They are also open maps; in fact, they are local diffeomorphisms. Thus, all the theory developed in Chapter 13 for Gibbs states of open, distance expanding maps, applies to them. Let us now investigate some consequences of this theory. Normally, one would only be working on X, and thus with open balls within X (i. e., the projection onto X of open balls in ℝd ). However, the use of the mean value theorem in the forthcoming Lemma 16.2.9 requires that Corollary 16.2.8, and accordingly Lemma 16.2.6, apply to sufficiently small open balls in ℝd centered on points of X (and correspondingly the preimages of those balls). The distinction is somewhat sub-

16.2 Conformal expanding repellers | 609

tle but it is for this sole reason that we proved in Theorem 4.1.5 that expanding repellers are actually distance expanding on B(X, κ) = {y ∈ ℝd : d(y, X) < κ}, a κ-neighborhood of X. Accordingly, all balls are from this point on open balls in B(X, κ) centered on points of X (as opposed to balls in X). To be more precise, the proof of Theorem 4.1.5 shows that expanding repellers are in Definition 4.1.1; distance expanding with constants δ = κ/2 and λ replaced by λ+1 2 λ+1 alternatively, see (13.1). To lighten notation, we will still denote 2 by λ. Let ξ be the constant arising in relations (4.29), (4.30) and (4.31); alternatively, see (13.2). Let φ(x) := − log |T ′ (x)| for all x ∈ U, where |T ′ (x)| is the operator norm of the derivative of T at x. This potential function plays a special role for conformal expanding repellers. First, note that φ is locally Hölder continuous with exponent ϵ since T ∈ C 1+ϵ (U). Moreover, because of the conformality of T, the ergodic sums of this potential enjoy the following crucial property. Lemma 16.2.5. Let (X, U, T) be a conformal expanding repeller and let φ(x) −j − log |T ′ (x)| for each x ∈ U. For all n ≥ 0 and all x ∈ ⋂n−1 j=0 T (U), we have that

:=

n−1

′ 󵄨 󵄨 󵄨 󵄨 Sn φ(x) = ∑ − log󵄨󵄨󵄨T ′ (T j (x))󵄨󵄨󵄨 = − log󵄨󵄨󵄨(T n ) (x)󵄨󵄨󵄨. j=0

Proof. The first equality is simply the definition of the ergodic sum. The second equality, and really the crucial one, is a direct consequence of the chain rule and the conformality of T (see (16.3)). One key property of conformal expanding repellers is that the distortion their iterates create near one point is comparable to that created near any other point. Lemma 16.2.6 (Distortion lemma). Let (X, U, T) be a conformal expanding repeller. There exists a constant K ≥ 1 such that for all n ≥ 0, all z ∈ X and all x, y ∈ Tz−n (B(T n (z), ξ )), we have that 󵄨󵄨 n ′ 󵄨󵄨 󵄨(T ) (x)󵄨 K −1 ≤ 󵄨󵄨 n ′ 󵄨󵄨 ≤ K. 󵄨󵄨(T ) (y)󵄨󵄨 󵄨 󵄨 Proof. Let υ, w ∈ B(T n (z), ξ ) be such that x = Tz−n (υ) and y = Tz−n (w). Let also φ(x) := − log |T ′ (x)|. It follows from Lemmas 16.2.5 and 13.1.1 that 󵄨󵄨 󵄨 n ′ 󵄨 󵄨 󵄨󵄨log󵄨󵄨(T ) (x)󵄨󵄨 − log󵄨󵄨󵄨(T n )′ (y)󵄨󵄨󵄨󵄨󵄨󵄨 = 󵄨󵄨󵄨Sn φ(x) − Sn φ(y)󵄨󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨 −n = 󵄨󵄨Sn φ(Tz (υ)) − Sn φ(Tz−n (w))󵄨󵄨󵄨 ϵ

≤ Cϵ (φ)(diam(X)) .

Rearranging and taking the exponential yields the result.

610 | 16 Conformal expanding repellers Remark 16.2.7. Lemma 13.1.1 was derived for balls in X. In the case of expanding repellers, using the full strength of Theorem 4.1.5, the reader ought to convince themself that Lemma 13.1.1 holds true for balls in B(X, κ) centred on points of X. Corollary 16.2.8 (Bounded distortion property (BDP)). Let (X, U, T) be a conformal expanding repeller. There exists a constant K ≥ 1 such that for all n ≥ 0, all z ∈ X and all x, y ∈ B(T n (z), ξ ), we have 󵄨󵄨 −n ′ 󵄨󵄨 󵄨(T ) (x)󵄨 K −1 ≤ 󵄨󵄨 z−n ′ 󵄨󵄨 ≤ K. 󵄨󵄨(T ) (y)󵄨󵄨 󵄨 z 󵄨 Proof. Since (T n )′ (Tz−n (x)) ∘ (Tz−n )′ (x) = Idℝd for all x ∈ B(T n (z), ξ ), the conformality of T implies that |(T n )′ (Tz−n (x))| ⋅ |(Tz−n )′ (x)| = 1. So |(Tz−n )′ (x)| = |(T n )′ (Tz−n (x))|−1 . The result ensues from Lemma 16.2.6. Thanks to the bounded distortion, the preimages of balls are distorted uniformly, in the sense that the nth preimages of balls contain, and are contained in, balls of radii that are comparable to the reciprocal of the norm of the derivative of the nth iterate, and this uniformly in n. This is the object of the next lemma. Lemma 16.2.9. Let (X, U, T) be a conformal expanding repeller. There exists a constant K ≥ 1 such that for all n ≥ 0 and all z ∈ X, we have B(z, K −1 |(T n )′ (z)|−1 ξ ) ⊆ Tz−n (B(T n (z), ξ )) ⊆ B(z, K|(T n )′ (z)|−1 ξ ). Proof. For any u, υ ∈ ℝd , let [u, υ] ⊆ ℝd denote the line segment joining u and υ. For the first inclusion, since Tz−n (B(T n (z), ξ )) is an open set, let R > 0 be the maximal radius such that B(z, R) ⊆ Tz−n (B(T n (z), ξ )). Then 𝜕B(z, R) ∩ 𝜕Tz−n (B(T n (z), ξ )) ≠ 0. The continuity of T also ensures that T n (B(z, R)) ⊆ B(T n (z), ξ ). So T n (B(z, R)) ∩ 𝜕B(T n (z), ξ ) ≠ 0. Therefore, for all ε > 0 there exists x ∈ B(z, R) such that ‖T n (x) − T n (z)‖ ≥ ξ − ε. Thus, by the mean value theorem and Lemma 16.2.6, 󵄩 󵄩 ξ − ε ≤ 󵄩󵄩󵄩T n (x) − T n (z)󵄩󵄩󵄩 󵄨 󵄨 ≤ sup 󵄨󵄨󵄨(T n )′ (w)󵄨󵄨󵄨 ⋅ ‖x − z‖ w∈[x,z]

󵄨 󵄨 ≤ K 󵄨󵄨󵄨(T n )′ (z)󵄨󵄨󵄨R.

16.2 Conformal expanding repellers | 611

Letting ε → 0, this implies that 󵄨−1 󵄨 K −1 󵄨󵄨󵄨(T n )′ (z)󵄨󵄨󵄨 ξ ≤ R. Hence, B(z, K −1 |(T n )′ (z)|−1 ξ ) ⊆ B(z, R) ⊆ Tz−n (B(T n (z), ξ )).

For the second inclusion, let x ∈ B(T n (z), ξ ). By the mean value theorem and Corollary 16.2.8, we have 󵄩󵄩 −n 󵄩 󵄩 −n 󵄩 −n n 󵄩󵄩Tz (x) − z 󵄩󵄩󵄩 = 󵄩󵄩󵄩Tz (x) − Tz (T (z))󵄩󵄩󵄩 ′ 󵄨 󵄨 󵄩 󵄩 ≤ sup 󵄨󵄨󵄨(Tz−n ) (y)󵄨󵄨󵄨 ⋅ 󵄩󵄩󵄩x − T n (z)󵄩󵄩󵄩 n y∈[x,T (z)]

′ 󵄨 󵄨 ≤ K 󵄨󵄨󵄨(Tz−n ) (T n (z))󵄨󵄨󵄨ξ ′ 󵄨 󵄨−1 = K 󵄨󵄨󵄨(T n ) (z)󵄨󵄨󵄨 ξ .

Thus, Tz−n (B(T n (z), ξ )) ⊆ B(z, K|(T n )′ (z)|−1 ξ ). Let x ∈ X and 0 < r < ξ . Set k = k(x, r) ≥ 0 to be the largest integer such that B(x, r) ⊆ Tx−k (B(T k (x), ξ ))

(16.8)

and set ℓ = ℓ(x, r) ≥ 0 to be the smallest integer for which Tx−ℓ (B(T ℓ (x), ξ )) ⊆ B(x, r).

(16.9)

Indeed, the sequence of sets (Tx−n (B(T n (x), ξ )))n=0 is descending and the diameters of the sets Tx−n (B(T n (x), ξ )) decrease to 0 as n increases to ∞, because T is distance expanding, and hence its inverse branches are contracting by a factor 0 < λ−1 < 1 for each iteration of T. Therefore, k = k(x, r) and ℓ = ℓ(x, r) are well defined. Observe that Tx−ℓ (B(T ℓ (x), ξ )) ⊆ Tx−k (B(T k (x), ξ )) and thereby ℓ ≥ k. The next result reveals that r is comparable to both |(T k )′ (x)|−1 and |(T ℓ )′ (x)|−1 . ∞

Corollary 16.2.10. Let (X, U, T) be a conformal expanding repeller. There exists a coñ ≥ 1 such that stant K ̃−1 ≤ K

r

|(T k )′ (x)|−1

̃ ≤K

̃−1 ≤ and K

r |(T ℓ )′ (x)|−1

̃ ≤ K,

∀x ∈ X, ∀0 < r < ξ ,

where k = k(x, r) and ℓ = ℓ(x, r) were defined just before this corollary. Proof. By the choice of k and Lemma 16.2.9, it follows that B(x, r) ⊆ Tx−k (B(T k (x), ξ )) ⊆ B(x, K|(T k )′ (x)|−1 ξ )

612 | 16 Conformal expanding repellers while B(x, r) ⊈ Tx−(k+1) (B(T k+1 (x), ξ )) ⊇ B(x, K −1 |(T k+1 )′ (x)|−1 ξ ). From these two relations, we deduce that ′ ′ 󵄨−1 󵄨 󵄨−1 󵄨 K −1 󵄨󵄨󵄨(T k+1 ) (x)󵄨󵄨󵄨 ξ < r ≤ K 󵄨󵄨󵄨(T k ) (x)󵄨󵄨󵄨 ξ .

Using the chain rule, we conclude that 󵄩 󵄩−1 K −1 󵄩󵄩󵄩T ′ 󵄩󵄩󵄩∞ ξ
0, where htop (T) is the topological entropy of T : X → X. (e) There exists a unique h ∈ ℝ such that P(h) = 0. In fact, h > 0. (f) The pressure function ℝ ∋ t 󳨃→ P(t) ∈ ℝ is convex. Proof. (a) By Corollary 12.1.11, we obtain that 󵄨󵄨 󵄨 󵄩 ′ 󵄩 󵄨󵄨P(t) − P(s)󵄨󵄨󵄨 ≤ ‖φt − φs ‖∞ = 󵄩󵄩󵄩 log |T | 󵄩󵄩󵄩∞ ⋅ |s − t|.

That is, the function t 󳨃→ P(t) is Lipschitz continuous, with Lipschitz constant L := ‖ log |T ′ | ‖∞ = log ‖T ′ ‖∞ . (b, c) By Proposition 13.7.12, there exists a unique equilibrium state μt for the Hölder continuous potential φt . Suppose that s ≤ t. Then, by the variational principle (Theorem 12.1.1), P(t) = hμt (T) + ∫ tφ dμt = hμt (T) + ∫ sφ dμt + (s − t) ∫ log |T ′ | dμt ≤ P(s) + (s − t) log λ, X

X

X

where λ comes from condition (d) of the definition of a conformal expanding repeller (cf. Definition 16.2.1). So lim P(t) ≤ P(s) + log λ ⋅ lim (s − t) = −∞

t→∞

t→∞

while lim P(s) ≥ P(t) + log λ ⋅ lim (t − s) = ∞.

s→−∞

s→−∞

Moreover, if s < t, then P(t) ≤ P(s) + (s − t) log λ < P(s).

16.3 Bowen’s formula

| 615

(d) By Remark 11.1.19, we know that P(0) = htop (T). Moreover, htop (T) > 0 according to Exercise 7.6.12, which states that the topological entropy of any transitive, open, expansive dynamical system is positive. (e) This immediately follows from properties (a)–(d). (f) This statement follows from the more general fact that the function C(X) ∋ ψ 󳨃→ P(T, ψ) is convex for any dynamical system T : X → X. This more general case can easily be deduced from the variational principle (Theorem 12.1.1). A direct proof, i. e., from the definition of the pressure function, also exists. Indeed, let ψ, χ ∈ C(X) and p ∈ [0, 1]. Fix momentarily n ∈ ℕ. For any finite subset E of X, it follows from Hölder’s inequality that p

1−p

∑ exp(pSn ψ(x) + (1 − p)Sn χ(x)) ≤ ( ∑ exp(Sn ψ(x))) ( ∑ exp(Sn χ(x)))

x∈E

x∈E

x∈E

.

Given any ε > 0, this inequality holds for all (n, ε)-separated sets. Passing to the supremum over all such sets on both sides yields Pn (T, pψ + (1 − p)χ, ε) ≤ Pn (T, ψ, ε)p Pn (T, χ, ε)1−p . Therefore, 1 log Pn (T, pψ + (1 − p)χ, ε) n 1 1 ≤ p lim sup log Pn (T, ψ, ε) + (1 − p) lim sup log Pn (T, χ, ε) n→∞ n n→∞ n

P(T, pψ + (1 − p)χ, ε) := lim sup n→∞

= pP(T, ψ, ε) + (1 − p)P(T, χ, ε).

Consequently, P(T, pψ + (1 − p)χ) := lim P(T, pψ + (1 − p)χ, ε) ε→0

≤ p lim P(T, ψ, ε) + (1 − p) lim P(T, χ, ε) ε→0

ε→0

= pP(T, ψ) + (1 − p)P(T, χ).

The next theorem is the main result of this chapter. Before stating it, let us recall that geometric measures were introduced in Definition 15.6.12. Theorem 16.3.2 (Bowen’s formula). Let (X, U, T) be a conformal expanding repeller. Per Proposition 16.3.1, let h be the unique zero of the pressure function t 󳨃→ P(t). Let φh : X → ℝ be the corresponding Hölder continuous potential φh (x) = −h log |T ′ (x)|. By virtue of Theorem 13.6.2 and Proposition 13.7.12, let mh be the eigenmeasure of the transfer operator’s dual ℒ∗φh : C(X)∗ → C(X)∗ and μh be the unique equilibrium state and unique T-invariant Gibbs state for the potential φh . Then mh and μh are geometric

616 | 16 Conformal expanding repellers measures with exponent h. In particular, (16.12)

HD(X) = PD(X) = BD(X) = h. Moreover, mh =

1 1 Hh = P . Hh (X) Ph (X) h

(16.13)

In particular, Hh (X) and Ph (X) are positive and finite. Proof. Recall that Proposition 13.6.16 asserts that mh is a Gibbs state for φh . Moreover, mh and μh are equivalent according to Theorem 13.7.7. Therefore, it suffices to prove the statement for one of these measures. We will use mh . Let x ∈ X and 0 < r < ξ . Let also k = k(x, r) be as defined just before Corollary 16.2.10. First, it follows from Lemma 16.2.5 that Sk φh (x) = −h log |(T k )′ (x)|. Lemma 16.2.12 applies with ψ = φh since P(T, ψ) = P(T, φh ) = P(h) = 0. Using ̂ ≥ 1 such that m = mh , there exists a constant K ̂ − h log󵄨󵄨󵄨(T k )′ (x)󵄨󵄨󵄨 log mh (B(x, r)) − log K ̂ − h log󵄨󵄨󵄨(T k )′ (x)󵄨󵄨󵄨 log K 󵄨 󵄨 ≤ 󵄨 󵄨. ≤ ̂ − log󵄨󵄨󵄨(T k )′ (x)󵄨󵄨󵄨 ̂ − log󵄨󵄨󵄨(T k )′ (x)󵄨󵄨󵄨 log r − log K log K 󵄨 󵄨 󵄨 󵄨 Equivalently, ̂ log K log |(T k )′ (x)| ̂ log K

log K log mh (B(x, r)) − log |(T k )′ (x)| − h ≤ ≤ , ̂ log K log r −1 −1 log |(T k )′ (x)|

−h

− log |(T k )′ (x)|

̂

󵄨 󵄨 Letting r ↘ 0 forces k ↗ ∞, and hence 󵄨󵄨󵄨(T k )′ (x)󵄨󵄨󵄨 ↗ ∞, so log mh (B(x, r)) −0 − h 0−h ≤ lim ≤ , −0 − 1 r→0 log r 0−1 log m (B(x,r))

h = h. Thus, mh is a geometric measure with exponent h. The i. e., limr→0 log r measure μh also has this property since μh and mh are equivalent, as stated by Theorem 13.7.7. Then (16.12) immediately follows from Theorem 15.6.14. Finally, according to Lemma 15.2.25, the measures Hh |X and Ph |X are (Λφh e−φh )-conformal since Λφh = exp(P(T, φh )) = 1 (cf. Definition 13.6.12). So, like mh , both Hh |X and Ph |X are fixed points of the dual operator ℒ∗φh per Lemma 13.6.13. Then (16.13) can be inferred from Proposition 13.8.10(a).

Example 16.3.3. Consider the middle-third Cantor set C (cf. Example 4.1.6). Recall that U = (− 91 , 49 ) ∪ ( 95 , 10 ) and T : U → ℝ is defined by 9 T(x) = {

3x 3x − 2

if x ∈ (− 91 , 49 ) if x ∈ ( 95 , 10 ). 9

16.3 Bowen’s formula

| 617

By setting B = (− 31 , 43 ), it follows from Example 16.2.2 that (C, U, T : U → ℝ) is a conformal expanding repeller. It is easy to see that the map T is topologically conjugate to the shift map σ : {0, 2}∞ → {0, 2}∞ via the conjugacy map h : {0, 2}∞ → C defined ωj by h(ω) = ∑∞ j=1 3j . Thus, P(0) = htop (T) = htop (σ) = log 2 by Proposition 7.2.18 and Example 7.2.26. By Proposition 11.3.1, we get 󵄨 󵄨 P(t) := P(T, −t log󵄨󵄨󵄨T ′ 󵄨󵄨󵄨) = P(T, −t log 3) = P(0) − t log 3 = log 2 − t log 3. So the unique t for which P(t) = 0 is t = by Bowen’s formula (Theorem 16.3.2).

log 2 . Therefore, HD(C) log 3

= PD(C) = BD(C) =

log 2 log 3

16.3.1 Special case of Hutchinson’s formula In the previous example, the calculation of the pressure function was simplified by the fact that the derivative of T was constant. Using Bowen’s formula, we now derive Hutchinson’s formula for iterated function systems (IFSs) consisting of finitely many contracting similarities satisfying the strong separation condition (SSC; see Example 16.2.2). The solution to this formula is simultaneously the Hausdorff, packing and box dimensions of the limit set X. Among others, this demonstrates that Bowen’s formula is a far-reaching extension of Hutchinson’s formula. Note that Hutchinson’s formula holds under the sole open set condition (OSC). We will deduce it from Bowen’s formula in an even more general context in Section 19.6.3. Theorem 16.3.4. Let {Fi (x) := ri Ai (x) + bi }ki=1 be an IFS consisting of contracting similarities satisfying the strong separation condition (SSC). The limit set X of this IFS satisfies HD(X) = PD(X) = BD(X) = h > 0, where h is the unique number such that k

∑ rih = 1. i=1

This equation is called Hutchinson’s formula. Proof. Let (X, U, f ) be the conformal expanding repeller engendered by this IFS, as outlined in Example 16.2.2. For every 1 ≤ i ≤ k and xi ∈ Fi (B), observe that ′ 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨−1 φt (xi ) = −t log󵄨󵄨󵄨f ′ (xi )󵄨󵄨󵄨 = −t log󵄨󵄨󵄨(Fi−1 ) (xi )󵄨󵄨󵄨 = −t log󵄨󵄨󵄨Fi′ (Fi−1 (xi ))󵄨󵄨󵄨 = log rit .

Thus, the function φt is piecewise constant on U := ⋃ki=1 Fi (B) ⊇ X. According to (4.25) in Section 4.1.2, the conformal expanding repeller (U, X, f ) is topologically conjugate to the symbolic system (E ∞ , σ), where E = {1, . . . , k}, via a codk ing map π : E ∞ → X, where {π(ω)} = ⋂∞ n=1 φω|n (X0 ) with X0 = ⋃i=1 Fi (B). More precisely, π ∘ σ = f ∘ π. Corollary 11.1.21 and Example 11.4.1 then jointly affirm that the

618 | 16 Conformal expanding repellers pressure function is k

k

i=1

i=1

t ? P(t) = P(f , φt ) = P(σ, φt ∘ π) = log ∑ exp(φ t ∘ π(e)) = log ∑ exp(φt (xi )) = log ∑ ri . e∈E

By Bowen’s formula (Theorem 16.3.2), we know that HD(X) = PD(X) = BD(X) = h, where h is the unique zero of the pressure function t 󳨃→ P(t). Thus, P(t) = 0 when ∑ki=1 rit = 1, and Hutchinson’s formula holds. Alternatively, since f : X → X is expansive, Theorem 11.1.26 guarantees that P(f , φt ) = P(f , φt , 𝒰 ) for any finite open partition 𝒰 of X with diam(𝒰 ) ≤ δ, where δ is an expansive constant for f . Let 𝒰 = {Fi (B) ∩ X : 1 ≤ i ≤ k}. By the OSC and since X ⊆ U := ⋃ki=1 Fi (B), it is obvious that 𝒰 is a finite open partition of X. However, it is unclear whether diam(𝒰 ) ≤ δ. Nevertheless, observe that 𝒰 n = {Fω (B) ∩ X : ω ∈ E n } and diam(𝒰 n ) ≤ λ−(n−1) diam(𝒰 ) for all n ∈ ℕ. Thus there exists N ∈ ℕ such that diam(𝒰 N ) ≤ δ. Lemma 11.1.15 affirms that P(f , φt , 𝒰 N ) = P(f , φt , 𝒰 ). So it suffices to compute P(f , φt , 𝒰 ). By Definition 11.1.11, the topological pressure of the potential φt with respect to 𝒰 is P(f , φt , 𝒰 ) := lim

n→∞

1 log Zn (φt , 𝒰 ), n

where, by Definition 11.1.2, the nth partition function of 𝒰 is Zn (φt , 𝒰 ) = inf{ ∑ exp(Sn φt (V)) : 𝒱 is a subcover of 𝒰 n }. V∈𝒱

As 𝒰 is a partition of X, so is 𝒰 n , and thus Zn (φt , 𝒰 ) reduces to Zn (φt , 𝒰 ) = ∑ exp(Sn φt (V)) = ∑ exp(Sn φt (Fω (B) ∩ X)). V∈𝒰 n

ω∈E n

Recall that φt (x) = log rit for all x ∈ Fi (B) and all 1 ≤ i ≤ k. Then S|ω| φt (x) = log rωt for t all x ∈ Fω (B) and all ω ∈ E ∗ , where rω := ∏|ω| j=1 rωj . Therefore, S|ω| φt (Fω (B) ∩ X) = log rω ∗ for all ω ∈ E . So k

n

Zn (φt , 𝒰 ) = ∑ rωt = (∑ rit ) . ω∈E n

i=1

Consequently, n

k k 1 log(∑ rit ) = log ∑ rit . n→∞ n i=1 i=1

P(t) = P(f , φt ) = P(f , φt , 𝒰 ) = lim

Thus, P(t) = 0 when ∑ki=1 rit = 1, and Hutchinson’s formula holds.

16.4 Real-analytic dependence of Hausdorff dimension of repellers in ℂ

|

619

Remark 16.3.5. Hutchinson’s formula does not generally hold if the OSC is not satisfied. See Exercise 16.7.4.

16.4 Real-analytic dependence of Hausdorff dimension of repellers in ℂ Consider a conformal expanding repeller (X, T, U) in ℂ (i. e. U ⊆ ℂ). Denote by J (rather than X) the associated compact invariant repelling set ∞

J := ⋂ T −n (U), n=0

i. e., the limit set of T. (The letter J is reminiscent of the Julia set for complex-valued functions of a complex variable.) We are primarily interested in analytic perturbations of T, defined as follows. Definition 16.4.1. Let (J, T, U) be a conformal expanding repeller in ℂ. Let also Λ be an open subset of ℂ. A family of conformal maps {Tλ : U → ℂ}λ∈Λ is called an holomorphic (or analytic) perturbation of T if the function Λ × U ∋ (λ, z) 󳨃󳨀→ Tλ (z) ∈ ℂ is analytic and T = Tλ0 for some λ0 ∈ Λ. ̂ with Our goal is to show that on some neighborhood of λ0 , the triples (Jλ , Tλ , U) ̂ yet-to-be-identified sets Jλ but a common neighborhood U of those sets Jλ , constitute conformal expanding repellers, and that the maps Tλ |Jλ and T|J are topologically conjugate (and more). For this, we need the powerful tools of holomorphic motion and quasisymmetric (or quasiconformal) homeomorphisms. Definition 16.4.2. Let (Y, dY ) and (Z, dZ ) be metric spaces. A map f : Y → Z is M-quasisymmetric for some M > 0 if dY (y1 , y2 ) ≤ dY (y1 , y3 )

󳨐⇒

dZ (f (y1 ), f (y2 )) ≤ MdZ (f (y1 ), f (y3 )).

The number M is called a quasisymmetric constant for f . Clearly, any M ′ > M is also a quasisymmetric constant for f . Moreover, if f is not constant then the infimum of all quasisymmetric constants for f is positive and is also a quasisymmetric constant for f . We will call it the quasisymmetric constant for f . When Y is an open subset of a Euclidean space of dimension at least 2, the term quasiconformal is usually preferred. In that case, if f is in addition a homeomorphism, several equivalent characterizations of quasiconformality are known and frequently used (for more information, see [1, 3, 14, 35, 55, 66, 67]).

620 | 16 Conformal expanding repellers We now define the concept of holomorphic motion, one of the key notions in holomorphic dynamics. ̂ Denote the open unit disk in ℂ by 𝔻 := {z : |z| < 1} and the Riemann sphere by ℂ. ̂ A family of maps Definition 16.4.3. Let A be a subset of ℂ. ̂ {τλ : A → ℂ} λ∈𝔻 is called a holomorphic motion of A if the following conditions are satisfied: (a) τ0 is the identity map; ̂ is injective; and (b) For every λ ∈ 𝔻, the map τλ : A → ℂ ̂ is holomorphic. (c) For every z ∈ A, the map 𝔻 ∋ λ 󳨃→ τλ (z) ∈ ℂ The result which makes this notion so amazingly useful is the following “lemma” due to Mañé, Sad and Sullivan [70]. ̂ λ∈𝔻 is a holomorphic motion of a set A ⊆ ℂ, ̂ Theorem 16.4.4 (λ-lemma). If {τλ : A → ℂ} ̂ then for every λ ∈ 𝔻 the map τλ : A → ℂ has an injective quasisymmetric extension ̂ τλ : A → ℂ. Furthermore: ̂ is holomorphic. (a) For every z ∈ A, the map 𝔻 ∋ λ 󳨃→ τλ (z) ∈ ℂ ̂ (b) The map 𝔻 × A ∋ (λ, z) 󳨃→ τλ (z) ∈ ℂ is continuous. (c) For every compact set K ⊆ 𝔻, there is 0 < M(K) < ∞ such that the maps τλ , λ ∈ K, are all M(K)-quasisymmetric. That is, the quasisymmetric constants of the maps τλ , λ ∈ K, are uniformly bounded above. (d) The quasisymmetric constants of the maps τλ converge uniformly to 1 as λ → 0. Proof. If the set A is finite, the theorem is trivially satisfied. So we may assume that A is infinite. The proof is based on the following well-known fact from hyperbolic geomê \ {0, 1, ∞} does not try: any holomorphic map from 𝔻 to the thrice-punctured sphere ℂ ̂ \ {0, 1, ∞}. Its proof is an easy consequence increase hyperbolic distances on 𝔻 and ℂ of Schwarz lemma (Lemma A.3.5 in Appendix A) and the fact that the universal cover of the thrice-punctured sphere is conformally the open unit disk 𝔻. Fix three points ̂ λ ∈ 𝔻, so that the images of these three of A and normalize the maps τλ : A → ℂ, points under each τλ are 0, 1 and ∞. This can be done by composing each map τλ with ̂ ∋ z 󳨃→ az+b ∈ ℂ). ̂ We do not lose any gensome appropriate Möbius transformation (ℂ cz+d erality in this way since Möbius transformations are uniformly Lipschitz, and hence ̂ uniformly continuous, with respect to the spherical metric on ℂ. For any three points x, y, z ∈ A, disjoint from those previously fixed, consider the ̂ following four holomorphic functions from 𝔻 to ℂ: x(λ) := τλ (x),

y(λ) := τλ (y),

z(λ) := τλ (z),

w(λ) :=

y(λ) − x(λ) . y(λ)

16.4 Real-analytic dependence of Hausdorff dimension of repellers in ℂ

|

621

By design, these four functions avoid the points 0, 1 and ∞. Fix any numbers 0 < m < 1 < M < ∞. Let A(m, M) := {ξ ∈ ℂ : m ≤ |ξ | ≤ M} be the closed annulus centered at 0 with inner radius m and outer radius M. Fix ̂ \ {0, 1, ∞} by dist∗hyp . Note that R, ε > 0. Denote the hyperbolic distance on ℂ Δ(M, R) := sup{|z| : z ∈ Bhyp (A(0, M), R)} < ∞. ∗

There exists δ > 0 so small that ̂ \ {0, 1, ∞}, |a| < [a, b ∈ ℂ

ε δ and dist∗hyp (a, b) ≤ R] 󳨐⇒ |b| < . m Δ(M, R)

Let dist0hyp be the hyperbolic distance on the unit disk 𝔻. Suppose that y ∈ A∩A(m, M), 0

|x − y| < δ and λ ∈ Bhyp (0, R) := {z ∈ 𝔻 : dist0hyp (z, 0) ≤ R}. Then |w(0)| < δ/m. Moreover, as holomorphic maps do not increase hyperbolic distances (this boils down to Schwarz lemma for holomorphic maps of the unit disk), we know that dist∗hyp (w(λ), w(0)) ≤ R. Hence, |w(λ)| < ε/Δ(M, R). On the other hand, like for w we have dist∗hyp (y(λ), y(0)) ≤ R and y(0) = y ∈ A(m, M). Thus |y(λ)| ≤ Δ(M, R). So 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨τλ (x) − τλ (y)󵄨󵄨󵄨 = 󵄨󵄨󵄨x(λ) − y(λ)󵄨󵄨󵄨 = 󵄨󵄨󵄨w(λ)󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨y(λ)󵄨󵄨󵄨 < ε. Thus we have proved that the family ̂ {τλ : A ∩ A(m, M) → ℂ}

0

λ∈Bhyp (0,R)

is uniformly equicontinuous for every R > 0. Letting R → ∞, we deduce that each map ̂ τλ : A ∩ A(m, M) → ℂ,

λ ∈ 𝔻,

is uniformly continuous. Since the annulus A(m, M) contains 1, by exchanging the roles of 0, 1, and ∞ (i. e., by considering the annuli A′ (m, M) := {ξ ∈ ℂ : m ≤ |ξ − 1| ≤ M} and ̂ : m ≤ dsph (ξ , ∞) ≤ M}, where dsph is the spherical metric A′′ (m, M) := {ξ ∈ ℂ ̂ ̂ In consequence, each map on ℂ), we know that finitely many of such annuli cover ℂ. τλ , λ ∈ 𝔻, has a continuous extension to A. Furthermore, the family ̂ {τλ : A → ℂ}

0

λ∈Bhyp (0,R)

is uniformly equicontinuous for each R > 0.

622 | 16 Conformal expanding repellers ̂ λ ∈ 𝔻, are injective. Fix again Now we shall prove that all maps τλ : A → ℂ, 0 < m < 1 < M < ∞ and R, Δ > 0. There exists δ > 0 so small that ̂ \ {0, 1, ∞}, |a| > [a, b ∈ ℂ

δ Δ and dist∗hyp (a, b) ≤ R] 󳨐⇒ |b| > . M m 0

Assume that y ∈ A ∩ A(m, M), |x − y| > δ and λ ∈ Bhyp (0, R). Then |w(0)| < δ/M and dist∗hyp (w(λ), w(0)) ≤ R. Hence, |w(λ)| > Δ/m. Therefore, 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨τλ (x) − τλ (y)󵄨󵄨󵄨 = 󵄨󵄨󵄨x(λ) − y(λ)󵄨󵄨󵄨 > Δ. Now, letting first m → 0 and M → ∞, and then exchanging the roles of 0, 1 and ∞, ̂ λ ∈ B0hyp (0, R), are injective. Letting R → ∞ leads we deduce that all maps τλ : A → ℂ, ̂ λ ∈ 𝔻, are injective. to the conclusion that the maps τλ : A → ℂ, ̂ λ ∈ 𝔻, are quasisymmetric, and to In order to prove that the maps τλ : A → ℂ, ̂ given by the formula prove parts (c) and (d), consider the function g : 𝔻 → ℂ g(λ) :=

x(λ) − y(λ) . x(λ) − z(λ)

This function omits 0, 1 and ∞. Fix R > 0. Assume further that x, y, z are chosen such that |g(0)| = 1. Then dist∗hyp (g(λ), 𝜕𝔻) ≤ R,

0

∀λ ∈ Bhyp (0, R).

(16.14)

Therefore, 0

0

0 < inf{|g(λ)| : λ ∈ Bhyp (0, R)} ≤ sup{|g(λ)| : λ ∈ Bhyp (0, R)} < ∞, ̂ λ ∈ B0hyp (0, R), are quasisymmetric with their yielding that all maps τλ : A → ℂ, quasisymmetric constants bounded above by 0

M(R) :=

sup{|g(λ)| : λ ∈ Bhyp (0, R)} 0

inf{|g(λ)| : λ ∈ Bhyp (0, R)}

< ∞.

(16.15)

So, item (c) is proved. In addition, by letting R → ∞, we can infer that the maps ̂ λ ∈ 𝔻, are quasisymmetric. On the other hand, we deduce from (16.14) τλ : A → ℂ, that lim M(R) = 1,

R→0

finishing the proof of item (d).

16.4 Real-analytic dependence of Hausdorff dimension of repellers in ℂ

|

623

Finally, pick any x ∈ A. There exists a sequence (xn )∞ n=1 of points in A such that limn→∞ xn = x. Then τλ (x) = lim τλ (xn ), n→∞

∀λ ∈ 𝔻.

This, along with the first part of the proof, implies that the map ̂ 𝔻 ∋ λ 󳨃→ τλ (x) ∈ ℂ ̂ n ∈ ℕ, converging uniformly on is a limit of holomorphic maps 𝔻 ∋ λ 󳨃→ τλ (xn ) ∈ ℂ, all compact subsets of 𝔻. Thus, it is holomorphic, proving item (a). Due to the already proved equicontinuity, the map in item (b) is then continuous. Remark 16.4.5. Note that the hypothesis that the domain of λ’s is an open subset of the complex plane ℂ is substantial. If, for instance, the motion is only for λ ∈ ℝ, then the lemma does not hold. Indeed, consider the holomorphic motion of A = ℂ \ ℝ such that the lower half-plane shifts horizontally in one direction (τλ (z) = z −λ, λ ∈ ℝ) while the upper half-plane moves in the opposite direction (τλ (z) = z + λ, λ ∈ ℝ). Clearly, ̂ this motion cannot be extended injectively nor continuously to A = ℂ. ̂ of the holomorphic motion in Theorem 16.4.4 Remark 16.4.6. The maps τλ : A → ℂ are Hölder continuous. Moreover, for every compact set K ⊆ 𝔻 these maps have a common Hölder exponent α(K) and a uniformly bounded above Hölder norm in Hα(K) (A) and the Hölder exponent converges to 1 when λ → 0. This follows from Slodkowski’s ̂ λ ∈ 𝔻, extheorem (Theorem A.3.18 or [119]), which asserts that the motion τλ : A → ℂ, tends from A to the whole Riemann sphere with the same quasisymmetric constants (see also [3]). Having this, we invoke the classical fact that each M-quasisymmetric (M-quasiconformal) homeomorphism is Hölder continuous with exponent 1/M and the corresponding Hölder norm uniformly bounded above (see [1, 67] for instance). We now study the analytic perturbations of a conformal expanding repeller. The family of conjugating homeomorphisms we are after is provided by the next result. Theorem 16.4.7. If {Tλ : U → ℂ}λ∈Λ is an analytic perturbation of a conformal expand̂ of λ0 and an open neighing repeller (Jλ0 , Tλ0 , U) in ℂ, then there exist a neighborhood Λ ̂ ⊆ U of Jλ such that for every λ ∈ Λ ̂ there are a conformal expanding repeller borhood U 0 ̂ and a homeomorphism τλ : Jλ → Jλ with the following properties: (Jλ , Tλ , U) 0 (a) For every fixed z ∈ Jλ0 , the map ̂ ∋ λ 󳨃󳨀→ τλ (z) Λ is holomorphic. ̂ the map τλ , i. e. (b) The map τλ0 is the identity map, and for every λ ∈ Λ Jλ0 ∋ z 󳨃󳨀→ τλ (z) ∈ Jλ ,

624 | 16 Conformal expanding repellers is quasiconformal and Hölder continuous with exponent converging to 1 uniformly as λ → λ0 . (c) The homeomorphism τλ conjugates the maps Tλ0 and Tλ , i. e., 󵄨 Tλ ∘ τλ = τλ ∘ Tλ0 󵄨󵄨󵄨J . λ 0

̂ × Jλ ∋ (λ, z) 󳨃󳨀→ τλ (z) ∈ U ̂ is continuous. (d) The map Λ 0 Proof. As this proof is long, it will be divided into parts and claims. ̂ ⊆ U and Λ ̂ ⊆ Λ. The following are necessary technicalities to The neighborhoods U ̂ ⊆ U and Λ ̂ ⊆ Λ. define conveniently the neighborhoods U First, observe that the triple (Jλ0 , Tλ0 , V) is a conformal expanding repeller for any open set V such that Jλ0 ⊆ V ⊆ U. Second, in Theorem 4.1.5, we proved that any expanding repeller such as (Jλ0 , Tλ0 , U), is distance expanding not only on Jλ0 but also on one of its neighborhoods, say B(Jλ0 , κ) for some κ > 0. We will reuse part of that argument to show that this property carries over to all λ in a neighborhood of λ0 . By shrinking U or κ if necessary, we may assume that U = B(Jλ0 , κ). To avoid any confusion, the expansion constant λ from Definition 16.2.1(d) will be replaced by L. Then 󵄨󵄨 ′ 󵄨 1+L , 󵄨󵄨Tλ0 (z)󵄨󵄨󵄨 ≥ 2

∀z ∈ U

(16.16)

and 󵄨󵄨 󵄨 1+L |z − w|, 󵄨󵄨Tλ0 (z) − Tλ0 (w)󵄨󵄨󵄨 ≥ 2

∀z, w ∈ U with |z − w| < κ.

(16.17)

As the set Jλ0 is compact and (16.16) holds, the analyticity of the perturbation (λ, z) 󳨃→ Tλ (z) ensures, by downsizing κ (correspondingly U) and Λ if necessary, that there exist two constants Γ ≥ γ > 1 such that 󵄨 󵄨 γ ≤ 󵄨󵄨󵄨Tλ′ (z)󵄨󵄨󵄨 ≤ Γ,

∀z ∈ U, ∀λ ∈ Λ.

(16.18)

Using (16.17) and after shrinking κ and Λ even further if necessary, the converse to Hurwitz’s theorem (Theorem A.3.15) asserts that the map Tλ : B(z, κ) → Tλ (B(z, κ)) is univalent for every z ∈ Jλ0 and every λ ∈ Λ. Denote its inverse by (Tλ )−1 z : Tλ (B(z, κ)) → B(z, κ).

(16.19)

By the Koebe quarter theorem (Theorem A.3.7) and (16.18), we know that 1 1 󵄨 󵄨 Tλ (B(z, κ)) ⊇ B(Tλ (z), κ 󵄨󵄨󵄨Tλ′ (z)󵄨󵄨󵄨) ⊇ B(Tλ (z), κγ), 4 4

∀z ∈ Jλ0 , ∀λ ∈ Λ.

16.4 Real-analytic dependence of Hausdorff dimension of repellers in ℂ

|

625

Alternatively, the proof of Lemma 4.2.2 can be generalized to yield q > 0 such that B(Tλ (z), q) ⊆ Tλ (B(z, κ)),

∀z ∈ Jλ0 , ∀λ ∈ Λ.

(16.20)

By reducing Λ appropriately, the continuity of (λ, z) 󳨃→ Tλ (z) then guarantees that there exists 0 < η ≤ κ such that Tλ (B(z, η)) ⊆ B(Tλ (z), q),

∀z ∈ Jλ0 , ∀λ ∈ Λ.

(16.21)

Combining (16.19)–(16.21), we observe that B(z, η) ⊆ (Tλ )−1 z (B(Tλ (z), q)) ⊆ B(z, κ),

∀z ∈ Jλ0 , ∀λ ∈ Λ.

(16.22)

Let λ ∈ Λ and let z1 , z2 ∈ B(Jλ0 , η/2) be such that |z1 − z2 | < η/2. Then there exists z ∈ Jλ0 such that z1 , z2 ∈ B(z, η). Consequently, Tλ (z1 ), Tλ (z2 ) ∈ B(Tλ (z), q) according to (16.21). Let S = [Tλ (z1 ), Tλ (z2 )] be the line segment joining Tλ (z1 ) and Tλ (z2 ). Due to the convexity of discs in ℂ, we know that S ⊆ B(Tλ (z), q). By (16.22), the curve (Tλ )−1 z (S) −1 joins the points (Tλ )−1 (T (z )) = z and (T ) (T (z )) = z from within B(z, κ). Hence, λ 1 1 λ z λ 2 2 z 󵄨󵄨 󵄨󵄨 −1 ′ |z1 − z2 | ≤ ℓ((Tλ )−1 z (S)) = ∫󵄨󵄨((Tλ )z ) (w)󵄨󵄨 dw,

(16.23)

S

where ℓ(C) stands for the length of the curve C. By (16.18), 󵄨 󵄨󵄨 −1 ′ −1 ′ 󵄨󵄨 󵄨󵄨 1 = 󵄨󵄨󵄨Tλ′ ((Tλ )−1 z (w)) ⋅ ((Tλ )z ) (w)󵄨󵄨 ≥ γ󵄨󵄨((Tλ )z ) (w)󵄨󵄨,

∀w ∈ S.

Consequently, (16.23) yields that 󵄨 󵄨 |z1 − z2 | ≤ γ−1 ℓ(S) = γ−1 󵄨󵄨󵄨Tλ (z1 ) − Tλ (z2 )󵄨󵄨󵄨. Letting κ = η/2, we infer that 󵄨󵄨 󵄨 󵄨󵄨Tλ (z) − Tλ (w)󵄨󵄨󵄨 ≥ γ|z − w|,

∀λ ∈ Λ,

∀z, w ∈ U with |z − w| < κ.

(16.24)

Moreover, the shadowing property of expanding maps (Corollary 4.3.6) states that there is 0 < α < κ/3 such that every infinite α-pseudo-orbit in Jλ0 is (κ/3)-shadowed by a unique point in Jλ0 . Finally, thanks to the continuity of the perturbation, we may shrink Λ and select 0 < κ̂ < α/(3Γ) such that 󵄨󵄨 󵄨 α 󵄨󵄨Tλ (z) − Tλ0 (z)󵄨󵄨󵄨 < , 3

∀z ∈ B(Jλ0 , κ̂), ∀λ ∈ Λ.

(16.25)

̂ := B(Jλ , κ̂). The neighborhood Λ ̂ will be identified later. All these techniWe set U 0 ̂ are conformal calities will be used at the end of the proof to establish that (Jλ , Tλ , U) expanding repellers. In the meantime, we can simply work with the sets U and Λ.

626 | 16 Conformal expanding repellers The map (λ, z) 󳨃→ τλ (z). By definition, the conformal expanding repeller Tλ0 : Jλ0 → Jλ0 is a transitive, open and distance expanding map. By Corollary 4.3.10, the set of periodic points of Tλ0 is dense in Jλ0 . In particular, there exists at least one periodic orbit P in Jλ0 . Denote its prime period by p. As Jλ0 is a compact subset of the open

set U, there exists δ > 0 such that B(Jλ0 , 2δ) ⊆ U. Given the continuity of the map Tλ0 and the openness of B(Jλ0 , δ), the set ⋂pk=0 Tλ−k (B(Jλ0 , δ)) is open. As Tλ0 (Jλ0 ) = Jλ0 , we 0

know that Jλ0 ⊆ ⋂pk=0 Tλ−k (B(Jλ0 , δ)). Since Jλ0 is compact, there is 0 < δ′ < δ such that 0

B(Jλ0 , δ′ ) ⊆ ⋂pk=0 Tλ−k (B(Jλ0 , δ)). Therefore 0 p

⋃ Tλk0 (B(Jλ0 , δ′ )) ⊆ B(Jλ0 , δ) ⊆ B(Jλ0 , 2δ) ⊆ U.

k=0

As {Tλ }λ∈Λ is an analytic perturbation of Tλ0 , the continuity of (λ, z) 󳨃→ Tλ (z) signifies that there exists R′ > 0 such that p

⋃ Tλk (B(Jλ0 , δ′ )) ⊆ B(Jλ0 , 2δ),

k=0

∀λ ∈ B(λ0 , R′ ).

Fix ξ ∈ P and consider the function F : B(Jλ0 , δ′ ) × B(λ0 , R′ ) → ℂ defined by F(z, λ) := Tλp (z) − z. Then F is well defined and (separately) holomorphic with respect to each of the variables z and λ and, therefore, is (jointly) holomorphic by virtue of Hartogs’ theorem (Theorem A.4.1 in Appendix A). As Tλ0 is a conformal expanding repeller, the periodic point ξ is repelling, i. e., 󵄨󵄨 p 󵄨 󵄨󵄨 󵄨󵄨 𝜕Tλ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 > 1. 󵄨󵄨 𝜕z 󵄨󵄨 󵄨 󵄨(ξ ,λ0 ) 󵄨󵄨󵄨 󵄨󵄨 Hence, 𝜕T p 󵄨󵄨󵄨 𝜕F 󵄨󵄨󵄨󵄨 = λ 󵄨󵄨󵄨 − 1 ≠ 0. 󵄨󵄨 𝜕z 󵄨󵄨(ξ ,λ0 ) 𝜕z 󵄨󵄨(ξ ,λ0 ) So the implicit function theorem (Theorem A.3.16) applies and yields the existence of two numbers δξ ∈ (0, δ′ ] and Rξ ∈ (0, R′ ] as well as a unique holomorphic function B(λ0 , Rξ ) ∋ λ 󳨃󳨀→ τλ (ξ ) ∈ B(ξ , δξ ) such that τλ0 (ξ ) = ξ

(16.26)

16.4 Real-analytic dependence of Hausdorff dimension of repellers in ℂ

|

627

and for every λ ∈ B(λ0 , Rξ ), the number τλ (ξ ) is the only point z ∈ B(ξ , δξ ) for which F(z, λ) = 0.

(16.27)

In particular, Tλp (τλ (ξ )) = τλ (ξ ),

∀λ ∈ B(λ0 , Rξ ).

(16.28)

Doing this for each ξ ∈ P, we set R := min{Rξ : ξ ∈ P} ∈ (0, R′ ].

(16.29)

Now fix an arbitrary point ∞

ζ ∈ ⋃ Tλ−n (P) \ P ⊆ Jλ0 . 0

(16.30)

n=1

This last inclusion ensues from the fact that Tλ−10 (Jλ0 ) = Jλ0 per Definition 16.2.1(c,e). Let n(ζ ) be the smallest n ∈ ℕ such that Tλn0 (ζ ) ∈ P.

(16.31)

Denote n(ζ )

ξ := Tλ

0

(16.32)

(ζ ).

Since ζ ∈ Jλ0 = Tλ0 (Jλ0 ) and since Tλ0 : U → ℂ is continuous, there is δζ′ > 0 such that Tλk0 (B(ζ , δζ′ )) ⊆ B(Jλ0 , δ),

∀k = 0, 1, . . . , n(ζ ).

Because {Tλ }λ∈Λ is an analytic perturbation of Tλ0 , there exists R′ζ ∈ (0, R] such that Tλk (B(ζ , δζ′ )) ⊆ B(Jλ0 , 2δ),

∀λ ∈ B(λ0 , R′ζ ), ∀k = 0, 1, . . . , n(ζ ).

Hence, the function G : B(ζ , δζ′ ) × B(λ0 , R′ζ ) → ℂ given by n(ζ )

G(z, λ) := Tλ

n(ζ )

(z) − τλ (ξ ) = Tλ

is well defined, holomorphic and such that n(ζ ) 󵄨

𝜕T 𝜕G 󵄨󵄨󵄨󵄨 = λ 󵄨󵄨 󵄨 𝜕z 󵄨(ζ ,λ0 ) 𝜕z

n(ζ )

(z) − τλ (Tλ

󵄨󵄨 󵄨󵄨 ≠ 0. 󵄨󵄨 󵄨(ζ ,λ0 )

0

(ζ ))

(16.33)

628 | 16 Conformal expanding repellers So the implicit function theorem (Theorem A.3.16) applies, yielding the existence of two numbers δζ ∈ (0, δζ′ ] and Rζ ∈ (0, R′ζ ] as well as a unique holomorphic function B(λ0 , Rζ ) ∋ λ 󳨃󳨀→ τλ (ζ ) ∈ B(ζ , δζ ) such that τλ0 (ζ ) = ζ

(16.34)

and for every λ ∈ B(λ0 , Rζ ), the number τλ (ζ ) is the only point z ∈ B(ζ , δζ ) such that G(z, λ) = 0.

(16.35)

In particular, n(ζ )

Tλ

n(ζ )

(τλ (ζ )) = τλ (ξ ) = τλ (Tλ

0

∀λ ∈ B(λ0 , Rζ ).

(ζ )),

(16.36)

We shall prove the following assertion. Claim 1. The function B(λ0 , Rζ ) ∋ λ 󳨃󳨀→ τλ (ζ ) ∈ B(ζ , δζ ) extends holomorphically to B(λ0 , R) and is uniquely determined by the relation n(ζ )

Tλ

n(ζ )

(τλ (ζ )) = τλ (ξ ) = τλ (Tλ

0

(ζ )),

∀λ ∈ B(λ0 , R).

Proof of Claim 1. Write the holomorphic function B(λ0 , Rζ ) ∋ λ 󳨃󳨀→ τλ (ζ ) ∈ B(ζ , δζ ) as a power series around λ0 , i. e., ∞

τλ (ζ ) = ∑ an (ζ )(λ − λ0 )n , n=0

∀λ ∈ B(λ0 , Rζ ).

Let ρ ≥ Rζ > 0 be the minimum of R and the radius of convergence of this series. Set n(ζ )

ρ′ := sup{t ∈ [0, ρ] : Tλ

(τλ (ζ )) ∈ B(Jλ0 , δ′ ), ∀λ ∈ B(λ0 , t)}.

We will show that ρ′ = ρ.

(16.37) n(ζ )

Then, by the continuity of the functions (λ, z) 󳨃󳨀→ Tλ we will deduce that n(ζ )

Tλ

(τλ (ζ )) = τλ (ξ ),

(z), λ 󳨃󳨀→ τλ (ζ ) and λ 󳨃󳨀→ τλ (ξ ),

∀λ ∈ B(λ0 , ρ).

(16.38)

16.4 Real-analytic dependence of Hausdorff dimension of repellers in ℂ

|

629

Seeking a contradiction, suppose that ρ′ < ρ. By (16.36) and the continuity of the n(ζ ) functions (λ, z) 󳨃󳨀→ Tλ (z), λ 󳨃󳨀→ τλ (ζ ) and λ 󳨃󳨀→ τλ (ξ ), we would then infer that n(ζ )

Tλ

(τλ (ζ )) = τλ (ξ ),

∀λ ∈ B(λ0 , ρ′ ).

(16.39)

As τλ (ξ ) ∈ B(ξ , δξ ) ⊆ B(ξ , δ′ ) for all λ ∈ B(λ0 , R) ⊇ B(λ0 , ρ′ ), we would deduce that n(ζ )

Tλ

(τλ (ζ )) ∈ B(Jλ0 , δ′ ),

∀λ ∈ B(λ0 , ρ′ ).

As the image of the compact set B(λ0 , ρ′ ) under the composition of the continuous n(ζ ) n(ζ ) functions (λ, z) 󳨃󳨀→ Tλ (z) and λ 󳨃󳨀→ τλ (ζ ), the set {Tλ (τλ (ζ )) : λ ∈ B(λ0 , ρ′ )} is compact. There would then exist r ∈ (ρ′ , ρ] such that n(ζ )

Tλ

(τλ (ζ )) ∈ B(Jλ0 , δ′ ),

∀λ ∈ B(λ0 , r),

contrary to the definition of ρ′ . Thus, (16.37) is established. Now we shall prove that ρ = R. Again seeking a contradiction, suppose that ρ < R. Fix any γ ∈ ℂ with |γ − λ0 | = ρ. Let (γk )∞ k=1 be any sequence in B(λ0 , ρ) converging to γ and such that the sequence ̂ (τγk (ζ ))∞ k=1 also converges. Denote this latter limit by ζ . By definition of λ 󳨃→ τλ (ζ ), we know that ζ̂ ∈ B(ζ , δ ) ⊆ B(ζ , δ′ ). It then follows from (16.33), (16.38), the convergence

ζ ζ ∞ ̂ of (τγk (ζ ))k=1 to ζ and the continuity of the maps (λ, z)

n(ζ )

󳨃󳨀→ Tλ

(z) and λ 󳨃󳨀→ τλ (ξ ), that

Tγn(ζ ) (ζ̂) = lim Tγn(ζk ) (τγk (ζ )) = lim τγk (ξ ) = τγ (ξ ). k→∞

k→∞

So we can apply the implicit function theorem (Theorem A.3.16) in exactly the same ̂ > 0, and a unique holomorphic function way as before to get two numbers δ̂ > 0 and R ̂ ̂ ∋ λ 󳨃󳨀→ τ̂λ (ζ ) ∈ B(ζ̂, δ) B(γ, R) such that τ̂γ (ζ ) = ζ̂

(16.40)

̂ such that ̂ the number τ̂λ (ζ ) is the only point z ∈ B(ζ̂, δ) and for every λ ∈ B(γ, R), n(ζ )

Tλ

(τ̂λ (ζ )) = τλ (ξ ).

But then the uniqueness part of the implicit function theorem affirms that τ̂λ (ζ ) = τλ (ζ ),

̂ ∀λ ∈ B(λ0 , ρ) ∩ B(γ, R).

(16.41)

630 | 16 Conformal expanding repellers We deduce that the holomorphic function B(λ0 , ρ) ∋ λ 󳨃󳨀→ τλ (ζ ) ∈ ℂ extends holomorphically to some larger ball B(λ0 , ρ′′ ) for some ρ′′ > ρ. This contradicts the definition of ρ, and finishes the proof of Claim 1. ◼ We now establish the following fact. Claim 2. The maps τλ satisfy the following commutativity property: Tλ ∘ τλ (z) = τλ ∘ Tλ0 (z),

∞

∀λ ∈ B(λ0 , R), ∀z ∈ ⋃ Tλ−n (P). 0 n=0

(16.42)

−n Proof of Claim 2. Consider first z = ζ ∈ ⋃∞ n=0 Tλ0 (P) \ P. Then n(ζ ) ≥ 1 and n(Tλ0 (ζ )) = n(ζ ) − 1 ≥ 0. By Claim 1, we have n(Tλ0 (ζ ))

Tλ

n(ζ )

(Tλ ∘τλ (ζ )) = Tλ

n(ζ )

(τλ (ζ )) = τλ (Tλ

0

n(Tλ0 (ζ ))

(ζ )) = τλ (Tλ

0

(Tλ0 (ζ ))), ∀λ ∈ B(λ0 , R).

Using Claim 1 once again but this time with Tλ0 (ζ ) instead of ζ , we get n(Tλ0 (ζ ))

Tλ

n(Tλ0 (ζ ))

(τλ ∘ Tλ0 (ζ )) = τλ (Tλ

0

∀λ ∈ B(λ0 , R).

(Tλ0 (ζ ))),

In addition, since the function B(λ0 , R) ∋ λ 󳨃󳨀→ Tλ ∘ τλ (ζ ) ∈ ℂ is continuous, there ̂ ζ ∈ (0, R] such that exists R Tλ ∘ τλ (ζ ) ∈ B(Tλ0 ∘ τλ0 (ζ ), δTλ

0

∘τλ0 (ζ ) )

= B(Tλ0 (ζ ), δTλ

0

(ζ ) ),

̂ ζ ). ∀λ ∈ B(λ0 , R

It thereby follows from the uniqueness part of Claim 1 for Tλ0 (ζ ) that Tλ ∘ τλ (ζ ) = τλ ∘ Tλ0 (ζ ),

̂ ζ ). ∀λ ∈ B(λ0 , R

Since both functions B(λ0 , R) ∋ λ 󳨃󳨀→ Tλ ∘ τλ (ζ ) ∈ ℂ and B(λ0 , R) ∋ λ 󳨃󳨀→ τλ ∘ Tλ0 (ζ ) ∈ ℂ are holomorphic, we conclude that Tλ ∘ τλ (ζ ) = τλ ∘ Tλ0 (ζ ),

∞

(P) \ P. ∀λ ∈ B(λ0 , R), ∀ζ ∈ ⋃ Tλ−n 0 n=0

(16.43)

Now, suppose that z = ξ ∈ P. Then both points ξ and Tλ0 (ξ ) are periodic with prime period p under the map Tλ0 . Hence, Tλp (τλ (Tλ0 (ξ ))) = τλ (Tλ0 (ξ ))

and Tλp (Tλ (τλ (ξ ))) = Tλ (Tλp (τλ (ξ ))) = Tλ (τλ (ξ )).

16.4 Real-analytic dependence of Hausdorff dimension of repellers in ℂ

|

631

So, in a similar way to the previous case, the uniqueness part of the implicit function theorem (see (16.26)–(16.28) with Tλ0 (ξ ) in lieu of ξ ) and the continuity of the function B(λ0 , R) ∋ λ 󳨃→ Tλ ∘ τλ (z) ∈ ℂ lead to the conclusion that Tλ ∘ τλ (ξ ) = τλ ∘ Tλ0 (ξ ). Along with (16.43), this completes the proof of Claim 2. ◼ −n Since both sets P and Tλ−10 (P) are finite and for every z ∈ ⋃∞ n=0 Tλ0 (P) the function ̂ ∈ (0, R] such that B(λ0 , R) ∋ λ 󳨃→ τλ (z) is continuous, there exists R

max

w∈Tλ−1 (P) 0

󵄨 󵄨 1 sup 󵄨󵄨󵄨τλ (w) − τλ0 (w)󵄨󵄨󵄨 < min{|ξ − ζ | : ξ ∈ P, ζ ∈ Tλ−10 (P) \ P}. 2 ̂

λ∈B(λ0 ,R)

(16.44)

̂ the function Claim 3. For every λ ∈ B(λ0 , R), ∞

(P) ∋ z 󳨃󳨀→ τλ (z) ∈ ℂ ⋃ Tλ−n 0

n=0

is injective. Proof of Claim 3. Suppose that τγ (z) = τγ (w)

(16.45)

̂ and z, w ∈ ⋃∞ T −n (P) with z ≠ w. We will consider several for some γ ∈ B(λ0 , R) n=0 λ0 cases. Assume first that z, w ∈ P, i. e., n(z) = n(w) = 0. Since τλ0 (z) = z ≠ w = τλ0 (w), we may assume that τλ (z) ≠ τλ (w),

∀λ ∈ B(λ0 , |γ − λ0 |).

(16.46)

Let y = τγ (z) = τγ (w). By Claim 2, Tλp (τλ (z)) = τλ (Tλp (z)) = τλ (z) and Tλp (τλ (w)) = τλ (w) 0 for every λ ∈ B(λ0 , R). In particular, Tγp (y) = y. It follows from the uniqueness part of the implicit function theorem that τλ (z) = τλ (w) for every λ belonging simultaneously to B(λ0 , R) and some sufficiently small neighborhood of γ. This, however, contradicts (16.46) and we are done in this case. −n Now assume that z ∈ P and w ∈ ⋃∞ n=0 Tλ0 (P) \ P, i. e., n(z) = 0

and

n(w) ≥ 1.

Because we are seeking a contradiction, we may assume without loss of generality that z, w are such that n(w) is the least n ∈ ℕ for which (16.45) holds. By Claim 2, τγ (Tλ0 (w)) = Tγ (τγ (w)) = Tγ (τγ (z)) = τγ (Tλ0 (z)).

(16.47)

632 | 16 Conformal expanding repellers −n But Tλ0 (z) ∈ P while Tλ0 (w) ∈ ⋃∞ n=0 Tλ0 (P). So (16.47) states that the pair Tλ0 (z), Tλ0 (w) satisfies (16.45). Since n(Tλ0 (w)) = n(w) − 1, we surmise from the minimality of n(w) that n(Tλ0 (w)) = 0, i. e.,

n(w) = 1.

(16.48)

Hence, w ∈ Tλ−10 (P) \ P. Since z ∈ P ⊆ Tλ−10 (P), it follows from (16.44) that 1 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨τγ (w) − τγ (z)󵄨󵄨󵄨 ≥ |z − w| − 󵄨󵄨󵄨τγ (z) − z 󵄨󵄨󵄨 − 󵄨󵄨󵄨τγ (w) − w󵄨󵄨󵄨 > |z − w| − 2 ⋅ |z − w| = 0. 2 This contradicts (16.45) and we are done in this case, too. Finally, assume that min{n(z), n(w)} ≥ 1. We may assume without loss of generality that n(z) ≤ n(w). By Claim 2, τγ ∘ Tλn(z) (z) = Tγn(z) ∘ τγ (z) = Tγn(z) ∘ τγ (w) = τγ ∘ Tλn(z) (w). 0

0

But n(Tλn(z) (z)) = 0, whence it follows from the previous case that 0

Tλn(z) (z) = Tλn(z) (w). 0

0

Since Tλn(z) (z) ∈ P, this implies that n(z) = n(w). As τγ (z) = τγ (w), it ensues from 0 Claim 1 that there exists δγ > 0 such that both functions B(γ, δγ ) ∋ λ 󳨃󳨀→ τλ (z), τλ (w) are well defined and τλ (z) = τλ (w) for all λ ∈ B(γ, δγ ). Since these two functions are ̂ In particular, z = τλ (z) = analytic, we conclude that τλ (z) = τλ (w) for all λ ∈ B(λ0 , R). 0 τλ0 (w) = w. The proof of Claim 3 is complete. ◼ We have thus proved that the family of maps ∞

{τλ : ⋃ Tλ−n (P) → ℂ} 0 n=0

̂ λ∈B(λ0 ,R)

is a (shifted and rescaled) holomorphic motion in the sense of Definition 16.4.3. As Tλ0 : Jλ0 → Jλ0 is transitive, open and distance expanding, it is very strongly transitive by Lemma 4.2.10 and Definition 1.5.14 affirms that the backward orbit of any point in −n Jλ0 is dense in Jλ0 . So Jλ0 = ⋃∞ n=0 Tλ0 (P). It follows from the λ-lemma (Theorem 16.4.4) ̂ have quasisymmetric extensions to Jλ . Furthermore, that all the maps τλ , λ ∈ B(λ0 , R), 0

16.4 Real-analytic dependence of Hausdorff dimension of repellers in ℂ

|

633

all of them are injective and enjoy properties (a)–(d) of Theorem 16.4.4. We keep for them the notation τλ : Jλ0 → ℂ. Property (a) of the current theorem follows immediately from property (a) of Theô are continuous per item (b) of rem 16.4.4. As the maps τλ : Jλ0 → ℂ, λ ∈ B(λ0 , R), Theorem 16.4.4, property (c) of the current theorem is a direct consequence of (16.42), −n the density of ⋃∞ n=0 Tλ0 (P) in Jλ0 , and the continuity of the maps τλ . Property (b) follows from item (d) of Theorem 16.4.4 and Remark 16.4.6. Property (d) comes directly from item (b) of the λ-lemma. ̂ := B(λ0 , R) ̂ define Finally, for every λ ∈ Λ Jλ := τλ (Jλ0 ). As the image of a compact set under a continuous map, this set is compact. ̂ is a conformal expanding repeller. Recall that U = B(Jλ , κ) for some κ (Jλ , Tλ , U) 0 small enough that (16.18) and (16.24) hold. Then a number 0 < α < κ/3 is selected so that every infinite α-pseudo-orbit in Jλ0 is (κ/3)-shadowed by a unique point in Jλ0 . ̂ = B(Jλ , κ̂), where 0 < κ̂ < α/(3Γ) is chosen so that (16.25) is valid. Finally, U 0 ̂ This Given that Tλ : U → ℂ is holomorphic, it is C 1+ϵ (U) and conformal on U ⊇ U. establishes properties (a) and (b) of Definition 16.2.1. Since τλ : Jλ0 → Jλ is a continuous bijection from a compact set to a Hausdorff space, it is a homeomorphism. Because Tλ0 (Jλ0 ) = Jλ0 , using property (c) proved above, it is easy to see that Tλ (Jλ ) = Jλ . So property (c) of Definition 16.2.1 holds. This also implies that τλ is a topological conjugacy between Tλ0 : Jλ0 → Jλ0 and Tλ : Jλ → Jλ . The continuity of the function (λ, z) 󳨃→ τλ (z) and (16.24) mean that Tλ : Jλ → Jλ is expanding, i. e., property (d) of Definition 16.2.1 is satisfied, provided λ is close enough to λ0 . Observe that the expansion is uniform in λ. −n ̂ Given that Tλ (Jλ ) ⊆ Jλ , it immediately follows that ⋂∞ n=0 Tλ (U) ⊇ Jλ . For the op∞ −n ̂ n ̂ := B(Jλ , κ̂) for all posite inclusion, let z0 ∈ ⋂n=0 Tλ (U). This means that Tλ (z0 ) ∈ U 0 n ≥ 0. That is, for every n ≥ 0 there exists zn ∈ Jλ0 such that Tλn (z0 ) ∈ B(zn , κ̂),

and thus

κ α 󵄨󵄨 n 󵄨 < . 󵄨󵄨Tλ (z0 ) − zn 󵄨󵄨󵄨 < κ̂ < 3Γ 9Γ

It ensues from this, (16.18) and (16.25) that 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 n+1 󵄨 󵄨 󵄨 n+1 󵄨󵄨zn+1 − Tλ0 (zn )󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨zn+1 − Tλ (z0 )󵄨󵄨󵄨 + 󵄨󵄨󵄨Tλ (z0 ) − Tλ (zn )󵄨󵄨󵄨 + 󵄨󵄨󵄨Tλ (zn ) − Tλ0 (zn )󵄨󵄨󵄨 α α 󵄨 󵄨 α < + Γ󵄨󵄨󵄨Tλn (z0 ) − zn 󵄨󵄨󵄨 + < 3 ⋅ = α. 3Γ 3 3

(16.49)

634 | 16 Conformal expanding repellers So the sequence (zn )∞ n=0 is an α-pseudo-orbit for Tλ0 : Jλ0 → Jλ0 . By definition of α, there exists a (unique) point w ∈ Jλ0 which (κ/3)-shadows this pseudo-orbit, i. e., 󵄨󵄨 n 󵄨 κ 󵄨󵄨Tλ0 (w) − zn 󵄨󵄨󵄨 < , 3

∀n ≥ 0.

(16.50)

Shrinking Λ if necessary, the continuity of the function (λ, z) 󳨃→ τλ (z) guarantees that 󵄨 󵄨󵄨 󵄨󵄨τλ (z) − τλ0 (z)󵄨󵄨󵄨 < κ̂,

∀z ∈ Jλ0 , ∀λ ∈ Λ.

(16.51)

It follows from (16.49)–(16.51) that 󵄨󵄨 n 󵄨 󵄨 󵄨 n n n 󵄨󵄨Tλ (τλ (w)) − Tλ (z0 )󵄨󵄨󵄨 = 󵄨󵄨󵄨τλ (Tλ0 (w)) − Tλ (z0 )󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨τλ (Tλn0 (w)) − τλ0 (Tλn0 (w))󵄨󵄨󵄨 + 󵄨󵄨󵄨Tλn0 (w) − zn 󵄨󵄨󵄨 + 󵄨󵄨󵄨zn − Tλn (z0 )󵄨󵄨󵄨 κ κ < κ̂ + + κ̂ = 2κ̂ + < κ 3 3 ̂ ⊆ U for all n ≥ 0. Moreover, Tλn (τλ (w)) and Tλn (z0 ) lie both in U. Indeed, Tλn (z0 ) ∈ U 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 κ n while 󵄨󵄨Tλ (τλ (w)) − zn 󵄨󵄨 ≤ 󵄨󵄨Tλ (τλ (w)) − Tλ (z0 )󵄨󵄨 + 󵄨󵄨Tλ (z0 ) − zn 󵄨󵄨 < 2κ̂ + 3 + κ̂ < κ and zn ∈ Jλ0 . Hence, Tλn (τλ (w)) ∈ B(Jλ0 , κ) = U. Per (16.24), we then know that 󵄨󵄨 n+1 󵄨 󵄨 n 󵄨 n+1 n 󵄨󵄨Tλ (τλ (w)) − Tλ (z0 )󵄨󵄨󵄨 ≥ γ󵄨󵄨󵄨Tλ (τλ (w)) − Tλ (z0 )󵄨󵄨󵄨,

∀n ≥ 0.

By recurrence, 󵄨󵄨 󵄨 󵄨 −n 󵄨 n n −n 󵄨󵄨τλ (w) − z0 󵄨󵄨󵄨 ≤ γ 󵄨󵄨󵄨Tλ (τλ (w)) − Tλ (z0 )󵄨󵄨󵄨 ≤ κγ ,

∀n ≥ 0.

−n ̂ Thus, |τλ (w) − z0 | = 0. So z0 = τλ (w) ∈ τλ (Jλ0 ) = Jλ . This shows that ⋂∞ n=0 Tλ (U) ⊆ Jλ . Therefore, property (e) of Definition 16.2.1 carries over to Tλ . Finally, recall that transitivity is a topological invariant (see Remark 1.5.7). Since τλ is a topological conjugacy between Tλ0 : Jλ0 → Jλ0 and Tλ : Jλ → Jλ and since Tλ0 : Jλ0 → Jλ0 is transitive, it follows that Tλ : Jλ → Jλ is transitive, too. So property (f) of Definition 16.2.1 holds.

We need one more technical result, which is a well-known fact in complex analysis. The proof we provide is taken from Lemma 9.1 in [84]. Fix an integer d ∈ ℕ. Embed ℂd into ℂ2d by the formula z = (z1 , z2 , . . . , zd ) 󳨃󳨀→ z̃ := (Re z1 , Im z1 , Re z2 , Im z2 , . . . , Re zd , Im zd ) ∈ ℝ2d ⊆ ℂ2d . For every z ∈ ℂd and every r > 0, denote by Dℂd (z, r) := ∏dk=1 Bℂ (zk , r) the d-dimensional polydisk in ℂd centered at z and of “radius” r (not to be confused with the open ball Bℂd (z, r) in ℂd centered at z and of radius r). Moreover, |z|2 := ∑dk=1 |zk |2 .

16.4 Real-analytic dependence of Hausdorff dimension of repellers in ℂ

|

635

Lemma 16.4.8. For every M ≥ 0, every R > 0, every λ0 ∈ ℂd , and every analytic function ψ : Dℂd (λ0 , R) → ℂ whose modulus is bounded above by M, there exists a unique analytic function ̃ : D 2d (λ̃0 , R/4) → ℂ ψ ℂ whose modulus is bounded above by 4d M and whose restriction to the polydisk Dℂd (λ0 , R/4) ≡ D d? (λ0 , R/4) = D 2d (λ̃0 , R/4) ∩ ℝ2d coincides with Re ψ, the real part of ψ. ℂ

ℂ

Proof. Denote by ℕ0 the set of all nonnegative integers. Write the analytic function ψ : Dℂd (λ0 , R) → ℂ in the form of its Taylor series expansion ψ(λ1 , λ2 , . . . , λd ) = ∑ aα (λ1 − λ10 ) 1 (λ2 − λ20 ) α

α2

α∈ℕd0

⋅ ⋅ ⋅ (λd − λd0 ) d . α

By Cauchy’s estimates (Theorem A.3.2 in Appendix A), M

|aα | ≤

R‖α‖1

,

∀α ∈ ℕd0 ,

(16.52)

where ‖α‖1 := ∑dj=1 |αj | = ∑dj=1 αj . Then α1

α1 p α −p )(Re λ1 − Re λ10 ) (Im λ1 − Im λ10 ) 1 iα1 −p ) p

Re ψ(λ1 , λ2 , . . . , λd ) = ∑ Re [aα ( ∑ ( p=0

α∈ℕd0

α2

α2 p α −p )(Re λ2 − Re λ20 ) (Im λ2 − Im λ20 ) 2 iα2 −p ) ⋅ ⋅ ⋅ p

⋅ (∑ ( p=0 αd

αd p α −p )(Re λd − Re λd0 ) (Im λd − Im λd0 ) d iαd −p )] p

⋅ (∑ ( p=0 d

βj(1) + βj(2)

j=1

βj(1)

d

βj(1) + βj(2)

j=1

βj(1)

= ∑ Re[aβ̂ ∏ ( β∈ℕ2d 0

= ∑ Re[aβ̂ ∏ ( β∈ℕ2d 0

)iβj (Re λj − Re λj0 ) (2)

βj(1)

(Im λj − Im λj0 )

β(1)

βj(2)

)iβj ](Re λj − Re λj0 ) j (Im λj − Im λj0 ) (2)

]

βj(2)

,

(1) (2) (1) (2) (1) (2) (1) ̂ where β ∈ ℕ2d 0 is written in the form (β1 , β1 , β2 , β2 , . . . , βd , βd ) and β = (β1 + ̂ . Set β1(2) , β2(1) + β2(2) , . . . , βd(1) + βd(2) ) ∈ ℕd0 . Note that ‖β‖1 = ‖β‖ 1 d

βj(1) + βj(2)

j=1

βj(1)

cβ = Re[aβ̂ ∏ (

)iβj ]. (2)

636 | 16 Conformal expanding repellers Using (16.52), we get βj(1) + βj(2)

d

|cβ | ≤ |aβ̂ | ∏ ( j=1

βj(1)

d

) ≤ MR−‖β‖1 ∏ 2βj ̂

(1)

+βj(2)

j=1

= MR−‖β‖1 2‖β‖1 .

Thus, the formula d

0 βj 0 βj ̃ ,z ,...,z ψ(z 1 2 2d−1 , z2d ) = ∑ cβ ∏(z2j−1 − Re λj ) (z2j − Im λj ) β∈ℕ2d 0

(1)

(2)

j=1

defines an analytic function on Dℂ2d (λ̃0 , R/4) and 󵄨󵄨̃ 󵄨 d 󵄨󵄨ψ(z1 , z2 , . . . , z2d−1 , z2d )󵄨󵄨󵄨 ≤ 4 M. ̃ Obviously, ψ| Dℂd (λ0 ,R/4) = Re ψ|Dℂd (λ0 ,R/4) , and we are done with the existence part of our lemma. The uniqueness part follows from the fact that the set where any two holomorphic functions are equal is a complex manifold and the smallest complex manifold containing ℝ2d is ℂ2d . We can now prove the real analyticity of the pressure function. Theorem 16.4.9. If {Tλ : U → ℂ}λ∈Λ is an analytic perturbation of a conformal expanding repeller (Jλ0 , Tλ0 , U) in ℂ, then the pressure function ̂ × ℝ ∋ (λ, t) 󳨃󳨀→ P(λ, t) := P(Tλ , −t log󵄨󵄨󵄨T ′ 󵄨󵄨󵄨) = P(Tλ , −t log󵄨󵄨󵄨T ′ 󵄨󵄨󵄨 ∘ τλ ) ∈ ℝ Λ 󵄨 λ󵄨 󵄨 λ󵄨 0

(16.53)

̂ and τλ arise from Theorem 16.4.7. is real analytic, where Λ Proof. By virtue of Theorem 16.4.7, it suffices to prove real analyticity on some open ̂ × ℝ of every point (λ0 , s), s ∈ ℝ. By Theorem 16.4.7(a), for every neighborhood in Λ z ∈ Jλ0 the function ̂ ∋ λ 󳨃󳨀→ Ψz (λ) := log󵄨󵄨󵄨T ′ (τλ (z))󵄨󵄨󵄨 − log󵄨󵄨󵄨T ′ (z)󵄨󵄨󵄨 Λ 󵄨 λ 󵄨 󵄨 λ0 󵄨 ̂ is harmonic with Ψz (λ0 ) = 0 (per Theorem 16.4.7(b)). Fix r > 0 such that B(λ0 , 8r) ⊆ Λ. The continuity of (λ, z) 󳨃→ τλ (z) (per Theorem 16.4.7(d)) combined with the continuity of (λ, z) 󳨃→ Tλ′ (z) (as {Tλ }λ∈Λ is an analytic perturbation of Tλ0 ) implies that 󵄨 󵄨 M := sup{󵄨󵄨󵄨Ψz (λ)󵄨󵄨󵄨 : (z, λ) ∈ Jλ0 × B(λ0 , 4r)} < ∞. ̃ z : B 2 (λ̃0 , r) → ℂ By Lemma 16.4.8, each function Ψz has an holomorphic extension Ψ ℂ such that ̃ ̃ ̃ ̃ 󵄨󵄨 ̃ := sup{󵄨󵄨󵄨Ψ M 󵄨 z (λ)󵄨󵄨 : (z, λ) ∈ Jλ0 × Bℂ2 (λ0 , r)} ≤ 4M < ∞.

16.4 Real-analytic dependence of Hausdorff dimension of repellers in ℂ

|

637

Since all the functions τλ , λ ∈ Bℂ (λ0 , 4r), are Hölder continuous with a common Hölder exponent, say α, and common Hölder norm for the exponent α (see Remark 16.4.6), an application of Cauchy’s estimates (Theorem A.3.2) affirms that for all λ̃ ∈ Bℂ2 (λ̃0 , r) ̃ is Hölder continuous with exponent α and the corresponding ̃ z (λ) the function z 󳨃→ Ψ ̃α . Hölder norms are uniformly bounded by, say, M Given any bounded open set V ⊆ ℂ, consider the two-complex-parameter family of potentials φλ,w ̃ : Jλ0 → ℂ defined by 󵄨󵄨 ′ 󵄨󵄨 ̃ ̃ φλ,w ̃ (z) := −w[Ψz (λ) + log󵄨󵄨Tλ0 (z)󵄨󵄨],

̃ w) ∈ B 2 (λ̃ , r) × V. (λ, 0 ℂ

Recall that Tλ0 ∈ C 1+ϵ (U) for some ϵ > 0 by Definition 16.2.1(a) of a repeller. Then 󵄨 󵄨 󵄨 󵄨 󵄩 󵄩 Tλ′0 , 󵄨󵄨󵄨Tλ′0 󵄨󵄨󵄨 and log󵄨󵄨󵄨Tλ′0 󵄨󵄨󵄨 are Hölder continuous with exponent ϵ. Let L := 󵄩󵄩󵄩log |Tλ′0 |󵄩󵄩󵄩ϵ . ̃α and L accordingly, the Replacing α and ϵ by min{α, ϵ} and adjusting the constants M above family satisfies 󵄩 󵄩󵄩 ̃ ̃ ̃ sup{󵄩󵄩󵄩φλ,w ̃ 󵄩 󵄩α : (λ, w) ∈ Bℂ2 (λ0 , r) × V} ≤ |V|(Mα + L) < ∞, where |V| := sup{|w| : w ∈ V}. Moreover, for every z ∈ Jλ0 the function Bℂ2 (λ̃0 , r) × V ∋ ̃ w) 󳨃󳨀→ φ̃ (z) ∈ ℂ is jointly holomorphic since it is separately holomorphic in each (λ, λ,w

variable λ̃ and w. So the hypotheses of Theorem 13.10.2 are fulfilled and the function ̃ t) 󳨃󳨀→ ℒ ∈ L(H (J )) Bℂ2 (λ̃0 , r) × V ∋ (λ, α λ0 φ̃λ,t

is holomorphic. For all (λ, t) ∈ Bℂ (λ0 , r) × (V ∩ ℝ), observe that 󵄨󵄨 ′ 󵄨󵄨 󵄨󵄨 ′ 󵄨󵄨 󵄨󵄨 ′ 󵄨󵄨 ̃ ̃ φλ,t ̃ (z) := −t[Ψz (λ) + log󵄨󵄨Tλ0 (z)󵄨󵄨]= −t[Ψz (λ) + log󵄨󵄨Tλ0 (z)󵄨󵄨] = −t log󵄨󵄨Tλ ∘ τλ (z)󵄨󵄨 ∈ ℝ. Thus all the hypotheses of Theorem 13.10.3 are satisfied, too, and hence the function 󵄨 󵄨 Bℂ (λ0 , r) × (V ∩ ℝ) ∋ (λ, t) 󳨃󳨀→ P(Tλ0 , −t log󵄨󵄨󵄨Tλ′ 󵄨󵄨󵄨 ∘ τλ ) ∈ ℝ ̂ and of V ⊆ ℂ as a bounded open set, is real analytic. Given the arbitrariness of λ0 ∈ Λ the localness of real analyticity allows us to infer that ̂ × ℝ ∋ (λ, t) 󳨃󳨀→ P(Tλ , −t log󵄨󵄨󵄨T ′ 󵄨󵄨󵄨 ∘ τλ ) ∈ ℝ Λ 󵄨 λ󵄨 0 is real analytic. By Theorem 16.4.7(c) and Corollary 11.1.21, it turns out that 󵄨 󵄨 󵄨 󵄨 P(λ, t) := P(Tλ , −t log󵄨󵄨󵄨Tλ′ 󵄨󵄨󵄨) = P(Tλ0 , −t log󵄨󵄨󵄨Tλ′ 󵄨󵄨󵄨 ∘ τλ ) and the result follows.

638 | 16 Conformal expanding repellers We now identify the partial derivative with respect to t of the pressure function ̂ × ℝ ∋ (λ, t) 󳨃󳨀→ P(λ, t). Λ Theorem 16.4.10. If {Tλ : U → ℂ}λ∈Λ is an analytic perturbation of a conformal expanding repeller (Jλ0 , Tλ0 , U) in ℂ, then 𝜕 󵄨 󵄨 P(λ, t) = − ∫ log󵄨󵄨󵄨Tλ′ 󵄨󵄨󵄨 ∘ τλ dμλ,t , 𝜕t

̂ × ℝ, ∀(λ, t) ∈ Λ

Jλ0

where μλ,t is the unique equilibrium state and unique Tλ0 -invariant Gibbs state for the potential φλ,t = −t log |Tλ′ | ∘ τλ . Proof. By virtue of Theorems 16.4.7 and 16.4.9 as well as Proposition 13.7.12, for every ̂ × ℝ the potential φλ,t = −t log |T ′ | ∘ τλ admits a unique equilibrium state (λ, t) ∈ Λ λ and unique Tλ0 -invariant Gibbs state μλ,t . Let s ∈ ℝ. By the variational principle (Theorem 12.1.1), we deduce that 󵄨 󵄨 P(λ, s) = P(Tλ0 , −s log󵄨󵄨󵄨Tλ′ 󵄨󵄨󵄨 ∘ τλ ) 󵄨 󵄨 ≥ hμλ,t (Tλ0 ) + ∫ −s log󵄨󵄨󵄨Tλ′ 󵄨󵄨󵄨 ∘ τλ dμλ,t Jλ0

󵄨 󵄨 󵄨 󵄨 = hμλ,t (Tλ0 ) + ∫ −t log󵄨󵄨󵄨Tλ′ 󵄨󵄨󵄨 ∘ τλ dμλ,t + (s − t) ∫ − log󵄨󵄨󵄨Tλ′ 󵄨󵄨󵄨 ∘ τλ dμλ,t Jλ0

Jλ0

󵄨 󵄨 󵄨 󵄨 = P(Tλ0 , −t log󵄨󵄨󵄨Tλ′ 󵄨󵄨󵄨 ∘ τλ ) + (s − t) ∫ − log󵄨󵄨󵄨Tλ′ 󵄨󵄨󵄨 ∘ τλ dμλ,t Jλ0

󵄨 󵄨 = P(λ, t) + (s − t) ∫ − log󵄨󵄨󵄨Tλ′ 󵄨󵄨󵄨 ∘ τλ dμλ,t . Jλ0

On one hand, P(λ, s) − P(λ, t) 𝜕 󵄨 󵄨 P(λ, t) = lim+ ≥ − ∫ log󵄨󵄨󵄨Tλ′ 󵄨󵄨󵄨 ∘ τλ dμλ,t . 𝜕t s−t s→t Jλ0

On the other hand, 𝜕 P(λ, s) − P(λ, t) 󵄨 󵄨 P(λ, t) = lim− ≤ − ∫ log󵄨󵄨󵄨Tλ′ 󵄨󵄨󵄨 ∘ τλ dμλ,t . 𝜕t s−t s→t Jλ0

In conclusion,

𝜕 P(λ, t) 𝜕t

= − ∫J log |Tλ′ | ∘ τλ dμλ,t . λ0

Finally, we demonstrate the real analyticity of the Hausdorff dimension function.

16.4 Real-analytic dependence of Hausdorff dimension of repellers in ℂ

|

639

Theorem 16.4.11. If {Tλ : U → ℂ}λ∈Λ is an analytic perturbation of a conformal expanding repeller (Jλ0 , Tλ0 , U) in ℂ, then the Hausdorff dimension function ̂ ∋ λ 󳨃󳨀→ HD(Jλ ) ∈ (0, 2) Λ is real analytic. Proof. Denote hλ := HD(Jλ ). By Bowen’s formula (Theorem 16.3.2) and formula (16.53), we know that 󵄨 󵄨 P(Tλ0 , −hλ log󵄨󵄨󵄨Tλ′ 󵄨󵄨󵄨 ∘ τλ ) = 0.

(16.54)

By Theorem 16.4.10, we further have that 𝜕 󵄨 󵄨 󵄨 󵄨 P(Tλ0 , −t log󵄨󵄨󵄨Tλ′ 󵄨󵄨󵄨 ∘ τλ ) = − ∫ log󵄨󵄨󵄨Tλ′ 󵄨󵄨󵄨 ∘ τλ dμλ,t < 0. 𝜕t Jλ0

It follows from the implicit function theorem, applied to the equation 󵄨 󵄨 P(Tλ0 , −t log󵄨󵄨󵄨Tλ′ 󵄨󵄨󵄨 ∘ τλ ) = 0, that hλ is the unique real-analytic function satisfying equation (16.54) on some sufficiently small neighborhood of λ0 . As real analyticity is a local property, the proof is complete. Remark 16.4.12. Let hλ := HD(Jλ ). One can wonder whether a result analogous to Theorem 16.4.11 holds on a deeper level, i. e., whether the function ̂ ∋ λ 󳨃󳨀→ Hh (Jλ ) Λ λ

(16.55)

is real analytic. This issue is very subtle. It is easy to see that the function ̂ ∋ λ 󳨃󳨀→ mh ∈ M(U) Λ λ is continuous, as mhλ , the hλ -conformal measure for the map Tλ : Jλ → Jλ , is also the normalized hλ -dimensional Hausdorff measure on Jλ . Indeed, one can just notice that a weak∗ limit of conformal measures is a conformal measure and apply the uniqueness of conformal measures. The much more involved continuity of the function in (16.55) was established in [128]. In fact, that continuity was proved therein in a more general setting. This work stemmed from the seminal paper of Olsen [89]. It was furthermore shown in [128] that the function in (16.55) is Hölder continuous. However, the question of its real analyticity remains open; even Lipschitz continuity is open. Nevertheless, piecewise real analyticity was proved in [128] in the context of repellers contained in the real line.

640 | 16 Conformal expanding repellers

16.5 Dimensions of measures, Lyapunov exponents and measure-theoretic entropy To describe the dimensions of invariant measures, we now introduce the concept of Lyapunov exponent. Definition 16.5.1. Let (X, U, T) be a conformal expanding repeller. For any measure μ ∈ M(T), i. e., any T-invariant Borel probability measure μ, the quantity 󵄨 󵄨 χμ (T) := ∫ log󵄨󵄨󵄨T ′ 󵄨󵄨󵄨 dμ X

is called the (characteristic) Lyapunov exponent of μ. The Lyapunov exponent of any measure μ is positive since χμ (T) ≥ log λ > 0 per Definition 16.2.1(d). Conformal expanding repellers exhibit a beautiful property: each of their ergodic invariant Borel probability measures has the same pointwise dimension almost everywhere, and this dimension is the ratio of the measure-theoretic entropy and the Lyapunov exponent of that measure. Theorem 16.5.2 (Volume lemma for ergodic invariant measures). Let (X, U, T) be a conformal expanding repeller and μ an ergodic T-invariant Borel probability measure. Then for μ-a. e. x ∈ X, dμ (x) =

hμ (T) χμ (T)

= HD(μ) = PD(μ),

where dμ (x) is the pointwise dimension of μ at x and hμ (T) is the measure-theoretic entropy of T with respect to μ. Moreover, as a geometric measure, the unique equilibrium state and unique T-invariant Gibbs state μHD(X) associated with the Hölder continuous potential φHD(X) (x) := −HD(X) log |T ′ (x)|, has the property that for all x ∈ X, dμHD(X) (x) =

hμHD(X) (T) χμHD(X) (T)

=

−μHD(X) (φHD(X) ) χμHD(X) (T)

= HD(X) = PD(X) = BD(X) = HD(μHD(X) ) = PD(μHD(X) ). Proof. Let x ∈ X. Let also ξ , 0 < r < ξ , k = k(x, r), and ℓ = ℓ(x, r) be as in the proof of Bowen’s formula (Theorem 16.3.2). For every n ≥ 0, we then have that Bn+1 (x, ξ ) = Tx−n (B(T n (x), ξ )),

(16.56)

where Bn+1 (x, ξ ) := ⋂nj=0 T −j (B(T j (x), ξ )) denotes the dynamical (n + 1, ξ )-ball at x (cf. Section 7.3). Indeed, the inclusion Tx−n (B(T n (x), ξ )) ⊆ Bn+1 (x, ξ ) is immediate while

16.5 Dimensions of measures, Lyapunov exponents and measure-theoretic entropy | 641

Bn+1 (x, ξ ) ⊆ Tx−n (B(T n (x), ξ )) follows by induction. By definition (cf. (16.8)–(16.9)), Tx−ℓ(x,r) (B(T ℓ(x,r) (x), ξ )) ⊆ B(x, r) ⊆ Tx−k(x,r) (B(T k(x,r) (x), ξ ))

(16.57)

lim k(x, r) = lim ℓ(x, r) = ∞.

(16.58)

and r→0

r→0

From this point on, we use the lighter notation k for k(x, r) and ℓ for ℓ(x, r). Combining (16.56)–(16.57) yields Bℓ+1 (x, ξ ) ⊆ B(x, r) ⊆ Bk+1 (x, ξ ). So μ(Bℓ+1 (x, ξ )) ≤ μ(B(x, r)) ≤ μ(Bk+1 (x, ξ )). Taking logarithms on both sides, multiplying by −1 and dividing by − log r, we get − log μ(B(x, r)) − log μ(Bℓ+1 (x, ξ )) ℓ + 1 − log μ(Bk+1 (x, ξ )) k + 1 ⋅ ≤ ≤ ⋅ . (16.59) k+1 − log r − log r ℓ+1 − log r ̃ ≥ 1 such that By Corollary 16.2.10, we know that there is K ̃ + log󵄨󵄨󵄨(T ℓ )′ (x)󵄨󵄨󵄨 ≤ − log r ≤ log K ̃ + log󵄨󵄨󵄨(T k )′ (x)󵄨󵄨󵄨. − log K 󵄨 󵄨 󵄨 󵄨 Inserting this into (16.59), we deduce that log μ(B(x, r)) − log μ(Bℓ+1 (x, ξ )) ℓ+1 ≤ ⋅ ̃ + log |(T ℓ )′ (x)| log r ℓ+1 − log K − log μ(Bℓ+1 (x, ξ )) 1 ⋅ ≤ ̃ − log K log |(T ℓ )′ (x)| ℓ+1 + ℓ+1

(16.60)

ℓ+1

and log μ(B(x, r)) − log μ(Bk+1 (x, ξ )) k+1 ≥ ⋅ ̃ + log |(T k )′ (x)| log r k+1 log K − log μ(Bk+1 (x, ξ )) 1 ≥ ⋅ . ̃ log K log |(T k )′ (x)| k+1 + k+1 k+1

(16.61)

Since μ is ergodic, we can apply the ergodic case of Birkhoff’s ergodic theorem (Corollary 8.2.14) to extract a Borel set X1 ⊆ X such that μ(X1 ) = 1 and 1 1 n−1 󵄨 ′ 󵄨 󵄨 󵄨 󵄨 󵄨 log󵄨󵄨󵄨(T n ) (x)󵄨󵄨󵄨 = lim ∑ log󵄨󵄨󵄨T ′ (T i (x))󵄨󵄨󵄨 = ∫ log󵄨󵄨󵄨T ′ 󵄨󵄨󵄨 dμ = χμ (T), ∀x ∈ X1 . (16.62) n→∞ n n→∞ n i=0 lim

X

642 | 16 Conformal expanding repellers Being distance expanding, the map T : X → X is expansive. It follows from Theorem 9.6.4 (Brin–Katok local entropy formula for expansive maps) that there exists a Borel set X2 ⊆ X such that μ(X2 ) = 1 and lim

n→∞

− log μ(Bn (x, ξ )) = hμ (T), n

∀x ∈ X2 .

Inserting this and (16.62) into (16.60)–(16.61) and invoking (16.58), we get for every x ∈ X1 ∩ X2 that lim sup r→0

log μ(B(x, r)) hμ (T) log μ(B(x, r)) ≤ ≤ lim inf . r→0 log r χμ (T) log r

Hence, log μ(B(x, r)) hμ (T) = . r→0 log r χμ (T)

dμ (x) := lim

Since μ(X1 ∩ X2 ) = 1, the proof of the first equality in the first part of Theorem 16.5.2 is complete. The second and third equalities in the first part are a direct application of Corollary 15.6.11. The second part of the statement follows directly from Bowen’s formula (Theorem 16.3.2), the first part of the current theorem, as well as the characteristic of μHD(X) of being an equilibrium state, for then hμHD(X) (T) = P(T, φHD(X) ) − ∫ φHD(X) dμHD(X) = 0 − μHD(X) (φHD(X) ). X

Remark 16.5.3. Recall that the almost everywhere existence and constantness of the pointwise dimension of a measure is usually referred to as the exact dimensionality of the measure (cf. Corollary 15.6.11). There is a more general version of the volume lemma for nonnecessarily ergodic but T-invariant probability measures, though it is much more complicated to derive. We will need several intermediate results. Lemma 16.5.4. Every increasing function k : I → ℝ defined on a compact interval I ⊆ ℝ is Lipschitz continuous at Lebesgue almost every point in I. In other words, for every ε > 0 there exist L > 0 and a set J ⊆ I such that Leb(I \ J) < ε and at each r ∈ J the function k is Lipschitz continuous with Lipschitz constant L. Proof. Suppose, on the contrary, that the set B∞ := {x ∈ I : sup y∈I y=x̸

|k(x) − k(y)| = ∞} |x − y|

16.5 Dimensions of measures, Lyapunov exponents and measure-theoretic entropy | 643

has positive Lebesgue measure. Write I = [a, b]. Because the Lebesgue measure is (inner) regular and every singleton is a null set, there is a compact set B ⊆ B∞ ∩ (a, b) of positive measure. For every x ∈ B, choose x′ ∈ (a, b) \ {x} such that |k(x) − k(x′ )| k(b) − k(a) >2 . |x − x′ | Leb(B)

(16.63)

Replace each pair x, x′ by a pair y, y′ such that (y, y′ ) ⊇ [min{x, x ′ }, max{x, x ′ }] and such that y, y′ is so close to x, x′ that (16.63) still holds for y, y′ . Note that y < y′ . We shall use for y, y′ the old notation x, x′ , so x < x′ . From the family of intervals (x, x′ ) choose a finite cover of the compact set B. It is possible to find a subcover ℐ of B which consists of two subfamilies ℐ 1 and ℐ 2 , each made up of pairwise disjoint intervals. Indeed, start with I1 = (x1 , x1′ ) ∈ ℐ with minimal possible x = x1 and maximal in ℐ in the sense of inclusion. Having found ′ I1 = (x1 , x1′ ), . . . , In = (xn , xn′ ), pick In+1 = (xn+1 , xn+1 ) as follows. Consider the family n

ℐn+1 := {(x, x ) ∈ ℐ : x ∈ ⋃ Ii and x > max xi }. ′

′

i=1

1≤i≤n

′

′ If ℐn+1 ≠ 0, set In+1 ∈ ℐn+1 so that xn+1 = max{x′ : (x, x ′ ) ∈ ℐn+1 }. If ℐn+1 = 0, set In+1 ∈ ℐ so that xn+1 is minimal satisfying xn+1 ≥ max1≤i≤n xi′ and maximal in ℐ . In this construction, the intervals In with even n’s are pairwise disjoint, as In+2 ∉ ℐn+1 . The same is true for odd n’s. We define ℐ 1 (resp., ℐ 2 ) as the family of (xn , xn′ ) for odd (resp., even) n’s. In view of the pairwise disjointness of the intervals in the families ℐ 1 and ℐ 2 , the monotonicity of k and (16.63), we obtain that

k(b) − k(a) ≥ ∑ (k(xn′ ) − k(xn )) > 2 n odd

k(b) − k(a) ∑ (x′ − xn ) Leb(B) n odd n

and a corresponding inequality for even n’s. Given that ℐ = ℐ 1 ∪ ℐ 2 covers B, we deduce that 2(k(b) − k(a)) > 2

k(b) − k(a) k(b) − k(a) Leb(B) = 2(k(b) − k(a)), ∑ (x′ − xn ) > 2 Leb(B) n∈ℕ n Leb(B)

which is a contradiction. Corollary 16.5.5. For every Borel probability measure ν on a compact metric space (X, d) and for every r > 0, there exists a finite partition 𝒫 = {Pt }M t=1 of X into Borel sets of positive ν-measure with diam(𝒫 ) < r and for which there is C > 0 such that ν(𝜕𝒫,a ) ≤ Ca, where 𝜕𝒫,a = ⋂M t=1 (⋃s=t̸ B(Ps , a)).

∀a > 0,

(16.64)

644 | 16 Conformal expanding repellers Proof. Let {x1 , . . . , xN } be a finite (r/4)-spanning set in X. Fix ε ∈ (0, r/4N). For each function t 󳨃→ ki (t) := ν(B(xi , t)), t ∈ I = [r/4, r/2], apply Lemma 16.5.4 and find appropriate Li and Ji for all 1 ≤ i ≤ N. Let L = max{Li : 1 ≤ i ≤ N} and let J = ⋂Ni=1 Ji . The set J has positive Lebesgue measure by the choice of ε. So we can choose a point r0 ∈ J with r/4 < r0 < r/2. For all 0 < a < a0 := min{r0 − r/4, r/2 − r0 } and for all 1 ≤ i ≤ N, we have ν(B(xi , r0 + a) \ B(xi , r0 − a)) ≤ 2La. Setting N

Δ(a) = ⋃(B(xi , r0 + a) \ B(xi , r0 − a)), i=1

we get ν(Δ(a)) ≤ 2LNa. Define 𝒫 = {⋂Ni=1 Bκ(i) (xi , r0 )} as a family over all functions κ : {1, . . . , N} → {+, −}, where B+ (xi , r0 ) := B(xi , r0 ) and B− (xi , r0 ) := X\B(xi , r0 ), but excluding functions κ that yield sets of measure zero. After removing from X a set of measure zero, the partition 𝒫 covers X. As r0 ≥ r/4, the balls {B(xi , r0 )}Ni=1 cover X. Hence, for each nonempty Pt ∈ 𝒫 , at least one value of κ is +. Thus, diam(Pt ) ≤ 2r0 < r. Now note that 𝜕𝒫,a ⊆ Δ(a). Indeed, let x ∈ 𝜕𝒫,a . Since 𝒫 covers X there exists t0 such that x ∈ Pt0 , so x ∉ Pt for all t ≠ t0 . However, as x ∈ ⋃t =t̸ 0 B(Pt , a), there exists t1 ≠ t0 such that dist(x, Pt1 ) < a. Let B = B(xi , r0 ) be such that Pt0 ⊆ B+ and Pt1 ⊆ B− , or vice versa. In the case where x ∈ Pt0 ⊆ B+ , by the triangle inequality d(x, xi ) > r0 − a, and since d(x, xi ) < r0 , we get x ∈ Δ(a). In the case where x ∈ Pt1 ⊆ B− , we have x ∈ B(xi , r0 + a) \ B(xi , r0 ) ⊆ Δ(a). We conclude that ν(𝜕𝒫,a ) ≤ ν(Δ(a)) ≤ 2LNa for all a < a0 . For all a ≥ a0 , it suffices to take C ≥ 1/a0 . Set C = max{2LN, 1/a0 }. Remark 16.5.6. If X is embedded in a compact manifold Y, then we can view ν as a measure on Y; we find a partition 𝒫 of Y, and then 𝜕𝒫,a = B(⋃M t=1 𝜕Pt , a), provided M ≥ 2. This justifies the notation 𝜕𝒫,a . Corollary 16.5.7. Let ν be a Borel probability measure on a compact metric space (X, d), and let f : X → X be a measure-preserving transformation. For every r > 0, the partition 𝒫 constructed in Corollary 16.5.5 is such that for every δ > 0 and ν-a. e. x ∈ X there exists N(x) ∈ ℕ for which B(f n (x), e−nδ ) ⊆ 𝒫 (f n (x)),

∀n ≥ N(x).

Proof. Fix δ > 0. By Corollary 16.5.5, ∞

∞

n=0

n=0

∑ ν(𝜕𝒫,exp(−nδ) ) ≤ ∑ C exp(−nδ) < ∞.

By the f -invariance of ν, we deduce that ∞

∑ ν(f −n (𝜕𝒫,exp(−nδ) )) < ∞.

n=0

(16.65)

16.5 Dimensions of measures, Lyapunov exponents and measure-theoretic entropy | 645

Applying the Borel–Cantelli lemma (Lemma A.1.20) to (f −n (𝜕𝒫,exp(−nδ) ))∞ n=1 , we conclude that for ν-a. e. x ∈ X there exists N(x) ∈ ℕ such that x ∉ f −n (𝜕𝒫,exp(−nδ) ) for every n ≥ N(x). So f n (x) ∉ 𝜕𝒫,exp(−nδ) for all n ≥ N(x). Hence, by definition of 𝜕𝒫,exp(−nδ) , if f n (x) ∈ Pt ∈ 𝒫 then f n (x) ∉ ⋃s=t̸ B(Ps , exp(−nδ)). Thus, B(f n (x), exp(−nδ)) ⊆ Pt = 𝒫 (f n (x)). Finally, we can establish a generalization of the volume lemma (Theorem 16.5.2). For this, we will need a slight refinement of the definition of Hausdorff dimension of a measure (cf. Definition 15.6.1). Definition 16.5.8. Let μ be a Borel probability measure on a metric space (X, d). Define 󵄨 HD⋆ (μ) := inf{HD(Y) 󵄨󵄨󵄨 μ(Y) > 0}. Observe that HD⋆ (μ) ≤ HD(μ). Theorem 16.5.9. HD⋆ (μ) = ess inf(dμ )

and

HD(μ) = ess sup(dμ ).

Proof. The statement about HD(μ) is just Corollary 15.6.7. Let θ1 = ess inf(dμ ). Applying Lemma 15.6.6(a), we have μ({x : dμ (x) ≥ θ1 }) = 1. For every Y with μ(Y) > 0, there then is Y ′ ⊆ Y with μ(Y ′ ) = μ(Y) such that dμ (x) ≥ θ1 for every x ∈ Y ′ . Hence, HD(Y) ≥ HD(Y ′ ) ≥ θ1 by Theorem 15.6.3(a), and thereby HD⋆ (μ) ≥ θ1 . On the other hand, by Lemma 15.6.6(a) we know that μ({x : dμ (x) < θ}) > 0 for every θ > θ1 , and thus HD({x : dμ (x) < θ}) ≤ θ by Theorem 15.6.3(b). Therefore, HD⋆ (μ) ≤ θ. Letting θ ↘ θ1 we get HD⋆ (μ) ≤ θ1 . We conclude from the above two paragraphs that HD⋆ (μ) = θ1 . The following is a generalization of Theorem 16.5.2.

Theorem 16.5.10 (Volume lemma for arbitrary invariant measures). Let (X, U, T) be a conformal expanding repeller and μ a T-invariant Borel probability measure on X. Then HD⋆ (μ) ≤

hμ (T) χμ (T)

≤ HD(μ).

(16.66)

In addition, if μ is ergodic then dμ (x) =

hμ (T) χμ (T)

= HD(μ) = PD(μ) for μ-a. e. x ∈ X.

(16.67)

Proof. Let r = ξ and 𝒫 be the corresponding partition constructed in Corollary 16.5.5. As we saw in Lemma 13.2.8, 𝒫

n+1

(x) ⊆ Tx−n (B(T n (x), ξ )),

∀x ∈ X, ∀n ≥ 0.

(16.68)

646 | 16 Conformal expanding repellers We now work toward obtaining some sort of opposite inclusion. Consider an arbitrary δ > 0, and x ∈ X so that Corollary 16.5.7 applies for all n ≥ N(x). For every N(x) ≤ i ≤ n, log ξ define k(i) = [i logδ λ + log λ ] + 1, where λ > 1 is an expanding constant for T and [⋅] is the integer part function. Then exp(−iδ) ≥ ξ λ−k(i) . Therefore,

i+k(i) TT−k(i) (x), ξ )) ⊆ B(T i (x), exp(−iδ)). i (x) (B(T

Using Corollary 16.5.7 with i in place of n, we deduce that Tx−(i+k(i)) (B(T i+k(i) (x), ξ )) ⊆ Tx−i (𝒫 (T i (x))),

∀N(x) ≤ i ≤ n.

From this inclusion over all N(x) ≤ i ≤ n, we conclude that n+1 Tx−(n+k(n)) (B(T n+k(n) (x), ξ )) ⊆ 𝒫N(x) (x).

(16.69)

Notice that for μ-a. e. x ∈ X there is a > 0 such that B(x, a) ⊆ 𝒫 N(x) (x), by definition of 𝜕𝒫,a in Corollary 16.5.5. It follows from this and (16.69) that for all n ≥ N(x) large enough (depending on x), Tx−(n+k(n)) (B(T n+k(n) (x), ξ )) ⊆ 𝒫 n+1 (x).

(16.70)

It ensues from (16.70), and (16.68) with n + k(n) in lieu of n, that − log μ(Tx−(n+k(n)) (B(T n+k(n) (x), ξ ))) 1 lim − log μ(𝒫 n (x)) ≤ lim inf n→∞ n→∞ n n −(n+k(n)) − log μ(Tx (B(T n+k(n) (x), ξ ))) ≤ lim sup n n→∞ 1 n+k(n)+1 ≤ lim − log μ(𝒫 (x)). n→∞ n The limits on the far left- and far right-hand sides of these inequalities exist for μ-a. e. x ∈ X by the Shannon–McMillan–Breiman theorem (Theorem 9.5.4), and their ratio is n equal to limn→∞ n+k(n) = 1/(1 + logδ λ ). Letting δ → 0, we obtain the existence of the limit for the middle terms and the equality − log μ(Tx−n (B(T n (x), ξ ))) 1 hμ (T, 𝒫 , x) := lim − log μ(𝒫 n (x)) = lim . n→∞ n n→∞ n

(16.71)

In view of Birkhoff’s ergodic theorem (Theorem 8.2.11), the limit χμ (T, x) := lim

n→∞

exists for μ-a. e. x ∈ X.

1 ′ 󵄨 󵄨 log󵄨󵄨󵄨(T n ) (x)󵄨󵄨󵄨 n

(16.72)

16.5 Dimensions of measures, Lyapunov exponents and measure-theoretic entropy | 647

Let K ≥ 1 be the bounded distortion constant from Lemma 16.2.9. Fix r ∈ (0, K −1 ξ ), and let k = k(x, r) ≥ 0 be the largest k ∈ ℕ such that 󵄨󵄨 k ′ 󵄨󵄨 󵄨󵄨(T ) (x)󵄨󵄨rK ≤ ξ .

(16.73)

By this and Lemma 16.2.9, we have that Tx−k (B(T k (x), ξ )) ⊇ B(x, K −1 |(T k )′ (x)|−1 ξ ) ⊇ B(x, r).

(16.74)

Take N ∈ ℕ such that λN−1 ≥ K 2 . We then obtain that ′ 󵄨󵄨 k+N ′ 󵄨󵄨 󵄨󵄨 N−1 ′ k+1 󵄨 󵄨 󵄨 ) (x)󵄨󵄨 = 󵄨󵄨(T ) (T (x))󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨(T k+1 ) (x)󵄨󵄨󵄨 ≥ λN−1 ⋅ (rK)−1 ξ ≥ r −1 Kξ . 󵄨󵄨(T

(16.75)

Hence, by Lemma 16.2.9, Tx−(k+N) (B(T k+N (x), ξ )) ⊆ B(x, K|(T k+N )′ (x)|−1 ξ ) ⊆ B(x, r).

(16.76)

It follows from (16.73)–(16.76) that log μ(Tx−k (B(T k (x), ξ )))

log(ξK) − log |(T k+N )′ (x)|

≤

log μ(B(x, r)) log μ(Tx−(k+N) (B(T k+N (x), ξ ))) ≤ . log r log(ξK −1 ) − log |(T k )′ (x)|

(16.77)

Given that N is independent of k and x and that λN ≤ |(T N )′ (y)| ≤ ‖T ′ ‖N∞ for all y ∈ X, the terms log |(T k )′ (x)| and log |(T k+N )′ (x)| are comparable with a constant of comparability independent of k and x. Passing to the limit r → 0 (so k → ∞) and using (16.71)–(16.72), we deduce that log μ(B(x, r)) hμ (T, 𝒫 , x) = . r→0 log r χμ (T, x) lim

By the Shannon–McMillan–Breiman theorem and Birkhoff’s ergodic theorem, ∫X hμ (T, 𝒫 , x) dμ(x) ∫X χμ (T, x) dμ(x)

=

hμ (T, 𝒫 ) χμ (T)

=

hμ (T) χμ (T)

.

The latter equality holds since T is expansive and diam(𝒫 ) is less than an expansive h (T,𝒫,x) h (T) constant for T. There thus exists a set of positive measure where μχ (T,x) ≤ χ μ(T) , and μ

a set of positive measure where the opposite inequality holds. Therefore,

μ

log μ(B(x, r)) hμ (T) ≤ r→0 log r χμ (T) lim

on a set of positive measure, and the opposite inequality holds on a positive measure set as well. The inequalities (16.66) follow from this and Theorem 16.5.9. In the ergodic case, hμ (T, 𝒫 , x) = hμ (T) and χμ (T, x) = χμ (T) for μ-a. e. x ∈ X. So (16.67) holds.

648 | 16 Conformal expanding repellers

16.6 Multifractal analysis of Gibbs states In Section 16.3, we derived Bowen’s formula for conformal expanding repellers. This formula states that the Hausdorff (as well as the packing and box) dimension of the limit set of such a repeller is equal to the unique zero of the repeller’s pressure function. The Hausdorff dimension of a fractal set gives us an idea of the global structure of that set. More precisely, it describes the complexity there is in covering the entire set at infinitesimal scales. Nevertheless, two fractal sets with similar global structures may have substantially different local structures. The pointwise or local dimension of a Borel measure contributes to capturing the local structure of a set, and potentially helps in distinguishing that set from another one which may be globally similar. Multifractal analysis consists in: (1) determining the range of local dimensions that a set admits; and (2) for each value in that range, determining the size of the subset of all points of the set where the local dimension is equal to that value. The resulting map is called multifractal spectrum, as it describes the relative sizes of a multitude of fractal subsets which form a splitting of the original fractal set. Multifractal formalism origins from physics and mathematics (among others, see [34, 42, 45, 68]). In this latter paper, strong hints of parallels between multifractal theory and the theory of statistical physics were suggested. Some of the first rigorous mathematical results on multifractals can be found in [15, 100]. Since then, many papers have been written on this subject. For instance, see [87, 88, 94–96]. In particular, Pesin [95] developed a general framework in which multifractal formalism can be derived. Formally, let ν be a Borel probability measure on a metric space X. Recall that ν has pointwise or local dimension α at a point x ∈ X if (cf. Definition 15.6.2) dν (x) := lim

r→0

log ν(B(x, r)) = α. log r

For each number 0 ≤ α ≤ ∞, let 󵄨 Xν (α) := {x ∈ X 󵄨󵄨󵄨 dν (x) = α} be the subset of all points x ∈ X where the measure ν has local dimension α. The domain of the pointwise dimension function dν , namely the set Xν := is called the regular part of X.

⋃ Xν (α),

0≤α≤∞

16.6 Multifractal analysis of Gibbs states | 649

Its complement 󵄨 Xνc := X \ Xν = {x ∈ X 󵄨󵄨󵄨 dν (x) ∄} is called the singular part of X. The decomposition (i. e., the partition) of the set X as X=

⋃ Xν (α) ∪ Xνc

0≤α≤∞

is called the multifractal decomposition of X with respect to the pointwise dimension. Let Fν (α) := HD(Xν (α)) be the Hausdorff dimension of the fractal subset Xν (α). The function α 󳨃󳨀→ Fν (α), whose domain is {0 ≤ α ≤ ∞ : Xν (α) ≠ 0}, is called (fine Hausdorff ) multifractal spectrum of the measure ν. Alternatively, it is called spectrum for pointwise dimensions or pointwise dimension spectrum. If X ⊆ ℝd , then for every q ∈ ℝ \ {1}, let Rq (ν) :=

q log ∑N(r) 1 i=1 ν(Bi ) lim , q − 1 r→0 log r

(16.78)

where N(r) is the total number of boxes Bi of the form Bi = {(x1 , . . . , xd ) ∈ ℝd : rkj ≤ xj ≤ r(kj + 1), ∀j = 1, . . . , d} for integers kj = kj (i) such that ν(Bi ) > 0. The function Rq , whose domain consists of all q for which the limit (16.78) exists, is called Rényi dimension spectrum. It is easy to check that it is equal to the Hentschel–Procaccia spectrum (often abbreviated HP-spectrum), which is defined by

HPq (ν) :=

1 lim q − 1 r→0

log inf 𝒢r

∑

B(xi ,r)∈𝒢r

ν(B(xi , r))q ,

log r

where the infimum is taken over all countable covers 𝒢r of the topological support of ν by balls of radius r centered at xi ∈ X, or log ∫X ν(B(x, r)) 1 HPq (ν) := lim q − 1 r→0 log r provided those limits exist.

q−1

dν(x)

650 | 16 Conformal expanding repellers For q = 1, we define the information dimension I(ν) in two steps. First, set Hν (r) := inf(− ∑ ν(B) log ν(B)), ℱr

B∈ℱr

where the infimum is taken over all partitions ℱr of a set of full ν-measure into Borel sets B of diameter at most r. Second, define I(ν) := lim

r→0

Hν (r) − log r

(16.79)

provided this limit exists. Note that for Rényi and HP dimensions it does not make any difference whether we consider covers of the topological support (the smallest closed set of full measure) of a measure or any set of full measure, since all balls have the same radius r, so we can always choose locally finite (number independent of r) subcovers. These are “box type” dimension quantities. To avoid any confusion with the temperature function that we will introduce later and that will be denoted by T, the conformal repellers will be denoted from this point on by (X, U, f ) rather than (X, U, T). Returning to the pointwise dimension spectrum, a priori there is no reason for the function Fν to behave nicely. Its domain of definition obviously depends on ν. If ν is a geometric measure (see Definition 15.6.12), then the domain of Fν is a singleton, namely, the exponent t of the geometric measure. Its range is the Hausdorff dimension of the underlying space X. For instance, we know by the volume lemma (Theorem 16.5.2) that the domain and the range of Fν are both {HD(X)} for the geometric measure ν = μ−HD(X) log |f ′ | associated to a conformal expanding repeller (X, U, f ). If ν is an ergodic f -invariant measure, then the volume lemma states that dν (x) = α0 for ν-a. e. x ∈ X, where α0 = HD(ν). In particular, we know that the domain of Fν is nonempty. However, for any α ≠ α0 the set Xν (α) is not visible to the measure ν since ν(Xν (α)) = 0. Whereas the function HPq (ν) can be determined by statistical properties of ν-typical (a. e.) trajectories, the function Fν (α) seems intractable. Nevertheless, if ν = μφ is the unique f -invariant Gibbs state and unique equilibrium state associated with a Hölder continuous potential φ on a conformal expanding repeller (X, U, f ), then miraculously the dimension spectra α 󳨃→ Fμφ (α) and q 󳨃→ HPq (μφ ) happen to be real-analytic functions and −Fμφ (−p) and HPq (μφ ) are mutual Legendre transforms. Accordingly, let φ : X → ℝ be a Hölder continuous potential. For q, t ∈ ℝ, consider the two-parameter family of auxiliary Hölder continuous potentials φq,t : X → ℝ defined by 󵄨 󵄨 φq,t := qφ − t log󵄨󵄨󵄨f ′ 󵄨󵄨󵄨.

16.6 Multifractal analysis of Gibbs states | 651

This family naturally generates a two-variable pressure function ℝ2 ∋ (q, t) 󳨃󳨀→ P(q, t) := P(f , φq,t ) = P(f , qφ − t log |f ′ |) ∈ ℝ. Lemma 16.6.1. Let (X, U, f ) be a conformal expanding repeller and let φ : X → ℝ be a Hölder continuous potential. For every q ∈ ℝ, the one-variable pressure function ℝ ∋ t 󳨃󳨀→ P(q, t) ∈ ℝ possesses the following properties: (a) It is Lipschitz with Lipschitz constant L := log ‖f ′ ‖∞ , where ‖f ′ ‖∞ := supx∈X |f ′ (x)|. (b) It is strictly decreasing. (c) lim P(q, t) = ∞ whereas lim P(q, t) = −∞. t→−∞

t→∞

(d) P(q, 0) = P(f , qφ). (e) There is a unique T(q) ∈ ℝ such that P(q, T(q)) = 0. The resulting function ℝ ∋ q 󳨃󳨀→ T(q) ∈ ℝ is called the temperature function. Moreover, (f) The two-variable pressure function (q, t) 󳨃󳨀→ P(q, t) is convex.

Proof. The proof is left to the reader. It goes along almost the same lines as that of the properties of the pressure function when q = 0 in Proposition 16.3.1. Per Proposition 13.7.12, let μφ be the unique f -invariant Gibbs state and unique equilibrium state for the potential φ. The unique f -invariant Gibbs state and unique equilibrium state for the potential φq,t , normally denoted by μφq,t , will be commonly abbreviated to μq,t . The unique f -invariant Gibbs state and unique equilibrium state for the potential φq,T(q) , normally denoted by μφq,T(q) or μq,T(q) , will be further abbreviated to μq . The relation between the measures {μq }q∈ℝ and the measure μφ is explained in the following result. Lemma 16.6.2. Let (X, U, f ) be a conformal expanding repeller and let φ : X → ℝ be a Hölder continuous potential such that P(f , φ) = 0. For every q ∈ ℝ there exists Cq ≥ 1 such that μq (B(x, r))

Cq−1 ≤

r T(q) μφ (B(x, r))

q

≤ Cq ,

∀x ∈ X, ∀0 < r < ξ .

(16.80)

Proof. Let k = k(x, r) be defined as just before Corollary 16.2.10. By (16.10) in Lemma 16.2.12 (with m equal to μφ and μq successively) and by Corollary 16.2.10, the ratios μφ (B(x, r))

exp(Sk φ(x))

,

μq (B(x, r)) , 󵄨󵄨 k ′ 󵄨󵄨−T(q) exp(qSk φ(x)) 󵄨󵄨(f ) (x)󵄨󵄨

and

r 󵄨󵄨 k ′ 󵄨󵄨−1 󵄨󵄨(f ) (x)󵄨󵄨

are comparable, i. e., they are all bounded from below and above by positive constants independent of x and r. This yields (16.80).

652 | 16 Conformal expanding repellers We now derive a generalization of the volume lemma (Theorem 16.5.2) for the pointwise dimension of the measure μφ . Lemma 16.6.3. Let (X, U, f ) be a conformal expanding repeller and let φ : X → ℝ be a Hölder continuous potential such that P(f , φ) = 0. For any ergodic f -invariant Borel probability measure ν on X, we have dμφ (x) =

−ν(φ) = HD(μφ ) = PD(μφ ) χν (f )

for ν-a. e. x ∈ X.

Proof. Using Lemma 16.2.12 and the ergodic case of Birkhoff’s ergodic theorem (Corollary 8.2.14), we get dμφ (x) = lim

r→0

=

log μφ (B(x, r)) log r

Sk φ(x) 󵄨 󵄨 k→∞ − log󵄨󵄨(f k )′ (x)󵄨󵄨 󵄨 󵄨

= lim

∫X φ dν limk→∞ k1 Sk φ(x) ν(φ) 󵄨󵄨 k ′ 󵄨󵄨 = − ∫ log |f ′ | dν = −χ (f ) 1 − limk→∞ k log󵄨󵄨(f ) (x)󵄨󵄨 ν X

for ν-a. e. x ∈ X. The proof of the first equality is complete. The second and third equalities are a direct application of Corollary 15.6.11. We can further conclude from Lemma 16.6.3 that the singular part Xμcφ of X has zero ν-measure with respect to every ergodic invariant measure ν, including μφ . Yet the set Xμcφ is usually geometrically big (see Exercise 16.7.15). On the Legendre transform Let k = k(q) : I → ℝ := ℝ ∪ {−∞, ∞} be a continuous convex function on an interval I = [α1 (k), α2 (k)] with −∞ ≤ α1 (k) ≤ α2 (k) ≤ ∞, excluding the degenerate cases where I is the singleton {−∞} or {∞}. So I is either (1) a point in ℝ; or (2) a closed real interval; or (3) a closed real half-line jointly with one of −∞ or ∞; or (4) ℝ. We also assume that k is finite on (α1 , α2 ). The Legendre transform of k is the function g of a new variable p defined by g(p) = sup{pq − k(q)}. q∈I

Its domain is the closure in ℝ of the set of points p ∈ ℝ where g(p) is finite, with g extended continuously to the boundary. It can be easily proved that the domain of g is also either (1) a real point; or (2) a closed real interval; or (3) a closed real semiline with one of −∞ or ∞; or (4) ℝ. More precisely, the domain is [α1 (g), α2 (g)], where α1 (g) = −∞ if α1 (k) is finite, or α1 (g) = limx→−∞ k ′ (x) if α1 (k) = −∞. The derivative here is a one-sided derivative; it does not matter whether it is left or right. A similar description of α2 (g) is obtained by replacing −∞ with ∞.

16.6 Multifractal analysis of Gibbs states | 653

It is further easy to show that g is a continuous convex function and that the Legendre transform is involutive. We then say that the functions k and g form a Legendre transform pair. Proposition 16.6.4. If two convex functions k and g form a Legendre transform pair then g(k ′ (q)) = qk ′ (q) − k(q), where k ′ (q) is any number between the left- and righthand side derivatives of k at q, defined as −∞ at q = α1 (k) if α1 (k) is finite and defined as ∞ at q = α2 (k) if α2 (k) is finite. Note that if k is C 2 with k ′′ > 0 and therefore is strictly convex, then g ′′ > 0 at all points k ′ (q) for α1 (k) < q < α2 (k) and thus g is strictly convex on [k ′ (α1 (k)), k ′ (α2 (k))]. Outside this interval, g is affine on the rest of its domain. If the domain of k is a singleton, then g is affine on ℝ and vice versa. We now formulate the main theorem of this section. Theorem 16.6.5. Let (X, U, f ) be an infinite conformal expanding repeller (infinite means #X = ∞). Let φ : X → ℝ be a Hölder continuous potential such that P(f , φ) = 0. (a) For μφ -a. e. x ∈ X, dμφ (x) =

−μφ (φ) χμφ (f )

=

hμφ (f ) χμφ (f )

= HD(μφ ) = PD(μφ ).

(b) The temperature function ℝ ∋ q 󳨃󳨀→ T(q) ∈ ℝ is real analytic and satisfies T(0) = HD(X),

T(1) = 0,

T ′ (q) =

μq (φ)

χμq (f )

≤

−hμq (f ) χμq (f )

< 0,

and

T ′′ (q) ≥ 0,

where μq is the f -invariant Gibbs measure for the potential φq,T(q) . (c) For all q ∈ ℝ, we have μq (Xμφ (−T ′ (q))) = 1. Moreover, HD(μq ) = HD(Xμφ (−T ′ (q))).

(d) For every q ∈ ℝ, we have Fμφ (−T ′ (q)) = T(q) − qT ′ (q), i. e., p 󳨃󳨀→ −Fμφ (−p) is the Legendre transform of T(q). In particular, Fμφ is continuous at the boundary points −T ′ (±∞) of its domain, as the Legendre transform is. Furthermore, the sets Xμφ (α) are empty for all α ∉

[−T ′ (∞), −T ′ (−∞)], so these α’s do not lie in the domain of Fμφ , as they do not belong to the domain of the Legendre transform. (This emptyness property is called the completeness of the Fμφ -spectrum.) If μφ = μ−HD(X) log |f ′ | (equivalently, if φ is cohomologous to −HD(X) log |f ′ | in the class of bounded functions on X), then T(q) is affine and the domain of Fμφ (α) is

one point, namely HD(X) = −T ′ (q) for all q ∈ ℝ. For this sole paragraph, suppose that the repeller (X, U, f ) is topologically exact. If μφ ≠ μ−HD(X) log |f ′ | , then T ′′ (q) > 0 and Fμ′′φ (α) < 0, i. e., the functions T(q) and Fμφ (α) are respectively strictly convex on ℝ and strictly concave on [−T ′ (∞), −T ′ (−∞)], which is a bounded interval in ℝ+ = {α ∈ ℝ : α > 0}.

654 | 16 Conformal expanding repellers (e) For every q ≠ 1, the HP and Rényi spectra exist (i. e., limits in their definitions exist) = HPq (μφ ) = Rq (μφ ). For q = 1, the information dimension I(μφ ) exists and T(q) 1−q and lim

q→1, q=1̸

T(q) = −T ′ (1) = HD(μφ ) = PD(μφ ) = I(μφ ). 1−q

For outlines of the graphs of T(q) and Fμφ (α), see Figures 16.1 and 16.2. See also Exercise 16.7.17. Compare with [95, Figures 17(a,b) on pp. 219–220].

Figure 16.1: Graph of T .

Figure 16.2: Graph of Fμφ .

16.6 Multifractal analysis of Gibbs states | 655

Proof. (a) As P(f , φ) = 0, the first equality in part (a) is an immediate consequence of the volume lemma (or of Lemma 16.6.3 with ν = μφ ) and the fact that μφ is an equilibrium state. The second and third equalities follow from Corollary 15.6.11. (b) The equality T(0) = HD(X) ensues from Bowen’s formula (Theorem 16.3.2) while T(1) = 0 follows from the equality P(f , φ) = 0. Concerning the real analyticity of q 󳨃→ T(q), let β be a Hölder exponent for the function φ. The function ℂ2 ∋ (q, t) 󳨃→ φq,t = qφ − t log |f ′ | ∈ Hβ (X) is affine. By Theorem 13.10.3, the pressure function ℝ2 ∋ (q, t) 󳨃→ P(q, t) := P(f , φq,t ) ∈ ℝ is real analytic. Thus, the real analyticity of q 󳨃→ T(q) will follow from the implicit function theorem once we verify the nondegeneracy assumption. For this, Theorem 13.10.6 (with T = f , φ = − log |f ′ | and ψ = φ) confirms that for every (q0 , t0 ) ∈ ℝ2 , 𝜕P(q, t) 󵄨󵄨󵄨󵄨 󵄨 󵄨 = − ∫ log󵄨󵄨󵄨f ′ 󵄨󵄨󵄨 dμq0 ,t0 = −χμq ,t (f ) < 0, 󵄨 0 0 𝜕t 󵄨󵄨󵄨(q0 ,t0 )

(16.81)

X

where μq0 ,t0 is the f -invariant Gibbs state for the function φq0 ,t0 , and the real analyticity of q 󳨃→ T(q) ensues from the implicit function theorem. Differentiating the equality P(q, t) = 0 with respect to q, we obtain 𝜕P(q, t) 󵄨󵄨󵄨󵄨 𝜕P(q, t) 󵄨󵄨󵄨󵄨 + ⋅ T ′ (q) = 0. 󵄨󵄨 󵄨 𝜕q 󵄨󵄨(q,T(q)) 𝜕t 󵄨󵄨󵄨(q,T(q))

(16.82)

Therefore, we get the standard formula T ′ (q) = −

𝜕P(q, t) 󵄨󵄨󵄨󵄨 𝜕P(q, t) 󵄨󵄨󵄨󵄨 / . 󵄨󵄨 󵄨 𝜕q 󵄨󵄨(q,T(q)) 𝜕t 󵄨󵄨󵄨(q,T(q))

Using Theorem 13.10.6, we infer expressions for the numerator and denominator of T ′ (q): 𝜕P(q, t) 󵄨󵄨󵄨󵄨 = ∫ φ dμq 󵄨 𝜕q 󵄨󵄨󵄨(q,T(q))

and

X

𝜕P(q, t) 󵄨󵄨󵄨󵄨 󵄨 󵄨 = − ∫ log󵄨󵄨󵄨f ′ 󵄨󵄨󵄨 dμq . 󵄨 𝜕t 󵄨󵄨󵄨(q,T(q))

(16.83)

X

So T ′ (q) =

μq (φ)

χμq (f )

(16.84)

.

As P(f , φ) = 0 and μq is f -invariant (i. e., μq ∈ M(f )), the variational principle (Theorem 12.1.1) implies that 0 = P(f , φ) = sup {hμ (f ) + ∫ φ dμ} ≥ hμq (f ) + ∫ φ dμq . μ∈M(f )

X

X

656 | 16 Conformal expanding repellers Hence, T ′ (q) ≤

−hμq (f ) χμq (f )

< 0.

The strict inequality holds since χμq (f ) ≥ log λ > 0 and hμq (f ) > 0 by Theorem 13.7.21.

Finally, the inequality T ′′ (q) ≥ 0 follows from the convexity of P(q, t) (see Lemma 16.6.1). Indeed, the fact that the part of ℝ3 above the graph of P(q, t) is convex |(q0 ,t0 ) = implies that its intersection with the plane (q, t) is also convex. Since 𝜕P(q,t) 𝜕t −χμq ,t (f ) < 0, this is the part of the plane above the graph of T. Hence, T is a convex 0 0 function. This completes the proof of (b). We now establish the last two paragraphs in part (d). Theorem 13.7.17 asserts that μφ = μ−HD(X) log |f ′ | if and only if φ + HD(X) log |f ′ | is cohomologous to a constant R in the class of bounded functions on X. By Bowen’s formula (Theorem 16.3.2), Lemma 13.2.6 and the hypothesis P(f , φ) = 0, this constant R is R = P(f , −HD(X) log |f ′ |) + R = P(f , φ) = 0. In other words, μφ = μ−HD(X) log |f ′ | if and only if φ is cohomologous to −HD(X) log |f ′ | via a bounded coboundary υ : X → ℝ. Using this and Lemma 13.2.6 once again, for every q ∈ ℝ we get P(q, T(q)) = 0 = P(f , φ) = P(f , −HD(X) log |f ′ |)

= P(f , −HD(X) log |f ′ | + qυ − qυ ∘ f ) = P(f , q(−HD(X) log |f ′ | + υ − υ ∘ f ) + (q − 1)HD(X) log |f ′ |)

= P(f , qφ + (q − 1)HD(X) log |f ′ |) = P(f , qφ − (1 − q)HD(X) log |f ′ |)

= P(q, (1 − q)HD(X)).

By the uniqueness of T(q), we infer that T(q) = (1 − q)HD(X). This proves the second last paragraph in (d). In part (b)’s proof of convexity of T, we avoided an explicit expression of T ′′ . However, to discuss strict convexity it is necessary to compute T ′′ . Differentiating the formula 0=

dP 𝜕P 𝜕P (q, T(q)) = (q, T(q)) + (q, T(q)) ⋅ T ′ (q) dq 𝜕q 𝜕t

with respect to q and using (16.81), we obtain that T (q) = ′′

𝜕2 P (q, T(q)) 𝜕q2

2

𝜕P + 2T ′ (q) 𝜕q𝜕t (q, T(q)) + (T ′ (q))

χμq (f )

2 𝜕2 P (q, T(q)) 𝜕t 2

.

(16.85)

16.6 Multifractal analysis of Gibbs states | 657

If the repeller f is topologically exact, then it follows from Theorem 13.10.7 that 𝜕2 P (q, T(q)) = σμ2 q ,∞ (φ) 𝜕q2

𝜕2 P 󵄨 󵄨 (q, T(q)) = σμ2 q ,∞ (− log󵄨󵄨󵄨f ′ 󵄨󵄨󵄨). 𝜕t 2

and

(16.86)

From Exercise 13.11.16 and Theorem 13.9.1, one can deduce the analog of Theorem 13.10.7 for the mixed second derivative 𝜕2 P 󵄨 󵄨 (q, T(q)) = σμ2 q ,∞ (− log󵄨󵄨󵄨f ′ 󵄨󵄨󵄨, φ). 𝜕q𝜕t

(16.87)

To shorten notation, let F := − log |f ′ |. So μq (F) = −χμq (f ). Using (13.87), (13.148) and (16.84), we can write 𝜕2 P 𝜕2 P P ′ (q, T(q)) + 2T (q) (q, T(q)) + (q, T(q)) 𝜕q𝜕t 𝜕t 2 𝜕q2 2

2𝜕

(T ′ (q))

2

∞

2

2

= (T ′ (q)) [μq ((F + χμq (f )1) ) + 2 ∑ [μq (F ⋅ F ∘ T k ) − (χμq (f )) ]] k=1

+ 2T (q)μq ((F + χμq (f )1)(φ − μq (φ)1)) ′

∞

+ 2T ′ (q) ∑ (μq (F ⋅ φ ∘ T k ) + χμq (f )μq (φ)) k=1 ∞

+ 2T ′ (q) ∑ (μq (F ∘ T k ⋅ φ) + χμq (f )μq (φ)) k=1

2

∞

2

+ [μq ((φ − μq (φ)1) ) + 2 ∑ [μq (φ ⋅ φ ∘ T k ) − (μq (φ)) ]] k=1

2

= μq ([T ′ (q)(F + χμq (f )1) + (φ − μq (φ)1)] ) 2

∞

2

+ 2(T ′ (q)) ∑ [μq (F ⋅ F ∘ T k ) − (χμq (f )) ] k=1

∞

+ 2T (q) ∑ (μq (F ⋅ φ ∘ T k ) + χμq (f )μq (φ)) ′

k=1 ∞

+ 2T ′ (q) ∑ (μq (F ∘ T k ⋅ φ) + χμq (f )μq (φ)) k=1

∞

2

+ 2 ∑ [μq (φ ⋅ φ ∘ T k ) − (μq (φ)) ] k=1

2

= μq ([(T ′ (q)F + φ) − μq (T ′ (q)F + φ)1] ) ∞

+ 2 ∑ μq (T ′ (q)F ⋅ (T ′ (q)F ∘ T k + φ ∘ T k )) k=1

658 | 16 Conformal expanding repellers ∞

2

∞

+ 2 ∑ μq (φ ⋅ (T ′ (q)F ∘ T k + φ ∘ T k )) − 2 ∑ (T ′ (q)χμq (f ) − μq (φ)) k=1

2

k=1

= μq ([(T ′ (q)F + φ) − μq (T ′ (q)F + φ)1] ) ∞

∞

k=1

k=1

+ 2 ∑ μq ((T ′ (q)F + φ) ⋅ (T ′ (q)F + φ) ∘ T k ) − 2 ∑ (0)2 = σμ2 q ,∞ (T ′ (q)F + φ) ≥ 0,

(16.88)

where the second last equality relies on (16.84) while the last equality follows from (13.87) for the Hölder function T ′ (q)F + φ = −T ′ (q) log |f ′ | + φ. Substituting (16.88) into (16.85), we get T ′′ (q) =

σμ2 q ,∞ (−T ′ (q) log |f ′ | + φ) χμq (f )

≥ 0.

(16.89)

In particular, this provides another proof of (b), albeit under the additional assumption that the repeller is topologically exact. By Theorem 13.9.14, we know that σμ2 q ,∞ (−T ′ (q) log |f ′ | + φ) = 0 if and only if the function −T ′ (q) log |f ′ | + φ is cohomologous to a constant R in the class of Hölder continuous functions. It then follows from (16.84) that R = ∫ R dμq = ∫(φ − T ′ (q) log |f ′ |) dμq = 0. Therefore, T ′ (q) log |f ′ | is cohomologous to φ, and hence P(f , T ′ (q) log |f ′ |) = P(f , φ) = 0 by Lemma 13.2.6 and the hypothesis P(f , φ) = 0. Thus, T ′ (q) = −HD(X) by Bowen’s formula (Theorem 16.3.2). Consequently, φ is cohomologous to the function −HD(X) log |f ′ |. This means that μφ = μ−HD(X) log |f ′ | . In view of (16.89), we conclude that if μφ ≠ μ−HD(X) log |f ′ | then T ′′ (q) > 0 for all q ∈ ℝ. This proves the last paragraph in (d), which is the only portion of the result where topological exactness is needed. (c) By Lemma 16.6.3 (applied with ν = μq ) and part (b), there exists a set Xq ⊆ X of full μq -measure such that dμφ (x) =

−μq (φ) χμq (f )

= −T ′ (q),

∀x ∈ Xq .

Hence, Xq ⊆ Xμφ (−T ′ (q)) and thereby μq (Xμφ (−T ′ (q))) = 1. Let x ∈ X and 0 < r < ξ . By Lemma 16.6.2, 󵄨󵄨 󵄨 󵄨󵄨log μq (B(x, r)) − T(q) log r − q log μφ (B(x, r))󵄨󵄨󵄨 < log Cq . 󵄨󵄨 󵄨󵄨 Therefore, 󵄨󵄨 log μφ (B(x, r)) 󵄨󵄨󵄨󵄨 󵄨󵄨 log μq (B(x, r)) 󵄨󵄨 = 0. − T(q) − q lim󵄨󵄨󵄨 󵄨󵄨 r→0󵄨󵄨 log r log r 󵄨󵄨 󵄨

(16.90)

16.6 Multifractal analysis of Gibbs states | 659

Using this, observe that for every x ∈ Xμφ (−T ′ (q)), dμq (x)=lim

log μq (B(x, r)) log r

r→0

=T(q) + q lim

log μφ (B(x, r))

r→0

log r

=T(q) + qdμφ (x)=T(q) − qT ′ (q).

In particular, Corollary 15.6.11 tells us that dμq (x) = T(q) − qT ′ (q) = HD(μq ),

∀x ∈ Xμφ (−T ′ (q)) ⊇ Xq .

(16.91)

Although Xq can be much smaller than Xμφ (−T ′ (q)), surprisingly they have the same

Hausdorff dimension. Indeed, the measure μq restricted to either Xq or Xμφ (−T ′ (q)) satisfies the hypotheses of Theorem 15.6.3 with θ1 = θ2 = T(q) − qT ′ (q). Therefore, HD(Xq ) = HD(Xμφ (−T ′ (q))) = T(q) − qT ′ (q)

(16.92)

Fμφ (−T ′ (q)) = T(q) − qT ′ (q).

(16.93)

and thus

Relations (16.91)–(16.92) establish (c) while (16.93) is the first part of (d). Remarks (i) Replacing the pointwise dimension dν by the lower pointwise dimension dν and replacing accordingly the sets Xν (α) := {x ∈ X | dν (x) = α} by the larger sets X ν (α) := {x ∈ X | dν (x) = α} in the definition of Fν (α), we obtain the same Hausdorff dimension spectra, again by virtue of Theorem 15.6.3. This means that the Fν -spectrum is the same; in particular, it is given by the same Legendre transform formula when ν = μφ . There is no singular part. (ii) Notice that (16.92) means that HD(Xμφ (−T ′ (q))) is the vertical intercept of the tangent line to the graph of T at the point (q, T(q)). (iii) We only used the f -invariance of μφ when estimating HD(Xμφ (−T ′ (q))) from below (through Birkhoff’s ergodic theorem). In the estimate from above, we solely used the Gibbs property of μφ . In a more general setting, it is sufficient that μφ be conformal. In the next steps of the proof, the following will be useful. Claim (Variational principle for q 󳨃→ T(q)). For every ergodic f -invariant Borel probability measure ν on X, consider the linear equation in the two variables q, t hν (f ) + ∫ φq,t dν = 0 X

or equivalently

t = tν (q) =

hν (f ) ν(φ) +q . χν (f ) χν (f )

(16.94)

660 | 16 Conformal expanding repellers Then for every q ∈ ℝ it turns out that T(q) = sup tν (q) = tμq (q), ν

where the supremum is taken over all ergodic f -invariant Borel probability measures ν. Proof of Claim. Fix q ∈ ℝ. Recall from Lemma 16.6.1 that t 󳨃→ P(q, t) is strictly decreasing, and that P(q, T(q)) = 0 by definition. Take any t > T(q). Then P(q, t) < 0. By the variational principle (Theorem 12.1.1), we know that hν (f ) + ∫ φq,t dν ≤ P(q, t) < 0. So the solution tν (q) to the equation hν (f ) + ∫ φq,t dν = 0 cannot be any t > T(q), i. e., tν (q) ≤ T(q). Consequently, sup tν (q) ≤ T(q).

(16.95)

ν

Recall further that μq is the unique equilibrium state for φq,T(q) . Thus, hμq (f ) + ∫ φq,T(q) dμq = P(q, T(q)) = 0. So the solution tμq (q) to the equation hμq (f ) +

∫ φq,t dμq = 0 is precisely T(q). Therefore,

T(q) = tμq (q) ≤ sup tν (q).

(16.96)

ν

The claim follows from (16.95)–(16.96).

◼

(d) We resume the proof of Theorem 16.6.5 by showing the missing part of (d). We have already proved that (cf. (16.91)–(16.93)) Fμφ (−T ′ (q)) = T(q) − qT ′ (q) = HD(Xμφ (−T ′ (q))) = HD(Xq ) = HD(μq ). Note that [−T ′ (∞), −T ′ (−∞)] ⊆ ℝ+ ∪ {0, ∞} since T ′ (q) < 0 for all q ∈ ℝ according to part (b). Notice further that −T ′ (−∞) := − lim T ′ (q) = lim q→−∞

q→−∞

−μq (φ) χμq (f )

≤

‖φ‖∞ 0 for all q ∈ ℝ, it suffices to show that

−μq (φ) χμq (f )

is bounded away from 0 as q → ∞. By (b), we know that

T(1) = 0 and T is strictly decreasing. Choose q0 > 1. Then T(q0 ) < 0. By the claim

16.6 Multifractal analysis of Gibbs states | 661

(variational principle for T), we have tμq (q0 ) ≤ T(q0 ). Since tμq (0) = −q0 Hence,

−μq (φ) χμq (f )

μq (φ)

χμq (f )

hμq (f ) χμq (f )

≥ 0, we get

󵄨 󵄨 = tμq (0) − tμq (q0 ) ≥ 0 − T(q0 ) = 󵄨󵄨󵄨T(q0 )󵄨󵄨󵄨.

≥ |T(q0 )|/q0 > 0 for all q ∈ ℝ.

To end the proof of (d), we need to (d.1) derive the formula for Fμφ at −T ′ (±∞)

when T is not affine and (d.2) prove that Xμφ (α) = 0 for each α ∉ [−T ′ (∞), −T ′ (−∞)]. First, note the following. (d*) For any ergodic f -invariant Borel probability measure ν on X, there exists q ∈ ℝ ∪ {±∞} such that ν(φ) μq (φ) = , χν (f ) χμq (f )

(16.97)

with limq→±∞ in the ±∞ cases. Indeed, by the claim (variational principle for T) the graphs of the functions tν (q) and T(q) do not intersect transversally; they can only be tangent when/if they intersect. Hence, the graph of the linear function tν (q) coincides with the tangent to the graph of T at some point (q0 , T(q0 )), or with one of the asymptotes of T at −∞ or ∞. Now (16.97) follows from (16.94) for ν = μq0 , since the graph of tμq is tan0 gent to the graph of T at (q0 , T(q0 )). (Note that the previous sentence proves the forμ (φ) mula T ′ (q) = χ q (f ) in a different way than in (b), namely via the variational principle μq

for T.) (d.2) Proof that Xμφ (α) = 0 for α ∉ [−T ′ (∞), −T ′ (−∞)]. Suppose there exists

x ∈ X with α := dμφ (x) ∉ [−T ′ (∞), −T ′ (−∞)]. Consider any sequence of integers nk → ∞ and real numbers b1 , b2 such that (cf. Lemma 16.2.12) lim

k→∞

1 S φ(x) = b1 , nk nk

lim −

k→∞

1 ′ 󵄨 󵄨 log󵄨󵄨󵄨(f nk ) (x)󵄨󵄨󵄨 = b2 , nk

and

b1 /b2 = α.

Let τ be any weak∗ -limit of the sequence of measures τnk :=

n −1

1 k ∑δj , nk j=0 f (x)

where δf j (x) is the Dirac measure supported at f j (x). Straightforward calculations show that τ(φ) := ∫ φ dτ = b1 X

Therefore,

τ(φ) −χτ (f )

= b1 /b2 = α.

and

󵄨 󵄨 − χτ (f ) := − ∫ log󵄨󵄨󵄨f ′ 󵄨󵄨󵄨 dτ = b2 . X

662 | 16 Conformal expanding repellers Due to Choquet’s representation theorem or the ergodic decomposition theorem (Theorem 8.2.26), we can assume that τ is ergodic. Indeed, τ is an “average” of ergodic measures. So among the ergodic measures ν comprised in that “average”, ν (φ) ν (φ) there are ν1 such that −χ1 (f ) ≤ α and ν2 such that −χ2 (f ) ≥ α. If α < −T ′ (∞), take ν1

ν2

ν1 as the ergodic τ; if α > −T ′ (−∞), pick ν2 instead. For the ergodic τ found in this way, the limit α can be different than for the original τ, but it will not belong to [−T ′ (∞), −T ′ (−∞)] and we shall use the same symbol α to denote it. Applying Lemma 16.6.3, we get α = dμφ (x) =

τ(φ) −χτ (f )

for τ-a. e. x ∈ X.

But by (16.97) there then exists q ∈ ℝ such that

μq (φ) −χμq (f )

=

τ(φ) −χτ (f )

= α, whence

α ∈ [−T (∞), −T (−∞)]. This contradiction finishes the proof of (d.2). ′

′

Remark. In fact, we proved that for all x ∈ X any limit number of the quotients log μφ (B(x, r))/ log r as r → 0 lies in [−T ′ (∞), −T ′ (−∞)], a statement stronger than dμφ (x) ∈ [−T ′ (∞), −T ′ (−∞)] for all x in the regular part of X. (d.1) On Fμφ (−T ′ (±∞)). Consider any weak∗ -limit τ of a subsequence of measures μq as q → ∞ (a similar argument holds when q → −∞). We shall try to proceed with τ just like we did with μq , though we will meet some difficulties. In particular, we do not know whether τ is ergodic and choosing an ergodic one from its ergodic decomposition might result in the loss of the convergence μq → τ. Nevertheless, using Birkhoff’s ergodic theorem (Theorem 8.2.11) and proceeding as in the proof of Lemma 16.6.3, we get ∫X limn→∞ n1 Sn φ(x) dτ(x)

− ∫X limn→∞ n1

log |(f n )′ (x)| dτ(x)

=

∫ φ dτ

− ∫ log |f ′ | dτ

= lim

q→∞

∫ φ dμq

− ∫ log |f ′ | dμq

= lim (−T ′ (q)) = −T ′ (∞), q→∞

with the convergence over a subsequence of q’s. Since we know by (d.2) that dμφ (x) =

limn→∞ n1 Sn φ(x)

− limn→∞

1 n

log |(f n )′ (x)|

≥ −T ′ (∞),

there must be a set Xτ of full τ-measure over which dμφ (x) = −T ′ (∞) for all x ∈ Xτ .

We conclude that Xτ ⊆ Xμφ (−T ′ (∞)) and hence HD(Xμφ (−T ′ (∞))) ≥ HD(Xτ ) ≥ HD(τ). Now we use the volume lemma with the measure τ. There is no reason for τ to be Gibbs or ergodic, so we rely on the version of the volume lemma in Theorem 16.5.10.

16.7 Exercises | 663

Moreover, thanks to the expansiveness of the repeller, we can use the upper semicontinuity of the measure-theoretic entropy function ν 󳨃→ hν (f ) at τ (see Theorem 12.2.5). We then obtain HD(Xμφ (−T ′ (∞))) ≥ HD(τ) ≥

hμq (f ) hτ (f ) ≥ lim sup = lim (T(q)−qT ′ (q)) = Fμφ (−T ′ (∞)). χτ (f ) q→∞ χμ (f ) q→∞ q

It only remains to estimate HD(Xμφ (−T ′ (∞))) from above. By (16.90), for every

q ∈ ℝ and x ∈ Xμφ (−T ′ (∞)) we have that dμq (x) := lim

log μq (B(x, r))

r→0

log r

= T(q) + q lim

log μφ (B(x, r)) log r

r→0

= T(q) − qT ′ (∞) ≤ T(q) − qT ′ (q).

= T(q) + qdμφ (x)

Thus, HD(Xμφ (−T ′ (∞))) ≤ T(q) − qT ′ (q). Letting q → ∞, we get HD(Xμφ (−T ′ (∞))) ≤

Fμφ (−T ′ (∞)). (e) HP and Rényi spectra. Recall that μφ and μq both have X for topological support, since these measures are Gibbs states and thus do not vanish on open subsets of X (for instance, see Proposition 13.6.14 and recall that μφ (resp., μq ) is equivalent to mφ (resp., mq )). For every finite or countable cover ℬr of X by balls of radius r and of multiplicity at most C, we have 1 ≤ ∑ μq (B) ≤ C. B∈ℬr

By Lemma 16.6.2, q

(Cq C)−1 ≤ r T(q) ∑ (μφ (B)) ≤ Cq C. B∈ℬr

Assume q ≠ 1. Taking logarithms and dividing by (1 − q) log r yields (e) for q ≠ 1. = −T ′ (1) by definition Information dimension. For q = 1, we have limq→1,q=1̸ T(q) 1−q of the derivative. It is equal to HD(μφ ) = PD(μφ ) by (a) and (b), and equal to I(μφ ) by Exercise 16.7.19. This is (e) when q = 1.

16.7 Exercises Exercise 16.7.1. Prove that a C 1 local diffeomorphism T : U → ℝd is conformal if and only if at every point x ∈ U its derivative T ′ (x) : ℝd → ℝd is a similarity map without a translation component, i. e., is of the form rA, where r > 0 and A is an orthogonal matrix.

664 | 16 Conformal expanding repellers Exercise 16.7.2. By providing a specific example, show that if condition (2) is not satisfied in Example 16.2.2, then the limit set X may be empty. Exercise 16.7.3. By providing an example, show that if condition (2) is not satisfied in Example 16.2.2, then the limit set X may not be compact. Exercise 16.7.4. Construct an IFS that satisfies condition (2) but not condition (1) in the SSC (cf. Example 16.2.2) and for which Hutchinson’s formula does not hold. Exercise 16.7.5. By applying Hutchinson’s formula to the conformal expanding repeller from Example 16.3.3, show that the Hausdorff dimension of the middle-third Cantor set is log 2/ log 3. Exercise 16.7.6. Prove Remark 16.2.13. Exercise 16.7.7. Fix d ∈ ℕ. Let 𝒦d be the space of nonempty compact subsets of the unit cube [0, 1]d . In some contrast to Theorem 16.4.11, show that the function 𝒦d ∋ K 󳨃󳨀→ HD(K) ∈ [0, d]

has no continuity point. Exercise 16.7.8. Let T : X → X be a conformal expanding repeller and φ : X → ℝ a Hölder continuous potential. Prove that the following three conditions are equivalent: (a) HD(μφ ) = HD(X). (b) μφ = μ−HD(X) log |T ′ | . (c) The potentials φ and −HD(X) log |T ′ | are cohomologous modulo a constant in the class of continuous (equivalently, Hölder continuous) functions from X to ℝ. Exercise 16.7.9. Strengthening Exercise 16.7.8, show that if T : X → X is a conformal expanding repeller and μ ∈ M(T), then HD(μ) = HD(X) if and only if μ = μ−HD(X) log |T ′ | . Exercise 16.7.10. Prove that if T : X → X is a conformal expanding repeller, then the function Hα (X, ℝ) ∋ φ 󳨃󳨀→ HD(μφ ) is continuous. Exercise 16.7.11. Show that if T : X → X is a conformal expanding repeller and φ : X → ℝ is a Hölder potential, then the function ℝ ∋ t 󳨃󳨀→ HD(μtφ ) is real analytic. Exercise 16.7.12. Under the hypotheses of Theorem 16.4.11, suppose that φ : U → ℝ is a Hölder continuous function. Prove that the map ̂ ∋ λ 󳨃󳨀→ HD(μφ| ) Λ J λ

is real analytic. Exercise 16.7.13. Prove the equality of the Rényi and Hentschel–Procaccia spectra.

16.7 Exercises | 665

Exercise 16.7.14. Show that Lemma 16.6.3 is a generalization of the volume lemma (Theorem 16.5.2) for the pointwise dimension of the measure μφ . Exercise 16.7.15. Show that if φ is not cohomologous to −HD(X) log |T ′ | then the singular part Xμcφ of X is nonempty. Moreover, HD(Xμcφ ) = HD(X). Hint: Using the shadowing lemma (Corollary 4.3.6), find trajectories that have blocks close to blocks of trajectories typical for μ−HD(X) log |T ′ | of length N interchanging with blocks close to blocks typical for μφ of length εN, for N arbitrarily large and ε > 0 arbitrarily small. Exercise 16.7.16. Prove Proposition 16.6.4 about Legendre transform pairs and the remarks preceding and following it. Exercise 16.7.17. For α = −T ′ (1), show that Fμφ (α) = α, Fμ′ φ (α) = 1, and that Fμ′ φ (−T ′ (±∞)) = ±∞. See Figure 16.2. Exercise 16.7.18. Prove (16.87). Exercise 16.7.19. Define the lower and upper information dimensions I(ν) and I(ν) by replacing in the definition of I(ν) (see (16.79)) the limit by the liminf and limsup, respectively. Prove that HD⋆ (ν) ≤ I(ν) ≤ I(ν) ≤ PD⋆ (ν). Exercise 16.7.20. Theorem 16.6.5 applies to infinite conformal expanding repellers. Write a corresponding theorem for finite repellers.

17 Countable state thermodynamic formalism In Chapter 3, we discussed symbolic dynamics over a finite alphabet. In Chapters 11 and 12, we developed thermodynamic formalism for topological dynamical systems, i. e., for self-maps of a compact metrizable phase space, subject to (Hölder) continuous potentials. In Chapter 13, we described more specific formalism for open, distance expanding maps. In this chapter, we present thermodynamic formalism for countablealphabet subshifts of finite type EA∞ generated by finitely irreducible transition matrices A : E × E → {0, 1} on a countable alphabet E. If the alphabet E is finite, then the symbolic space EA∞ is compact, and thus all the knowledge acquired in Chapters 11–13 about pressure, variational principle, equilibrium states, Gibbs states and transfer operators, applies to the shift map σ : EA∞ → EA∞ . If the alphabet E is infinite and the incidence matrix A is finitely irreducible, then the phase space EA∞ is not compact and essentially none of the results proved in the compact case holds directly: thermodynamic formalism in the case of a countably infinite alphabet must be developed anew, although its exposition can be guided and motivated by the compact case to a certain extent. The compact case may also provide valuable hints for some proofs in the noncompact case but there are many important issues specific to the infinite-alphabet case that have no analogue in the finite-alphabet case and must be dealt with entirely on their own. Thermodynamic formalism for countable state subshifts of finite type originated in the papers [43, 44]. However, the incidence matrix A was therein assumed to be such that the resulting symbolic space EA∞ is locally compact. This permitted the performance of a one-point (Alexandroff) compactification of the space EA∞ and the use of the methods developed for compact dynamical systems. Finite irreducibility of A, which is the underlying assumption in this and the next chapter, rules out local compactness of EA∞ and appropriate thermodynamic formalism, which will be unveiled in those two chapters, must be developed independently. Our exposition stems from [77] and [81]. It expands these two sources, in particular improving some results and/or their proofs, or providing alternative arguments. The primary goal of this chapter is to introduce the appropriate concept of topological pressure and to prove the existence and uniqueness of equilibrium states and invariant Gibbs states for potentials that are summable and Hölder continuous on cylinders. The definitions of pressure, equilibrium state and Gibbs state reduce to those in [11] in the case of a finite alphabet. These definitions allow us to outline a rich thermodynamic formalism that serves well in the geometric applications detailed in Chapters 19–21 about countable-alphabet conformal graph directed Markov systems and their special subclass of conformal iterated function systems. In Section 17.1, we give the basic definitions of finite irreducibility and finite primitivity. Unless otherwise stated, the results mentioned in this paragraph assume that the subshift is finitely irreducible. In Section 17.2, we define the pressure. In order to speak of equilibrium states, one needs a variational principle for the pressure and we https://doi.org/10.1515/9783110702699-017

668 | 17 Countable state thermodynamic formalism prove three such principles in Section 17.3. In Section 17.4, we study Gibbs states and prove the existence of a unique invariant Gibbs state. This state is ergodic, and even totally ergodic when the subshift is finitely primitive. In Section 17.5, we identify a simple condition under which the unique invariant Gibbs state is an equilibrium state. Moreover, we show that equilibrium states are unique. Consequently, we deduce that invariant Gibbs states and equilibrium states exist, are unique, and coincide. In Section 17.6, we examine the transfer operator (also called Perron–Frobenius operator) and we demonstrate that its dual admits at most one eigenmeasure, which turns out to be a (nonnecessarily invariant) Gibbs state. In Section 17.7, the existence of such an eigenmeasure is established. So the dual of the transfer operator has a unique eigenmeasure. Its eigenvalue is the exponential of the pressure. The aforementioned facts are summarized in Corollary 17.7.5. Finally, in Section 17.8 we confirm the presence of a pressure gap and, consequently, the positivity of the measure-theoretic entropy of the subshift with respect to the unique equilibrium state. There are two more aspects we would like to emphasize. First, unlike the case of a finite alphabet, in the realm of countable-alphabet subshifts of finite type there are two disjoint classes of Hölder continuous functions that are involved with the transfer operator: those that are summable (and thus are necessarily unbounded from below) and those that are bounded. The former class serves as the source of potentials for Gibbs and equilibrium states while the latter is the Banach space on which the transfer operator acts. Secondly, the method for proving the existence of an eigenmeasure for the transfer operator is to truncate the subshift to larger and larger finite subsets of the alphabet. The transfer operators associated with the finite-subalphabet subshifts each have an eigenmeasure according to Chapter 13. The resulting sequence of Borel probability measures is tight. Prohorov’s theorem then affirms that it has a weak∗ limit point. This limit point is the eigenmeasure of the transfer operator.

17.1 Finitely irreducible subshifts Let E be a countable alphabet and let σ : E ∞ → E ∞ be the shift map, i. e., the map cutting off the first coordinate from any word. It is given by the formula ∞ σ((ωn )∞ n=1 ) := (ωn+1 )n=1 .

We let E 0 := {ϵ}, where ϵ is the empty word, and the set of finite words ∞

E ∗ := ⋃ E n . n=0

For every ω ∈ E ∗ , we denote by |ω| the only integer n ≥ 0 such that ω ∈ E n . We call |ω| the length of ω. If ω ∈ E ∞ and n ∈ ℕ, we set ω|n := ω1 ω2 ⋅ ⋅ ⋅ ωn ∈ E n .

17.1 Finitely irreducible subshifts | 669

Given ω, τ ∈ E ∞ , we define ω ∧ τ ∈ E ∞ ∪ E ∗ to be the longest initial block common to ω and τ. Two words ω, τ ∈ E ∞ ∪ E ∗ are said to be comparable if one of them is the extension of the other, i. e., if |ω ∧ τ| = min{|ω|, |τ|}. Otherwise, they are incomparable. For each 0 < s < 1, we define a metric ds on E ∞ by setting ds (ω, τ) := s|ω∧τ| . These metrics are all topologically equivalent, and so induce the same topology and hence the same Borel sets on E ∞ . A function is uniformly continuous with respect to one of these metrics if and only if it is uniformly continuous with respect to all. If no metric is specifically mentioned, we take it to be d(ω, τ) := de−1 (ω, τ) = e−|ω∧τ| . A matrix A : E ×E → {0, 1} will act as an incidence matrix, i. e., it will dictate which letter(s) may follow any given letter. By definition, letter e2 ∈ E can follow letter e1 ∈ E precisely when Ae1 e2 = 1. Accordingly, we set 󵄨 EA∞ := {ω ∈ E ∞ 󵄨󵄨󵄨 Aωj ωj+1 = 1, ∀j ∈ ℕ}. The elements of EA∞ are called infinite A-admissible words. For each n ≥ 0, let 󵄨 EAn := {ω ∈ E n 󵄨󵄨󵄨 Aωj ωj+1 = 1, ∀1 ≤ j < n}

and set

∞

EA∗ := ⋃ EAn . n=0

The elements of EAn are called A-admissible n-letter words or more simply n-words, while the elements of EA∗ are called finite A-admissible words. For every ω ∈ EA∗ , let [ω] = {τ ∈ EA∞ : τ||ω| = ω}. These sets are called initial A-cylinders. They are both the open and the closed balls in EA∞ , and thus they form a countable base for the topology on EA∞ . Let also 󵄨 EA∞,ω := {γ ∈ EA∞ 󵄨󵄨󵄨 ωγ ∈ EA∞ } =

⋃

[e]

e∈E: Aω|ω| e =1

be the set of all infinite A-admissible suffixes of ω, whereas 󵄨 󵄨 EA∗,ω := {γ ∈ EA∗ 󵄨󵄨󵄨 ωγ ∈ EA∗ } = {γ ∈ EA∗ 󵄨󵄨󵄨 Aω|ω| γ1 = 1} will denote the set of all finite A-admissible suffixes of ω and 󵄨 󵄨 EAn,ω := {γ ∈ EAn 󵄨󵄨󵄨 ωγ ∈ EA∗ } = {γ ∈ EAn 󵄨󵄨󵄨 Aω|ω| γ1 = 1} will be the set of all A-admissible n-letter suffixes of ω.

670 | 17 Countable state thermodynamic formalism The following fact is obvious. Proposition 17.1.1. EA∞ is a closed invariant subset of E ∞ , and hence σ : EA∞ → EA∞ is a subshift of the full E-shift σ : E ∞ → E ∞ . The subshifts we will restrict our attention to are generated by a finitely irreducible matrix. They are a generalization of irreducible subshifts over finite alphabets. Definition 17.1.2. The matrix A : E × E → {0, 1} is said to be finitely irreducible if there is a finite set Λ ⊆ EA∗ such that for all e1 , e2 ∈ E there exists a word ω = ω(e1 , e2 ) ∈ Λ such that e1 ωe2 ∈ EA∗ . It will sometimes be helpful to replace the original set Λ by a finite set of words of a specific minimal length. Lemma 17.1.3. Suppose that a matrix A : E × E → {0, 1} is finitely irreducible and let Λ ⊆ EA∗ be a finite set which witnesses the finite irreducibility of A. For any L ∈ ℕ, there exists a finite set ΛL ⊆ EA∗ which witnesses the finite irreducibility of A and whose words have length at least L. Proof. Let L ∈ ℕ. Recall that ϵ is the empty word, the only word of length 0. If Λ = {ϵ}, then all transitions are allowed and one can thus set Λ = {eL } for any e ∈ E. Assume now that Λ ⊇ {ϵ, η} for some η ∈ EA∗ \ {ϵ}. Consider ̃ = {αηω ∈ E ∗ : α, ω ∈ Λ}. Λ A ̃ ≤ (#Λ)2 < ∞. Let e1 , e2 ∈ E. As Λ witnesses the finite irreducibility of A, there Then #Λ exists α, ω ∈ Λ such that e1 αηωe2 ∈ EA∗ . Moreover, |αηω| = |α| + |η| + |ω| ≥ |η| ≥ 1. ̃ is a finite set of words of length at least Since e1 , e2 ∈ E are arbitrary, it follows that Λ 1 witnessing the finite irreducibility of A. Take ̃ L ∩ E∗. Λ′ = (Λ) A ̃ L < ∞ and words in Λ′ have length at least L. It remains to show Clearly, #Λ′ ≤ (#Λ) ̃ that this set witnesses the finite irreducibility of A. Let e1 , e2 ∈ E. There exists α1 ∈ Λ ∗ ∗ ̃ such that e1 α1 ∈ EA . In turn, there is α2 ∈ Λ such that α1 α2 ∈ EA . Continuing in this ̃ such that αk αk+1 ∈ E ∗ , and this for every 1 ≤ k < L − 1. way, one can find αk+1 ∈ Λ A ̃ such that αL−1 αL e2 ∈ E ∗ . Then the word Once αL−1 has been found, there is αL ∈ Λ A α′ = α1 α2 . . . αL ∈ Λ′ is such that e1 α′ e2 ∈ EA∗ . Since e1 , e2 ∈ E are arbitrary, the set Λ′ witnesses the finite irreducibility of A. 17.1.1 Finitely primitive subshifts At times, we will restrict our attention to subshifts that are generated by a finitely primitive matrix. This is a stronger requirement than finite irreducibility.

17.2 Topological pressure

| 671

Definition 17.1.4. The matrix A : E × E → {0, 1} is said to be finitely primitive if there is a finite set Ω ⊆ EA∗ of words of equal length such that for all e1 , e2 ∈ E there is a word ω = ω(e1 , e2 ) ∈ Ω for which e1 ωe2 ∈ EA∗ .

17.2 Topological pressure We will concentrate our attention on a family of potentials that are natural generalizations of Hölder continuous potentials on compact spaces.

17.2.1 Potentials: acceptability and Hölder continuity on cylinders Let us recall the notion of Birkhoff (or ergodic) sum, this time in the context of symbolic systems. Given a potential f : EA∞ → ℝ, the nth Birkhoff sum of f at a word τ ∈ EA∞ is n−1

Sn f (τ) = ∑ f (σ j (τ)). j=0

This is the sum of the values of potential f at the first n points in the forward orbit of τ under the shift map σ : EA∞ → EA∞ . For every G ⊆ EA∞ , let Sn f (G) := sup Sn f (τ). τ∈G

We shall be interested in one special class of potentials. Definition 17.2.1. A potential f : EA∞ → ℝ is said to be acceptable provided that it is uniformly continuous and osc(f ) := osc(f , 𝒰E ) = sup[sup(f |[e] ) − inf(f |[e] )] < ∞. e∈E

That is, a potential is acceptable if it is uniformly continuous and has finite oscillation over the partition 𝒰E := {[e] : e ∈ E} of EA∞ into its initial 1-cylinders. Note that an acceptable potential need not be bounded when E is infinite. A particularly interesting subclass of acceptable potentials consists of those which are Hölder continuous on cylinders. Definition 17.2.2. A function f : EA∞ → ℝ is said to be Hölder continuous on cylinders if there exist constants β > 0 and c ≥ 0 for which β 󵄨 󵄨 |ρ ∧ ζ | ≥ 1 󳨐⇒ 󵄨󵄨󵄨f (ρ) − f (ζ )󵄨󵄨󵄨 ≤ c[d(ρ, ζ )] .

672 | 17 Countable state thermodynamic formalism The constant β is called an Hölder exponent of f and the least constant c with this property will be denoted by υβ (f ). Note that a function is Hölder continuous on cylinders with respect to the metric d(ρ, ζ ) = e−|ρ∧ζ | if and only if it is Hölder continuous on cylinders with respect to all metrics ds (ρ, ζ ) = s|ρ∧ζ | , 0 < s < 1. Of course, the Hölder exponent β depends on the metric ds . Functions that are Hölder continuous on cylinders are obviously bounded on every cylinder but they need not be (globally) bounded when E is infinite. In fact, Hölder continuity on cylinders and (global) boundedness are, together, equivalent to global Hölder continuity. The ergodic sums of functions that are Hölder continuous on cylinders have bounded variation on cylinders (cf. Lemma 13.1.1). This flexibility is crucial when E is infinite. Lemma 17.2.3 (Bounded variation principle for ergodic sums). If a potential f : EA∞ → ℝ is Hölder continuous on cylinders with exponent β, then for all n ∈ ℕ, all ω ∈ EAn , and all ρ, γ ∈ EA∞,ω , we have υβ (f ) −β|ρ∧γ| β 󵄨󵄨 󵄨 υβ (f ) [d(ρ, γ)] = β e . 󵄨󵄨Sn f (ωρ) − Sn f (ωγ)󵄨󵄨󵄨 ≤ β e −1 e −1 In particular, given ξ , ζ ∈ EA∞ , 󵄨 󵄨 υβ (f ) |ξ ∧ ζ | ≥ n 󳨐⇒ 󵄨󵄨󵄨Sn f (ξ ) − Sn f (ζ )󵄨󵄨󵄨 ≤ β . e −1 Consequently, for any τω ∈ EA∗ , S|τ| f ([τ]) + S|ω| f ([ω]) −

υβ (f )

eβ − 1

≤ S|τω| f ([τω]) ≤ S|τ| f ([τ]) + S|ω| f ([ω]).

Proof. Let n ∈ ℕ, ω ∈ EAn and ρ, γ ∈ EA∞,ω . Then n−1

󵄨󵄨 󵄨 󵄨 j 󵄨 j 󵄨󵄨Sn f (ωρ) − Sn f (ωγ)󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨f (σ (ωρ)) − f (σ (ωγ))󵄨󵄨󵄨 j=0

n−1

β

≤ ∑ υβ (f )[d(σ j (ωρ), σ j (ωγ))] j=0

n−1

β

= υβ (f ) ∑ e−β(n−j) [d(ρ, γ)] j=0

≤

υβ (f )

eβ − 1

β

[d(ρ, γ)] ≤

υβ (f )

eβ − 1

.

17.2 Topological pressure

| 673

For any τω ∈ EA∗ , this implies that S|τω| f ([τω]) = sup S|τ|+|ω| f (ξ ) = sup [S|τ| f (ξ ) + S|ω| f (σ |τ| (ξ ))] ξ ∈[τω]

ξ ∈[τω]

≥ inf S|τ| f (ξ ) + sup S|ω| f (σ |τ| (ξ )) ξ ∈[τω]

ξ ∈[τω]

≥ inf S|τ| f (ζ ) + sup S|ω| f (χ) ζ ∈[τ]

χ∈[ω]

≥ sup S|τ| f (ζ ) − ζ ∈[τ]

υβ (f )

eβ − 1

+ sup S|ω| f (χ) χ∈[ω]

= S|τ| f ([τ]) + S|ω| f ([ω]) −

υβ (f )

eβ − 1

.

On the other hand, S|τω| f ([τω]) = sup [S|τ| f (ξ ) + S|ω| f (σ |τ| (ξ ))] ξ ∈[τω]

≤ sup S|τ| f (ξ ) + sup S|ω| f (σ |τ| (ξ )) ξ ∈[τω]

ξ ∈[τω]

≤ sup S|τ| f (ζ ) + sup S|ω| f (χ) χ∈[ω]

ζ ∈[τ]

= S|τ| f ([τ]) + S|ω| f ([ω]). 17.2.2 Partition functions Subalphabets will later play an important role since some properties of infinite systems can be derived from the corresponding properties of their finite subsystems. Given a subalphabet F ⊆ E, let F ∞ = {ω ∈ E ∞ : ωi ∈ F, ∀i ∈ ℕ}

and FA∞ = F ∞ ∩ EA∞ .

Similarly, let FAn = F n ∩ EAn

and FA∗ = F ∗ ∩ EA∗ .

Then σF := σ|F ∞ : FA∞ → FA∞ is a subshift of σ : EA∞ → EA∞ . For any ω ∈ FA∗ , let A [ω]F = [ω] ∩ FA∞ . Notice that σE = σ and [ω]E = [ω]. Note also that if A : E × E → {0, 1} is finitely irreducible and Λ ⊆ EA∗ is a finite set witnessing the finite irreducibility of A, then A : F × F → {0, 1} is finitely irreducible when ΛE ⊆ F, where ΛE = {ωk : ω ∈ Λ and 1 ≤ k ≤ |ω|}. If 𝒰F = {[e]F : e ∈ F} is the open partition (and hence an open cover) of FA∞ into −j n ∞ its initial 1-cylinders, then 𝒰Fn := ⋁n−1 j=0 σF (𝒰F ) = {[ω]F : ω ∈ FA } is the partition of FA into its initial n-cylinders. Definition 11.1.2 states that the nth partition function of 𝒰F (or to lighten terminology and notation, the nth partition function of F) with respect

674 | 17 Countable state thermodynamic formalism to a potential f is Zn (f , F) := Zn (f , 𝒰F ) = ∑ exp(Sn f ([ω]F )). ω∈FAn

Remark 17.2.4. (a) By convention, S|ω| f ([ω]F ) = −∞ when [ω]F = 0, and e−∞ = 0. This is justified by the expectation that Zn (f , F) be the number of elements in the cover 𝒰Fn when the potential f ≡ 0. (b) 0 ≤ Zn (f , F) ≤ ∞ for all n ∈ ℕ. (c) If #F < ∞, then Zn (f , F) < ∞ for all n ∈ ℕ. (d) ∃n ∈ ℕ, Zn (f , F) = 0 ⇐⇒ Zn (f , F) = 0, ∀n ∈ ℕ 󳨐⇒ FA∞ = 0. (e) If F = E, we suppress the subscript and write simply Zn (f ) for Zn (f , E). That is, Zn (f ) = ∑ exp(Sn f ([ω])).

(17.1)

ω∈EAn

We now address the question of bounded sub- and supermultiplicativity of the partition functions. A sequence (an )∞ n=1 of positive real numbers is said to be boundedly submultiplicative with constant B > 0 if am+n ≤ Bam an ,

∀m, n ∈ ℕ.

If 0 < B ≤ 1, then the sequence is submultiplicative. Similarly, a sequence (an )∞ n=1 of positive real numbers is boundedly supermultiplicative with constant C > 0 if C −1 am an ≤ am+n ,

∀m, n ∈ ℕ.

If 0 < C ≤ 1, then the sequence is supermultiplicative. Lemma 17.2.5. For any F ⊆ E, the sequence (Zn (f , F))∞ n=1 is submultiplicative. Moreover, if f : EA∞ → ℝ is acceptable and A is finitely irreducible, then there exists a constant R > 0 such that Zn+1 (f ) ≥ RZn (f ) for all n ∈ ℕ. In particular, Z1 (f ) = ∞ ⇐⇒ ∃n ∈ ℕ,

Zn (f ) = ∞ ⇐⇒ Zn (f ) = ∞,

∀n ∈ ℕ.

If, in addition, f is Hölder continuous on cylinders, then the sequence (Zn (f ))∞ n=1 is boundedly supermultiplicative. Proof. Let us first prove the submultiplicativity. We must show that Zm+n (f , F) ≤ Zm (f , F)Zn (f , F) for all m, n ∈ ℕ. And indeed, Zm+n (f , F) = ≤

∑ exp(Sm+n f ([ω]F ))

ω∈FAm+n

∑ exp(Sm f ([ω]F ) + Sn f ([σ m (ω)]F ))

ω∈FAm+n

17.2 Topological pressure

| 675

≤ ∑ ∑ exp(Sm f ([χ]F ) + Sn f ([ρ]F )) χ∈FAm ρ∈FAn

= ∑ exp(Sm f ([χ]F )) ∑ exp(Sn f ([ρ]F )) χ∈FAm

ρ∈FAn

= Zm (f , F)Zn (f , F). Now, suppose that A is finitely irreducible, and that infζ ∈[e] f (ζ ) > −∞ for all e ∈ E (this is the case when f is acceptable). Let Λ ⊆ EA∗ be a finite set which witnesses the finite irreducibility of A. Let also ΛE := {ωk : ω ∈ Λ and 1 ≤ k ≤ |ω|}. Set m := min{inf f |[e] : e ∈ ΛE } and observe that m > −∞. For every ω ∈ EA∗ , let eω ∈ ΛE be such that eω ω ∈ EA∗ . Then Zn+1 (f ) = ∑ exp(Sn+1 f ([τ])) τ∈EAn+1

≥ ∑ exp(Sn+1 f ([eω ω])) ω∈EAn

= ∑ exp( sup (f (ρ) + Sn f (σ(ρ)))) ω∈EAn

ρ∈[eω ω]

≥ ∑ exp( inf f (ζ ) + sup Sn f (σ(ρ))) ω∈EAn

ζ ∈[eω ω]

ρ∈[eω ω]

≥ ∑ exp( inf f (ζ )) exp( sup Sn f (χ)) ω∈EAn

ζ ∈[eω ]

χ∈[ω]

≥ em ∑ exp(Sn f ([ω])) ω∈EAn

= em Zn (f ). Together with submultiplicativity, this implies that Z1 (f )e(n−1)m ≤ Zn (f ) ≤ (Z1 (f ))n for all n ∈ ℕ. In particular, Z1 (f ) = ∞ if and only if Zn (f ) = ∞ for some n ∈ ℕ if and only if Zn (f ) = ∞ for all n ∈ ℕ. Finally, we show that the partition functions of the system are boundedly supermultiplicative when the potential f is additionally Hölder continuous on cylinders. We must thus show that there exists 0 < Q < ∞ such that Zm (f )Zn (f ) ≤ QZm+n (f ),

∀m, n ∈ ℕ.

Since Z1 (f ) = ∞ if and only if Zn (f ) = ∞ for all n ∈ ℕ, bounded supermultiplicativity is trivial if Z1 (f ) = ∞. So assume that Z1 (f ) < ∞. Let Λ ⊆ EA∗ be a finite set which makes the matrix A finitely irreducible. Set |Λ| := max{|ω| : ω ∈ Λ} and M := min{S|ω| f ([ω]) : ω ∈ Λ}, and notice that −∞ < M < ∞ according to Lemma 17.2.3. For every couple (e1 , e2 ) ∈ E 2 , let ω(e1 , e2 ) ∈ Λ be such that e1 ω(e1 , e2 )e2 ∈ EA∗ . More generally, for every

676 | 17 Countable state thermodynamic formalism couple (τ, χ) ∈ (EA∗ \{ϵ})2 define ω(τ, χ) := ω(τ|τ| , χ1 ). Fix m, n ∈ ℕ. The map |Λ|

l : EAm × EAn 󳨀→ ⋃ EAm+n+k k=0

(τ, χ) 󳨃󳨀→ τω(τ, χ)χ is clearly injective. Using the principle of bounded variation (Lemma 17.2.3), we obtain Zm (f )Zn (f ) = ∑ exp(Sm f ([τ])) ∑ exp(Sn f ([χ])) χ∈EAn

τ∈EAm

≤

1 ∑ ∑ exp(Sm f ([τ]) + S|ω(τ,χ)| f ([ω(τ, χ)]) + Sn f ([χ])) eM τ∈E m χ∈E n A

≤

e

≤e

2

eβ −1

eM

2

A

υβ (f )

υβ (f ) eβ −1

∑ ∑ exp(S|l(τ,χ)| f ([l(τ, χ)]))

τ∈EAm χ∈EAn −M

|Λ|

∑

∑

k=0 ω∈E m+n+k

exp(Sm+n+k f ([ω]))

A

=e ≤e

where Q = e

2

υβ (f ) eβ −1

−M

2

υβ (f ) eβ −1

−M

|Λ|

∑ Zm+n+k (f )

k=0 2

υβ (f ) eβ −1

−M

|Λ|

∑ Zk (f ) ⋅ Zm+n (f ) = QZm+n (f ),

k=0

Z (f ) ≤ e ∑|Λ| k=0 k

2

υβ (f ) −M eβ −1

(|Λ| + 1) max{1, (Z1 (f ))|Λ| }.

We now aim at proving that every partition function of the system coincides with the supremum of the corresponding partition functions of its finite subsystems for acceptable potentials. Lemma 17.2.6. If f : EA∞ → ℝ is acceptable and A is finitely irreducible, then Zn (f ) = sup{Zn (f , F) : F ⊆ E and #F < ∞} for every n ∈ ℕ. Proof. Fix n ∈ ℕ. The inequality Zn (f ) ≥ sup{Zn (f , F)} is obvious. To prove the converse, suppose first that Zn (f ) < ∞. Let ε > 0. There exists a finite set Ω = Ω(ε) ⊆ EAn such that Zn (f ) ≤ eε ∑ exp(Sn f ([ω])). ω∈Ω

(17.2)

17.2 Topological pressure

| 677

Since f is acceptable, Sn f ([ω]) < ∞ for every ω ∈ EAn . So for every ω ∈ Ω, there is ρ(ω) ∈ [ω] such that 󵄨󵄨 󵄨 󵄨󵄨Sn f (ρ(ω)) − Sn f ([ω])󵄨󵄨󵄨 ≤ ε. It follows from (17.2) that Zn (f ) ≤ e2ε ∑ exp(Sn f (ρ(ω))). ω∈Ω

(17.3)

By the uniform continuity of f , there exists ℓ = ℓ(ε) ∈ ℕ such that 󵄨 󵄨 ε |χ ∧ ζ | ≥ ℓ 󳨐⇒ 󵄨󵄨󵄨f (χ) − f (ζ )󵄨󵄨󵄨 ≤ . n Let Λ ⊆ EA∗ be a finite set which makes the matrix A finitely irreducible. Let F = F(ε) = {(ρ(ω))j : ω ∈ Ω

and

1 ≤ j ≤ ℓ + n} ∪ ΛE ,

where ΛE := {τk : τ ∈ Λ and 1 ≤ k ≤ |τ|}. Clearly, #F < ∞ and FA∞ ≠ 0. In fact, A : F × F → {0, 1} is (finitely) irreducible, so [χ]F ≠ 0 for all χ ∈ FA∗ \{ϵ}. Note also that F ⊇ ΩE , where ΩE := {ωk : ω ∈ Ω and 1 ≤ k ≤ |ω|}. Therefore, FAn ⊇ Ω. For each ω ∈ Ω, let ρ̃(ω) ∈ [(ρ(ω))1 ⋅ ⋅ ⋅ (ρ(ω))ℓ+n ]F ⊆ [(ρ(ω))1 ⋅ ⋅ ⋅ (ρ(ω))n ]F = [ω]F .

(17.4)

Then |ρ(ω) ∧ ρ̃(ω)| ≥ ℓ + n, and thus |σ j (ρ(ω)) ∧ σ j (ρ̃(ω))| ≥ ℓ + n − j ≥ ℓ for all 0 ≤ j ≤ n. Consequently, n−1 n−1 ε 󵄨󵄨 󵄨 󵄨 j 󵄨 j 󵄨󵄨Sn f (ρ(ω)) − Sn f (ρ̃(ω))󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨f (σ (ρ(ω))) − f (σ (ρ̃(ω)))󵄨󵄨󵄨 ≤ ∑ = ε. n j=0 j=0

It follows from this, (17.3) and (17.4) that Zn (f ) ≤ e3ε ∑ exp(Sn f (ρ̃(ω))) ≤ e3ε ∑ exp(Sn f ([ω]F )) = e3ε Zn (f , F). ω∈Ω

ω∈FAn

Therefore Zn (f ) ≤ e3ε sup{Zn (f , F)}. Since ε > 0 was chosen arbitrarily, we deduce that Zn (f ) ≤ sup{Zn (f , F)}. A similar proof applies to the case Zn (f ) = ∞. 17.2.3 The pressure function Theorem 11.1.26 suggests defining the topological pressure PF (f ) of σF : FA∞ → FA∞ when subject to the potential f as the topological pressure of σF with respect to the par-

678 | 17 Countable state thermodynamic formalism tition 𝒰F = {[e]F : e ∈ F}, i. e., P(σF , f , 𝒰F ) as in Definition 11.1.11. This would amount to defining PF (f ) as the asymptotic exponential growth rate of the sequence (Zn (f , F))∞ n=1 , 1 ∞ i. e., as the limit of the sequence ( n log Zn (f , F))n=1 . But despite being submultiplicative, the partition functions (Zn (f , F))∞ n=1 may be infinite and Lemma 3.2.17 may not apply. That lemma affirms that for any submul1 ∞ tiplicative sequence (an )∞ n=1 of positive real numbers, the sequence ( n log an )n=1 converges and lim

n→∞

1 1 log an = inf log an . n∈ℕ n n

(17.5)

However, this result does not generally apply to submultiplicative sequences of extended real numbers (e. g., consider ∞, 1, ∞, 1, . . .). We need to be more cautious. Definition 17.2.7. The topological pressure PF (f ) of the shift map σF : FA∞ → FA∞ under a potential f is defined to be PF (f ) := lim sup n→∞

1 log Zn (f , F). n

If F = E, we suppress the subscript and simply write P(f ) for PE (f ). We now prove that the finite or infinite nature of the pressure of a symbolic system is determined by the nature of any of its partition functions. Theorem 17.2.8. If f : EA∞ → ℝ is acceptable and A is finitely irreducible, then the following statements hold: (a) P(f ) < ∞ ⇐⇒ Z1 (f ) < ∞ ⇐⇒ Zn (f ) < ∞ for some n ∈ ℕ ⇐⇒ Zn (f ) < ∞, ∀n ∈ ℕ. (b) P(f ) = ∞ ⇐⇒ Z1 (f ) = ∞ ⇐⇒ Zn (f ) = ∞ for some n ∈ ℕ ⇐⇒ Zn (f ) = ∞, ∀n ∈ ℕ. 1 1 (c) P(f ) = lim log Zn (f ) = inf log Zn (f ). n→∞ n n∈ℕ n Proof. Lemma 17.2.5 asserts that Z1 (f ) = ∞ if and only if Zn (f ) = ∞ for some n ∈ ℕ if and only if Zn (f ) = ∞ for all n ∈ ℕ. So if Z1 (f ) = ∞, then P(f ) = lim

n→∞

1 1 log Zn (f ) = inf log Zn (f ) = ∞. n∈ℕ n n

On the other hand, Z1 (f ) < ∞ if and only if Zn (f ) < ∞ for some n ∈ ℕ if and only if Zn (f ) < ∞ for all n ∈ ℕ. So if Z1 (f ) < ∞, since the Zn (f ), n ∈ ℕ, are submultiplicative, relation (17.5) applies and gives that P(f ) = lim

n→∞

1 1 log Zn (f ) = inf log Zn (f ) < ∞. n∈ℕ n n

We now demonstrate that, as for the partition functions, the pressure of the system is the supremum of the pressures of all its finite subsystems.

17.2 Topological pressure

| 679

Theorem 17.2.9. If f : EA∞ → ℝ is acceptable and A is finitely irreducible, then P(f ) = sup{PF (f ) : F ⊆ E and #F < ∞}. Proof. The inequality P(f ) ≥ sup{PF (f )} is obvious. To prove the converse, suppose first that P(f ) < ∞. Fix ε > 0. By the uniform continuity of f , there exists ℓ = ℓ(ε) ∈ ℕ such that 󵄨 󵄨 |ρ ∧ ζ | ≥ ℓ 󳨐⇒ 󵄨󵄨󵄨f (ρ) − f (ζ )󵄨󵄨󵄨 ≤ ε. Let M = (ℓ − 1)osc(f ), and k ≥ ℓ be such that kε ≥ M. For any ω ∈ EAk and ρ, ζ ∈ [ω], we have that |ρ ∧ ζ | ≥ |ω| = k, and thus |σ j (ρ) ∧ σ j (ζ )| ≥ k − j ≥ ℓ whenever 0 ≤ j ≤ k − ℓ. Therefore, k−1

󵄨󵄨 󵄨 󵄨 j 󵄨 j 󵄨󵄨Sk f (ρ) − Sk f (ζ )󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨f (σ (ρ)) − f (σ (ζ ))󵄨󵄨󵄨 j=0

k−ℓ

k−1

j=0

j=k−ℓ+1

󵄨 󵄨 = ∑ 󵄨󵄨󵄨f (σ j (ρ)) − f (σ j (ζ ))󵄨󵄨󵄨 +

󵄨 󵄨 ∑ 󵄨󵄨󵄨f (σ j (ρ)) − f (σ j (ζ ))󵄨󵄨󵄨

≤ (k − ℓ + 1)ε + (ℓ − 1)osc(f ) ≤ kε + M ≤ 2kε. Thus, 0 ≤ Sk f ([ω]) − Sk f ([ω]) ≤ 2kε,

(17.6)

where Sk f ([ω]) = supζ ∈[ω] Sk f (ζ ) and Sk f ([ω]) = infζ ∈[ω] Sk f (ζ ). Let Λ ⊆ EA∗ be a finite set which witnesses the finite irreducibility of A. Let |Λ| := max{|ω| : ω ∈ Λ} < ∞. By Theorem 17.2.8, we know that k1 log Zk (f ) ≥ P(f ). 1 Choosing k large enough, we may suppose that k+|Λ| log Zk (f ) > P(f ) − ε. According to Lemma 17.2.6, there then exists a finite set F ⊆ E such that 1 log Zk (f , F) ≥ P(f ) − ε. k + |Λ|

(17.7)

Enlarging F if necessary, we may assume that F ⊇ ΛE , where ΛE := {ωi : ω ∈ Λ and 1 ≤ i ≤ |ω|}. Then FA∗ ⊇ Λ and for each τ = (τ(1) , τ(2) , . . . , τ(n) ) ∈ (FAk )n (not necessarily in FAkn ), there are words α(1) , α(2) , . . . , α(n−1) ∈ Λ such that τ := τ(1) α(1) τ(2) α(2) ⋅ ⋅ ⋅ τ(n−1) α(n−1) τ(n) ∈ FA∗ . Note that kn ≤ |τ| ≤ kn + |Λ|(n − 1) and that the mapping τ 󳨃→ τ is at most un−1 -to-1, where u is the number of different lengths among the words composing Λ. Indeed, suppose that τ(1) α(1) τ(2) α(2) ⋅ ⋅ ⋅ τ(n−1) α(n−1) τ(n) = ω(1) β(1) ω(2) β(2) ⋅ ⋅ ⋅ ω(n−1) β(n−1) ω(n) ∈ FA∗ ,

680 | 17 Countable state thermodynamic formalism where τ(i) , ω(i) ∈ FAk for all 1 ≤ i ≤ n and α(j) , β(j) ∈ Λ for all 1 ≤ j < n. Comparing initial subwords, it is clear that τ(1) = ω(1) since |τ(1) | = |ω(1) | = k. If |α(1) | = |β(1) |, then α(1) = β(1) . It follows that τ(2) = ω(2) . If additionally |α(2) | = |β(2) |, then α(2) = β(2) . It follows that τ(3) = ω(3) . More generally, if |α(j) | = |β(j) | for all j, then τ(i) = ω(i) for all i and α(j) = β(j) for all j. This means that for each (n − 1)-tuple of lengths for the α(j) ’s, the mapping τ 󳨃→ τ has at most one preimage. Since there are u different potential lengths for each α(j) , there are at most un−1 different (n − 1)-tuples of lengths, and thus the mapping τ 󳨃→ τ is at most un−1 -to-1. Setting −∞ < m := min{S|ω| f ([ω]) : ω ∈ Λ} < ∞, for every n ∈ ℕ we obtain that un−1

kn+|Λ|(n−1)

∑ j=kn

Zj (f , F) = un−1

kn+|Λ|(n−1)

∑ exp(S|ω| f ([ω]F ))

∑ j=kn

ω∈EAj

≥ ∑ exp(S|τ| f ([τ]F )) τ∈(FAk )n

≥ ∑ exp(S|τ| f ([τ]F )) τ∈(FAk )n

n

≥ ∑ exp(∑ Sk f ([τ(i) ]F ) + m(n − 1)) i=1

τ∈(FAk )n

n

≥ em(n−1) ∑ exp ∑(Sk f ([τ(i) ]F ) − 2kε) by (17.6) i=1

τ∈(FAk )n

n

= em(n−1)−2kεn ∑ ⋅ ⋅ ⋅ ∑ ∏ exp(Sk f ([τ(i) ]F )) τ(1) ∈FAk

τ(n) ∈FAk i=1

= em(n−1)−2kεn ∑ exp(Sk f ([τ(1) ]F )) ⋅ τ(1) ∈FAk

⋅ ⋅ ⋅ ⋅ ∑ exp(Sk f ([τ(n) ]F )) τ(n) ∈FAk

n

= e−m en(m−2kε) ( ∑ exp(Sk f ([τ′ ]F ))) τ′ ∈FAk

n

= e−m en(m−2kε) (Zk (f , F)) . Hence, there exists kn ≤ jn ≤ kn + |Λ|(n − 1) ≤ (k + |Λ|)n such that Zjn (f , F) ≥

1 n ue−m en(m−2kε−log u) (Zk (f , F)) . |Λ|(n − 1) + 1

17.3 Variational principles and equilibrium states | 681

Using (17.7), we obtain that PF (f ) = lim

n→∞

1 |m| + 2kε + log u log Zjn (f , F) ≥ − + P(f ) − ε ≥ P(f ) − 5ε jn k

as long as k is chosen large enough. Letting ε ↘ 0, the theorem follows in the case P(f ) < ∞. The case P(f ) = ∞ can be treated similarly.

17.3 Variational principles and equilibrium states The reader may remember the variational principle (Theorem 12.1.1). This principle applies to all dynamical systems living on compact spaces but not necessarily to those which subsist on noncompact spaces. We shall obtain three variational principles that rule in the current symbolic setting. Definition 17.3.1. A Borel probability measure μ on EA∞ is said to be finitely supported provided that there exists a finite set F ⊆ E such that supp(μ) ⊆ FA∞ . Let f : EA∞ → ℝ be a continuous potential. The variational principle (Theorem 12.1.1), which applies to finitely supported measures since FA∞ is compact whenever F is finite, tells us that PF (f ) = sup{hμ (σ) + ∫ f dμ} μ

(17.8)

EA∞

for every finite set F ⊆ E, where the supremum is taken over the set of all σ-invariant Borel probability measures μ on EA∞ that are supported on FA∞ . In fact, according to Corollary 12.1.10, this supremum can be restricted to the subset of those measures that are ergodic. Applying Theorem 17.2.9, we readily obtain the following principle for acceptable potentials (cf. Definition 17.2.1). Theorem 17.3.2 (1st variational principle). If f : EA∞ → ℝ is an acceptable potential and A is a finitely irreducible matrix, then P(f ) = sup{hμ (σ) + ∫ f dμ}, μ

EA∞

where the supremum is taken over the set of all finitely supported σ-invariant Borel probability measures μ on EA∞ . In fact, the supremum can be restricted to the subset of those measures that are ergodic. Now, let 𝒰E := {[e] : e ∈ E} be the partition of EA∞ into its initial cylinders of −j ∞ length 1. Then 𝒰En = ⋁n−1 j=0 σ (𝒰E ) is the partition of EA into its initial cylinders of length n.

682 | 17 Countable state thermodynamic formalism Theorem 17.3.3 (2nd variational principle). If f : EA∞ → ℝ is an acceptable potential and A is a finitely irreducible matrix, then P(f ) ≥ sup{hμ (σ) + ∫ f dμ}, μ

EA∞

where the supremum is taken over all σ-invariant Borel probability measures μ on EA∞ such that ∫ f dμ > −∞. In addition, if P(f ) < ∞ then Hμ (𝒰En ) < ∞ for all n ∈ ℕ and all such measures μ. Proof. If P(f ) = ∞, there is nothing to prove. So suppose that P(f ) < ∞. By Theorem 17.2.8, this means that Zn (f ) < ∞ for every n ∈ ℕ. Also, we have ∑ μ([ω])Sn f ([ω]) ≥ ∫ Sn f dμ = n ∫ f dμ > −∞.

ω∈EAn

EA∞

EA∞

Therefore, using the concavity of the function k(x) = −x log x, we obtain that Hμ (𝒰En ) + ∫ Sn f dμ ≤ ∑ μ([ω])(Sn f ([ω]) − log μ([ω])) ω∈EAn

EA∞

= Zn (f ) ∑ Zn (f )−1 eSn f ([ω]) k(μ([ω])e−Sn f ([ω]) ) ω∈EAn

≤ Zn (f )k( ∑ Zn (f )−1 eSn f ([ω]) μ([ω])e−Sn f ([ω]) ) ω∈EAn

= Zn (f )k(Zn (f )−1 ) = log Zn (f ). Hence, Hμ (𝒰En ) ≤ log Zn (f ) − n ∫E ∞ f dμ < ∞ for every n ∈ ℕ. Since the partition 𝒰E is A

a generator (cf. Definition 9.4.19), upon dividing both sides by n, passing to the limit as n → ∞ and using a straightforward generalization of Corollary 9.4.18 to countable measurable partitions, we conclude that hμ (σ) + ∫ f dμ = lim EA∞

n→∞

1 1 (H (𝒰 n ) + ∫ Sn f dμ) ≤ lim log Zn (f ) = P(f ). n→∞ n n μ E ∞ EA

Observe that if f is an acceptable potential and μ is a finitely supported Borel probability measure, then ∫E ∞ f dμ > −∞. Hence, as a direct consequence of TheoA

rems 17.3.2–17.3.3, we deduce the following.

17.4 Gibbs states | 683

Theorem 17.3.4 (3rd variational principle). If f : EA∞ → ℝ is an acceptable potential and A is a finitely irreducible matrix, then P(f ) = sup{hμ (σ) + ∫ f dμ}, μ

EA∞

where the supremum is taken over the set of all σ-invariant Borel probability measures μ on EA∞ such that ∫E ∞ f dμ > −∞. In fact, the supremum can be restricted to the subset A

of those measures that are ergodic.

This latter result is the analog of the usual variational principle (Theorem 12.1.1). In light of this, we make the following definition (cf. Definition 12.2.1 and the sentence that follows it). Definition 17.3.5. Let f : EA∞ → ℝ be an acceptable potential and A a finitely irreducible matrix. A σ-invariant Borel probability measure μ on EA∞ is said to be an equilibrium state for f if ∫ f dμ > −∞ EA∞

and

P(f ) = hμ (σ) + ∫ f dμ. EA∞

17.4 Gibbs states Let f : EA∞ → ℝ be an acceptable potential. For a symbolic system, a Borel probability measure m on EA∞ is called a Gibbs state for f (cf. Definition 13.2.1) if there exist a number P ∈ ℝ and a constant C ≥ 1 such that for every ω ∈ EA∗ and every ρ ∈ [ω] the following holds: C −1 ≤

m([ω]) ≤ C. exp(S|ω| f (ρ) − P|ω|)

(17.9)

This means that the m-measure of any cylinder [ω] is comparable with the exponential of the difference between: (1) the sum of the values of f at the first |ω| iterates of any word in the cylinder, and (2) P times the length |ω| of the cylinder. If additionally m is σ-invariant, then m is called an invariant Gibbs state. Remark 17.4.1. Since osc(f ) < ∞, the ergodic sum S|ω| f (ρ) in (17.9) can be replaced by S|ω| f ([ω]) := sup(S|ω| f |[ω] ) or by S|ω| f ([ω]) := inf(S|ω| f |[ω] ). We start with the following observations, which are analog to Proposition 13.2.4.

684 | 17 Countable state thermodynamic formalism Proposition 17.4.2. Let f : EA∞ → ℝ be an acceptable potential. (a) For every Gibbs state, P = P(f ). (b) If m is a Gibbs state for the potential f , then a Borel probability measure μ is a Gibbs state for f if and only if m and μ are boundedly equivalent, i. e., have Radon– Nikodym derivatives bounded away from zero and infinity. Proof. (a) Fix n ∈ ℕ. Recall that ∑ω∈E n m([ω]) = 1. Replacing S|ω| f (ρ) by S|ω| f ([ω]) A in (17.9) and taking the sum over all words ω ∈ EAn , we get C −1 e−Pn ∑ exp(Sn f ([ω])) ≤ 1 ≤ Ce−Pn ∑ exp(Sn f ([ω])). ω∈EAn

ω∈EAn

Applying the natural logarithm to all three terms, dividing by n and taking the limsup as n → ∞, we obtain −P + P(f ) ≤ 0 ≤ −P + P(f ), which means that P = P(f ). (b) Suppose m and μ are Gibbs states for a same potential f . Part (a) and the Gibbs property guarantee the existence of a constant D ≥ 1 such that D−1 ≤

m([ω]) ≤ D, μ([ω])

∀ω ∈ EA∗ .

Since the initial cylinders are the open balls of EA∞ and since both measures are regular, it ensues that D−1 μ(B) ≤ m(B) ≤ Dμ(B) for all Borel sets B ⊆ EA∞ . That is, m and μ are boundedly equivalent and D−1 ≤

dm dμ

≤ D.

The proof of the converse implication consists in walking back the argument just given. As an immediate consequence of (17.9) and Remark 17.4.1, we get the following. Proposition 17.4.3. Any uniformly continuous function f : EA∞ → ℝ that has a Gibbs state is acceptable. Throughout the remainder of this chapter, our main considerations will pertain to the existence, the uniqueness and the properties of Gibbs states. Proposition 17.4.3 reveals that we must then restrict ourselves to potentials which are at least acceptable. The next major result in this direction is due to Sarig [115]. It asserts that if an invariant Gibbs state exists and the incidence matrix is irreducible, then it must be finitely irreducible. To establish this fact, we first need an auxiliary lemma. Lemma 17.4.4. Suppose that the incidence matrix A : E × E → {0, 1} is irreducible and m is a shift-invariant Gibbs state for some acceptable potential f : EA∞ → ℝ. Then there is a constant K > 0 such that for every e ∈ E, min{

∑

a∈E: Aea =1

exp(sup(f |[a] )),

∑

a∈E: Aae =1

exp(sup(f |[a] ))} ≥ K.

17.4 Gibbs states | 685

Proof. Fix any a, b ∈ E such that ab ∈ EA∗ . Since m is a Gibbs state for f , we have by virtue of (17.9) and Proposition 17.4.2 that C −1 e−2P(f ) einf(f |[a] )+inf(f |[b] ) ≤ m([ab]) ≤ Ce−2P(f ) esup(f |[a] )+sup(f |[b] ) .

(17.10)

Therefore, m([a]) =

∑

c∈E: Aac =1

m([ac]) ≤ Ce−2P(f ) esup(f |[a] )

∑

c∈E: Aac =1

exp(sup(f |[c] )).

(17.11)

On the other hand, the Gibbs property of m yields that m([a]) ≥ C −1 e−P(f ) exp(sup(f |[a] )).

(17.12)

It ensues from (17.12) and (17.11) that C −2 eP(f ) ≤

∑

c∈E: Aac =1

exp(sup(f |[c] )).

(17.13)

This is half of the inequality sought. Similarly, as m is shift-invariant, we know that m([a]) = m(σ −1 ([a])) =

∑

c∈E: Aca =1

m([ca]).

Using this, (17.12) and (17.10), we deduce that C −1 e−P(f ) exp(sup(f |[a] )) ≤ Ce−2P(f )

∑

c∈E: Aca =1

exp(sup(f |[c] )) exp(sup(f |[a] )).

Consequently, C −2 eP(f ) ≤

∑

c∈E: Aca =1

exp(sup(f |[c] )).

Along with (17.13), this completes the proof. Theorem 17.4.5. Assume that an incidence matrix A : E × E → {0, 1} is irreducible. If an acceptable potential f : EA∞ → ℝ has an invariant Gibbs state, then the incidence matrix A is finitely irreducible. Proof. Since A is irreducible, it suffices to find a finite set of letters F ⊆ E such that for every letter e ∈ E there exist a, b ∈ F such that Aea = Abe = 1. Indeed, as a set Λ yielding the finite irreducibility of A we can then take {iωi,j j : i, j ∈ F}

686 | 17 Countable state thermodynamic formalism where the ωi,j are words such that iωi,j j ∈ EA∗ . Their existence is guaranteed by the irreducibility of A. Without loss of generality, we may assume that E = ℕ. Since f has a Gibbs state, we have by virtue of (17.9) and Proposition 17.4.2(a) that ∑ exp(sup(f |[n] )) ≤ CeP(f ) < ∞.

n∈ℕ

So there exists some q ∈ ℕ such that ∑ exp(sup(f |[j] )) < K,

j>q

(17.14)

where K is the constant in Lemma 17.4.4. Let F := {1, 2, . . . , q}. It follows from (17.14) and Lemma 17.4.4 that every letter from E is followed by some letter in F and that every letter from E is preceded by some letter in F. As irreducibility is the weakest form of topological mixing, from this point on we are fully justified to restrict our attention to finitely irreducible matrices. This is thus what we will do henceforth. We now address the question of uniqueness and ergodicity of invariant Gibbs states. Theorem 17.4.6. If an acceptable potential f : EA∞ → ℝ has a Gibbs state and if the incidence matrix A is finitely irreducible, then f has a unique invariant Gibbs state μf , and this latter is ergodic. Furthermore, if A is finitely primitive then μf is totally ergodic, that is, all the iterates of the shift map are ergodic with respect to μf , including the shift map σ itself. Proof. Given ω ∈ EA∗ and n ∈ ℕ, let EAn (ω) = {τ ∈ EAn : Aτn ω1 = 1} be the set of A-admissible n-letter prefixes of ω. Let m be a Gibbs state for f . Using (17.9), Remark 17.4.1 and Proposition 17.4.2(a), we get m(σ −n ([ω])) = ≤ ≤

∑ m([τω]) ≤

τ∈EAn (ω)

∑ C exp(Sn+|ω| f ([τω]) − (n + |ω|)P(f ))

τ∈EAn (ω)

∑ C exp(Sn f ([τ]) − nP(f )) exp(S|ω| f ([ω]) − |ω|P(f ))

τ∈EAn (ω)

∑ C ⋅ Cm([τ]) ⋅ Cm([ω]) ≤ C 3 m([ω]).

τ∈EAn (ω)

This inequality actually holds for any union of cylinders, since every union of cylinders can be expressed as a union of mutually disjoint cylinders. Furthermore, as

17.4 Gibbs states | 687

cylinders form an algebra generating the Borel σ-algebra on EA∞ , it ensues from Lemma A.1.32 in Appendix A that this inequality extends to all Borel sets, i. e., m(σ −n (B)) ≤ C 3 m(B),

∀B ∈ ℬ(EA∞ ),

(17.15)

∀n ∈ ℕ.

On the other hand, let Λ ⊆ EA∗ be a finite set of words witnessing the finite irreducibility of the matrix A and let |Λ| be the maximal length of these words. Since f is acceptable, ℓ := min{S|α| f ([α]) − |α|P(f ) : α ∈ Λ} > −∞. For each τ, ω ∈ EA∗ , let α = α(τ, ω) ∈ Λ be such that ταω ∈ EA∗ . For all ω ∈ EA∗ and all n ∈ ℕ, we then have that n+|Λ|

n+|Λ|

∑ m(σ −j ([ω])) = ∑

j=n

∑ m([βω]) ≥ ∑ m([τα(τ, ω)ω]) τ∈EAn

j=n β∈E j (ω) A

≥ ∑ C −1 exp(S|τ|+|α|+|ω| f ([ταω]) − (|τ| + |α| + |ω|)P(f )) τ∈EAn

≥ C −1 ∑ exp(Sn f ([τ]) − nP(f )) ⋅ exp(S|α| f ([α]) − |α|P(f )) τ∈EAn

⋅ exp(S|ω| f ([ω]) − |ω|P(f )) ≥ C −1 eℓ exp(S|ω| f ([ω]) − |ω|P(f )) ∑ exp(Sn f ([τ]) − nP(f )) τ∈EAn

≥ C −1 eℓ C −1 m([ω]) ∑ C −1 m([τ]) = C −3 eℓ m([ω]). τ∈EAn

By the same argument as above, this inequality extends to all Borel sets, i. e., n+|Λ|

∑ m(σ −j (B)) ≥ C −3 eℓ m(B),

j=n

∀B ∈ ℬ(EA∞ ),

∀n ∈ ℕ.

(17.16)

Let L be a Banach limit defined on the Banach space of all bounded sequences of real numbers. It is not difficult to check that the formula μ(B) = L( (

n+|Λ|

∞

1 ∑ m(σ −j (B))) ) |Λ| + 1 j=n n=0

defines a finite, nonzero, nonnegative, σ-invariant, finitely additive function on the Borel subsets of EA∞ . In fact, μ(EA∞ ) = L((1)∞ n=0 ) = 1. It further follows from (17.15)

688 | 17 Countable state thermodynamic formalism and (17.16) that C −3 eℓ m(B) ≤ μ(B) ≤ C 3 m(B), |Λ| + 1

∀B ∈ ℬ(EA∞ ).

(17.17)

This shows that μ is equivalent to m. So Lemma A.1.31 asserts that μ is a countably additive function and hence is a σ-invariant Borel probability measure. Moreover, by Proposition 17.4.2, the measure μ is then, like m, a Gibbs state, albeit with a different constant Cμ . Let us prove the ergodicity of μ. Let ω ∈ EA∗ . Going along almost the same lines as before (replacing m by μ), for each τ ∈ EA∗ we find that |ω|+|Λ|

∑ μ(σ −j ([τ]) ∩ [ω]) ≥ μ([ωα(ω, τ)τ]) ≥ Cμ−3 eℓμ μ([τ])μ([ω]).

j=|ω|

(17.18)

Take an arbitrary Borel set B ⊆ EA∞ . Fix ε > 0. Since the initial cylinders {[τ] : τ ∈ EA∗ } are the open balls in EA∞ and since μ is regular, for every ω ∈ EA∗ there is a set Z = Z(ω) ⊆ EA∗ consisting of mutually incomparable words such that B ⊆ ⋃τ∈Z [τ] and μ(⋃ [τ] \ B) ≤ τ∈Z

ε . |Λ| + 1

It follows from this and the σ-invariance of μ that ∑ μ(σ −j ([τ]) ∩ [ω]) ≤ μ(σ −j (B) ∩ [ω]) +

τ∈Z

ε , |Λ| + 1

∀|ω| ≤ j ≤ |ω| + |Λ|.

Using also (17.18), we then infer that |ω|+|Λ|

|ω|+|Λ|

∑ μ(σ −j (B) ∩ [ω]) + ε ≥ ∑ ∑ μ(σ −j ([τ]) ∩ [ω]) j=|ω| τ∈Z

j=|ω|

≥ ∑ Cμ−3 eℓμ μ([τ])μ([ω]) ≥

τ∈Z Cμ−3 eℓμ μ(B)μ([ω]).

Letting ε → 0, we obtain |ω|+|Λ|

∑ μ(σ −j (B) ∩ [ω]) ≥ Cμ−3 eℓμ μ(B)μ([ω]).

j=|ω|

From this inequality, we deduce that |ω|+|Λ|

∑ μ(σ −j (EA∞ \ B) ∩ [ω]) ≤ [|Λ| + 1 − Cμ−3 eℓμ μ(B)]μ([ω]).

j=|ω|

17.4 Gibbs states | 689

Replacing EA∞ \ B by B, we thus have |ω|+|Λ|

∑ μ(σ −j (B) ∩ [ω]) ≤ [|Λ| + 1 − Cμ−3 eℓμ (1 − μ(B))]μ([ω])

j=|ω|

(17.19)

for every Borel set B ⊆ EA∞ and every ω ∈ EA∗ . In order to conclude the proof of the ergodicity of σ with respect to μ, suppose that σ −1 (B) = B and that 0 < μ(B) < 1 for some B ∈ ℬ(EA∞ ). Let ζ = 1−Cμ−3 eℓμ (1−μ(B))/(|Λ|+1). It follows from (17.19) and μ(B) < 1 that 0 < ζ < 1 and that for every ω ∈ EA∗ there exists |ω| ≤ jω ≤ |ω| + |Λ| such that μ(σ −jω (B) ∩ [ω]) ≤ ζ μ([ω]). The σ-invariance of B implies that μ(B ∩ [ω]) ≤ ζ μ([ω]) for every ω ∈ EA∗ . Pick 1 < η < 1/ζ and choose a set R ⊆ EA∗ consisting of mutually incomparable words such that B ⊆ ⋃ω∈R [ω] and ∑ω∈R μ([ω]) ≤ ημ(B). Then μ(B) = ∑ μ(B ∩ [ω]) ≤ ∑ ζ μ([ω]) ≤ ζημ(B) < μ(B). ω∈R

ω∈R

This contradiction finishes the proof of the existence of an ergodic invariant Gibbs state. According to Theorem 8.2.21, any two distinct ergodic invariant measures must be mutually singular. The uniqueness of an invariant Gibbs state readily follows from the ergodicity of any invariant Gibbs state and from Proposition 17.4.2(b). Finally, let us prove the complete ergodicity of μ when A is finitely primitive. In essence, we repeat the argument just given. Let Λ be a finite set of words of equal length which witnesses the finite primitivity of A. Fix r ∈ ℕ. Let ω ∈ EA∗ . For each τ ∈ EA∗ , we find the following improvement of (17.18): μ(σ −(|ω|+|Λ|r) ([τ]) ∩ [ω]) ≥

∑

μ([ωατ])

α∈Λr ∩EA|Λ|r : Aω|ω| α1 =Aα|Λ|r τ1 =1

≥ Cμ−3 erℓμ μ([τ])μ([ω]).

(17.20)

Take an arbitrary Borel set B ⊆ EA∞ . Fix ε > 0. For every ω ∈ EA∗ , there exists a set Z ⊆ EA∗ consisting of mutually incomparable words such that B ⊆ ⋃τ∈Z [τ] and ∑ μ(σ −(|ω|+|Λ|r) ([τ]) ∩ [ω]) ≤ μ(σ −(|ω|+|Λ|r) (B) ∩ [ω]) + ε.

τ∈Z

Using (17.20), we then get μ(σ −(|ω|+|Λ|r) (B) ∩ [ω]) + ε ≥ ∑ Cμ−3 erℓμ μ([τ])μ([ω]) ≥ Cμ−3 erℓμ μ(B)μ([ω]). τ∈Z

Letting ε → 0, we obtain μ(σ −(|ω|+|Λ|r) (B) ∩ [ω]) ≥ C(r)μ(B)μ([ω]),

690 | 17 Countable state thermodynamic formalism where C(r) = Cμ−3 erℓμ . By setting B = EA∞ , it follows from this last inequality that C(r) ≤ 1. From this same inequality, we also find that μ(σ −(|ω|+|Λ|r) (EA∞ \ B) ∩ [ω]) ≤ (1 − C(r)μ(B))μ([ω]). Thus, for every ω ∈ EA∗ we have μ(σ −(|ω|+|Λ|r) (B) ∩ [ω]) ≤ (1 − C(r)(1 − μ(B)))μ([ω]).

(17.21)

In order to conclude the proof of the complete ergodicity of σ with respect to μ, suppose that σ −r (B) = B and 0 < μ(B) < 1. Set ζ = 1 − C(r)(1 − μ(B)). It follows from (17.21) and nr −(|ω|+|Λ|r) μ(B) < 1 that 0 < ζ < 1 and for every ω ∈ ⋃∞ (B)∩ n=1 EA we have μ(B∩[ω]) = μ(σ ∞ nr [ω]) ≤ ζ μ([ω]). Take 1 < η < 1/ζ and choose a set R ⊆ ⋃n=1 EA consisting of mutually incomparable words such that B ⊆ ⋃ω∈R [ω] and ∑ω∈R μ([ω]) ≤ ημ(B). Then μ(B) = ∑ μ(B ∩ [ω]) ≤ ∑ ζ μ([ω]) ≤ ζημ(B) < μ(B). ω∈R

ω∈R

This contradiction completes the proof of the complete ergodicity of μ. We shall now characterize the condition ∫ f dμf > −∞, where μf is the unique invariant Gibbs state (if it exists) for an acceptable potential f . Recall that this condition is part of the definition of an equilibrium state. Lemma 17.4.7. Suppose that the incidence matrix A is finitely irreducible and that an acceptable potential f : EA∞ → ℝ has a Gibbs state. Denote by μf its unique invariant Gibbs state per Theorem 17.4.6. Then the following three conditions are equivalent: (a) ∫E ∞ f dμf > −∞. A

(b) ∑e∈E inf(−f |[e] ) exp(inf f |[e] ) < ∞. (c) Hμf (𝒰E ) < ∞, where 𝒰E = {[e] : e ∈ E} is the partition of EA∞ into its initial cylinders of length 1.

Proof. [(a)⇒(b)] Suppose that ∫E ∞ f dμf > −∞, i. e. ∑e∈E ∫[e] (−f ) dμf < ∞. As μf satisA

fies the Gibbs property (17.9) (see also Remark 17.4.1), we have

∞ > ∑ inf(−f |[e] )μf ([e]) ≥ Cμ−1f ∑ inf(−f |[e] ) exp(inf(f |[e] ) − P(f )) =

e∈E Cμ−1f e−P(f )

e∈E

∑ inf(−f |[e] ) exp(inf(f |[e] )).

e∈E

[(b)⇒(c)] Assume that ∑e∈E inf(−f |[e] ) exp(inf(f |[e] )) < ∞. Then Hμf (𝒰E ) = ∑ −μf ([e]) log μf ([e]) ≤ ∑ −μf ([e])(inf(f |[e] ) − P(f ) − log Cμf ) e∈E

e∈E

= P(f ) + log Cμf + ∑ −μf ([e]) inf(f |[e] ). e∈E

17.5 Gibbs states versus equilibrium states | 691

Thus, it remains to show that ∑e∈E −μf ([e]) inf(f |[e] ) < ∞. And indeed, ∑ −μf ([e]) inf(f |[e] ) = ∑ μf ([e]) sup(−f |[e] )

e∈E

e∈E

≤ ∑ μf ([e])(inf(−f |[e] ) + osc(f )) e∈E

= osc(f ) + ∑ μf ([e]) inf(−f |[e] ). e∈E

Since osc(f ) < ∞, we need to show that ∑e∈E μf ([e]) inf(−f |[e] ) < ∞. Without loss of generality, we may assume that E = ℕ. As μf is a probability measure, lime→∞ μf ([e]) = 0. Then it follows from the Gibbs property that lime→∞ (sup f |[e] − P(f )) = −∞. Since P(f ) < ∞, we deduce that lime→∞ sup f |[e] = −∞. It is thus possible to reorder the set E so that there is k ∈ E such that sup f |[e] ≥ 0 for all e < k while sup f |[e] < 0 for all e ≥ k. Hence inf(−f |[e] ) ≤ 0 for all e < k whereas inf(−f |[e] ) > 0 for all e ≥ k. Using the Gibbs property again, we then get ∑ μf ([e]) inf(−f |[e] ) ≤ ∑ Cμf exp(inf(f |[e] ) − P(f )) inf(−f |[e] )

e∈E

e≥k

= Cμf e−P(f ) ∑ exp(inf(f |[e] )) inf(−f |[e] ), e≥k

and this latter sum is finite by assumption. Therefore, Hμf (𝒰E ) < ∞. [(c)⇒(a)] Suppose that Hμf (𝒰E ) < ∞. Then

∞ > Hμf (𝒰E ) = ∑ −μf ([e]) log μf ([e]) e∈E

≥ ∑ −μf ([e])(inf(f |[e] ) − P(f ) + log Cμf ) e∈E

= P(f ) − log Cμf + ∑ −μf ([e]) inf(f |[e] ). e∈E

Hence, ∑e∈E −μf ([e]) inf(f |[e] ) < ∞, i. e. ∑e∈E μf ([e]) inf(f |[e] ) > −∞. Therefore, ∫ f dμf = ∑ ∫ f dμf ≥ ∑ inf(f |[e] )μf ([e]) > −∞. EA∞

e∈E

[e]

e∈E

Remark 17.4.8. Observe that the shift-invariance of μf was not used anywhere in above proof and the previous lemma actually holds for any Gibbs state.

17.5 Gibbs states versus equilibrium states The next theorem shows that the condition ∫ f dμf > −∞ is sufficient for the unique invariant Gibbs state μf to be the unique equilibrium state for any acceptable potential f .

692 | 17 Countable state thermodynamic formalism Theorem 17.5.1. Suppose that the incidence matrix A is finitely irreducible. Suppose also that f : EA∞ → ℝ is an acceptable potential which admits a Gibbs state and that ∫E ∞ f dμf > −∞, where μf is the unique invariant Gibbs state for the potential f per A

Theorem 17.4.6. Then μf is the unique equilibrium state for f .

Proof. In order to show that μf is an equilibrium state for the potential f , consider the partition 𝒰E = {[e] : e ∈ E} of EA∞ into its initial cylinders of length 1. By Lemma 17.4.7, Hμf (𝒰E ) < ∞. Using the ergodic case of Shannon–McMillan–Breiman’s theorem (Corollary 9.5.5), the Gibbs property (17.9) and the ergodic case of Birkhoff’s theorem (Corollary 8.2.14), we get for μf -a. e. ρ ∈ EA∞ , 1 hμf (σ) ≥ hμf (σ, 𝒰E ) = lim − log μf ([ρ|n ]) n→∞ n 1 ≥ lim − (log Cμf + Sn f (ρ) − nP(f )) n→∞ n 1 = − lim Sn f (ρ) + P(f ) = − ∫ f dμf + P(f ). n→∞ n ∞ EA

In conjunction with the 2nd variational principle (Theorem 17.3.3), this implies that μf is an equilibrium state for the potential f . To prove the uniqueness of the equilibrium state, suppose that ν is an equilibrium state for the potential f : EA∞ → ℝ and that ν ≠ μf . Applying the ergodic decomposition theorem (Theorem 8.2.26), we may assume that ν is ergodic. Using the Gibbs property (17.9), we then have that for every n ∈ ℕ, 0 = n(hν (σ) + ∫ (f − P(f )) dν) ≤ Hν (𝒰En ) + ∫ (Sn f − nP(f )) dν EA∞

EA∞

= − ∑ ν([ω])(log ν([ω]) − ω∈EAn

1 ∫ (Sn f − nP(f )) dν) ν([ω]) [ω]

≤ − ∑ ν([ω])(log ν([ω]) − (Sn f ([ω]) − nP(f ))) ω∈EAn

= − ∑ ν([ω]) log[ν([ω]) exp(nP(f ) − Sn f ([ω]))] ω∈EAn

−1

≤ − ∑ ν([ω]) log[ν([ω])(Cμf μf ([ω])) ] ω∈EAn

= log Cμf − ∑ ν([ω]) log ω∈EAn

ν([ω]) . μf ([ω])

Therefore, in order to complete the proof by contradiction, it suffices to show that lim − ∑ ν([ω]) log

n→∞

ω∈EAn

ν([ω]) = −∞. μf ([ω])

17.6 Transfer operator

| 693

Since both measures ν and μf are ergodic and ν ≠ μf , the measures ν and μf must be mutually singular. According to Lemma 13.7.14, ν([ω|n ]) ≤ t}) = 0 μf ([ω|n ])

lim ν({ω ∈ EA∞ :

n→∞

for every t > 0. For every n ∈ ℕ and every j ∈ ℤ, set Fn,j = {ω ∈ EA∞ : e−j ≤

ν([ω|n ]) < e−j+1 }. μf ([ω|n ])

Then Fn,j is a union of cylinders of length n and ν(Fn,j ) = ∫ Fn,j

ν([ω|n ]) dμ (ω) ≤ e−j+1 . μf ([ω|n ]) f

Notice that − ∑ ν([ω]) log ω∈EAn

ν([ω|n ]) ν([ω]) = − ∫ log dν(ω) μf ([ω]) μ f ([ω|n ]) ∞ EA

= − ∑ ∫ log j∈ℤ F

n,j

ν([ω|n ]) dν(ω) ≤ ∑ jν(Fn,j ). μf ([ω|n ]) j∈ℤ

Thus, for each k ∈ −ℕ we obtain that − ∑ ν([ω]) log ω∈EAn

ν([ω]) ≤ k ∑ ν(Fn,j ) + ∑ je−j+1 μf ([ω]) j≥1 j≤k = kν({ω ∈ EA∞ : e−k ≤

ν([ω|n ]) }) + ∑ je−j+1 . μf ([ω|n ]) j≥1

Hence, for each k ∈ −ℕ we get that lim sup − ∑ ν([ω]) log n→∞

ω∈EAn

ν([ω]) ≤ k + ∑ je−j+1 . μf ([ω]) j≥1

Letting k → −∞ completes the proof.

17.6 Transfer operator We will now describe basic properties of the transfer operator. This operator is ultimately used to prove the existence of Gibbs states (see Theorem 17.7.4 and Corollary 17.7.5). It is also used to demonstrate several stochastic laws that Gibbs states obey to and the real analyticity of the topological pressure, though the discussion of these issues is deferred to Chapters 18 and 20, respectively.

694 | 17 Countable state thermodynamic formalism Let Cb (EA∞ ) denote the space of all bounded continuous functions on EA∞ equipped with the supremum norm. Given a potential f : EA∞ → ℝ and a function g ∈ Cb (EA∞ ), define the transfer operator ℒf by ℒf (g)(ρ) :=

∑

e∈E: Aeρ1 =1

g(eρ) exp(f (eρ)).

(17.22)

In our generally noncompact symbolic setting, this definition is in line with the one given in (13.26). To make sure that the transfer operator preserves the space Cb (EA∞ ), we will assume not only that the potential f is acceptable but also summable. Definition 17.6.1. A potential f : EA∞ → ℝ is said to be summable if ∑ exp(sup f |[e] ) < ∞.

e∈E

Note that if f has a Gibbs state, then f is summable. Lemma 17.6.2. If f : EA∞ → ℝ is a summable and acceptable potential, then the linear operator ℒf : Cb (EA∞ ) → Cb (EA∞ ) is well-defined and is bounded with ‖ℒf ‖∞ ≤ ∑e∈E exp(sup f |[e] ). Proof. The argument is somewhat similar to that of Lemma 13.6.1. It is easy to see that the nth iterate of this operator is given by n

ℒf (g)(ρ) =

∑

τ∈EAn : Aτn ρ1 =1

g(τρ) exp(Sn f (τρ)).

(17.23)

The dual operator ℒ∗f : Cb (EA∞ )∗ → Cb (EA∞ )∗ has the following form: ∗

ℒf (μ)(g) := μ(ℒf (g)) := ∫ ℒf (g) dμ.

(17.24)

EA∞

Throughout the rest of this subsection, we assume that m is a Borel probability eigenmeasure of ℒ∗f . The corresponding eigenvalue is denoted by λ. Since ℒf is a positive operator, λ ≥ 0. Then, for every n ∈ ℕ, n

(ℒ∗f ) (m) = λn m. The integral version of this equality is ∫

∑

n EA∞ τ∈EA : Aτn ρ1 =1

g(τρ)eSn f (τρ) dm(ρ) = λn ∫ g dm, EA∞

∀g ∈ Cb (EA∞ ).

(17.25)

In fact, formulas (17.22) to (17.25) extend to the space of all bounded Borel-measurable functions on EA∞ . In particular, taking ω ∈ EAn , a Borel set B ⊆ EA∞,ω and g = 1ωB , where

17.6 Transfer operator

| 695

EA∞,ω := {γ ∈ EA∞ | ωγ ∈ EA∞ } and ωB := {ωρ : ρ ∈ B}, we obtain from (17.25) that m(ωB) = λ−n ∫ eSn f (ωρ) dm(ρ).

(17.26)

B

Remark 17.6.3. If (17.26) holds, then by representing a Borel set B ⊆ EA∞ as the union ⋃ω∈E n (ωBω ), where Bω = {χ ∈ EA∞,ω : ωχ ∈ B}, a straightforward calculation based A on (17.26) demonstrates that (17.25) is satisfied for the characteristic function 1B . It follows from standard approximation arguments that (17.25) is then satisfied for all m-integrable functions. Thus, m is an eigenmeasure of ℒ∗f if and only if formula (17.26) holds with n = 1. We now show that any eigenmeasure of the transfer operator’s dual is a Gibbs state under the condition that the potential is summable and Hölder continuous on cylinders. Note that if Hölder continuity held globally, then the potential would be bounded and hence would not be summable. In turn, this would imply that such a potential would not have a Gibbs state. Theorem 17.6.4. If the incidence matrix A is finitely irreducible and if a potential f is summable and Hölder continuous on cylinders, then any eigenmeasure m of the dual operator ℒ∗f is a Gibbs state for f . In addition, its eigenvalue is λ = eP(f ) . Furthermore, υβ (f )

Cf := e eβ −1 max{1, #Λ exp(2|Λ| ⋅ |P(f )| − 2 min S|α| f ([α]))} α∈Λ

is a Gibbs constant for m, where #Λ denotes the cardinality of Λ while |Λ| = max{|α| : α ∈ Λ}. Proof. Let ω ∈ EA∗ \ {ϵ} and n = |ω| ∈ ℕ. Since f is summable and Hölder continuous on cylinders, relation (17.26) and Lemma 17.2.3 imply that for every ρ ∈ [ω], m([ω]) = m(ωEA∞,ω ) = λ−n ∫ eSn f (ωγ) dm(γ) EA∞,ω

≤ λ−n ⋅ e υβ (f )

Sn f (ρ)+

υβ (f ) eβ −1

m(EA∞,ω )

≤ e eβ −1 ⋅ exp(Sn f (ρ) − n log λ) ≤ Cf exp(Sn f (ρ) − n log λ).

(17.27)

On the other hand, let Λ ⊆ EA∗ be a nonempty finite set which witnesses the finite irreducibility of A. By the definition of Λ, ⋃ EA∞,α = EA∞ .

α∈Λ

696 | 17 Countable state thermodynamic formalism ∞,α

Hence, there exists α0 ∈ Λ such that m(EA 0 ) ≥ 1/#Λ. Then there exists ξ ∈ Λ such that ωξ α0 ∈ EA∗ . Let ρ ∈ [ω] and write n = |ω|, p = |ξ | and q = |α0 |. Using (17.26) and Lemma 17.2.3 once again, we obtain that ∞,α0

m([ω]) ≥ m(ωξ α0 EA

) = λ−(n+p+q) ∫ eSn+p+q f (ωξ α0 γ) dm(γ) ∞,α0

EA

= λ−(n+p+q) ∫ eSn f (ωξ α0 γ) eSp f (ξ α0 γ) eSq f (α0 γ) dm(γ) ∞,α0

EA

≥ λ−(p+q) exp(2 min S|α| f ([α])) ∫ eSn f (ωξ α0 γ) dm(γ) ⋅ λ−n α∈Λ

∞,α0

EA

≥ min{1, λ−2|Λ| } exp(2 min S|α| f ([α])) ⋅ e

Sn f (ρ)−

α∈Λ

≥e

−

υβ (f ) eβ −1

υβ (f ) eβ −1

∞,α0

m(EA

) ⋅ λ−n

min{1, λ−2|Λ| } exp(2 min S|α| f ([α])) ⋅ (1/#Λ) α∈Λ

⋅ exp(Sn f (ρ) − n log λ).

(17.28)

Combining (17.27) and (17.28), we conclude that m is a Gibbs state for f . The equality λ = eP(f ) follows immediately from Proposition 17.4.2(a). We can infer from this and (17.27)–(17.28) that Cf is a Gibbs constant for m. Let us now introduce the normalized transfer operator ̂f = e ℒ

−P(f )

ℒf .

Theorem 17.6.5. If the incidence matrix A is finitely irreducible and if a potential f is ̂∗ fixes at most summable and Hölder continuous on cylinders, then the dual operator ℒ f one Borel probability measure. ̂∗ . In view of Theorem 17.6.4 and Proof. Suppose that m and m1 are fixed points under ℒ f Proposition 17.4.2(b), the measures m and m1 are boundedly equivalent. Consider the Radon–Nikodym derivative ρ = dm1 /dm. Temporarily fix ω ∈ EA∗ with n = |ω| ≥ 2. We deduce from (17.26) and Theorem 17.6.4 that m([ω]) = ∫ exp(Sn f (ωγ) − nP(f ))dm(γ) EA∞,ω

= ∫ exp(Sn−1 f (σ(ωγ)) − (n − 1)P(f )) exp(f (ωγ) − P(f ))dm(γ) EA∞,ω

=

∫ exp(Sn−1 f (σ(ωγ)) − (n − 1)P(f )) exp(f (ωγ) − P(f ))dm(γ). EA∞,σ(ω)

17.7 Existence & uniqueness of eigenmeasures, Gibbs states and equilibrium states | 697

Thus, inf(exp(f |[ω] − P(f )))m([σ(ω)]) ≤ m([ω]) ≤ sup(exp(f |[ω] − P(f )))m([σ(ω)]). Since f : EA∞ → ℝ is continuous, we conclude that for every ω ∈ EA∞ , lim

n→∞

m([ω|n ]) = exp(f (ω) − P(f )). m([σ(ω)|n−1 ])

(17.29)

Obviously, the same formula holds with m replaced by m1 . But according to Theorem 17.4.6 and Proposition 17.4.2(b), there is a σ-invariant Borel probability measure which is equivalent to m. Using this and Exercise 9.7.14, we infer that there exists a set of points ω ∈ EA∞ with full m-measure on which ρ(ω) = lim

n→∞

m1 ([ω|n ]) m([ω|n ])

and ρ(σ(ω)) = lim

n→∞

m1 ([σ(ω)|n ]) . m([σ(ω)|n ])

From this, (17.29) and its version for m1 , we then obtain ρ(ω) = lim [ n→∞

m ([σ(ω)|n−1 ]) m([σ(ω)|n−1 ]) m1 ([ω|n ]) ⋅ 1 ⋅ ] m1 ([σ(ω)|n−1 ]) m([σ(ω)|n−1 ]) m([ω|n ]) −1

= exp(f (ω) − P(f )) ⋅ ρ(σ(ω)) ⋅ [exp(f (ω) − P(f ))]

= ρ(σ(ω))

on a set of full m-measure. But according to Theorem 17.4.6, the shift map σ is ergodic with respect to a σ-invariant Borel probability measure equivalent to m. By a short argument relying on Theorem 8.2.18, the function ρ is m-almost everywhere constant. So there exists a real number ρ ≥ 0 such that ρ(ω) = ρ for m-a.e. ω ∈ EA∞ . Therefore, 1 = m(EA∞ ) = ∫ ρ(ω)dm(ω) = ρm(EA∞ ) = ρ. EA∞

In consequence, m1 = m.

17.7 Existence and uniqueness of eigenmeasures of the dual transfer operator, of Gibbs states and of equilibrium states So far the character of this chapter has been somewhat conditional. To remedy this, we prove in this section the existence of eigenmeasures of the dual transfer operator. We begin with the finite case, which is due to Bowen [11]. Lemma 17.7.1. If the alphabet E is finite and the incidence matrix A is irreducible, then for every continuous potential f : EA∞ → ℝ the dual operator ℒ∗f admits a Borel probability eigenmeasure mf . Proof. This is a special case of Theorem 13.6.2.

698 | 17 Countable state thermodynamic formalism Remark 17.7.2. In Lemma 17.7.1, it suffices that the matrix contain at least one 1 in every row and every column. Irreducibility will ensure that the eigenmeasure is a Gibbs state per Theorem 17.6.4. The following simple fact about irreducible matrices will be needed to generalize Lemma 17.7.1 to the infinite case. Lemma 17.7.3. If A : E → E is an irreducible matrix, then there exists an ascending sequence (Fn )∞ n=1 of finite subsets of E whose union is E and such that A|Fn ×Fn is an irreducible matrix. Lemma 17.7.1 can be generalized to a countably infinite alphabet as follows. Theorem 17.7.4. If the incidence matrix A is finitely irreducible and if a potential f : EA∞ → ℝ is summable and Hölder continuous on cylinders, then the dual operator ℒ∗f has a Borel probability eigenmeasure mf . Proof. Without loss of generality, we may assume that E = ℕ. Since the incidence matrix is irreducible, Lemma 17.7.3 affirms that we can reorder the set ℕ so that there exists a strictly increasing sequence (ℓn )∞ n=1 in ℕ such that the matrix A|{1,...,ℓn }×{1,...,ℓn } is irreducible for every n ∈ ℕ. Let Eℓ∞n := EA∞ ∩ {1, . . . , ℓn }∞ = {1, . . . , ℓn }∞ A . Consider the potential fn := f |Eℓ∞ , and the corresponding transfer operator ℒfn : n

C(Eℓ∞n ) → C(Eℓ∞n ) and its dual ℒ∗fn : C(Eℓ∞n )∗ → C(Eℓ∞n )∗ . As f is continuous (being Hölder on cylinders), Lemma 17.7.1 guarantees that for every n ∈ ℕ there is a Borel probability eigenmeasure mn : ℬ(Eℓ∞n ) → [0, 1] of ℒ∗fn . Our first goal is to show that the sequence (mn )∞ n=1 is tight when the mn ’s are treated ∞ as Borel probability measures on EA . That is, for every ε > 0 we must find a compact set Kε ⊆ EA∞ such that mn (Kε ) ≥ 1 − ε for all n ∈ ℕ. For every k ∈ ℕ, let πk : EA∞ → E be the projection onto the kth coordinate, i. e., πk (ω) = ωk . Let also Pn = P(σ|Eℓ∞ , f |Eℓ∞ ). n

n

Obviously, Pn+1 ≥ Pn for all n ∈ ℕ. By Theorem 17.6.4, the eigenvalue of ℒ∗fn correspond-

ing to the eigenmeasure mn is ePn . This measure is also a Gibbs state for the potential f |Eℓ∞ applied to the finite-state subsystem σ|Eℓ∞ . We thus obtain for every n ∈ ℕ, every n n k ∈ ℕ, and every e ∈ E = ℕ that mn (πk−1 (e)) =

∑ ω∈Eℓkn : ωk =e

mn ([ω]) ≤

≤ exp(−Pn k)

∑ ω∈Eℓkn : ωk =e

∑ ω∈Eℓkn : ωk =e

exp(Sk−1 f ([ω]) + sup f |[e] ) k−1

≤ exp(−P1 k)(∑ exp(sup f |[i] )) i∈E

exp(Sk f ([ω]) − Pn k)

exp(sup f |[e] ).

17.7 Existence & uniqueness of eigenmeasures, Gibbs states and equilibrium states | 699

Therefore, k−1

mn (πk−1 ([e + 1, ∞))) ≤ e−P1 k (∑ esup f |[i] ) i∈E

∑ esup f |[j] .

j>e

Fix ε > 0. Since f is summable, for every k ∈ ℕ choose nk ∈ ℕ such that k−1

e−P1 k (∑ esup f |[i] ) i∈E

∑ esup f |[j] ≤

j>nk

ε . 2k

Then mn (πk−1 ([nk + 1, ∞))) ≤ ε/2k for every n ∈ ℕ and every k ∈ ℕ. Hence, ∞

∞

k=1

k=1

∞

ε =1−ε k 2 k=1

mn (EA∞ ∩ ∏[1, nk ]) ≥ 1 − ∑ mn (πk−1 ([nk + 1, ∞))) ≥ 1 − ∑

∞ for every n ∈ ℕ. As EA∞ ∩ ∏∞ k=1 [1, nk ] is a compact subset of EA , the tightness of the sequence (mn )∞ n=1 is proved. Prohorov’s theorem (see Chapter 1, Section 5 in Billings∗ ley [9]) then affirms that (mn )∞ n=1 has a weak limit point m. Naturally, m is our candidate as eigenmeasure of ℒ∗f . To simplify calculations, we ̂f = e−P(f ) ℒf and ℒ ̂f = e−Pn ℒf , and their respective will use the normalized operators ℒ n

duals. Fix g ∈ Cb (EA∞ ). Let gn = g|Eℓ∞ . By the triangle inequality,

n

n

󵄨󵄨 ̂∗ 󵄨 󵄨 ̂∗ 󵄨󵄨 󵄨󵄨 ̂∗ 󵄨󵄨 ̂∗ ̂∗ 󵄨󵄨ℒf (m)(g) − m(g)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨ℒ f (m)(g) − ℒf (mn )(g)󵄨󵄨 + 󵄨󵄨ℒf (mn )(g) − ℒfn (mn )(g)󵄨󵄨 󵄨 ̂∗ 󵄨󵄨 󵄨󵄨 󵄨󵄨 + 󵄨󵄨󵄨ℒ fn (mn )(g) − mn (gn )󵄨󵄨 + 󵄨󵄨mn (gn ) − mn (g)󵄨󵄨 󵄨 󵄨 + 󵄨󵄨󵄨mn (g) − m(g)󵄨󵄨󵄨. (17.30) Observe that ∗

∗

∗

̂ (mn )(g) = ∫ g dℒ ̂ (mn ) = ∫ g dℒ ̂ (mn ) ℒ fn fn fn EA∞

Eℓ∞n

̂∗ (mn ) = ℒ ̂∗ (mn )(gn ) = mn (gn ) = ∫ gn dℒ fn fn

(17.31)

Eℓ∞n

and mn (gn ) − mn (g) = ∫ (gn − g)dmn = ∫ 0 dmn = 0. Eℓ∞n

(17.32)

Eℓ∞n

Thus, the right-hand side of (17.30) reduces to the first, second and fifth terms. Let ∗ ε > 0. Recall that (mn )∞ n=1 weak converges to m, that f is summable, and that

700 | 17 Countable state thermodynamic formalism limn→∞ Pn = P(f ) by Theorem 17.2.9. Take n ∈ ℕ so large that 󵄨 ε 󵄨󵄨 󵄨󵄨m(g) − mn (g)󵄨󵄨󵄨 ≤ , 3 󵄨󵄨 ε 󵄨󵄨 ̂ ̂ 󵄨󵄨m(ℒf (g)) − mn (ℒf (g))󵄨󵄨 ≤ , 3 ε −P(f ) ‖g‖∞ e ∑ exp(sup f |[i] ) ≤ , 6 i>ℓ n

∑ ‖g‖∞ exp(sup f |[i] − P1 ) ⋅ (P(f ) − Pn ) ≤

i∈E

ε . 6

(17.33) (17.34) (17.35) (17.36)

Relations (17.34) and (17.33) state that the first and last summands on the right-hand side of (17.30) are each bounded above by ε/3. Let us look at the second summand. First, from (17.31) notice that ∗

∗

̂ (mn )(g) = ℒ ̂ (mn )(gn ) = mn (ℒ ̂f (gn )) ℒ fn fn n = ∫

∑

gn (iω) exp(fn (iω) − Pn )dmn (ω)

∑

g(iω) exp(f (iω) − Pn )dmn (ω).

Eℓ∞n i≤ℓn : Aiω1 =1

= ∫ EA∞

i≤ℓn : Aiω1 =1

Using the mean value theorem, (17.36) and (17.35), we then obtain 󵄨󵄨 󵄨󵄨 ̂∗ ̂∗ (mn )(g)󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∫ 󵄨󵄨ℒf (mn )(g) − ℒ fn 󵄨 󵄨󵄨 󵄨 ∞ EA

∑

i≤ℓn : Aiω1 =1

g(iω)(exp(f (iω) − P(f ))

− exp(f (iω) − Pn ))dmn (ω) 󵄨󵄨 󵄨 + ∫ ∑ g(iω) exp(f (iω) − P(f ))dmn (ω)󵄨󵄨󵄨 󵄨󵄨 ∞ i>ℓn : Aiω =1 EA

1

≤ ∑ ‖g‖∞ exp(sup f |[i] − P1 ) ⋅ (P(f ) − Pn ) i≤ℓn

+ ∑ ‖g‖∞ exp(sup f |[i] − P(f )) i>ℓn

ε ε ε ≤ + = . 6 6 3 So the second summand on the right-hand side of (17.30) is also bounded above by ε/3. In summary, (17.30) reduces to ε 󵄨󵄨 ̂∗ 󵄨 ε ε 󵄨󵄨ℒf (m)(g) − m(g)󵄨󵄨󵄨 ≤ + + 0 + 0 + = ε. 3 3 3

17.7 Existence & uniqueness of eigenmeasures, Gibbs states and equilibrium states | 701

̂∗ (m)(g) = m(g), i. e., ℒ∗ (m)(g) = eP(f ) m(g). As Letting ε → 0, we deduce that ℒ f f g ∈ Cb (EA∞ ) was chosen arbitrarily, we conclude that ℒ∗f (m) = eP(f ) m.

As an immediate consequence of Theorems 17.7.4, 17.6.5, 17.6.4, 17.4.6 and 17.5.1, we deduce the following fundamental result. Corollary 17.7.5. Suppose that a potential f : EA∞ → ℝ is summable and Hölder continuous on cylinders and that the incidence matrix A is finitely irreducible. (a) There exists a unique Borel probability eigenmeasure mf of the dual transfer operator ℒ∗f and its corresponding eigenvalue is eP(f ) . (b) The eigenmeasure mf is a Gibbs state for f . (c) The potential f has a unique σ-invariant Gibbs state μf , and this state is ergodic. If ∫E ∞ f dμf > −∞, then μf is also the unique equilibrium state for f . Finally, if A is A

finitely primitive, then μf is totally ergodic.

The last result of this section explains the real dynamical meaning of the fixed points of the normalized transfer operator. Proposition 17.7.6. Suppose that a potential f : EA∞ → ℝ is summable and Hölder continuous on cylinders and that the incidence matrix A is finitely irreducible. Let mf ̂∗ of the normalized transfer be the unique Borel probability measure fixed by the dual ℒ f ̂f . Let operator ℒ ̂f ) = {g ∈ L1 (mf ) : ℒ ̂f (g) = g, ∫ g dmf = 1, and g ≥ 0} Fix(ℒ EA∞

and AI(mf ) = {g ∈ L1 (mf ) : gmf ∘ σ −1 = gmf , ∫ g dmf = 1, and g ≥ 0}. EA∞

̂f ) = AI(mf ). Then Fix(ℒ Proof. It follows from (17.29) that for every e ∈ E and every ω ∈ EA∞ with Aeω1 = 1, we have dmf ∘ e dmf

(ω) = exp(f (eω) − P(f )),

where the map e : {ω ∈ EA∞ : Aeω1 = 1} → EA∞ on the left-hand side is defined by e(ω) := eω. So if g ∈ AI(mf ), then d(gmf ∘ e) dmf

(ω) = g(eω) exp(f (eω) − P(f )).

702 | 17 Countable state thermodynamic formalism ̂f sends the Radon-Nikodym derivative of a measure m absolutely continTherefore, ℒ uous with respect to mf to the Radon-Nikodym derivative of the measure m ∘ σ −1 with respect to mf and the result ensues.

17.8 The invariant Gibbs state has positive entropy; pressure gap First, we shall prove the following strengthening of Proposition 13.7.19. Proposition 17.8.1. Let (X, d) be a separable complete metric space and μ a Borel finite measure on X. For every r > 0, define Sμ (r) := sup{μ(B(x, r)) : x ∈ X}. If the measure μ is atomless, then lim Sμ (r) = 0.

r→0

Proof. Fix ε > 0. Since μ is inner regular, there exists a compact set K ⊆ X such that μ(X\K) < ε/2. It then follows from Proposition 13.7.19 that there is rε > 0 such that μ(B(z, 2r) ∩ K) < ε/2,

∀z ∈ K, ∀r ≤ rε .

Fix x ∈ X and 0 < r ≤ rε . If B(x, r) ∩ K = 0, then μ(B(x, r)) ≤ μ(X\K) < ε/2 < ε. If B(x, r) ∩ K ≠ 0, let z ∈ B(x, r) ∩ K. Then B(x, r) ⊆ B(z, 2r), whence μ(B(x, r)) = μ(B(x, r) ∩ K) + μ(B(x, r)\K) ≤ μ(B(z, 2r) ∩ K) + μ(X\K) < 2 ⋅

ε = ε. 2

Having this proposition, we can prove the following theorem in an entirely analogous way to Theorem 13.7.20. Theorem 17.8.2. If E is a countable set with at least two elements and A : E × E → {0, 1} is a finitely irreducible incidence matrix such that EA∞ is not a single periodic orbit of the shift map σ : EA∞ → EA∞ , then each potential f : EA∞ → ℝ which is summable and Hölder continuous on cylinders has a pressure gap with respect to the shift map σ : EA∞ → EA∞ . Proof. Since the invariant Gibbs state μf is ergodic and has full support supp(μf ) = EA∞ , and since EA∞ is infinite, we infer that μf is atomless. Let Cμf ≥ 1 be a constant C from (17.9) ascribed to μf and let P = P(f ) = P(σ, f ). Since the metric space (EA∞ , ds ) is separable and complete, it follows from Proposition 17.8.1 that there exists k ∈ ℕ such

17.9 Exercises | 703

that μf ([ω]) ≤

1 , eCμf

∀ω ∈ EAk .

Given any ω ∈ EAk and ρ ∈ [ω], it follows from this and (17.9) that Sk f (ρ) − kP(σ, f ) ≤ log Cμf + log μf ([ω]) ≤ log Cμf + log 1 − log e − log Cμf = −1. Using this and Exercise 17.9.9, we conclude that Sk f (ρ) ≤ kP(σ, f ) − 1 = P(σ k , Sk f ) − 1 for all ρ ∈ [ω] and all ω ∈ EAk . That is, there is a pressure gap. The existence of a pressure gap entails that the system has strictly positive entropy with respect to the unique σ-invariant Gibbs state μf . Theorem 17.8.3. Let E be a countable set with at least two elements and A : E × E → {0, 1} a finitely irreducible matrix such that EA∞ is not a single periodic orbit of the shift map σ : EA∞ → EA∞ . If a potential f : EA∞ → ℝ is summable and Hölder continuous on cylinders and ∫E ∞ f dμf > −∞, then hμf (σ) > 0. A

Proof. The proof is similar to the alternative argument given in the proof of Theorem 13.7.21. Use Corollary 17.7.5(c), the 3rd variational principle (Theorem 17.3.4), Exercise 17.9.9 and Theorem 17.8.2.

17.9 Exercises Exercise 17.9.1. Let E be a countably infinite alphabet and A : E × E → {0, 1} be a finitely irreducible incidence matrix. Prove that the symbolic space EA∞ is not locally compact nor even σ-compact. Exercise 17.9.2. Show that any potential which is Hölder continuous on cylinders, is acceptable. Exercise 17.9.3. Let f : EA∞ → ℝ. Show that f is Hölder continuous if and only if it is Hölder continuous on cylinders and bounded. Exercise 17.9.4. A sequence (an )∞ n=1 of positive real numbers is boundedly submultiplicative with constant B > 0 if am+n ≤ Bam an ,

∀m, n ∈ ℕ.

704 | 17 Countable state thermodynamic formalism (If 0 < B ≤ 1, then the sequence is submultiplicative.) Prove that lim

n→∞

1 1 log an = inf log(Ban ). n∈ℕ n n

Show also that the above limit may be −∞ but if infn∈ℕ an > 0, then the limit is nonnegative. N. B.: This result is due to Fekete. Exercise 17.9.5. A sequence (an )∞ n=1 of positive real numbers is boundedly supermultiplicative with constant C > 0 if C −1 am an ≤ am+n ,

∀m, n ∈ ℕ.

(If 0 < C ≤ 1, then the sequence is supermultiplicative.) Prove that lim

n→∞

1 1 log an = sup log(C −1 an ). n n n∈ℕ

Show also that the limit may be ∞ but if supn∈ℕ an < ∞, then the limit is nonpositive. Exercise 17.9.6. The normalization of a boundedly submultiplicative sequence (an )∞ n=1 ̃ n )∞ is the sequence (a n=1 , where ̃ n = an e−Pn a

and

P = lim

n→∞

1 log an . n

Prove that the normalization of a boundedly submultiplicative sequence with constant B is a boundedly submultiplicative sequence with the same constant B but that lim

n→∞

1 1 ̃ n = inf log(Ba ̃ n ) = 0. log a n∈ℕ n n

Exercise 17.9.7. The normalization of a boundedly supermultiplicative sequence ̃ n )∞ (an )∞ n=1 is the sequence (a n=1 , where ̃ n = an e−Pn a

and

P = lim

n→∞

1 log an . n

Show that the normalization of a boundedly supermultiplicative sequence with constant C is a boundedly supermultiplicative sequence with the same constant C but lim

n→∞

1 1 ̃ n = sup log(C −1 a ̃ n ) = 0. log a n n∈ℕ n

̃ n )∞ Exercise 17.9.8. Prove that the normalization (a n=1 of a boundedly supermultiplicative and boundedly submultiplicative sequence (an )∞ n=1 of positive numbers such that −∞ < P < ∞ is a bounded sequence. More precisely, it is bounded from below by B−1 , where B is any constant of submultiplicativity, and from above by any constant C of supermultiplicativity.

17.9 Exercises | 705

Exercise 17.9.9. Suppose that f : EA∞ → ℝ is an acceptable potential and that A is a finitely irreducible matrix. Show that P(σ k , Sk f ) = kP(σ, f ) for any k ∈ ℕ. (This is a generalization of Theorem 11.1.22 to the countable state symbolic setting.) Exercise 17.9.10. Let E be a countable alphabet and A : E × E → {0, 1} be a finitely irreducible matrix. Denote by Ca (EA∞ ) the set of all acceptable potentials on EA∞ . (a) Show that Ca (EA∞ ) is a convex set. (b) Using the 1st variational principle (Theorem 17.3.2), prove that the pressure function P(∙) : Ca (EA∞ ) → ℝ is convex. (c) Using rather the 3rd variational principle (Theorem 17.3.4), give an alternative demonstration of the convexity of the pressure function. Exercise 17.9.11. Prove Proposition 17.4.3. Exercise 17.9.12. Show that any potential f : EA∞ → ℝ that admits a Gibbs state must be summable. Exercise 17.9.13. Prove Lemma 17.6.2. Exercise 17.9.14. Prove that a Hölder continuous potential is not summable. Infer that such a potential does not admit a Gibbs state. Conclude that it is of no interest to study Hölder continuous potentials for countable state symbolic systems. Exercise 17.9.15. Two functions φ, ψ : EA∞ → ℝ are said to be cohomologous modulo a constant (or, equivalently, φ − ψ is cohomologous to a constant) in the additive group of bounded functions on EA∞ if there exist a bounded function u : EA∞ → ℝ and a constant c ∈ ℝ such that φ − ψ = u ∘ σ − u + c. (cf. Definition 13.2.5) The objective of this exercise is to establish results analogous to Lemma 13.2.6 and Exercise 12.4.5. Suppose that φ, ψ : EA∞ → ℝ are acceptable potentials and that A is finitely irreducible. (a) Show that P(φ) = P(ψ) + c. (b) Prove that 𝒢φ = 𝒢ψ , i. e., φ and ψ share the same set of Gibbs states. (c) Demonstrate that φ and ψ share the same set of equilibrium states. Exercise 17.9.16. Repeat Exercise 13.11.13 but this time with a countable alphabet E. Exercise 17.9.17. Repeat Exercise 13.11.14 but with a countable alphabet E.

18 Countable state thermodynamic formalism: finer properties In this chapter, we continue the development of thermodynamic formalism for countable-alphabet subshifts of finite type σ : EA∞ → EA∞ . This time, we deal with refined properties of σ-invariant Gibbs states μf for potentials f : EA∞ → ℝ that are summable and Hölder continuous on cylinders. We show that the measure-theoretic dynamical systems (EA∞ , σ, μf ) deserve to be called random or chaotic by proving in Section 18.3 that they obey classical stochastic laws such as the exponential decay of correlations and the central limit theorem. Although we do not prove it, they also behave in accordance with the law of the iterated logarithm. Another property essential for randomness is the existence of asymptotic variances for Hölder continuous functions. We prove two very useful formulas for them. All of this is achieved via a refined study of transfer operators acting on the Banach space of Hölder continuous functions and especially by establishing their spectral properties, such as quasicompactness (spectral gap) and simplicity of the leading eigenvalue eP(f ) . To set the table, we prove in Section 18.1 the famous Ionescu-Tulcea and Marinescu inequality. In Section 18.2, we establish the continuity (1) of the pressure function f 󳨃→ P(f ); (2) of the unique σ-invariant Gibbs states f 󳨃→ μf ; (3) of the unique eigenmeasures of the dual operators f 󳨃→ mf ; and (4) of the Radon–Nikodym derivative of μf with respect to mf . Finally, in Section 18.4 we characterize, primarily by means of cohomological equations, those potentials that have the same invariant Gibbs state.

18.1 Ionescu-Tulcea and Marinescu inequality In this section, we prove the famous Ionescu-Tulcea and Marinescu inequality for countable state subshifts of finite type, with the aim of obtaining a spectral picture and the rate of convergence of the iterates of the transfer operator. Unless otherwise stated, E is a countable alphabet and A : E × E → {0, 1} a finitely irreducible incidence matrix. Let β ∈ (0, 1]. Denote by 𝒦β (EA∞ ) the set of all complexvalued functions that are Hölder continuous on cylinders with exponent β (cf. Definition 17.2.2). Denote by 𝒦β (EA∞ , ℝ) the subspace of 𝒦β (EA∞ ) consisting of those functions that are real valued. Set (cf. Definition 17.6.1) s

𝒦β (EA ) := {f ∈ 𝒦β (EA ) : Re(f ) is summable} ∞

∞

= {f ∈ 𝒦β (EA∞ ) : ∑ exp(sup Re(f |[e] )) < ∞}. e∈E

Notice that the real part of the functions in 𝒦βs (EA∞ ) is always bounded from above, but is not bounded from below when E is infinite. https://doi.org/10.1515/9783110702699-018

708 | 18 Countable state thermodynamic formalism: finer properties

and

Let Cb (EA∞ ) be the set of all bounded continuous complex-valued functions on EA∞ b

𝒦β (EA ) := 𝒦β (EA ) ∩ Cb (EA ). ∞

∞

∞

Recall that a function f : EA∞ → ℂ is Hölder continuous if and only if it is Hölder continuous on cylinders and bounded. Thus, 𝒦βb (EA∞ ) is the set of all complex-valued functions on EA∞ that are Hölder with exponent β. When endowed with the natural norm ‖f ‖β := υβ (f ) + ‖f ‖∞ ,

(18.1)

the set 𝒦βb (EA∞ ) becomes a Banach space. Note that 𝒦βb (EA∞ ) = 𝒦βs (EA∞ ) when E is finite, while 𝒦βb (EA∞ ) ∩ 𝒦βs (EA∞ ) = 0 when E is infinite. For any f ∈ 𝒦βb (EA∞ ) and F ⊆ EA∞ , define

‖f ‖β,F = υβ,F (f ) + ‖f ‖∞,F , where 󵄨󵄨 󵄨 󵄨f (ω) − f (τ)󵄨󵄨󵄨 υβ,F (f ) := υβ (f |F ) = sup 󵄨 β d (ω, τ) ω=τ∈F: ̸ ω1 =τ1

and

󵄩 󵄩 󵄨 󵄨 ‖f ‖∞,F := 󵄩󵄩󵄩f |F 󵄩󵄩󵄩∞ = sup󵄨󵄨󵄨f (ω)󵄨󵄨󵄨. ω∈F

Clearly, υβ,F (f ) ≤ υβ (f ) and ‖f ‖∞,F ≤ ‖f ‖∞ , so that ‖f ‖β,F ≤ ‖f ‖β . It is further easy to see that (cf. Exercise 18.5.4) ‖fg‖β,F ≤ ‖f ‖β,F ‖g‖β,F ,

∀f , g ∈ 𝒦βb (EA∞ ).

(18.2)

We briefly look at the iterates of the normalized transfer operator. Lemma 18.1.1. Let E be a countable alphabet and A : E ×E → {0, 1} a finitely irreducible incidence matrix. Let f ∈ 𝒦βs (EA∞ , ℝ) be a potential and let C be a ratio bounding constant for a Gibbs state for f . Then 󵄩󵄩 ̂n 󵄩󵄩 󵄩󵄩ℒf (1)󵄩󵄩∞ ≤ C,

∀n ∈ ℕ

and hence

󵄩󵄩 ̂n 󵄩󵄩 󵄩󵄩ℒf (g)󵄩󵄩∞ ≤ C‖g‖∞ ,

∀n ∈ ℕ, ∀g ∈ Cb (EA∞ ).

Proof. First, a Gibbs state exists according to Corollary 17.7.5. Denote by m the Gibbs state with ratio bounding constant C. By (17.22) and (17.9), for every ρ ∈ EA∞ we have n

̂ (1)(ρ) = ℒ f

∑

ω∈EAn : Aωn ρ1 =1

exp(Sn f (ωρ) − P(f )n) ≤ C

∑

ω∈EAn : Aωn ρ1 =1

m([ω]) ≤ C.

Lemma 18.1.2. Suppose Λ is a finite set of words witnessing the finite irreducibility of the incidence matrix A. Let |Λ| be the maximal length of words in Λ. If f ∈ 𝒦βs (EA∞ , ℝ),

18.1 Ionescu-Tulcea and Marinescu inequality | 709

then there exists a constant D > 0 such that n+|Λ|

̂k (1)(ω) ≥ D, ∑ ℒ f

k=n

∀n ∈ ℕ, ∀ω ∈ EA∞ .

̂k (1) dmf = 1 Proof. As f ∈ 𝒦βs (EA∞ , ℝ), it follows from Corollary 17.7.5 and (17.25) that ∫ ℒ f ∞ k ̂ (1)(γ(k)) ≥ 1. Since Λ for all k ∈ ℕ. So, for every k ∈ ℕ there is γ(k) ∈ E such that ℒ A

is finite and f ∈ 𝒦β (EA∞ , ℝ), we have

f

ℓ := min exp(inf(S|α| f |[α] ) − P(f )|α|) > 0. α∈Λ

Let n ∈ ℕ and τ ∈ EA∞ . Then n+|Λ|

n+|Λ|

̂k (1)(τ) = ∑ ∑ ℒ f

k=n

∑

k=n ω∈EAk : Aω τ =1 k 1

exp(Sk f (ωτ) − P(f )k).

For each ω ∈ EAn , choose α(ω, τ) ∈ Λ such that ωα(ω, τ)τ ∈ EA∞ . Then n+|Λ|

̂k (1)(τ) ≥ ∑ exp(Sn+|α(ω,τ)| f (ωα(ω, τ)τ) − P(f )(n + 󵄨󵄨󵄨α(ω, τ)󵄨󵄨󵄨)) ∑ ℒ f 󵄨 󵄨 ω∈EAn

k=n

= ∑ exp(Sn f (ωα(ω, τ)τ) − P(f )n) ω∈EAn

󵄨 󵄨 × exp(S|α(ω,τ)| f (α(ω, τ)τ) − P(f )󵄨󵄨󵄨α(ω, τ)󵄨󵄨󵄨)

≥ ℓ ∑ exp(Sn f (ωα(ω, τ)τ) − P(f )n). ω∈EAn

If Aωn γ(n)1 = 1, then by Lemma 17.2.3 we obtain that exp(Sn f (ωα(ω, τ)τ) − P(f )n) ≥ e

−

υβ (f ) eβ −1

exp(Sn f (ωγ(n)) − P(f )n).

Therefore, n+|Λ|

υβ (f )

̂k (1)(τ) ≥ ℓe− eβ −1 ∑ ℒ f

k=n

≥ ℓe

−

υβ (f ) eβ −1

∑

ω∈EAn : Aωn γ(n)1 =1 n

exp(Sn f (ωγ(n)) − P(f )n)

̂ (1)(γ(n)) ≥ ℓe ℒ f

−

υβ (f ) eβ −1

.

Remark 18.1.3. If the underlying matrix A in Lemma 18.1.2 is finitely primitive, then ̂n (1)(ω) ≥ D for all n ∈ ℕ and all ω ∈ E ∞ . there is a constant D > 0 such that ℒ A f The main technical result of this section is the following.

710 | 18 Countable state thermodynamic formalism: finer properties Lemma 18.1.4 (Ionescu-Tulcea and Marinescu inequality). Let E be a countable alphabet and A : E × E → {0, 1} a finitely irreducible incidence matrix. If f ∈ 𝒦βs (EA∞ , ℝ), then ̂f : Cb (E ∞ ) → Cb (E ∞ ) preserves the space 𝒦b (E ∞ ), the normalized transfer operator ℒ A

A

̂f (𝒦b (E ∞ )) ⊆ 𝒦b (E ∞ ). Moreover, there exists a constant Q > 1 such that i. e. ℒ β A β A ̂n (g)) ≤ C(e−βn ‖g‖β + (Q − 1)‖g‖∞ ) and υβ (ℒ f

A

β

󵄩󵄩 ̂n 󵄩󵄩 󵄩󵄩ℒf (g)󵄩󵄩∞ ≤ C‖g‖∞

for all n ∈ ℕ and all g ∈ 𝒦βb (EA∞ ), where C is a ratio bounding constant for a Gibbs state for f and C, Q are independent of n and g. Consequently, 󵄩󵄩 ̂n 󵄩󵄩 −βn 󵄩󵄩ℒf (g)󵄩󵄩β ≤ C(e ‖g‖β + Q‖g‖∞ ),

∀n ∈ ℕ, ∀g ∈ 𝒦βb (EA∞ ).

Proof. Let ρ, τ ∈ EA∞ be such that |ρ ∧ τ| ≥ 1. For every n ∈ ℕ, n

n

̂ (g)(ρ) − ℒ ̂ (g)(τ) = ℒ f f

∑

ω∈EAn : Aωn ρ1 =1

− =

∑

ω∈EAn : Aωn ρ1 =1

∑

ω∈EAn : Aωn ρ1 =1

+

g(ωρ) exp(Sn f (ωρ) − P(f )n) g(ωτ) exp(Sn f (ωτ) − P(f )n)

(g(ωρ) − g(ωτ)) exp(Sn f (ωρ) − P(f )n)

∑

ω∈EAn : Aωn ρ1 =1

g(ωτ)[exp(Sn f (ωρ) − P(f )n) − exp(Sn f (ωτ) − P(f )n)].

(18.3)

But 󵄨󵄨 󵄨 β −βn β 󵄨󵄨g(ωρ) − g(ωτ)󵄨󵄨󵄨 ≤ υβ (g)d (ωρ, ωτ) = e υβ (g)d (ρ, τ). Using Lemma 18.1.1, we obtain that ∑

ω∈EAn : Aωn ρ1 =1

󵄨󵄨 󵄨 −βn β ̂n (1))(ρ) 󵄨󵄨g(ωρ) − g(ωτ)󵄨󵄨󵄨 exp(Sn f (ωρ) − P(f )n) ≤ e υβ (g)d (ρ, τ) (ℒ f ≤ Ce−βn ‖g‖β dβ (ρ, τ).

(18.4)

Notice that there exists a constant Q0 ≥ 1 such that |1 − ex | ≤ Q0 |x| for all x with |x| ≤

υβ (f ) . eβ −1

Employing Lemma 17.2.3, we get

󵄨󵄨 󵄨 󵄨󵄨exp(Sn f (ωρ) − P(f )n) − exp(Sn f (ωτ) − P(f )n)󵄨󵄨󵄨 󵄨 󵄨 = exp(Sn f (ωρ) − P(f )n)󵄨󵄨󵄨1 − exp(Sn f (ωτ) − Sn f (ωρ))󵄨󵄨󵄨 󵄨 󵄨 ≤ Q0 exp(Sn f (ωρ) − P(f )n)󵄨󵄨󵄨Sn f (ωρ) − Sn f (ωτ)󵄨󵄨󵄨 υβ (f ) β ≤ Q0 exp(Sn f (ωρ) − P(f )n) β d (ρ, τ). e −1

18.1 Ionescu-Tulcea and Marinescu inequality |

711

Therefore, using Lemma 18.1.1, we have that ∑

ω∈EAn : Aωn ρ1 =1

󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨g(ωτ)󵄨󵄨󵄨󵄨󵄨󵄨exp(Sn f (ωρ) − P(f )n) − exp(Sn f (ωτ) − P(f )n)󵄨󵄨󵄨 󵄨 󵄨

≤ ‖g‖∞ Q0 = ‖g‖∞ Q0

υβ (f )

eβ − 1 υβ (f )

eβ

−1

dβ (ρ, τ)

∑

ω∈EAn : Aωn ρ1 =1

exp(Sn f (ωρ) − P(f )n)

̂n (1)(ρ) ≤ CQ0 dβ (ρ, τ)ℒ f

υβ (f )

eβ − 1

‖g‖∞ dβ (ρ, τ).

Combining this inequality with (18.4) and (18.3), we get υ (f ) 󵄨󵄨 ̂n ̂n (g)(τ)󵄨󵄨󵄨 ≤ [Ce−βn ‖g‖β + CQ0 β ‖g‖∞ ]dβ (ρ, τ), 󵄨󵄨ℒf (g)(ρ) − ℒ f 󵄨 eβ − 1 ̂n (g)) ≤ Ce−βn ‖g‖β + CQ0 υββ(f ) ‖g‖∞ . Joining this with Lemma 18.1.1, we coni. e., υβ (ℒ f e −1 clude that υβ (f ) 󵄩󵄩 ̂n 󵄩󵄩 −βn + 1)‖g‖∞ . 󵄩󵄩ℒf (g)󵄩󵄩β ≤ Ce ‖g‖β + C(Q0 β e −1 υ (f )

Set Q = Q0 eββ −1 + 1. Remark 18.1.5. The proof of Lemma 18.1.4 used only a “weaker” property of Gibbs states, namely only the right inequality of (17.9). Remark 18.1.6. If the unit ball in 𝒦βb (EA∞ ) was a compact subset of the Banach space (Cb (EA∞ ), ‖⋅‖∞ ), we could use the famous Ionescu-Tulcea and Marinescu theorem (Theorem B.1.1 in Appendix B) to establish some useful spectral properties of the normal̂f . But this ball is compact only in the topology of uniform conized transfer operator ℒ vergence on compact subsets of EA∞ and we thus need to prove these spectral properties directly. ̂f . Next, we introduce an important fixed point ψf of ℒ Theorem 18.1.7. Let E be a countable alphabet and A : E × E → {0, 1} a finitely irreducible incidence matrix. Let f ∈ 𝒦βs (EA∞ , ℝ) be a potential and mf be the unique Borel probability eigenmeasure mf of the dual transfer operator ℒ∗f or, equivalently, the ̂∗ per Corollary 17.7.5. Then ℒ ̂f has a fixed point ψf ∈ 𝒦b (E ∞ ) such unique fixed point of ℒ that

f

∫ ψf dmf = 1 and EA∞

β

0
0. As f is summable, the series ∑e∈E exp(sup(f |[e] ) − P(f )) converges to, say, S > 0. Consequently, there exists a finite set Eε ⊆ E such that ε 2B ∑ exp(sup(f |[e] ) − P(f )) < . 2 e∈E\E

(18.8)

ε

Fix any compact set K ⊆ EA∞ . For every e ∈ E, the set eK := {eω : ω ∈ K and Aeω1 = 1} is also compact and so is the set ∪e∈Eε eK. Since (gk )∞ k=1 converges uniformly on compact sets to g, there exists N ∈ ℕ such that ‖gn − g‖∞,⋃e∈E

ε

eK

≤

ε , 2S

∀n ≥ N.

(18.9)

Using (18.8)–(18.9), we get for every n ≥ N and every ω ∈ K that 󵄨󵄨 ̂ 󵄨󵄨 ̂f (gn )(ω)󵄨󵄨󵄨 = 󵄨󵄨󵄨ℒ ̂ 󵄨󵄨ℒf (g)(ω) − ℒ 󵄨 󵄨 f (g − gn )(ω)󵄨󵄨 󵄨 󵄨 ≤ ∑ 󵄨󵄨󵄨gn (eω) − g(eω)󵄨󵄨󵄨 exp(f (eω) − P(f )) e∈Eε : Aeω1 =1

+ ≤

e∈E\Eε : Aeω1 =1

∑

e∈Eε : Aeω1

+ ≤

∑

󵄨󵄨 󵄨 󵄨󵄨gn (eω) − g(eω)󵄨󵄨󵄨 exp(f (eω) − P(f ))

ε exp(f (eω) − P(f )) 2S =1

∑

e∈E\Eε : Aeω1 =1

2B exp(f (eω) − P(f ))

ε ε + = ε. 2 2

̂f (gk ))∞ converges uniformly on compact subsets of E ∞ to ℒ ̂f (g) whenever Hence, (ℒ A k=1 ∞ ∞ a sequence (gk )k=1 in Cb (EA ) is uniformly bounded and converges uniformly on comnk −1 ̂j pact subsets of EA∞ to a function g. As the sequence ( n1 ∑j=0 ℒf (1))k∈ℕ is uniformly k bounded and converges uniformly on compact subsets of EA∞ to ψf : EA∞ → ℝ, we ̂f (ψf ) = ψf . conclude from (18.7) that ℒ

714 | 18 Countable state thermodynamic formalism: finer properties Corollary 18.1.8. Let E be a countable alphabet and A : E × E → {0, 1} a finitely irreducible incidence matrix. Let f ∈ 𝒦βs (EA∞ , ℝ) be a potential. Per Corollary 17.7.5, let mf be ̂∗ and let μf be the unique σ-invariant the unique Borel probability measure fixed by ℒ f ̂f identified Gibbs state for the potential f . Finally, let ψf ∈ 𝒦b (E ∞ ) be the fixed point of ℒ β

in Theorem 18.1.7. Then

A

μf = ψf mf . Moreover, μf and mf are equivalent with Radon–Nikodym derivatives bounded away ̂f , is nonnegfrom 0 and ∞. In addition, ψf is the unique ψ ∈ Cb (EA∞ ) which is fixed by ℒ ative and such that ∫ ψ dmf = 1. Proof. The proof is in some respect similar to that of Theorem 13.7.7. However, we do know that f admits a unique σ-invariant Gibbs state, and that state is μf . By virtue of its definition and (18.5) in Theorem 18.1.7, we know that ψf mf is a Borel probability measure and that it is equivalent to mf with Radon–Nikodym derivatives bounded away from 0 and ∞. Since mf is a Gibbs state for f and since ψf mf is boundedly equivalent to mf , we know by Proposition 17.4.2 that ψf mf is a Gibbs state for f , too. Let us show that ψf mf is σ-invariant. Let g ∈ Cb (EA∞ ). Then ̂f (ψf ) dmf ψf mf (g) = ∫ g d(ψf mf ) = ∫ g ⋅ ψf dmf = ∫ g ⋅ ℒ EA∞

EA∞

EA∞

̂∗ (mf ) ̂f ((g ∘ σ) ⋅ ψf ) dmf = ∫ (g ∘ σ) ⋅ ψf dℒ = ∫ℒ f EA∞

EA∞

= ∫ (g ∘ σ) ⋅ ψf dmf = ∫ g ∘ σ d(ψf mf ) = ψf mf (g ∘ σ). EA∞

EA∞

As Cb (EA∞ ) is dense in L1 (ψf mf ), we deduce that ψf mf is σ-invariant. In summary, ψf mf is a σ-invariant Gibbs state and by the uniqueness of such a state we conclude that μf = ψf mf . It remains to prove that ψf is unique. Suppose that two such functions ψ1 , ψ2 ∈ Cb (EA∞ ) exist but that ψ1 ≠ ψ2 . By the uniqueness of the σ-invariant Gibbs state for f , we would then have that ψ1 mf = μf = ψ2 mf . But there would also exist a point ω ∈ EA∞ and δ > 0 such that ψ1 (ω) ≤ ψ2 (ω) − 2δ (without loss of generality; otherwise, interchange ψ1 and ψ2 ). By continuity, there would then exist n ∈ ℕ such that ψ1 (τ) ≤ ψ2 (τ) − δ for all τ ∈ [ω|n ]. Upon integrating over [ω|n ], we would get μf ([ω|n ]) = ψ1 mf ([ω|n ]) ≤ (ψ2 − δ)mf ([ω|n ]) = μf ([ω|n ]) − δmf ([ω|n ]) < μf ([ω|n ]), which is impossible.

18.1 Ionescu-Tulcea and Marinescu inequality |

715

As ψf is uniformly bounded away from 0, we can mimick the main idea in Section 13.8.1 by introducing the function ̂f = f + log ψ − log ψ ∘ σ f f ̂̂ : Cb (E ∞ ) → Cb (E ∞ ) and observing that the weighted normalized transfer operator ℒ A A f is well-defined and satisfies ̂̂ (g) = ℒ f

1 ̂ ℒ (gψf ). ψf f

(18.10)

̂̂ (𝒦b (E ∞ )) ⊆ 𝒦b (E ∞ ), as 1/ψf ∈ 𝒦b (E ∞ ) and the It is straightforward to check that ℒ β A β A β A f

product of any two functions in 𝒦βb (EA∞ ) is in 𝒦βb (EA∞ ); see Exercise 18.5.4. The basic ̂f and ℒ ̂̂ , which follow from Lemma 18.1.4, Lemma 18.1.2 properties of the operators ℒ f and Theorem 18.1.7, are listed in the next two theorems. Theorem 18.1.9. Let E be a countable alphabet and A : E × E → {0, 1} a finitely irreducible incidence matrix. Let f ∈ 𝒦βs (EA∞ , ℝ) be a potential. The following statements hold: ̂f (ψf ) = ψf for some positive ψf ∈ 𝒦b (E ∞ ) whose values are uniformly bounded (a) ℒ β A away from 0 and such that mf (ψf ) = 1. ̂∗ (mf ) = mf , where mf is the unique Borel probability eigenmeasure mf of the dual (b) ℒ f

transfer operator ℒ∗f per Corollary 17.7.5. (c) The vector subspaces ℝψf , 0,mf

Cb

b,0,mf

(EA∞ ) := {g ∈ Cb (EA∞ ) : mf (g) = 0} and 𝒦β 0,mf

̂f –invariant. Moreover, C are ℒ b

is closed in (𝒦βb (EA∞ ), ‖ ⋅ ‖β ). 󵄩 ̂n 󵄩󵄩 (d) sup󵄩󵄩󵄩ℒ 󵄩β < ∞. f󵄩 n∈ℕ

b,0,mf

(e) 𝒦βb (EA∞ ) = ℝψf ⨁ 𝒦β

0,mf

(EA∞ ) := 𝒦β (EA∞ )∩Cb

b,0,mf

(EA∞ ) is closed in (Cb (EA∞ ), ‖⋅‖∞ ) while 𝒦β

(EA∞ ) (EA∞ )

(EA∞ ) as g = mf (g)ψf + (g − mf (g)ψf ).

(f) If the spaces Cb (EA∞ ) and 𝒦β (EA∞ ) are extended to complex-valued functions in the obvious and natural way, then the above statements with ℝ replaced by ℂ hold. ̂f : 𝒦b (E ∞ ) → 𝒦b (E ∞ ) is the union of the spectrum Consequently, the spectrum of ℒ β A β A ̂f restricted to ℂψf and of the spectrum of ℒ ̂f restricted to the complex version of ℒ b,0,mf

of 𝒦β

(EA∞ ).

Theorem 18.1.10. Let E be a countable alphabet and A : E × E → {0, 1} a finitely irreducible incidence matrix. Let f ∈ 𝒦βs (EA∞ , ℝ) be a potential. For every n ∈ ℕ, define the

716 | 18 Countable state thermodynamic formalism: finer properties balancing function un : EA∞ → ℝ by un (ω) := exp(Sn f (ω) − P(f )n)

ψf (ω)

ψf (σ n (ω))

.

The following statements hold: (a) For all g ∈ 𝒦βb (EA∞ ) and all n ∈ ℕ, we have n

̂̂ (g)(ω) = ℒ f

1 ̂n ℒ (gψf )(ω) = ∑ ψf (ω) f τ∈E n : A A

un (τω)g(τω).

τn ω1 =1

̂̂ (1) = 1 and 󵄩󵄩󵄩ℒ ̂n 󵄩󵄩 (b) ℒ 󵄩 ̂f 󵄩󵄩∞ = 1 for all n ∈ ℕ. f

̂̂∗ (μf ) = μf , where μf is the unique (ergodic) σ-invariant Gibbs state for f , per Corol(c) ℒ f lary 17.7.5. (d) The vector subspaces ℝ1, 0,μ

b,0,μf

Cb f (EA∞ ) := {g ∈ Cb (EA∞ ) : μf (g) = 0} and

𝒦β

0,μ

(EA∞ ) := 𝒦β (EA∞ ) ∩ Cb f (EA∞ )

0,μ

b,0,μf

̂̂ –invariant. Moreover, C f (E ∞ ) is closed in (Cb (E ∞ ), ‖⋅‖∞ ) while 𝒦 are ℒ A A f b β

is closed in (𝒦βb (EA∞ ), ‖ ⋅ ‖β ). 󵄩 ̂n 󵄩󵄩 (e) M := sup󵄩󵄩󵄩ℒ ̂f 󵄩 󵄩β < ∞. n∈ℕ

b,0,μf

(f) 𝒦βb (EA∞ ) = ℝ1 ⨁ 𝒦β

(EA∞ )

(EA∞ ) as g = μf (g)1 + (g − μf (g)1).

(g) If the spaces Cb (EA∞ ) and 𝒦β (EA∞ ) are extended to complex-valued functions in the obvious and natural way, then the above statements with ℝ replaced by ℂ hold. This ̂̂ : 𝒦b (E ∞ ) → 𝒦b (E ∞ ) is the union of the spectrum implies that the spectrum of ℒ β A β A f ̂̂ restricted to ℂ1 and of the spectrum of ℒ ̂̂ restricted to the complex version of of ℒ f b,0,μf

𝒦β

f

(EA∞ ).

Denote b,0,μf

𝒦β,1

b,0,μf

(EA∞ ) := {g ∈ 𝒦β

(EA∞ ) : ‖g‖β ≤ 1}.

We prove the following. Lemma 18.1.11. Let E be a countable alphabet and A : E × E → {0, 1} a finitely primitive incidence matrix. Let f ∈ 𝒦βs (EA∞ , ℝ) be a potential. For each n ≥ 0, let b,0,μf 󵄩 ̂n 󵄩󵄩 ∞ bn = sup{󵄩󵄩󵄩ℒ ̂f (g)󵄩 󵄩β : g ∈ 𝒦β,1 (EA )}.

Then lim bn = 0. n→∞

18.1 Ionescu-Tulcea and Marinescu inequality |

717

Proof. For every n ≥ 0, define b,0,μf 󵄩 ̂n 󵄩󵄩 ∞ an := sup{󵄩󵄩󵄩ℒ ̂f (g)󵄩 󵄩∞ : g ∈ 𝒦β,1 (EA )} ≥ 0.

It follows from Theorem 18.1.10(b) that the sequence (an )∞ n=1 is nonincreasing and a1 ≤ 1. We first show that a := limn→∞ an = 0. Suppose on the contrary that a > 0. By Theorem 18.1.10(e), we deduce that ̂̂n (g)) : g ∈ 𝒦b,0,μf (E ∞ ), n ∈ ℕ} ≤ M < ∞. Therefore, there exists k ∈ ℕ such sup{υβ (ℒ A β,1 f that 󵄨 ̂n 󵄨󵄨 ̂n |ω ∧ τ| ≥ k 󳨐⇒ 󵄨󵄨󵄨ℒ ̂f (g)(ω) − ℒ̂f (g)(τ)󵄨󵄨 ≤ a/2,

b,0,μf

∀g ∈ 𝒦β,1

(EA∞ ), ∀n ≥ 0.

(18.11)

Let μf be the unique ergodic σ-invariant Gibbs state for f , per Corollary 17.7.5. As μf is b,0,μ

regular, there exists a compact set K ⊆ EA∞ such that μf (K) ≥ 1−a/8. Fix g ∈ 𝒦β,1 f (EA∞ ) ̂̂n (g) dμf = 0 by Theô̂n (g)(τ) > a/4 for all τ ∈ K. Since ∫ ℒ and n ≥ 0. Suppose that ℒ f f 󵄩 ̂n 󵄩󵄩 rem 18.1.10(d) and since 󵄩󵄩󵄩ℒ (g) ≤ ‖g‖ ≤ 1, we find that 󵄩 ∞ ̂f 󵄩∞ ̂̂n (g) dμf + ∫ ℒ ̂̂n (g) dμf ≥ a (1 − a ) − a = a (1 − a ) > 0. 0 = ∫ℒ f f 4 8 8 8 4 ∞ K

EA \K

̂̂n (g)(τ) ≤ a/4 for some τ ∈ K. Fix this τ. Let Λ ⊆ E ∗ This contradiction shows that ℒ A f be a finite set of finite words witnessing the finite primitivity of the incidence matrix A and let |Λ| be the length of these words. For every α ∈ Λ, pick a word α ∈ EA∞ such that α||α| = α. For every ω ∈ EA∞ , there exists α(ω) ∈ Λ such that ρ(ω) := τ|k α(ω)ω ∈ EA∞ . Since ρ(ω)|k = τ|k , it follows from (18.11) that n

n

̂̂ (g)(ρ(ω)) ≤ ℒ ̂̂ (g)(τ) + ℒ f f

a 3 a ≤ a ≤ an − . 2 4 4

Moreover, let p := k + |Λ|. Using Theorem 18.1.7 (with the notation DΛ := Lemma 17.2.3, we get up (ρ(ω)) up (τ)

= exp(Sp f (ρ(ω)) − Sp f (τ))

ψf (ρ(ω))

ψf (σ p (ρ(ω)))

(18.12) D ) |Λ|+1

ψf (σ p (τ)) ψf (τ)

≥ exp(Sk f (ρ(ω)) − Sk f (τ)) exp(S|Λ| f (σ k (ρ(ω))) − S|Λ| f (σ k (τ))) ⋅ ≥e

−

υβ (f ) eβ −1

⋅ exp(S|Λ| f (α(ω)ω) − |Λ| sup(f )) ⋅

≥ (DΛ /Cf )2 e

−2

≥ (DΛ /Cf )2 e

−2

υβ (f ) eβ −1

υβ (f ) eβ −1

D2Λ Cf2

exp(S|Λ| f (α(ω))) exp(−|Λ| sup(f )) exp(min{S|Λ| f (α) : α ∈ Λ}) exp(−|Λ| sup(f )).

D2Λ Cf2

and

718 | 18 Countable state thermodynamic formalism: finer properties Denote the last (constant) expression appearing in this formula by U. As up is positive and continuous and K is compact, we infer that up (ρ(ω)) ≥ U inf(up |K ) > 0. Using Theorem 18.1.10(a,b) and (18.12), we deduce that p f

n

n

̂ (ℒ ̂̂ (g))(ω) = up (ρ(ω))ℒ ̂̂ (g)(ρ(ω)) + ℒ ̂ f f

∑

η∈σ −p (ω)\{ρ(ω)}

̂̂n (g)(η) up (η)ℒ f

a )u (ρ(ω)) + an up (η) ∑ 4 p η∈σ −p (ω)\{ρ(ω)} ̂̂n (1)(ω) − a up (ρ(ω)) ≤ an − a U inf(up |K ). = an ℒ f 4 4 ≤ (an −

̂p (ℒ ̂̂n (g))(ω) ≥ −an + (a/4)U inf(up |K ). Consequently, we have Similarly, we get ℒ ̂f f 󵄩󵄩 ̂p+n 󵄩󵄩 󵄩󵄩ℒ̂ (g)󵄩󵄩∞ ≤ an − (a/4)U inf(up |K ) or, equivalently, f a 󵄩󵄩 ̂n 󵄩󵄩 󵄩󵄩ℒ̂f (g)󵄩󵄩∞ ≤ an−p − U inf(up |K ), 4

∀n ≥ p.

b,0,μ

Taking the supremum over all g ∈ 𝒦β,1 f (EA∞ ), we find an ≤ an−p − (a/4)U inf(up |K ). So a := limn→∞ an ≤ limn→∞ an−p − (a/4)U inf(up |K ) = a − (a/4)U inf(up |K ) < a. This contradiction shows that a := limn→∞ an = 0. Now, fix ε > 0. Choose N ∈ ℕ so large that aN ≤ ε/(2Cf Q) and Cf e−βN M ≤ ε/2. By b,0,μf

virtue of Lemma 18.1.4, for every n ≥ 2N and every g ∈ 𝒦β,1

(EA∞ ) we get

󵄩󵄩 ̂n 󵄩󵄩 󵄩 ̂n−N ̂N 󵄩 󵄩 ̂N 󵄩󵄩 󵄩󵄩 ̂N 󵄩󵄩 (ℒ̂f (g))󵄩󵄩󵄩β ≤ Cf (e−β(n−N) 󵄩󵄩󵄩ℒ 󵄩󵄩ℒ̂f (g)󵄩󵄩β = 󵄩󵄩󵄩ℒ ̂f (g)󵄩 ̂f 󵄩β + Q󵄩󵄩ℒ̂f (g)󵄩󵄩∞ ) ≤ Cf e−βN M + Cf QaN ≤ ε.

So bn ≤ ε for all n ≥ 2N. The arbitrariness of ε implies that limn→∞ bn = 0. We now describe the rates of convergence of the iterates of the normalized transfer operator and of its weighted counterpart. Theorem 18.1.12. Let E be a countable alphabet and A : E × E → {0, 1} a finitely primitive incidence matrix. Let f ∈ 𝒦βs (EA∞ , ℝ) be a potential. Let mf be the unique Borel ̂f and let ̂∗ of the normalized transfer operator ℒ probability measure fixed by the dual ℒ f μf be the unique ergodic σ-invariant Gibbs state for f , per Corollary 17.7.5. Finally, let ψf ̂f identified in Theorem 18.1.7 and ℒ ̂̂ be the weighted normalized be the fixed point of ℒ f transfer operator defined by (18.10). Then there exist constants c > 0 and 0 < ζ < 1 such that for every g ∈ 𝒦βb (EA∞ ) and every n ≥ 0, (a) 󵄩󵄩 󵄩󵄩 󵄩󵄩 ̂n 󵄩 n 󵄩󵄩ℒ̂f (g) − ( ∫ g dμf )1󵄩󵄩󵄩 ≤ cζ ‖g‖β . 󵄩󵄩 󵄩󵄩β ∞ EA

18.2 Continuity of Gibbs states | 719

(b) 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ̂n n 󵄩󵄩ℒf (g) − ( ∫ g dmf )ψf 󵄩󵄩󵄩 ≤ cζ ‖g‖β . 󵄩󵄩β 󵄩󵄩 ∞ EA

(c) Upon complexifying 𝒦βb (EA∞ ) in the natural way, the above relations also hold for

all complex-valued functions g ∈ 𝒦βb (EA∞ ). Additionally, the number 1 is a simple ̂f and ℒ ̂̂ . More precisely, apart from number isolated eigenvalue of the operators ℒ f 1, the rest of the spectrum of each of these operators is contained in Bℂ (0, ζ ).

󵄩̂ n Proof. Lemma 18.1.11 says that limn→∞ 󵄩󵄩󵄩ℒ ̂f | b,0,μf

󵄩󵄩 󵄩󵄩β = 0. Thus, there exists q ∈ ℕ 󵄩̂ q 󵄩󵄩 󵄩󵄩 ̂ pq 󵄩󵄩 p such that 󵄩󵄩󵄩ℒ ̂f | b,0,μf ∞ 󵄩 󵄩β ≤ 1/2. By induction, we have 󵄩󵄩ℒ̂f |𝒦b,0,μf (E ∞ ) 󵄩󵄩β ≤ (1/2) for all 𝒦 (E ) 𝒦β

(EA∞ )

A

β

β

A

p ∈ ℤ+ . Consider an arbitrary n ≥ 0 and write n = pq + r, with 0 ≤ r < q and p ∈ ℤ+ b,0,μf

uniquely determined. Using Theorem 18.1.10(d,e), we get for every h ∈ 𝒦β

(EA∞ ) that

n−r n 󵄩󵄩 ̂n 󵄩󵄩 󵄩 ̂pq ̂r 󵄩󵄩 p󵄩 󵄩 ̂r 󵄩󵄩 q q 󵄩󵄩ℒ̂f (h)󵄩󵄩β = 󵄩󵄩󵄩ℒ 󵄩β ≤ (1/2) 󵄩󵄩ℒ̂f (h)󵄩󵄩β ≤ (1/2) M‖h‖β ≤ 2M(1/2) ‖h‖β . ̂f (ℒ̂f (h))󵄩

󵄩̂ n Therefore, 󵄩󵄩󵄩ℒ ̂f | b,0,μf

∈ 𝒦βb (EA∞ ),

then g − μf (g) ∈ Theorem 18.1.10(b), we obtain that

Using also

𝒦β

󵄩󵄩 󵄩󵄩β ≤ 2Mζ n for every n ≥ 0, where ζ = (1/2)1/q . If g (EA∞ ) b,0,μ 𝒦β f (EA∞ ) and ‖g − μf (g)‖β ≤ ‖g‖β + ‖μf (g)‖β ≤ 2‖g‖β .

󵄩󵄩 ̂n 󵄩 󵄩 ̂n 󵄩󵄩 󵄩󵄩 ̂ n 󵄩󵄩 n 󵄩󵄩ℒ̂f (g) − μf (g)1󵄩󵄩󵄩β = 󵄩󵄩󵄩ℒ ̂f (g − μf (g))󵄩 󵄩β ≤ 󵄩󵄩ℒ̂f |𝒦b,0,μf (E ∞ ) 󵄩󵄩β ‖g − μf (g)‖β ≤ 4Mζ ‖g‖β β

A

for every n ≥ 0 and g ∈ 𝒦βb (EA∞ ). This proves (a). Part (b) is a consequence of (a). For ̂f . A similar argument holds for ℒ ̂̂ . By part (c), let us concentrate on the operator ℒ f ̂f : 𝒦b (E ∞ ) → 𝒦b (E ∞ ) is the union of the specTheorem 18.1.9(f), the spectrum of ℒ β

A

β

A

̂f restricted to ℂψf and of ℒ ̂f restricted to the complex version of 𝒦b,0,mf (E ∞ ). tra of ℒ A β The spectrum of the former is 1 by Theorem 18.1.9(a). The spectrum of the latter is contained in Bℂ (0, ζ ) by the just proved part (b). It then clear that 1 is a simple isolated ̂f . eigenvalue for ℒ

18.2 Continuity of Gibbs states In this section, we shall prove continuity results analogous to those for open distance expanding maps from Subsection 13.8.3. However, the situation is now much more delicate for several reasons. First of all, when the alphabet is infinite the potentials are not bounded from below, and thus do not exhibit a natural metric space structure. Furthermore, the symbolic space EA∞ is not compact, so we must use Arzèla–Ascoli’s

720 | 18 Countable state thermodynamic formalism: finer properties theorem with special care. Moreover, the space M(EA∞ ) of all Borel probability measures on EA∞ , endowed with the weak∗ topology, is no longer compact, thus we make use of the concept of tightness in a somewhat similar manner to the proof of Theorem 17.7.4. Fix β > 0 and equip the set 𝒦βs (EA∞ , ℝ) with the metric ρ∞ (f , g) := {

min{1, ‖f − g‖∞ } 1

if f − g ∈ 𝒦βb (EA∞ , ℝ)

if f − g ∉ 𝒦βb (EA∞ , ℝ).

Also endow 𝒦βs (EA∞ , ℝ) with the pseudometric ρβ (f , g) := υβ (f − g). Nevertheless, convergence in 𝒦βs (EA∞ , ℝ) and its subspaces will be solely determined by the metric ρ∞ . That is, convergence is uniform convergence. The pseudometric ρβ will be strictly used to restrict the subspace over which the next result holds, namely, to concentrate on the subspace 𝒦βs (EA∞ , ℝ) ∩ Bρβ (0, R). We shall prove the following. Theorem 18.2.1. Let E be a countable alphabet and A : E × E → {0, 1} a finitely irreducible incidence matrix. For every β > 0 and every R > 0, the following statements hold: (a) The function 𝒦βs (EA∞ , ℝ) ∋ f 󳨃󳨀→ P(f ) ∈ ℝ is continuous.

(b) The function 𝒦βs (EA∞ , ℝ) ∩ Bρβ (0, R) ∋ f 󳨃󳨀→ mf ∈ M(EA∞ ) is continuous, where mf is the unique eigenmeasure of the dual transfer operator ℒ∗f (and also a Gibbs state), per Corollary 17.7.5. (c) The function 𝒦βs (EA∞ , ℝ) ∩ Bρβ (0, R) ∋ f 󳨃󳨀→ ψf ∈ Cb (EA∞ ) is continuous, where ψf is ̂f identified in Theorem 18.1.7 (see also Corollary 18.1.8). the fixed point of ℒ (d) The function 𝒦βs (EA∞ , ℝ) ∩ Bρβ (0, R) ∋ f 󳨃󳨀→ μf ∈ M(EA∞ ) is continuous, where μf is the unique ergodic σ-invariant Gibbs state for f , per Corollary 17.7.5.

Proof. Part (a) follows from a straightforward calculation from the very definition of topological pressure; alternatively, it follows by an even easier calculation from Theorem 17.3.4 (3rd variational principle). The proof of part (b) is motivated by and similar to that of Theorem 17.7.4. Let (fk )∞ k=1 be a sequence of functions in 𝒦βs (EA∞ , ℝ) ∩ Bρβ (0, R) converging in the ρ∞ -metric (i. e.

uniformly) to a function f ∈ 𝒦βs (EA∞ , ℝ) ∩ Bρβ (0, R). We must show that the sequence of ∗ measures (mfk )∞ k=1 converges weak to the measure mf . Let ε > 0. Disregarding finitely many terms in the sequence (fk )∞ k=1 , we may assume without loss of generality that ρ∞ (fk , f ) ≤ 1/2,

∀k ∈ ℕ

(18.13)

18.2 Continuity of Gibbs states | 721

and, invoking part (a), that 󵄨 󵄨󵄨 󵄨󵄨P(fk ) − P(f )󵄨󵄨󵄨 ≤ 1,

(18.14)

∀k ∈ ℕ.

A somewhat long but natural series of estimates then shows that the constant Cfk arising from Theorem 17.6.4 for the potential fk satisfies R

Cfk ≤ e eβ −1 max{1, #Λ exp(2|Λ| ⋅ (|P(f )| + 1) + |Λ| − 2 min S|α| f ([α]))} =: Cf ,R α∈Λ

(18.15)

for every k ∈ ℕ. We first establish the following fact. Claim 1. The sequence (fk )∞ k=1 is tight. Proof of Claim 1. Employing Corollary 17.7.5(b) and inequalities (18.13)–(18.15), we get for every k ∈ ℕ, every n ∈ ℕ, and every e ∈ E = ℕ, that mfk (πn−1 (e)) =

∑

ω∈EAn : ωn =e

mfk ([ω]) ≤ Cfk

≤ Cf ,R e−nP(fk )

∑

ω∈EAn : ωn =e

∑

ω∈EAn : ωn =e

exp(sup(Sn fk |[ω] ) − nP(fk ))

exp(sup(Sn−1 fk |[ω] ) + sup(fk |[e] )) n−1

≤ Cf ,R e−n(P(f )−1) exp(sup(fk |[e] )) ∑ exp( ∑ sup(fk |[ωj ] )) j=1

ω∈EAn−1

n−1

≤ Cf ,R e−n(P(f )−1) exp(sup(fk |[e] ))[∑ exp(sup(fk |[i] ))] i∈E

n−1

≤ Cf ,R e−n(P(f )−3/2) exp(sup(f |[e] ))[∑ exp(sup(f |[i] ))] i∈E

,

which is finite since the potential f is summable. Therefore, n−1 ∞

mfk (πn−1 ([e + 1, ∞))) ≤ Cf ,R e−n(P(f )−3/2) [∑ exp(sup(f |[i] ))] i∈E

∑ exp(sup(f |[j] )).

j=e+1

Since f is summable, there exists qn ∈ ℕ such that n−1

Cf ,R e−n(P(f )−3/2) [∑ exp(sup(f |[i] ))] i∈E

∞

∑ exp(sup(f |[j] ))
0. By Claim 2, there exists N1 ∈ ℕ such that 󵄩 󵄩󵄩 󵄩󵄩ℒk − ℒf 󵄩󵄩󵄩∞ ≤ ε,

∀k ≥ N1 .

(18.17)

Using (18.16)–(18.17), we get 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩ℒk ψk − ℒf ψk 󵄩󵄩󵄩∞ = 󵄩󵄩󵄩(ℒk − ℒf )ψk 󵄩󵄩󵄩∞ ≤ 󵄩󵄩󵄩ℒk − ℒf 󵄩󵄩󵄩∞ ⋅‖ψk ‖∞ ≤ C(f , R)ε,

∀k ≥ N1 . (18.18)

Since the function f is summable, there exists N2 ∈ ℕ such that ∞

∑ exp(sup(f |[n] )) ≤ ε.

n=N2 +1

(18.19)

Fix ω ∈ EA∞ . Since the set {nω ∈ EA∞ : n ≤ N2 } is finite and limk→∞ ρ∞ (ψk , ψ) = 0, there is N3 ≥ N1 such that 󵄨󵄨 󵄨 󵄨󵄨ψk (ω) − ψ(ω)󵄨󵄨󵄨 < ε

(18.20)

and 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∑ ef (nω) ψk (ω) − ∑ ef (nω) ψ(ω)󵄨󵄨󵄨 ≤ ∑ ef (nω) 󵄨󵄨󵄨ψk (ω) − ψ(ω)󵄨󵄨󵄨 < ε 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨 n≤N2 : 󵄨󵄨 n≤N2 : n≤N2 : ∞ ∞ ∞ nω∈EA

nω∈EA

(18.21)

nω∈EA

for every k ≥ N3 . Per part (a), there exists N4 ≥ N3 such that 󵄨󵄨 P(f ) 󵄨 − eP(fk ) 󵄨󵄨󵄨 < ε, 󵄨󵄨e

∀k ≥ N4 .

(18.22)

Using (18.16), (18.18), (18.20) and (18.22), for all k ≥ N4 we get that 󵄨󵄨 P(f ) 󵄨 󵄨󵄨e ψ(ω) − ℒf ψ(ω)󵄨󵄨󵄨 󵄨󵄨 = 󵄨󵄨󵄨eP(f ) (ψ(ω) − ψk (ω)) + (eP(f ) − eP(fk ) )ψk (ω) 󵄨

󵄨󵄨 + (eP(fk ) ψk (ω) − ℒk ψk (ω)) + (ℒk ψk (ω) − ℒf ψk (ω)) + (ℒf ψk (ω) − ℒf ψ(ω))󵄨󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 P(f ) 󵄨󵄨 P(f ) 󵄨󵄨 P(fk ) 󵄨󵄨 󵄨󵄨 ≤ e 󵄨󵄨ψ(ω) − ψk (ω)󵄨󵄨 + 󵄨󵄨e −e 󵄨󵄨 ⋅ 󵄨󵄨ψk (ω)󵄨󵄨 󵄨󵄨 P(fk ) 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 + 󵄨󵄨e ψk (ω) − ℒk ψk (ω)󵄨󵄨 + 󵄨󵄨ℒk ψk (ω) − ℒf ψk (ω)󵄨󵄨󵄨 + 󵄨󵄨󵄨ℒf ψk (ω) − ℒf ψ(ω)󵄨󵄨󵄨 󵄨 󵄨 󵄩 󵄩 󵄨 󵄨 ≤ eP(f ) ε + 󵄨󵄨󵄨eP(f ) − eP(fk ) 󵄨󵄨󵄨 ⋅ ‖ψk ‖∞ + 0 + 󵄩󵄩󵄩ℒk ψk − ℒf ψk 󵄩󵄩󵄩∞ + 󵄨󵄨󵄨ℒf ψk (ω) − ℒf ψ(ω)󵄨󵄨󵄨 󵄨 󵄨 ≤ eP(f ) ε + ε ⋅ C(f , R) + C(f , R)ε + 󵄨󵄨󵄨ℒf ψk (ω) − ℒf ψ(ω)󵄨󵄨󵄨 󵄨 󵄨 ≤ (eP(f ) + 2C(f , R))ε + 󵄨󵄨󵄨ℒf ψk (ω) − ℒf ψ(ω)󵄨󵄨󵄨. (18.23)

724 | 18 Countable state thermodynamic formalism: finer properties This last term can be estimated using (18.19) and (18.20). For all k ≥ max{N2 , N4 }, we obtain that 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 f (nω) ψk (ω) + ∑ ef (nω) ψk (ω)) 󵄨󵄨ℒf ψk (ω) − ℒf ψ(ω)󵄨󵄨󵄨 = 󵄨󵄨󵄨( ∑ e 󵄨󵄨󵄨 n≤N2 : n>N2 : ∞ ∞ nω∈EA

nω∈EA

−( ∑ e

f (nω)

n≤N2 : nω∈EA∞

ψ(ω) + ∑ e n>N2 : nω∈EA∞

f (nω)

󵄨󵄨 󵄨󵄨 ψ(ω))󵄨󵄨󵄨 󵄨󵄨 󵄨

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨 ∑ ef (nω) ψk (ω) − ∑ ef (nω) ψ(ω)󵄨󵄨󵄨 + ∑ ef (nω) 󵄨󵄨󵄨ψk (ω) − ψ(ω)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 n≤N2 : 󵄨 n≤N2∞: 󵄨 n>N2∞: ∞ nω∈EA

≤ε+ ∑ e

f (nω)

n>N2 : nω∈EA∞

nω∈EA

nω∈EA

⋅ ε ≤ (1 + ε)ε.

Inserting this into (18.23), we have that 󵄨󵄨 P(f ) 󵄨 P(f ) + 2C(f , R) + 1 + ε)ε. 󵄨󵄨e ψ(ω) − ℒf ψ(ω)󵄨󵄨󵄨 ≤ (e As ε > 0 is arbitrary, we deduce that ℒf ψ(ω) = eP(f ) ψ(ω). Thus, ℒf ψ = e

P(f )

ψ.

Using part (b), we get that ∫ ψ dmf = lim ∫ ψk dmfk = lim 1 = 1. k→∞

EA∞

k→∞

EA∞

Hence, invoking Corollary 18.1.8, we conclude that ψ = ψf . So every subsequence of ∞ (ψfk )∞ k=1 has itself a subsequence which converges uniformly on compact subsets of EA ∞ to ψf . This implies that the sequence (ψfk )k=1 converges uniformly on compact subsets of EA∞ to ψf . That is, part (c) is proved. Finally, part (d) is a by-product of (b) and (c). Indeed, let (fk )∞ k=1 be a sequence of functions in 𝒦βs (EA∞ , ℝ) ∩ Bρβ (0, R) converging to a function f ∈ 𝒦βs (EA∞ , ℝ) ∩ Bρβ (0, R). ∗ ∞ By (b), the sequence (mfk )∞ k=1 converges to mf in the weak topology on M(EA ). By (c), ∞ ∞ the sequence (ψfk )∞ k=1 converges to ψf in Cb (EA ). For every function g ∈ Cb (EA ), we then have lim ∫ g dμfk = lim ∫ g ⋅ ψfk dmfk = ∫ g ⋅ ψf dmf = ∫ g dμf .

k→∞

EA∞

k→∞

EA∞

EA∞

EA∞

As Cb (EA∞ ) is dense in L1 (μf ), the above equality holds for all g ∈ L1 (μf ). That is, the ∗ ∞ sequence (μfk )∞ k=1 converges to μf in the weak topology on M(EA ).

18.3 Stochastic laws |

725

18.3 Stochastic laws In this section, we prove the basic stochastic laws for countable state subshifts of finite type and Hölder-on-cylinders summable potentials. This section, its content and proofs, is analogous to Section 13.9. It relies on all that has been proved in Section 18.1, the main tool being Theorem 18.1.12. 18.3.1 Exponential decay of correlations We start with a demonstration that, given a potential f which is summable and Hölder continuous on cylinders, for suitable functions g and h the correlations Cn (g, h) decay exponentially with respect to the σ-invariant Gibbs state μf . This is a counterpart to Theorem 13.9.1. Theorem 18.3.1. Let E be a countable alphabet, let A : E × E → {0, 1} be a finitely primitive incidence matrix, and let f ∈ 𝒦βs (EA∞ , ℝ) be a potential. Let μf be the unique σ-invariant Gibbs state for f , per Corollary 17.7.5. Then there exist constants c ≥ 0 and 0 < ζ < 1 such that for all g ∈ L1 (μf ) and all h ∈ 𝒦βb (EA∞ ) we have 󵄩 󵄩 󵄩 󵄩 Cμf ,n (g, h) ≤ cζ n 󵄩󵄩󵄩g − Eμf (g)󵄩󵄩󵄩L1 (μ ) 󵄩󵄩󵄩h − Eμf (h)󵄩󵄩󵄩β ,

∀n ∈ ℕ.

f

Proof. Let g ∈ L1 (μf ) and h ∈ 𝒦βb (EA∞ ). Replacing g by g − Eμf (g) and h by h − Eμf (h), we may assume without loss of generality that Eμf (g) = Eμf (h) = 0. Using (13.76) and Theorem 18.1.10(c), we obtain for every n ∈ ℕ that 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 ̂n 󵄨 󵄨 ̂̂n (h) dμf 󵄨󵄨󵄨. (18.24) Cμf ,n (g, h) = 󵄨󵄨󵄨 ∫ h ⋅ (g ∘ σ n ) dμf 󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∫ ℒ (h ⋅ (g ∘ σ n )) dμf 󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∫ g ⋅ ℒ ̂ 󵄨󵄨 f f 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 ∞ ∞ ∞ EA

EA

EA

Applying Theorem 18.1.12(a), we have that 󵄩󵄩 ̂n 󵄩󵄩 󵄩 ̂n 󵄩󵄩 n 󵄩󵄩ℒ̂f (h)󵄩󵄩∞ ≤ 󵄩󵄩󵄩ℒ ̂f (h)󵄩 󵄩β ≤ cζ ‖h‖β for some c ≥ 0 and 0 < ζ < 1 independent of n. Inserting this into (18.24), we conclude that 󵄨 ̂n 󵄨󵄨 󵄩󵄩 ̂n 󵄩󵄩 n Cμf ,n (g, h) ≤ ∫ 󵄨󵄨󵄨g ⋅ ℒ ̂f (h)󵄨󵄨 dμf ≤ 󵄩 󵄩ℒ̂f (h)󵄩󵄩∞ ∫ |g| dμf ≤ cζ ‖h‖β ‖g‖L1 (μf ) . EA∞

EA∞

18.3.2 Asymptotic variance An immediate consequence of Theorems 18.3.1 and 13.9.5 is the following counterpart of Corollary 13.9.6.

726 | 18 Countable state thermodynamic formalism: finer properties Theorem 18.3.2. Let E be a countable alphabet, let A : E × E → {0, 1} be a finitely primitive incidence matrix, and let a potential f : EA∞ → ℝ be summable and Hölder continuous on cylinders. Let μf be the unique σ-invariant Gibbs state for f , per Corollary 17.7.5. If g : EA∞ → ℝ is Hölder continuous, then the asymptotic variance σμ2 f ,∞ (g) exists and the following three conditions are equivalent: (a) σμ2 f ,∞ (g) = 0. 󵄩 󵄩 (b) sup󵄩󵄩󵄩Sn (g − μf (g)1)󵄩󵄩󵄩L2 (μ ) < ∞. f n∈ℕ

(c) g is cohomologous to a constant in the Hilbert space L2 (μf ).

As a direct consequence of Theorem 18.3.1 and Lemma 13.9.9, we obtain the following generalization of Lemma 13.9.11 (cf. Lemma 13.9.13). Lemma 18.3.3. Let E be a countable alphabet, let A : E×E → {0, 1} be a finitely primitive s incidence matrix, and let f ∈ 𝒦α (EA∞ , ℝ) be a potential. Let μf be the unique σ-invariant Gibbs state for f , per Corollary 17.7.5. If g ∈ 𝒦βb (EA∞ , ℝ) and σμ2 f ,∞ (g) = 0, then there exists b (EA∞ , ℝ) such that u ∈ 𝒦min{α,β}

g = μf (g) + u − u ∘ σ, b i. e., g is cohomologous to the constant μf (g) in the additive group 𝒦min{α,β} (EA∞ , ℝ).

18.3.3 Central limit theorem The celebrated central limit theorem also holds for countable state subshifts of finite type. This is the counterpart of Theorem 13.9.15. Theorem 18.3.4 (Central limit theorem). Let E be a countable alphabet, let A : E × E → {0, 1} be a finitely primitive incidence matrix, and let a potential f : EA∞ → ℝ be summable and Hölder continuous on cylinders. Let μf be the unique σ-invariant Gibbs state for f , per Corollary 17.7.5. If g : EA∞ → ℝ is Hölder continuous but is not cohomological to a constant in the class of Hölder continuous functions on EA∞ , then for every Lebesgue measurable set B ⊆ ℝ with boundary of Lebesgue measure zero we have that lim μf ({ω ∈ EA∞ :

n→∞

Sn g(ω) − μf (g)n √n

∈ B}) = 𝒩0 (σμ2 f ,∞ (g))(B) =

1 √2πσμ2 f ,∞ (g)

∫ exp( B

−u2

2σμ2 f ,∞ (g)

)du,

where σμ2 f ,∞ (g) is the asymptotic variance of g whose existence is guaranteed by Theorem 18.3.2 and whose positivity follows from Lemma 18.3.3.

18.4 Potentials with the same Gibbs states | 727

Proof. The proof is very similar to that of Theorem 13.9.15. It is based on Theorem 18.1.12 in place of Theorem 13.8.18.

18.3.4 Law of the iterated logarithm Finally, the law of the iterated logarithm also has a version for countable state subshifts of finite type. For the sake of completeness, we state it, though like its counterpart Theorem 13.9.17 a full proof necessitates deep results from probability theory and is too involved to be presented in this book. We instead refer the reader to [41, 98]. Theorem 18.3.5 (Law of the iterated logarithm). Let E be a countable alphabet, let A : E × E → {0, 1} be a finitely primitive incidence matrix, and let a potential f : EA∞ → ℝ be summable and Hölder continuous on cylinders. Let μf be the unique σ-invariant Gibbs state for f , per Corollary 17.7.5. If g : EA∞ → ℝ is Hölder continuous but is not cohomological to a constant in the class of Hölder continuous functions on EA∞ , then lim sup n→∞

Sn g(ω) − n ∫E ∞ g dμf A

√n log log n

= √2σμ2 f ,∞ (g)

for μf -a. e. ω ∈ EA∞ .

18.4 Potentials with the same Gibbs states We are ready to give a complete characterization of the potentials which have the same invariant Gibbs state. The next result is analogous to Theorem 13.9.14. Theorem 18.4.1. Let E be a countable alphabet and A : E × E → {0, 1} a finitely irreducible incidence matrix. If two potentials φ, ψ : EA∞ → ℝ are summable and Hölder continuous on cylinders, then the following statements are equivalent: (a) The σ-invariant Gibbs states of φ and ψ coincide, i. e., μφ = μψ . (b) There exists a constant R1 ∈ ℝ such that Sn φ(ω) − Sn ψ(ω) = R1 n whenever n ∈ ℕ and ω ∈ EA∞ is a periodic point of σ of period n. (c) There exist constants R2 ∈ ℝ and C ∈ [0, ∞) such that 󵄨󵄨 󵄨 󵄨󵄨Sn φ(ω) − Sn ψ(ω) − R2 n󵄨󵄨󵄨 ≤ C,

∀n ∈ ℕ, ∀ω ∈ EA∞ .

(d) φ − ψ is cohomologous to a constant R3 ∈ ℝ in the additive group of bounded functions on EA∞ . (e) φ − ψ is cohomologous to a constant R4 ∈ ℝ in the additive group of bounded continuous functions on EA∞ .

728 | 18 Countable state thermodynamic formalism: finer properties (f) φ − ψ is cohomologous to a constant R5 ∈ ℝ in the additive group of Hölder continuous functions on EA∞ . (g) 𝒢φ = 𝒢ψ . (h) 𝒢φ ∩ 𝒢ψ ≠ 0. If any of the above conditions holds and A is finitely primitive, then (i) σμ2 f ,∞ (φ − ψ) = 0 for all potentials f : EA∞ → ℝ which are summable and Hölder continuous on cylinders. If, in addition to the hypotheses of the present theorem, the function φ − ψ is bounded and A is finitely primitive then (j) the equality σμ2 f ,∞ (φ − ψ) = 0 for some potential f : EA∞ → ℝ which is summable and Hölder continuous on cylinders, entails all of the above items (a)–(g). Finally, if any of the above conditions holds, then all the constants R1 –R5 are the same and equal to P(σ, φ) − P(σ, ψ). Proof. Obviously, (a)⇒(g)⇒(h) and (f)⇒(e)⇒(d). It is also straightforward to see that (d)⇒(c). To see that (c)⇒(b), take a periodic point ω of period n, write the formula in (c) with n replaced by kn for an arbitrary k ∈ ℕ, and let k → ∞ to conclude that the formula in (b) holds. It follows in turn that R5 = R4 = R3 = R2 = R1 (cf. Exercise 13.11.5). The implication (d)⇒(a) is a consequence of two facts: (1) φ and ψ share the same set of Gibbs states according to Exercise 17.9.15(b); and (2) each potential has a unique σ-invariant Gibbs state by virtue of Corollary 17.7.5. Let us now prove that (h)⇒(b). Let χ := [φ − P(σ, φ)] − [ψ − P(σ, ψ)] = [φ − ψ] − [P(σ, φ) − P(σ, ψ)]. As there is a Gibbs state common to both φ and ψ (cf. (17.9)), there exists a constant D ≥ 1 such that D−1 ≤ exp([Sn φ(τ) − nP(σ, φ)] − [Sn ψ(τ) − nP(σ, ψ)]) = eSn χ(τ) ≤ D

(18.25)

for all n ∈ ℕ and all τ ∈ EA∞ . In particular, if τ is a periodic point of period n then Sn χ(τ) = 0. Otherwise, we would have Skn χ(τ) = kSn χ(τ) for all k ∈ ℕ and (18.25) would not hold. Thus, item (b) holds with R1 = P(σ, φ) − P(σ, ψ). With this value for R1 , let us establish that (b) implies (f). Keep χ the same as above. Since the shift map σ is transitive, let τ ∈ EA∞ be such that its ω-limit set is ω(τ) = EA∞ . Define u(σ n (τ)) := Sn χ(τ) for every n ≥ 0, where we define S0 χ ≡ 0. This function u is well-defined on 𝒪+ (τ). Indeed, if τ is periodic then u is well-defined since SN χ(τ) = 0, where N is the prime period of τ. If τ is not periodic, then u is well defined simply because σ m (τ) ≠ σ n (τ) whenever m ≠ n. Observe further that χ(σ n (τ)) = Sn+1 χ(τ) − Sn χ(τ) = u(σ n+1 (τ)) − u(σ n (τ)) = u ∘ σ(σ n (τ)) − u(σ n (τ)) = (u ∘ σ − u)(σ n (τ))

(18.26)

18.4 Potentials with the same Gibbs states | 729

for all n ≥ 0, that is, χ is cohomologous to 0 on 𝒪+ (τ) with u as a coboundary. Equivalently, φ − ψ is cohomologous to P(σ, φ) − P(σ, ψ) on 𝒪+ (τ) with u as a coboundary. If τ is periodic, then 𝒪+ (τ) = EA∞ and u is the sought coboundary. If τ is not periodic, then 𝒪+ (τ) ≠ EA∞ and we need to extend u continuously to 𝒪+ (τ) = EA∞ . In order to do this (and more), assume that |σ k (τ) ∧ σ ℓ (τ)| ≥ 1 for some k < ℓ. Let η := τ|k (σ k (τ)|ℓ−k )∞ ∈ EA∞ . Then σ k (η) = (σ k (τ)|ℓ−k )∞ is a periodic point of period ℓ − k. So Sℓ−k χ(σ k (η)) = 0. Hence, u(σ ℓ (τ)) − u(σ k (τ)) = Sℓ χ(τ) − Sk χ(τ) = Sℓ−k χ(σ k (τ)) = Sℓ−k χ(σ k (τ)) − Sℓ−k χ(σ k (η))

= Sℓ−k χ(σ k (τ)|ℓ−k σ ℓ (τ)) − Sℓ−k χ(σ k (η)|ℓ−k σ ℓ (η)). Observe that |τ ∧ η| = ℓ + |σ k (τ) ∧ σ ℓ (τ)|, and thus |σ ℓ (τ) ∧ σ ℓ (η)| = |σ k (τ) ∧ σ ℓ (τ)|. Let β be an Hölder-on-cylinders exponent of χ (see Exercise 18.5.5). Using Lemma 17.2.3 with f = χ, n = ℓ − k and ω = σ k (τ)|ℓ−k , we deduce that 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ℓ k k ℓ k ℓ 󵄨󵄨u(σ (τ)) − u(σ (τ))󵄨󵄨󵄨 = 󵄨󵄨󵄨Sℓ−k χ(σ (τ)|ℓ−k σ (τ)) − Sℓ−k χ(σ (η)|ℓ−k σ (η))󵄨󵄨󵄨 󵄨 󵄨 υβ (χ) β ℓ ℓ [d(σ (τ), σ (η))] ≤ β e −1 υβ (φ) + υβ (ψ) β [d(σ k (τ), σ ℓ (τ))] . ≤ eβ − 1

(18.27)

That is, β 󵄨󵄨 󵄨 υβ (φ) + υβ (ψ) [d(γ, ρ)] 󵄨󵄨u(γ) − u(ρ)󵄨󵄨󵄨 ≤ eβ − 1

(18.28)

whenever γ, ρ ∈ 𝒪+ (τ) and γ1 = ρ1 . In particular, it follows from this that u is uniformly continuous on 𝒪+ (τ). Since 𝒪+ (τ) is a dense subset of EA∞ , we therefore conclude that u has a unique continuous extension on EA∞ , which we also denote by u. Moreover, (18.28) extends to all γ, ρ ∈ EA∞ . In particular, the function u is Hölder continuous on cylinders. Hence, (18.26) holds on all of EA∞ , that is, χ is cohomologous to 0 with u as a coboundary. We are left to prove that u is bounded. The proof is essentially identical to the end of the proof of Lemma 13.9.9. Let Λ ⊆ EA∗ be a finite set witnessing the finite irreducibility of A. Let |Λ| := max{|α| : α ∈ Λ}

and

Λ1 := {α1 : α ∈ Λ}.

Since both functions χ and u are Hölder continuous on cylinders, sup(|χ||[e] ) and sup(|u||[e] ) are finite for all e ∈ E. So 󵄨 Mχ := max{sup(|χ|󵄨󵄨󵄨[e] ) : e ∈ Λ1 } < ∞

and

󵄨 Mu := max{sup(|u|󵄨󵄨󵄨[e] ) : e ∈ Λ1 } < ∞.

730 | 18 Countable state thermodynamic formalism: finer properties Pick an arbitrary ρ ∈ EA∞ and choose any ρ̂ ∈ Λ such that ρ̂ρ ∈ EA∞ . Let k := |ρ̂| ≤ |Λ|. As χ = u ∘ σ − u, it ensues that Sk χ(ρ̂ρ) = u(σ k (ρ̂ρ)) − u(ρ̂ρ) = u(ρ) − u(ρ̂ρ). Therefore, 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨u(ρ)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨Sk χ(ρ̂ρ)󵄨󵄨󵄨 + 󵄨󵄨󵄨u(ρ̂ρ)󵄨󵄨󵄨 ≤ Sk |χ|(ρ̂ρ) + sup(|u|󵄨󵄨󵄨[̂ρ1 ] ) ≤ kMχ + Mu ≤ |Λ|Mχ + Mu . So ‖u‖∞ ≤ |Λ|Mχ + Mu < ∞, and the proof of the implication (b)⇒(f) is complete. If any of the conditions (a)–(h) holds, then we already know that all of them hold, whence (i) directly follows from (h) under the additional assumption that A is finitely primitive according to Theorem 18.3.2. Obviously, (i)⇒(j). Finally, if (j) holds then (h) follows from Lemma 18.3.3.

18.5 Exercises Exercise 18.5.1. Let E be a countably infinite alphabet and A : E × E → {0, 1} be a finitely irreducible incidence matrix. Show that the closed unit ball in the Banach space 𝒦βb (EA∞ ) is not compact when treated as a subset of Cb (EA∞ ). Exercise 18.5.2. Let f , g ∈ 𝒦β (EA∞ ), F ⊆ EA∞ and c ∈ ℂ. Prove the following statements: (a) υβ,F (f + g) ≤ υβ,F (f ) + υβ,F (g). (b) υβ,F (cf ) = |c| υβ,F (f ). (c) If m := infω∈F |f | > 0, then υβ,F (1/f ) ≤ υβ,F (f )/m2 . Exercise 18.5.3. Let f , g ∈ Cb (EA∞ ), F ⊆ EA∞ and c ∈ ℂ. Prove the following statements: (a) ‖f + g‖∞,F ≤ ‖f ‖∞,F + ‖g‖∞,F . (b) ‖cf ‖∞,F = |c| ‖f ‖∞,F . (c) ‖fg‖∞,F ≤ ‖f ‖∞,F ‖g‖∞,F . (d) If m := infω∈F |f | > 0, then ‖1/f ‖∞,F ≤ 1/m. Exercise 18.5.4. Let f , g ∈ 𝒦βb (EA∞ ), F ⊆ EA∞ and c ∈ ℂ. Prove the following statements:

b (a) f + g ∈ 𝒦β,F (EA∞ ) and ‖f + g‖β,F ≤ ‖f ‖β,F + ‖g‖β,F . b (b) cf ∈ 𝒦β,F (EA∞ ) and ‖cf ‖β,F = |c| ‖f ‖β,F . (c) υβ,F (fg) ≤ ‖f ‖∞,F υβ,F (g) + υβ,F (f )‖g‖∞,F .

b (d) fg ∈ 𝒦β,F (EA∞ ) and ‖fg‖β,F ≤ ‖f ‖β,F ‖g‖β,F .

b (e) If m := infω∈F |f | > 0, then 1/f ∈ 𝒦β,F (EA∞ ) and ‖1/f ‖β,F ≤ ‖f ‖β,F /m2 .

Exercise 18.5.5. Let α, β ∈ (0, 1] and 0 < γ ≤ min{α, β}. Let also c ∈ ℂ and let f ∈

𝒦α (EA∞ ) while g ∈ 𝒦β (EA∞ ). Finally, let F ⊆ EA∞ . Prove the following statements:

18.5 Exercises | 731

(a) f ∈ 𝒦γ (EA∞ ) and υγ,F (f ) ≤ υα,F (f ). (b) υγ,F (f + g) ≤ υα,F (f ) + υβ,F (g). (c) Deduce similar statements for cf , fg and 1/f using the previous exercises. Exercise 18.5.6. For any ψ ∈ 𝒦β (EA∞ ) and e ∈ E, show that 󵄨 󵄨 󵄩 󵄩󵄩 󵄩󵄩ψ|[e] 󵄩󵄩󵄩∞ ≤ 󵄨󵄨󵄨sup ψ|[e] 󵄨󵄨󵄨 + υβ (ψ). Exercise 18.5.7. Prove Theorems 18.1.9–18.1.10.

19 Conformal graph directed Markov systems In this chapter, we describe a powerful method for constructing and studying the geometric and dynamical properties of quite general fractal sets. This method arises from the theory of countable-alphabet conformal iterated function systems, or more generally from the theory of countable-alphabet conformal graph directed Markov systems, as defined and developed in the paper [75] and the book [81]. Many sets (not only fractal ones) can be viewed as limit sets of conformal (or sometimes even more simply, of similarity) iterated function systems (IFSs). Among them are illustrious fractal sets such as the middle-third Cantor set, the Sierpiński triangle, the Sierpiński carpet, the Koch curve, various kinds of Menger sets and the limit set of real and complex continued fractions. Despite being less glamorous, the unit interval [0, 1], every cube [0, 1]n and the set of irrational numbers in [0, 1], are also limit sets of conformal IFSs. On the other hand, the unit circle 𝕊1 ⊆ ℝ2 is not the limit set of any conformal IFS; it is, however, the limit set of a graph directed Markov system (GDMS). A GDMS is a generalization of an IFS, with a graph comprising vertices and edges and with an edge-transition matrix. Historically, an iterative scheme for defining sets with desired topological and geometric properties, came into use in the 19th century works of Georg Cantor and others, and has been pretty common in topology ever since. It was formalized in John Hutchinson’s groundbreaking paper [56]. Michael Barnsley popularized the scheme and coined its name in [4], and a flow of research ensued. At the early stage, the work on these so-called IFSs focused mainly on finite-alphabet systems consisting of similarities and satisfying the celebrated Open Set Condition (OSC). This condition was introduced in [56] where a formula, nowadays called Hutchinson’s formula, was discovered. This formula describes the value of the Hausdorff dimension of the limit set/attractor of such an IFS solely in terms of the scaling factors of its generating similarities. A leap forward in the advancement of the theory occurred with Rufus Bowen’s seminal paper [12]. Although this article did not concern IFSs per se, an analogous formula was nevertheless proved in the context of quasi-Fuchsian groups, which are conformal (but nonsimilarity) IFSs. According to Bowen’s formula (cf. Chapter 16), the Hausdorff dimension of the limit set of a conformal expanding repeller is equal to the unique zero of the pressure function for a particular, geometric potential. Following Bowen’s paper, research on Bowen’s formula and its variations in many settings, abounded. Bedford proved an appropriate counterpart for conformal IFSs in [5]. Mauldin and Williams introduced GDMSs consisting of similarities with finite sets of edges and vertices in [82]. A proper version of Hutchinson’s formula was proved therein. Countable-alphabet conformal IFSs were introduced in Mauldin and Urbański’s paper [75], where among the main results a suitable form of Bowen’s formula was demonstrated. It is in that article that the intriguing possibility of a pressure function having no zero was first observed, leading to the concept of irregular systems. https://doi.org/10.1515/9783110702699-019

734 | 19 Conformal graph directed Markov systems It was further established that if h is the Hausdorff dimension of the limit set then, although this set’s h-dimensional Hausdorff measure is always finite and this set’s h-dimensional packing measure is usually positive (whenever the Strong Open Set Condition (SOSC) is satisfied), the former may vanish while the latter may be infinite. In the book [81], conformal GDMSs with a finite set of vertices but a countable set of edges, were introduced and systematically studied. Some elements of this theory were presented in the books [50] and [18]. The presentation in this chapter stems from [81] but develops and improves it in many respects. Turning our attention to the contents of this chapter, the first five sections concentrate on the foundations of the theory of conformal graph directed Markov systems. In Section 19.1, we start with the introduction of basic notions and notation for GDMSs: in particular, the underlying symbolic space and the shift map acting on it, the limit set and the projection (coding map) from the symbolic space onto the limit set. Section 19.2 collects properties of conformal maps in Euclidean spaces that will be crucial in deriving in Section 19.3 the core properties of conformal GDMSs (CGDMSs). In Section 19.4, we define an appropriate version of the topological pressure, the finiteness parameter, and the Bowen parameter. Based on the behavior of the pressure function, we then classify in Section 19.5 CGDMSs as irregular or regular, and in this latter case a finer splitting into strongly regular and critically regular systems is described. Cofinite or hereditary regularity is also defined. In Section 19.6, a variation of Bowen’s formula is derived. Among its many consequences, we get an almost cost-free, effective and nontrivial lower estimate for the Hausdorff dimension of the limit sets of finitely irreducible CGDMSs. We further deduce from it the classical version of Hutchinson’s formula. A particularly important tool to study the fractal geometry of limit sets of CGDMSs is the concept of conformal measure. We introduce and study these measures at length in Section 19.8, and we give a formula, called volume lemma, for the Hausdorff dimension of projections of ergodic invariant measures from the underlying symbolic space. This formula is expressed in the familiar form of the ratio of the measure-theoretic entropy and the Lyapunov exponent. Beforehand, though, we examine in Section 19.7 different separation and geometric conditions: the Strong Open Set Condition (SOSC), the Boundary Separation Condition (BSC), the Strong Separation Condition (SSC) and the Cone Condition (CC). Any one of those, when satisfied, has a profound impact on conformal measures, invariant measures and the whole dynamics and geometry of CGDMSs. Finally, we provide a variety of examples in Section 19.9. Returning to the all important topic of Section 19.8, conformal measures were first defined by Samuel Patterson in his influential paper [92] (see also [93]) in the context of Fuchsian groups. Dennis Sullivan extended this concept to all Kleinian groups in [122, 124, 126]. In the papers [123, 125], he then introduced and proved the existence of conformal measures for all rational functions of the Riemann sphere. Both Patterson and Sullivan created the concept of conformal measure to get an understanding of geometric measures, i. e., of Hausdorff and packing measures. Though

19.1 Graph directed Markov systems | 735

Sullivan already noticed that there are conformal measures for Kleinian groups that are not equal, nor even equivalent, to any Hausdorff or packing measure, their main purpose is to comprehend Hausdorff and packing measures. Conformal measures for rational functions, in the sense of Sullivan, have been studied in greater detail by Denker and Urbański in [24] where, in particular, the structure of the set of their exponents was examined. Subsequently, the concept of conformal measure, still in the sense of Sullivan, was extended to countable-alphabet conformal IFSs (CIFSs) in [75] and to CGDMSs in [81]. Additional information about conformal graph directed Markov systems can be found in many papers and books, including [16–19, 36–39, 50, 73, 74, 76, 78–80, 83, 102, 132, 133].

19.1 Graph directed Markov systems Graph directed Markov systems are based on a directed multigraph 𝒢 and an incidence matrix A. 19.1.1 The underlying multigraph 𝒢 The directed graph 𝒢 = (V, E, i, t) consists of a finite set V of vertices, a countable (finite or infinite) set E of directed edges, and two functions i, t : E → V that indicate for each directed edge e ∈ E its initial vertex i(e) and its terminal vertex t(e), respectively. In symbolic terms, the set E of edges will play the role of the alphabet. Accordingly, we will use the terminology and notation from Chapter 17. n The functions i and t can naturally be extended to E ∗ \{ϵ} = ⋃∞ n=1 E by defining ∞ i(ω) := i(ω1 ) and t(ω) := t(ω|ω| ). The function i also extends to E . The directed graph 𝒢 imposes some restrictions on the edges that can follow any given edge. To reflect these constraints, we set 󵄨 E𝒢∞ := {ω ∈ E ∞ 󵄨󵄨󵄨 t(ωj ) = i(ωj+1 ), ∀j ∈ ℕ}. The elements of E𝒢∞ are called infinite 𝒢 -admissible words. For each n ≥ 0, let 󵄨 E𝒢n := {ω ∈ E n 󵄨󵄨󵄨 t(ωj ) = i(ωj+1 ), ∀1 ≤ j < n} be the set of all 𝒢 -admissible n-letter words (note: E𝒢0 = {ϵ} and E𝒢1 = E). Let ∞

E𝒢∗ := ⋃ E𝒢n n=0

736 | 19 Conformal graph directed Markov systems be the set of all finite 𝒢 -admissible words. For every ω ∈ E𝒢∗ , let [ω]𝒢 := {τ ∈ E𝒢∞ : τ||ω| = ω}. These sets are called initial 𝒢 -cylinders. For every ω ∈ E𝒢∗ , let also 󵄨 E𝒢∞,ω := {γ ∈ E𝒢∞ 󵄨󵄨󵄨 ωγ ∈ E𝒢∞ } =

⋃

[e]𝒢

e∈E: t(ω)=i(e)

be the set of all infinite 𝒢 -admissible suffixes of ω, whereas 󵄨 󵄨 E𝒢∗,ω := {γ ∈ E𝒢∗ 󵄨󵄨󵄨 ωγ ∈ E𝒢∗ } = {γ ∈ E𝒢∗ 󵄨󵄨󵄨 t(ω) = i(γ)} will denote the set of all finite 𝒢 -admissible suffixes of ω and 󵄨 󵄨 E𝒢n,ω := {γ ∈ E𝒢n 󵄨󵄨󵄨 ωγ ∈ E𝒢∗ } = {γ ∈ E𝒢n 󵄨󵄨󵄨 t(ω) = i(γ)} be the set of all 𝒢 -admissible n-letter suffixes of ω. 19.1.2 The underlying matrix A The matrix A : E × E → {0, 1} is an edge incidence matrix, i. e., it dictates which edge(s) may follow any given edge. By definition, edge e2 ∈ E can follow edge e1 ∈ E when Ae1 e2 = 1 but not otherwise. The matrix A must further respect the graph 𝒢 , meaning that if Ae1 e2 = 1 then t(e1 ) = i(e2 ). More precisely, the incidence matrix A : E ×E → {0, 1} must respect the underlying graph 𝒢 (i. e., Ae1 e2 = 0 whenever t(e1 ) ≠ i(e2 )) but potentially imposes additional restrictions on the admissibility of words (i. e., it may be such that Ae1 e2 = 0 even though t(e1 ) = i(e2 )). Accordingly, 󵄨 EA∞ := {ω ∈ E ∞ 󵄨󵄨󵄨 Aωj ωj+1 = 1, ∀j ∈ ℕ} ⊆ E𝒢∞ . Complete metrizability and separability of EA∞ can be achieved using, for instance, one of the classical metrics ds (ω, τ) = s|ω∧τ| , where s ∈ (0, 1) and |ω ∧ τ| is the length of the initial block that ω and τ have in common (see Chapter 17). However, note that EA∞ is generally not compact when E is infinite. For each n ≥ 0, we also have that 󵄨 EAn := {ω ∈ E n 󵄨󵄨󵄨 Aωj ωj+1 = 1, ∀1 ≤ j < n} ⊆ E𝒢n , and ∞

EA∗ := ⋃ EAn ⊆ E𝒢∗ . n=0

19.1 Graph directed Markov systems | 737

Similarly, for every ω ∈ EA∗ the following hold: [ω] = [ω]A := {τ ∈ EA∞ : τ||ω| = ω} = [ω]𝒢 ∩ EA∞ 󵄨 EA∞,ω := {γ ∈ EA∞ 󵄨󵄨󵄨 ωγ ∈ EA∞ } = [e] ⊆ E𝒢∞,ω ⋃ e∈E: Aω|ω| e =1

󵄨 󵄨 EA∗,ω := {γ ∈ EA∗ 󵄨󵄨󵄨 ωγ ∈ EA∗ } = {γ ∈ EA∗ 󵄨󵄨󵄨 Aω|ω| γ1 = 1} ⊆ E𝒢∗,ω 󵄨 󵄨 EAn,ω := {γ ∈ EAn 󵄨󵄨󵄨 ωγ ∈ EA∗ } = {γ ∈ EAn 󵄨󵄨󵄨 Aω|ω| γ1 = 1} ⊆ E𝒢n,ω . 19.1.3 The system itself A graph directed Markov system (GDMS) Φ consists of a directed graph 𝒢 and a matrix A as described above, together with a set of nonempty compact subsets {Xυ }υ∈V of a common Euclidean space ℝd and a set of one-to-one contractions {φe : Xt(e) → Xi(e) }e∈E with common contraction ratio s < 1. Note that the contraction φe “goes in the opposite direction” to the edge e. This is because each contraction is conceived as an inverse branch of a distance expanding dynamical system. A GDMS is said to be finite if the set of edges E is finite. Otherwise, a GDMS is said to be infinite. A GDMS is called a graph directed system (GDS) if the matrix A solely provides the information already carried by the graph 𝒢 , i. e., if the transition (passage) from one edge to another only depends on the graph, i. e., provided that Ae1 e2 = 1 if and only if t(e1 ) = i(e2 ). Since the matrix A must always respect the graph 𝒢 , every GDMS is a subsystem of a unique GDS. This latter is called the underlying GDS. A GDS is called an iterated function system (IFS) if the set V is a singleton. The hyperbolic Cantor sets studied in Section 4.1.2 (see also Example 16.2.2 and Remark 16.2.3) are classical examples of IFSs. In a GDMS Φ = {φe }e∈E , the composition of maps is first subject to the graph admissibility of the underlying symbolic words. For ω ∈ E𝒢∗ \{ϵ}, we define φω := φω1 ∘ φω2 ∘ ⋅ ⋅ ⋅ ∘ φω|ω| : Xt(ω) → Xi(ω) and φϵ = IdX , where X := ⨆υ∈V Xυ is the disjoint union of the compact sets Xυ . The main object of interest generated by a GDMS Φ is its limit set J. This set is the image of the symbolic subspace EA∞ under a coding map π. Indeed, given any ω ∈ EA∞ , the sets φω|n (Xt(ω|n ) ), n ∈ ℕ, form a descending sequence of nonempty compact sets. Their diameters do not exceed sn max{diam(Xυ ) | υ ∈ V}, and thus converge to zero as n → ∞. Therefore, their intersection ∞

⋂ φω|n (Xt(ω|n ) )

n=1

738 | 19 Conformal graph directed Markov systems is a singleton, and we denote its element by π(ω). This defines the coding map (or projection) π : EA∞ → X. The limit set of the GDMS Φ is ∞

J := π(EA∞ ) = ⋃ ⋂ φω|n (Xt(ω|n ) ). ω∈EA∞ n=1

Clearly, π is a continuous map when EA∞ is equipped with the topology generated by the initial A-cylinders [e], e ∈ E. This is the case under any of the metrics ds (ω, τ) = s|ω∧τ| , where s ∈ (0, 1). Like EA∞ , the limit set J is generally not compact when E is infinite. Nevertheless, since J is a continuous image of the Polish (i. e., the completely metrizable and separable) space EA∞ , we have the following. Theorem 19.1.1. The limit set of any GDMS is an analytic set in the sense of descriptive set theory. For more information on analytic sets and descriptive set theory, see [120]. When a GDMS is pointwise finite, there is an alternative description of its limit set. Definition 19.1.2. A GDMS Φ = {φe : Xt(e) → Xi(e) }e∈E is said to be pointwise finite if every point of the phase space belongs to finitely many first-level sets, i. e., if #{e ∈ E : x ∈ φe (Xt(e) )} < ∞,

∀x ∈ ℝd .

It is straightforward to see that the limit set of a pointwise finite GDMS is the intersection of its level sets. Lemma 19.1.3. If Φ = {φe }e∈E is a pointwise finite GDMS, then ∞

J = ⋂ ⋃ φω (Xt(ω) ). n=0 ω∈EAn

In particular, J is a Fσδ –set. The limit set J can also be defined in terms of “vertexwise components”: for every υ ∈ V, define Jυ := J ∩ Xυ to be the section of the limit set at the vertex υ. Observe that J = ⋃υ∈V Jυ . It is further worth noticing that for any ω ∈ EA∞ , ∞

∞

∞

n=1 ∞

n=1 ∞

n=1

n=1

n=1

{π(σ(ω))} = ⋂ φσ(ω)|n (Xt(σ(ω)|n ) ) = ⋂ φω2 ω3 ⋅⋅⋅ωn+1 (Xt(ωn+1 ) ) = ⋂ φω2 ω3 ⋅⋅⋅ωn ∘ φωn+1 (Xt(ωn+1 ) ) ⊆ ⋂ φω2 ω3 ⋅⋅⋅ωn (Xi(ωn+1 ) ) = ⋂ φω2 ω3 ⋅⋅⋅ωn (Xt(ωn ) ).

19.2 Properties of conformal maps in ℝd , d ≥ 2

| 739

In fact, ∞

{π(σ(ω))} = ⋂ φω2 ω3 ⋅⋅⋅ωn (Xt(ωn ) ) n=1

since both sides are singleton sets. It follows that ∞

∞

n=1

n=1

{φω1 (π(σ(ω)))} = φω1 ( ⋂ φω2 ω3 ⋅⋅⋅ωn (Xt(ωn ) )) ⊆ ⋂ φω1 ω2 ω3 ⋅⋅⋅ωn (Xt(ωn ) ) = {π(ω)}. Hence, φω1 (π(σ(ω))) = π(ω). By induction on n ∈ ℕ, φω|n (π(σ n (ω))) = π(ω).

(19.1)

By the chain rule, φ′ω|n (π(σ n (ω))) = φ′ω1 (π(σ(ω))) ∘ φ′ω2 (π(σ 2 (ω))) ∘ ⋅ ⋅ ⋅ ∘ φ′ωn (π(σ n (ω))).

(19.2)

Finally, we introduce natural terminology. Definition 19.1.4. A GDMS Φ is said to be finitely irreducible if its underlying matrix A is finitely irreducible (see Definition 17.1.2). In the same vein, a GDMS Φ is said to be finitely primitive if its underlying matrix A is finitely primitive (see Definition 17.1.4).

19.2 Properties of conformal maps in ℝd , d ≥ 2 As we will later concentrate on GDMSs whose generators are conformal maps, in this section we shall study the analytic, geometric, and particularly the distortion properties of conformal maps in ℝd , d ≥ 2. Let U be a connected, open subset of ℝd . From Section 16.1, a C 1 diffeomorphism φ : U → ℝd is conformal if and only if its derivative at every point of U is a similarity map without a translation component. Recall that such maps satisfy properties (16.1)–(16.3). When d = 1, every C 1 diffeomorphism is conformal since T ′ (x) = α(x)⋅Idℝ for some α(x) ∈ ℝ. When d = 2, the conformal C 1 diffeomorphisms, as functions of a complex variable, are the holomorphic and antiholomorphic univalent functions. When d ≥ 3, we have the following characterization. Theorem 19.2.1 (Liouville’s theorem). If d ≥ 3 and U is a connected, open subset of ℝd , then every conformal C 1 diffeomorphism T : U → ℝd is a Möbius transformation, i. e.,

740 | 19 Conformal graph directed Markov systems takes the form φ = λA ∘ i + b,

(19.3)

where λ > 0 and b ∈ ℝd are constants, A is an orthogonal matrix and i is either the x−a d identity map Idℝd or the inversion ia (x) = ‖x−a‖ 2 in the unit sphere centered at a ∈ ℝ . The proof is long and we will not reproduce it here. For details, we refer the reader to Blair [10] and Benedetti and Petronio [6]. Details of the two-dimensional case can also be found there. It is easy to compute that λ 󵄨󵄨 ′ 󵄨󵄨 󵄨󵄨φ (x)󵄨󵄨 = ‖x − a‖2

φ′ (x) = Idℝd − 2Q(x − a), |φ′ (x)|

and

(19.4)

where Q(x) is the matrix whose entries are (Q(x))ij =

xi xj

(19.5)

‖x‖2

and where 󵄨󵄨 ′ 󵄨󵄨 󵄩 ′ 󵄩 󵄨󵄨φ (x)󵄨󵄨 := sup 󵄩󵄩󵄩φ (x)υ󵄩󵄩󵄩 ‖υ‖≤1

is the operator norm of the derivative φ′ (x). In dimension d = 2, we will make extensive use of the part of Koebe distortion theorem about the derivative of univalent functions (see Theorems A.3.8–A.3.10 in Appendix A). We will also need the following reformulation of that theorem (for a proof, see the arguments leading to formula (17.4.21) on p. 353 of [53]). Theorem 19.2.2. Fix 0 < γ ≤ 1/2 so small that max{[

1−γ 1+γ ] , } ≤ √2. 3 (1 + γ) (1 − γ)3 −1

There exists a constant K2 ≥ 1 such that if x ∈ ℂ, R > 0 and φ : B(x, R) → ℂ is a univalent holomorphic function, then 󵄨󵄨 ′ 󵄨 K 󵄨 ′ 󵄨 ′ 󵄨󵄨φ (y) − φ (x)󵄨󵄨󵄨 ≤ 2 󵄨󵄨󵄨φ (x)󵄨󵄨󵄨 ⋅ |y − x|, R

∀y ∈ B(x, δ2 R),

where δ2 = min{γ, π/24}. In dimension d ≥ 3, we have the following corresponding result. Theorem 19.2.3. Suppose that Y is a bounded subset of ℝd , where d ≥ 3, and that W is an open subset of ℝd containing Y. Then there exists a constant K3 ≥ 1 such that if

19.2 Properties of conformal maps in ℝd , d ≥ 2

|

741

φ : ℝd → ℝd is a conformal C 1 diffeomorphism with φ(W) ⊆ ℝd , then 󵄨 󵄨󵄨 ′ ′ ′ 󵄨󵄨|φ (y)| − |φ (x)|󵄨󵄨󵄨 ≤ K3 |φ (x)| ⋅ ‖y − x‖, where δ3 =

1 2

∀x ∈ Y, ∀y ∈ B(Y, δ3 ),

min{diam(Y), dist(Y, 𝜕W)}. In particular, |φ′ (y)| ≤ 1 + K3 diam(Y), |φ′ (x)|

∀x ∈ Y, ∀y ∈ B(Y, δ3 ).

Proof. In view of (19.3), there exist λ > 0, a linear isometry A : ℝd → ℝd , an inversion i = ia or the identity map i = Idℝd , and a vector b ∈ ℝd such that φ = λA ∘ i + b. When i is the identity map, we have |φ′ (y)| = |φ′ (x)| = λ for all x, y ∈ ℝd and the statement of the theorem is obviously true. Assume now that i is an inversion. Since φ(W) ⊆ ℝd , we know that a ∉ W. Therefore, for all x ∈ Y and all y ∈ B(Y, δ3 ) we have ‖x − a‖ ‖x − y‖ + ‖y − a‖ ‖x − y‖ ≤ =1+ ≤1+ ‖y − a‖ ‖y − a‖ ‖y − a‖

1 2

3 2

diam(Y)

dist(Y, 𝜕W)

.

(19.6)

Using (19.4), we deduce that 2

|φ′ (y)| ‖x − a‖2 3 diam(Y) = ≤ (1 + ). |φ′ (x)| ‖y − a‖2 dist(Y, 𝜕W) The second part of the statement of the theorem follows easily. In order to prove the first part, we may assume without losing generality that |φ′ (x)| ≤ |φ′ (y)|. Using (19.4) and (19.6), we then get |φ′ (y)| − |φ′ (x)| = |φ′ (x)|( = |φ′ (x)|(

|φ′ (y)| ‖x − a‖2 − 1) = |φ′ (x)|( − 1) ′ |φ (x)| ‖y − a‖2

‖x − a‖ ‖x − a‖ + 1)( − 1) ‖y − a‖ ‖y − a‖

≤ |φ′ (x)|(2 + ≤ |φ′ (x)|(2 + = [(2 +

3 diam(Y) ‖x − y‖ ) dist(Y, 𝜕W) ‖y − a‖

3 diam(Y) ‖x − y‖ )1 dist(Y, 𝜕W) dist(Y, 𝜕W) 2

3 diam(Y) 2 ) ]|φ′ (x)| ⋅ ‖x − y‖. dist(Y, 𝜕W) dist(Y, 𝜕W)

As a consequence of Theorems 19.2.2 and 19.2.3, we obtain the following result on local bounded distortion. Corollary 19.2.4. Suppose that Y is a bounded subset of ℝd , d ≥ 2, and W is an open subset of ℝd containing Y. For every ε > 0, there exists 0 < δ < dist(Y, ℝd \ W) such that

742 | 19 Conformal graph directed Markov systems if φ : ℝd → ℝd is a conformal C 1 diffeomorphism such that φ(W) ⊆ ℝd , then (1 + ε)−1 ≤

|φ′ (y)| ≤ 1 + ε, |φ′ (x)|

∀x ∈ Y, ∀y ∈ B(x, δ).

19.3 Conformal graph directed Markov systems Definition 19.3.1. A GDMS Φ = {φe }e∈E is called conformal (and thereby a CGDMS) if its underlying GDS satisfies the following four conditions: (a) For every vertex υ ∈ V, the set Xυ is a compact subset of ℝd which is the closure of its interior, i. e., Xυ = Intℝd (Xυ ). (b) (Open Set Condition (OSC)) For all edges e1 , e2 ∈ E, e1 ≠ e2 , φe1 (Int(Xt(e1 ) )) ∩ φe2 (Int(Xt(e2 ) )) = 0. (c) For every vertex υ ∈ V, there exists an open, connected and bounded set Wυ such that Xυ ⊆ Wυ ⊆ ℝd and such that for every e ∈ E with t(e) = υ, the map φe extends to a contracting conformal diffeomorphism of Wυ into ℝd , and all of these extensions have a common contraction ratio s < 1. (d) There exist constants L ≥ 1 and α > 0 such that 󵄨󵄨 ′ 󵄨 ′ ′ α 󵄨󵄨|φe (y)| − |φe (x)|󵄨󵄨󵄨 ≤ L|φe (x)| ⋅ ‖y − x‖ for every e ∈ E and every pair of points x, y ∈ Wt(e) . Remark 19.3.2. Let Φ = {φe }e∈E be a CGDMS. (a) For all incomparable words ω, τ ∈ E𝒢∗ , it readily follows from the OSC (condition (b) of a CGDMS) that φω (Int(Xt(ω) )) ∩ φτ (Int(Xt(τ) )) = 0 and thus φω (Xt(ω) ) ∩ φτ (Xt(τ) ) = φω (𝜕Xt(ω) ) ∩ φτ (𝜕Xt(τ) ). (b) The boundedness of the sets {Wυ }υ∈V is required in condition (c) of a CGDMS to ensure that the system exhibits bounded distortion. If unbounded sets {Wυ }υ∈V otherwise satisfy condition (c), then there exist open, connected and bounded sets {Wυ′ }υ∈V that fulfill condition (c). In fact, we have the following. Claim. For every vertex υ ∈ V, there exists an open, connected and bounded set Wυ′ such that Xυ ⊆ Wυ′ ⊆ Wυ′ ⊆ Wυ .

19.3 Conformal graph directed Markov systems | 743

Proof of Claim. Fix υ ∈ V. Let δυ := dist(Xυ , ℝd \Wυ ) > 0. The neighborhood B(Xυ , δυ /2) := ⋃x∈Xυ B(x, δυ /2) of Xυ is open and bounded but not necessarily connected. However, each open ball B(x, δυ /2), x ∈ X, is connected, and thus every connected component of B(Xυ , δυ /2) contains at least one such ball. Since B(Xυ , δυ /2) is bounded and its connected components are mutually disjoint and each contains at least one ball of radius δυ /2, the neighborhood B(Xυ , δυ /2) has only finitely many connected components. Denote these components by (kυ ) (2) (j) (j) B(1) υ , Bυ , . . . , Bυ . For each 1 ≤ j ≤ kυ , choose xυ ∈ Bυ ∩ Xυ . Since Wυ is path1 (j) connected, for every 1 ≤ j < kυ there is a piecewise C curve γ(j) υ joining xυ and (j+1) (j) (j) xυ in Wυ . Since γυ is compact and Wυ is open, there is δυ > 0 such that kυ −1 (j) ′ (j) (j) B(γ(j) , υ δυ ) ⊆ Wυ . Set Wυ := B(Xυ , δυ /2) ∪ ⋃j=1 B(γυ , δυ /2). (c) When d ≥ 2, condition (d) of a CGDMS is respected with α = 1 provided that the {Wυ }υ∈V are replaced by the {Wυ′ }υ∈V . In the case d = 2, this follows from condition (c), remark (b) and Theorem 19.2.2, by covering each Wυ′ with a finite chain of open balls of radius smaller than dist(Wυ′ , ℝd \Wυ ) > 0. In the case d ≥ 3, this follows from condition (c), remark (b) and Theorem 19.2.3 with Y = Wυ′ and W = Wυ . (d) In [81], a cone condition was part of the definition of a CGDMS. However, this condition is not necessary for our purposes in most of this chapter. It will be described in Section 19.7 and used for establishing the existence of conformal measures and a volume lemma in Section 19.8. Conformality will play a central role in Chapter 21. Remark 19.3.3. Let Φ = {φe }e∈E be a CGDMS. By (19.2) and the conformality of the generators φe , e ∈ E, for every ω ∈ EA∞ and n ∈ ℕ we have that n

󵄨󵄨 ′ 󵄨 󵄨 ′ 󵄨 n j 󵄨󵄨φω|n (π(σ (ω)))󵄨󵄨󵄨 = ∏󵄨󵄨󵄨φωj (π(σ (ω)))󵄨󵄨󵄨. j=1

As a consequence of condition (d) of a CGDMS, we get the following. Lemma 19.3.4. Let Φ = {φe }e∈E be a CGDMS. For all ω ∈ E𝒢∗ and all x, y ∈ Wt(ω) , we have L 󵄨󵄨 󵄨 ′ ′ ‖y − x‖α . 󵄨󵄨log |φω (y)| − log |φω (x)|󵄨󵄨󵄨 ≤ 1 − sα Proof. Let ω ∈ E𝒢∗ , say ω ∈ E𝒢n , and let x, y ∈ Wt(ω) . For every 1 ≤ k ≤ n and z ∈ Wt(ω) , set zk := φωn−k+1 ∘ φωn−k+2 ∘ ⋅ ⋅ ⋅ ∘ φωn (z); set also z0 = z. The chain rule states that φ′ω (z) = φ′ω1 (zn−1 ) ∘ φ′ω2 (zn−2 ) ∘ ⋅ ⋅ ⋅ ∘ φ′ωn (z0 ). The conformality of the generators implies that n

󵄨󵄨 ′ 󵄨󵄨 󵄨 ′ 󵄨 󵄨󵄨φω (z)󵄨󵄨 = ∏󵄨󵄨󵄨φωk (zn−k )󵄨󵄨󵄨. k=1

744 | 19 Conformal graph directed Markov systems Thus, n

n

k=1 n

k=1

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 log󵄨󵄨󵄨φ′ω (y)󵄨󵄨󵄨 − log󵄨󵄨󵄨φ′ω (x)󵄨󵄨󵄨 = ∑ log󵄨󵄨󵄨φ′ωk (yn−k )󵄨󵄨󵄨 − ∑ log󵄨󵄨󵄨φ′ωk (xn−k )󵄨󵄨󵄨 󵄨 󵄨󵄨 ′ 󵄨󵄨φωk (yn−k )󵄨󵄨󵄨 = ∑ log 󵄨 ′ 󵄨󵄨φ (xn−k )󵄨󵄨󵄨 k=1 󵄨 󵄨 ωk 󵄨 󵄨 󵄨 󵄨󵄨 ′ n 󵄨󵄨φωk (yn−k )󵄨󵄨󵄨 − 󵄨󵄨󵄨φ′ωk (xn−k )󵄨󵄨󵄨 = ∑ log(1 + ) 󵄨󵄨 ′ 󵄨 󵄨󵄨φωk (xn−k )󵄨󵄨󵄨 k=1 󵄨󵄨 ′ 󵄨 n 󵄨󵄨|φωk (yn−k )| − |φ′ωk (xn−k )|󵄨󵄨󵄨 ) ≤ ∑ log(1 + |φ′ωk (xn−k )| k=1 󵄨󵄨 ′ n 󵄨󵄨󵄨|φ′ (y 󵄨 ωk n−k )| − |φωk (xn−k )|󵄨󵄨 ≤∑ , |φ′ωk (xn−k )| k=1 where the second-last inequality is just a consequence of the increasing behavior of the logarithm function while the last inequality follows from the fact that log(1 + t) ≤ t for all t ≥ 0. Using condition (d) of a CGDMS and the uniform contraction ratio of the generators, we deduce that n n L 󵄨 󵄨 󵄨 󵄨 ‖y − x‖α . log󵄨󵄨󵄨φ′ω (y)󵄨󵄨󵄨 − log󵄨󵄨󵄨φ′ω (x)󵄨󵄨󵄨 ≤ ∑ L‖yn−k − xn−k ‖α ≤ L ∑ sα(n−k) ‖y − x‖α ≤ α 1 − s k=1 k=1

Interchanging x and y leads to the conclusion. As an immediate consequence of Lemma 19.3.4, and thus as a by-product of condition (d) of a CGDMS, we get bounded distortion. Corollary 19.3.5 (Bounded Distortion Property (BDP)). Let Φ = {φe }e∈E be a CGDMS. There exists a constant K ≥ 1 such that for all ω ∈ E𝒢∗ and all x, y ∈ Wt(ω) we have 󵄨󵄨 ′ 󵄨󵄨 󵄨 ′ 󵄨 󵄨󵄨φω (y)󵄨󵄨 ≤ K 󵄨󵄨󵄨φω (x)󵄨󵄨󵄨. An immediate repercussion of BDP comes next. Corollary 19.3.6. Let Φ = {φe }e∈E be a CGDMS. For all ωτ ∈ E𝒢∗ , the following holds: 󵄩 󵄩 K −1 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩X

t(ω)

󵄩󵄩 ′ 󵄩󵄩 󵄩 ′ 󵄩 󵄩 ′󵄩 󵄩 ′󵄩 󵄩󵄩φτ 󵄩󵄩Xt(τ) ≤ 󵄩󵄩󵄩φωτ 󵄩󵄩󵄩Xt(ωτ) ≤ 󵄩󵄩󵄩φω 󵄩󵄩󵄩Xt(ω) 󵄩󵄩󵄩φτ 󵄩󵄩󵄩Xt(τ) .

N. B.: The Xυ ’s can be replaced by the Wυ ’s in this relationship. Proof. We will prove the lower bound and leave the upper bound to the reader. Let ωτ ∈ E𝒢∗ . Then 󵄩󵄩 ′ 󵄩󵄩 󵄨 ′ 󵄨 󵄩󵄩φωτ 󵄩󵄩Xt(ωτ) = sup 󵄨󵄨󵄨φωτ (x)󵄨󵄨󵄨 x∈Xt(ωτ)

󵄨 󵄨 = sup 󵄨󵄨󵄨φ′ω (φτ (x)) ∘ φ′τ (x)󵄨󵄨󵄨 x∈Xt(τ)

745

19.3 Conformal graph directed Markov systems |

󵄨 󵄨 󵄨 󵄨 = sup 󵄨󵄨󵄨φ′ω (φτ (x))󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨φ′τ (x)󵄨󵄨󵄨 x∈Xt(τ)

󵄨 󵄨 󵄨 󵄨 ≥ inf 󵄨󵄨󵄨φ′ω (φτ (x))󵄨󵄨󵄨 ⋅ sup 󵄨󵄨󵄨φ′τ (x)󵄨󵄨󵄨 x∈X t(τ)

x∈Xt(τ)

󵄨 󵄨 󵄩 󵄩 ≥ inf 󵄨󵄨󵄨φ′ω (z)󵄨󵄨󵄨 ⋅ 󵄩󵄩󵄩φ′τ 󵄩󵄩󵄩X t(τ) z∈X i(τ)

≥K

−1

󵄨 󵄨 󵄩 󵄩 sup 󵄨󵄨󵄨φ′ω (z)󵄨󵄨󵄨 ⋅ 󵄩󵄩󵄩φ′τ 󵄩󵄩󵄩X t(τ)

z∈Xt(ω)

by BDP

󵄩 󵄩 󵄩 󵄩 = K −1 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩X 󵄩󵄩󵄩φ′τ 󵄩󵄩󵄩X . t(ω) t(τ) It is easy to see that the Xυ ’s can be replaced by the Wυ ’s. We shall now prove some geometric consequences of conditions (a)–(d) of a CGDMS and BDP. We first obtain a metric upper bound on the size of the image of any convex set, as well as a set-theoretic upper bound on the image of any ball. Lemma 19.3.7. Let Φ = {φe }e∈E be a CGDMS. For all ω ∈ E𝒢∗ and convex sets C ⊆ Wt(ω) , we have 󵄩 󵄩 󵄩 󵄩 diam(φω (C)) ≤ 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩W diam(C) ≤ K 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩X diam(C). t(ω) t(ω)

(19.7)

Moreover, for all ω ∈ E𝒢∗ , all x ∈ Wt(ω) and all radii 0 < r ≤ dist(x, 𝜕Wt(ω) ), we have 󵄩 󵄩 φω (B(x, r)) ⊆ B(φω (x), 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩W r) ⊆ B(φω (x), K|φ′ω (x)|r). t(ω)

(19.8)

Proof. These results are simple consequences of the mean value theorem and the BDP (Corollary 19.3.5). We now aim at finding an upper bound on the size of the images of the Xυ ’s. Since these sets need not be convex (not even connected), we show that points in a same Xυ can be linked in Wυ by piecewise C 1 curves of uniformly bounded lengths. Lemma 19.3.8. Let Φ = {φe }e∈E be a CGDMS. There exists a constant ℓ > 0 such that for every υ ∈ V and every x, y ∈ Xυ there is a piecewise C 1 curve γx,y joining x and y in Wυ whose length is at most ℓ. Proof. Since the set of vertices V is finite, it suffices to prove the statement for one vertex υ. Fix z ∈ Wυ . Let ℓ > 0. Consider the set 󵄨󵄨 󵄨 ∃ a piecewise C 1 curve γz,w joining z and w in Wυ Wυ (ℓ) := {w ∈ Wυ 󵄨󵄨󵄨󵄨 }. 󵄨󵄨 and whose length is smaller than ℓ The set Wυ (ℓ) is open and connected, and thus path-connected, since Wυ is open and connected. Moreover, if 0 < ℓ1 ≤ ℓ2 then Wυ (ℓ1 ) ⊆ Wυ (ℓ2 ) and ⋃ℓ>0 Wυ (ℓ) = Wυ ⊇ Xυ . Since Xυ is compact, there exists ℓ > 0 such that Wυ (ℓ/2) ⊇ Xυ .

746 | 19 Conformal graph directed Markov systems Now, let x, y ∈ Xυ ⊆ Wυ (ℓ/2). Then there are piecewise C 1 curves γz,x joining z and x and γz,y joining z and y in Wυ whose lengths are smaller than ℓ/2. Gluing those curves at z, we obtain a piecewise C 1 curve γx,y joining x and y in Wυ and whose length is smaller than ℓ. Lemma 19.3.9. Let Φ = {φe }e∈E be a CGDMS. There exists a constant D ≥ 1 such that 󵄩 󵄩 󵄩 󵄩 diam(φω (Xt(ω) )) ≤ D󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩W ≤ KD󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩X , t(ω) t(ω)

∀ω ∈ E𝒢∗ .

(19.9)

Proof. Let ω ∈ E𝒢∗ and x, y ∈ Xt(ω) . By Lemma 19.3.8, there exists a piecewise C 1 curve γx,y : [0, 1] → Wt(ω) joining x and y and such that L(γx,y ) ≤ ℓ, where L(γ) denotes the length of a curve γ. Then φω (γx,y ) is a piecewise C 1 curve joining φω (x) and φω (y), and 󵄩󵄩 󵄩 󵄨 ′ 󵄨 󵄩󵄩φω (x) − φω (y)󵄩󵄩󵄩 ≤ L(φω (γx,y )) := ∫ 󵄨󵄨󵄨φω (γx,y (t))󵄨󵄨󵄨 dt [0,1]

󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩γ ⋅ L(γx,y ) ≤ ℓ󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩W ≤ ℓK 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩X . x,y t(ω) t(ω) Let D = ℓ. We shall now obtain lower bounds. First, we establish the counterpart of (19.8). Lemma 19.3.10. Let Φ = {φe }e∈E be a CGDMS. For all ω ∈ E𝒢∗ , all x ∈ Wt(ω) and all 0 < r ≤ dist(x, 𝜕Wt(ω) ), we have 󵄩 󵄩 φω (B(x, r)) ⊇ B(φω (x), K −1 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩W r) ⊇ B(φω (x), K −1 |φ′ω (x)|r). t(ω)

(19.10)

Proof. First, using BDP (Corollary 19.3.5), observe that for any ω ∈ E𝒢∗ and z ∈ Wt(ω) , 󵄨󵄨 −1 ′ 󵄨−1 󵄨 ′ 󵄨 −1 󵄩 ′ 󵄩 󵄨󵄨(φω ) (φω (z))󵄨󵄨󵄨 = 󵄨󵄨󵄨φω (z)󵄨󵄨󵄨 ≥ K 󵄩󵄩󵄩φω 󵄩󵄩󵄩Wt(ω) . Thus, 󵄩󵄩 −1 ′ 󵄩󵄩 󵄩 ′ 󵄩−1 󵄩󵄩(φω ) 󵄩󵄩φω (Wt(ω) ) ≤ K 󵄩󵄩󵄩φω 󵄩󵄩󵄩Wt(ω) .

(19.11)

Now, fix ω, x and r as in the statement. Let R > 0 be the maximal radius such that B(φω (x), R) ⊆ φω (B(x, r)).

(19.12)

By virtue of the mean value theorem and (19.11)–(19.12), we obtain 󵄩󵄩 −1 ′ 󵄩󵄩 φ−1 ω (B(φω (x), R)) ⊆ B(x, 󵄩 󵄩(φω ) 󵄩󵄩φ

ω (Wt(ω) )

󵄩 󵄩−1 R) ⊆ B(x, K 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩W R). t(ω)

This implies that B(φω (x), R) ⊆ φω (B(x, K‖φ′ω ‖−1 Wt(ω) R)). From the maximality of R

in (19.12) and the openness of φω , it ensues that K‖φ′ω ‖−1 Wt(ω) R ≥ r. That is, R ≥ K −1 ‖φ′ω ‖Wt(ω) r and (19.12) leads to (19.10).

19.3 Conformal graph directed Markov systems | 747

We shall now prove the counterpart of (19.9). Lemma 19.3.11. Let Φ = {φe }e∈E be a CGDMS. There exists a constant D ≥ 1 such that 󵄩 󵄩 diam(φω (Xt(ω) )) ≥ D−1 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩W

t(ω)

,

∀ω ∈ E𝒢∗ .

(19.13)

Proof. Let δ := min{dist(Xυ , ℝd \Wυ ) : υ ∈ V}. By condition (a) of a CGDMS, for each υ ∈ V there are xυ ≠ yυ ∈ Xυ such that yυ ∈ B(xυ , K −1 δ). For every ω ∈ E𝒢∗ , we have by (19.10) that 󵄩 󵄩 φω (B(xt(ω) , δ)) ⊇ B(φω (xt(ω) ), K −1 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩W δ), t(ω) while we have by (19.8) in Lemma 19.3.7 that 󵄩 󵄩 φω (yt(ω) ) ∈ B(φω (xt(ω) ), K −1 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩W δ). t(ω) Therefore, applying the mean value theorem to the map φ−1 ω restricted to the convex −1 ′ set B(φω (xt(ω) ), K ‖φω ‖Wt(ω) δ) followed by (19.11), we obtain 󵄩 −1 󵄩󵄩 ‖yt(ω) − xt(ω) ‖ = 󵄩󵄩󵄩φ−1 ω (φω (yt(ω) )) − φω (φω (xt(ω) ))󵄩 󵄩 ′󵄩 󵄩 󵄩󵄩 󵄩󵄩 󵄩 ≤ 󵄩󵄩󵄩(φ−1 ) ⋅ φ (y ) − φ 󵄩 󵄩 ω (xt(ω) )󵄩 ω 󵄩φω (Wt(ω) ) 󵄩 ω t(ω) 󵄩 󵄩󵄩 ′ 󵄩󵄩−1 󵄩󵄩 󵄩󵄩 ≤ K 󵄩󵄩φω 󵄩󵄩W 󵄩󵄩φω (yt(ω) ) − φω (xt(ω) )󵄩󵄩. t(ω) Thus, 󵄩 󵄩 󵄩 󵄩 diam(φω (Xt(ω) )) ≥ 󵄩󵄩󵄩φω (yt(ω) ) − φω (xt(ω) )󵄩󵄩󵄩 ≥ K −1 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩W ⋅ ‖yt(ω) − xt(ω) ‖ t(ω) 󵄩 ′ 󵄩󵄩 󵄩 󵄩 −1 󵄩 ≥ K min{‖yυ − xυ ‖ : υ ∈ V}󵄩󵄩φω 󵄩󵄩W = D−1 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩W , t(ω)

t(ω)

where D = K(min{‖yυ − xυ ‖ : υ ∈ V})−1 . We now make a simple geometric observation which follows from the OSC (condition (b) of a CGDMS) and some of the previous lemmas. Lemma 19.3.12. Let Φ = {φe }e∈E be a CGDMS. For all 0 < κ1 ≤ κ2 < ∞, all r > 0 and all x ∈ X := ⨆υ∈V Xυ , the cardinality of any collection of mutually incomparable words ω ∈ E𝒢∗ that satisfy the conditions φω (Xt(ω) ) ∩ B(x, r) ≠ 0

and

κ1 r ≤ diam(φω (Xt(ω) )) ≤ κ2 r,

is bounded above by the number d

((1 + κ2 )KD(Rκ1 )−1 ) , where R is the radius of the largest ball that can be inscribed in all the sets Xυ , υ ∈ V.

748 | 19 Conformal graph directed Markov systems Proof. Let Vd = λd (B(0, 1)) be the Lebesgue measure of the unit ball in ℝd . For every υ ∈ V, let xυ ∈ Xυ be such that B(xυ , R) ⊆ Xυ . Fix 0 < κ1 ≤ κ2 < ∞, r > 0 and x ∈ X. Let W be a collection of incomparable words as described in the statement. Then for every ω ∈ W, we have φω (Xt(ω) ) ⊆ B(x, r + diam(φω (Xt(ω) ))) ⊆ B(x, (1 + κ2 )r). As the sets {φω (Int(Xt(ω) ))}ω∈W are mutually disjoint due to the OSC, using Lemmas 19.3.10 and 19.3.9 successively we obtain that Vd (1 + κ2 )d r d = λd (B(x, (1 + κ2 )r)) ≥ λd ( ⋃ φω (Int(Xt(ω) ))) ω∈W

= ∑ λd (φω (Int(Xt(ω) ))) ω∈W

≥ ∑ λd (φω (B(xt(ω) , R))) ω∈W

󵄩 󵄩 ≥ ∑ λd (B(φω (xt(ω) ), K −1 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩W R)) t(ω) ω∈W

≥ ∑ λd (B(φω (xt(ω) ), K −1 D−1 diam(φω (Xt(ω) ))R)) ω∈W

≥ ∑ λd (B(φω (xt(ω) ), (KD)−1 κ1 rR)) ω∈W

d

= #W((KD)−1 Rκ1 r) Vd . d

Hence, #W ≤ ((1 + κ2 )(KD)(Rκ1 )−1 ) . We now look at the points in the limit set that lie in the boundary of the real space. First, we show that the set of admissible infinite words that generate points on the boundary of the real space is forward shift-invariant. Lemma 19.3.13. For any CGDMS Φ = {φe }e∈E , σ(π −1 (𝜕X)) ⊆ π −1 (𝜕X). Proof. Since the φe ’s are open maps, φe (Int(Xt(e) )) ⊆ Int(Xi(e) ) for every e ∈ E. Because 𝜕Xυ ∩ Int(Xυ ) = 0 for all υ ∈ V, it is then clear that φ−1 e (𝜕Xi(e) ) ∩ Xt(e) ⊆ 𝜕Xt(e) . If ω ∈ π −1 (𝜕X), then π(ω) ∈ 𝜕Xi(ω1 ) . As π(ω) = φω1 (π(σ(ω))), we deduce that −1 π(σ(ω)) = φ−1 ω1 (π(ω)) ∈ φω1 (𝜕Xi(ω1 ) ) ∩ Xt(ω1 ) ⊆ 𝜕Xt(ω1 ) .

Hence, σ(ω) ∈ π −1 (𝜕Xt(ω1 ) ) ⊆ π −1 (𝜕X). So σ(π −1 (𝜕X)) ⊆ π −1 (𝜕X).

19.4 Topological pressure, finiteness parameter, and Bowen parameter for CGDMSs | 749

Lemma 19.3.14. Let Φ = {φe }e∈E be a CGDMS. For every ω ∈ E𝒢∗ , we have σ |ω| (π −1 (φω (𝜕Xt(ω) ))) ⊆ π −1 (𝜕X). Proof. Set n = |ω|. Let ρ ∈ σ |ω| (π −1 (φω (𝜕Xt(ω) ))). Then ρ = σ n (γ) for some γ ∈ π −1 (φω (𝜕Xt(ω) )). Thus, φγ|n (Xt(γ|n ) ) ∋ φγ|n (π(σ n (γ))) = π(γ) ∈ φω (𝜕Xt(ω) ). It follows from Remark 19.3.2(a) that π(γ) ∈ φγ|n (𝜕Xt(γ|n ) ). Then π(ρ) = π(σ n (γ)) = −1 φ−1 γ|n (π(γ)) ∈ 𝜕Xt(γn ) ⊆ 𝜕X. Hence, ρ ∈ π (𝜕X).

19.4 Topological pressure, finiteness parameter, and Bowen parameter for CGDMSs Let Φ = {φe }e∈E be a CGDMS. We will now impose on the symbolic space EA∞ a family of potentials tΞ, t ≥ 0, which is intimately related to the dynamics taking place on the limit set J := π(EA∞ ) of Φ. Lemma 19.4.1. Let Φ = {φe }e∈E be a CGDMS. The potential Ξ : EA∞ → ℝ given by 󵄨 󵄨 Ξ(ω) := log󵄨󵄨󵄨φ′ω1 (π(σ(ω)))󵄨󵄨󵄨

(19.14)

is Hölder continuous on cylinders, and hence is acceptable. This potential induces the pressure function P(⋅ Ξ) : [0, ∞) → ℝ ∪ {∞}, where P(tΞ) := P(σ, tΞ) is the topological pressure of the shift map σ : EA∞ → EA∞ subjected to the potential tΞ, as described in Definition 17.2.7. If Φ is finitely irreducible, then this pressure function satisfies P(tΞ) = lim

n→∞

1 1 log Zn (tΞ) = inf log Zn (tΞ), n∈ℕ n n

with 󵄩 󵄩t 󵄩 󵄩t K −t ∑ 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩X ≤ Zn (tΞ) ≤ K t ∑ 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩X , t(ω) t(ω) ω∈EAn

ω∈EAn

∀n ∈ ℕ.

Proof. Using Lemma 19.3.4, we deduce for any ω, τ ∈ EA∞ with |ω ∧ τ| ≥ 1 that 󵄨󵄨 󵄨 󵄨󵄨 󵄨 ′ 󵄨 󵄨 ′ 󵄨󵄨󵄨 󵄨󵄨Ξ(ω) − Ξ(τ)󵄨󵄨󵄨 = 󵄨󵄨󵄨log󵄨󵄨󵄨φω1 (π(σ(ω)))󵄨󵄨󵄨 − log󵄨󵄨󵄨φω1 (π(σ(τ)))󵄨󵄨󵄨󵄨󵄨󵄨 󵄨 󵄨 L 󵄩󵄩 󵄩󵄩α ≤ 󵄩π(σ(ω)) − π(σ(τ))󵄩󵄩 1 − sα 󵄩 L Ls−α L α |σ(ω)∧σ(τ)|α (|ω∧τ|−1)α s = s = [d (ω, τ)] . ≤ 1 − sα 1 − sα 1 − sα s

750 | 19 Conformal graph directed Markov systems This proves the Hölder continuity on cylinders. Then, by Theorem 17.2.8, P(tΞ) = lim

n→∞

1 1 log Zn (tΞ) = inf log Zn (tΞΦ ). n∈ℕ n n

Using Remark 19.3.3, we observe that Zn (tΞ) = ∑ exp(Sn (tΞ)([ω])) ω∈EAn

n−1

= ∑ exp( sup ∑ tΞ(σ j (τ))) ω∈EAn

τ∈[ω] j=0

ω∈EA

j=0

n−1 󵄨󵄨 󵄨󵄨 = ∑ exp( sup ∑ t log󵄨󵄨󵄨φ′(σj (τ)) (π(σ(σ j (τ))))󵄨󵄨󵄨) 1 󵄨 󵄨 n τ∈[ω] n−1 󵄨 󵄨󵄨t 󵄨 = ∑ exp( sup log ∏󵄨󵄨󵄨φ′τj+1 (π(σ j+1 (τ)))󵄨󵄨󵄨 ) 󵄨 󵄨 n τ∈[ω] ω∈EA

j=0 n 󵄨

ω∈EA

j=1

󵄨󵄨t 󵄨 = ∑ exp(log sup ∏󵄨󵄨󵄨φ′τj (π(σ j (τ)))󵄨󵄨󵄨 ) 󵄨 󵄨 n τ∈[ω] 󵄨󵄨 󵄨󵄨t 󵄨󵄨 󵄨󵄨t = ∑ sup 󵄨󵄨󵄨φ′τ|n (π(σ n (τ)))󵄨󵄨󵄨 = ∑ sup 󵄨󵄨󵄨φ′ω (π(σ |ω| (τ)))󵄨󵄨󵄨 󵄨 󵄨 n τ∈[ω]󵄨 n τ∈[ω]󵄨 ω∈EA

ω∈EA

󵄩 󵄩t ≍ ∑ 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩X , t(ω) ω∈EAn

where a constant of comparability is K t , the t-th power of the constant K of bounded distortion from Corollary 19.3.5. As a result, we make the following definition. Definition 19.4.2. Let n ∈ ℕ. The nth partition function Zn : [0, ∞) → [0, ∞] of a CGDMS Φ = {φe }e∈E is defined at every parameter t ≥ 0 by 󵄩 󵄩t Zn (t) := ∑ 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩X . t(ω) ω∈EAn

Every partition function is nonincreasing and may take the value ∞ when #E = ∞. Accordingly, for each n ∈ ℕ let Finn := {t ≥ 0 : Zn (t) < ∞} be the finiteness parameter set for the nth partition function Zn . Thereafter, let θn := inf Finn be the finiteness parameter of that function.

19.4 Topological pressure, finiteness parameter, and Bowen parameter for CGDMSs | 751

Definition 19.4.3. The topological pressure function P : [0, ∞) → (−∞, ∞] of a finitely irreducible CGDMS Φ = {φe }e∈E is defined at the parameter t ≥ 0 by P(t) := lim

n→∞

1 log Zn (t). n

Note that K −t Zn (t) ≤ Zn (tΞ) ≤ K t Zn (t) for all n ∈ ℕ, according to Lemma 19.4.1. Thus, P(t) = P(tΞ) for all t ≥ 0 and the pressure function is thereby well defined. The finiteness parameter set for the topological pressure function Fin := {t ≥ 0 : P(t) < ∞} is called the finiteness parameter set of the CGDMS Φ. The finiteness parameter θ of Φ is defined by θ := inf Fin. The partition functions enjoy the following properties. Proposition 19.4.4. The partition functions Zn : [0, ∞) → [0, ∞], n ∈ ℕ, of a finitely irreducible CGDMS Φ = {φe }e∈E have the following properties: (a) For every t ≥ 0, the sequence (Zn (t))∞ n=1 is submultiplicative and boundedly supermultiplicative. (b) Finn = Fin for all n ∈ ℕ. In particular, θn = θ for all n ∈ ℕ. (c) Zn is nonincreasing on [0, ∞), is strictly decreasing to −∞ and convex on [θ, ∞), is continuous on (θ, ∞), and is right-continuous at θ. Proof. (a) Fix t ≥ 0. Since Zn (t) is comparable to Zn (tΞ) according to Lemma 19.4.1 and since the sequence (Zn (tΞ))∞ n=1 is submultiplicative according to Lemma 17.2.5, the ∞ sequence (Zn (t))n=1 is boundedly submultiplicative, i. e., there exists a constant B(t) > 0 such that Zm+n (t) ≤ B(t)Zm (t)Zn (t) for all m, n ∈ ℕ. One such constant is K 3t since K −t Zn (t) ≤ Zn (tΞ) ≤ K t Zn (t). In fact, we even have submultiplicativity since 󵄩 󵄩t Zm+n (t) := ∑ 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩X ≤ t(ω) ω∈EAm+n

󵄩 󵄩t 󵄩󵄩 ′ 󵄩t ∑ 󵄩󵄩󵄩φ′ω|m 󵄩󵄩󵄩X 󵄩φ m+n 󵄩󵄩 t(ω|m ) 󵄩 ω|m+1 󵄩Xt(ω|m+n )

ω∈EAm+n

m+1

󵄩 󵄩t 󵄩 󵄩t ≤ ∑ ∑ 󵄩󵄩󵄩φ′τ 󵄩󵄩󵄩X 󵄩󵄩󵄩φ′ρ 󵄩󵄩󵄩X = Zm (t)Zn (t). t(τ) t(ρ) τ∈EAm ρ∈EAn

Similarly, as Zn (t) is comparable to Zn (tΞ) and as the sequence (Zn (tΞ))∞ n=1 is boundedly supermultiplicative according to Lemma 17.2.5, the sequence (Zn (t))∞ n=1 is boundedly supermultiplicative, i. e., there exists a constant C(t) > 0 such that Zm+n (t) ≥ C(t)Zm (t)Zn (t) for all m, n ∈ ℕ. (b) Since Zn (t) is comparable to Zn (tΞ), Theorem 17.2.8 affirms that Fin = Finn for all n ∈ ℕ. It follows immediately that θ = θn for all n ∈ ℕ.

752 | 19 Conformal graph directed Markov systems (c) The nonincreasing property of Zn is clear. The strictly decreasing behavior of Zn on [θ, ∞) can be more precisely described as follows. Let θ ≤ t1 < t2 . Then Zn (t2 ) ≤ sn(t2 −t1 ) Zn (t1 ) for all n ∈ ℕ. The convexity of Zn on Fin follows from the fact that it is the sum of the convex functions t 󳨃→ ‖φ′ω ‖tXt(ω) , ω ∈ EAn . Its continuity on (θ, ∞) is a direct consequence of its convexity on that interval. The right-continuity at θ follows immediately from the convexity on [θ, ∞) if θ ∈ Fin. However, if θ ∉ Fin, then Zn (θ) = ∞, or equivalently, Zn (θΞ) = ∞. By Lemma 17.2.6, this means that sup{Zn (θΞ, F) : F ⊆ E and #F < ∞} = ∞. If Zn were not right-continuous at θ, then we would have supt>θ Zn (t) < ∞. Therefore, there would exist a finite set F ⊆ E such that Zn (θΞ, F) > 2K θ supt>θ Zn (t). Since K −θ Zn (θ, F) ≤ Zn (θΞ, F) ≤ K θ Zn (θ, F), this implies that Zn (θ, F) > 2 supt>θ Zn (t). As F is finite, Zn (⋅, F) is strictly decreasing and continuous on ℝ (recall that Zn (t2 , F) ≤ sn(t2 −t1 ) Zn (t1 , F)), and thus supt>θ Zn (t, F) = Zn (θ, F). Therefore, supt>θ Zn (t) ≥ supt>θ Zn (t, F) = Zn (θ, F) > 2 supt>θ Zn (t). This is a contradiction. So Zn is right-continuous at θ. Finally, the convexity of Zn on [0, ∞) follows from its (already established) convexity on Fin and its right-continuity at θ. We now turn our attention to the topological pressure function. The following is a rewriting of Theorem 17.2.8 for P(t). Proposition 19.4.5. If Φ = {φe }e∈E is a finitely irreducible CGDMS, then: (a) P(t) < ∞ ⇐⇒ Z1 (t) < ∞ ⇐⇒ Zn (t) < ∞ for some n ∈ ℕ ⇐⇒ Zn (t) < ∞, ∀n ∈ ℕ. (b) P(t) = ∞ ⇐⇒ Z1 (t) = ∞ ⇐⇒ Zn (t) = ∞ for some n ∈ ℕ ⇐⇒ Zn (t) = ∞, ∀n ∈ ℕ. 1 1 (c) P(t) = lim log Zn (t) = inf log Zn (t). n→∞ n n∈ℕ n Proof. Statements (a) and (b) follow from the fact that Zn (t) is comparable to Zn (tΞ). (Recall that K −t Zn (t) ≤ Zn (tΞ) ≤ K t Zn (t).) Statement (c) is a consequence of (a) and (b), Proposition 19.4.4 and (17.5). It ensues that the pressure function has the following properties. Proposition 19.4.6. The pressure function P : [0, ∞) → ℝ ∪ {∞} of a finitely irreducible CGDMS Φ = {φe }e∈E , exhibits the following basic properties: (a) P(t) = P(tΞ). (b) P(t) = sup{PF (t) : F ⊆ E and #F < ∞}, where PF (t) is the pressure of the subsystem Φ|F := {φe }e∈F . (c) P is nonincreasing on [0, ∞), strictly decreasing to −∞ and convex on [θ, ∞), continuous on (θ, ∞), and right-continuous at θ. (d) If #E < ∞ then P(t) < ∞ for all t ∈ [0, ∞). Furthermore, P(0) ≤ log #E. On the other hand, if #E = ∞ then P(0) = ∞. Proof. (a) Since Zn (t) is comparable to Zn (tΞ) (recall that K −t Zn (t) ≤ Zn (tΞ) ≤ K t Zn (t)), it follows from Theorem 17.2.8 that P(tΞ) = lim

n→∞

1 1 log Zn (tΞ) = lim log Zn (t) = P(t). n→∞ n n

19.5 Classification of CGDMSs | 753

(b) By (a), we obtain P(t) = P(tΞ) =

sup PF (tΞΦ ) =

F⊆E finite

sup PF (t).

F⊆E finite

(c) The nonincreasing property of the pressure function P follows from the nonincreasing property of the partition functions Zn , n ∈ ℕ. The strictly decreasing behavior of P on [θ, ∞) can be more specifically described as follows. Let θ ≤ t1 < t2 . Since Zn (t2 ) ≤ sn(t2 −t1 ) Zn (t1 ) for all n ∈ ℕ, we deduce that P(t2 ) ≤ (t2 − t1 ) log s + P(t1 ). The convexity of the pressure on [θ, ∞) follows from the convexity of the partition functions on [θ, ∞). The continuity of P on (θ, ∞) and its right-continuity at θ is a direct consequence of its convexity, except at θ when θ ∉ Fin, in which case an argument similar to that given in the proof of Proposition 19.4.4(c) can be formulated for the pressure. (d) If #E < ∞, then #(EAn ) ≤ (#E)n < ∞ for all n ∈ ℕ, and thus Zn (t) < ∞ for all n ∈ ℕ and all t ≥ 0. It follows from Proposition 19.4.5 that P(t) < ∞ for all t ≥ 0. In particular, Z1 (0) = #E, and thus P(0) ≤ log #E. If #E = ∞, then Z1 (0) = ∞ and P(0) = ∞ by Proposition 19.4.5. Some parameters t have a special meaning. One such parameter is the finiteness parameter θ. Another one is the Bowen parameter h. Definition 19.4.7. The number h := inf{t ≥ 0 : P(t) ≤ 0} is called the Bowen parameter of the CGDMS Φ. By virtue of Proposition 19.4.6, we know that θ ≤ h and that P(h) ≤ 0. In Section 19.6, we will show that the Bowen parameter h of a finitely irreducible CGDMS Φ is equal to the Hausdorff dimension of the limit set J of Φ. This is accordingly called Bowen’s formula.

19.5 Classification of CGDMSs Infinite CGDMSs naturally break into two main classes: regular and irregular systems. This dichotomy is determined by the existence of a zero of the pressure function. Definition 19.5.1. A CGDMS Φ is called regular if there is some t ≥ 0 such that P(t) = 0. Otherwise, the system is said to be irregular. For finitely irreducible CGDMSs, it follows from the nonincrease of P and its strict decrease on [θ, ∞) (see Proposition 19.4.6) that such a t is unique (if it exists) and that we have the following. Lemma 19.5.2. A finitely irreducible CGDMS Φ is regular if and only if P(h) = 0.

754 | 19 Conformal graph directed Markov systems Note that every finite CGDMS is regular. There are natural subclasses of regular systems. Definition 19.5.3. A CGDMS Φ is said to be (a) strongly regular if 0 < P(t) < ∞ for some t ≥ 0; (b) critically regular if P(θ) = 0. In this case, θ = h. It is an immediate consequence of the continuity and strict decrease of P on [θ, ∞) that the class of finitely irreducible, strongly regular systems consists of those systems whose Bowen parameter is strictly larger than their finiteness parameter. Lemma 19.5.4. A finitely irreducible CGDMS Φ is strongly regular if and only if h > θ. Within the subclass of strongly regular systems can be found yet another subclass. Definition 19.5.5. A CGDMS Φ is called cofinitely regular provided it has infinite pressure at its finiteness parameter, i. e., if P(θ) = ∞.

19.6 Bowen’s formula for CGDMSs We will now derive a formula for the Hausdorff dimension of the limit set of a finitely irreducible CGDMS. It is entirely expressed in dynamical terms. Since it stems from the seminal work of Bowen [12], it is referred to as Bowen’s formula. 19.6.1 The finite case Assume that E is a finite alphabet and that A : E × E → {0, 1} is an irreducible matrix. Then the symbolic space EA∞ is compact due to the finiteness of E. Moreover, Theorem 3.2.14 states that the shift map σ : EA∞ → EA∞ is transitive precisely when A is irreducible. According to Theorem 3.2.12, the shift map is also open. Furthermore, as witnessed in Example 4.1.3, this map is distance expanding. (These properties can be proved directly.) Thus, the theory developed in Chapter 13 applies with X = EA∞ and T = σ. Of particular interest are Sections 13.5, 13.6 and 13.7. Lemma 19.6.1. Let Φ = {φe }e∈E be a finite and irreducible CGDMS. Let also M(EA∞ ) be the space of all Borel probability measures on EA∞ . For every t ≥ 0, there exists a unique ̃ t ∈ M(EA∞ ) for the Hölder continuous σ-invariant Gibbs state and equilibrium state m ∞ potential ft : EA → ℝ defined by ft = tΞ, where Ξ is the potential introduced in (19.14). ̃ t is also ergodic. The state m In particular, there exists a constant C(t) ≥ 1 such that C(t)−1 ≤

̃ t ([ω]) m 󵄩 󵄩t −|ω| e P(t) 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩X t(ω)

≤ C(t),

∀ω ∈ EA∗

(19.15)

19.6 Bowen’s formula for CGDMSs | 755

and 󵄨󵄨 ̃ 󵄨󵄨 ′ P(t) = hm ̃ t (σ) + t ∫ log󵄨󵄨φρ1 (π(σ(ρ)))󵄨󵄨 dm t (ρ).

(19.16)

EA∞

Proof. Let t ≥ 0. By Lemma 19.4.1, we know that ft is Hölder continuous on cylinders since Ξ is. As EA∞ is compact, the potential ft is bounded and it is then easy to see that Hölder continuity on cylinders is equivalent to global Hölder continuity. By Proposition 13.7.12, there thus exists a unique σ-invariant Gibbs state and equilibrium state ̃ t ∈ M(EA∞ ) associated to ft . This state is also ergodic. m ̃ t (cf. Definition 13.2.1), observe that the initial Regarding the Gibbs property of m cylinders are the open balls of the symbolic space EA∞ (they are also the closed balls of that space). Let us use the metric d1/2 (ω, τ) := (1/2)|ω∧τ| on EA∞ . As the image under the shift map of an initial cylinder of length k is an initial cylinder of length k −1 whenever k ≥ 2, we can choose δ = ξ = 1/2 in (13.1), (13.2) and (13.13). Let τ ∈ EA∞ and n ∈ ℕ. Then B(σ n (τ), ξ ) = [τn+1 τn+2 ]. Consequently, στ−n (B(σ n (τ), ξ )) = [τ1 . . . τn τn+1 τn+2 ] = [τ|n+2 ]. Moreover, 󵄨 󵄨 󵄨 󵄨t 󵄩 󵄩t eSn ft (τ) = etSn Ξ(τ) = exp(t log󵄨󵄨󵄨φ′τ|n (π(σ n (τ)))󵄨󵄨󵄨) = 󵄨󵄨󵄨φ′τ|n (π(σ n (τ)))󵄨󵄨󵄨 ≍ 󵄩󵄩󵄩φ′τ|n 󵄩󵄩󵄩X

t(τ|n )

,

where a constant of comparability is K t , the t-th power of the constant K of bounded distortion from Corollary 19.3.5. Finally, recall that P(t) = P(ft ) according to Proposition 19.4.6(a). Rewriting (13.13) in our symbolic setting, we deduce that there exists a constant C(t) ≥ 1 such that C(t)−1 ≤

̃ t ([τ|n+2 ]) m 󵄩 󵄩t −n e P(t) 󵄩󵄩󵄩φ′τ| 󵄩󵄩󵄩X t(τ|n ) n

≤ C(t),

∀n ∈ ℕ, ∀τ ∈ EA∞ .

There is a mismatch between n + 2 in the numerator and n in the denominator. Coñ t ([τ|n ]) ≥ m ̃ t ([τ|n+2 ]). So cerning the lower bound, observe that m C(t)−1 ≤

̃ t ([τ|n ]) m , 󵄩 󵄩t e−n P(t) 󵄩󵄩󵄩φ′τ| 󵄩󵄩󵄩X t(τ|n ) n

∀n ∈ ℕ, ∀τ ∈ EA∞ .

Deriving the upper bound is trickier. First, e−(n+2) P(t) = e−n P(t) e−2 P(t) . Moreover, by Corollary 19.3.6, 󵄩󵄩 ′ 󵄩󵄩t 󵄩 ′ 󵄩t −t 󵄩 ′ 󵄩t −t 󵄩 ′ 󵄩t 󵄩󵄩φτ|n+2 󵄩󵄩Xt(τ| ) ≥ 󵄩󵄩󵄩φτ|n 󵄩󵄩󵄩Xt(τ| ) ⋅ K 󵄩󵄩󵄩φτn+1 󵄩󵄩󵄩Xt(τ ) ⋅ K 󵄩󵄩󵄩φτn+2 󵄩󵄩󵄩Xt(τ ) n n+1 n+2 n+2 2t 󵄩 󵄩t 󵄩 󵄩 ≥ 󵄩󵄩󵄩φ′τ|n 󵄩󵄩󵄩X ⋅ K −2t (min󵄩󵄩󵄩φ′e 󵄩󵄩󵄩X ) t(τ| ) t(e) n

̃2t 󵄩󵄩󵄩φ′ 󵄩󵄩󵄩t =K 󵄩 τ|n 󵄩Xt(τ| ) , n

e∈E

756 | 19 Conformal graph directed Markov systems ̃ := K −1 mine∈E ‖φ′ ‖X . Therefore, where K e t(e) ̃ t ([τ|n+2 ]) m −(n+2) P(t) 󵄩 󵄩󵄩φ′ 󵄩󵄩󵄩t e 󵄩 τ|n+2 󵄩Xt(τ|n+2 )

̃−2t , ≤ C(t)e2 P(t) K

∀n ∈ ℕ, ∀τ ∈ EA∞ .

In summary, we have proved the lower bound in (19.15) for all cylinders and the upper bound for all cylinders of length at least 3. Since there are only finitely many cylinders of length smaller than 3, the upper bound holds for all cylinders by increasing the constant if necessary. Finally, relation (19.16) is just the definition of an equilibrium state for the potential ft (see Definition 12.2.1). Since the system Φ is finite, its Bowen parameter h satisfies the equation P(h) = 0.

(19.17)

The previous lemma has the following immediate ramification. Corollary 19.6.2. Let Φ = {φe }e∈E be a finite and irreducible CGDMS, and h be its Bowen parameter. The unique ergodic σ-invariant Gibbs state and equilibrium state ̃ h ∈ M(EA∞ ) for the Hölder continuous potential fh = hΞ satisfies m h ̃ h ([ω]) ≍ 󵄩󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩󵄩X . m t(ω)

(19.18)

Now, let ̃ h ∘ π −1 mh := m ̃ h under the coding map π : be the pushdown onto the limit set J of the measure m EA∞ → J. We shall prove the following result, whose last part is Bowen’s formula for finite and irreducible CGDMSs. In the case of finite-alphabet conformal IFSs, this result was first obtained by Bedford [5]. Theorem 19.6.3. Let Φ be a finite and irreducible CGDMS, and h be its Bowen parameter. The following statements hold: (a) mh is a geometric measure with exponent h, i. e., there exists a constant C(h) ≥ 1 such that C(h)−1 ≤

mh (B(x, r)) rh

≤ C(h)

for all x ∈ J and all sufficiently small r > 0. (b) 0 < Hh (J), Ph (J) < ∞, where Hh (J) and Ph (J) respectively denote the h-dimensional Hausdorff and packing measures of the limit set J.

19.6 Bowen’s formula for CGDMSs | 757

(c) (Bowen’s formula) HD(J) = PD(J) = BD(J) = h, where HD(J), PD(J) and BD(J) respectively denote the Hausdorff, packing and box-counting dimensions of J. Proof. Let x ∈ J and 0 < r < 21 min{diam(Xυ ) : υ ∈ V}. Then x = π(ω) for some ω ∈ EA∞ . Take the smallest n ∈ ℕ such that φω|n (Xt(ωn ) ) ⊆ B(x, r). By (19.18), we find that ̃ 󵄩󵄩󵄩φ′ 󵄩󵄩󵄩h ̃ h ∘ π −1 (φω|n (Xt(ωn ) )) ≥ m ̃ h ([ω|n ]) ≥ C(h) mh (B(x, r)) ≥ m 󵄩 ω|n 󵄩Xt(ω

n)

(19.19)

̃ for some constant C(h) independent of ω and n, i. e., independent of x and r. Since #E < ∞, we know that 󵄩 󵄩 ξ ′ := min{󵄩󵄩󵄩φ′e 󵄩󵄩󵄩X : e ∈ E} > 0. t(e) By the definition of n, by Lemma 19.3.9 and Corollary 19.3.6, we also have that 󵄩 󵄩 r ≤ diam(φω|n−1 (Xt(ωn−1 ) )) ≤ KD󵄩󵄩󵄩φ′ω|n−1 󵄩󵄩󵄩X

t(ωn−1 )

󵄩󵄩 ′ 󵄩󵄩 󵄩󵄩φωn 󵄩󵄩Xt(ω ) 󵄩󵄩 ′ 󵄩󵄩 n ≤ KD󵄩󵄩φω|n−1 󵄩󵄩X t(ωn−1 ) ξ′ 󵄩 󵄩 ≤ (ξ ′ )−1 K 2 D󵄩󵄩󵄩φ′ω|n 󵄩󵄩󵄩X . t(ωn )

Using (19.19), it ensues that ′ −1 2 −h ̃ mh (B(x, r)) ≥ [C(h)((ξ ) K D) ] ⋅ r h .

(19.20)

In order to prove a similar inequality in the opposite direction, let W be the family of all minimal (in the sense of length) words τ ∈ EA∗ such that φτ (Xt(τ) ) ∩ B(x, r) ≠ 0

and

diam(φτ (Xt(τ) )) < 2r.

(19.21)

Let τ ∈ W. Then diam(φτ||τ|−1 (Xt(τ||τ|−1 ) )) ≥ 2r and it follows from Lemma 19.3.11, Corollary 19.3.6 and Lemma 19.3.9 that 󵄩 󵄩 diam(φτ (Xt(τ) )) ≥ D−1 󵄩󵄩󵄩φ′τ 󵄩󵄩󵄩X t(τ) 󵄩 󵄩 󵄩 󵄩 ≥ (KD)−1 󵄩󵄩󵄩φ′τ||τ|−1 󵄩󵄩󵄩X ⋅ 󵄩󵄩φ′ 󵄩󵄩 t(τ||τ|−1 ) 󵄩 τ|τ| 󵄩Xt(τ|τ| ) ≥ (KD)−1 ⋅ (KD)−1 diam(φτ||τ|−1 (Xt(τ||τ|−1 ) )) ⋅ ξ ′ ≥ 2(KD)−2 ξ ′ r.

By definition, the family W consists of mutually incomparable words. Therefore, in virtue of Lemma 19.3.12 with κ1 = 2(KD)−2 ξ ′ and κ2 = 2, we have −1 d

d

#W ≤ (3KD(R ⋅ 2(KD)−2 ξ ′ ) ) = (3(KD)3 (2ξ ′ R)−1 ) .

(19.22)

758 | 19 Conformal graph directed Markov systems As π −1 (B(x, r)) ⊆ ⋃τ∈W [τ], we get from (19.18), Lemma 19.3.11, (19.21) and (19.22) that ̃ h ∘ π −1 (B(x, r)) ≤ m ̃ h ( ⋃ [τ]) mh (B(x, r)) = m τ∈W

̃ 󵄩󵄩󵄩φ′ 󵄩󵄩󵄩h ̃ h ([τ]) ≤ ∑ C(h) ≤ ∑ m 󵄩 τ 󵄩X τ∈W

τ∈W

t(τ)

h

̃ ≤ C(h) ∑ (D diam(φτ (Xt(τ) ))) τ∈W

̃ ≤ C(h) ∑ (D ⋅ 2r)h τ∈W

̃ = C(h) ⋅ (2D)h r h ⋅ #W d h ̃ ≤ [C(h)(2D) (3(KD)3 (2ξ ′ R)−1 ) ] ⋅ r h . Along with (19.20), this proves that mh is a geometric measure with exponent h, i. e., statement (a) is proved. Statement (b) is then an immediate consequence of Theorem 15.6.14(a), while statement (c) follows directly from Theorem 15.6.14(b). Using the communication classes introduced in Definition 3.2.23, a generalization of Bowen’s formula for all finite CGDMSs was later derived by Ghenciu, Mauldin and Roy [36].

19.6.2 The general case We are now ready to provide a short and simple proof of the main theorem of this chapter. The following characterization of the Hausdorff dimension HD(J) of the limit set J, which is a variation of the classical Bowen’s formula, was proved for finitely irreducible CGDMSs by Mauldin and Urbański [81]. A further generalization of that formula was subsequently found by Roy [102]. For every F ⊆ E, we write Φ|F for the subsystem {φe }e∈F of Φ, and JF for the limit set of Φ|F . Theorem 19.6.4 (Bowen’s formula). If Φ is a finitely irreducible CGDMS, then its Bowen parameter h satisfies h := inf{t ≥ 0 : P(t) ≤ 0} = HD(J) = sup{HD(JF ) : F ⊆ E, #F < ∞} ≥ θ. Moreover, if P(t) = 0 then t is the only zero of the pressure function and t = h = HD(J). Proof. Set H = HD(J) and h∞ = sup{HD(JF ) : F ⊆ E, #F < ∞}. Fix t > h. Then P(t) = limn→∞ n1 log Zn (t) < 0. Therefore, for all n ∈ ℕ large enough, 󵄩 󵄩t ∑ 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩X =: Zn (t) ≤ exp(n P(t)/2). t(ω)

ω∈EAn

19.6 Bowen’s formula for CGDMSs | 759

Using Lemma 19.3.9, we obtain that t 󵄩 󵄩t ∑ (diam(φω (Xt(ω) ))) ≤ (KD)t ∑ 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩X ≤ (KD)t exp(n P(t)/2). t(ω)

ω∈EAn

ω∈EAn

Given that the family {φω (Xt(ω) )}ω∈E n forms a cover of J of diameter at most A sn max{diam(Xυ ) : υ ∈ V}, by letting n → ∞ we get that Ht (J) = 0. This implies that t ≥ H for all t > h, and consequently h ≥ H. Obviously h∞ ≤ H, and hence h∞ ≤ H ≤ h. We are thus left to show that h ≤ h∞ . If F is a finite subset of E such that A : F × F → {0, 1} is irreducible, then hF = HD(JF ) according to Theorem 19.6.3(c). So hF ≤ h∞ . Moreover, PF (hF ) = 0 by (19.17). Then, by virtue of Proposition 19.4.6(c), it follows that PF (h∞ ) ≤ PF (hF ) = 0. So, in view of Theorem 17.2.9, P(h∞ ) = sup{PF (h∞ ) : F ⊆ E, #F < ∞} ≤ 0. Therefore, h ≤ h∞ . Combining this theorem with Section 19.5, we deduce the following result. Corollary 19.6.5. Let Φ be a finitely irreducible CGDMS. Then: (a) Φ is strongly regular if and only if HD(J) = h > θ. (b) Φ is critically regular or irregular if and only if HD(J) = h = θ. Remark 19.6.6. The finiteness parameter θ is an important lower bound for the Hausdorff dimension of J since it requires no iteration: θ can be expressed in terms of Z1 (t) alone.

19.6.3 Hutchinson’s formula From Bowen’s formula (more precisely, Theorem 19.6.3(c)), we deduce Hutchinson’s formula in its full generality. A special case of that formula was obtained in Section 16.3.1. Theorem 19.6.7. Let {Fe (x) = re Ae (x)+be }e∈E (with 2 ≤ #E < ∞) be an IFS of contracting similarities satisfying the open set condition. Then its limit set J has the property that HD(J) = PD(J) = BD(J) = h > 0, where h is the unique number such that ∑ reh = 1.

e∈E

This equation is called Hutchinson’s formula.

760 | 19 Conformal graph directed Markov systems

19.7 Other separation conditions and cone condition The Open Set Condition (OSC) is one of the conditions defining a CGDMS (see Definition 19.3.1). Some CGDMSs satisfy stricter separation or other geometric conditions. We describe some of these conditions as well as some of their consequences in this section. In the next section, we will demonstrate that some of these conditions guarantee the existence and uniqueness of conformal measures, measures which will underpin the analysis of the local multifractal structure of the limit sets of CGDMSs in Chapter 21.

19.7.1 The strong open set condition Definition 19.7.1. A CGDMS Φ satisfies the strong open set condition (SOSC) if J ∩ Int(X) ≠ 0, i. e., if ⋃υ∈V (Jυ ∩ Int(Xυ )) ≠ 0. All finite-alphabet CGDMSs satisfy the SOSC (see [116]). However, there are infinite-alphabet CGDMSs which do not fulfill the SOSC (see [129]). The SOSC has several interesting and important consequences. First, it implies that the set of infinite A-admissible words that generate points on the boundary of the real space is negligible under any ergodic shift-invariant measure that has full topological support. Theorem 19.7.2. Let Φ = {φe }e∈E be a CGDMS satisfying the SOSC. For any ergodic σ-invariant Borel probability measure μ on EA∞ such that supp(μ) = EA∞ , the following statements hold: −n −1 (a) μ ∘ π −1 (𝜕X) = μ(⋃∞ n=0 σ (π (𝜕X))) = 0. (b) μ ∘ π −1 (φω (𝜕Xt(ω) )) = 0 for all ω ∈ EA∗ . (c) If ω, τ ∈ EA∗ are incomparable words, then μ ∘ π −1 (φω (Xt(ω) ) ∩ φτ (Xt(τ) )) = 0. Proof. (a) Since Φ fulfills the SOSC, there exists ω ∈ EA∞ such that π(ω) ∈ Int(Xi(ω1 ) ). Then there is n ∈ ℕ such that φω|n (Xt(ωn ) ) ⊆ Int(Xi(ω1 ) ). As π([ω|n ]) ⊆ φω|n (Xt(ωn ) ) and supp(μ) = EA∞ , we deduce that μ ∘ π −1 (Int(X)) ≥ μ ∘ π −1 (Int(Xi(ω1 ) )) ≥ μ ∘ π −1 (φω|n (Xt(ωn ) )) ≥ μ([ω|n ]) > 0. Therefore, μ ∘ π −1 (𝜕X) < 1. It follows from Lemma 19.3.13 and the ergodicity of μ (see −n −1 Exercise 8.5.36) that μ ∘ π −1 (𝜕X) = μ(⋃∞ n=0 σ (π (𝜕X))) = 0. (b) Let ω ∈ EA∗ . It ensues from part (a), Lemma 19.3.14 and the σ-invariance of μ that 0 = μ ∘ π −1 (𝜕X) ≥ μ(σ |ω| (π −1 (φω (𝜕Xt(ω) )))) ≥ μ(π −1 (φω (𝜕Xt(ω) ))).

19.7 Other separation conditions and cone condition

| 761

(c) If ω, τ ∈ EA∗ are incomparable words, then ω = γaα and τ = γbβ for some γ, α, β ∈ EA∗ and a, b ∈ E with a ≠ b. Using the OSC, we obtain that φω (Xt(ω) ) ∩ φτ (Xt(τ) ) ⊆ φγ (φa (Xt(a) ) ∩ φb (Xt(b) ))

= φγ (φa (𝜕Xt(a) ) ∩ φb (𝜕Xt(b) )) ⊆ φγa (𝜕Xt(a) ).

So, by (b), μ ∘ π −1 (φω (Xt(ω) ) ∩ φτ (Xt(τ) )) ≤ μ ∘ π −1 (φγa (𝜕Xt(a) )) = 0. According to part (a) of the above result, the completely σ-invariant set 󵄨 󵄨 {ω ∈ EA∞ 󵄨󵄨󵄨 π(𝒪+ (ω)) ∩ 𝜕X ≠ 0} = {ω ∈ EA∞ 󵄨󵄨󵄨 ∃N ≥ 0 with π(σ N (ω)) ∈ 𝜕Xi(σN (ω)) } 󵄨 = {ω ∈ EA∞ 󵄨󵄨󵄨 ∃N ≥ 0 with π(σ n (ω)) ∈ 𝜕Xi(σn (ω)) , ∀n ≥ N} is negligible under any ergodic shift-invariant measure that has full topological support. There is an even bigger similar set with this property, namely 󵄨󵄨 {ω ∈ EA∞ 󵄨󵄨󵄨 lim dist(π(σ n (ω)), 𝜕Xi(σn (ω)) ) = 0}. 󵄨 n→∞ In Chapter 21, we will develop a multifractal analysis on the complement of this set, which can be described by 󵄨󵄨 E0∞ := {ω ∈ EA∞ 󵄨󵄨󵄨 lim sup dist(π(σ n (ω)), 𝜕Xi(σn (ω)) ) > 0}. 󵄨 n→∞

(19.23)

Clearly, this set is completely σ-invariant. We now prove that it is of full measure under any ergodic shift-invariant measure with full topological support. Theorem 19.7.3. Let Φ = {φe }e∈E be a CGDMS satisfying the SOSC. For every ergodic σ-invariant Borel probability measure μ on EA∞ such that supp(μ) = EA∞ , we have μ(E0∞ ) = 1. Proof. Since E0∞ is completely σ-invariant, the ergodicity of μ implies that μ(E0∞ ) is 0 or 1. We shall now show that this latter possibility always prevails. Since Φ satisfies the SOSC, there exists x ∈ J ∩ Int(X). Let ω ∈ EA∞ be such that π(ω) = x. Let also 0 < r < dist(x, 𝜕X). Recall that for any τ ∈ EA∗ , we have π([τ]) ⊆ φτ (Xt(τ) ). Since φω|k (Xt(ωk ) ) ⊆ B(π(ω), r) for all k ∈ ℕ large enough, we obtain that [ω|k ] ⊆ π −1 (B(x, r)) for some k ∈ ℕ. Then μ(π −1 (B(x, r))) ≥ μ([ω|k ]) > 0 since supp(μ) = EA∞ . It follows from Corollary 8.2.15 to Birkhoff’s ergodic theorem (Theorem 8.2.11) that the set of infinite A-admissible words whose iterates’ projections visit infinitely many times the ball B(x, r) has measure 1, i. e., 󵄨 μ({ρ ∈ EA∞ 󵄨󵄨󵄨 π(σ n (ρ)) ∈ B(x, r) for infinitely many n}) = 1.

762 | 19 Conformal graph directed Markov systems Thus, μ(E0∞ ) = 1. (Note: Once we know that μ(E0∞ ) is 0 or 1 and μ(π −1 (B(x, r))) > 0, the conclusion can also be drawn by means of Poincaré’s recurrence theorem (Theorem 8.1.2).) Under the OSC or even under the SOSC, the limit set J of a CGDMS Φ is generally only an analytic set in the sense of descriptive set theory (see Theorem 19.1.1 and [120]). If the alphabet E is finite, it is considerably more: J is compact. To avoid problems with boundary intersections (and in particular with nonunique coding), we sometimes consider the Borel subsets ̃J := J \ ⋃ φω (𝜕Xt(ω) ) ω∈EA∗

and

−1 ̃ ∞ ̃ E A := π (J).

∞ ∞ ̃ ̃ ̃ ̃ Observe that σ(E ∞ . Notice also that for every x ∈ J there A ) ⊆ EA . So let σ := σ|Ẽ A ∞ . Accordingly, ̃ exists a unique ω(x) ∈ E ∞ such that x = π(ω(x)). Obviously, ω(x) ∈ E A

A

∞ ̃ ̃ ̃ : E ̃ −1 let π̃ := π|Ẽ ∞ . Then π A → J is a continuous bijection, with π (x) = ω(x). The A ∞ →E ∞ induces a map ̃ ̃ ̃ restricted shift map σ̃ : E f : ̃J → ̃J via the formula A

A

̃f (x) := π̃ ∘ σ̃ ∘ π̃ −1 (x) = φ−1 (x), ω(x)1 so that the diagram σ̃ ∞ ̃ E A 󳨀󳨀󳨀󳨀󳨀󳨀→ ↑ ↑ ̃↑ π ↑ ↓ ̃J 󳨀󳨀󳨀󳨀󳨀󳨀→ ̃f

∞ ̃ E A ↑ ↑ ↑ ↑π̃ ↓ ̃J

̃ := μ|Ẽ commutes. Let μ ∞ . If (σ, μ) is a measure-preserving dynamical system, then A ̃ ̃ ̃ ∘ π̃ −1 ) is so is the system (σ , μ). By definition of ̃f , it ensues immediately that (̃f , μ also measure-preserving. This discussion and Theorem 19.7.2 lead to the following result. Theorem 19.7.4. Let Φ = {φe }e∈E be a CGDMS satisfying the SOSC. For any ergodic σ-invariant Borel probability measure μ on EA∞ such that supp(μ) = EA∞ , it turns out that μ ∘ π −1 (̃J) = 1 and the measure-preserving dynamical systems (σ, μ) and (̃f , μ ∘ π −1 ) are isomorphic. −1 ̃ ∞ ̃ Proof. We have already observed that the set E A := π (J) is forward σ-invariant. As ∞ ̃ μ is σ-invariant and σ-ergodic, it ensues that μ(EA ) ∈ {0, 1}. Theorem 19.7.2(a) reveals ∞ ̃ that μ(E A ) = 1. The rest of the statement was proved in the discussion preceding this lemma.

19.7 Other separation conditions and cone condition

| 763

19.7.2 The boundary separation condition A condition considerably stricter than the SOSC is the following. Definition 19.7.5. A CGDMS Φ is said to satisfy the Boundary Separation Condition (BSC) if min dist(𝜕Xυ , ⋃ φe (Xt(e) )) > 0. υ∈V

e∈E: i(e)=υ

In other words, a CGDMS satisfies the BSC if its first-level sets stay uniformly away from the boundary of the phase space. 19.7.3 The strong separation condition Definition 19.7.6. A CGDMS Φ is said to satisfy the Strong Separation Condition (SSC) if φe1 (Xt(e1 ) ) ∩ φe2 (Xt(e2 ) ) = 0,

∀e1 , e2 ∈ E, e1 ≠ e2 .

Clearly, the SSC is a stricter requirement than the OSC. It does not necessarily imply the SOSC, though. The BSC and the SSC are related in the following way. Lemma 19.7.7. If a CGDMS Φ = {φe : Xt(e) → Xi(e) }e∈E satisfies the BSC, then the CGDMS ′ ′ Φ′ = {φe : Xt(e) → Xi(e) }e∈E satisfies the SSC, where Xυ′ = Xυ \ B(𝜕Xυ , δ)

for any 0 < δ < min dist(𝜕Xυ , ⋃ φe (Xt(e) )). υ∈V

e∈E: i(e)=υ

The CGDMSs Φ and Φ′ have the same coding map π : EA∞ → X, and hence the same limit set J. Loosely speaking, we can thus say that a CGDMS that satisfies the BSC also satisfies the SSC. Proof. The proof is left to the reader as an exercise. It is not difficult to observe the following. Theorem 19.7.8. If a CGDMS satisfies the BSC or the SSC, then the coding map π : EA∞ → J is one-to-one. In particular, if E is finite (so EA∞ is compact), then the coding map π is a homeomorphism. In this case, the limit set J, as a compact, perfect, totally disconnected, metrizable set, is a topological Cantor set, i. e., it is homeomorphic to the middle-third Cantor set. Proof. The proof is left to the reader as an exercise.

764 | 19 Conformal graph directed Markov systems 19.7.4 The cone condition Given a point x ∈ ℝd , an angle α ∈ (0, π/2) and a vector u ∈ ℝd , let Con(x, α, u) = {y ∈ ℝd : ∠(y − x, u) ≤ α and ‖y − x‖ ≤ ‖u‖}

= {y ∈ ℝd : ⟨y − x, u⟩ ≥ ‖y − x‖ ⋅ ‖u‖ cos α and ‖y − x‖ ≤ ‖u‖},

where ⟨υ, u⟩ := ‖υ‖ ⋅ ‖u‖ cos ∠(υ, u) is the usual inner product in ℝd . The set Con(x, α, u) is called the (closed) cone with vertex x, central angle α and direction vector u. We aim to prove a geometrical fact about the images of cones under conformal maps, like we did for balls in Lemma 19.3.7. For this, we need two intermediate results. First, as an immediate consequence of Theorem 19.2.2 we obtain the following result. Theorem 19.7.9. If x ∈ ℂ, R > 0 and φ : B(x, R) → ℂ is a univalent holomorphic function, then 󵄨󵄨 󵄨 K 󵄨 ′ 󵄨 2 ′ 󵄨󵄨φ(y) − φ(x) − φ (x)(y − x)󵄨󵄨󵄨 ≤ 2 󵄨󵄨󵄨φ (x)󵄨󵄨󵄨 ⋅ |y − x| , R

∀y ∈ B(x, δ2 R),

where δ2 = min{γ, π/24}. Proof. In view of Theorem 19.2.2, we have 󵄨󵄨 y 󵄨󵄨 󵄨󵄨 K2 󵄨 ′ 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 ′ ′ ′ 󵄨 󵄨 󵄨󵄨 ≤ φ(y) − φ(x) − φ (x)(y − x) = (φ (z) − φ (x)) dz ∫ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 R 󵄨󵄨φ (x)󵄨󵄨 ⋅ |y − x| ⋅ |y − x|. 󵄨󵄨󵄨 x 󵄨󵄨 A similar result holds in higher-dimensional Euclidean spaces. Theorem 19.7.10. Suppose that Y is a bounded subset of ℝd , d ≥ 3, and that W is an open subset of ℝd containing Y. Then there exists a constant K4 ≥ 1 such that if φ : ℝd → ℝd is a conformal C 1 diffeomorphism such that φ(W) ⊆ ℝd , then for all x ∈ Y and all y ∈ B(Y, 21 dist(Y, 𝜕W)) we have 󵄩󵄩 󵄩 󵄨 ′ 󵄨 ′ 2 󵄩󵄩φ(y) − φ(x) − φ (x)(y − x)󵄩󵄩󵄩 ≤ K4 󵄨󵄨󵄨φ (x)󵄨󵄨󵄨 ⋅ ‖y − x‖ . Proof. Per Liouville’s Theorem (Theorem 19.2.1), write φ = λA ∘ i + b. If i is the identity map, then the left-hand side of the claimed inequality is equal to 0 and we are done. If i = ia , then setting ∇φ := φ(y) − φ(x) − φ′ (x)(y − x) and using (19.4) we get ∇φ = λA ∘ i(y) − λA ∘ i(x) − λA ∘ i′ (x)(y − x) = λA(i(y) − i(x) − i′ (x)(y − x))

= λA(

x−a 1 y−a − − (Id d − 2Q(x − a))(y − x)) =: λA(Δ). (19.24) ‖y − a‖2 ‖x − a‖2 ‖x − a‖2 ℝ

19.7 Other separation conditions and cone condition

| 765

By (19.5), we have Q(x − a)u =

⟨x − a, u⟩ (x − a), ‖x − a‖2

∀u ∈ ℝd .

Hence, Δ=

y−x x−a x−a y−x ⟨x − a, y − x⟩ + − − +2 (x − a) ‖x − a‖4 ‖y − a‖2 ‖y − a‖2 ‖x − a‖2 ‖x − a‖2

=(

1 1 1 ⟨x − a, y − x⟩ 1 − )(y − x) + ( − +2 )(x − a). ‖x − a‖4 ‖y − a‖2 ‖x − a‖2 ‖y − a‖2 ‖x − a‖2

Now decompose x − a = α(y − x) + β(y − x)⊥ , where (y − x)⊥ is a vector perpendicular to y − x with ‖(y − x)⊥ ‖ = ‖y − x‖. Then ‖x − a‖2 = (α2 + β2 )‖y − x‖2 ,

‖y − a‖2 = ((α + 1)2 + β2 )‖y − x‖2

and

⟨x − a, y − x⟩ = α‖y − x‖2 .

(19.25)

Using these expressions, we get Δ=(

1 1 − )(y − x) ‖y − a‖2 ‖x − a‖2

+ α( + β( =

1 1 2α‖y − x‖2 − + )(y − x) 2 2 ‖x − a‖4 ‖y − a‖ ‖x − a‖

1 1 2α‖y − x‖2 − + )(y − x)⊥ 2 2 ‖x − a‖4 ‖y − a‖ ‖x − a‖

(1 + α)(α2 + β2 ) 1 2α2 [( − (1 + α) + 2 )(y − x) 2 2 2 ‖x − a‖ (α + 1) + β α + β2 + β(

α2 + β2 2α −1+ 2 )(y − x)⊥ ]. (α + 1)2 + β2 α + β2

(19.26)

Since φ(W) ⊆ ℝd , we know that a ∉ W. A direct calculation using (19.25) shows that 󵄨󵄨 (1 + α)(α2 + β2 ) 2α2 󵄨󵄨󵄨󵄨 󵄨󵄨 − (1 + α) + 󵄨󵄨 󵄨= 󵄨󵄨 (α + 1)2 + β2 α2 + β2 󵄨󵄨󵄨 ≤ ≤

󵄨󵄨 3 󵄨 󵄨󵄨α + α2 − 3αβ2 − β2 󵄨󵄨󵄨 (α2 + β2 )((α + 1)2 + β2 )

‖y − x‖4 (|α|α2 + α2 + 3|α|β2 + β2 ) ‖x − a‖2 ‖y − a‖2

‖y − x‖4 (3|α|(α2 + β2 ) + (α2 + β2 )) ‖x − a‖2 ‖y − a‖2

766 | 19 Conformal graph directed Markov systems

≤ = ≤ = ≤ where R =

1 2

‖x − a‖3 ‖x − a‖2 ‖y − x‖4 (3 + ) 2 2 ‖y − x‖3 ‖y − x‖2 ‖x − a‖ ‖y − a‖ ‖y − x‖ ‖x − a‖ ‖y − x‖ (3 + ) ‖y − a‖ ‖y − a‖ ‖y − a‖

‖y − x‖ ‖y − x‖ + ‖y − a‖ ‖y − x‖ (3 + ) ‖y − a‖ ‖y − a‖ ‖y − a‖ ‖y − x‖ ‖y − x‖ (4 + 3) ‖y − a‖ ‖y − a‖

diam(Y) + R 1 (4 + 3)‖y − x‖, R R

dist(Y, 𝜕W), while

󵄨󵄨 󵄨󵄨 α2 + β 2 2α 󵄨󵄨 󵄨󵄨 − 1 + ) 󵄨󵄨β( 󵄨 󵄨󵄨 (α + 1)2 + β2 α2 + β2 󵄨󵄨󵄨 |β|(3α2 + β2 + 2|α|) |β| ⋅ |3α2 − β2 + 2α| ‖y − x‖4 ≤ = 2 ‖x − a‖2 ‖y − a‖2 (α + β2 )((α + 1)2 + β2 )

‖x − a‖3 ‖x − a‖2 +2 )‖y − x‖4 3 ‖y − x‖ ‖x − − ‖y − x‖2 ‖y − x‖ ‖y − x‖ (3‖x − a‖ + 2‖y − x‖) ≤ (3‖y − a‖ + 5‖y − x‖) = 2 ‖y − a‖ ‖y − a‖2

≤

=

1

a‖2 ‖y

a‖2

(3

‖y − x‖ ‖y − x‖ 1 diam(Y) + R (3 + 5 ) ≤ (3 + 5 )‖y − x‖. ‖y − a‖ ‖y − a‖ R R

Combining the previous two estimates with (19.26), we find a constant K4 > 0 such that ‖Δ‖ ≤

1 K ‖y − x‖2 . ‖x − a‖2 4

It follows from this, along with (19.24) and (19.4), that ‖∇φ‖ ≤

λ 󵄨 󵄨 K ‖y − x‖2 = K4 󵄨󵄨󵄨φ′ (x)󵄨󵄨󵄨 ⋅ ‖y − x‖2 . ‖x − a‖2 4

We can now derive the main fact we are after, which pertains to the images of cones under conformal maps. Theorem 19.7.11. Suppose that Y is a bounded subset of ℝd , d ≥ 2, and that W is an open subset of ℝd containing Y. Then for every ε ∈ (0, π/2) there exists δ > 0 such that if x ∈ Y, α ∈ (0, π/2 − ε), ‖u‖ ≤ δ, and φ : ℝd → ℝd is a conformal C 1 diffeomorphism such that φ(W) ⊆ ℝd , then φ(Con(x, α, u)) ⊆ Con(φ(x), α + ε, 2φ′ (x)u).

19.7 Other separation conditions and cone condition

| 767

Proof. Let ε ∈ (0, π/2) and α ∈ (0, π/2 − ε). If d = 2, then φ is either holomorphic or antiholomorphic. Since complex conjugacy is an isometry, we may assume that φ is holomorphic when d = 2. In view of Theorems 19.7.9 and 19.7.10, setting K5 = max{K4 , 2K2 / dist(Y, 𝜕W)} we see that if x ∈ Y and ‖y − x‖ < δ2 dist(Y, 𝜕W), then 󵄨 ′ 󵄨 󵄩 󵄩󵄩 2 ′ 󵄩󵄩Δφ − φ (x)(y − x)󵄩󵄩󵄩 ≤ K5 󵄨󵄨󵄨φ (x)󵄨󵄨󵄨 ⋅ ‖y − x‖ ,

(19.27)

where Δφ := φ(y) − φ(x). Fix η > 0. By virtue of Corollary 19.2.4, there exists a constant 0 < δ < δ2 dist(Y, 𝜕W), independent of φ, such that (1 + η)−1 ≤

|φ′ (y)| ≤ 1 + η, |φ′ (x)|

∀x ∈ Y, ∀y ∈ B(x, δ).

(19.28)

Fix u ∈ ℝd with ‖u‖ ≤ δ, and let y ∈ Con(x, α, u). This means that ‖y − x‖ ≤ ‖u‖ ≤ δ and ⟨y − x, u⟩ ≥ ‖y − x‖ ⋅ ‖u‖ cos α. By the mean value theorem, (19.28) and (16.1), we get 󵄨 󵄨 󵄨 󵄨 󵄩 󵄩 ‖Δφ‖ ≤ (1 + η)󵄨󵄨󵄨φ′ (x)󵄨󵄨󵄨 ⋅ ‖y − x‖ ≤ (1 + η)󵄨󵄨󵄨φ′ (x)󵄨󵄨󵄨 ⋅ ‖u‖ = 󵄩󵄩󵄩(1 + η)φ′ (x)u󵄩󵄩󵄩.

(19.29)

Moreover, ⟨Δφ, (1 + η)φ′ (x)u⟩ = ⟨φ′ (x)(y − x), (1 + η)φ′ (x)u⟩

+ ⟨Δφ − φ′ (x)(y − x), (1 + η)φ′ (x)u⟩.

(19.30)

Using (16.1), (16.2) and (19.29), we obtain the following lower bound for the first term on the right-hand side of (19.30): 󵄨 󵄨2 ⟨φ′ (x)(y − x), (1 + η)φ′ (x)u⟩ = (1 + η)󵄨󵄨󵄨φ′ (x)󵄨󵄨󵄨 ⟨y − x, u⟩ 󵄨 󵄨2 ≥ (1 + η)󵄨󵄨󵄨φ′ (x)󵄨󵄨󵄨 ‖y − x‖ ⋅ ‖u‖ cos α 󵄩 󵄩 ≥ ‖Δφ‖ ⋅ 󵄩󵄩󵄩φ′ (x)u󵄩󵄩󵄩 cos α.

(19.31)

Concerning the second term, it follows from (19.27) that 󵄨󵄨 󵄨 󵄩 󵄩 󵄩 󵄩 ′ ′ ′ ′ 󵄨󵄨⟨Δφ − φ (x)(y − x), (1 + η)φ (x)u⟩󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩Δφ − φ (x)(y − x)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩(1 + η)φ (x)u󵄩󵄩󵄩 󵄨 󵄨 󵄩 󵄩 ≤ (1 + η)K5 󵄨󵄨󵄨φ′ (x)󵄨󵄨󵄨 ⋅ ‖y − x‖2 ⋅ 󵄩󵄩󵄩φ′ (x)u󵄩󵄩󵄩. Assuming δ > 0 to be smaller than (2K5 )−1 , from (19.27) we have 󵄨 󵄨 󵄨 󵄨 ‖Δφ‖ ≥ 󵄨󵄨󵄨φ′ (x)󵄨󵄨󵄨 ⋅ ‖y − x‖ − K5 󵄨󵄨󵄨φ′ (x)󵄨󵄨󵄨 ⋅ ‖y − x‖2 󵄨 󵄨 󵄨 󵄨 ≥ 󵄨󵄨󵄨φ′ (x)󵄨󵄨󵄨 ⋅ ‖y − x‖ − K5 󵄨󵄨󵄨φ′ (x)󵄨󵄨󵄨 ⋅ ‖y − x‖(2K5 )−1 1󵄨 󵄨 = 󵄨󵄨󵄨φ′ (x)󵄨󵄨󵄨 ⋅ ‖y − x‖. 2

(19.32)

768 | 19 Conformal graph directed Markov systems Consequently, we can continue (19.32) as follows and obtain an upper estimate of the absolute value of the second term on the right-hand side of (19.30): 󵄩 󵄩 ′ 󵄨 󵄨󵄨 ′ ′ 󵄨󵄨⟨Δφ − φ (x)(y − x), (1 + η)φ (x)u⟩󵄨󵄨󵄨 ≤ (1 + η)K5 ⋅ 2‖Δφ‖ ⋅ ‖y − x‖ ⋅ 󵄩󵄩󵄩φ (x)u󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 ′ (19.33) ≤ 2(1 + η)K5 ‖Δφ‖ ⋅ δ󵄩󵄩φ (x)u󵄩󵄩. Combining (19.31) with (19.33), we obtain a lower estimate of the left-hand side of (19.30): 󵄩 󵄩 ⟨Δφ, (1 + η)φ′ (x)u⟩ ≥ (cos α − 2(1 + η)K5 δ)󵄩󵄩󵄩φ′ (x)u󵄩󵄩󵄩 ⋅ ‖Δφ‖ 󵄩󵄩 󵄩 −1 = ((1 + η) cos α − 2K5 δ) ⋅ 󵄩󵄩(1 + η)φ′ (x)u󵄩󵄩󵄩 ⋅ ‖Δφ‖. Therefore, cos ∠(Δφ, (1 + η)φ′ (x)u) ≥ (1 + η)−1 cos α − 2K5 δ. Taking 0 < η ≤ 1 and δ > 0 so small that (1 + η)−1 cos α − 2K5 δ ≥ cos(α + ε), the above inequality implies that ∠(Δφ, (1 + η)φ′ (x)u) ≤ α + ε while (19.29) yields that ‖Δφ‖ ≤ ‖(1 + η)φ′ (x)u‖. That is, φ(y) ∈ Con(φ(x), α + ε, (1 + η)φ′ (x)u). Hence, φ(Con(x, α, u)) ⊆ Con(φ(x), α + ε, (1 + η)φ′ (x)u) ⊆ Con(φ(x), α + ε, 2φ′ (x)u). Recall from Section 19.1.1 that 󵄨 E𝒢∗ := {ω ∈ E ∗ 󵄨󵄨󵄨 t(ωj ) = i(ωj+1 ), ∀1 ≤ j < |ω|} is the set of all finite words that are permissible based solely on the underlying graph 𝒢 , i. e., making abstraction of the matrix A. Obviously, E𝒢∗ ⊇ EA∗ . The next proposition follows immediately from Theorem 19.7.11 when d ≥ 2 and is a direct consequence of the Bounded Distortion Property (BDP) when d = 1 (see Corollary 19.3.5). Proposition 19.7.12. Let Φ = {φe }e∈E be a CGDMS. For every ε > 0, there exists δ > 0 such that if ω ∈ E𝒢∗ , x ∈ Xt(ω) , ‖u‖ ≤ δ and α ∈ (0, π/2 − ε), then φω (Con(x, α, u)) ⊆ Con(φω (x), α + ε, 2φ′ω (x)u). In the next section, we will see that the following purely geometric condition ensures the existence and uniqueness of a conformal measure. Definition 19.7.13. A CGDMS Φ is said to satisfy the Cone Condition (CC) if there exist ψ, ℓ > 0 such that for every υ ∈ V and every x ∈ Xυ there is a cone with vertex x, central angle ψ, altitude ℓ and unit direction vector ux such that Con(x, ψ, ℓux )\{x} ⊆ Int(Xυ ). This condition has the following important geometrical repercussion.

19.7 Other separation conditions and cone condition

| 769

Theorem 19.7.14. Let Φ be a CGDMS satisfying the cone condition. There exists β > 0 such that if D ≥ 1 is sufficiently large, then for all ω ∈ E𝒢∗ and x ∈ Xt(ω) we have φω (Xt(ω) ) ⊇ Con(φω (x), β, D−1 φ′ω (x)ux ) ⊇ Con(φω (x), β, D−2 diam(φω (Xt(ω) ))ux ), where ux is a unit direction vector from the cone condition and ux =

(19.34)

φ′ω (x) u . |φ′ω (x)| x

Proof. When d = 1, the first inclusion is a consequence of Lemma 19.3.10, which clearly applies to all words in E𝒢∗ . So we may assume that d ≥ 2 when proving this inclusion. Let 0 < ψ < π/2 be an allowable central angle from the cone condition. Then there is q ∈ ℕ such that for every x ∈ X there exist unit vectors u1 (x), . . . , uq (x) ∈ ℝd so that all q-tuples (ux , u1 (x), . . . , uq (x)), x ∈ X, are linearly isometric to one another and q

Con(x, ψ, ℝ+ ux ) ∪ ⋃ Con(x, ψ, ℝ+ uj (x)) = ℝd j=1

(19.35)

while q

Con(x, ψ/2, ℝ+ ux ) ∩ ⋃ Con(x, ψ, ℝ+ uj (x)) = 0. j=1

(19.36)

Then there exists ε > 0 such that q

̃ j (x)) = 0 ̃ x ) ∩ ⋃ Con(x, ψ + ε, ℝ+ u Con(x, ψ/4, ℝ+ u j=1

(19.37)

̃x , u ̃ 1 (x), . . . , u ̃ q (x)) that are equivalent, via a similarity for all x ∈ X and all q-tuples (u map, to (ux , u1 (x), . . . , uq (x)). Let ℓ > 0 be an allowable altitude according to the cone condition and let 0 < δ < min{ℓ, dist(X, 𝜕W)} be ascribed to the above ε by virtue of Proposition 19.7.12. In view of that proposition, relation (19.37) and the fact that φ′ω (x) is a similarity map, we get q

Con(φω (x), ψ/4, ℝ+ φ′ω (x)ux ) ∩ ⋃ φω (Con(x, ψ, δuj (x))) = 0 j=1

(19.38)

for all ω ∈ E𝒢∗ and x ∈ Xt(ω) . In view of (19.35) and Lemma 19.3.10, which applies to all words in E𝒢∗ , we find that q

󵄩 󵄩 φω (Con(x, ψ, δux )) ∪ ⋃ φω (Con(x, ψ, δuj (x))) ⊇ φω (B(x, δ)) ⊇ B(φω (x), K −1 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩W δ). t(ω) j=1

770 | 19 Conformal graph directed Markov systems This and (19.38) imply that 󵄩 󵄩 󵄨 󵄨−1 φω (Con(x, ψ, δux )) ⊇ Con(φω (x), ψ/4, K −1 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩W δ󵄨󵄨󵄨φ′ω (x)󵄨󵄨󵄨 φ′ω (x)ux ) t(ω) ⊇ Con(φω (x), ψ/4, K −1 δφ′ω (x)ux ).

Taking any D ≥ Kδ−1 and β = ψ/4 achieves the first inclusion in (19.34). The second inclusion holds if D is chosen large enough in (19.9) of Lemma 19.3.9, which applies to all words in E𝒢∗ . It ensues from this theorem that each point x of the space belongs to a finite number of n-th level sets, and that number is actually uniformly bounded in n and x. Corollary 19.7.15. If Φ is a CGDMS satisfying the cone condition, then sup sup #{ω ∈ E𝒢n : x ∈ φω (Xt(ω) )} ≤ n∈ℕ x∈X

λd−1 (Sd−1 ) < ∞, β

where Sd−1 is the (d − 1)-dimensional unit sphere and λd−1 is the (d − 1)-dimensional Lebesgue measure on Sd−1 , while β arises from Theorem 19.7.14. Proof. Let x ∈ X and n ∈ ℕ. By Theorem 19.7.14, any set φω (Xt(ω) ) ∋ x contains a cone with vertex x and central angle β. But by the OSC, the sets φτ (Xt(τ) ), τ ∈ E𝒢n , are nonoverlapping beyond their boundaries. Therefore the sets φω (Xt(ω) ), with ω ∈ E𝒢n and x ∈ φω (Xt(ω) ), contain mutually disjoint open cones with vertex x and central angle β. There are at most λd−1 (Sd−1 )/β such cones. We next establish a result similar to Lemma 19.3.12. Lemma 19.7.16. If Φ is a CGDMS satisfying the cone condition, then for every κ > 0, every r > 0, and every x ∈ X, the cardinality of any collection of mutually incomparable words ω ∈ E𝒢∗ that satisfy the conditions φω (Xt(ω) ) ∩ B(x, r) ≠ 0

and

κr ≤ diam(φω (Xt(ω) ))

is bounded above by the number d

Vd κ −d D2d β−1 (1 + κD−2 ) , where Vd is the volume of a unit ball in ℝd while D and β originate from Theorem 19.7.14. Proof. Fix κ > 0, r > 0 and x ∈ X, and let W be a collection as described in the statement of the lemma. For each ω ∈ W, let xω ∈ Xt(ω) be such that φω (xω ) ∈ φω (Xt(ω) ) ∩ B(x, r). It follows from Theorem 19.7.14 that φω (Xt(ω) ) ⊇ Con(φω (xω ), β, D−2 κruxω ) ⊆ B(x, (1 + D−2 κ)r).

19.7 Other separation conditions and cone condition

| 771

Due to the OSC (see Definition 19.3.1), the cones Con(φω (xω ), β, D−2 κruxω ), ω ∈ W, intersect at most on their boundaries. Therefore, d

λd (B(x, (1 + D−2 κ)r)) ≥ ∑ λd (Con(φω (xω ), β, D−2 κruxω )) = #W ⋅ β(D−2 κr) . ω∈W

So #W ≤ β−1 (D−2 κr)−d λd (B(x, (1 + D−2 κ)r)) = β−1 D2d (κr)−d Vd (1 + D−2 κ)d r d . 19.7.5 Conformal-likeness Definition 19.7.17. Given q ∈ ℕ, we say that two distinct words ω, τ ∈ E𝒢∗ of equal length n > q form a pair of q-codes for a point x ∈ X if x ∈ φω (Xt(ω) ) ∩ φτ (Xt(τ) )

and

ω|n−q = τ|n−q .

A CGDMS Φ is said to be conformal-like if for every q ∈ ℕ there is no point in J with arbitrarily long pairs of q-codes. We now make a simple observation. Lemma 19.7.18. Each conformal-like CGDMS is pointwise finite, and hence its associated coding map π : EA∞ → J is bounded-to-one and its limit set J is a Fσδ –set. Proof. The reader can convince themself of this by looking at Definitions 19.7.17 and 19.1.2 as well as Lemma 19.1.3. Later on, we will observe that conformal-likeness implies the existence and uniqueness of a conformal measure. However, conformal-likeness can be challenging to demonstrate directly. We thus identify sufficient conditions for it to hold. Proposition 19.7.19. Every CGDMS satisfying the Boundary Separation Condition (BSC), the Strong Separation Condition (SSC) or the Cone Condition (CC) is conformal-like. Proof. Suppose that a CGDMS Φ satisfying the CC is not conformal-like. This means that there exist a point x ∈ X, an integer q ∈ ℕ and a strictly increasing sequence (k) (nk )∞ ≠ τ(k) ∈ E𝒢∗ of length nk such that k=1 along with words ω x ∈ φω(k) (Xt(ω(k) ) ) ∩ φτ(k) (Xt(τ(k) ) )

and

ω(k) |nk −q = τ(k) |nk −q .

Passing to a subsequence if necessary, we may assume that nk+1 − nk ≥ q for every k ∈ ℕ. We shall construct by induction on k a sequence (Ck )∞ k=1 such that Ck consists of at least k + 1 incomparable words from {ω(j) , τ(j) : j ≤ k}. In particular, note that this will imply that |Ck | := max{|ρ| : ρ ∈ Ck } ≤ nk . In fact, we will see that |Ck | = nk . Set C1 = {ω(1) , τ(1) }. Suppose now that Ck has been defined. If ω(k+1) is comparable to a word ρ ∈ Ck , then ρ is unique. Indeed, if ω(k+1) is comparable to ρ1 , ρ2 ∈ Ck , then as

772 | 19 Conformal graph directed Markov systems |ω(k+1) | = nk+1 ≥ nk = |Ck | ≥ max{|ρ1 |, |ρ2 |} and as the words in Ck are incomparable, we deduce that ρ1 = ρ2 . This means that ω(k+1) (and similarly τ(k+1) ) is comparable to at most one word in Ck . If ω(k+1) is not comparable to any word in Ck , then we form Ck+1 by simply adjoining ω(k+1) to Ck . We can do a similar addition if τ(k+1) is not comparable to any word in Ck . If ω(k+1) and τ(k+1) are respectively comparable to words ρ and η in Ck , then in fact ρ and η are both extended by ω(k+1) since ω(k+1) |nk+1 −q = τ(k+1) |nk+1 −q and nk+1 − q ≥ nk = |Ck | ≥ max{|ρ|, |η|}. As the words in Ck are incomparable, we deduce that ρ = η. In this case, we form Ck+1 from Ck by taking away ρ and adding both ω(k+1) and τ(k+1) . This completes the inductive proof of the existence of the sequence (Ck )∞ k=1 . But Ck consists of at least k + 1 incomparable words ρ such that x ∈ φρ (Xt(ρ) ). This contradicts Corollary 19.7.15. The proof of the conformal-likeness of any CGDMS Φ satisfying the BSC or the SSC is left to the reader as an exercise. As an immediate consequence of Proposition 19.7.19 and Lemma 19.7.18, we obtain the following. Corollary 19.7.20. Every CGDMS satisfying the BSC, the SSC or the CC is pointwise finite and its limit set is an Fσδ –set. In Theorem 19.7.2(c), we saw that for every CGDMS satisfying the SOSC the intersection of any two level sets generated by incomparable words is negligible under any ergodic shift-invariant measure that has full topological support. For conformal-like CGDMSs, this is also true even when the measure is not ergodic or does not have full topological support. Theorem 19.7.21. Let Φ = {φe }e∈E be a conformal-like CGDMS. For any σ-invariant Borel probability measure μ on EA∞ , we have μ ∘ π −1 (φω (Xt(ω) ) ∩ φτ (Xt(τ) )) = 0 whenever ω, τ ∈ EA∗ are incomparable words. Proof. Suppose on the contrary that there are some incomparable words ω, τ ∈ EA∗ such that μ ∘ π −1 (φω (Xt(ω) ) ∩ φτ (Xt(τ) )) > 0. Then i(ω) = i(τ) =: υ ∈ V. We may assume without loss of generality that ω and τ are of the same length, say q ∈ ℕ. Let B = φω (Xt(ω) ) ∩ φτ (Xt(τ) ) and for every n ∈ ℕ let Bn =

⋃

α∈EAn : t(α)=υ

φα (B).

Since each element of Bn admits at least two q-codes of length n + q (namely, αω and ατ), we deduce from the conformal-likeness of Φ (see Definition 19.7.17) that ∞ ∞

J ∩ ⋂ ⋃ Bn = 0. k=1 n=k

(19.39)

19.8 Hölder families of functions and conformal measures | 773

On the other hand, π −1 (Bn ) ⊇ σ −n (π −1 (B)) for all n ∈ ℕ. Therefore, μ(π −1 (Bn )) ≥ μ(σ −n (π −1 (B))) = μ(π −1 (B)) for all n ∈ ℕ, and hence ∞ ∞

∞

μ ∘ π −1 (J ∩ ⋂ ⋃ Bn ) = lim μ ∘ π −1 ( ⋃ Bn ) ≥ lim inf μ(π −1 (Bk )) ≥ μ(π −1 (B)) > 0. k=1 n=k

k→∞

n=k

k→∞

∞ In particular, J ∩ ⋂∞ k=1 ⋃n=k Bn ≠ 0. This contradicts (19.39).

19.8 Hölder families of functions and conformal measures We now turn our attention to families of functions on the real space. By convention, such families shall be denoted by uppercase letters, while their members and amalgamation will be denoted by the corresponding lowercase letters. 19.8.1 Basic definitions and properties All of the notions that we introduce in this section for families of functions on the real space are counterparts to those that we studied earlier for potentials on the symbolic space. In fact, each family of functions on the real space generates a special potential on the symbolic space. Definition 19.8.1. Let Φ = {φe : Xt(e) → Xi(e) }e∈E be a CGDMS and let F = {fe : Xt(e) → ℝ}e∈E be a family of real-valued functions. The potential f : EA∞ → ℝ defined by f (ω) := fω1 (π(σ(ω))) is called the amalgamated function induced by the family F. Given ω ∈ EA∗ , define the function Sω F : Xt(ω) → ℝ by declaring |ω|

Sω F := ∑ fωj ∘ φσj (ω) . j=1

By convention, the empty word ϵ is the only word of length 0 and φϵ = IdX . Recall that EA∞,ω = EA

∞,ω|ω|

= {ρ ∈ EA∞ : ωρ ∈ EA∞ }

is the set of all infinite A-admissible suffixes of ω. The values of Sω F and S|ω| f are related as follows.

774 | 19 Conformal graph directed Markov systems Lemma 19.8.2. Let Φ = {φe : Xt(e) → Xi(e) }e∈E be a CGDMS and let f be the amalgamated function induced by a family of functions F = {fe : Xt(e) → ℝ}e∈E . For every ω ∈ EA∗ and every ρ ∈ EA∞,ω , we have Sω F(π(ρ)) = S|ω| f (ωρ). Proof. Let ω ∈ EA∗ , let 1 ≤ j ≤ |ω| and ρ ∈ EA∞,ω . By (19.1), φσj (ω) (π(ρ)) = φσj (ω) (π(σ |ω|−j (σ j (ωρ)))) = π(σ j (ωρ)). Hence, |ω|

|ω|

j=1

j=1

Sω F(π(ρ)) = ∑ fωj ∘ φσj (ω) (π(ρ)) = ∑ fωj (π(σ(σ j−1 (ωρ)))) |ω|

|ω|−1

j=1

j=0

= ∑ f (σ j−1 (ωρ)) = ∑ f (σ j (ωρ)) = S|ω| f (ωρ). Definition 19.8.3. Let Φ = {φe : Xt(e) → Xi(e) }e∈E be a CGDMS. A family of functions F = {fe : Xt(e) → ℝ}e∈E is called summable if 󵄩 󵄩 ∑ 󵄩󵄩󵄩exp(fe )󵄩󵄩󵄩X

t(e)

e∈E

< ∞.

The following fact is obvious (cf. Definition 17.6.1). Lemma 19.8.4. Let Φ = {φe : Xt(e) → Xi(e) }e∈E be a CGDMS. A family of functions F = {fe : Xt(e) → ℝ}e∈E is summable if and only if its amalgamated function f is summable. Now, for every β > 0 and n ∈ ℕ let 󵄨 󵄨 υβ,n (F) := sup sup 󵄨󵄨󵄨fω1 (φσ(ω) (x)) − fω1 (φσ(ω) (y))󵄨󵄨󵄨eβn . n ω∈EA x,y∈Xt(ω)

Thus, υβ,1 (F) < ∞ simply means that the diameters of the sets fe (Xt(e) ), e ∈ E, are uniformly bounded. Definition 19.8.5. Let Φ = {φe : Xt(e) → Xi(e) }e∈E be a CGDMS. A family of functions F = {fe : Xt(e) → ℝ}e∈E is said to be Hölder with exponent β if υβ (F) := sup υβ,n (F) < ∞. n∈ℕ

The following fact is fairly straightforward to establish and is left to the reader (cf. Definition 17.2.2).

19.8 Hölder families of functions and conformal measures | 775

Lemma 19.8.6. Let Φ = {φe }e∈E be a CGDMS. If F = {fe }e∈E is a Hölder family with exponent β, then its amalgamated function f is Hölder continuous on cylinders with exponent β. Hölder families obey to a bounded variation principle (cf. Lemma 17.2.3). Lemma 19.8.7 (Bounded variation principle). Let Φ = {φe }e∈E be a CGDMS and F = {fe }e∈E a Hölder family of functions with exponent β. If ω ∈ EA∗ and x, y ∈ Xt(ω) ∩ φτ (Xt(τ) ) for some τ ∈ EA∗ , then υβ (F) −β|τ| 󵄨󵄨 󵄨 e . 󵄨󵄨Sω F(x) − Sω F(y)󵄨󵄨󵄨 ≤ 1 − e−β Proof. There exist x̃, ỹ ∈ Xt(τ) such that x = φτ (x̃) and y = φτ (ỹ). Then 󵄨󵄨 |ω| 󵄨󵄨 |ω| 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 ̃ ̃ 󵄨󵄨Sω F(x) − Sω F(y)󵄨󵄨 = 󵄨󵄨∑ fωj ∘ φσj (ω) (φτ (x)) − ∑ fωj ∘ φσj (ω) (φτ (y))󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 j=1 󵄨j=1 󵄨 |ω|

󵄨 󵄨 ≤ ∑󵄨󵄨󵄨f(ωτ)j ∘ φσj (ωτ) (x̃) − f(ωτ)j ∘ φσj (ωτ) (ỹ)󵄨󵄨󵄨 j=1 |ω|

≤ ∑ υβ (F)e−β(|ω|+|τ|−j) ≤ j=1

υβ (F)

1 − e−β

e−β|τ| .

Not surprisingly, the pressure of a family of functions is defined as follows. Definition 19.8.8. Let Φ = {φe }e∈E be a CGDMS. The topological pressure P(F) of a summable family of functions F = {fe }e∈E is defined by P(F) := lim

n→∞

1 󵄩 󵄩 log ∑ 󵄩󵄩󵄩exp(Sω F)󵄩󵄩󵄩X . t(ω) n n ω∈E A

The limit exists because the sequence (∑ω∈E n ‖ exp(Sω F)‖Xt(ω) )∞ n=1 is submultiplicaA tive. Moreover, P(F) ≤ ∑e∈E ‖ exp(fe )‖Xt(e) < ∞. The next result asserts that the topological pressure of a family of functions coincides with the topological pressure of the amalgamated function it induces. Proposition 19.8.9. Let Φ = {φe }e∈E be a CGDMS. If F = {fe }e∈E is a summable Hölder family of functions and f is the amalgamated function induced by F, then P(F) = P(f ). Proof. Let ω ∈ EA∗ . By taking the supremum over all ρ ∈ EA∞,ω in Lemma 19.8.2, we get 󵄩󵄩 󵄩 ∞,ω 󵄩󵄩exp(Sω F)󵄩󵄩󵄩Xt(ω) ≥ exp(sup{S|ω| f (ωρ) : ρ ∈ EA }) = exp(S|ω| f ([ω])). Consequently, P(F) ≥ P(f ) (cf. Definitions 19.8.8 and 17.2.7).

776 | 19 Conformal graph directed Markov systems υ (F)

β On the other hand, let ρ ∈ EA∞,ω and x ∈ Xt(ω) . Set Vβ (F) = exp( 1−e −β ). By Lemmas 19.8.7 and 19.8.2, we obtain that

exp(Sω F(x)) ≤ Vβ (F) exp(Sω F(π(ρ))) = Vβ (F) exp(S|ω| f (ωρ)) ≤ Vβ (F) exp(S|ω| f ([ω])). 󵄩 󵄩 Thus, 󵄩󵄩󵄩exp(Sω F)󵄩󵄩󵄩X

t(ω)

≤ Vβ (F) exp(S|ω| f ([ω])) and thereby P(F) ≤ P(f ).

One family of functions will play a special role for CGDMSs, as it induces the potential Ξ, which is at the heart of the definition of the topological pressure of a CGDMS (see Section 19.4). Proposition 19.8.10. For any CGDMS Φ = {φe }e∈E , the family LogΦ := {log |φ′e |}e∈E has for amalgamated function the potential Ξ. The family LogΦ is Hölder. Moreover, for any t ∈ Fin, the family t LogΦ is summable and P(t LogΦ ) = P(t). Proof. It is easy to see that the potential Ξ is induced by the LogΦ family. This family is Hölder with exponent −α log s by Lemma 19.3.4. For any t ∈ Fin, the family t LogΦ is summable since Z1 (t) < ∞ according to Proposition 19.4.5(a). Finally, P(t LogΦ ) = P(tΞ) = P(t) by Proposition 19.8.9 and the note subsequent to Definition 19.4.3. 19.8.2 Conformal measures for summable Hölder families of functions In Chapter 13, we witnessed the usefulness of conformal measures associated with potentials imposed on open, distance expanding systems. We now introduce this concept for families of functions imposed on CGDMSs (cf. Definition 13.6.12). Definition 19.8.11. Let Φ = {φe }e∈E be a CGDMS and F = {fe }e∈E be a summable Hölder family of functions. A Borel probability measure m on X is said to be F-conformal provided it is supported on the limit set J and the following two conditions are satisfied: (1) For every e ∈ E and for every Borel set B ⊆ π(EA∞,e ), we have m(φe (B)) = ∫ exp(fe − P(F)) dm.

(19.40)

B

(2) For all e1 ≠ e2 ∈ E, we have m(φe1 (Xt(e1 ) ) ∩ φe2 (Xt(e2 ) )) = 0.

(19.41)

A simple inductive argument shows that it follows from this that for every ω ∈ EA∗ and for every Borel set B ⊆ π(EA∞,ω ) we get m(φω (B)) = ∫ exp(Sω F − |ω| P(F)) dm, B

(19.42)

19.8 Hölder families of functions and conformal measures | 777

and for all incomparable words ω, τ ∈ EA∗ , m(φω (Xt(ω) ) ∩ φτ (Xt(τ) )) = 0.

(19.43)

This last property implies that for all incomparable words ω, τ ∈ EA∗ , m(φω (Xt(ω) ) ∩ π([τ])) = 0

(19.44)

m(π([ω]) ∩ π([τ])) = 0.

(19.45)

and

It might be tempting to define conformality of a measure in terms of all Borel sets B ⊆ Xt(ω) . However, it is subtle but crucial that conformality be defined solely in terms of the Borel subsets of π(EA∞,ω ), as (19.42) does not generally hold for all Borel sets B ⊆ Xt(ω) . Indeed, we have the following. Lemma 19.8.12. Let Φ = {φe }e∈E be a CGDMS and F = {fe }e∈E be a summable Hölder family of functions. If Φ admits an F-conformal measure m, then for every ω ∈ EA∗ and every Borel set B ⊆ Xt(ω) we have that m(φω (B)) = m(φω (B ∩ π(EA∞,ω ))) =

∫

exp(Sω F − |ω| P(F)) dm.

B∩π(EA∞,ω )

Proof. Recall that EA1,ω := {e ∈ E : ωe ∈ EA∗ }. Then m(φω (B)) = ∑ m(φω (B) ∩ π([τ]))

by (19.45) and Lemma A.1.19(g)

τ∈EA|ω|

= m(φω (B) ∩ π([ω]))

by (19.44)

= m(φω (B) ∩ ⋃ π([ωe])) e∈EA1,ω

= ∑ m(φω (B) ∩ π([ωe])) by (19.45) and Lemma A.1.19(g) e∈EA1,ω

= ∑ m(φω (B) ∩ φω (π([e]))) e∈EA1,ω

= ∑ m(φω (B ∩ π([e]))) by injectivity of φω e∈EA1,ω

=

∫ ⋃ B∩π([e]) e∈EA1,ω

exp(Sω F − |ω| P(F)) dm by (19.45) and Lemma A.1.19(g)

778 | 19 Conformal graph directed Markov systems

=

∫

exp(Sω F − |ω| P(F)) dm

B∩π(EA∞,ω )

= m(φω (B ∩ π(EA∞,ω ))) by conformality of m. Remark 19.8.13. Let Φ = {φe }e∈E be a CGDMS and F = {fe }e∈E be a summable Hölder family of functions. Suppose that Φ admits an F-conformal measure m. Let ω ∈ EA∗ and B ⊆ Xt(ω) a Borel set. According to Lemma 19.8.12, m(φω (B)) ≤ ∫ exp(Sω F − |ω| P(F)) dm B

but the equality does not hold in general. We now establish the existence and uniqueness of a conformal measure for any summable Hölder family of functions associated to a conformal-like CGDMS. We further describe its form. Theorem 19.8.14. Let Φ = {φe }e∈E be a finitely irreducible, conformal-like CGDMS. For any summable Hölder family of functions F = {fe }e∈E , the CGDMS Φ has a unique F-conformal measure mF . Moreover, mF = mf ∘ π −1 , where f is the amalgamated function induced by F and mf is the eigenmeasure of the dual transfer operator ℒ∗f . Proof. According to Lemmas 19.8.4 and 19.8.6, the amalgamated function f : EA∞ → ℝ induced by the family F is summable and Hölder continuous on cylinders. By Corollary 17.7.5(a), there then exists an eigenmeasure mf of the dual operator ℒ∗f whose

corresponding eigenvalue is eP(f ) . Let mF = mf ∘ π −1 . We shall show that mF is an F-conformal measure. First, we establish (19.43). Per Corollary 17.7.5(c) there exists an ergodic σ-invariant Gibbs state μf for f which, by Proposition 17.4.2(b), is boundedly equivalent with mf . Therefore, mF = mf ∘ π −1 is boundedly equivalent with μf ∘ π −1 . As (19.43) holds for μf ∘ π −1 according to Theorem 19.7.21, it must also hold for mF . It remains to prove (19.42). Fix ω ∈ EA∗ , say ω ∈ EAn , and a Borel set B ⊆ π(EA∞,ω ). Set 󵄨 πω−1 (B) := π −1 (B) ∩ EA∞,ω = {ρ ∈ π −1 (B) 󵄨󵄨󵄨 Aωn ρ1 = 1}. Then π(πω−1 (B)) = B

(19.46)

19.8 Hölder families of functions and conformal measures | 779

and π −1 (B) \ πω−1 (B) ⊆ EA∞ \ EA∞,ω ⊆

⋃ [e] =

e∈E\EA1,ω

⋃

e∈E: Aωn e =0

[e].

So, using (19.46), we obtain π(π −1 (B) \ πω−1 (B)) ⊆ B ∩ π(

[b])

⋃

b∈E: Aωn b =0

⊆[ ⊆

⋃

a∈E: Aωn a =1

⋃

φa (Xt(a) )] ∩ [ ⋃

a∈E: Aωn a =1 b∈E: Aωn b =0

⋃

b∈E: Aωn b =0

φb (Xt(b) )]

φa (Xt(a) ) ∩ φb (Xt(b) ).

Since property (19.43) holds for mF := mf ∘ π −1 , we deduce that mF (π(π −1 (B) \ πω−1 (B))) = 0, and hence mf (π −1 (B) \ πω−1 (B)) = 0.

(19.47)

Now, ρ ∈ π −1 (φω (B)) if and only if π(ρ) ∈ φω (B), i. e., φρ|n (π(σ n (ρ))) ∈ φω (B). Thus, ωπω−1 (B) ⊆ π −1 (φω (B)) ⊆ ωπω−1 (B) ∪ π −1 ( ⋃

τ∈EAn \{ω}

φτ (Xt(τ) ) ∩ φω (Xt(ω) )).

Using (19.43) and the definition of the measure mF , it ensues that mf (π −1 (φω (B))) = mf (ωπω−1 (B)). Combining this equality with (17.26), Lemma 19.8.2, Proposition 19.8.9, (19.47) and (19.46), we get mF (φω (B)) = mf ∘ π −1 (φω (B)) = mf (ωπω−1 (B)) = ∫ exp(Sn f (ωρ) − n P(f )) dmf (ρ) πω−1 (B)

= ∫ exp(Sω F(π(ρ)) − n P(F)) dmf (ρ) πω−1 (B)

= ∫ exp(Sω F(π(ρ)) − n P(F)) dmf (ρ) π −1 (B)

780 | 19 Conformal graph directed Markov systems = ∫ exp(Sω F(x) − n P(F)) dmf ∘ π −1 (x) B

= ∫ exp(Sω F − n P(F)) dmF . B

This establishes (19.42) and completes the proof that mF is an F-conformal measure. To prove uniqueness, suppose that νF is an F-conformal measure. It is not difficult to show that ν := νF ∘ π is a Borel probability measure on EA∞ . (The set of all unions of mutually disjoint initial cylinders forms an algebra on which ν can be shown to be countably additive using (19.43). Then Carathéodory’s extension theorem (Theorem A.1.28) affirms that ν is a Borel probability measure.) Since π is surjective, note that νF = ν ∘ π −1 . Fix n ∈ ℕ, a word ω ∈ EAn and a Borel set B ⊆ EA∞,ω . Using (19.47) with mf replaced by ν and B replaced by π(B), observe that ν(B) = νF (π(B)) = ν(π −1 (π(B))) = ν(πω−1 (π(B))).

(19.48)

Then, using successively (19.42), Lemma 8.1.2, Lemma 19.8.2 and (19.48), we obtain that ν(ωB) = νF (π(ωB)) = νF (φω (π(B))) = ∫ exp(Sω F − n P(F)) dν ∘ π −1 π(B)

=

∫

exp(Sω F ∘ π − n P(F)) dν

π −1 (π(B))

=

∫

exp(Sω F ∘ π − n P(F)) dν

πω−1 (π(B))

= ∫ exp(Sn f (ωρ) − n P(f )) dν(ρ). B

Therefore, in view of Remark 17.6.3, ν is an eigenmeasure of the operator ℒ∗f corresponding to the eigenvalue exp(P(f )). Hence, ν = mf by Corollary 17.7.5(a). Consequently, νF = ν ∘ π −1 = mf ∘ π −1 = mF . Remark 19.8.15. We would like to draw the reader’s attention to the fact that the uniqueness part of the proof demonstrates an even stronger reality. Namely, if a measure m supported on J satisfies (19.43) and (19.42) with an arbitrary constant P replacing the constant P(F), then m ∘ π is an eigenmeasure of the dual transfer operator ℒ∗f corresponding to the eigenvalue exp(P). It then follows from Corollary 17.7.5(a) that P = P(f ) = P(F), and consequently m is F-conformal. The following result is a direct consequence of Theorem 19.8.14 and Proposition 19.7.19.

19.8 Hölder families of functions and conformal measures | 781

Corollary 19.8.16. Let Φ = {φe }e∈E be a finitely irreducible CGDMS satisfying the Boundary Separation Condition (BSC), the Strong Separation Condition (SSC) or the Cone Condition (CC). For any summable Hölder family of functions F = {fe }e∈E , the CGDMS Φ has a unique F-conformal measure mF . Moreover, mF = mf ∘ π −1 , where f is the amalgamated function induced by F and mf is the eigenmeasure of the dual transfer operator ℒ∗f . Conformal measures have nice properties. In the next few results, we describe some of them. First, conformal measures confer a positive measure to the image of any cylinder. Lemma 19.8.17. Let Φ = {φe }e∈E be a finitely irreducible, conformal-like CGDMS. Let also F = {fe }e∈E be a summable Hölder family of functions and mF the unique F-conformal measure. Then mF (π([ω])) > 0 for every ω ∈ EA∗ . Proof. First, observe that 1 = mF (J) = mF (π(EA∞ )) = mF (⋃ π([e])) ≤ ∑ mF (π([e])). e∈E

e∈E

So there exists e0 ∈ E for which mF (π([e0 ])) > 0. Now, let Λ ⊆ EA∗ be a finite set of words witnessing the finite irreducibility of Φ. Pick any ω ∈ EA∗ . Then there is τ = τ(ω, e0 ) ∈ Λ such that ωτe0 ∈ EA∗ . It follows that [e0 ] ⊆ EA∞,ωτ and it ensues from the conformality of mF (more precisely, (19.42)) that mF (π([ω])) ≥ mF (π([ωτe0 ])) = mF (φωτ (π([e0 ]))) =

∫ eSωτ F−|ωτ| P(F) dmF . π([e0 ])

This latter term is positive since the set π([e0 ]) has positive mF -measure and the integrand is positive and bounded below on π([e0 ]). Indeed, P(F) < ∞ since F is summable while Sωτ F is bounded below over π([e0 ]) because F is Hölder. In fact, the set of followers to any finite word has positive measure, and that measure is uniformly bounded away from zero. Lemma 19.8.18. Let Φ = {φe }e∈E be a finitely irreducible, conformal-like CGDMS. Let also F = {fe }e∈E be a summable Hölder family of functions and mF the unique F-conformal measure. Let Λ ∈ EA∗ be a finite set of words witnessing the finite irreducibility of the system and denote by ΛE the set of letters in words from Λ. Then inf mF (π(EA∞,ω )) ≥ min mF (π([e])).

ω∈EA∗

e∈ΛE

782 | 19 Conformal graph directed Markov systems Proof. For every ω ∈ EA∗ , there is eω ∈ ΛE such that ωeω ∈ EA∗ . Then EA∞,ω ⊇ [eω ] and mF (π(EA∞,ω )) ≥ mF (π([eω ])) ≥ min mF (π([e])). e∈ΛE

The positivity of this last term is a consequence of Lemma 19.8.17. We now show that the measure of the closure of the image of a union of cylinder sets that are uniformly bounded in length, is equal to the measure of the image itself. Lemma 19.8.19. Let Φ = {φe }e∈E be a finitely irreducible, conformal-like CGDMS satisfying the SOSC. Let also F = {fe }e∈E be a summable Hölder family of functions and mF the unique F-conformal measure. For any family W ⊆ EA∗ of words of uniformly bounded lengths, we have mF ( ⋃ π([ω])) = mF ( ⋃ π([ω])). ω∈W

ω∈W

Proof. Let n < ∞ be the maximum length of words in W and let W ′ be the set of words of length n that are comparable to a word in W. Then ⋃τ∈W ′ π([τ]) = ⋃ω∈W π([ω]). We can thus assume without loss of generality that all words in W have the same length n. Let x ∈ J ∩ ⋃ω∈W π([ω]) \ ⋃ω∈W π([ω]). On one hand, as x ∈ J\ ⋃ω∈W π([ω]) and J = ⋃ρ∈E n π([ρ]) we know that A

x∈

⋃ π([τ]) ⊆

τ∈EAn \W

⋃ φτ (Xt(τ) ).

τ∈EAn \W

So x ∈ φτ (Xt(τ) ) for some τ ∈ EAn \W. In fact, x ∉ Int(φτ (Xt(τ) )). Otherwise, there would be ε > 0 such that B(x, ε) ⊆ Int(φτ (Xt(τ) )). But since x ∈ ⋃ω∈W π([ω]) there would also be ω ∈ W such that B(x, ε) ∩ π([ω]) ≠ 0. Therefore, Int(φτ (Xt(τ) )) ∩ Int(φω (Xt(ω) )) ≠ 0. This would mean that τ = ω, which is impossible. Hence, x ∈ φτ (Xt(τ) ) \ Int(φτ (Xt(τ) )) = 𝜕φτ (Xt(τ) ) = φτ (𝜕Xt(τ) ). We readily deduce that J ∩ ⋃ π([ω]) \ ⋃ π([ω]) ⊆ ω∈W

ω∈W

⋃ φτ (𝜕Xt(τ) ).

τ∈EAn \W

Consequently, μF ( ⋃ π([ω]) \ ⋃ π([ω])) ≤ ω∈W

ω∈W

∑ μf ∘ π −1 (φτ (𝜕Xt(τ) )) = 0

τ∈EAn \W

by Theorem 19.7.2(b). The conclusion follows from the equivalence of mF and μF .

19.8 Hölder families of functions and conformal measures | 783

In the same vein, we have the following property. Lemma 19.8.20. Let Φ = {φe }e∈E be a finitely irreducible, conformal-like CGDMS satisfying the SOSC. Let also F = {fe }e∈E be a summable Hölder family of functions and mF the unique F-conformal measure. For all ω ∈ EA∗ , mF (π(EA∞,ω )) = mF (π(EA∞,ω )). Proof. Let x ∈ J ∩ π(EA∞,ω ) \ π(EA∞,ω ). On one hand, x = π(ρ) for some ρ ∈ EA∞ with Aω|ω| ρ1 = 0 since x ∈ J \ π(EA∞,ω ). Then x ∈ φρ1 (Xt(ρ1 ) ).

On the other hand, x = limn→∞ π(ω(n) ) for some ω(n) ∈ EA∞,ω since x ∈ π(EA∞,ω ). Then x ∈ ⋃n∈ℕ φω(n) (Xt(ω(n) ) ). Also, Aω ω(n) = 1 and thus ω(n) 1 ≠ ρ1 for all n ∈ ℕ. 1

1

|ω|

1

In summary, x ∈ φρ1 (Xt(ρ1 ) ) ∩ ⋃e∈E\{ρ1 } φe (Xt(e) ). The OSC then imposes that x ∈ 𝜕φρ1 (Xt(ρ1 ) ). Therefore, J ∩ π(EA∞,ω ) \ π(EA∞,ω ) ⊆ ⋃ 𝜕φe (Xt(e) ). e∈E

Thus, mF (J ∩ π(EA∞,ω ) \ π(EA∞,ω )) ≤ ∑ mF (𝜕φe (Xt(e) )). e∈E

But in view of Corollary 17.7.5(c), there exists an ergodic σ-invariant Gibbs state μf for f which, by Proposition 17.4.2(b), is boundedly equivalent with mf . By Theorem 19.7.2(b), it ensues that mF (𝜕φe (Xt(e) )) = mf ∘ π −1 (𝜕φe (Xt(e) )) ≍ μf ∘ π −1 (𝜕φe (Xt(e) )) = 0,

∀e ∈ E.

Hence, mF (J ∩ π(EA∞,ω ) \ π(EA∞,ω )) = 0. Consequently, mF (π(EA∞,ω )) = mF (J ∩ π(EA∞,ω )) = mF (π(EA∞,ω )). As the reader may recall from Lemma 13.6.15, if μ is a Borel probability measure on a compact metric space X and supp(μ) = X, then inf μ(B(x, r)) > 0

x∈X

for any r > 0. We now show a refinement of this property for conformal measures.

784 | 19 Conformal graph directed Markov systems Lemma 19.8.21. Let Φ = {φe }e∈E be a finitely irreducible, conformal-like CGDMS satisfying the SOSC. Let also F = {fe }e∈E be a summable Hölder family of functions and mF the unique F-conformal measure. For every r > 0, ω ∈ EA∗ and family W ⊆ EA∗ of words uniformly bounded in length: (a)

(b)

inf

x∈⋃ω∈W π([ω])

mF (B(x, r) ∩ ⋃ π([ω])) > 0. ω∈W

inf∞,ω mF (B(x, r) ∩ π(EA∞,ω )) > 0.

x∈π(EA

)

Proof. As ⋃ω∈W π([ω]) is closed, and thus compact (⋃ω∈W π([ω]) might not be) and as mF (⋃ω∈W π([ω])) > 0 according to Lemma 19.8.17, it follows from Lemma 13.6.15 that inf

x∈⋃ω∈W π([ω])

mF (B(x, r) ∩ ⋃ π([ω])) > 0. ω∈W

However, it ensues from Lemma 19.8.19 that mF (B(x, r) ∩ ⋃ π([ω])) = mF (B(x, r) ∩ ⋃ π([ω])), ω∈W

ω∈W

∀x ∈ ⋃ π([ω]). ω∈W

From the above two relations, we deduce that inf

x∈⋃ω∈W π([ω])

mF (B(x, r) ∩ ⋃ π([ω])) ≥ ω∈W

inf

x∈⋃ω∈W π([ω])

mF (B(x, r) ∩ ⋃ π([ω])) > 0. ω∈W

This establishes the first statement. Using Lemmas 19.8.20 and 19.8.18, we obtain along similar lines that inf

x∈π(EA∞,ω )

mF (B(x, r) ∩ π(EA∞,ω )) ≥

inf

x∈π(EA∞,ω )

mF (B(x, r) ∩ π(EA∞,ω )) > 0.

19.8.3 Conformal measures for CGDMSs We have already seen that infinite CGDMSs naturally break into two main classes called irregular and regular systems. This dichotomy arises from the existence or nonexistence of a zero for the pressure function or, as we shall soon see, from the existence or nonexistence of a conformal measure (cf. Definitions 13.6.12 and 19.8.11). Definition 19.8.22. Let Φ = {φe }e∈E be a CGDMS. A Borel probability measure m on X is said to be t-conformal provided that m(J) = 1 and the following two conditions are satisfied: (1) For every e ∈ E and for every Borel set B ⊆ π(EA∞,e ), we have 󵄨 󵄨t m(φe (B)) = ∫󵄨󵄨󵄨φ′e 󵄨󵄨󵄨 dm. B

(19.49)

19.8 Hölder families of functions and conformal measures | 785

(2) For all e1 ≠ e2 ∈ E, we have m(φe1 (Xt(e1 ) ) ∩ φe2 (Xt(e2 ) )) = 0.

(19.50)

Note that the conformality of a measure depends on the measure of first-level sets and subsets. By induction and using the change-of-variables formula (13.20) in Proposition 13.3.1, the conformality cascades down to all levels. Indeed, a Borel probability measure m on X is t-conformal if and only if m(J) = 1, for every ω ∈ EA∗ and for every Borel set B ⊆ π(EA∞,ω ) 󵄨 󵄨t m(φω (B)) = ∫󵄨󵄨󵄨φ′ω 󵄨󵄨󵄨 dm,

(19.51)

B

and for all incomparable words ω, τ ∈ EA∗ m(φω (Xt(ω) ) ∩ φτ (Xt(τ) )) = 0.

(19.52)

The notion of t-conformality is reminiscent of that of F-conformality (compare Definitions 19.8.22 and 19.8.11). The two concepts are related. Indeed, in light of Proposition 19.8.10 and Theorem 19.8.14, any finitely irreducible, conformal-like CGDMS Φ = {φe }e∈E admits for every t ∈ Fin a unique t LogΦ -conformal measure mt LogΦ . The family t LogΦ := {t log |φ′e |}e∈E has for amalgamated function the potential tΞ and P(t LogΦ ) = P(t). A simple calculation reveals that the measure mt LogΦ is characterized by the following properties: (1) For every e ∈ E and for every Borel set B ⊆ π(EA∞,e ), 󵄨 󵄨t mt LogΦ (φe (B)) = e− P(t) ∫󵄨󵄨󵄨φ′e 󵄨󵄨󵄨 dmt LogΦ .

(19.53)

mt LogΦ (φe1 (Xt(e1 ) ) ∩ φe2 (Xt(e2 ) )) = 0.

(19.54)

B

(2) For all e1 ≠ e2 ∈ E,

It is therefore not surprising that the existence of a t-conformal measure is equivalent to t being a zero of the pressure function (compare (19.53) with (19.49)). Theorem 19.8.23. Let Φ = {φe }e∈E be a finitely irreducible, conformal-like CGDMS. Then Φ is regular if and only if it admits a t-conformal measure. If such a measure exists, then it is unique, P(t) = 0 and the t-conformal measure m is t LogΦ -conformal, so m = mt LogΦ = mtΞ ∘ π −1 . Proof. Recall that regularity means the existence of a parameter t such that P(t) = 0.

786 | 19 Conformal graph directed Markov systems Suppose that a t-conformal measure m exists. Then by Remark 19.8.15 and Proposition 19.8.10, we know that P(t) = 0 and m is t LogΦ -conformal. The uniqueness and the form of m are guaranteed by Theorem 19.8.14. It therefore remains show that if P(t) = 0, then a t-conformal measure exists. If P(t) = 0, then t ∈ Fin. By Proposition 19.8.10 and Theorem 19.8.14, there exists a unique t LogΦ -conformal measure. Since P(t LogΦ ) = P(t) = 0, this measure is t-conformal. An immediate consequence of Theorem 19.8.23 and Proposition 19.7.19 is the following. Corollary 19.8.24. Let Φ = {φe }e∈E be a finitely irreducible CGDMS satisfying the BSC, the SSC or the CC condition. Then Φ is regular if and only if it admits a t-conformal measure. If such a measure exists, then it is unique, P(t) = 0 and the t-conformal measure m is t LogΦ -conformal, so m = mt LogΦ = mtΞ ∘ π −1 . For a general measure m and map f , the composition m ∘ f is not a measure even if m is. Nevertheless, by (19.50), we have the following. Lemma 19.8.25. Let Φ = {φe }e∈E be a finitely irreducible CGDMS. If t ≥ 0 and m is a t-conformal measure, then m ∘ π : EA∞ → [0, 1] is a measure such that m = (m ∘ π) ∘ π −1 . Proof. Because of the continuity of the map π, we know that π(W) is a Lebesgue measurable set in X for any Borel set W ⊆ EA∞ . Let W1 and W2 be two disjoint Borel subsets of EA∞ . If x ∈ π(W1 ) ∩ π(W2 ), then there are w(1) ∈ W1 and w(2) ∈ W2 such that 󵄨 󵄨 x = π(w(1) ) = π(w(2) ). Let n = 󵄨󵄨󵄨w(1) ∧ w(2) 󵄨󵄨󵄨 + 1. Then x ∈ φw(1) |n (Xt(w(1) ) ) ∩ φw(2) |n (Xt(w(2) ) ). n n We infer from this that π(W1 ) ∩ π(W2 ) ⊆

⋃

ω,τ∈EA∗ ω,τ incomparable

φτ (Xt(τ) ) ∩ φω (Xt(ω) ).

By (19.52), we deduce that m(π(W1 ) ∩ π(W2 )) ≤

∑

ω,τ∈EA∗ ω,τ incomparable

m(φτ (Xt(τ) ) ∩ φω (Xt(ω) )) = 0.

Consequently, m ∘ π(W1 ∪ W2 ) = m ∘ π(W1 ) + m ∘ π(W2 ). This argument easily generalizes to a sequence of disjoint sets (Wk )∞ k=1 . So m ∘ π is a measure.

19.8 Hölder families of functions and conformal measures | 787

We now uncover a counterpart of Lemma 13.6.13. Lemma 19.8.26. Let Φ = {φe }e∈E be a finitely irreducible CGDMS. If m is a t-conformal measure for some t ≥ 0, then P(t) = 0 and ℒ∗t (m ∘ π)(g ∘ π) = m(g) for all g ∈ C(X), where ℒt := ℒtΞ . Proof. We have already established that P(t) = 0 in Theorem 19.8.23 and the conformallikeness was not used for this. A more direct way to prove this fact is, using the definition of t-conformality and the BDP (Corollary 19.3.5), to observe that 󵄩 󵄩t 1 ≤ ∑ 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩X ≤ K t , t(ω) ω∈EAn

∀n ∈ ℕ.

So P(t) = 0. Now, let g ∈ C(X). For every e ∈ E and every Borel set B ⊆ π(EA∞,e ), it follows from (19.49) that 󵄨 󵄨t ∫ g dm = ∫(g ∘ φe )󵄨󵄨󵄨φ′e 󵄨󵄨󵄨 dm.

(19.55)

B

φe (B)

It is clear that π −1 (π(EA∞,e )) ⊇ EA∞,e . Moreover, if τ ∈ π −1 (π(EA∞,e )) \ EA∞,e then π(τ) ∈ π(EA∞,e ) and Aeτ1 = 0, i. e., there is ω ∈ EA∞,e such that π(τ) = π(ω) and Aeτ1 = 0. This means that π(τ) = π(ω) with Aeω1 = 1 and Aeτ1 = 0. So τ1 ≠ ω1 , and thus π(τ) = π(ω) ∈ φτ1 (Xt(τ1 ) ) ∩ φω1 (Xt(ω1 ) ). By (19.50), we deduce that m ∘ π(π −1 (π(EA∞,e )) \ EA∞,e ) ≤ ∑ m(φa (Xt(a) ) ∩ φb (Xt(b) )) = 0. a=b∈E ̸

Using the t-conformality of m as well as (19.55) and (19.56), we obtain that m(g) = ∫ g dm J

=

g dm

∫ ⋃e∈E φe (π(EA∞,e ))

=∑

e∈E

=∑

e∈E

=∑

e∈E

g dm

∫ φe (π(EA∞,e ))

∫ π(EA∞,e )

󵄨 󵄨t g ∘ φe (x)󵄨󵄨󵄨φ′e (x)󵄨󵄨󵄨 dm(x)

∫ π −1 (π(EA∞,e ))

󵄨 󵄨t g ∘ φe ∘ π(ω)󵄨󵄨󵄨φ′e ∘ π(ω)󵄨󵄨󵄨 dm ∘ π(ω)

(19.56)

788 | 19 Conformal graph directed Markov systems 󵄨 󵄨t = ∑ ∫ g ∘ φe ∘ π(ω)󵄨󵄨󵄨φ′e ∘ π(ω)󵄨󵄨󵄨 dm ∘ π(ω) e∈E

EA∞,e

= ∫

∑

󵄨 󵄨t g ∘ φe ∘ π(ω)󵄨󵄨󵄨φ′e ∘ π(ω)󵄨󵄨󵄨 dm ∘ π(ω)

∑

g ∘ π(eω) exp(tΞ(eω)) dm ∘ π(ω)

EA∞ e∈E: Aeω1 =1

= ∫ EA∞

e∈E: Aeω1 =1

= ∫ ℒtΞ (g ∘ π) dm ∘ π EA∞

= ℒ∗t (m ∘ π)(g ∘ π). For CGDSs (conformal graph directed systems, i. e., when Aab = 1 if and only if t(a) = i(b)), it is possible to define a transfer operator directly on the phase space ̃t : C(X) → C(X), where X := ⨆υ∈V Xυ , namely ℒ ̃t (g)(x) := ℒ

∑

e∈E: t(e)=υ

󵄨 󵄨t g(φe (x))󵄨󵄨󵄨φ′e (x)󵄨󵄨󵄨 ,

∀g ∈ C(X), ∀x ∈ Xυ , ∀υ ∈ V.

A closer analogue of Lemma 13.6.13 holds. Lemma 19.8.27. Let Φ = {φe }e∈E be a CGDS. If m is a t-conformal measure for some ̃∗ (m) = m. t ≥ 0, then P(t) = 0 and ℒ t Proof. The proof is a little simpler than that of Lemma 19.8.26: m(g) = ∫ g dm =

∫

X

⋃e∈E φe (Xt(e) )

=∑

e∈E

∫ φe (Xt(e) )

= ∑

g dm

󵄨 󵄨t g dm = ∑ ∫ g ∘ φe 󵄨󵄨󵄨φ′e 󵄨󵄨󵄨 dm

∑

e∈E X

t(e)

󵄨 󵄨t ∫ g ∘ φe 󵄨󵄨󵄨φ′e 󵄨󵄨󵄨 dm

υ∈V e∈E: t(e)=υ X

υ

= ∑∫

∑

υ∈V X e∈E: t(e)=υ

󵄨 󵄨t g ∘ φe 󵄨󵄨󵄨φ′e 󵄨󵄨󵄨 dm

υ

̃t (g) dm = ℒ ̃∗ (m)(g). = ∫ℒ t X

19.8.4 Dimensions of measures for CGDMSs The argument given to establish (19.43) in the proof of Theorem 19.8.14 also yields the following remarkable fact, which can be seen as a measure-theoretic open set condition.

19.8 Hölder families of functions and conformal measures | 789

Theorem 19.8.28. Let Φ = {φe }e∈E be a finitely irreducible, conformal-like CGDMS. If μ is a σ-invariant Borel probability measure on EA∞ , then μ ∘ π −1 (φω (Xt(ω) ) ∩ φτ (Xt(τ) )) = 0

(19.57)

for all incomparable words ω, τ ∈ EA∗ . In turn, we observe the following interesting fact. Corollary 19.8.29. Let Φ = {φe }e∈E be a finitely irreducible, conformal-like CGDMS. If μ is a σ-invariant Borel probability measure on EA∞ , then μ ∘ π −1 (φω (Xt(ω) )) = μ([ω]),

∀ω ∈ EA∗ .

(19.58)

Proof. Clearly, π −1 (φω (Xt(ω) )) ⊇ [ω] and thus μ ∘ π −1 (φω (Xt(ω) )) ≥ μ([ω]) for all ω ∈ EA∗ . Suppose that μ ∘ π −1 (φτ (Xt(τ) )) > μ([τ]) for some τ ∈ EA∗ . Then μ ∘ π −1 (φτ (Xt(τ) )) = μ([τ]) + ε for some ε > 0. Using Theorem 19.8.28, we get 1 = μ ∘ π −1 (J) = μ ∘ π −1 ( ⋃ φω (Xt(ω) )) = ∑ μ ∘ π −1 (φω (Xt(ω) )) ω∈EA|τ|

≥

∑

ω∈EA|τ| \{τ}

ω∈EA|τ|

μ([ω]) + μ([τ]) + ε = μ( ⋃ [ω]) + ε = μ(EA∞ ) + ε = 1 + ε. ω∈EA|τ|

This contradiction shows that μ ∘ π −1 (φω (Xt(ω) )) = μ([ω]) for all ω ∈ EA∗ . As before, the partition of EA∞ into its initial cylinders of length 1 is denoted by α = {[e] : e ∈ E}. If μ is a σ-invariant Borel probability measure on EA∞ , then the entropy of the partition α with respect to μ is denoted by Hμ (α), the entropy of μ with respect to the shift map σ : EA∞ → EA∞ by hμ (σ) and its (characteristic) Lyapunov exponent (cf. Definition 16.5.1) by χμ (σ) := − ∫ ΞΦ dμ > 0, EA∞

where 󵄨 󵄨 Ξ(ω) := log󵄨󵄨󵄨φ′ω1 (π(σ(ω)))󵄨󵄨󵄨 is the potential at the heart of the definition of the topological pressure of a CGDMS (see Section 19.4). This potential is also the amalgamated function induced by the family LogΦ . Recall from Definition 15.6.1 that the Hausdorff dimension HD(μ ∘ π −1 ) is the infimum of the Hausdorff dimensions of the sets of full μ ∘ π −1 -measure.

790 | 19 Conformal graph directed Markov systems We now derive a volume lemma for measures associated with CGDMSs (cf. Lemma 16.5.2). Theorem 19.8.30 (Volume lemma). Let Φ = {φe }e∈E be a finitely irreducible, conformallike CGDMS. If μ is an ergodic σ-invariant Borel probability measure on EA∞ such that χμ (σ) < ∞, then HD(μ ∘ π −1 ) =

hμ (σ) χμ (σ)

.

Proof. By hypothesis, Ξ ∈ L1 (μ). Let α = {[e] : e ∈ E}. Assume first that Hμ (α) < ∞. Since α is a generating partition, we have that hμ (σ) = hμ (σ, α) ≤ Hμ (α) < ∞. Thus, in view of the ergodic case of Birkhoff’s ergodic theorem (Corollary 8.2.14) and of Shannon–McMillan–Breiman’s theorem (Corollary 9.5.5) there exists a set Z ⊆ EA∞ such that μ(Z) = 1 and for all ω ∈ Z, 1 n−1 ∑ Ξ ∘ σ j (ω) = −χμ (σ) and n→∞ n j=0 lim

lim

n→∞

log μ([ω|n ]) = −hμ (σ). n

(19.59)

Fix ω ∈ Z and 0 < ε < χμ (σ). For each r > 0, let n = n(ω, r) ∈ ℕ be the least integer n such that φω|n (Xt(ωn ) ) ⊆ B(π(ω), r). By (19.59), for every r > 0 small enough (which implies that n = n(ω, r) is large enough) we then have log μ ∘ π −1 (B(π(ω), r)) ≥ log μ ∘ π −1 (φω|n (Xt(ωn ) )) ≥ log μ([ω|n ]) ≥ −(hμ (σ) + ε)n and diam(φω|n−1 (Xt(ωn−1 ) )) ≥ r. This last inequality, a slight adjustment to (19.9) in Lemma 19.3.9, and (19.59) imply that 󵄨 󵄨 log r ≤ log diam(φω|n−1 (Xt(ωn−1 ) )) ≤ log(KD󵄨󵄨󵄨φ′ω|n−1 (π(σ n−1 (ω)))󵄨󵄨󵄨) n−1

󵄨 󵄨 ≤ log(KD) + ∑ log󵄨󵄨󵄨φ′ωj (π(σ j (ω)))󵄨󵄨󵄨 ≤ log(KD) + (n − 1)(−χμ (σ) + ε) j=1

for all r > 0 small enough (and corresponding n). Therefore, for r small enough we get −(hμ (σ) + ε)n log μ ∘ π −1 (B(π(ω), r)) ≤ = log r log(KD) + (n − 1)(−χμ (σ) + ε)

hμ (σ) + ε

− log(KD) n

+

n−1 (χμ (σ) n

− ε)

.

19.8 Hölder families of functions and conformal measures | 791

Letting r → 0 (and consequently n → ∞), the upper local dimension of μ ∘ π −1 at π(ω) has for upper bound dμ∘π −1 (π(ω)) := lim sup r→0

log μ ∘ π −1 (B(π(ω), r)) hμ (σ) + ε ≤ . log r χμ (σ) − ε

As ε > 0 is arbitrary, we deduce that dμ∘π −1 (π(ω)) ≤

hμ (σ) χμ (σ)

,

∀ω ∈ Z.

Theorem 15.6.3(b) then asserts that HD(π(Z)) ≤ hμ (σ)/χμ (σ). As μ ∘ π −1 (π(Z)) = 1, we conclude that HD(μ ∘ π −1 ) ≤ hμ (σ)/χμ (σ). If hμ (σ) = 0, then this inequality suffices to establish the theorem. To demonstrate the opposite inequality when hμ (σ) > 0, let J1 ⊆ J be a Borel set such that μ ∘ π −1 (J1 ) > 0. Fix 0 < ε < hμ (σ). In view of (19.59) and Egorov’s theorem (Theorem A.1.45), there exist n0 ∈ ℕ and a Borel set ̃J1 ⊆ π −1 (J1 ) such that μ(̃J1 ) > μ(π −1 (J1 ))/2 > 0, μ([ω|n ]) ≤ exp((−hμ (σ) + ε)n)

(19.60)

󵄨 󵄨 and 󵄨󵄨󵄨φ′ω|n (π(σ n (ω)))󵄨󵄨󵄨 ≥ exp((−χμ (σ)−ε)n) for all ω ∈ ̃J1 and all n ≥ n0 . By Lemma 19.3.11, this last inequality implies that there exists n1 ≥ n0 such that diam(φω|n (Xt(ωn ) )) ≥ D−1 e(−χμ (σ)−ε)n ≥ e−(χμ (σ)+2ε)n

(19.61)

for all n ≥ n1 and all ω ∈ ̃J1 . Given 0 < r < exp(−(χμ (σ) + 2ε)n1 ) and ω ∈ ̃J1 , let n(ω, r) be the least integer n such that diam(φω|n+1 (Xt(ωn+1 ) )) < r.

(19.62)

Using (19.61), we know that n(ω, r) ≥ n1 and diam(φω|n(ω,r) (Xt(ωn(ω,r) ) )) ≥ r.

(19.63)

By virtue of Lemma 19.7.16 (with κ = 1), there is a universal constant L ≥ 1 such that for every ω ∈ ̃J1 and 0 < r < exp(−(χμ (σ) + 2ε)n1 ) there exist words ω(1) , . . . , ω(k) ∈ ̃J1 , where k = k(ω) ≤ L, such that k

π(̃J1 ) ∩ B(π(ω), r) ⊆ ⋃ φω(j) | j=1

n(ω(j) ,r)

(Xt(ω(j) |

) n(ω(j) ,r)

).

(19.64)

792 | 19 Conformal graph directed Markov systems ̃ := μ|̃J be the restriction of the measure μ to the set ̃J1 . Using successively (19.64), Let μ 1 (19.58), (19.60), (19.61) and (19.62) we get ̃ ∘ π −1 (B(π(ω), r)) = μ(̃J1 ∩ π −1 (B(π(ω), r))) μ ≤ μ ∘ π −1 (π(̃J1 ) ∩ B(π(ω), r)) k

≤ ∑ μ ∘ π −1 (φω(j) |

n(ω(j) ,r)

j=1

(Xt(ω(j) |

) n(ω(j) ,r)

k

k

j=1

j=1

))

= ∑ μ([ω(j) |n(ω(j) ,r) ]) ≤ ∑ exp((−hμ (σ) + ε)n(ω(j) , r)) k

= ∑ exp(−(χμ (σ) + 2ε)(n(ω(j) , r) + 1) ⋅ j=1

n(ω(j) , r)

k

≤ ∑(diam(φω(j) |

n(ω(j) ,r)+1

j=1 k

n(ω(j) ,r)

≤ ∑ r n(ω(j) ,r)+1

hμ (σ)−ε μ (σ)+2ε

⋅χ

j=1

(Xt(ω(j) |

n(ω(j) ,r)+1

n(ω(j) , r)

) )))

⋅

−hμ (σ) + ε

+ 1 −(χμ (σ) + 2ε)

)

n(ω(j) ,r) hμ (σ)−ε ⋅ n(ω(j) ,r)+1 χμ (σ)+2ε

hμ (σ)−2ε

≤ Lr χμ (σ)+2ε ,

where the last inequality holds assuming n1 to be so that Consequently,

n1 (h (σ) − ε) n1 +1 μ

≥ hμ (σ) − 2ε.

h (σ)−2ε

μ ̃ ∘ π −1 (B(π(ω), r)) log L + χμ (σ)+2ε log r log L hμ (σ) − 2ε log μ ≥ = + . log r log r log r χμ (σ) + 2ε

̃ ∘ π −1 at π(ω) has for lower bound Letting r → 0, the lower local dimension of μ dμ̃∘π −1 (π(ω)) := lim inf r→0

̃ ∘ π −1 (B(π(ω), r)) hμ (σ) − 2ε log μ ≥ . log r χμ (σ) + 2ε

Letting ε → 0, we deduce that dμ̃∘π −1 (π(ω)) ≥

hμ (σ) χμ (σ)

,

∀ω ∈ ̃J1 .

Theorem 15.6.3(a) then affirms that HD(J1 ) ≥ HD(π(̃J1 )) ≥ hμ (σ)/χμ (σ). Since J1 is any Borel subset of J such that μ ∘ π −1 (J1 ) > 0, we conclude that HD(μ ∘ π −1 ) ≥ hμ (σ)/χμ (σ) and the proof is complete in the case where Hμ (α) < ∞. If Hμ (α) = ∞, then the above considerations imply that HD(μ ∘ π −1 ) = ∞, which is impossible. A direct consequence of Theorem 19.8.30 (volume lemma) and Proposition 19.7.19 is the following.

19.9 Examples | 793

Corollary 19.8.31 (Volume lemma). Let Φ = {φe }e∈E be a finitely irreducible CGDMS satisfying the BSC, the SSC or the CC condition. If μ is an ergodic σ-invariant Borel probability measure on EA∞ such that χμ (σ) < ∞, then HD(μ ∘ π −1 ) =

hμ (σ) χμ (σ)

.

19.9 Examples Examples of CGDMSs abound. It is easy to construct them and the flexibility of some constructions can be enormous. Example 19.9.1. Fix d ∈ ℕ. Let V be a finite set and E = ℕ. Choose a collection {Xυ }υ∈V of mutually disjoint compact subsets of ℝd satisfying Xυ = Intℝd (Xυ ). Pick a collection {Wυ }υ∈V of mutually disjoint, open, connected and bounded subsets of ℝd such that Xυ ⊆ Wυ for all υ ∈ V. Fix any countable set ξ := {ξn }n∈ℕ ⊆ ⋃ Xυ υ∈V

such that for every υ ∈ V the set Xυ ∩ ξ is discrete; it is straightforward to construct such a set inductively. For every n ∈ ℕ, let un ∈ V be such that ξn ∈ Xun and also select arbitrarily υn ∈ V. Proceeding inductively with a different induction process, suppose that for some n ∈ ℕ a closed ball B(ξn , rn ) has been chosen such that n−1

B(ξn , 2rn ) ∩ ( ⋃ B(ξk , 2rk ) ∪ {ξℓ : ℓ ≥ n + 1}) = 0. k=1

Choose any rn+1 > 0 such that n

B(ξn+1 , 2rn+1 ) ∩ ( ⋃ B(ξk , 2rk ) ∪ {ξℓ : ℓ ≥ n + 2}) = 0. k=1

Then for every n ∈ ℕ choose any conformal map φn : Wυn → Wun such that φn (Xυn ) ⊆ Xun

󵄩 󵄩 and 󵄩󵄩󵄩φ′n 󵄩󵄩󵄩W ≤ 1/2. υn

For instance, the conformal map φn could simply be affine ℝd ∋ x 󳨃󳨀→ an x + bn ,

794 | 19 Conformal graph directed Markov systems with an ∈ (0, 1/2) and bn ∈ ℝd judiciously chosen. Then the maps {φn }n∈ℕ form a conformal IFS satisfying the Strong Open Set Condition (SOSC). In the next few examples, we describe systems with specific features or of special interest. Example 19.9.2. The limit set J is an Fσδ but not a Gδ . Denote by Q the set of all rational numbers in [0, 1]. Let X = [0, 1] × [0, 1] and Δ = {(x, x) : x ∈ [0, 1]} be the diagonal of X. Let E = {−1} ∪ Q. Consider a conformal IFS {φe : X → X}e∈E consisting of linear mappings such that: (a) φe (X) ∩ Δ = {φe (0, 1)} = {(e, e)} for all e ∈ Q; (b) φ−1 (x, y) = (x/2, (y + 1)/2); (c) The sets φe (X), e ∈ E, are mutually disjoint. Then J ∩ Δ = Q is not a Gδ –set, so neither is J. Note also that this system is pointwise finite but not locally finite (see Definition 19.1.2 and Exercise 19.10.18). Example 19.9.3. An irregular similarity IFS. Let 2

E = {(n, k) : n ∈ ℕ and 1 ≤ k ≤ 2n −1 }. Let X = [0, 1] and let Φ = {φn,k : X → X | (n, k) ∈ E} be a system consisting of similarity maps φn,k such that 󵄩󵄩 ′ 󵄩󵄩 −(n2 +n) 󵄩󵄩φn,k 󵄩󵄩 = 2 and such that the intervals φn,k (X) are mutually disjoint. This last requirement can be satisfied since ∞

2 2 󵄩 󵄩 Z1 (1) = ∑ 󵄩󵄩󵄩φ′n,k 󵄩󵄩󵄩 = ∑ 2n −1 2−(n +n) = 1/2 < 1.

n=1

(n,k)∈E

As the generators are similarities, this computation further reveals that P(1) = log(1/2) < 0. On the other hand, observe that ∞

2

2

∞

2

Z1 (t) = ∑ 2n −1 2−(n +n)t = ∑ 2n (1−t)−nt−1 = ∞ n=1

n=1

for all 0 ≤ t < 1. Thus, Φ is irregular and HD(J) = h = θ = 1 by Bowen’s formula (Theorem 19.6.4 or Corollary 19.6.5). Example 19.9.4. A critically regular similarity IFS.

19.9 Examples | 795

This example is very similar to Example 19.9.3. The only difference in its construction is that we now take 2

E = {(n, k) : n ∈ ℕ and 1 ≤ k ≤ 2n }. Then the same computations as in Example 19.9.3 show that Z1 (1) = 1, whence P(1) = 0, and Z1 (t) = ∞ for all 0 ≤ t < 1. Hence, Φ is critically regular, the only conformal measure is the Lebesgue measure on [0, 1], and HD(J) = h = θ = 1. Example 19.9.5. Real continued fractions. This conformal IFS (or CIFS), call it 𝒢 , is given by the maps φn : [0, 1] → [0, 1], n ∈ ℕ, defined by the formula φn (x) =

1 . x+n

It is easy to see that the corresponding coding map π : ℕ∞ → [0, 1] is given by the continued fractions formula π(ω) =

ω1 +

1

1

ω2 + ω 1+⋅⋅⋅ 3

(19.65)

and that its range, i. e., the limit set J𝒢 of the system 𝒢 is the set of all irrational numbers in [0, 1]. In particular, HD(J𝒢 ) = 1 and the Lebesgue measure on ℝ restricted to J𝒢 is the 1-conformal measure for the system 𝒢 . Furthermore, since 1 󵄨󵄨 ′ 󵄨󵄨 , 󵄨󵄨φn (x)󵄨󵄨 = (x + n)2 it is straightforward to see that Z1 (t) < ∞ for all t > 1/2 and Z1 (t) = ∞ for all t ∈ [0, 1/2]. Thus, the finiteness parameter θ𝒢 is equal to 1/2 and the system 𝒢 is cofinitely regular. All theorems proved in this chapter for CGDMSs apply to 𝒢 and all its subsystems. Note that the maps φn : [0, 1] → [0, 1], n ∈ ℕ, are the continuous inverse branches of the Gauss map G : [0, 1) → [0, 1) given by the formula G(x) :=

1 1 − [ ]. x x

Its G-invariant Borel probability measure νG , absolutely continuous with respect to the Lebesgue measure on ℝ restricted to J𝒢 , is given by the formula νG (B) =

1 1 dx. ∫ log 2 1 + x B

(19.66)

796 | 19 Conformal graph directed Markov systems This measure is known as Gauss measure. It is the projection under the coding map π of the σ-invariant Gibbs state μ𝒢 of the system 𝒢 corresponding to the potential 󵄨 󵄨 ℕ∞ ∋ ω 󳨃󳨀→ Ξ(ω) := log󵄨󵄨󵄨φ′ω1 (π(σ(ω)))󵄨󵄨󵄨 ∈ ℝ, i. e. νG = μ𝒢 ∘ π −1 . According to Theorem 19.7.4, the coding map π : ℕ∞ → [0, 1] is a measurepreserving isomorphism establishing a measure-theoretic conjugacy between the shift map (σ : ℕ∞ → ℕ∞ , μ𝒢 ) and the Gauss map (̃f = G : [0, 1] → [0, 1], μ𝒢 ∘ π −1 = νG ). As an immediate consequence of this conjugacy and of Theorem 17.4.6, we deduce the following. Theorem 19.9.6. The Gauss measure νG , which is equivalent to the Lebesgue measure on ℝ restricted to J𝒢 , is ergodic with respect to the Gauss map G. Moreover, all theorems proved in Chapter 17 for Gibbs measures and finitely primitive subshifts apply to μ𝒢 and 𝒢 . Today, quite a lot is known about the CIFS 𝒢 , its subsystems, and the fractal geometry of their limit sets. It is impossible to cite all the relevant works here. We list only a few: [16, 17, 23, 49, 50, 58, 59, 76, 136]. Example 19.9.7. Complex continued fractions. Let 󵄨 E = {m + ni 󵄨󵄨󵄨 (m, n) ∈ ℕ × ℤ},

X = B(1/2, 1/2) ⊆ ℂ

and W = B(1/2, 3/4) ⊆ ℂ.

For every e ∈ E, define the map φe : W → W by φe (z) :=

1 . e+z

Then φe (X) ⊆ X for all e ∈ E and the system ℂ𝒢 := {φe }e∈E is a conformal IFS. It is again easy to see that the corresponding coding map π : E ∞ → X is given by the complex version of the continued fractions formula (19.65). Furthermore, since 1 󵄨󵄨 ′ 󵄨󵄨 , 󵄨󵄨φe (z)󵄨󵄨 = |z + e|2 it is straightforward to see that Z1 (t) < ∞ for all t > 1 and Z1 (t) = ∞ for all t ∈ [0, 1]. Thus, the finiteness parameter θℂ𝒢 is equal to 1 and the system ℂ𝒢 , like its real counterpart, is cofinitely regular. In consequence, HD(Jℂ𝒢 ) = hℂ𝒢 > θℂ𝒢 = 1. All theorems proved in this chapter for CGDMSs apply to ℂ𝒢 and all its subsystems. As for real continued fractions, a lot is known about complex continued fractions; see [16, 17, 31, 75].

19.10 Exercises | 797

19.10 Exercises Exercise 19.10.1. Show that the unit circle 𝕊1 ⊆ ℝ2 is not the limit set of any conformal IFS. Exercise 19.10.2. Prove Lemma 19.1.3. Exercise 19.10.3. Prove that if Φ is a conformal IFS, then {HD(JF ) : F ⊆ E, #F < ∞} ⊇ [0, θ]. (See [75].) Exercise 19.10.4. Show that if Φ is a CGDMS, then the set {HD(JF ) : F ⊆ E, #F < ∞} is compact. (See [16, 75].) Exercise 19.10.5. Build a CGDMS which satisfies the Strong Separation Condition (SSC) but not the Strong Open Set Condition (SOSC). Exercise 19.10.6. Prove Lemma 19.7.7. Exercise 19.10.7. Prove Theorem 19.7.8. Exercise 19.10.8. Show that there is q ∈ ℕ such that for every x ∈ X there exist unit vectors u1 (x), . . . , uq (x) ∈ ℝd so that all q-tuples (ux , u1 (x), . . . , uq (x)), x ∈ X, are mutually linearly isometric and satisfy (19.35)–(19.36). Exercise 19.10.9. Prove Proposition 19.7.19 in the case of a CGDMS satisfying the SSC. Deduce that the result also holds for any CGDMS satisfying the BSC. Exercise 19.10.10. Prove Lemma 19.8.6. Exercise 19.10.11. Prove that the pressure function is well defined in Definition 19.8.8. Exercise 19.10.12. Using an inductive argument, prove that the pair of formulae (19.42) and (19.43) follow from the pair of formulae (19.40) and (19.41). Exercise 19.10.13. Let Φ be a CGDMS and f ∈ 𝒦βs (EA∞ , ℝ) be a potential such that χμf (σ) < ∞. Show that the following three conditions are equivalent:

(a) HD(μf ∘ π −1 ) = HD(J). (b) μf = μHD(J)Ξ . (c) The potentials f and HD(J)Ξ are cohomologous modulo a constant in the class of continuous (equivalently, Hölder continuous) functions from EA∞ to ℝ. Exercise 19.10.14. Let Φ be a CGDMS. Strengthening Exercise 19.10.13, show that if μ ∈ M(σ) is such that χμ (σ) < ∞, then HD(μ ∘ π −1 ) = HD(J) if and only if μ = μHD(J)Ξ .

798 | 19 Conformal graph directed Markov systems Exercise 19.10.15. Let Φ be a CGDMS. Prove that the function 𝒦βs (EA∞ , ℝ) ∋ f 󳨃→ HD(μf ∘ π −1 ) is continuous.

∞ Exercise 19.10.16. Let σ : ℕ∞ A → ℕA be a finitely irreducible subshift of finite type and f ∈ 𝒦βs (ℕ∞ A , ℝ) a potential. Prove that the sequence ∞

(μf |{1,2,...,n}∞ )n=1 A

weak∗ converges to μf . Exercise 19.10.17. Let E be a countable set and p : E → (0, 1] be a function such that ∑ p(e) = 1.

e∈E

Show that the Bernoulli measure μp on E ∞ is the Gibbs state for the potential E ∞ ∋ ω 󳨃→ log p(ω1 ). Exercise 19.10.18. A GDMS Φ = {φe : Xt(e) → Xi(e) }e∈E is said to be locally finite if every point of the phase space X has a neighbourhood that intersects only finitely many first-level sets, i. e., if for every x ∈ X there is a neighbourhood Vx of x such that #{e ∈ E : Vx ∩ φe (Xt(e) )} < ∞. (a) Prove that every locally finite GDMS is pointwise finite (cf. Definition 19.1.2). (b) Show that the conformal IFS in Example 19.9.2 is pointwise finite but not locally finite. Exercise 19.10.19. Prove that the Gauss measure νG in (19.66) is invariant under the Gauss map G. Then show that the Lebesgue measure on [0, 1] is not invariant under the Gauss map.

20 Real analyticity of topological pressure and Hausdorff dimension The first three sections of this chapter are related to our considerations in Section 13.10. However, the forthcoming considerations are subtler and more technically involved. These sections are a continuation of Chapters 17 and 18, as they exclusively deal with finitely irreducible/primitive countable-alphabet subshifts of finite type. In these three sections, we show that under proper hypotheses the pressure function P(φ + g) depends real analytically on g (in an appropriately defined way). In all of these sections, the potential φ is assumed to be Hölder continuous on cylinders and strongly summable (see Definition 20.1.1). In Section 20.1, the Hölder-on-cylinders perturbation g, therein denoted by γz , need not be bounded but must fulfill several technical assumptions (see Definition 20.1.6). Section 20.2 is dedicated to the special case where g is of the form zγ. This case is very important, as it is frequently encountered in applications. Finally, in Section 20.3 we assume that the Hölder-on-cylinders perturbation g is essentially bounded. However, we impose no other technical condition. Each of these sections culminates with the establishment of the real-analytic dependence of the topological pressure g 󳨃→ P(φ + g). The methods of proof are of complex analytic and functional analytic character. The notorious difficulty in demonstrating real-analytic dependence of the topological pressure is that the limit of a uniformly convergent sequence of real-analytic functions need not be real analytic (look at the Stone–Weierstrass theorem, for example), and the pressure is defined as such a limit. Nevertheless, the limit of a uniformly convergent sequence of complex-analytic (i. e. holomorphic) functions is complex analytic. Accordingly, our strategy consists in complexifying potentials first, and thus Perron–Frobenius (transfer) operators, too. Then we prove that these complexified operators depend holomorphically on the perturbation g. This is where infinite-dimensional complex analysis on Banach spaces is used. Next enters functional analysis. Indeed, we know that the exponential of the pressure function is a simple isolated eigenvalue of the transfer operator. So the Kato–Rellich perturbation theorem yields holomorphic dependence of appropriate eigenvalues of the complexified transfer operators. Restricting these operators to the real realm gives real analyticity of the exponential of the pressure function, and hence of the pressure itself. Once we know that the topological pressure is real analytic, it is natural to ask what its derivative is. Sections 20.2 and 20.3 provide an answer: the pressure’s derivative is the integral of the perturbation against the appropriate Gibbs state. The first results on the real analyticity of the topological pressure were obtained by David Ruelle in [111] and [113] for finite-alphabet subshifts of finite type and Hölder continuous potentials and perturbations. We followed here Ruelle’s general strategy. In comparison to Section 13.10, the additional difficulties we face are caused by the fact that transfer operators are given as the sum of an infinite convergent series https://doi.org/10.1515/9783110702699-020

800 | 20 Real analyticity of topological pressure and Hausdorff dimension (rather than a finite sum) and that neither the potentials nor the perturbations need be bounded. In Section 20.4, we consider families of CGDMSs in the complex plane ℂ. We assume that these families depend in a suitable way on a complex parameter λ. In particular, we define a (somewhat technical) notion of strong parameter. We want to emphasize that at first we do not assume the open set condition to be satisfied when we show the real-analytic dependence on λ of the Bowen parameter h. This is achieved in two steps. First, we establish the real analyticity of the topological pressure by verifying that the conditions in Definition 20.1.6 are fulfilled. Then the use of the implicit function theorem yields the real analyticity of the Bowen parameter on some sufficiently small neighborhood of any strong parameter. As an immediate consequence, we conclude that if the open set condition holds, then the Hausdorff dimension of the limit set varies real analytically near any strong parameter. Our considerations up to that point in this section find their origin in the paper [134]. We somewhat drifted away from that paper, developing and improving it. The rest of the section is a detailed and improved account of [118]. It is devoted to providing easy-to-check sufficient conditions for an analytic family of CGDMSs to admit a strong parameter. For this, we first introduce the concept of Hölder stability of such families at their parameters and prove that this entails such parameters to be strong. Then we show that if such an analytic family is engendered by a holomorphic motion, then the parameters for which the corresponding CGDMSs are strongly regular, turn out to be parameters that are Hölder stable and hence strong. Finally, we describe a mild, easy-to-verify condition, called periodic separation, which guarantees that an analytic family of CGDMSs is generated by a holomorphic motion. However, this section is not the end of the story. In the paper [106], we proved a meaningful real-analytic dependence for CGDMSs living in higher-dimensional Euclidean spaces. Other analyticity and continuity results of a similar type can be found in [103, 104, 105].

20.1 Real analyticity of pressure: part I Let E be a countable alphabet and A : E × E → {0, 1} a finitely irreducible incidence matrix. Let also β ∈ (0, 1]. Recall that 𝒦β (EA∞ ) represents the set of all complex-valued functions that are Hölder continuous on cylinders with exponent β while its subset comprising those functions that are real-valued is denoted by 𝒦β (EA∞ , ℝ). Moreover, those functions whose real part is summable are designated by 𝒦βs (EA∞ ). The real part of summable functions is always bounded from above, but it is not bounded from below when #E = ∞. Remember that Cb (EA∞ ) is the set of all bounded continuous complex-valued functions on EA∞ and the set 𝒦βb (EA∞ ) := 𝒦β (EA∞ ) ∩ Cb (EA∞ ) of all globally Hölder continuous functions, equipped with the norm ‖f ‖β := υβ (f ) + ‖f ‖∞ , forms a Banach space. Recall that 𝒦βb (EA∞ ) = 𝒦βs (EA∞ ) when #E < ∞, while 𝒦βb (EA∞ ) ∩ 𝒦βs (EA∞ ) = 0 when #E = ∞. (For more information, see the beginning of the Section 18.1.)

20.1 Real analyticity of pressure: part I

| 801

We now introduce a new subclass of 𝒦βs (EA∞ , ℝ). Definition 20.1.1. A function φ ∈ 𝒦βs (EA∞ , ℝ) is strongly summable provided that there exists ν ∈ (0, 1) such that νφ is summable, i. e., νφ ∈ 𝒦βs (EA∞ , ℝ). We enumerate a few properties of summability and strong summability. Lemma 20.1.2. (a) If φ ∈ 𝒦βs (EA∞ , ℝ) and t > 1, then tφ ∈ 𝒦βs (EA∞ , ℝ). (b) If φ is strongly summable, then uφ is strongly summable for any u > 0 large enough. More precisely, if νφ ∈ 𝒦βs (EA∞ , ℝ) for some ν ∈ (0, 1), then uφ is strongly summable for any u > ν. (c) If h ∈ 𝒦βb (EA∞ , ℝ) and φ ∈ 𝒦βs (EA∞ , ℝ), then φ + h ∈ 𝒦βs (EA∞ , ℝ). (d) Let h ∈ 𝒦βb (EA∞ , ℝ). If φ ∈ 𝒦βs (EA∞ , ℝ) is strongly summable, then so is φ + h.

We now turn our attention to the exponential of functions that are Hölder continuous on cylinders. We start with the following elementary observation. Lemma 20.1.3. For any z, ζ ∈ ℂ, we have that 󵄨󵄨 z ζ󵄨 󵄨󵄨e − e 󵄨󵄨󵄨 ≤ exp(max{Re(z), Re(ζ )})|z − ζ |. Proof. It follows from the mean value inequality that 󵄨󵄨 z ζ󵄨 Re(w) ⋅ |z − ζ | = exp(max{Re(z), Re(ζ )})|z − ζ |. 󵄨󵄨e − e 󵄨󵄨󵄨 ≤ sup e w∈[z,ζ ]

A fairly immediate application of the previous lemma leads to the following result. Lemma 20.1.4. If f ∈ 𝒦β (EA∞ ) is bounded above, then ef ∈ 𝒦βb (EA∞ ) and for any F ⊆ EA∞ , 󵄩󵄩 f 󵄩󵄩 sup Re(f |F ) . 󵄩󵄩e 󵄩󵄩β,F ≤ (1 + υβ,F (f ))e Proof. It follows from the definition of 𝒦β (EA∞ ) and the upper boundedness of f that ef ∈ Cb (EA∞ ) and 󵄩󵄩 f 󵄩󵄩 sup Re(f |F ) < ∞. 󵄩󵄩e 󵄩󵄩∞,F ≤ e

(20.1)

If ω, τ ∈ F and ω1 = τ1 , we get from Lemma 20.1.3 that 󵄨󵄨 f (ω) 󵄨 󵄨 󵄨 − ef (τ) 󵄨󵄨󵄨 ≤ exp(max{Re(f (ω)), Re(f (τ))})󵄨󵄨󵄨f (ω) − f (τ)󵄨󵄨󵄨 ≤ esup Re(f |F ) υβ,F (f )dβ (ω, τ). 󵄨󵄨e Therefore, υβ,F (ef ) ≤ esup Re(f |F ) υβ,F (f ). Together, (20.1) and (20.2) yield the result.

(20.2)

802 | 20 Real analyticity of topological pressure and Hausdorff dimension Given e ∈ E and h : EA∞ 󳨀→ ℂ, define the function h ∘ e : EA∞ 󳨀→ ℂ by h(eω) if Aeω1 = 1

h ∘ e(ω) := {

0

if Aeω1 = 0.

One can then rewrite the transfer operator ℒf in the form (cf. (17.22)) ℒf (g)(ω) = ∑ g ∘ e(ω) exp(f ∘ e(ω)). e∈E

Moreover, we have the following basic properties. Lemma 20.1.5. For every e ∈ E, the following statements hold: (a) If f ∈ 𝒦β (EA∞ ), then f ∘ e ∈ 𝒦β (EA∞ ) and υβ (f ∘ e) = υβ,[e] (f ) ≤ υβ (f ). (b) If f ∈ 𝒦βb (EA∞ ), then f ∘ e ∈ 𝒦βb (EA∞ ) and ‖f ∘ e‖β = ‖f ‖β,[e] ≤ ‖f ‖β . (c) If f , g : EA∞ → ℂ and h = f + g, then h ∘ e = f ∘ e + g ∘ e. In the sequel, we will make use of complex analysis on Banach spaces. For more information on this topic, we refer the reader to [25, 86, 110]. Definition 20.1.6. Consider the following conditions: (A) E is a countable alphabet and A : E × E → {0, 1} is a finitely irreducible matrix. (B) φ ∈ 𝒦β (EA∞ ) for some β ∈ (0, 1]. (C) ψ ∈ 𝒦βs (EA∞ , ℝ) is strongly summable, i. e., there exists ν ∈ (0, 1) such that νψ ∈ 𝒦βs (EA∞ , ℝ). (D) There are numbers r0 ∈ ℝ and δ > 0, and a function Bℂ (r0 , 4δ) ∋ z 󳨃󳨀→ (γz : EA∞ → ℂ) with the following properties: (D1)γz ∈ 𝒦β (EA∞ ) for every z ∈ Bℂ (r0 , 4δ) and Q := sup{υβ (γz ) : z ∈ Bℂ (r0 , 4δ)} < ∞. (D2)For every ω ∈ EA∞ , the function Bℂ (r0 , 4δ) ∋ z 󳨃󳨀→ γz (ω) ∈ ℂ is holomorphic. (E) There exists κ ≥ 1 such that for every z ∈ Bℂ (r0 , 4δ), the function γz satisfies |γz | ≤ κ(|ψ| + 1). (F) For every z ∈ Bℂ (r0 , 4δ), the function φz := φ + γz satisfies Re(φz ) ≤ ψ.

20.1 Real analyticity of pressure: part I

| 803

Under these conditions, the functions {φz }z∈Bℂ (r0 ,4δ) are said to form a complexparametrized family of complex-valued potentials. As an immediate consequence of Cauchy’s integral formula (Theorem A.3.2 in Appendix A), we get from conditions (D2) and (E) that 󵄨󵄨 ′ 󵄨󵄨 κ 󵄨󵄨γz 󵄨󵄨 ≤ (|ψ| + 1), δ

∀z ∈ Bℂ (r0 , 3δ).

(20.3)

We shall now prove that sup{υβ (γ′z ) : z ∈ Bℂ (r0 , 3δ)} ≤

Q < ∞. δ

(20.4)

Indeed, condition (D1) guarantees that 󵄨 󵄨 |ω ∧ τ| ≥ 1 󳨐⇒ 󵄨󵄨󵄨γz (ω) − γz (τ)󵄨󵄨󵄨 ≤ Qdβ (ω, τ),

∀z ∈ Bℂ (r0 , 4δ).

It follows from Cauchy’s integral formula and condition (D2) that 󵄨 󵄨 Q |ω ∧ τ| ≥ 1 󳨐⇒ 󵄨󵄨󵄨γ′z (ω) − γ′z (τ)󵄨󵄨󵄨 ≤ dβ (ω, τ), δ

∀z ∈ Bℂ (r0 , 3δ).

This means that γ′z ∈ 𝒦β (EA∞ ) and υβ (γ′z ) ≤ Q/δ for all z ∈ Bℂ (r0 , 3δ). Relation (20.4) ensues. Note that φ′z = γ′z for all z ∈ Bℂ (r0 , 4δ). So (20.3)–(20.4) hold with γ′z replaced by φ′z , i. e., 󵄨󵄨 ′ 󵄨󵄨 κ 󵄨󵄨φz 󵄨󵄨 ≤ (|ψ| + 1) δ z∈Bℂ (r0 ,3δ)

and

sup

sup

z∈Bℂ (r0 ,3δ)

υβ (φ′z ) ≤

Q . δ

(20.5)

For every z ∈ Bℂ (r0 , 4δ) and e ∈ E, define the linear operator ℒφz ,e : Cb (EA∞ ) 󳨀→ Cb (EA∞ ) by ℒφz ,e (g) := (g ∘ e) ⋅ exp(φz ∘ e).

(20.6)

As a direct repercussion of Lemmas 20.1.4–20.1.5, relation (18.2) with F = [e], and conditions (D1) and (F), this operator has the following properties. Lemma 20.1.7. Let {φz }z∈Bℂ (r0 ,4δ) be a complex-parametrized family of complex-valued potentials. For every z ∈ Bℂ (r0 , 4δ) and e ∈ E, the linear operator ℒφz ,e preserves the Banach space 𝒦βb (EA∞ ), i. e., b

∞

b

∞

ℒφz ,e (𝒦β (EA )) ⊆ 𝒦β (EA ).

Furthermore, this operator is bounded and ‖ℒφz ,e ‖β ≤ (1 + υβ (φz )) exp(sup Re(φz |[e] )) ≤ (1 + υβ (φ) + Q) exp(sup ψ|[e] ).

804 | 20 Real analyticity of topological pressure and Hausdorff dimension We now investigate the dependence on z of the operator ℒφz ,e . Lemma 20.1.8. Let {φz }z∈Bℂ (r0 ,4δ) be a complex-parametrized family of complex-valued potentials. For every e ∈ E, the function Bℂ (r0 , 2δ) ∋ z 󳨃󳨀→ ℒφz ,e ∈ L(𝒦βb (EA∞ )) is continuous. In fact, there exists a constant M ≥ 1 such that 󵄩󵄩 󵄩 󵄩󵄩ℒφz ,e − ℒφζ ,e 󵄩󵄩󵄩β ≤ M exp(ν sup ψ|[e] )|z − ζ | for every e ∈ E and all z, ζ ∈ Bℂ (r0 , 2δ), where ν comes from Definition 20.1.6(C). Proof. By a standard calculus argument, there is a constant Dψ ≥ 1 such that κ max{1, 2(υβ (φ) + Q)} ⋅ (|x| + υβ (ψ) + 1)ex ≤ Dψ eνx , δ

∀x ≤ sup ψ.

(20.7)

Fix e ∈ E and z, ζ ∈ Bℂ (r0 , 2δ). We first seek an upper bound on the supremum norm of eφz ∘e −eφζ ∘e . Using successively Lemma 20.1.3, condition (F), the mean value inequality, relation (20.5), Exercise 18.5.6 and relation (20.7), we obtain that 󵄩󵄩 φz ∘e 󵄩 − eφζ ∘e 󵄩󵄩󵄩∞ ≤ exp( sup max{Re(φz (ω)), Re(φζ (ω))}) ⋅ ‖φz − φζ ‖∞,[e] 󵄩󵄩e ω∈[e]

≤ exp(sup ψ|[e] ) ⋅

sup

ξ ∈Bℂ (r0 ,2δ)

󵄩󵄩 ′ 󵄩󵄩 󵄩󵄩φξ |[e] 󵄩󵄩∞ ⋅ |z − ζ |

κ 󵄩 󵄩 ≤ (󵄩󵄩󵄩ψ|[e] 󵄩󵄩󵄩∞ + 1) exp(sup ψ|[e] )|z − ζ | δ κ 󵄨 󵄨 ≤ (󵄨󵄨󵄨sup ψ|[e] 󵄨󵄨󵄨 + υβ (ψ) + 1) exp(sup ψ|[e] )|z − ζ | δ ≤ Dψ exp(ν sup ψ|[e] )|z − ζ |.

(20.8)

We now find a Hölder constant for eφz ∘e −eφζ ∘e . For every ω ∈ EA∞ , the fundamental theorem of calculus affirms that z

eφz (ω) − eφζ (ω) = ∫ φ′ξ (ω)eφξ (ω) dξ , ζ

where the integration is along the closed line segment from ζ to z. Let ω, τ ∈ EA∞ . It ensues that (e

φz (ω)

−e

φζ (ω)

) − (e

φz (τ)

−e

φζ (τ)

z

) = ∫(φ′ξ (ω)eφξ (ω) − φ′ξ (τ)eφξ (τ) ) dξ ζ z

= ∫[(φ′ξ (ω) − φ′ξ (τ))eφξ (ω) + φ′ξ (τ)(eφξ (ω) − eφξ (τ) )] dξ . ζ

20.1 Real analyticity of pressure: part I

| 805

Assume that τ1 = ω1 and that ψ(ω) ≤ ψ(τ). Using Lemma 20.1.3, conditions (F) and (D1), as well as relations (20.5) and (20.7), we get for every ξ ∈ [ζ , z] that 󵄨󵄨 ′ 󵄨 󵄨󵄨(φ (ω) − φ′ (τ))eφξ (ω) + φ′ (τ)(eφξ (ω) − eφξ (τ) )󵄨󵄨󵄨 ξ ξ 󵄨󵄨 ξ 󵄨󵄨 󵄨 󵄨󵄨 ′ 󵄨󵄨 Re(φξ (ω)) 󵄨󵄨 ′ 󵄨󵄨 󵄨󵄨 φξ (ω) ′ − eφξ (τ) 󵄨󵄨󵄨 ≤ 󵄨󵄨φξ (ω) − φξ (τ)󵄨󵄨e + 󵄨󵄨φξ (τ)󵄨󵄨 ⋅ 󵄨󵄨e

≤ υβ (φ′ξ )dβ (ω, τ)eRe(φξ (ω)) 󵄨 󵄨 󵄨 󵄨 + 󵄨󵄨󵄨φ′ξ (τ)󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨φξ (ω) − φξ (τ)󵄨󵄨󵄨 ⋅ exp(max{Re(φξ (ω)), Re(φξ (τ))}) 󵄨 󵄨 ≤ υβ (φ′ξ )dβ (ω, τ)eψ(ω) + 󵄨󵄨󵄨φ′ξ (τ)󵄨󵄨󵄨 ⋅ υβ (φξ )dβ (ω, τ) ⋅ exp(max{ψ(ω), ψ(τ)}) 󵄨 󵄨 ≤ (υβ (φ′ξ ) + υβ (φξ )󵄨󵄨󵄨φ′ξ (τ)󵄨󵄨󵄨)eψ(τ) dβ (ω, τ) κ Q ≤ ( + υβ (φξ ) ⋅ (|ψ(τ)| + 1))eψ(τ) dβ (ω, τ) δ δ Q κ ≤ ( + (υβ (φ) + Q)(|ψ(τ)| + 1))eψ(τ) dβ (ω, τ) δ δ 2κ ≤ (υβ (φ) + Q)(|ψ(τ)| + 1)eψ(τ) dβ (ω, τ) δ ≤ Dψ eνψ(τ) dβ (ω, τ) ≤ Dψ exp(sup(νψ|[ω1 ] ))dβ (ω, τ).

Hence, 󵄨󵄨 φz (ω) 󵄨 − eφζ (ω) ) − (eφz (τ) − eφζ (τ) )󵄨󵄨󵄨 ≤ Dψ exp(sup(νψ|[ω1 ] ))|z − ζ |dβ (ω, τ). 󵄨󵄨(e Thus, for every e ∈ E we deduce that 󵄨󵄨 φz ∘e(ω) 󵄨 − eφζ ∘e(ω) ) − (eφz ∘e(τ) − eφζ ∘e(τ) )󵄨󵄨󵄨 ≤ Dψ exp(ν sup ψ|[e] )|z − ζ |dβ (ω, τ). 󵄨󵄨(e This holds for all ω, τ ∈ EA∞ such that ω1 = τ1 . Therefore, υβ (eφz ∘e − eφζ ∘e ) ≤ Dψ exp(ν sup ψ|[e] )|z − ζ |. Combining this with (20.8), we conclude that 󵄩󵄩 φz ∘e 󵄩 − eφζ ∘e 󵄩󵄩󵄩β ≤ 2Dψ exp(ν sup ψ|[e] )|z − ζ |. 󵄩󵄩e Applying Lemma 20.1.5(c) and (18.2) with F = EA∞ , we get for every g ∈ 𝒦βb (EA∞ ) that 󵄩󵄩 󵄩 󵄩 φ ∘e 󵄩 φ ∘e 󵄩󵄩ℒφz ,e (g) − ℒφζ ,e (g)󵄩󵄩󵄩β = 󵄩󵄩󵄩(e z ) ⋅ (g ∘ e) − (e ζ ) ⋅ (g ∘ e)󵄩󵄩󵄩β 󵄩 󵄩 = 󵄩󵄩󵄩(eφz ∘e − eφζ ∘e ) ⋅ (g ∘ e)󵄩󵄩󵄩β ≤ 2Dψ exp(ν sup ψ|[e] )|z − ζ | ⋅ ‖g‖β .

806 | 20 Real analyticity of topological pressure and Hausdorff dimension Thus, 󵄩 󵄩󵄩 󵄩󵄩ℒφz ,e − ℒφζ ,e 󵄩󵄩󵄩β ≤ 2Dψ exp(ν sup ψ|[e] )|z − ζ |. Choose M = 2Dψ . We now look at the operator which is the sum over e ∈ E of the operators ℒφz ,e . Proposition 20.1.9. Let {φz }z∈Bℂ (r0 ,4δ) be a complex-parametrized family of complexvalued potentials. (a) The series Bℂ (r0 , 2δ) ∋ z 󳨃󳨀→ ℒφz := ∑ ℒφz ,e e∈E

converges absolutely uniformly in 𝒦βb (EA∞ ). (b) In particular: (b1) The linear operator ℒφz preserves the Banach space 𝒦βb (EA∞ ), i. e., b

∞

b

∞

ℒφz (𝒦β (EA )) ⊆ 𝒦β (EA ),

∀z ∈ Bℂ (r0 , 2δ).

(b2) The operator ℒφz : 𝒦βb (EA∞ ) 󳨀→ 𝒦βb (EA∞ ) is bounded and 󵄩󵄩 󵄩 󵄩󵄩ℒφz 󵄩󵄩󵄩β ≤ (1 + υβ (φ) + Q) ∑ exp(sup ψ|[e] ), e∈E

∀z ∈ Bℂ (r0 , 2δ).

(b3) The function Bℂ (r0 , 2δ) ∋ z 󳨃󳨀→ ℒφz ∈ L(𝒦βb (EA∞ )) is continuous. (c) Moreover, 󵄩󵄩 󵄩 󵄩󵄩ℒφz − ℒφζ 󵄩󵄩󵄩β ≤ M( ∑ exp(ν sup ψ|[e] ))|z − ζ |, e∈E

∀z, ζ ∈ Bℂ (r0 , 2δ).

Proof. Items (a) and (b) are an immediate consequence of Lemma 20.1.7, condition (C) and the first part of Lemma 20.1.8. By invoking the second part of Lemma 20.1.8 and item (a), we get 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩ℒφz − ℒφζ 󵄩󵄩󵄩β ≤ ∑ 󵄩󵄩󵄩ℒφz ,e − ℒφζ ,e 󵄩󵄩󵄩β ≤ M( ∑ exp(ν sup ψ|[e] ))|z − ζ |. e∈E

e∈E

So item (c) is also established. More can be said about the function z 󳨃󳨀→ ℒφz .

20.1 Real analyticity of pressure: part I

| 807

Lemma 20.1.10. Let {φz }z∈Bℂ (r0 ,4δ) be a complex-parametrized family of complex-valued potentials. Then the function Bℂ (r0 , δ) ∋ z 󳨃󳨀→ ℒφz ∈ L(𝒦βb (EA∞ )) is holomorphic. Proof. Let Γ ⊆ Bℂ (r0 , δ) be a closed rectifiable curve. Fix g ∈ 𝒦βb (EA∞ ) and ω ∈ EA∞ . As the function Bℂ (r0 , δ) ∋ z 󳨃󳨀→ (ℒφz ,e (g))(ω) ∈ ℂ is holomorphic for every e ∈ E, it follows from Proposition 20.1.9(a) that the function Bℂ (r0 , δ) ∋ z 󳨃󳨀→ (ℒφz (g))(ω) ∈ ℂ is holomorphic. Hence, by Cauchy’s theorem (Theorem A.3.1 in Appendix A), ∫(ℒφz (g))(ω) dz = 0. Γ

Since the function Bℂ (r0 , δ) ∋ z 󳨃󳨀→ ℒφz (g) ∈ 𝒦βb (EA∞ ) is continuous according to Proposition 20.1.9(c), the integral ∫Γ ℒφz (g) dz exists and ∫ ℒφz (g) dz(ω) := ∫(ℒφz (g))(ω) dz = 0, Γ

Γ

∀ω ∈ EA∞ .

Thus, ∫ ℒφz (g) dz = 0. Γ

As the function Bℂ (r0 , δ) ∋ z 󳨃󳨀→ ℒφz ∈ L(𝒦βb (EA∞ )) is continuous by Proposition 20.1.9(c), the integral ∫Γ ℒφz dz exists and ∫ ℒφz dz(g) := ∫ ℒφz (g) dz = 0, Γ

Γ

∀g ∈ 𝒦βb (EA∞ ).

Therefore, ∫ ℒφz dz = 0. Γ

By Morera’s theorem, the map Bℂ (r0 , δ) ∋ z 󳨃󳨀→ ℒφz ∈ L(𝒦βb (EA∞ )) is holomorphic. (Note: Morera’s theorem for Banach space-valued functions is a direct consequence

808 | 20 Real analyticity of topological pressure and Hausdorff dimension of the classical, complex-valued version of Morera’s theorem (Theorem A.3.3) and the equivalence of weak and strong holomorphicity, which is for instance proved in Theorem 3.31 of Rudin [110]). As an immediate reverberation of this lemma and Hartogs’ theorem (Theorem A.4.1 in Appendix A), we obtain the following. Lemma 20.1.11. Let {φz }z∈B

ℂd

(r0 ,4δ)

be a d-dimensional complex-parametrized family of

complex-valued potentials with ℂ replaced by ℂd and r0 ∈ ℝd . Then the function Bℂd (r0 , δ) ∋ z 󳨃󳨀→ ℒφz ∈ L(𝒦βb (EA∞ )) is holomorphic.

Finally, we can establish a first result on the real analyticity of the pressure function. Theorem 20.1.12. Let {φz }z∈B

ℂd

(r0 ,4δ)

be a d-dimensional complex-parametrized family

of complex-valued potentials. If r0 ∈ ℝd and φr ∈ 𝒦β (EA∞ , ℝ) for all r ∈ Bℝd (r0 , δ), then the function Bℝd (r0 , δ) ∋ r 󳨃󳨀→ P(φr ) ∈ ℝ is real analytic. Proof. Fix u ∈ Bℝd (r0 , δ). Then there exists R > 0 such that Bℂd (u, R) ⊆ Bℂd (r0 , δ).

By Theorem 18.1.12, for all r ∈ Bℝd (u, R) the number exp(P(φr )) is a simple isolated eigenvalue of the operator ℒφr ∈ L(𝒦βb (EA∞ )). (For more information on the spectral theory of bounded linear operators, see [21, 26].) It follows from Lemma 20.1.11 and the Kato–Rellich theorem for perturbations of linear operators (Theorem A.2.5) that there exist R′ ∈ (0, R] and a holomorphic function ρ : Bℂd (u, R′ ) 󳨀→ ℂ such that ρ(z) is an eigenvalue of the operator ℒφz for all z ∈ Bℂd (u, R′ ) and ρ(r) = exp(P(φr )),

∀r ∈ Bℝd (u, R′ ).

Consequently, the function Bℝd (u, R′ ) ∋ r 󳨃󳨀→ P(φr ) ∈ ℝ is real analytic. Since real analyticity is a local property, we are done.

20.2 Real analyticity of pressure: part II

| 809

20.2 Real analyticity of pressure: part II Let us look at the special case of a complex-parametrized family of complex-valued potentials {φz }z∈Bℂ (r0 ,4δ) originating from a “linear” family φz = φ + zγ

(20.9)

with φr0 strongly summable. The functions φ and γ are assumed to take the form φ = bφ + mφ f

and γ = bγ + mγ f ,

(20.10)

where bφ , bγ ∈ 𝒦βb (EA∞ , ℝ),

mφ , mγ ∈ ℝ,

and

f ∈ 𝒦β (EA∞ , ℝ).

(20.11)

Under these assumptions, we will check that the family {φz }z∈Bℂ (r0 ,4δ) fulfills conditions (A)–(F) in Definition 20.1.6 for appropriate choices of ψ, κ, and δ. Condition (A) is met de facto. From (20.10)–(20.11), it is immediate that φ, γ ∈ 𝒦β (EA∞ , ℝ). In particular, condition (B) holds. Regarding condition (C), observe that φr = (bφ + rbγ ) + (mφ + rmγ )f ,

∀r ∈ B(r0 , 4δ).

As bφ + rbγ ∈ 𝒦βb (EA∞ , ℝ), Lemma 20.1.2(d) asserts that the strong summability of φr is equivalent to the strong summability of (mφ + rmγ )f . But then it follows from Lemma 20.1.2(b) that the strong summability of φr0 (or, equivalently, of (mφ + r0 mγ )f ) entails the existence of an open interval I ⊆ ℝ centered on r0 such that the function φr (or, equivalently, (mφ + rmγ )f ) is strongly summable for every r ∈ I. If necessary, reduce δ > 0 so that B(r0 , 8δ) ⊆ I. Let m = min{mφ + rmγ : r ∈ B(r0 , 4δ)} and M = max{mφ + rmγ : r ∈ B(r0 , 4δ)}. By reducing δ if necessary, it is possible to make sure that mM ≠ 0. Set ψ = [‖bφ ‖∞ + (|r0 | + 4δ)‖bγ ‖∞ ] + max{mf , Mf }. By definition of m and M, the functions mf and Mf are both strongly summable. It readily ensues that their maximum is strongly summable. The term between square brackets being constant, it follows from Lemma 20.1.2(d) that the function ψ is strongly summable. Condition (C) is satisfied. Concerning condition (D), by comparing the expression of φz in (20.9) with its definition in condition (F), we must set γz = zγ in condition (D). As a multiple of γ ∈ 𝒦β (EA∞ , ℝ), it is clear that γz ∈ 𝒦β (EA∞ ) for every z ∈ Bℂ (r0 , 4δ) and Q :=

sup

z∈Bℂ (r0 ,4δ)

υβ (γz ) =

sup

z∈Bℂ (r0 ,4δ)

|z| ⋅ υβ (γ) = (|r0 | + 4δ)υβ (γ) < ∞.

810 | 20 Real analyticity of topological pressure and Hausdorff dimension Furthermore, for every ω ∈ EA∞ the function Bℂ (r0 , 4δ) ∋ z 󳨃󳨀→ γz (ω) = zγ(ω) ∈ ℂ is obviously holomorphic. Thus, condition (D) is fulfilled. For condition (E), let z ∈ Bℂ (r0 , 4δ). First, note that the summability of ψ imposes that the set E ′ := {e ∈ E : ψ|[e] ≤

1 max{mf |[e] , Mf |[e] } < 0} 2

is cofinite. On ⋃e∈E ′ [e], it turns out that 1󵄨 󵄨 1 |ψ| ≥ 󵄨󵄨󵄨max{mf , Mf }󵄨󵄨󵄨 ≥ |f | min{|m|, |M|} > 0 2 2 and hence 󵄨 󵄨 |γz | = |z||γ| ≤ (|r0 | + 4δ)󵄨󵄨󵄨bγ + mγ f 󵄨󵄨󵄨 ≤ (|r0 | + 4δ)(‖bγ ‖∞ + |mγ ||f |) ≤ (|r0 | + 4δ)[‖bγ ‖∞ + ≤ (|r0 | + 4δ)[‖bγ ‖∞ +

1 2

1 min{|m|, |M|}|f |] 2 min{|m|, |M|} |mγ |

2|mγ |

⋅

min{|m|, |M|}

|ψ|].

On the complement ⋃e∈E\E ′ [e], it is easy to see that |γz | ≤ (|r0 | + 4δ)U, where U := 2|m |

γ supe∈E\E ′ ‖γ‖∞,[e] < ∞. Choosing κ = (|r0 | + 4δ) max{U, ‖bγ ‖∞ , min{|m|,|M|} }, condition (E) holds. Finally, as Re(φz ) = φRe(z) , condition (F) reduces to demonstrating that φr ≤ ψ for every r ∈ B(r0 , 4δ). But

ψ ≥ [‖bφ ‖∞ + |r| ⋅ ‖bγ ‖∞ ] + (mφ + rmγ )f ≥ (bφ + rbγ ) + (mφ + rmγ )f = φr . That is, condition (F) is respected. Therefore, the following result is a consequence of Proposition 20.1.9(b). Lemma 20.2.1. Let σ : EA∞ → EA∞ be a finitely irreducible subshift of finite type over a countable alphabet E. Let {φz }z∈Bℂ (r0 ,4δ) be a complex-parametrized family of complexvalued potentials of the form (20.9)–(20.11). If the potential φr0 is strongly summable, then there exists 0 < δr0 < δ such that for every z ∈ Bℂ (r0 , 2δr0 ) we have b

∞

b

∞

ℒφ+zγ (𝒦β (EA )) ⊆ 𝒦β (EA )

(20.12)

and 󵄩󵄩 󵄩 󵄩󵄩ℒφ+zγ 󵄩󵄩󵄩β ≤ (1 + υβ (φ) + (|r0 | + 4δr0 )υβ (γ)) ∑ exp(sup ψr0 |[e] ) < ∞, e∈E

(20.13)

20.2 Real analyticity of pressure: part II

| 811

where ψr0 = [‖bφ ‖∞ + (|r0 | + 4δr0 )‖bγ ‖∞ ] + max{mr0 f , Mr0 f } and mr0 = min{mφ + rmγ : r ∈ B(r0 , 4δr0 )} and Mr0 = max{mφ + rmγ : r ∈ B(r0 , 4δr0 )}. Similarly, as a direct consequence of Lemma 20.1.10 we get the following. Lemma 20.2.2. Let σ : EA∞ → EA∞ be a finitely irreducible subshift of finite type over a countable alphabet E. Let {φz }z∈Bℂ (r0 ,4δ) be a complex-parametrized family of complexvalued potentials of the form (20.9)–(20.11). If the potential φr0 is strongly summable, then there exists 0 < δr0 < δ such that the function Bℂ (r0 , δr0 ) ∋ z 󳨃󳨀→ ℒφ+zγ ∈ L(𝒦βb (EA∞ )) is holomorphic. Here is the second result on the real analyticity of pressure. It is a direct application of Theorem 20.1.12. Theorem 20.2.3. Let σ : EA∞ → EA∞ be a finitely irreducible subshift of finite type over a countable alphabet E. Let {φz }z∈Bℂ (r0 ,4δ) be a complex-parametrized family of complexvalued potentials of the form (20.9)–(20.11). If the potential φr0 is strongly summable, then there exists an open real interval I ∋ r0 over which the function I ∋ r 󳨃󳨀→ P(φr ) = P(φ + rγ) ∈ ℝ is real analytic. We now examine the derivative of the pressure function. Theorem 20.2.4. Let σ : EA∞ → EA∞ be a finitely irreducible subshift of finite type over a countable alphabet E. Let {φz }z∈Bℂ (r0 ,4δ) be a complex-parametrized family of complexvalued potentials of the form (20.9)–(20.11). If the potential φr0 is strongly summable, then dP(φ + rγ) 󵄨󵄨󵄨󵄨 = ∫ γ dμφ+r0 γ , 󵄨󵄨 󵄨󵄨r=r0 dr ∞ EA

where μφ+r0 γ = μφr is the unique ergodic σ-invariant Gibbs state and the unique equi0 librium state for φr0 . Proof. The proof goes along similar lines to that of Theorem 16.4.10. By Corollary 17.7.5, the potential φr0 has a unique σ-invariant Gibbs state μφr . This 0

state is ergodic. The strong summability of φr0 implies that ∫E ∞ φr dμφr A

0

> −∞ for

all r in some open real interval containing r0 . In particular, ∫E ∞ φr0 dμφr > −∞ and A

0

812 | 20 Real analyticity of topological pressure and Hausdorff dimension Corollary 17.7.5 guarantees that μφr is the unique equilibrium state for φr0 . The 3rd 0 variational principle (Theorem 17.3.4) then entails that for every r in the said interval, we have P(φ + rγ) ≥ hμφ+r γ (σ) + ∫ (φ + rγ) dμφ+r0 γ 0

EA∞

= hμφ+r γ (σ) + ∫ (φ + r0 γ) dμφ+r0 γ + (r − r0 ) ∫ γ dμφ+r0 γ 0

EA∞

EA∞

= P(φ + r0 γ) + (r − r0 ) ∫ γ dμφ+r0 γ . EA∞

On one hand, P(φ + rγ) − P(φ + r0 γ) dP(φ + rγ) (r0 ) = lim+ ≥ ∫ γ dμφ+r0 γ . dr r − r0 r→r0 ∞ EA

On the other hand, P(φ + rγ) − P(φ + r0 γ) dP(φ + rγ) (r0 ) = lim− ≤ ∫ γ dμφ+r0 γ . dr r − r0 r→r0 ∞ EA

In conclusion,

dP(φ+rγ) (r0 ) dr

= ∫E ∞ γ dμφ+r0 γ . A

An alternative proof analogous to that of Theorem 13.10.6 is left to the reader as an exercise.

20.3 Real analyticity of pressure: part III We now want to prove a result similar to Theorem 20.1.12, in a sense stronger but in another sense weaker. This time, the complex-parametrized family of complex-valued potentials {φz }z∈Bℂ (0,4δ) is assumed to originate from a “linear” family φz = f + h + zγ,

(20.14)

with a strongly summable potential f ∈ 𝒦β (EA∞ , ℝ) and h, γ ∈ 𝒦βb (EA∞ ). Under these assumptions, we will check that the family {φz }z∈Bℂ (0,4δ) fulfills conditions (A)–(F) in Definition 20.1.6 for any δ > 0 and appropriate choices of φ, ψ, and κ. Condition (A) is met de facto. Let φ = f + h. Then φ ∈ 𝒦β (EA∞ ) since f , h ∈ 𝒦β (EA∞ ). So condition (B) is satisfied.

(20.15)

20.3 Real analyticity of pressure: part III

| 813

Regarding condition (C), set ψ = f + [‖h‖∞ + 4δ‖γ‖∞ ]. The term between square brackets being constant, it follows from Lemma 20.1.2(d) that the function ψ is strongly summable, like f . Condition (C) holds. Concerning condition (D), by comparing the expression of φz in (20.14) with its definition in condition (F) and taking into account (20.15), we must set γz = zγ in condition (D). As a multiple of γ ∈ 𝒦βb (EA∞ ), it is clear that γz ∈ 𝒦βb (EA∞ ) for every z ∈ Bℂ (0, 4δ) and Q :=

sup

z∈Bℂ (0,4δ)

υβ (γz ) =

sup

z∈Bℂ (0,4δ)

|z| ⋅ υβ (γ) = 4δυβ (γ) < ∞.

Furthermore, for every ω ∈ EA∞ the function Bℂ (0, 4δ) ∋ z 󳨀 󳨃 → γz (ω) = zγ(ω) ∈ ℂ is obviously holomorphic. Thus, condition (D) is fulfilled. For condition (E), let z ∈ Bℂ (0, 4δ) and notice that |γz | = |z||γ| ≤ 4δ‖γ‖∞ . Choosing κ = 4δ‖γ‖∞ , condition (E) holds. Finally, for any z ∈ Bℂ (0, 4δ) we have ψ = f + [‖h‖∞ + 4δ‖γ‖∞ ] ≥ f + [|h| + |z||γ|] ≥ f + Re(h) + Re(zγ) = Re(φz ). That is, condition (F) is respected. Therefore, as a direct consequence of Proposition 20.1.9(b) we obtain the following. Lemma 20.3.1. Let σ : EA∞ → EA∞ be a finitely irreducible subshift of finite type over a countable alphabet E. If f ∈ 𝒦β (EA∞ , ℝ) is strongly summable and h, γ ∈ 𝒦βb (EA∞ ), then b

∞

b

∞

ℒf +h+zγ (𝒦β (EA )) ⊆ 𝒦β (EA ),

∀z ∈ ℂ

(20.16)

and 󵄩󵄩 󵄩 ‖h‖ +4δ‖γ‖∞ ∑ exp(sup f |[e] ) < ∞ (20.17) 󵄩󵄩ℒf +h+zγ 󵄩󵄩󵄩β ≤ (1 + υβ (f ) + υβ (h) + 4δυβ (γ))e ∞ e∈E

for all z ∈ Bℂ (0, 2δ). Similarly, as a direct consequence of Lemma 20.1.10 we get the following. Lemma 20.3.2. Let σ : EA∞ → EA∞ be a finitely irreducible subshift of finite type over a countable alphabet E. If f ∈ 𝒦β (EA∞ , ℝ) is strongly summable and h, γ ∈ 𝒦βb (EA∞ ), then the function ℂ ∋ z 󳨃󳨀→ ℒf +h+zγ ∈ L(𝒦βb (EA∞ ))

814 | 20 Real analyticity of topological pressure and Hausdorff dimension is holomorphic. This means that the function b

b

𝒦β (EA ) ∋ h 󳨃󳨀→ ℒf +h ∈ L(𝒦β (EA )) ∞

∞

is G (Goursat)-holomorphic in the sense of [86] and [25]. Here is the third result on the real analyticity of pressure. Theorem 20.3.3. Let σ : EA∞ → EA∞ be a finitely irreducible subshift of finite type over a countable alphabet E. If f ∈ 𝒦β (EA∞ , ℝ) is strongly summable, then the function b

𝒦β (EA , ℝ) ∋ h 󳨃󳨀→ P(f + h) ∈ ℝ ∞

is real analytic. Proof. Fix g ∈ 𝒦βb (EA∞ , ℝ). As real analyticity is a local property, it suffices to prove the real analyticity of the function h 󳨃→ P(f + h) at the point g. By Theorem 18.1.12, the number exp(P(f +g)) is a simple isolated eigenvalue of the operator ℒf +g ∈ L(𝒦βb (EA∞ )). So, it follows from Lemma 20.3.2 and the Kato–Rellich theorem for perturbations of linear operators (Theorem A.2.5) that there exist r > 0 and a holomorphic function B𝒦b (E ∞ ) (g, r) ∋ h 󳨃󳨀→ ρh ∈ 𝒦βb (EA∞ ) A

β

such that ρh is an eigenvalue of the operator ℒf +h for all h ∈ B𝒦b (E ∞ ) (g, r), and ρh =

exp(P(f + h)) for those h ∈ B𝒦b (E ∞ ,ℝ) (g, r). Therefore the function

β

A

A

β

B𝒦b (E ∞ ,ℝ) (g, r) ∋ h 󳨃󳨀→ exp(P(f + h)) ∈ ℝ β

A

is real analytic, whence so is the function B𝒦b (E ∞ ,ℝ) (g, r) ∋ h 󳨃󳨀→ P(f + h) ∈ ℝ. β

A

We now examine the derivative of the pressure function. Theorem 20.3.4. Let σ : EA∞ → EA∞ be a finitely irreducible subshift of finite type over a countable alphabet E. Let h ∈ 𝒦βb (EA∞ , ℝ). Suppose that q0 ∈ ℝ and f ∈ 𝒦β (EA∞ , ℝ) are such that q0 f is strongly summable. Then dP(qf + h) 󵄨󵄨󵄨󵄨 = ∫ f dμq0 f +h , 󵄨󵄨 󵄨󵄨q=q0 dq ∞ EA

where μq0 f +h is the unique ergodic σ-invariant Gibbs state and the unique equilibrium state for q0 f + h. Proof. The proof goes along the same lines as that of Theorem 20.2.4.

20.4 Real analyticity of Hausdorff dimension for CGDMSs in ℂ

| 815

20.4 Real analyticity of Hausdorff dimension for CGDMSs in ℂ The results of this subsection form a far-reaching strengthening of existing theorems about real analyticity (see [106] and references therein) and even continuity (see [104]) of the Hausdorff dimension of limit sets of conformal graph directed Markov systems whose phase space is contained in ℂ. It is worth emphasizing that most of the results in this subsection do not require that the Open Set Condition (OSC) hold, as they are primarily formulated in terms of the real analyticity of the Bowen parameter. Real analyticity of the Hausdorff dimension of limit sets will be obtained as an immediate corollary in the case where the OSC holds. Let Λ ⊆ ℂd be an open set. As in Chapter 19, let 𝒢 = (V, E, i, t) be a directed multigraph and A : E × E → {0, 1} an incidence matrix. Let Φ = {Φλ }λ∈Λ be a family of CGDMSs (not necessarily satisfying the OSC; see Definition 19.3.1) with the following properties: λ λ 󵄨󵄨 (i) Every CGDMS Φλ = {φλe : Xt(e) → Xi(e) 󵄨󵄨 e ∈ E} is generated over the (same) multigraph 𝒢 and the (same) matrix A and compact sets Xυλ ⊆ ℂ for all υ ∈ V and λ ∈ Λ. (ii) Though the seed sets (Xυλ )υ∈V, λ∈Λ may depend on λ, the open, connected and bounded sets (Wυ )υ∈V to which the generators {φλe }e∈E, λ∈Λ extend, are independent of λ. 󵄨 (iii) inf{dist(Xυλ , 𝜕Wυ ) 󵄨󵄨󵄨 υ ∈ V, λ ∈ Λ} > 0. (iv) There exists 0 ≤ s < 1 such that 󵄩󵄩 λ ′ 󵄩󵄩 󵄩󵄩(φe ) 󵄩󵄩Wt(e) ≤ s,

∀λ ∈ Λ, ∀e ∈ E.

Definition 20.4.1. A family Φ = {Φλ }λ∈Λ of CGDMSs in ℂ (not necessarily satisfying the OSC) is called analytic if for any e ∈ E and every z ∈ Wt(e) , the function Λ ∋ λ 󳨃󳨀→ φλe (z) ∈ ℂ is holomorphic. As in (19.14), for every λ ∈ Λ let fλ := ΞΦλ : EA∞ → ℝ be the potential given by ′ 󵄨 󵄨 fλ (ω) := ΞΦλ (ω) = log󵄨󵄨󵄨(φλω1 ) (πλ (σ(ω)))󵄨󵄨󵄨,

∀ω ∈ EA∞ ,

where πλ : EA∞ → JΦλ =: Jλ is the canonical coding map induced by the CGDMS Φλ (cf. Subsection 19.1.3). For every υ ∈ V, choose any xυ ∈ Wυ . Because it is a uniform limit of the holomorphic functions λ 󳨃󳨀→ φλω|n (xt(ωn ) ), the function λ 󳨃󳨀→ πλ (ω) is holomorphic for each ω ∈ EA∞ . Definition 20.4.2. An analytic family Φ = {Φλ }λ∈Λ of CGDMSs in ℂ (not necessarily satisfying the OSC) is called strong at a parameter λ0 ∈ Λ if (a) the CGDMS Φλ0 is strongly regular (then we also say that the parameter λ0 is strongly regular); and

816 | 20 Real analyticity of topological pressure and Hausdorff dimension (b) 󵄨󵄨 f (ω) 󵄨󵄨 󵄨󵄨 󵄨 ∞ sup{󵄨󵄨󵄨 λ 󵄨 : λ ∈ Λ, ω ∈ EA } < ∞. 󵄨󵄨 fλ (ω) 󵄨󵄨󵄨 0 Note that this latter condition is equivalent to 󵄨󵄨 fλ (ω) − fλ (ω) 󵄨󵄨 󵄨 󵄨󵄨 ∞ 0 sup{󵄨󵄨󵄨 󵄨󵄨 : λ ∈ Λ, ω ∈ EA } < ∞. 󵄨󵄨 󵄨󵄨 fλ0 (ω)

(20.18)

For every λ ∈ Λ, let θλ := inf{t ≥ 0 : PΦλ (t) < ∞}

and hλ := inf{t ≥ 0 : PΦλ (t) ≤ 0}

be respectively the finiteness parameter and the Bowen parameter of the CGDMS Φλ . The main result of this section asserts that the Bowen parameter of an analytic family varies real analytically in the vicinity of any strong parameter. Theorem 20.4.3. Let Λ ⊆ ℂd be an open set. If {Φλ }λ∈Λ is an analytic family of finitely irreducible CGDMSs (not necessarily satisfying the OSC), then the Bowen parameter function Λ ∋ λ 󳨃󳨀→ hλ ∈ ℝ is real analytic on some neighborhood of every strong parameter. Proof. Let λ0 ∈ Λ be a strong parameter. For every ω ∈ EA∞ , let ηω : Λ → ℂ \ {0} be the holomorphic function ηω (λ) :=

(φλω1 ) (πλ (σ(ω))) ′

λ

′

(φω01 ) (πλ0 (σ(ω)))

.

Shrinking Λ if necessary, we may assume without loss of generality that Λ is simply connected. Then for every ω ∈ EA∞ the logarithm of ηω has an holomorphic branch, which is denoted by logω ηω , such that logω ηω (λ0 ) = logω 1 = 0. Define the holomorphic function Zω : Λ → ℂ by Zω (λ) =

logω ηω (λ) . fλ0 (ω)

Claim 1. For every M > 0, there exists RM > 0 such that Bℂd (λ0 , RM ) ⊆ Λ and 󵄨 󵄨 sup{󵄨󵄨󵄨Zω (λ)󵄨󵄨󵄨 : ω ∈ EA∞ , λ ∈ Bℂd (λ0 , RM )} < M. Proof of Claim 1. Fix M > 0. Let B > 0 be the supremum in (20.18). For all ω ∈ EA∞ and all λ ∈ Λ, we then have that 󵄨󵄨 fλ (ω) − fλ (ω) 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 B 0 󵄨󵄨 ≤ e . 󵄨󵄨exp(Zω (λ))󵄨󵄨󵄨 ≤ exp󵄨󵄨󵄨Re(Zω (λ))󵄨󵄨󵄨 = exp󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 fλ0 (ω)

(20.19)

20.4 Real analyticity of Hausdorff dimension for CGDMSs in ℂ

| 817

Fix an arbitrary r > 0 so small that Bℂd (λ0 , 2r) ⊆ Λ and let λ ∈ Bℂd (λ0 , r). Set Γr = {γ ∈ Λ : |γj − λj | = r, ∀1 ≤ j ≤ d}. Using Cauchy’s integral formula (Theorem A.3.2) and (20.19), we obtain that 󵄨󵄨 d 󵄨󵄨 󵄨󵄨 󵄨󵄨 1 exp(Zω (γ)) 󵄨󵄨 󵄨󵄨 󵄨 󵄨 dγ ⋅ ⋅ ⋅ dγ ∫ 󵄨󵄨 exp(Zω (λ))󵄨󵄨󵄨 = 󵄨󵄨󵄨 󵄨󵄨 1 d 2 2 󵄨󵄨 dλ 󵄨󵄨 󵄨󵄨 󵄨󵄨 (2πi)d (γ1 − λ1 ) ⋅ ⋅ ⋅ (γd − λd ) Γr

≤

󵄨󵄨 󵄨󵄨 1 󵄨󵄨exp(Zω (γ))󵄨󵄨 |dγ1 | ⋅ ⋅ ⋅ |dγd | ∫ (2π)d |γ1 − λ1 |2 ⋅ ⋅ ⋅ |γd − λd |2 Γr

1 󵄨 󵄨 = ∫󵄨󵄨exp(Zω (γ))󵄨󵄨󵄨|dγ1 | ⋅ ⋅ ⋅ |dγd | (2πr 2 )d 󵄨 ≤

Γr

eB e (2πr) = . (2πr 2 )d rd B

d

As Zω (λ0 ) = 0, we therefore get for all λ ∈ Bℂd (λ0 , r) that B 󵄨󵄨 󵄨 󵄨 󵄨 e 󵄨󵄨exp(Zω (λ)) − 1󵄨󵄨󵄨 = 󵄨󵄨󵄨exp(Zω (λ)) − exp(Zω (λ0 ))󵄨󵄨󵄨 ≤ d |λ − λ0 |. r

Taking RM ∈ (0, r] sufficiently small and recalling that Zω (λ0 ) = 0, we deduce that 󵄨 󵄨 sup{󵄨󵄨󵄨Zω (λ)󵄨󵄨󵄨 : ω ∈ EA∞ , λ ∈ Bℂd (λ0 , RM )} < M. This completes the proof of Claim 1.

◼

Now, let Dℂn (z, r) := ∏nk=1 Bℂ (zk , r) be the n-dimensional polydisk in ℂn centered at z = (z1 , . . . , zn ) and of “radius” r, not to be confused with the open ball Bℂn (z, r) in ℂn centered at z and of real radius r. Indeed, Bℂn (z, r) ⊆ Dℂn (z, r) ⊆ Bℂn (z, √nr) but the equality does not hold when n > 1 and r > 0. Fix any t > θλ0 . Pick any 0 < ε < t − θλ0 and any θλ0 < ̂t < t − ε. Thereafter take M = 4−d min{̂t, t−ε−̂t}/(t+ε) and its associated RM > 0 from Claim 1. For every ω ∈ EA∞ , let Z̃ω : Dℂ2d (λ̃0 , RM /4) → ℂ be the holomorphic function produced by Lemma 16.4.8 with ψ = Zω . Obviously, we can always assume that λ0 = (λ0,1 , . . . , λ0,d ) ∈ ℝd , so that = λ̃0 = (λ0,1 , 0, . . . , λ0,d , 0) ∈ ℝ2d . Let u ∈ Bℂ (t, ε). We will show that {φλ̃ }λ∈B ̃ (r ,4δ) ℂ2d 0 ̃ ̃ {ω 󳨃→ uf (ω)(1 + Z̃ (λ))} is a complex-parametrized family of complex̃ λ0

ω

λ∈Bℂ2d (λ0 ,RM /8)

valued potentials by checking that conditions (A)–(F) in Definition 20.1.6 are satisfied with ℂ replaced by ℂ2d and r0 = λ̃0 ,

δ=

RM , 32

φ(ω) = ufλ0 (ω),

̃ γλ̃ (ω) = ufλ0 (ω)Z̃ω (λ),

ψ(ω) = ̂tfλ0 (ω), κ = 1.

818 | 20 Real analyticity of topological pressure and Hausdorff dimension Conditions (A) and (B) are immediately fulfilled. Condition (C) holds because ̂t > θλ0 . To verify condition (D), first note that (D2) is valid because all functions Z̃ω , ω ∈ EA∞ , are holomorphic and Dℂ2d (λ̃0 , RM /4) ⊇ Bℂ2d (λ̃0 , 4δ). For (D1), fix ω, τ ∈ EA∞ with ω1 = τ1 . Define the holomorphic function Δω,τ : Λ → ℂ by Δω,τ (λ) := logω ηω (λ) − logτ ητ (λ).

(20.20)

Recall that the function λ 󳨃→ πλ (ω) is holomorphic for each ω ∈ EA∞ . Using Koebe distortion theorems (the reformulation in Theorem 19.2.2 and the version for the argument in Theorem A.3.11) and reducing RM if needed, there exists a constant K ′ > 0 such that 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 ′ 󵄨 󵄨󵄨Δω,τ (λ)󵄨󵄨󵄨 ≤ K (󵄨󵄨󵄨πλ (σ(ω)) − πλ (σ(τ))󵄨󵄨󵄨 + 󵄨󵄨󵄨πλ0 (σ(ω)) − πλ0 (σ(τ))󵄨󵄨󵄨),

∀λ ∈ Dℂd (λ0 , RM /2).

So 󵄨󵄨 󵄨 |ω∧τ|−1 , 󵄨󵄨Δω,τ (λ)󵄨󵄨󵄨 ≤ Ks

∀λ ∈ Dℂd (λ0 , RM /2),

(20.21)

̃ ω,τ : D 2d (λ̃0 , RM /8) → ℂ be the holomorphic function produced where K = 2K ′ . Let Δ ℂ by Lemma 16.4.8 with ψ = Δω,τ . By virtue of that lemma and (20.21), we have 󵄨󵄨 ̃ ̃ 󵄨󵄨󵄨 ≤ 4d Ks|ω∧τ|−1 , 󵄨󵄨Δω,τ (λ) 󵄨

∀λ̃ ∈ Dℂ2d (λ̃0 , RM /8).

(20.22)

Since Δω,τ (λ) = fλ0 (ω)Zω (λ) − fλ0 (τ)Zτ (λ),

∀λ ∈ Dℂd (λ0 , RM /2),

we conclude from the unique holomorphic extension theorem (Dℂd (λ0 , RM /8) = Dℝ2d (λ̃0 , RM /8) is a real, not a complex, submanifold and the least complex submanifold containing Dℝ2d (λ̃0 , RM /8) is Dℂ2d (λ̃0 , RM /8)) that ̃ = f (ω)Z̃ (λ) ̃ − f (τ)Z̃ (λ), ̃ ̃ ω,τ (λ) Δ λ0 ω λ0 τ

∀λ̃ ∈ Dℂ2d (λ̃0 , RM /8).

(20.23)

In particular, (20.22)–(20.23) hold for all λ̃ ∈ Bℂ2d (λ̃0 , 4δ). These relations yield 󵄨󵄨 󵄨 󵄨 ̃ d −1 −β|ω∧τ| ̃ 󵄨󵄨 , 󵄨󵄨γλ̃ (ω) − γλ̃ (τ)󵄨󵄨󵄨 = 󵄨󵄨󵄨uΔ ω,τ (λ)󵄨󵄨 ≤ (t + ε) ⋅ 4 Ks e

∀λ̃ ∈ Bℂ2d (λ̃0 , 4δ),

where β = − log s. This means that υβ (γλ̃ ) ≤ 4d Ks−1 (t + ε), whence condition (D1) is established.

∀λ̃ ∈ Bℂ2d (λ̃0 , 4δ),

(20.24)

20.4 Real analyticity of Hausdorff dimension for CGDMSs in ℂ

| 819

By Lemma 16.4.8, Claim 1 and the choice of M, observe further that 󵄨󵄨 󵄨 󵄨 ̃ 󵄨󵄨󵄨 ≤ (t + ε)󵄨󵄨󵄨f (ω)󵄨󵄨󵄨 ⋅ 4d M ≤ ̂t 󵄨󵄨󵄨f (ω)󵄨󵄨󵄨 = 󵄨󵄨󵄨ψ(ω)󵄨󵄨󵄨 󵄨󵄨γλ̃ (ω)󵄨󵄨󵄨 = 󵄨󵄨󵄨ufλ0 (ω)Z̃ω (λ) 󵄨 󵄨 󵄨 󵄨 λ0 󵄨 󵄨 λ0 󵄨

(20.25)

for all λ̃ ∈ Dℂ2d (λ̃0 , RM /4) ⊇ Bℂ2d (λ̃0 , 4δ). Condition (E) holds by setting κ = 1. Finally, let us check condition (F). For every ω ∈ EA∞ and every λ̃ ∈ Bℂ2d (λ̃0 , 4δ), we obtain from (20.25) and the choice of M that 󵄨 󵄨 Re(φλ̃ (ω)) ≤ Re(φ(ω)) + 󵄨󵄨󵄨γλ̃ (ω)󵄨󵄨󵄨

󵄨 󵄨 ≤ Re(u)fλ0 (ω) + 4d M(t + ε)󵄨󵄨󵄨fλ0 (ω)󵄨󵄨󵄨

≤ (t − ε)fλ0 (ω) − 4d M(t + ε)fλ0 (ω)

≤ ψ(ω).

(20.26)

So condition (F) is fulfilled. Thus, we have verified all the conditions (A)–(F) and hence ̃ ̃ {φλ̃ (ω) = ufλ0 (ω)(1 + Z̃ω (λ))} is a complex-parametrized family of complexλ∈B (λ̃ ,4δ) ℂ2d

0

valued potentials. This holds for all u ∈ Bℂ (t, ε) and δ is independent of u. For all ω ∈ EA∞ and λ̃ ∈ Bℂ2d (λ̃0 , 4δ), set ̃ ̃f̃ (ω) := f (ω)(1 + Z̃ (λ)). λ0 ω λ

(20.27)

Per Lemma 20.1.11, we then know the following. Claim 2. If Λ is an open subset of ℂd and Φ = {Φλ }λ∈Λ is an analytic family of CGDMSs (not necessarily satisfying the OSC) with a strong parameter λ0 ∈ Λ, then for every t > θλ0 and every 0 < ε < t − θλ0 there is δ > 0 such that for every u ∈ Bℂ (t, ε) the function b ∞ Bℂ2d (λ̃0 , δ) 󳨃󳨀→ ℒλ,u ̃ ∈ L(𝒦β (EA ))

̃ is holomorphic, where ℒλ,u ̃ is the transfer operator corresponding to the potential ufλ̃ . Now, using much of the work just done, it is not difficult to demonstrate that for ̃ every λ̃ ∈ Bℂ2d (λ̃0 , 4δ) the family {φu }u∈Bℂ (t,ε) = {ω 󳨃→ ufλ0 (ω)(1 + Z̃ω (λ))} u∈Bℂ (t,ε) is a complex-parametrized family of complex-valued potentials by checking that conditions (A)–(F) in Definition 20.1.6 are satisfied with r0 = t,

δ = ε/4,

φ ≡ 0,

̃ γu (ω) = φu (ω) = ufλ0 (ω)(1 + Z̃ω (λ)),

ψ = ̂tfλ0

and an appropriate κ. As before, conditions (A), (B), (C) and (D2) are fulfilled. For (D1), fix ω, τ ∈ EA∞ with ω1 = τ1 . Noting that γu = ufλ0 + γλ̃ , we deduce from (20.24) that υβ (γu ) ≤ |u|υβ (fλ0 ) + υβ (γλ̃ ) ≤ (υβ (fλ0 ) + 4d Ks−1 )(t + ε), whence condition (D) is established.

∀u ∈ Bℂ (t, ε),

820 | 20 Real analyticity of topological pressure and Hausdorff dimension Using (20.25), we also get d 󵄨 󵄨 (1 + 4 M)(t + ε) 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 d 󵄨󵄨γu 󵄨󵄨 ≤ 󵄨󵄨ufλ0 󵄨󵄨󵄨 + 󵄨󵄨󵄨γλ̃ 󵄨󵄨󵄨 ≤ (t + ε)󵄨󵄨󵄨fλ0 󵄨󵄨󵄨 + 4 M(t + ε)󵄨󵄨󵄨fλ0 󵄨󵄨󵄨 = 󵄨󵄨ψ󵄨󵄨, ̂t

∀u ∈ Bℂ (t, ε).

Condition (E) holds with κ = (1 + 4d M)(t + ε)/̂t. Finally, condition (F) is respected because φu = φλ̃ and (20.26) holds. ̃ Hence {φu (ω) = ufλ0 (ω)(1 + Z̃ω (λ))} u∈Bℂ (t,ε) is a complex-parametrized family of ̃ ̃ ̃ potentials. This holds for all λ ∈ B 2d (λ , 4δ) and ε is independent of λ. ℂ

0

Per Lemma 20.1.11, we then obtain the following result.

Claim 3. If Λ is an open subset of ℂd and Φ = {Φλ }λ∈Λ is an analytic family of CGDMSs (not necessarily satisfying the OSC) with a strong parameter λ0 ∈ Λ, then for every t > θλ0 and every 0 < ε < t − θ there is δ > 0 such that for every λ̃ ∈ B 2d (λ̃ , δ) the function λ0

ℂ

0

b ∞ Bℂ (t, ε) ∋ u 󳨃󳨀→ ℒλ,u ̃ ∈ L(𝒦β (EA ))

is holomorphic. Using Hartogs’ theorem (Theorem A.4.1), we finally establish the following fact. Claim 4. If Λ is an open subset of ℂd and Φ = {Φλ }λ∈Λ is an analytic family of CGDMSs (not necessarily satisfying the OSC) with a strong parameter λ0 ∈ Λ, then for every t > θλ0 and every 0 < ε < t − θλ0 there is δ > 0 such that the function ̃ u) 󳨃󳨀→ ℒ̃ ∈ L(𝒦b (E ∞ )) Bℂ2d (λ̃0 , δ) × Bℂ (t, ε) ∋ (λ, β A λ,u ̃ is holomorphic, where ℒλ,u ̃ is the transfer operator corresponding to the potential ufλ̃ . We can now provide the concluding argument to the proof of the theorem. Looking at (20.26), observe that all the complex-valued potentials ũfλ̃ , with λ̃ ∈ Bℂ2d (λ̃0 , δ) and u ∈ Bℂ (t, ε), have a real part which is summable and thus even strongly summable (see Definition 20.1.1). In particular, this includes the real-valued potentials sfλ , with λ ∈ Bℂd (λ0 , δ) and s ∈ Bℝ (t, ε). These real-valued potentials are also Hölder on cylinders since fλ := ΞΦλ . Theorem 18.1.12 then ensures that the number exp(P(sfλ )) is a simple isolated eigenvalue of the operator ℒλ,s for all λ ∈ Bℂd (λ0 , δ) and all s ∈ Bℝ (t, ε). It follows from Claim 4 and the Kato–Rellich theorem for perturbations of linear operators (Theorem A.2.5) that there exist 0 < η < min{δ, ε} and a holomorphic function ρ : Bℂ2d (λ̃0 , η) × Bℂ (t, η) → ℂ ̃ u) is an eigenvalue of the operator ℒ̃ for all (λ, ̃ u) ∈ B 2d (λ̃ , η)×B (t, η) such that ρ(λ, 0 ℂ ℂ λ,u and ρ(λ, s) = exp(P(sfλ )),

∀(λ, s) ∈ Bℂd (λ0 , η) × Bℝ (t, η).

20.4 Real analyticity of Hausdorff dimension for CGDMSs in ℂ

| 821

Hence, the function Bℂd (λ0 , η) × Bℝ (t, η) ∋ (λ, s) 󳨃󳨀→ exp(P(sfλ )) ∈ ℝ is real analytic. Consequently, the function Bℂd (λ0 , η) × Bℝ (t, η) ∋ (λ, s) 󳨃󳨀→ P(sfλ ) ∈ ℝ is real analytic. Moreover, it is easy to see that for each λ ∈ Bℂd (λ0 , η) and each s ∈ Bℝ (t, η) the family {φz }z∈Bℂ (s,η−|s−t|) = {ω 󳨃→ zfλ (ω)}z∈Bℂ (s,η−|s−t|) is a complexparametrized family of complex-valued potentials of the form (20.9)–(20.11), with φs strongly summable for all λ ∈ Bℂd (λ0 , η). Then Theorem 20.2.3 asserts that P(sfλ ) 𝜕 P(sfλ ) = depends real analytically on s for any λ and Theorem 20.2.4 affirms that 𝜕s ∫E ∞ fλ dμλ,s < 0 for all (λ, s) ∈ Bℂd (λ0 , η) × Bℝ (t, η). It follows from an application of A

the real-analytic implicit function theorem (a real-analytic version of Theorem A.3.16) to the equation P(sfλ ) = 0 that the solution function, namely, the Bowen parameter function Bℂd (λ0 , η) ∋ λ 󳨃󳨀→ hλ ∈ ℝ is real analytic.

Corollary 20.4.4. Let Λ ⊆ ℂd be an open set. If {Φλ }λ∈Λ is an analytic family of finitely irreducible CGDMSs (satisfying the OSC), then the Hausdorff dimension function Λ ∋ λ 󳨃→ HD(Jλ ) ∈ ℝ is real analytic on some neighborhood of every strong parameter. Proof. By Theorem 19.6.4, the Bowen parameter of each CGDMS in this family is equal to the Hausdorff dimension of the corresponding limit set. So the result follows from Theorem 20.4.3. We now provide a useful, sufficient condition for a parameter to be strong. Definition 20.4.5. An analytic family of CGDMSs {Φλ }λ∈Λ (not necessarily satisfying the OSC) is Hölder stable at some parameter λ0 ∈ Λ if (a) the CGDMS Φλ0 is strongly regular; and (b) there are two constants C > 0 and β ∈ (0, 1] such that for every λ ∈ Λ there exists a homeomorphism Hλ : Jλ0 → Jλ with the properties 󵄨 󵄨 C −1 |z − w|1/β ≤ 󵄨󵄨󵄨Hλ (z) − Hλ (w)󵄨󵄨󵄨 ≤ C|z − w|β , and φλe ∘ Hλ = Hλ ∘ φλe0 ,

∀e ∈ E.

∀z, w ∈ Jλ0

822 | 20 Real analyticity of topological pressure and Hausdorff dimension Proposition 20.4.6. If an analytic family of CGDMSs {Φλ }λ∈Λ (not necessarily satisfying the OSC) is Hölder stable at some parameter λ0 ∈ Λ, then λ0 is a strong parameter for that family. Proof. It suffices to check condition (20.18). From condition (b) in Definition 20.4.5, it follows that C −1 diam1/β (φλe0 (Jλ0 )) ≤ diam(φλe (Jλ )) ≤ C diamβ (φλe0 (Jλ0 )),

∀e ∈ E, ∀λ ∈ Λ.

But by Koebe’s distortion theorem (Theorem A.3.8 or A.3.9), the diameter of the image of a compact set (having at least two elements) under a univalent function is comparable to the derivative of the univalent function at any point of the domain multiplied ̂ ≥ 1 such that by the diameter of the set. So there exists C ̂ −1 󵄨󵄨󵄨(φλ0 )′ (z)󵄨󵄨󵄨1/β ≤ 󵄨󵄨󵄨(φλ )′ (w)󵄨󵄨󵄨 ≤ C ̂ 󵄨󵄨󵄨(φλ0 )′ (z)󵄨󵄨󵄨β C 󵄨 e 󵄨 󵄨 e 󵄨 󵄨 e 󵄨

(20.28)

for all z ∈ Jλ0 , all w ∈ Jλ , all e ∈ E and all λ ∈ Λ. Hence, ̂ + ( 1 − 1) log󵄨󵄨󵄨(φλ0 )′ (z)󵄨󵄨󵄨 ≤ log󵄨󵄨󵄨(φλ )′ (w)󵄨󵄨󵄨 − log󵄨󵄨󵄨(φλ0 )′ (z)󵄨󵄨󵄨 − log C 󵄨 e 󵄨 󵄨 e 󵄨 󵄨 e 󵄨 β ̂ + (β − 1) log󵄨󵄨󵄨(φλ0 )′ (z)󵄨󵄨󵄨. ≤ log C 󵄨 e 󵄨 Therefore, 󵄨󵄨 log󵄨󵄨󵄨(φλ )′ (w)󵄨󵄨󵄨 − log󵄨󵄨󵄨(φλ0 )′ (z)󵄨󵄨󵄨 󵄨󵄨 ̂ 󵄨󵄨 󵄨 e 󵄨 󵄨 e 󵄨 󵄨󵄨󵄨 ≤ max{1 − β, 1 − 1} − log C < ∞. 󵄨󵄨 󵄨󵄨 󵄨󵄨 λ0 ′ 󵄨󵄨 󵄨󵄨 β log s 󵄨 log󵄨󵄨(φe ) (z)󵄨󵄨 An immediate consequence of Theorem 20.4.3 and Proposition 20.4.6 is the following. Corollary 20.4.7. If an analytic family of finitely irreducible CGDMSs {Φλ }λ∈Λ (not necessarily satisfying the OSC) is Hölder stable at some parameter λ0 ∈ Λ, then the Bowen parameter function Λ ∋ λ 󳨃→ hλ is real analytic in some neighborhood of λ0 . A straightforward repercussion of this and Theorem 19.6.4 is the following. Corollary 20.4.8. If an analytic family of finitely irreducible CGDMSs {Φλ }λ∈Λ (satisfying the OSC) is Hölder stable at some parameter λ0 ∈ Λ, then the Hausdorff dimension function Λ ∋ λ 󳨃→ HD(Jλ ) ∈ ℝ is real analytic in some neighborhood of λ0 . As the following result asserts, the existence of a holomorphic motion which is a conjugation of the generators of an analytic family, ensures the real analyticity of the Bowen parameter function in a neighborhood of any strongly regular system.

20.4 Real analyticity of Hausdorff dimension for CGDMSs in ℂ

| 823

Theorem 20.4.9. Let 𝔻 be the unit disk in ℂ. Let {Φλ }λ∈𝔻 be an analytic family of finitely irreducible CGDMSs (not necessarily satisfying the OSC) such that Φ0 is a strongly reĝ→ℂ ̂ such that ular system. If there exists an holomorphic motion H : 𝔻 × ℂ φλe (H(λ, z)) = H(λ, φ0e (z)),

∀λ ∈ 𝔻, ∀z ∈ J0 ,

then the Bowen parameter function 𝔻 ∋ λ → hλ is real analytic on a neighborhood of 0. Proof. By the λ-lemma (Theorem 16.4.4) and Remark 16.4.6, the analytic family {Φλ }λ∈𝔻 is Hölder stable at 0. A direct application of Corollary 20.4.7 yields the result. Below is a direct consequence of this and Theorem 19.6.4. Corollary 20.4.10. Let 𝔻 be the unit disk in ℂ. Let {Φλ }λ∈𝔻 be an analytic family of finitely irreducible CGDMSs (satisfying the OSC). If Φ0 is a strongly regular system and ̂→ℂ ̂ such that if there exists an holomorphic motion H : 𝔻 × ℂ φλe (H(λ, z)) = H(λ, φ0e (z)),

∀λ ∈ 𝔻, ∀z ∈ J0 ,

then the Hausdorff dimension function Λ ∋ λ 󳨃→ HD(Jλ ) ∈ ℝ is real analytic on a neighborhood of 0. We now provide some mild, quite general, sufficient conditions for an analytic family of CGDMSs to admit a holomorphic motion. Definition 20.4.11. Let Φ = {φe }e∈E be a CGDMS (not necessarily satisfying the OSC) and ∗ Eper = {ω ∈ EA∗ : Aω|ω| ω1 = 1 and ω ≠ τk for any τ ∈ EA∗ and k ≥ 2}. ∗ For every ω ∈ Eper , let xω ∈ Xt(ω) be the fixed point of the map φω : Wt(ω) → Wt(ω) . ∗ Then Φ is said to be periodically separated if xω ≠ xτ for all ω, τ ∈ Eper such that ω ≠ τ.

The following lemma asserts that periodic separation is a well-defined concept. Lemma 20.4.12. Let Φ = {φe }e∈E be a CGDMS (not necessarily satisfying the OSC). Dis∗ tinct words in Eper generate distinct periodic points. ∗ Proof. If ω, χ ∈ Eper are not comparable, then ωi ≠ χi for some i ≤ min{|ω|, |χ|} and ∗ hence ω∞ ≠ χ ∞ . So suppose that ω, ωρ ∈ Eper with ρ ≠ ϵ. There are unique m, n ∈ ℤ+ with n < |ω| such that |ρ| = m|ω| + n. In turn, there are unique q ∈ ℕ and 0 ≤ r < n such that |ω| = qn + r. If ω∞ = (ωρ)∞ , then it can be shown that ω = τk for some τ ∈ EA∗ ∗ with |τ| = gcd(n, r). Obviously, k = |ω|/ gcd(n, r) ∈ ℕ \{1}. But this means that ω ∈ ̸ Eper . ∞ ∞ Thus, ω ≠ (ωρ) .

There is one obvious separation condition under which a CGDMS is periodically separated.

824 | 20 Real analyticity of topological pressure and Hausdorff dimension Proposition 20.4.13. If a CGDMS Φ = {φe }e∈E (not necessarily satisfying the OSC) satisfies the Strong Separation Condition (SSC), then Φ is periodically separated. Analytic families of periodically separated, finitely irreducible CGDMSs admit holomorphic motions. Lemma 20.4.14. If Λ ⊆ ℂ is an open simply connected set and {Φλ }λ∈Λ is an analytic family of periodically separated, finitely irreducible CGDMSs (not necessarily satisfying the OSC), then for every λ0 ∈ Λ there exists a holomorphic motion H : Λ × Jλ0 → ℂ such that φλe (H(λ, z)) = H(λ, φλe0 (z)),

∀λ ∈ Λ, ∀z ∈ Jλ0 .

In addition, H({λ} × Jλ0 ) = Jλ

and

H({λ} × Jλ0 ) = Jλ ,

∀λ ∈ Λ.

∗ Proof. Fix momentarily ω ∈ Eper . Since the map

Λ × Wt(ω) ∋ (λ, z) 󳨃󳨀→ φλω (z) ∈ ℂ is holomorphic and since (φλω )′ (z) ≠ 1 for all (λ, z) ∈ Λ × Wt(ω) , it follows from the implicit function theorem (Theorem A.3.16 with F(z, λ) = φλω (z)−z) that for every λ0 ∈ Λ and rλ0 ,ω > 0 small enough there exists a unique holomorphic function B(λ0 , rλ0 ,ω ) ∋ λ 󳨃󳨀→ xλλ0 ,ω ∈ Wt(ω) such that φλω (xλλ0 ,ω ) = xλλ0 ,ω , i. e., xλλ0 ,ω is the unique fixed point of φλω : Wt(ω) → Wt(ω) .

The uniqueness of that function imposes that if λ ∈ B(λ0′ , rλ0′ ,ω )∩B(λ0 , rλ0 ,ω ), then xλλ′ ,ω = 0

xλλ0 ,ω and thus all the maps {B(λ0 , rλ0 ,ω ) ∋ λ 󳨃→ xλλ0 ,ω ∈ Wt(ω) }λ0 ∈Λ glue together into a

unique holomorphic function Λ ∋ λ 󳨃→ xωλ ∈ Wt(ω) such that φλω (xωλ ) = xωλ ,

∀λ ∈ Λ.

∗ We now consider the different ω ∈ Eper altogether. For each λ ∈ Λ, let ∗ Yλ = {xωλ : ω ∈ Eper }.

Since the system Φλ is periodically separated, the map ∗ Fλ : Eper 󳨀→ Yλ ω 󳨃󳨀→ Fλ (ω) := xωλ

(20.29)

20.4 Real analyticity of Hausdorff dimension for CGDMSs in ℂ

| 825

is bijective. Consequently, so is the map Yλ0 󳨀→ Yλ z 󳨃󳨀→ Fλ ∘ Fλ−10 (z) = xFλ −1 (z) . λ0

Therefore, the map H : Λ × Yλ0 → ℂ given by H(λ, z) = xFλ −1 (z) λ0

is a holomorphic motion. By the λ-lemma (Theorem 16.4.4), this map uniquely extends to a holomorphic motion H : Λ × Yλ0 → ℂ. Moreover, Yλ = Jλ for each λ ∈ Λ due to the finite irreducibility of A. Furthermore, by continuity of H and (20.29) it ensues that φλω (H(λ, z)) = H(λ, φλω0 (z)),

∀λ ∈ Λ, ∀z ∈ Jλ0 .

(20.30)

In addition, it follows from (20.30) that πλ (ω) = H(λ, πλ0 (ω)),

∀λ ∈ Λ, ∀ω ∈ EA∞ .

Therefore, Jλ = πλ (EA∞ ) = H({λ} × πλ0 (EA∞ )) = H({λ} × Jλ0 ),

∀λ ∈ Λ.

By a standard argument, one deduces from all the above that Jλ = Yλ = H({λ} × Yλ0 ) = H({λ} × Yλ0 ) = H({λ} × Jλ0 ),

∀λ ∈ Λ.

Using this lemma and Slodkowski’s theorem (Theorem A.3.18), we obtain the following. Theorem 20.4.15. Let Λ ⊆ ℂ be an open simply connected set whose complement ℂ\Λ contains at least two points. If {Φλ }λ∈Λ is an analytic family of periodically separated, finitely irreducible CGDMSs (not necessarily satisfying the OSC), then for every λ0 ∈ Λ ̂→ℂ ̂ such that there exists a holomorphic motion H : Λ × ℂ φλe (H(λ, z)) = H(λ, φλe0 (z)),

∀λ ∈ Λ, ∀z ∈ Jλ0 .

In addition, H({λ} × Jλ0 ) = Jλ

and

H({λ} × Jλ0 ) = Jλ

∀λ ∈ Λ.

Proof. By virtue of Lemma 20.4.14 there exists a holomorphic motion on λ × Jλ0 satisfying the required properties. By Slodkowski’s theorem, this motion can be extended ̂ with uniformly bounded dilatation. to an holomorphic motion on Λ × ℂ

826 | 20 Real analyticity of topological pressure and Hausdorff dimension The following result is an immediate consequence from Theorem 20.4.15 and Proposition 20.4.9. Corollary 20.4.16. Let Λ ⊆ ℂ be an open simply connected set whose complement ℂ\Λ contains at least two points. Let {Φλ }λ∈Λ be an analytic family of periodically separated, finitely irreducible CGDMSs (not necessarily satisfying the OSC). If the system Φλ0 is strongly regular, then the Bowen parameter function λ 󳨃→ hλ is real analytic on some neighborhood of λ0 . Corollary 20.4.17. Let Λ ⊆ ℂ be an open simply connected set whose complement ℂ\Λ contains at least two points. Let {Φλ }λ∈Λ be an analytic family of periodically separated, finitely irreducible CGDMSs (satisfying the OSC). If the system Φλ0 is strongly regular, then the Hausdorff dimension function Λ ∋ λ 󳨃→ HD(Jλ ) ∈ ℝ is real analytic on some neighborhood of λ0 .

20.5 Exercises Exercise 20.5.1. Let Φ be a CGDMS and f ∈ 𝒦βs (EA∞ , ℝ) a strongly summable potential. Show that the function t 󳨃→ HD(μtf ∘π −1 ), defined on a sufficiently small neighborhood of 1, is real analytic.

Exercise 20.5.2. Let Λ ⊆ ℂd be an open set. If {Φλ }λ∈Λ is an analytic family of finitely irreducible CGDMSs (not necessarily satisfying the OSC) and if f ∈ 𝒦βs (EA∞ , ℝ) is a potential such that χμf (σ) < ∞, prove that the function Λ ∋ λ 󳨃󳨀→ HD(μf ∘ πλ−1 ) is real analytic.

21 Multifractal analysis for conformal graph directed Markov systems In this chapter, we perform a multifractal analysis of the Hausdorff dimension of the level sets of Gibbs states for any Hölder family of functions of the form ge + u log |φ′e |, where {ge }e∈E is a bounded Hölder family of functions and Φ = {φe }e∈E is a CGDMS (conformal graph directed Markov system). This is a fertile field of applications of the theory of CGDMSs and more largely of this book. In addition to its intrinsic interest and importance, the multifractal analysis conducted in this chapter bears witness to the strength and sophistication of the tools and methods developed in this book. To our best knowledge, rigorous mathematical research on multifractal analysis began in 1989 with David Rand’s work [100] on cookie-cutter Cantor sets, which nowadays can be viewed as limit sets of finite(-alphabet) IFSs (iterated function systems) generated by similarities. Shortly afterwards, a multifractal analysis was done for general finite similarity IFSs by Cawley and Mauldin [15]. Roughly a decade later, a much more general and, we have to say, quite tedious multifractal analysis for countable(-alphabet) CIFSs (conformal IFSs) was achieved by Hanus, Mauldin and Urbański [46] and reproduced in [81] in the context of CGDMSs. That analysis relies on cylinders. Nearly a decade thereafter, a multifractal analysis was completed by Roy and Urbański [107] using balls. Our exposition stems from that paper but is more detailed and improved. Recall that the Hausdorff dimension of a fractal set gives us an idea of the global structure of that set. Nevertheless, two fractal sets with similar global structures may have substantially different local structures. In Section 16.6, we saw that the pointwise or local dimension of a Borel measure seizes the local structure of a set and potentially helps in distinguishing that set from another one which may be globally similar. This is accomplished through a multifractal analysis, i. e., the examination of the multifractal spectrum, which describes the relative sizes of a multitude of fractal subsets which form a splitting of the original fractal set. More precisely, let μ be a Borel probability measure on a metric space X. For each number α ≥ 0, let Xμ (α) := {x ∈ X | dμ (x) = α} be the subset of all points x ∈ X where the measure μ has local dimension α. Let also Xμc := {x ∈ X | dμ (x)∄} = X \ ⋃α≥0 Xμ (α) be the subset of all points x ∈ X where the measure μ does not have a local dimension. In general, Xμc ≠ 0 and a study of that set may be of interest. However, we will see that in our case Xμc = 0. Then ⋃α≥0 Xμ (α) = X. Let dμ (α) = HD(Xμ (α)) be the Hausdorff dimension of the fractal subset Xμ (α). (Note: In Section 16.6, we used the more classical notation Fμ (α). In this chapter, dμ (α) is more convenient since F will bear a different meaning.) The map α 󳨃→ dμ (α) is the (fine Hausdorff ) multifractal spectrum of the measure μ. In [46, 81], the multifractal analysis used at each point of the limit set J the natural filtration generated by the initial cylinders associated to the word that encodes the point. In brief, the analysis was carried out using cylinders. Aiming to give the https://doi.org/10.1515/9783110702699-021

828 | 21 Multifractal analysis for conformal graph directed Markov systems multifractal analysis a transparent geometrical meaning, we derive in this chapter an analysis using as the filtration a base of balls centred at the given point. We will conduct the analysis of the conformal measure (or, equivalently, of the invariant measure) associated to a family of weights imposed upon a (finite or infinite) CGDMS. We will also conduct this analysis over the full set of parameters on which it can be expected to hold. However, we will restrict our attention to cofinitely regular, finitely irreducible, conformal-like CGDMSs that satisfy the SOSC (see Definitions 19.5.5, 17.1.2, 19.7.17 and 19.7.1, resp.) and to a large and dynamically significant subset J0 of the limit set J, as opposed to the entire limit set, though the sets J0 and J coincide under a mild boundary separation condition. An additional geometric flavor of this analysis results from the fact that we concentrate on a geometrically meaningful family of Hölder weights. The results apply to large classes of CGDMSs, one- and multi-dimensional alike, including real and complex continued fractions. Let us describe the content of the chapter more precisely. In Section 21.1, we describe the Hölder families of weights F and Fq,t we shall work with, and we study the properties of the pressure function P(q, t) and of the temperature function T(q) they determine. Recall that the general properties of Hölder families of functions and the amalgamated function they induce were studied in Section 19.8. In Section 21.2, we carry out the multifractal analysis of the conformal measure mF (or, equivalently, of the invariant measure μF ) associated to the family of weights F. This analysis is done over a set J0 ⊆ J of full μF -measure and conducted by means of balls. In particular, we show that for each α there is an auxiliary measure that witnesses the Hausdorff dimension of the set J0,μ (α) and that the d0,μ (α) curve is the Legendre transform of the temperature function T(q). Part of the analysis is restricted to a subclass called almost-CIFSs. In Section 21.3, we derive the multifractal analysis of mF under additional conditions on the almost-CIFSs. In Subsection 21.3.1, we observe that J0 = J for all CGDMSs which satisfy the boundary separation condition. Real continued fractions with the digit 1 deleted are an important example of such systems. In Subsection 21.3.2, we derive the multifractal analysis over J under three conditions and show that there are families of one-dimensional CIFSs that meet these three conditions. Real continued fractions (with or without the digit 1) are a fundamental example of such families. Finally, in Section 21.4 we conduct a similar analysis on smaller subsets of points of the limit set J (they have a symbolic coding with at least one infinitely-seen letter from a finite subalphabet) but for all CGDMSs that are cofinitely regular, finitely irreducible, conformal-like, and satisfy the SOSC.

21.1 Pressure and temperature In this section, we study the concept of temperature. This notion is defined in terms of the pressure of a two-parameter family of functions.

21.1 Pressure and temperature

| 829

Let Φ = {φe : Xt(e) → Xi(e) | e ∈ E} be a finitely irreducible CGDMS. Let FinΦ := {t ≥ 0 : P(t) < ∞} and θ := inf FinΦ be the finiteness set and finiteness parameter of Φ, respectively. Let u ∈ FinΦ . Let also G = {ge : Xt(e) → ℝ | e ∈ E} be a bounded Hölder family of functions, and let β be an Hölder exponent for that family. Bounded simply means that ‖G‖ := sup ‖ge ‖Xt(e) < ∞. e∈E

The amalgamated function g : EA∞ → ℝ induced by the family G is clearly bounded, and Hölder continuous on cylinders with exponent β by Lemma 19.8.6. Per Proposition 19.8.10, the family LogΦ := {log |φ′e | : Xt(e) → ℝ | e ∈ E} is Hölder with exponent −α log s and the family t LogΦ is summable for any t ∈ FinΦ . In particular, u LogΦ is summable and its amalgamated function uΞΦ , where ΞΦ was defined in (19.14), is summable and Hölder on cylinders with exponent −α log s. The family F := G + u LogΦ , i. e., F = {fe : Xt(e) → ℝ | e ∈ E}, where 󵄨 󵄨 fe = ge + u log󵄨󵄨󵄨φ′e 󵄨󵄨󵄨, is then summable and Hölder with exponent ζ := min{β, −α log s}. Its amalgamated function f : EA∞ → ℝ satisfies f = g + uΞΦ and is summable and Hölder continuous on cylinders with exponent ζ . Note also that F and f are bounded above by ‖G‖. Recall that the topological pressure P(F) of the family F is defined by P(F) := lim

n→∞

1 󵄩 󵄩 log ∑ 󵄩󵄩󵄩exp(Sω F)󵄩󵄩󵄩X < ∞. t(ω) n n ω∈E A

Replacing the family G by G − P(F), we may assume without loss of generality that the family F is such that P(F) = 0. This will be a standing assumption throughout this chapter. Equivalently, P(f ) = 0 by Proposition 19.8.9, where P(f ) = lim

n→∞

1 1 log Zn (f ) = lim ∑ exp(Sn f ([ω])). n→∞ n n ω∈E n A

Now, for every ordered pair of parameters (q, t) ∈ ℝ2 define the family Fq,t := qF + t LogΦ = qG + (qu + t) LogΦ

830 | 21 Multifractal analysis for conformal graph directed Markov systems and its corresponding amalgamated function fq,t = qf + tΞΦ = qg + (qu + t)ΞΦ . Observe that F1,0 = F and f1,0 = f . Note also that F0,t = t LogΦ and f0,t = tΞΦ . It is easy to see that all the Fq,t are Hölder families of functions with exponent ζ (the exponent of F), and thus the fq,t are Hölder continuous on cylinders with exponent ζ (like f ). By Proposition 19.8.10, Fq,t and fq,t are summable if and only if qu + t ∈ FinΦ . In particular, as Φ is finitely irreducible, Corollary 17.7.5 guarantees that for every (q, t) ∈ ℝ2 such that qu + t > θ the dual transfer operator ℒ∗fq,t has a unique eigenmeasure mfq,t . This measure is also a Gibbs state for fq,t and fq,t admits a unique σ-invariant Gibbs state μfq,t according to that same corollary. As two Gibbs states for fq,t , the measures μfq,t and mfq,t are boundedly equivalent. By the same corollary still, μfq,t is ergodic, and even totally ergodic if Φ is finitely primitive. The measure μfq,t is also the unique equilibrium state for fq,t . Indeed, note that fq,t ∈ L1 (μfq,t ) since g is bounded and ΞΦ ∈ L1 (μfq,t ), as we will now show.

To shorten notation, we shall write ‖φ′ω ‖ for ‖φ′ω ‖Xt(ω) . Let 0 < δ < (qu + t) − θ and

I := inf{‖φ′e ‖δ log ‖φ′e ‖ : e ∈ E}. Notice that −∞ < I < 0 because limx→0+ xδ log x = 0− . Let Cq,t ≥ 1 be a Gibbs constant for the Gibbs state μfq,t and let K ≥ 1 be a constant from the BDP (Corollary 19.3.5). Then 󵄨 󵄨 ∫ |ΞΦ |dμfq,t = ∑ ∫ (−ΞΦ ) dμfq,t = ∑ ∫ − log󵄨󵄨󵄨φ′e (π(σ(ω)))󵄨󵄨󵄨 dμfq,t (ω)

EA∞

e∈E [e]

e∈E [e]

󵄩 󵄩 ≤ ∑ ∫ − log(K −1 󵄩󵄩󵄩φ′e 󵄩󵄩󵄩)dμfq,t e∈E

[e]

󵄩 󵄩 = log K − ∑ log󵄩󵄩󵄩φ′e 󵄩󵄩󵄩μfq,t ([e]) e∈E

󵄩 󵄩 ≤ log K − Cq,t e−P(fq,t ) ∑ log󵄩󵄩󵄩φ′e 󵄩󵄩󵄩 exp(sup(fq,t |[e] )) e∈E

≤ log K − Cq,t e ≤ log K − Cq,t e

−P(fq,t )

󵄩 󵄩 󵄨 󵄨 ∑ log󵄩󵄩󵄩φ′e 󵄩󵄩󵄩 exp(sup(qge ) + (qu + t) sup log󵄨󵄨󵄨φ′e 󵄨󵄨󵄨)

e∈E −P(fq,t )+|q|⋅‖G‖

󵄩 󵄩 󵄩 󵄩 ∑ log󵄩󵄩󵄩φ′e 󵄩󵄩󵄩 exp((qu + t) log󵄩󵄩󵄩φ′e 󵄩󵄩󵄩)

e∈E

= log K − Cq,t e

−P(fq,t )+|q|⋅‖G‖

= log K − Cq,t e

−P(fq,t )+|q|⋅‖G‖

≤ log K − Cq,t e

−P(fq,t )+|q|⋅‖G‖

󵄩 󵄩qu+t 󵄩 󵄩 log󵄩󵄩󵄩φ′e 󵄩󵄩󵄩 ∑ 󵄩󵄩󵄩φ′e 󵄩󵄩󵄩

e∈E

󵄩 󵄩(qu+t)−δ 󵄩󵄩 ′ 󵄩󵄩δ 󵄩 ′󵄩 ∑ 󵄩󵄩󵄩φ′e 󵄩󵄩󵄩 󵄩󵄩φe 󵄩󵄩 log󵄩󵄩󵄩φe 󵄩󵄩󵄩

e∈E

󵄩 󵄩(qu+t)−δ I ∑ 󵄩󵄩󵄩φ′e 󵄩󵄩󵄩 e∈E

= log K − Cq,t e−P(fq,t )+|q|⋅‖G‖ I Z1 ((qu + t) − δ) < ∞.

(21.1)

21.1 Pressure and temperature

| 831

So ΞΦ , and hence fq,t , are in L1 (μfq,t ). By Corollary 17.7.5(c), the measure μfq,t is then the unique equilibrium state for fq,t . Finally, for every n ∈ ℕ observe that (cf. (17.1) and Definition 19.4.2) 󵄨 󵄨qu+t ] Zn (fq,t ) = ∑ sup [exp(qSn g(ρ)) ⋅ 󵄨󵄨󵄨φ′ω (π(σ n (ρ)))󵄨󵄨󵄨 ω∈EAn ρ∈[ω]

≥ exp(−n|q| ⋅ ‖G‖) ⋅ K −(qu+t) Zn (qu + t). Hence, P(fq,t ) ≥ −|q| ⋅ ‖G‖ + P(qu + t). A similar upper bound holds to yield P(qu + t) − |q| ⋅ ‖G‖ ≤ P(fq,t ) ≤ P(qu + t) + |q| ⋅ ‖G‖.

(21.2)

We now state fundamental properties of the pressure as a function of the two variables q and t. Theorem 21.1.1. Let Φ = {φe }e∈E be a finitely irreducible CGDMS. Let θ be the finiteness parameter of Φ and u > θ. The pressure function ℝ2 ∋ (q, t) 󳨃󳨀→ P(q, t) := P(Fq,t ) = P(fq,t ) ∈ (−∞, ∞] satisfies the following properties: (a) The quantity P(q, t) has the same (finite or infinite) nature as P(qu + t). That is, P(q, t) < ∞ if and only if qu + t ∈ FinΦ . Therefore, Φ is cofinitely regular if and only if P(q, t) = ∞ whenever qu + t = θ. (b) If q1 ≤ q2 and t1 ≤ t2 are such that (q2 − q1 )(sup G + u log s) + (t2 − t1 ) log s ≤ 0, then P(q2 , t2 ) ≤ P(q1 , t1 ). In particular, if sup G ≤ −u log s, then P(q, t) is decreasing with respect to both variables q and t. (c) t 󳨃󳨀→ P(q, t) is strictly decreasing on (θ − qu, ∞) for every q ∈ ℝ. (d) lim P(q, t) = −∞ for every q ∈ ℝ. t→∞

(e) If Φ is cofinitely regular, then (f)

lim

t→(θ−qu)+

P(q, t) = ∞ for every q ∈ ℝ.

𝜕P (q, t) = ∫ f dμfq,t ≤ 0 for every (q, t) ∈ ℝ2 such that qu + t > θ. 𝜕q ∞ EA

𝜕P (g) (q, t) = −χμf (σ) := ∫ ΞΦ dμfq,t < 0 for every (q, t) ∈ ℝ2 such that qu + t > θ. q,t 𝜕t ∞ EA

832 | 21 Multifractal analysis for conformal graph directed Markov systems (h) The pressure function t 󳨃󳨀→ P(q, t) is convex on the interval (θ − qu, ∞) for every q ∈ ℝ. (i) The pressure function (q, t) 󳨃󳨀→ P(q, t) is convex on the half-plane {(q, t) ∈ ℝ2 : t > θ − qu}. Proof. (a) This follows from (21.2), the definition of FinΦ , and the fact that cofinite regularity is characterized by P(θ) = ∞. (b) Let q1 ≤ q2 and t1 ≤ t2 . If q1 u + t1 ∉ FinΦ , then P(q1 , t1 ) = ∞ according to (a) and the statement thus holds. So suppose that q1 u + t1 ∈ FinΦ . Then q2 u + t2 ∈ FinΦ and the nth partition function of fq2 ,t2 satisfies Zn (fq2 ,t2 ) = ∑ exp( sup Sn fq2 ,t2 (ρ)) ω∈EAn

ρ∈[ω]

= ∑ sup exp(q2 Sn g(ρ) + (q2 u + t2 )Sn ΞΦ (ρ)) ω∈EAn ρ∈[ω]

󵄨 󵄨q u+t = ∑ sup [exp(q2 Sn g(ρ)) ⋅ 󵄨󵄨󵄨φ′ω (π(σ n (ρ)))󵄨󵄨󵄨 2 2 ] ω∈EAn ρ∈[ω]

󵄩 󵄩q u+t ≤ ∑ sup [exp(q2 Sn g(ρ))] ⋅ 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩X2 2 t(ω)

ω∈EAn ρ∈[ω]

= ∑ sup [exp(q1 Sn g(ρ) + (q2 − q1 )Sn g(ρ))] ω∈EAn ρ∈[ω]

󵄩 󵄩q u+t 󵄩 󵄩(q −q )u+(t2 −t1 ) ⋅ 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩X1 1 ⋅ 󵄩󵄩󵄩φ′ω 󵄩󵄩󵄩X 2 1 t(ω) t(ω)

≤ ∑ sup exp(q1 Sn g(ρ)) ⋅ en(q2 −q1 ) sup G ω∈EAn ρ∈[ω]

󵄨 󵄨q u+t ⋅ K q1 u+t1 inf 󵄨󵄨󵄨φ′ω (x)󵄨󵄨󵄨 1 1 ⋅ sn[(q2 −q1 )u+(t2 −t1 )] x∈X t(ω)

K q1 u+t1 󵄨 󵄨q u+t ⋅ ∑ sup [exp(q1 Sn g(ρ)) ⋅ 󵄨󵄨󵄨φ′ω (π(σ n (ρ)))󵄨󵄨󵄨 1 1 ]

≤e

n(q2 −q1 ) sup G n[(q2 −q1 )u+(t2 −t1 )]

s

ω∈EAn ρ∈[ω]

= en(q2 −q1 ) sup G sn[(q2 −q1 )u+(t2 −t1 )] K q1 u+t1 Zn (fq1 ,t1 ). Consequently, P(q2 , t2 ) ≤ (q2 − q1 ) sup G + [(q2 − q1 )u + (t2 − t1 )] log s + P(q1 , t1 ).

(21.3)

Part (b) follows immediately. (c) Letting q1 = q2 = q and t1 < t2 in (21.3) yields (c). (d) This also follows from (21.3) by setting q1 = q2 = q, t1 > θ and t2 = t, and then letting t → ∞.

21.1 Pressure and temperature

| 833

(e) Using (21.2) and the right-continuity of P(∙), we obtain that lim

t→(θ−qu)+

P(q, t) ≥ −|q|‖G‖ + P(θ) = ∞

when Φ is cofinitely regular. (f) First, note that f ∈ L1 (μfq,t ) for all (q, t) ∈ ℝ2 such that qu + t > θ since G is

bounded, u > θ and ΞΦ ∈ L1 (μfq,t ) by (21.1). The statement then follows from Theorem 20.2.4, by replacing r with q and setting φ = tΞΦ and γ = f , so that φq = φ + qγ = qf + tΞΦ = fq,t . Furthermore, by the 2nd or 3rd variational principle (Theorem 17.3.3 or 17.3.4), observe that ∫ f dμfq,t ≤ P(f ) − hμf (σ) = −hμf (σ) ≤ 0. q,t

EA∞

(21.4)

q,t

Recall that P(F) = P(f ) = 0 is a standing assumption in this chapter. (g) The proof goes along similar lines to (f), using the fact that χμf (σ) < ∞ whenq,t

ever qu + t > θ according to (21.1). More precisely, in Theorem 20.2.4, replace r with t and set φ = qf and γ = ΞΦ , so that φt = φ + tγ = qf + tΞΦ = fq,t . (h,i) These statements follow from Exercise 17.9.10 and the fact that the family of potentials fq,t is convex on the half-plane H = {(q, t) ∈ ℝ2 : t > θ − qu}, i. e. for any p ∈ [0, 1] and (q1 , t1 ), (q2 , t2 ) ∈ H we have fpq1 +(1−p)q2 , pt1 +(1−p)t2 = pfq1 ,t1 + (1 − p)fq2 ,t2 . Remark 21.1.2. In the proof of part (e), observe that P(θ) ≥ |q|‖G‖ suffices to guarantee the existence of a zero for the pressure function t 󳨃→ P(q, t). The assumption of cofinite regularity on Φ ensures that the pressure function t 󳨃→ P(q, t) has a zero for every q ∈ ℝ. Theorem 21.1.1 and Remark 21.1.2 lead to the next definition. Definition 21.1.3. Let Φ = {φe }e∈E be a cofinitely regular, finitely irreducible CGDMS. The function T : ℝ → ℝ defined for every q ∈ ℝ as the unique T(q) ∈ (θ − qu, ∞) such that P(q, T(q)) = 0, is called the temperature function. In order to lighten notation, we shall henceforth let fq := fq,T(q) ,

Fq := Fq,T(q)

and

mq := mfq,T(q) ,

μq := μfq,T(q) .

Let q ∈ ℝ. Recall that ΞΦ , f ∈ L1 (μq ). As qu + T(q) > θ, Theorem 21.1.1(f,g) asserts that 𝜕P (q, T(q)) = ∫E ∞ f dμq ≤ 0 while 𝜕P (q, T(q)) = −χμq (σ) = ∫E ∞ ΞΦ dμq < 0. 𝜕q 𝜕t Consequently,

A

0 ≤ α(q) :=

A

𝜕P (q, T(q)) 𝜕q 𝜕P (q, T(q)) 𝜕t

=

∫E ∞ f dμq A

−χμq (σ)

< ∞.

(21.5)

834 | 21 Multifractal analysis for conformal graph directed Markov systems We now enumerate basic features of the temperature function. Theorem 21.1.4. Let Φ = {φe }e∈E be a cofinitely regular, finitely irreducible CGDMS. The temperature function q 󳨃→ T(q) exhibits the following properties: (a) The function T : ℝ → ℝ is real analytic. (b) T(0) = HD(J) while T(1) = 0. (c) T ′ (q) = −α(q) < 0 for all q ∈ ℝ. (d) The function T : ℝ → ℝ is convex, i. e. T ′′ (q) ≥ 0 for all q ∈ ℝ. (e) If μf = μHD(J)ΞΦ or, equivalently, if f and HD(J)ΞΦ are cohomologous in the group of Hölder functions on EA∞ , then T is linear. More precisely, T(q) = HD(J)(1 − q). Proof. (a) By Theorem 21.1.1(g), 𝜕P (q, T(q)) = ∫ ΞΦ dμq = −χμq (σ) < 0, 𝜕t ∞

∀q ∈ ℝ.

EA

Since T(q) is uniquely determined by the condition P(q, T(q)) = 0, it follows from the implicit function theorem and Theorem 20.2.3 (applied in the same setting as in the proof of Theorem 21.1.1(g)) that T is real analytic on ℝ. (b) By definition, T(0) is the unique number t ∈ (θ, ∞) such that 0 = P(0, t) = P(f0,t ) = P(tΞΦ ) = P(t). However, P(HD(J)) = 0 since Φ is regular. So T(0) = HD(J). Similarly, T(1) is the unique number t ∈ (θ − u, ∞) such that P(1, t) = 0. However, since f1,0 = f and P(F) = P(f ) = 0 by standing assumption, we deduce that P(1, 0) = P(f1,0 ) = P(f ) = 0. Therefore, T(1) = 0. (c) It follows from the fact that P(q, T(q)) = 0 for all q ∈ ℝ and from Theorem 21.1.1(f,g) that 0=

𝜕P 𝜕P dP (q, T(q)) = (q, T(q)) + (q, T(q)) ⋅ T ′ (q). dq 𝜕q 𝜕t

Hence, 𝜕P (q, T(q)) 𝜕q

T ′ (q) = − 𝜕P

(q, T(q)) 𝜕t

=−

∫E ∞ f dμq A

−χμq (σ)

(21.6)

= −α(q).

Having already observed that α(q) ≥ 0, we thus know that T ′ (q) ≤ 0. In order to prove that T ′ (q) < 0, we need to show that ∫E ∞ f dμq ≠ 0. But since ∫E ∞ f dμq ≤ −hμq (σ) A

A

by (21.4), it suffices to show that hμq (σ) > 0. This follows immediately from Theo-

rem 17.8.3 since fq ∈ 𝒦ζs (EA∞ ) ∩ L1 (μq ). Also, EA∞ is not a single periodic orbit since Φ is cofinitely regular. (d) The proof of this part essentially goes along the same lines as that of the last paragraph in Theorem 16.6.5(b,d). We recap part of the argument. The inequality T ′′ (q) ≥ 0 follows from the convexity of P(q, t) (see Theorem 21.1.1(i)). Indeed, the fact

21.2 Multifractal analysis of the conformal measure mF over a subset of J

| 835

that the part of ℝ3 above the graph of P(q, t) is convex implies that its intersection with the (q, t)-plane is also convex. Since 𝜕P(q,t) = −χμf (σ) < 0 by Theorem 21.1.1(g), 𝜕t q,t this is the part of the plane above the graph of T. Hence, T is a convex function. (e) If μf = μHD(J)ΞΦ then Theorem 18.4.1 asserts that f − HD(J)ΞΦ is cohomologous to a constant R via a Hölder coboundary υ : EA∞ → ℝ. By an analog of Lemma 13.2.6 (whose proof carries over to the current setting), Bowen’s formula (Theorem 19.6.4) and our standing assumption on f , we deduce that 0 = P(f ) = P(HD(J)ΞΦ ) + R = R. Consequently, P(q, T(q)) = 0 = P(HD(J)ΞΦ ) = P(qυ − qυ ∘ σ + HD(J)ΞΦ ) = P(q(HD(J)ΞΦ + υ − υ ∘ σ) + HD(J)(1 − q)ΞΦ ) = P(qf + HD(J)(1 − q)ΞΦ )

= P(q, HD(J)(1 − q)).

By definition of T(q), we infer that T(q) = HD(J)(1 − q).

21.2 Multifractal analysis of the conformal measure mF over a subset of J Let Φ = {φe : Xt(e) → Xi(e) | e ∈ E} be a cofinitely regular, finitely irreducible, conformal-like CGDMS satisfying the Strong Open Set Condition (SOSC). Let θ be the finiteness parameter of Φ. Let also F be a family of functions of the form F = G+u LogΦ such that P(F) = 0, where G is a bounded Hölder family of functions and u > θ. If qu + t ∈ FinΦ , then Fq,t and fq,t are summable and Theorem 19.8.14 asserts that there is a unique Fq,t -conformal measure mFq,t supported on J. This measure is mFq,t = mfq,t ∘ π −1 . Like μfq,t and mfq,t , the measures μFq,t := μfq,t ∘ π −1 and mFq,t = mfq,t ∘ π −1 are boundedly equivalent. For this reason, μFq,t is called the Φ-invariant version of mFq,t . To abridge notation, we will write mFq := mFq,T(q)

and μFq := μFq,T(q) .

We now develop a multifractal analysis of the conformal measure mF (or, equivalently, of the invariant measure μF ) associated to the family F. We shall conduct this analysis by means of balls and we will restrict ourselves to the subset J0 := π(E0∞ ) of the limit set J of Φ. According to Theorem 19.7.3, the completely shift-invariant set 󵄨󵄨 E0∞ := {ω ∈ EA∞ 󵄨󵄨󵄨 lim sup dist(π(σ n (ω)), 𝜕Xi(σn (ω)) ) > 0} 󵄨 n→∞

836 | 21 Multifractal analysis for conformal graph directed Markov systems is of full measure under any ergodic σ-invariant Borel probability measure on EA∞ that has full topological support and we thus have the next result. Corollary 21.2.1. Let Φ = {φe }e∈E be a finitely irreducible CGDMS satisfying the SOSC. For all (q, t) ∈ ℝ2 such that qu + t > θ, we have μFq,t (J0 ) = μfq,t (E0∞ ) = 1. In particular, μF (J0 ) = μf (E0∞ ) = 1, i. e., the set J0 is a set of full μF -measure. If, additionally, Φ is cofinitely regular then for all q ∈ ℝ, μFq (J0 ) = μq (E0∞ ) = 1. For every ω ∈ E0∞ , let r(ω) := lim sup dist(π(σ n (ω)), 𝜕Xi(σn (ω)) ) > 0. n→∞

(21.7)

For every R > 0, define the set 󵄨 ER∞ := {ω ∈ E0∞ 󵄨󵄨󵄨 r(ω) ≥ R}

(21.8)

Observe that E0∞ = ⋃R>0 ER∞ . Like E0∞ , the sets ER∞ are completely σ-invariant. We now prove likewise that they are of full measure under any ergodic shift-invariant measure with full topological support, as long as R is small enough. Theorem 21.2.2. Let Φ = {φe }e∈E be a CGDMS satisfying the SOSC. For every ergodic σ-invariant Borel probability measure μ on EA∞ such that supp(μ) = EA∞ , it turns out that μ(ER∞ ) = μ ∘ π −1 ( ⋃ Xυ \ B(𝜕Xυ , R)) = 1 υ∈V

for all 0 < R < sup dist(x, 𝜕X). x∈J

Proof. Let 0 < R < D := supx∈J dist(x, 𝜕X) = supυ∈V supx∈Jυ dist(x, 𝜕Xυ ). The SOSC guarantees that D > 0, and thus there is x ∈ Jυ∗ \ B(𝜕Xυ∗ , (R + D)/2) for some υ∗ ∈ V. Let ω ∈ EA∞ be such that π(ω) = x. Then there is k ∈ ℕ such that π([ω|k ]) ⊆ φω|k (Xt(ωk ) ) ⊆ B(π(ω), (D − R)/2) ⊆ Xυ∗ \ B(𝜕Xυ∗ , R).

21.2 Multifractal analysis of the conformal measure mF over a subset of J

| 837

Therefore, μ ∘ π −1 ( ⋃ Xυ \B(𝜕Xυ , R)) ≥ μ ∘ π −1 (Xυ∗ \B(𝜕Xυ∗ , R)) ≥ μ ∘ π −1 (π([ω|k ])) ≥ μ([ω|k ]) > 0 υ∈V

since supp(μ) = EA∞ . Poincaré’s recurrence theorem (Theorem 8.1.2) states that 󵄨󵄨 󵄨 μ({ρ ∈ EA∞ 󵄨󵄨󵄨 σ n (ρ) ∈ π −1 ( ⋃ Xυ \B(𝜕Xυ , R)) for infinitely many n}) 󵄨󵄨 υ∈V = μ(π −1 ( ⋃ Xυ \B(𝜕Xυ , R))) > 0. υ∈V

But the set on the left-hand side of this relation is completely σ-invariant, so the ergodicity of μ imposes that its measure be 1. As that set is contained in ER∞ , the result ensues. We now refine the sets we look at by restricting some of the symbols they can have. For every ω ∈ EA∞ , every R > 0 and every e ∈ E, let (nj (ω, R, e))Nj=1 be the sequence of all n ≥ 0 such that σ n (ω) ∈ π −1 ( ⋃ Xυ \B(𝜕Xυ , R)) ∩ [e]. υ∈V

(21.9)

This means that nj (ω, R, e) is the jth time that an iterate of ω hits the set π −1 (⋃υ∈V Xυ \ B(𝜕Xυ , R)) ∩ [e]. In other words, nj (ω, R, e) is the jth time that an iterate of ω starts with the letter e and is projected to a point that lies at a distance greater than or equal to R from the boundary of X. This sequence may be empty (N = 0), nonempty and finite (N ∈ ℕ), or infinite (N = ∞), depending on ω, R and e. To shorten notation, we shall usually write nj (ω) instead of nj (ω, R, e) when R and e are already fixed. Define the sets 󵄨󵄨 Snj+1 (ω) ΞΦ (ω) nj+1 (ω) 󵄨 ∞ = 1 and lim = 1} ER,e := {ω ∈ ER∞ 󵄨󵄨󵄨 ωnj (ω)+1 = e, ∀j ∈ ℕ, lim 󵄨󵄨 j→∞ Sn (ω) ΞΦ (ω) j→∞ nj (ω) j and ∞ ∞ E0,e := ⋃ ER,e ⊆ E0∞ . R>0

Theorem 21.2.3. Let Φ = {φe }e∈E be a CGDMS satisfying the SOSC. For every e ∈ E, every 0 < R < supx∈J dist(x, 𝜕X) and every ergodic σ-invariant Borel probability measure μ on EA∞ such that supp(μ) = EA∞ and ∫E ∞ ΞΦ dμ > −∞, it holds that A

∞ μ(ER,e ) = 1,

and thus

∞ μ(E0,e ) = 1.

838 | 21 Multifractal analysis for conformal graph directed Markov systems Proof. Let e ∈ E and 0 < R < D := supx∈J dist(x, 𝜕X). By Theorem 21.2.2, we know that μ∘π −1 (⋃υ∈V Xυ \B(𝜕Xυ , R)) = 1. Since supp(μ) = EA∞ , we also know that μ([e]) > 0. Then μ(π −1 ( ⋃ Xυ \B(𝜕Xυ , R)) ∩ [e]) = μ([e]) > 0. υ∈V

Applying twice the ergodic case of Birkhoff’s ergodic theorem (Corollary 8.2.14), once with the indicator function of the set ⋃υ∈V π −1 (Xυ \B(𝜕Xυ , R)) ∩ [e] and once with the potential function ΞΦ , we obtain that for μ-a. e. ω ∈ EA∞ , lim

n→∞

1 #{0 ≤ k ≤ n : σ k (ω) ∈ π −1 ( ⋃ Xυ \B(𝜕Xυ , R)) ∩ [e]} n υ∈V

= μ(π −1 ( ⋃ Xυ \B(𝜕Xυ , R)) ∩ [e]) > 0

(21.10)

υ∈V

and lim

n→∞

1 S Ξ (ω) = −χμ (σ). n n Φ

(21.11)

From (21.10), we deduce that the sequence (nj (ω))Nj=1 is infinite, i. e., N = ∞, for μ-a. e. ω ∈ EA∞ . Moreover, lim

j→∞

1 j = μ(π −1 ( ⋃ Xυ \B(𝜕Xυ , R)) ∩ [e]) =: L > 0 nj (ω) υ∈V

(21.12)

for μ-a. e. ω ∈ EA∞ . Consequently, lim

j→∞

nj+1 (ω) nj (ω)

= lim

j→∞

j/nj (ω)

(j + 1)/nj+1 (ω)

=

L =1 L

(21.13)

for μ-a. e. ω ∈ EA∞ . From (21.11), we infer that lim

j→∞

1 S Ξ (ω) = −χμ (σ) < 0 nj (ω) nj (ω) Φ

(21.14)

for μ-a. e. ω ∈ EA∞ . It follows from (21.14) and (21.13) that lim

j→∞

Snj+1 (ω) ΞΦ (ω) Snj (ω) ΞΦ (ω)

= lim

j→∞

Snj+1 (ω) ΞΦ (ω)/nj+1 (ω) Snj (ω) ΞΦ (ω)/nj (ω)

=

−χμ (σ) −χμ (σ)

= 1.

(21.15)

Finally, by definition of nj (ω), we have that σ nj (ω) (ω) ∈ [e], i. e., ωnj (ω)+1 = e,

∀j ∈ ℕ.

∞ By (21.13), (21.15) and (21.16), the set ER,e has full measure.

(21.16)

21.2 Multifractal analysis of the conformal measure mF over a subset of J

| 839

As a direct consequence of this theorem, we get the following result. Corollary 21.2.4. Let Φ = {φe }e∈E be a finitely irreducible CGDMS satisfying the SOSC. For every e ∈ E, every 0 < R < supx∈J dist(x, 𝜕X) and every (q, t) ∈ ℝ2 such that qu+t > θ, we have that ∞ ∞ μFq,t (π(ER,e )) = μfq,t (ER,e ) = 1 and thus

∞ ∞ μFq,t (π(E0,e )) = μfq,t (E0,e ) = 1.

In particular, ∞ ∞ μF (π(ER,e )) = μf (ER,e )=1

and so

∞ ∞ μF (π(E0,e )) = μf (E0,e ) = 1.

If, additionally, Φ is cofinitely regular then for every q ∈ ℝ, ∞ ∞ μFq (π(ER,e )) = μq (ER,e ) = 1 and hence

∞ ∞ μFq (π(E0,e )) = μq (E0,e ) = 1.

We now recall a few basic definitions from multifractal analysis. While the Hausdorff dimension of a fractal set gives us an idea of the global structure of that set, the pointwise dimension of a measure on that set permits us to capture the local structure of the set. Let μ be a Borel probability measure on J0 := π(E0∞ ). Per Definition 15.6.2, the pointwise or local dimension dμ (x) of μ at a point x ∈ J0 is the power law behavior (if any) of μ(B(x, r)) for small r > 0, i. e., log μ(B(x, r)) . r→0 log r

dμ (x) := lim

The local structure of J0 is determined by the following partition of that set. For each number α ≥ 0, denote by J0,μ (α) the set of all points of J0 at which the pointwise dimension of μ is equal to α, i. e., 󵄨 J0,μ (α) := {x ∈ J0 󵄨󵄨󵄨 dμ (x) = α}. Denote the Hausdorff dimension of J0,μ (α) by d0,μ (α) := HD(J0,μ (α)). The multifractal analysis of J0 consists in: (1) determining the range of the function x 󳨃→ dμ (x), and (2) for each α in that range, determining d0,μ (α).

840 | 21 Multifractal analysis for conformal graph directed Markov systems The map α 󳨃→ d0,μ (α) is the (fine Hausdorff ) multifractal spectrum of the measure μ. We now prove that for every q ∈ ℝ the measure μFq confers full measure to the subset of points of J0 where the local dimension of the measure mF (or, equivalently, μF ) is α(q). In other terms, the measure μFq is entirely distributed over the set J0,mF (α(q)) and not at all over any of the other sets J0,mF (α), α ≠ α(q). Accordingly, the measure μFq (and its sister mFq ) will be used to determine the Hausdorff dimension of J0,mF (α(q)). Given e ∈ E, set 󵄨 S f (ω) ∞ ∞ ∞ 󵄨󵄨󵄨 = α} ⊆ E0,e ⊆ E0∞ . E0,e (α) := {ω ∈ E0,e lim n 󵄨󵄨 n→∞ 󵄨󵄨 Sn ΞΦ (ω) Theorem 21.2.5. Let Φ = {φe }e∈E be a cofinitely regular, finitely irreducible, conformallike CGDMS satisfying the SOSC. For every e ∈ E, the following two statements hold: ∞ (a) π(E0,e (α)) ⊆ J0,mF (α) for all α ≥ 0. ∞ (b) μq (E0,e (α(q))) = 1 for all q ∈ ℝ. Moreover, (c) μFq (J0,mF (α(q))) = 1 for all q ∈ ℝ. ∞ ∞ Proof. (a) Let x ∈ π(E0,e (α)). Then there is some ω ∈ E0,e (α) such that π(ω) = x. ∞ ∞ Therefore, ω ∈ ER,e for some R > 0. Since the sets ER,e get bigger as R gets smaller, we ∞ may assume that 0 < R < supy∈J dist(y, 𝜕X). Let (nj )∞ j=1 := (nj (ω, R, e))j=1 be the strictly increasing sequence of all n ∈ ℕ defined in (21.9), i. e., of all n ∈ ℕ such that

π(σ n (ω)) ∈ Xt(ωn ) \ B(𝜕Xt(ωn ) , R)

and

ωn+1 = e.

󵄨 󵄨 Let 0 < r ≤ K −1 R󵄨󵄨󵄨φ′ω|n (π(σ n1 (ω)))󵄨󵄨󵄨. Let j ∈ ℕ be the unique number such that 1

󵄨 󵄨 󵄨 󵄨 K −1 R󵄨󵄨󵄨φ′ω|n (π(σ nj+1 (ω)))󵄨󵄨󵄨 < r ≤ K −1 R󵄨󵄨󵄨φ′ω|n (π(σ nj (ω)))󵄨󵄨󵄨. j+1

j

(21.17)

Since B(π(σ nj (ω)), R) ⊆ Xt(ωn ) , the right inequality in (21.17) and Lemma 19.3.10 yield j

󵄨 󵄨 B(x, r) ⊆ B(π(ω), K −1 R󵄨󵄨󵄨φ′ω|n (π(σ nj (ω)))󵄨󵄨󵄨) ⊆ φω|n (B(π(σ nj (ω)), R)). j

j

(21.18)

Using successively (1) relation (21.18); (2) the conformality of mF (by means of Lemma 19.8.12); (3) the fact that P(F) = 0 by assumption; (4) Lemma 19.8.7 with Vγ (F) := υγ (F) ; 1−e−γ

and (5) Lemma 19.8.2, we obtain mF (B(x, r)) ≤ mF (φω|n (B(π(σ nj (ω)), R))) j

≤ exp( sup Sω|n F(y)) y∈Xt(ωn

j

)

j

≤ exp(Sω|n F(π(σ nj (ω))) + Vγ (F)) j

= eVγ (F) exp(Snj f (ω)).

(21.19)

21.2 Multifractal analysis of the conformal measure mF over a subset of J

|

841

On the other hand, by the left inequality in (21.17) and (19.8) in Lemma 19.3.7, we know that 󵄨 󵄨 B(x, r) ⊇ B(π(ω), K −1 R󵄨󵄨󵄨φ′ω|n (π(σ nj+1 (ω)))󵄨󵄨󵄨) ⊇ φω|n (B(π(σ nj+1 (ω)), K −2 R)). j+1

j+1

(21.20)

Using successively (1) relation (21.20); (2) the conformality of mF ; (3) P(F) = 0; (4) ∞,ωnj+1

Lemma 19.8.7; (5) the fact that EA

⊇ [ωnj+1 +1 ] = [e]; and (6) Lemma 19.8.2, we get

mF (B(x, r)) ≥ mF (φω|n (B(π(σ nj+1 (ω)), K −2 R))) j+1

≥ exp(

Sω|n F(y))mF (B(π(σ nj+1 (ω)), K −2 R) ∩ π(EA

∞,ωnj+1

inf

y∈Xt(ωn

j+1

)

j+1

))

≥ exp(Sω|n F(π(σ nj+1 (ω))) − Vγ (F)) j+1

⋅ mF (B(π(σ nj+1 (ω)), K −2 R) ∩ π(EA

∞,ωnj+1

))

≥ e−Vγ (F) exp(Sω|n F(π(σ nj+1 (ω))))mF (B(π(σ nj+1 (ω)), K −2 R) ∩ π([e])) j+1

≥e

−Vγ (F)

exp(Snj+1 f (ω))MF,e (K −2 R),

(21.21)

where MF,e (s) :=

inf mF (B(y, s) ∩ π([e])) > 0.

(21.22)

y∈π([e])

The positivity of MF,e (s) is guaranteed by Lemma 19.8.21. Moreover, we infer from (21.17) that 󵄨 󵄨 󵄨 󵄨 log(K −1 R)+log󵄨󵄨󵄨φ′ω|n (π(σ nj+1 (ω)))󵄨󵄨󵄨 < log r ≤ log(K −1 R)+log󵄨󵄨󵄨φ′ω|n (π(σ nj (ω)))󵄨󵄨󵄨. (21.23) j+1

j

We deduce from (21.21) and (21.23) that dmF (x) := lim sup r→0

Snj+1 f (ω) + log(MF,e (K −2 R)e−Vγ (F) ) log mF (B(x, r)) ≤ lim sup 󵄨 󵄨 log r log(K −1 R) + log󵄨󵄨󵄨φ′ω| (π(σ nj (ω)))󵄨󵄨󵄨 j→∞ nj

= lim sup

Snj+1 f (ω) + log(MF,e (K −2 R)e−Vγ (F) ) log(K −1 R) + Snj ΞΦ (ω)

j→∞

Snj+1 f (ω)

= lim sup

Snj+1 ΞΦ (ω)

j→∞

=

α+0 =α 0+1

+

log(MF,e (K −2 R)e−Vγ (F) ) Snj+1 ΞΦ (ω)

log(K −1 R) Snj+1 ΞΦ (ω)

+

Snj ΞΦ (ω)

Snj+1 ΞΦ (ω)

842 | 21 Multifractal analysis for conformal graph directed Markov systems ∞ ∞ since ω ∈ E0,e (α) ∩ ER,e and Sn ΞΦ ≤ n log s for all n. Similarly, we deduce from (21.19) and (21.23) that

dmF (x) := lim inf r→0

Snj f (ω) + Vγ (F) log mF (B(x, r)) ≥ lim inf 󵄨 󵄨󵄨 = α. nj+1 j→∞ log(K −1 R) + log󵄨󵄨φ′ log r 󵄨 ω| (π(σ (ω)))󵄨󵄨 nj+1

Hence, α ≤ dmF (x) ≤ dmF (x) ≤ α, i. e., dmF (x) = α. So x ∈ J0,mF (α). (b) Let q ∈ ℝ. According to Corollary 21.2.4 and the ergodic case of Birkhoff’s er∞ godic theorem (Corollary 8.2.14), there exists Eq ⊆ E0,e such that μq (Eq ) = 1 and so that for all ω ∈ Eq we have lim

n→∞

1 S f (ω) = ∫ f dμq n n ∞ EA

and

lim

n→∞

1 S Ξ (ω) = ∫ ΞΦ dμq . n n Φ ∞ EA

Using this and (21.5), we get ∫ f dμq Sn f (ω) = = α(q), n→∞ S Ξ (ω) ∫ ΞΦ dμq n Φ lim

∀ω ∈ Eq .

∞ ∞ Hence, Eq ⊆ E0,e (α(q)) and thus μq (E0,e (α(q))) ≥ μq (Eq ) = 1. (c) Let q ∈ ℝ and pick any e ∈ E. Using part (a) with α = α(q) and part (b), we deduce that ∞ ∞ μFq (J0,mF (α(q))) ≥ μq ∘ π −1 (π(E0,e (α(q)))) ≥ μq (E0,e (α(q))) = 1.

We shall now prove that d0,mF (α) and T(q) form a Legendre transform pair under an additional restriction on the CGDMS. (For more information on the Legendre transform, see the discussion around Proposition 16.6.4.) This means that the Hausdorff dimension of the set of points of J0 where the local dimension of the measure mF is α(q) is equal to qα(q) + T(q). Definition 21.2.6. A CGDMS Φ = {φe }e∈E is said to be an almost-CIFS if E = ℕ and there exists P ∈ ℕ such that Aab = 1 for all a ∈ ℕ and all b ≥ P. It is obvious that any CIFS is finitely primitive. However, an almost-CIFS may not even be irreducible. Nevertheless, every irreducible almost-CIFS is finitely primitive. Theorem 21.2.7. Let Φ = {φe }e∈E be a cofinitely regular, irreducible, conformal-like almost-CIFS satisfying the SOSC. For every q ∈ ℝ we have d0,mF (−T ′ (q)) = T(q) − qT ′ (q). Alternatively, d0,mF (α(q)) = qα(q) + T(q). Proof. Using Theorems 21.2.5(a,b), 19.8.30 and 17.5.1 (which guarantees that μq is an (in fact, the unique) equilibrium state for fq ), and the fact that 0 = P(q, T(q)) = P(Fq ) =

21.2 Multifractal analysis of the conformal measure mF over a subset of J

|

843

P(fq ) by definition of T(q), we obtain ∞ d0,mF (α(q)) := HD(J0,mF (α(q))) ≥ HD(π(E0,e (α(q)))) for any e ∈ E

≥ HD(μq ∘ π −1 ) = =

P(fq ) − ∫ fq dμq χμq (σ)

=q

∫ f dμq

∫ ΞΦ dμq

hμq (σ) χμq (σ) =

− ∫ fq dμq

− ∫ ΞΦ dμq

=

∫(qf + T(q)ΞΦ ) dμq ∫ ΞΦ dμq

+ T(q) = qα(q) + T(q).

To prove the opposite inequality, fix x ∈ J0,mF (α(q)). Then there is ω ∈ E0∞ such that π(ω) = x. Let 0 < R < min{K −1 , r(ω)}, where r(ω) comes from (21.7). Let also (nj )∞ j=1 be any subsequence of the strictly increasing sequence (nj (ω, R))∞ j=1 of all n such that π(σ n (ω)) ∈ Xt(ωn ) \B(𝜕Xt(ωn ) , R). Relation (19.8) in Lemma 19.3.7 asserts that 󵄨 󵄨 φω|n (B(π(σ n (ω)), R)) ⊆ B(π(ω), KR󵄨󵄨󵄨φ′ω|n (π(σ n (ω)))󵄨󵄨󵄨) for every n ∈ ℕ. Since Φ is an almost-CIFS, we know that EA n ⊇ [ωn+1 ] ∪ ⋃b≥P [b] for all n ∈ ℕ. In a similar way to (21.21), the conformality of mFq and the bounded variation principle for Fq give ∞,ω

󵄨 󵄨 mFq (B(x, KR󵄨󵄨󵄨φ′ω|n (π(σ n (ω)))󵄨󵄨󵄨)) ≥ mFq (φω|n (B(π(σ n (ω)), R))) ≥ e−Vγ (Fq ) exp(Sn fq (ω)) ⋅ mFq (B(π(σ n (ω)), K −2 R) ∩ π(EA

∞,ωn

))

̃F (K −2 R), ≥ e−Vγ (Fq ) exp(Sn fq (ω))M q

(21.24)

where ̃F (s) := min{min MF ,b (s), M q q b 0. b≥P

This last quantity is positive by Lemma 19.8.21 (MFq ,b (s) was defined in (21.22)). Hence, 󵄨 󵄨 ̃F (K −2 R)e−Vγ (Fq ) ) + Sn fq (ω) log mFq (B(x, KR󵄨󵄨󵄨φ′ω|n (π(σ n (ω)))󵄨󵄨󵄨)) log(M q ≤ 󵄨 󵄨 log(KR) + Sn ΞΦ (ω) log(KR󵄨󵄨󵄨φ′ω| (π(σ n (ω)))󵄨󵄨󵄨) n ̃F (K −2 R)e−Vγ (Fq ) ) log(M q

=

Sn ΞΦ (ω)

S f (ω) n Φ (ω)

+ q S nΞ

log(KR) Sn ΞΦ (ω)

+1

+ T(q)

(21.25)

844 | 21 Multifractal analysis for conformal graph directed Markov systems for all n ∈ ℕ. As in (21.19), we have 󵄨 󵄨 mF (B(x, K −1 R󵄨󵄨󵄨φ′ω|n (π(σ nj (ω)))󵄨󵄨󵄨)) ≤ eVγ (F) exp(Snj f (ω)). j

Thus, 󵄨 󵄨 log mF (B(x, K −1 R󵄨󵄨󵄨φ′ω|n (π(σ nj (ω)))󵄨󵄨󵄨)) Vγ (F) + Snj f (ω) j ≥ 󵄨 󵄨 log(K −1 R) + Snj ΞΦ (ω) log(K −1 R󵄨󵄨󵄨φ′ω| (π(σ nj (ω)))󵄨󵄨󵄨)

(21.26)

nj

for all j ∈ ℕ large enough. Using (21.25) and (21.26) as well as the facts that Sn ΞΦ ≤ n log s for all n and that x ∈ J0,mF (α(q)), we deduce that dmF (x) := lim inf q

log mFq (B(x, r))

log r 󵄨 󵄨 log mFq (B(x, KR󵄨󵄨󵄨φ′ω|n (π(σ nj (ω)))󵄨󵄨󵄨)) j ≤ lim inf 󵄨 󵄨 j→∞ log(KR󵄨󵄨󵄨φ′ω| (π(σ nj (ω)))󵄨󵄨󵄨) nj r→0

≤ q lim inf j→∞

= q lim inf

Snj f (ω)

Snj ΞΦ (ω)

+ T(q)

Vγ (F) + Snj f (ω)

+ T(q) log(K −1 R) + Snj ΞΦ (ω) 󵄨 󵄨 log mF (B(x, K −1 R󵄨󵄨󵄨φ′ω|n (π(σ nj (ω)))󵄨󵄨󵄨)) j + T(q) ≤ q lim inf 󵄨 󵄨 j→∞ log(K −1 R󵄨󵄨󵄨φ′ω| (π(σ nj (ω)))󵄨󵄨󵄨) nj j→∞

= qα(q) + T(q).

As dmF (x) ≤ qα(q) + T(q) for every x ∈ J0,mF (α(q)), we conclude by Theorem 15.6.3(b)

that

q

d0,mF (α(q)) := HD(J0,mF (α(q))) ≤ qα(q) + T(q). Finally, recall that T ′ (q) = −α(q) by Theorem 21.1.4. All of the above results give us an analog of Theorem 4.9.4 in [81] about the fine Hausdorff multifractal spectrum of the set J0 . Theorem 21.2.8. Let Φ = {φe }e∈E be a cofinitely regular, irreducible, conformal-like almost-CIFS satisfying the SOSC. Then the following statements hold: (a) For μF -a. e. x ∈ J0 , dμF (x) =

∫E ∞ f dμf A

∫E ∞ ΞΦ dμf A

= HD(μF ).

21.3 Multifractal analysis over J

| 845

(b) The temperature function T : ℝ → ℝ, defined by P(q, T(q)) = 0 for all q ∈ ℝ, is real analytic and such that – T(0) = HD(J) – T(1) = 0 –

∫E ∞ f dμq

T ′ (q) = −α(q) = − ∫

A

E∞ A

ΞΦ dμq

rnj ≥ C 󵄩󵄩󵄩φ′ω|n 󵄩󵄩󵄩X for all j ∈ ℕ; and t(ωn ) j j (c) lim rnj = 0. j→∞

Then dmF (α(q)) = qα(q) + T(q) for all q ∈ ℝ. Proof. Clearly, dmF (α(q)) ≥ d0,mF (α(q)) = qα(q) + T(q) using Theorem 21.2.7. We shall now prove the other inequality. Since J ′ is countable, according to Theorem 15.6.3(b) it suffices to show that dmF (x) ≤ qα(q) + T(q) for every x ∈ JmF (α(q)) \ J ′ . Accordingly, q

∞ fix x ∈ JmF (α(q)) \ J ′ . Let ω, C, (nj )∞ j=1 and (rnj )j=1 be as in the statement of the theorem.

21.3 Multifractal analysis over J

|

847

By (19.8) in Lemma 19.3.7, we know that 󵄨 󵄨 φω|n (B(π(σ n (ω)), R)) ⊆ B(π(ω), KR󵄨󵄨󵄨φ′ω|n (π(σ n (ω)))󵄨󵄨󵄨) for every n ∈ ℕ and 0 < R ≤ dist(X, 𝜕W) := min{dist(Xυ , 𝜕Wυ ) : υ ∈ V}. As in (21.24), the conformality of mFq and the bounded variation principle for Fq give 󵄨 󵄨 ̃F (K −2 R)e−Vγ (Fq ) exp(Sn fq (ω)). mFq (B(x, KR󵄨󵄨󵄨φ′ω|n (π(σ n (ω)))󵄨󵄨󵄨)) ≥ M q Hence, as in (21.25), 󵄨 󵄨 log mFq (B(x, KR󵄨󵄨󵄨φ′ω|n (π(σ n (ω)))󵄨󵄨󵄨)) ≤ 󵄨 󵄨 log(KR󵄨󵄨󵄨φ′ω| (π(σ n (ω)))󵄨󵄨󵄨) n

̃F (K −2 R)e−Vγ (Fq ) ) log(M q Sn ΞΦ (ω)

S f (ω) n Φ (ω)

+ q S nΞ

log(KR) Sn ΞΦ (ω)

+1

+ T(q)

(21.27)

for all n ∈ ℕ large enough. By assumptions (a) and (b), we obtain for all j ∈ ℕ, log mF (B(x, rnj )) log rnj

log C + Snj f (ω)

≥

log rnj log C + Snj f (ω) ≥ 󵄩 󵄩 log C + log󵄩󵄩󵄩φ′ω| 󵄩󵄩󵄩X t(ωn ) nj j log C + Snj f (ω) 󵄨 󵄨 log C + log󵄨󵄨󵄨φ′ω| (ω)󵄨󵄨󵄨 nj log C + Snj f (ω) = . log C + Snj ΞΦ (ω)

≥

Using (21.27) and (21.28), we deduce that dmF (x) := lim inf q

log mFq (B(x, r))

log r 󵄨 󵄨 log mFq (B(x, KR󵄨󵄨󵄨φ′ω|n (π(σ nj (ω)))󵄨󵄨󵄨)) j ≤ lim inf 󵄨 󵄨 j→∞ log(KR󵄨󵄨󵄨φ′ω| (π(σ nj (ω)))󵄨󵄨󵄨) nj Snj f (ω) + T(q) ≤ q lim inf j→∞ Sn ΞΦ (ω) j log C + Snj f (ω) = q lim inf + T(q) j→∞ log C + Sn ΞΦ (ω) j r→0

≤ q lim inf j→∞

log mF (B(x, rnj ))

= qα(q) + T(q).

log rnj

+ T(q)

(21.28)

848 | 21 Multifractal analysis for conformal graph directed Markov systems As dmF (x) ≤ qα(q) + T(q) for every x ∈ JmF (α(q)) \ J ′ , we infer from Theorem 15.6.3(b) q

that HD(JmF (α(q)) \ J ′ ) ≤ qα(q) + T(q). Since J ′ is countable, it ensues that dmF (α(q)) := HD(JmF (α(q))) = HD(JmF (α(q)) \ J ′ ) ≤ qα(q) + T(q). Theorem 21.3.4. Let Φ = {φe }e∈E be an almost-CIFS fulfilling the hypotheses of Theorem 21.3.3. Then the conclusion of Theorem 21.3.1 applies.

There are interesting families of CIFSs which satisfy the conditions imposed in Theorem 21.3.3. Among others, let us mention real continued fractions over the unit interval (cf. Example 19.9.5). Example 21.3.5. Let X = [0, 1]. Let Φ = {φn : X → X | n ∈ ℕ}, where φn (x) =

1 . n+x

Then Φ is a cofinitely regular CIFS which satisfies the SOSC and the Cone Condition (CC). This latter implies that Φ is conformal-like, according to Proposition 19.7.19. This CIFS also satisfies conditions (a)–(c) of Theorem 21.3.3 but not the BSC. Moreover, writing x=

x1 +

1

1 x2 +...

,

note that we can set J ′ = {x ∈ [0, 1] : lim infn→∞ xn < ∞}, and thus J \ J ′ = {x ∈ [0, 1] : limn→∞ xn = ∞}. In fact, there is a larger class of one-dimensional CIFSs for which the conditions of Theorem 21.3.3 are fulfilled and to which Theorem 21.3.4 thereby applies. By onedimensional, we simply mean that X is a subinterval of ℝ. Every one-dimensional CIFS obeys the cone condition (CC), and hence is conformal-like by Proposition 19.7.19. Every one-dimensional CIFS also satisfies the SOSC. Theorem 21.3.6. Let Φ = {φn : X → X | n ∈ ℕ} be a one-dimensional cofinitely regular CIFS satisfying the following properties: 󵄩 󵄩 󵄩 󵄩 (a) 󵄩󵄩󵄩φ′n 󵄩󵄩󵄩X is comparable to 󵄩󵄩󵄩φ′n+1 󵄩󵄩󵄩X , i. e., there exists a constant C ≥ 1 such that 󵄩󵄩 ′ 󵄩󵄩 󵄩φ 󵄩󵄩X ≤ C, C −1 ≤ 󵄩󵄩 n+1 󵄩󵄩φ′ 󵄩󵄩󵄩 󵄩 n 󵄩X

∀n ∈ ℕ.

(b) The generators φn of Φ either all preserve orientation (i. e., φ′n > 0 for all n ∈ ℕ) or all reverse orientation (i. e., φ′n < 0 for all n ∈ ℕ). (c) Either φn+1 > φn for all n ∈ ℕ or φn+1 < φn for all n ∈ ℕ. Then Φ satisfies the hypotheses of Theorem 21.3.3.

21.3 Multifractal analysis over J

|

849

Proof. It follows from the bounded distortion property (BDP) and (a) that 󵄨 󵄨󵄨 󵄨φ (X)󵄨󵄨 (CK)−1 ≤ 󵄨 󵄨 n+1 󵄨 󵄨 ≤ CK, 󵄨󵄨φn (X)󵄨󵄨 󵄨 󵄨

∀n ∈ ℕ,

(21.29)

where |I| denotes the length of an interval I. Moreover, a rather straightforward calculation shows that the amalgamated function ΞΦ satisfies sup

ω,τ∈EA∞ : |ω1 −τ1 |=1

󵄨󵄨 󵄨 󵄨󵄨ΞΦ (ω) − ΞΦ (τ)󵄨󵄨󵄨 ≤ log(CK) =: D.

−n ∞ ′ ∞ The set J ′ = π(∪∞ such n=1 σ (1 )) is clearly countable. Let x ∈ J \ J and ω ∈ ℕ that π(ω) = x. Let nj be the position of the jth letter in ω which is not a 1. Let rnj = 󵄨󵄨 󵄨 󵄨󵄨φωn (X)󵄨󵄨󵄨/(CK). By (21.29), we have j

󵄨 󵄨󵄨 󵄨󵄨 󵄨 rnj ≤ min{ 󵄨󵄨󵄨φωn −1 (X)󵄨󵄨󵄨, 󵄨󵄨󵄨φωn (X)󵄨󵄨󵄨, 󵄨󵄨󵄨φωn +1 (X)󵄨󵄨󵄨 } j j j and hence B(x, rnj ) ∩ J ⊆

ωnj +1

⋃ φω|n −1 k (X) ∩ J. j

k=ωnj −1

Recall that F = G + u LogΦ , so f = g + uΞΦ . Using the conformality of mF , the standing υγ (h) assumption P(F) = 0, and Lemmas 19.8.7 and 17.2.3 with Vγ (h) := 1−e −γ , we obtain that mF (B(x, rnj )) ≤

ωnj +1

∑ mF (φω|n −1 k (X)) j

k=ωnj −1 ωnj +1

∑ ∫ exp(Sω|n −1 k F(y) − nj P(F)) dmF (y)

≤

j

k=ωnj −1 J ωnj +1

∑ exp(sup Sω|n −1 k F(y))

≤

ωnj +1

≤

∑ eVγ (F) exp(

Vγ (F)

= eVγ (F)

inf

τ∈[ω|nj −1 k]

k=ωnj −1

≤e

j

y∈J

k=ωnj −1

Snj f (τ))

ωnj +1

∑ exp(Snj f (ω|nj −1 kω|∞ nj +1 ))

k=ωnj −1 ωnj +1

∞ ∑ exp(Snj −1 f (ω|nj −1 kω|∞ nj +1 ) + f (kω|nj +1 ))

k=ωnj −1

850 | 21 Multifractal analysis for conformal graph directed Markov systems

=e

Vγ (F)

ωnj +1

∑ exp(Snj −1 f (ω|nj −1 kω|∞ nj +1 ) − Snj −1 f (ω))

k=ωnj −1

∞ ∞ ⋅ exp(Snj −1 f (ω)) exp(f (kω|∞ nj +1 ) − f (ω|nj )) exp(f (ω|nj ))

≤ eVγ (F)+Vγ (f ) ≤ 3e

ωnj +1

∑ exp(Snj −1 f (ω))e

2‖g‖∞ +|k−ωnj |uD

k=ωnj −1

2‖g‖∞ +uD Vγ (F)+Vγ (f )

e

exp(f (ω|∞ nj ))

exp(Snj f (ω)).

Thus, condition (a) in Theorem 21.3.3 is satisfied. Condition (b) is fulfilled since rnj =

1 󵄩󵄩 ′ 󵄩󵄩 1 󵄨󵄨 󵄨 󵄨φ (X)󵄨󵄨󵄨 ≥ 󵄩φ 󵄩 . CK 󵄨 ωnj CK 2 󵄩 ωnj 󵄩X

Finally, condition (c) is obviously satisfied, as ∑∞ j=1 |φωn (X)| ≤ |X| < ∞, and hence j

rnj → 0 as j → ∞.

21.4 Multifractal analysis over another subset of J We will work in the general context of Section 21.2. Let Φ = {φe : Xt(e) → Xi(e) | e ∈ E} be a cofinitely regular, finitely irreducible, conformal-like CGDMS satisfying the SOSC. Let θ be the finiteness parameter of Φ. Let F = G + u LogΦ be a family of functions such that P(F) = 0, where G is a bounded Hölder family of functions and u > θ. For every ω ∈ EA∞ , every R > 0, and every finite subalphabet W ⊆ E, let (nj (ω, R, W))Nj=1 be the sequence of all n ≥ 0 such that (cf. (21.9)) σ n (ω) ∈ π −1 ( ⋃ Xυ \ B(𝜕Xυ , R)) ∩ ⋃ [e]. υ∈V

e∈W

(21.30)

This means that nj (ω, R, W) is the jth time that an iterate of ω hits the set π −1 (⋃υ∈V Xυ \ B(𝜕Xυ , R)) ∩ ⋃e∈W [e]. This sequence may be empty (N = 0), nonempty and finite (N ∈ ℕ), or infinite (N = ∞), depending on ω, R and W. We will correspondingly write N(ω, R, W). To shorten notation, we shall usually write nj (ω) instead of nj (ω, R, W) and N(ω) in place of N(ω, R, W) when R and W are already fixed. Recall the definition of ER∞ in (21.8). Define the sets 󵄨 ∞ ÊR,W := {ω ∈ ER∞ 󵄨󵄨󵄨 N(ω, R, W) = ∞} and 󵄨󵄨 Snj+1 (ω) ΞΦ (ω) nj+1 (ω) 󵄨 ∞ ∞ = 1; lim = 1} ⊆ ÊR,W . ER,W := {ω ∈ ER∞ 󵄨󵄨󵄨 ωnj (ω)+1 ∈ W, ∀j ∈ ℕ; lim 󵄨󵄨 j→∞ Sn (ω) ΞΦ (ω) j→∞ nj (ω) j

21.4 Multifractal analysis over another subset of J

|

851

Theorem 21.4.1. Let Φ = {φe }e∈E be a CGDMS satisfying the SOSC. For every 0 < R < supx∈J dist(x, 𝜕X), every finite subalphabet W ⊆ E and every ergodic σ-invariant Borel probability measure μ on EA∞ such that supp(μ) = EA∞ and ∫E ∞ ΞΦ dμ > −∞, it holds A

that

∞ μ(ER,W ) = 1 and thus

∞ ) = 1. μ(ÊR,W

Proof. Simply replace [e] by ⋃e∈W [e] in the proof of Theorem 21.2.3. As a consequence of this theorem, we get the following result (cf. Corollary 21.2.4). Corollary 21.4.2. Let Φ = {φe }e∈E be a finitely irreducible CGDMS satisfying the SOSC. For every 0 < R < supx∈J dist(x, 𝜕X), every finite subalphabet W ⊆ E, and every (q, t) ∈ ℝ2 such that qu + t > θ, we have that ∞ ∞ μFq,t (π(ER,W )) = μfq,t (ER,W ) = 1 and thus

∞ ∞ μFq,t (π(ÊR,W )) = μfq,t (ÊR,W ) = 1.

In particular, ∞ ∞ μF (π(ER,W )) = μf (ER,W )=1

and so

∞ ∞ μF (π(ÊR,W )) = μf (ÊR,W ) = 1.

If, additionally, Φ is cofinitely regular then for every q ∈ ℝ, ∞ ∞ μFq (π(ER,W )) = μq (ER,W ) = 1 and hence

∞ ∞ μFq (π(ÊR,W )) = μq (ÊR,W ) = 1.

Fix 0 < R < supx∈J dist(x, 𝜕X) and a finite subalphabet W ⊆ E. Let μ be a Borel probability measure on ∞ JR,W := π(ÊR,W ).

The local structure of JR,W is determined by the following partition of that set. For each number α ≥ 0, denote by JR,W,μ (α) the set of all points of JR,W at which the pointwise dimension of μ is equal to α, i. e., 󵄨 JR,W,μ (α) := {x ∈ JR,W 󵄨󵄨󵄨 dμ (x) = α}. Denote the Hausdorff dimension of JR,W,μ (α) by dR,W,μ (α) := HD(JR,W,μ (α)). The multifractal analysis of JR,W consists in: (1) determining the range of the function x 󳨃→ dμ (x), and (2) for each α in that range, determining dR,W,μ (α).

852 | 21 Multifractal analysis for conformal graph directed Markov systems The map α 󳨃→ dR,W,μ (α) is the (fine Hausdorff ) multifractal spectrum of the measure μ. We now prove that for every q ∈ ℝ the measure μFq confers full measure to the subset of points of JR,W where the local dimension of the measure mF (or, equivalently, μF ) is α(q). In other terms, the measure μFq is entirely distributed over the set JR,W,mF (α(q)) and not at all over any of the other sets JR,W,mF (α), α ≠ α(q). Accordingly, it is this measure (and its sister mFq ), which will be used to determine the Hausdorff dimension of JR,W,mF (α(q)). Set 󵄨 S f (ω) ∞ ∞ 󵄨󵄨󵄨 ER,W (α) := {ω ∈ ER,W lim n = α}. 󵄨󵄨 n→∞ 󵄨󵄨 Sn ΞΦ (ω) Theorem 21.4.3. Let Φ = {φe }e∈E be a cofinitely regular, finitely irreducible, conformallike CGDMS satisfying the SOSC. For every 0 < R < supx∈J dist(x, 𝜕X) and for every finite subalphabet W ⊆ E, the following statements hold: ∞ (a) π(ER,W (α)) ⊆ JR,W,mF (α) for all α ≥ 0. ∞ (b) μq (ER,W (α(q))) = 1 for all q ∈ ℝ. (c) μFq (JR,W,mF (α(q))) = 1 for all q ∈ ℝ. Proof. The proof goes along similar lines to that of Theorem 21.2.5. Next, we establish that dR,W,mF (α) and T(q) form a Legendre transform pair. This means that the Hausdorff dimension of the set of points of JR,W where the local dimension of the measure mF is α(q) is equal to qα(q) + T(q). Theorem 21.4.4. Let Φ = {φe }e∈E be a cofinitely regular, finitely irreducible, conformallike CGDMS satisfying the SOSC. Fix 0 < R < supx∈J dist(x, 𝜕X) and a finite subalphabet W ⊆ E. For every q ∈ ℝ, we have dR,W,mF (−T ′ (q)) = T(q) − qT ′ (q). Alternatively, dR,W,mF (α(q)) = qα(q) + T(q). Proof. The proof goes along similar lines to that of Theorem 21.2.7. All of the above results give us an analog of Theorem 4.9.4 in [81] about the fine Hausdorff multifractal spectrum of the set JR,W . Theorem 21.4.5. Let Φ = {φe }e∈E be a cofinitely regular, finitely irreducible, conformallike CGDMS satisfying the SOSC. For every 0 < R < supx∈J dist(x, 𝜕X) and for every finite subalphabet W ⊆ E, the following statements hold: (a) For μF -a. e. x ∈ JR,W , dμF (x) =

∫E ∞ f dμf A

∫E ∞ ΞΦ dμf

= HD(μF ).

A

(b) The temperature function T : ℝ → ℝ, defined by P(q, T(q)) = 0 for all q ∈ ℝ, is real analytic and such that

21.5 Exercises | 853

– –

T(0) = HD(J) T(1) = 0

–

T ′ (q) = −α(q) = − ∫

∫E ∞ f dμq A

E∞ A

ΞΦ dμq

0 and B ∈ σ(𝒜) there exists some A ∈ 𝒜 such that μ(A △ B) < ε. Proof. See Theorem 4.4 in Kingman and Taylor [62].

A.1.5 Integration Let us now briefly recollect some facts about integration. First, the definition of the integral of a measurable function with respect to a measure.

A.1 Measure theory | 867

Definition A.1.33. Let (X, 𝒜, μ) be a measure space and let A ∈ 𝒜. (a) If s : X → [0, ∞) is a measurable simple function of the form n

s = ∑ αi 1Ai , i=1

then the integral of the function s over the set A with respect to the measure μ is defined as n

∫ s dμ := ∑ αi μ(Ai ∩ A). i=1

A

We use the convention that 0 ⋅ ∞ = 0 in case it happens that αi = 0 and μ(Ai ∩ A) = ∞ for some 1 ≤ i ≤ n. (b) If f : X → [0, ∞] is a measurable function, then the integral of the function f over the set A with respect to the measure μ is defined as ∫ f dμ := sup ∫ s dμ, A

A

where the supremum is taken over all measurable simple functions 0 ≤ s ≤ f . Note that if f is simple, then definitions (a) and (b) coincide. (c) If f : X → ℝ is a measurable function, then the integral of the function f over the set A with respect to the measure μ is defined as ∫ f dμ := ∫ f+ dμ − ∫ f− dμ, A

A

A

as long as min{∫A f+ dμ, ∫A f− dμ} < ∞, where f+ and f− respectively denote the positive and negative parts of f . That is, f+ (x) := max{f (x), 0} whereas f− (x) := max{−f (x), 0}. (d) A measurable function f : X → ℝ is said to be integrable if ∫X |f | dμ < ∞. We denote this by f ∈ L1 (X, 𝒜, μ). If no confusion is possible, we simply write f ∈ L1 (μ). (e) A property is said to hold μ-almost everywhere (sometimes abbreviated μ-a. e.) if the property holds on the entire space except possibly on a set of μ-measure zero. The following properties follow from this definition. Lemma A.1.34. Let (X, 𝒜, μ) be a measure space. Let f , g ∈ L1 (X, 𝒜, μ), A, B ∈ 𝒜 and a, b ∈ ℝ. (a) If f ≤ g μ-a. e., then ∫A f dμ ≤ ∫A g dμ. Also, if f < g μ-a. e., then ∫A f dμ < ∫A g dμ. (b) If A ⊆ B and 0 ≤ f μ-a. e., then 0 ≤ ∫A f dμ ≤ ∫B f dμ.

868 | Appendix A – A selection of classical results (c) “Triangle inequality”: 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨∫ f dμ󵄨󵄨󵄨 ≤ ∫ |f | dμ. 󵄨󵄨 󵄨󵄨 A

A

(d) Linearity: ∫(af + bg) dμ = a ∫ f dμ + b ∫ g dμ. A

A

A

(e) If (An )∞ n=1 is a sequence of mutually disjoint measurable sets, then ∞

∫

f dμ = ∑ ∫ f dμ. n=1

⋃∞ n=1 An

An

(f) f = g μ-a. e. ⇐⇒ ∫A f dμ = ∫A g dμ, ∀A ∈ 𝒜. (g) The relation f = g μ-a. e. is an equivalence relation on the set L1 (X, 𝒜, μ). The equivalence classes generated by this relation form a Banach space also denoted by L1 (X, 𝒜, μ) (or L1 (μ), for short) with norm ‖f ‖1 := ∫ |f | dμ < ∞. X

A.1.6 Convergence theorems In measure theory, there are fundamental theorems that are especially helpful for finding the integral of functions that are the pointwise limit of sequences of functions. The first of these results applies to monotone sequences of functions. A sequence of functions (fn )∞ n=1 is monotone if it is increasing pointwise (fn+1 (x) ≥ fn (x) for all x ∈ X and all n ∈ ℕ) or decreasing pointwise (fn+1 (x) ≤ fn (x) for all x ∈ X and all n ∈ ℕ). We state the theorem for increasing sequences, but its counterpart for decreasing sequences can be easily deduced from it. Theorem A.1.35 (Monotone convergence theorem). Let (X, 𝒜, μ) be a measure space. If (fn )∞ n=1 is an increasing sequence of nonnegative measurable functions, then the integral of their pointwise limit is equal to the limit of their integrals, i. e., ∫ lim fn dμ = lim ∫ fn dμ. X

n→∞

Proof. See Theorem 1.26 in Rudin [109].

n→∞

X

A.1 Measure theory | 869

Note that this theorem holds for almost everywhere increasing sequences of almost everywhere nonnegative measurable functions with an almost everywhere pointwise limit. For general sequences of nonnegative functions, we have the following immediate consequence. Lemma A.1.36 (Fatou’s lemma). Let (X, 𝒜, μ) be a measure space. For any sequence (fn )∞ n=1 of nonnegative measurable functions, ∫ lim inf fn dμ ≤ lim inf ∫ fn dμ. X

n→∞

n→∞

X

Proof. For every x ∈ X and n ∈ ℕ, define gn (x) = inf{fi (x) : i ≥ n} and apply the monotone convergence theorem to the sequence (gn )∞ n=1 . For more detail, see Theorem 1.28 in Rudin [109]. The following lemma is another application of the monotone convergence theorem. It offers another way of integrating a nonnegative function. Lemma A.1.37. Let (X, 𝒜, μ) be a measure space. Let f be a nonnegative measurable function and A ∈ 𝒜. Then ∞

∫ f dμ = ∫ μ({x ∈ A : f (x) > r}) dr. A

0

Proof. Suppose that f = 1B for some B ∈ 𝒜. Then ∞

1

0

0

∫ μ({x ∈ A : f (x) > r}) dr = ∫ μ({x ∈ A : 1B (x) > r}) dr 1

= ∫ μ(A ∩ B) dr = μ(A ∩ B) 0

= ∫ 1A∩B dμ = ∫ 1A ⋅ 1B dμ = ∫ f dμ. X

X

A

So the equality holds for characteristic functions. We leave it to the reader to show that the equality prevails for all nonnegative measurable simple functions. If f is a general nonnegative measurable function, then by Theorem A.1.17 there exists an increasing sequence (sn )∞ n=1 of nonnegative measurable simple functions such that limn→∞ sn (x) = f (x) for every x ∈ X. For every r ≥ 0, let ̃f (r) = μ({x ∈ A : f (x) > r}).

870 | Appendix A – A selection of classical results This function is obviously nonnegative and decreasing. Hence, by Theorem A.1.15 it is Borel measurable since ̃f −1 ((t, ∞]) is an interval for all t ∈ [0, ∞] and the sets {(t, ∞]}t∈[0,∞] generate the Borel σ-algebra of [0, ∞]. Fix momentarily r ≥ 0. Since sn ↗ f , the sets ({x ∈ A : sn (x) > r})∞ n=1 form an ascending sequence such that ∞

⋃ {x ∈ A : sn (x) > r} = {x ∈ A : f (x) > r}.

n=1

Then by Lemma A.1.19(d), ∞

̃f (r) = μ( ⋃ {x ∈ A : s (x) > r}) = lim μ({x ∈ A : s (x) > r}) = lim ̃s (r). n n n n→∞

n=1

n→∞

Observe that (̃sn )∞ n=1 is an increasing sequence of nonnegative decreasing functions. So (̃sn )∞ is a sequence of nonnegative Borel measurable functions which increases n=1 ̃ pointwise to f . It follows from the monotone convergence theorem (Theorem A.1.35) that ∞

∞

∞

0

0

0

∫ μ({x ∈ A : f (x) > r}) dr = ∫ ̃f (r) dr = ∫ lim ̃sn (r) dr n→∞

∞

∞

= lim ∫ ̃sn (r) dr = lim ∫ μ({x ∈ A : sn (x) > r}) dr n→∞

n→∞

0

0

= lim ∫ sn dμ = ∫ lim sn dμ = ∫ f dμ. n→∞

A

A

n→∞

A

Pointwise convergence of a sequence of integrable functions does not guarantee convergence in L1 (see Exercise 8.5.14). However, under one relatively weak additional assumption, this becomes true. The second fundamental theorem of convergence applies to sequences of functions which have an almost everywhere pointwise limit and are dominated (i. e., uniformly bounded) almost everywhere by an integrable function. Theorem A.1.38 (Lebesgue’s dominated convergence theorem). If a sequence of measurable functions (fn )∞ n=1 on a measure space (X, 𝒜, μ) converges pointwise μ-a. e. to a function f and if there exists g ∈ L1 (μ) such that |fn (x)| ≤ g(x) for all n ∈ ℕ and μ-a. e. x ∈ X, then f ∈ L1 (μ) and lim ‖fn − f ‖1 = 0 and

n→∞

lim ∫ fn dμ = ∫ f dμ.

n→∞

X

X

Proof. Apply Fatou’s lemma to 2g − |fn − f | ≥ 0. See Theorem 1.34 in Rudin [109] for more detail.

A.1 Measure theory |

871

Note that lim ‖fn − f ‖1 = 0

lim ‖fn ‖1 = ‖f ‖1

󳨐⇒

n→∞

n→∞

󵄨 󵄨 since 󵄨󵄨󵄨‖fn ‖1 − ‖f ‖1 󵄨󵄨󵄨 ≤ ‖fn − f ‖1 and lim ‖fn − f ‖1 = 0

󳨐⇒

n→∞

lim ∫ fn dμ = ∫ f dμ

n→∞

X

X

by applying Lemma A.1.34(c) to fn −f . The opposite implications do not hold in general. Nevertheless, the following lemma states that any sequence of integrable functions (fn )∞ n=1 that converges pointwise almost everywhere to an integrable function f will also converge to that function in L1 if and only if their L1 norms converge to the L1 norm of f . Lemma A.1.39 (Scheffé’s lemma). Let (X, 𝒜, μ) be a measure space. If a sequence 1 1 (fn )∞ n=1 of functions in L (μ) converges pointwise μ-a. e. to a function f ∈ L (μ), then lim ‖fn − f ‖1 = 0

n→∞

lim ‖fn ‖1 = ‖f ‖1 .

⇐⇒

n→∞

In particular, if fn ≥ 0 μ-a. e. for all n ∈ ℕ, then f ≥ 0 μ-a. e. and lim ‖fn − f ‖1 = 0

⇐⇒

n→∞

lim ∫ fn dμ = ∫ f dμ.

n→∞

X

X

Proof. The direct implication ⇒ is trivial. For the converse implication, assume that limn→∞ ‖fn ‖1 = ‖f ‖1 . Suppose first that fn ≥ 0 for all n ∈ ℕ. Then f ≥ 0, and hence our assumption reduces to limn→∞ ∫X fn dμ = ∫X f dμ. Let ℓn = min{f , fn } and un = ∞ max{f , fn }. Then both (ℓn )∞ n=1 and (un )n=1 converge pointwise μ-a. e. to f . Also, |ℓn | = ℓn ≤ f for all n, so Lebesgue’s dominated convergence theorem asserts that lim ∫ ℓn dμ = ∫ f dμ.

n→∞

X

X

Observing that un = f + fn − ℓn , we also get that lim ∫ un dμ = ∫ f dμ + lim ∫ fn dμ − lim ∫ ℓn dμ = ∫ f dμ.

n→∞

X

n→∞

X

n→∞

X

X

X

Thus, lim ‖fn − f ‖1 = lim ∫ |fn − f | dμ = lim (∫ un dμ − ∫ ℓn dμ) = 0.

n→∞

n→∞

X

n→∞

X

X

872 | Appendix A – A selection of classical results So the implication ⇐ holds for nonnegative functions. As g = g+ −g− and ((fn )+ )∞ n=1 and 1 1 ((fn )− )∞ are sequences of functions in L (μ) converging pointwise μ-a. e. to f + ∈ L (μ) n=1 and f− ∈ L1 (μ), respectively, it is easy to see that the general case follows from the case for nonnegative functions. If g is a L1 function on a general measure space (X, 𝒜, μ), then the sequence of nonnegative measurable functions (gM )∞ M=1 , where gM = |g| ⋅ 1{|g|≥M} , decreases to 0 pointwise and is dominated by |g|. Therefore, the monotone convergence theorem (or, alternatively, Lebesgue’s dominated convergence theorem) affirms that lim

|g| dμ = 0.

∫

M→∞

{|g|≥M}

This suggests introducing the following concept. Definition A.1.40. Let (X, 𝒜, μ) be a measure space. A sequence of measurable functions (fn )∞ n=1 is uniformly integrable if lim sup

M→∞ n∈ℕ

∫

|fn | dμ = 0.

{|fn |≥M}

On finite measure spaces, there exists a generalization of Lebesgue’s dominated convergence theorem (Theorem A.1.38). Theorem A.1.41. Let (X, 𝒜, μ) be a finite measure space and (fn )∞ n=1 a sequence of measurable functions that converges pointwise μ-a. e. to a function f . 1 1 (a) If (fn )∞ n=1 is uniformly integrable, then fn ∈ L (μ) for all n ∈ ℕ and f ∈ L (μ). Moreover, lim ‖fn − f ‖1 = 0 and

n→∞

lim ∫ fn dμ = ∫ f dμ.

n→∞

X

X

(b) If f , fn ∈ L1 (μ) and fn ≥ 0 μ-a. e. for all n ∈ ℕ, then limn→∞ ∫X fn dμ = ∫X f dμ implies that (fn )∞ n=1 is uniformly integrable. Proof. See Theorem 16.14 in Billingsley [8]. Corollary A.1.42. Let (X, 𝒜, μ) be a finite measure space and (fn )∞ n=1 a sequence of integrable functions that converges pointwise μ-a. e. to an integrable function f . Then the following conditions are equivalent: (a) The sequence (fn )∞ n=1 is uniformly integrable. (b) limn→∞ ‖fn − f ‖1 = 0. (c) limn→∞ ‖fn ‖1 = ‖f ‖1 .

A.1 Measure theory |

873

󵄨 Proof. Part (a) of Theorem A.1.41 yields (a)⇒(b). That (b)⇒(c) follows from 󵄨󵄨󵄨‖fn ‖1 − 󵄨󵄨 ‖f ‖1 󵄨󵄨 ≤ ‖fn − f ‖1 . Finally, replacing fn by |fn − f | and f by 0 in part (b) of Theorem A.1.41 gives (c)⇒(a). It is obvious that uniform convergence of a sequence of functions implies pointwise convergence of the sequence. The converse is generally not true but the following partial converse holds. Theorem A.1.43 (Dini’s theorem). If X is a compact topological space and (fn )∞ n=1 is an increasing sequence of real-valued continuous functions on X converging pointwise to a continuous function f , then the convergence is uniform. The same conclusion holds if (fn )∞ n=1 is decreasing instead of increasing. It is also natural to ask whether, in some way, a pointwise convergent sequence converges “almost” uniformly. Definition A.1.44. Let (X, 𝒜, μ) be a measure space. A sequence (fn )∞ n=1 of measurable functions on X is said to converge μ-almost uniformly to a function f if for every ε > 0 there exists Y ∈ 𝒜 such that μ(Y) < ε and (fn )∞ n=1 converges uniformly to f on X \ Y. It is clear that almost uniform convergence implies almost everywhere pointwise convergence. The converse is not true in general but these two types of convergence are one and the same on any finite measure space. Theorem A.1.45 (Egorov’s theorem). Let (X, 𝒜, μ) be a finite measure space. A sequence (fn )∞ n=1 of measurable functions on X converges pointwise μ-almost everywhere to a limit function f if and only if that sequence converges μ-almost uniformly to f . Proof. See Chapter 3, Exercise 16 in Rudin [109]. The reader ought to convince themself that this result does not generally hold on infinite spaces. Convergence in measure is another interesting type of convergence. Definition A.1.46. Let (X, 𝒜, μ) be a measure space. A sequence (fn )∞ n=1 of measurable functions converges in measure to a measurable function f provided that for each ε > 0, 󵄨 󵄨 lim μ({x ∈ X : 󵄨󵄨󵄨fn (x) − f (x)󵄨󵄨󵄨 > ε}) = 0.

n→∞

Lemma A.1.47. Let (X, 𝒜, μ) be a measure space. If a sequence (fn )∞ n=1 of measurable 1 ∞ functions converges in L (μ) to a measurable function f , then (fn )n=1 converges in measure to f .

874 | Appendix A – A selection of classical results Proof. Let ε > 0. Then 󵄨 󵄨 μ({x ∈ X : 󵄨󵄨󵄨fn (x) − f (x)󵄨󵄨󵄨 > ε}) ≤

∫ {x∈X : |fn (x)−f (x)|>ε}

|fn − f | 1 dμ ≤ ‖fn − f ‖1 . ε ε

Taking the limit of both sides as n → ∞ completes the proof. When the measure is finite, there is a close relationship between pointwise convergence and convergence in measure. Theorem A.1.48. Let (X, 𝒜, μ) be a finite measure space and (fn )∞ n=1 a sequence of measurable functions. ∞ (a) If (fn )∞ n=1 converges pointwise μ-a. e. to a function f , then (fn )n=1 converges in measure to f . (b) If (fn )∞ n=1 converges in measure to a function f , then there exists a subsequence ∞ (fnk )k=1 which converges pointwise μ-a. e. to f . ∞ (c) (fn )∞ n=1 converges in measure to a function f if and only if each subsequence (fnk )k=1 ∞ admits a further subsequence (fnk )l=1 that converges pointwise μ-a. e. to f . l

Proof. See Theorem 20.5 in Billingsley [8]. The previous two results reveal that, on a finite measure space, a sequence of integrable functions that converges in L1 to an integrable function admits a subsequence which converges pointwise almost everywhere to that function. In general, the sequence itself might not converge pointwise almost everywhere (see Exercise 8.5.17). In some sense, the following result is a form of convergence theorem. It asserts that Borel measurable functions can be approximated by continuous functions on “arbitrarily large” portions of their domain. Theorem A.1.49 (Luzin’s theorem). Let (X, ℬ(X), μ) be a finite Borel measure space and let f : X → ℝ be a Borel measurable function. Given any ε > 0, for every B ∈ ℬ(X) there is a closed set E with μ(B \ E) < ε such that f |E is continuous. If B is locally compact, then the set E can be chosen to be compact and then there is a continuous function fε : X → ℝ with compact support that coincides with f on E and such that supx∈X |fε (x)| ≤ supx∈X |f (x)|. A.1.7 Mutual singularity, absolute continuity and equivalence of measures We now leave aside convergence of sequences of functions and recall the definitions of mutually singular, absolutely continuous and equivalent measures. Definition A.1.50. Let (X, 𝒜) be a measurable space, and μ and ν be two measures on (X, 𝒜).

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(a) The measures μ and ν are said to be mutually singular, denoted by μ⊥ν, if there exist disjoint sets Xμ , Xν ∈ 𝒜 such that μ(X \ Xμ ) = 0 = ν(X \ Xν ). (b) The measure μ is said to be absolutely continuous with respect to ν, denoted μ ≺≺ ν, if ν(A) = 0 󳨐⇒ μ(A) = 0. (c) The measures μ and ν are said to be equivalent if μ ≺≺ ν and ν ≺≺ μ. The Radon–Nikodym theorem provides a characterization of absolute continuity. Though it is valid for σ-finite measures, the following version for finite measures is sufficient for our purposes. Theorem A.1.51 (Radon–Nikodym theorem). Let (X, 𝒜) be a measurable space and let μ and ν be two finite measures on (X, 𝒜). The following statements are equivalent: (a) μ ≺≺ ν. (b) For every ε > 0, there exists δ > 0 such that ν(A) < δ 󳨐⇒ μ(A) < ε. (c) There exists a ν-a. e. unique function f ∈ L1 (ν) such that f ≥ 0 and μ(A) = ∫ f dν,

∀A ∈ 𝒜.

A

Proof. See relation (32.4) and Theorem 32.2 in Billingsley [8]. Remark A.1.52. The function f is often denoted by derivative of μ with respect to ν.

dμ dν

and called the Radon–Nikodym

Per Lemma A.1.34(f), two integrable functions are equal almost everywhere if and only if their integrals are equal over every measurable set. When the measure is finite, we can restrict our attention to any generating π-system. Corollary A.1.53. Let (X, 𝒜, ν) be a finite measure space and suppose that 𝒜 = σ(𝒫 ) for some π-system 𝒫 . Let f , g ∈ L1 (ν). Then f = g ν-a. e.

⇐⇒

∫ f dν = ∫ g dν, P

∀P ∈ 𝒫 .

P

Proof. The direct implication ⇒ is obvious. So let us assume that ∫P f dν = ∫P g dν for all P ∈ 𝒫 . The measures μf (A) := ∫A f dν and μg (A) := ∫A g dν are equal on the π-system 𝒫 . According to Lemma A.1.26, this implies that μf = μg . It follows from the uniqueness part of the Radon–Nikodym theorem that f = g ν-almost everywhere. A.1.8 The space C(X ), its dual C(X )∗ and the subspace M(X ) Another important result is Riesz representation theorem. Before stating it, we first establish some notation. Let X be a compact metrizable space. Let C(X) be the set of

876 | Appendix A – A selection of classical results all continuous real-valued functions on X. This set becomes a normed vector space when endowed with the supremum norm ‖f ‖∞ := sup{|f (x)| : x ∈ X}.

(A.3)

This norm defines a metric on C(X) in the usual way: d∞ (f , g) := ‖f − g‖∞ . The topology induced by the metric d∞ on C(X) is called the topology of uniform convergence on X. Indeed, limn→∞ d∞ (fn , f ) = 0 if and only if the sequence (fn )∞ n=1 converges to f uniformly on X. It is not hard to see that C(X) is a separable Banach space (i. e., a separable and complete normed vector space). Let C(X)∗ denote the dual space of C(X), i. e., 󵄨 C(X)∗ := {F : C(X) → ℝ 󵄨󵄨󵄨 F is continuous and linear}. Recall that a real-valued function F defined on C(X) is called a functional on C(X). It is well known that a linear functional F is continuous if and only if it is bounded, i. e., if and only if its operator norm ‖F‖ (sometimes denoted ‖F‖∞ ) is finite, where ‖F‖ := sup{|F(f )| : f ∈ C(X) and ‖f ‖∞ ≤ 1}.

(A.4)

So C(X)∗ can also be described as the normed vector space of all bounded linear functionals on C(X). The operator norm defines a metric on C(X)∗ in the usual manner: d(F, G) := ‖F − G‖. The topology induced by the metric d on C(X)∗ is called the operator norm topology, or strong topology, on C(X)∗ . It is not difficult to see that C(X)∗ is a separable Banach space. Furthermore, a linear functional F is said to be normalized if F(1) = 1 and is called positive if F(f ) ≥ 0 whenever f ≥ 0. Finally, we denote the set of all Borel probability measures on X by M(X). This set is clearly convex and can be characterized as follows. Theorem A.1.54 (Riesz representation theorem). Let X be a compact metrizable space, and let F be a normalized and positive linear functional on C(X). Then there exists a unique μ ∈ M(X) such that F(f ) = ∫ f dμ,

∀f ∈ C(X).

(A.5)

X

Conversely, any μ ∈ M(X) defines a normalized positive linear functional on C(X) via formula (A.5). This linear functional is bounded.

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Proof. The converse statement is straightforward to check. For the other direction, see Theorem 2.14 in Rudin [109]. It immediately follows from Riesz’ representation theorem that every Borel probability measure on a compact metrizable space is uniquely determined by the way it integrates continuous functions on that space. Corollary A.1.55. If μ and ν are two Borel probability measures on a compact metrizable space X, then μ=ν

⇐⇒

∫ f dμ = ∫ f dν, X

∀f ∈ C(X).

X

Let us now discuss the weak ∗ topology on the set M(X). Recall that if Z is a set and (Zα )α∈A is a family of topological spaces, then the weak topology induced on Z by a collection of maps {ψα : Z → Zα | α ∈ A} is the smallest topology on Z that makes each ψα continuous. Evidently, the sets ψ−1 α (Uα ), for Uα open in Zα , constitute a subbase for the weak topology. The weak∗ topology on M(X) is the weak topology induced by C(X) on its dual space C(X)∗ , where measures in M(X) and normalized positive linear functionals in C(X)∗ are identified via the Riesz representation theorem. Note that M(X) is metrizable, although C(X)∗ with the weak∗ topology usually is not. Indeed, both C(X) and its subspace C(X, [0, 1]) of continuous functions on X taking values in [0, 1], are separable since X is a compact metrizable space. Then for any dense subset {fn }∞ n=1 of C(X, [0, 1]), a metric on M(X) is 󵄨󵄨 1 󵄨󵄨󵄨󵄨 󵄨 󵄨󵄨∫ fn dμ − ∫ fn dν󵄨󵄨󵄨. n 󵄨 󵄨󵄨 2 󵄨 n=1 ∞

d(μ, ν) = ∑

X

X

In this book, we will sometimes denote the convergence of a sequence of measures ∗ ∗ (μn )∞ n=1 to a measure μ in the weak topology of M(X) by μn → μ. Remark A.1.56. Note that this notion is often presented as “weak convergence” of measures. This can be slightly confusing at first sight, but it helps to bear in mind that, as we have seen above, the weak∗ topology is just one instance of a weak topology. The following theorem gives several equivalent characterizations of weak∗ convergence of Borel probability measures. Theorem A.1.57 (Portmanteau theorem). Let (μn )∞ n=1 and μ be Borel probability measures on a compact metrizable space X. The following statements are equivalent: ∗ (a) μn → μ. (b) For all continuous functions f : X → ℝ, lim ∫ f dμn = ∫ f dμ.

n→∞

X

X

878 | Appendix A – A selection of classical results (c) For all closed sets F ⊆ X, lim sup μn (F) ≤ μ(F). n→∞

(d) For all open sets G ⊆ X, lim inf μn (G) ≤ μ(G). n→∞

(e) For all sets A ∈ ℬ(X) such that μ(𝜕A) = 0, lim μ (A) n→∞ n

= μ(A).

Proof. See Theorem 2.1 in Billingsley [9]. For us, the most important result concerning weak∗ convergence of measures is that the set M(X) of all Borel probability measures on a compact metrizable space X is a compact and convex set in the weak∗ topology. In order to establish this, we need to remember Banach–Alaoglu’s theorem. In this theorem, note that the boundedness and closedness are with respect to the operator norm on the dual space while the compactness is with respect to the weak∗ topology on the dual space. Theorem A.1.58 (Banach–Alaoglu’s theorem). The closed unit ball in the dual space B∗ of a Banach space B is compact in the weak∗ topology on B∗ . Furthermore, every closed, bounded subset of B∗ is compact in the weak∗ topology on B∗ . Proof. See Theorem V.4.2 and Corollary V.4.3 in Dunford and Schwartz [26]. Theorem A.1.59. Let X be a compact metrizable space. The set M(X) is compact and convex in the weak∗ topology of C(X)∗ . Proof. The set M(X) is closed with respect to the operator norm topology on C(X)∗ . ∗ Indeed, suppose that (μn )∞ n=1 is a sequence in M(X) which converges to a F ∈ C(X) in ∗ the operator norm topology of C(X) . In other words, suppose that limn→∞ ‖μn −F‖ = 0. By definition of the operator norm (see (A.4)) and thanks to the linearity of F, this implies that F(f ) = lim ∫ f dμn , n→∞

∀f ∈ C(X).

X

In particular, F is normalized (since F(1) = 1) and positive (as F(f ) ≥ 0 for all f ≥ 0). By Riesz representation theorem (Theorem A.1.54), there is μ ∈ M(X) that represents F. So F ∈ M(X), and thus M(X) is closed in the operator norm topology of C(X)∗ .

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The set M(X) is also bounded in that topology. Indeed, if μ ∈ M(X) then ‖μ‖ ≤ sup{∫ |f | dμ : f ∈ C(X), ‖f ‖∞ ≤ 1} = 1. X

Since X is a compact metrizable space, the space C(X) is a Banach space, as earlier mentioned. We can then infer from Banach–Alaoglu’s theorem that the set M(X) is compact in the weak∗ topology. The convexity of M(X) is obvious. Indeed, if μ, ν ∈ M(X) so is any convex combination m = αμ + (1 − α)ν, where α ∈ [0, 1]. A.1.9 Expected values and conditional expectation functions The mean or expected value of a function over a set is a straightforward generalization of the mean value of a real-valued function defined on an interval of the real line. Definition A.1.60. Let (X, 𝒜, μ) be a probability space and let φ ∈ L1 (μ). The mean or expected value E(φ|A) of the function φ over the set A ∈ 𝒜 is defined to be E(φ|A) := {

1 ∫ φ dμ μ(A) A

0

if μ(A) > 0 if μ(A) = 0.

Given that μ is a probability measure, the expected value of φ over the entire space is simply given by E(φ) := E(φ|X) = ∫ φ dμ =: μ(φ). X

Our next goal is to give the definition of the conditional expectation of a function with respect to a σ-algebra. Let (X, 𝒜, μ) be a probability space and ℬ be a sub-σalgebra of 𝒜. Let also φ ∈ L1 (X, 𝒜, μ). Notice that φ : X → ℝ is not necessarily measurable if X is endowed with the sub-σ-algebra ℬ instead of the σ-algebra 𝒜. In short, we say that φ is 𝒜-measurable but not necessarily ℬ-measurable. We aim to find a function E(φ|ℬ) ∈ L1 (X, ℬ, μ) such that ∫ E(φ|ℬ) dμ = ∫ φ dμ, B

∀B ∈ ℬ.

(A.6)

B

This condition means that the function E(φ|ℬ) has the same expected value as φ on every measurable set belonging to the sub-σ-algebra ℬ. Accordingly, E(φ|ℬ) is called the conditional expectation of φ with respect to ℬ.

880 | Appendix A – A selection of classical results We now demonstrate the existence and μ-a. e. uniqueness of the conditional expectation. Let us begin with the existence of that function. Suppose first that the function φ is nonnegative. If φ = 0 μ-a. e., then simply set E(φ|ℬ) = 0. If φ ≠ 0 μ-a. e., then the set function ν(A) := ∫A φ dμ defines a finite measure on (X, 𝒜) which is absolutely continuous with respect to μ. The restriction of ν to ℬ also determines a finite measure on (X, ℬ) which is absolutely continuous with respect to the restriction of μ to ℬ. So by the Radon–Nikodym theorem (Theorem A.1.51), there exists a μ-a. e. unique ̂ ∈ L1 (X, ℬ, μ) such that ν(B) = ∫B φ ̂ dμ for every B ∈ ℬ. Then nonnegative function φ ̂ dμ = ν(B) = ∫ φ dμ, ∫φ B

∀B ∈ ℬ.

B

The point here is that although it may look as if we have not really achieved anything, ̂ is ℬ-measurable, whereas φ may not be. Therefore, φ ̂ we have actually gained that φ is the sought-after conditional expectation E(φ|ℬ) of φ with respect to ℬ. If φ takes both negative and positive values, write φ = φ+ − φ− , where φ+ (x) := max{φ(x), 0} is the positive part of φ and φ− (x) := max{−φ(x), 0} is the negative part of φ. Then define the conditional expectation linearly, i. e., set E(φ|ℬ) := E(φ+ |ℬ) − E(φ− |ℬ). This proves the existence of the conditional expectation function. Its μ-a. e. uniqueness follows from its defining property (A.6) and Lemma A.1.34(f). The conditional expectation exhibits several natural properties. We mention a few of them in the next proposition. Proposition A.1.61. Let (X, 𝒜, μ) be a probability space, let ℬ and 𝒞 denote sub-σalgebras of 𝒜 and let φ ∈ L1 (X, 𝒜, μ). (a) If φ ≥ 0 μ-a. e., then E(φ|ℬ) ≥ 0 μ-a. e. (b) If φ1 ≥ φ2 μ-a. e., then E(φ1 |ℬ) ≥ E(φ2 |ℬ) μ-a. e. 󵄨 󵄨 󵄨 (c) 󵄨󵄨󵄨E(φ|ℬ)󵄨󵄨󵄨 ≤ E(|φ| 󵄨󵄨󵄨 ℬ). (d) The functional E(⋅|ℬ) is linear, i. e. for any c1 , c2 ∈ ℝ and φ1 , φ2 ∈ L1 (X, 𝒜, μ), 󵄨 E(c1 φ1 + c2 φ2 󵄨󵄨󵄨ℬ) = c1 E(φ1 |ℬ) + c2 E(φ2 |ℬ). 󵄨 (e) If φ is ℬ-measurable, then E(φ|ℬ) = φ. In particular, E( E(φ|ℬ) 󵄨󵄨󵄨 ℬ) = E(φ|ℬ). Also, if φ ≡ c ∈ ℝ is a constant function, then E(φ|ℬ) = φ ≡ c. 󵄨 (f) If 𝒞 ⊆ ℬ, then E(φ|𝒞 ) = E( E(φ|ℬ) 󵄨󵄨󵄨 𝒞 ). Proof. This is left as an exercise to the reader. We will now determine the conditional expectation of an arbitrary integrable function φ with respect to various sub-σ-algebras of particular interest.

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Example A.1.62. Let (X, 𝒜, μ) be a probability space. The family 𝒩 of all measurable sets that are either of null or of full measure constitutes a sub-σ-algebra of 𝒜. Let φ ∈ L1 (X, 𝒜, μ). Then the function E(φ|𝒩 ) has to belong to L1 (X, 𝒩 , μ) and must satisfy condition (A.6). In particular, E(φ|𝒩 ) must be 𝒩 -measurable. This means that for each Borel subset R of ℝ, the function E(φ|𝒩 ) must be such that E(φ|𝒩 )−1 (R) ∈ 𝒩 . Among others, for every t ∈ ℝ we must have E(φ|𝒩 )−1 ({t}) ∈ 𝒩 ; in other words, for each t ∈ ℝ the set E(φ|𝒩 )−1 ({t}) must be of measure zero or of measure one. Also bear in mind that X = E(φ|𝒩 )−1 (ℝ) = ⋃ E(φ|𝒩 )−1 ({t}). t∈ℝ

Since the above union consists of mutually disjoint sets of measure zero and one, it follows that only one of these sets can be of measure one. In other words, there exists a unique t ∈ ℝ such that E(φ|𝒩 )−1 ({t}) = A for some A ∈ 𝒜 with μ(A) = 1. Because the function E(φ|𝒩 ) is unique up to a set of measure zero, we may assume without loss of generality that A = X. Hence, E(φ|𝒩 ) is a constant function. More specifically, its value is E(φ|𝒩 ) = ∫ E(φ|𝒩 ) dμ = ∫ φ dμ. X

X

Example A.1.63. Let (X, 𝒜) be a measurable space and let α = {An }∞ n=1 be a countable measurable partition of X. That is, each An ∈ 𝒜, Ai ∩Aj = 0 for all i ≠ j and X = ⋃∞ n=1 An . The sub-σ-algebra of 𝒜 generated by α is the family of all sets which can be written as a union of elements of α, i. e., σ(α) = {A ⊆ X : A = ⋃ Aj for some J ⊆ ℕ}. j∈J

When α is finite, so is σ(α). When α is countably infinite, σ(α) is uncountable. Let μ be a probability measure on (X, 𝒜). Let φ ∈ L1 (X, 𝒜, μ) and set ℬ = σ(α). Then the conditional expectation E(φ|ℬ) : X → ℝ has to be a L1 (X, ℬ, μ) function that satisfies condition (A.6). In particular, E(φ|ℬ) must be ℬ-measurable. Thus, for any t ∈ ℝ we must have E(φ|ℬ)−1 ({t}) ∈ ℬ, i. e., the set E(φ|ℬ)−1 ({t}) must be a union of elements of α. This means that the conditional expectation function E(φ|ℬ) is constant on each element of α. Let An ∈ α. If μ(An ) = 0, then E(φ|ℬ)|An = 0. Otherwise, 󵄨 E(φ|ℬ)󵄨󵄨󵄨A = n

1 1 ∫ E(φ|ℬ) dμ = ∫ φ dμ = E(φ|An ). μ(An ) μ(An ) An

An

In summary, the conditional expectation E(φ|ℬ) of a function φ with respect to a sub-σ-algebra generated by a countable measurable partition is constant on each el-

882 | Appendix A – A selection of classical results ement of that partition. More precisely, on any given element of the partition, E(φ|ℬ) is equal to the mean value of φ on that element. The next result is a special case of a theorem originally due to Doob and called the martingale convergence theorem. But, first, let us define the martingale itself. Definition A.1.64. Let (X, 𝒜, μ) be a probability space. Let (𝒜n )∞ n=1 be a sequence of sub-σ-algebras of 𝒜. Let also (φn : X → ℝ)∞ be a sequence of 𝒜-measurable funcn=1 tions. The sequence ((φn , 𝒜n ))∞ is called a martingale if the following conditions are n=1 satisfied: (a) (𝒜n )∞ n=1 is an ascending sequence, i. e., 𝒜n ⊆ 𝒜n+1 for all n ∈ ℕ. (b) φn is 𝒜n -measurable for all n ∈ ℕ. (c) φn ∈ L1 (μ) for all n ∈ ℕ. (d) E(φn+1 |𝒜n ) = φn μ-a. e. for all n ∈ ℕ. Theorem A.1.65 (Martingale convergence theorem). Let (X, 𝒜, μ) be a probability space. If ((φn , 𝒜n ))∞ n=1 is a martingale such that sup ‖φn ‖1 < ∞, n∈ℕ

̂ ∈ L1 (X, 𝒜, μ) such that then there exists φ ̂(x) lim φn (x) = φ

n→∞

for μ-a. e. x ∈ X

and

̂‖1 ≤ sup ‖φn ‖1 . ‖φ n∈ℕ

Proof. See Theorem 35.5 in Billingsley [8]. One natural martingale is formed by the conditional expectations of a function with respect to an ascending sequence of sub-σ-algebras. Example A.1.66. Let (X, 𝒜, μ) be a probability space and (𝒜n )∞ n=1 be an ascending se1 quence of sub-σ-algebras of 𝒜. For any φ ∈ L (X, 𝒜, μ), the sequence ((E(φ|𝒜n ), 𝒜n ))∞ n=1 is a martingale. Indeed, set φn = E(φ|𝒜n ) for all n ∈ ℕ. Condition (a) in Definition A.1.64 is automatically fulfilled. Conditions (b) and (c) follow from the very definition of the conditional expectation function. Regarding condition (d), a straightforward application of Proposition A.1.61(f) gives 󵄨 E(φn+1 |𝒜n ) = E( E(φ|𝒜n+1 ) 󵄨󵄨󵄨 𝒜n ) = E(φ|𝒜n ) = φn μ-a. e.,

∀n ∈ ℕ.

So ((E(φ|𝒜n ), 𝒜n ))n=1 is a martingale. Using Proposition A.1.61(c), note that ∞

󵄨 󵄨 󵄨 sup ‖φn ‖1 = sup ∫󵄨󵄨󵄨E(φ|𝒜n )󵄨󵄨󵄨 dμ ≤ sup ∫ E(|φ| 󵄨󵄨󵄨 𝒜n ) dμ = ∫ |φ| dμ = ‖φ‖1 < ∞. n∈ℕ

n∈ℕ

X

n∈ℕ

X

X

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883

̂ ∈ L1 (X, 𝒜, μ) such that According to Theorem A.1.65, there thus exists φ ̂(x) lim E(φ|𝒜n )(x) = φ

for μ-a. e. x ∈ X

n→∞

and

̂‖1 ≤ ‖φ‖1 . ‖φ

̂? This is the question we will answer in Theorem A.1.68. What is φ Beforehand, we establish the uniform integrability of this martingale (see Definition A.1.40). Lemma A.1.67. Let (X, 𝒜, μ) be a probability space and let (𝒜n )∞ n=1 be a sequence of 1 sub-σ-algebras of 𝒜. For any φ ∈ L (X, 𝒜, μ), the sequence (E(φ|𝒜n ))∞ n=1 is uniformly integrable. Proof. Without loss of generality, we may assume that φ ≥ 0. Let ε > 0. Since ν(A) = ∫A φ dμ is absolutely continuous with respect to μ, it follows from the Radon–Nikodym theorem (Theorem A.1.51) that there exists δ > 0 such that A ∈ 𝒜, μ(A) < δ

󳨐⇒

∫ φ dμ < ε.

(A.7)

A

Set M > ∫X φ dμ/δ. For each n ∈ ℕ, let Xn (M) = {x ∈ X : E(φ|𝒜n )(x) ≥ M}. Observe that Xn (M) ∈ 𝒜n since E(φ|𝒜n ) is 𝒜n -measurable. Therefore, μ(Xn (M)) ≤

1 1 1 ∫ E(φ|𝒜n ) dμ = ∫ φ dμ ≤ ∫ φ dμ < δ M M M Xn (M)

Xn (M)

X

for all n ∈ ℕ. Consequently, by (A.7), ∫ E(φ|𝒜n ) dμ = ∫ φ dμ < ε Xn (M)

Xn (M)

for all n ∈ ℕ. Thus, sup n∈ℕ

E(φ|𝒜n ) dμ ≤ ε.

∫ {E(φ|𝒜n )≥M}

Since this holds for large enough M’s and since ε > 0 is arbitrary, we have lim sup

M→∞ n∈ℕ

∫

E(φ|𝒜n ) dμ = 0,

{E(φ|𝒜n )≥M}

i. e., the sequence (E(φ|𝒜n ))∞ n=1 is uniformly integrable.

884 | Appendix A – A selection of classical results Theorem A.1.68 (Martingale convergence theorem for conditional expectations). Let (X, 𝒜, μ) be a probability space and φ ∈ L1 (X, 𝒜, μ). Let (𝒜n )∞ n=1 be an ascending sequence of sub-σ-algebras of 𝒜 and ∞

𝒜∞ := σ( ⋃ 𝒜n ). n=1

Then 󵄩 󵄩 lim 󵄩󵄩E(φ|𝒜n ) − E(φ|𝒜∞ )󵄩󵄩󵄩1 = 0 and

n→∞󵄩

lim E(φ|𝒜n ) = E(φ|𝒜∞ ) μ-a. e. on X.

n→∞

Proof. Let φn = E(φ|𝒜n ). In Example A.1.66 and Lemma A.1.67, we have seen that ((φn , 𝒜n ))∞ n=1 is a uniformly integrable martingale such that ̂ lim φn = φ

n→∞

μ-a. e. on X

̂ ∈ L1 (X, 𝒜, μ). For all n ∈ ℕ the function φn is 𝒜∞ -measurable since it is for some φ ̂ is 𝒜∞ -measurable, too. Moreover, it follows 𝒜n -measurable and 𝒜n ⊆ 𝒜∞ . Thus, φ from Theorem A.1.41 that ̂‖1 = 0 lim ‖φn − φ

n→∞

and

̂ dμ, lim ∫ φn dμ = ∫ φ

n→∞

A

∀A ∈ 𝒜.

A

̂ = E(φ|𝒜∞ ). Therefore, it just remains to show that φ Let k ∈ ℕ and A ∈ 𝒜k . If n ≥ k, then A ∈ 𝒜n ⊆ 𝒜∞ , and thus ∫ φn dμ = ∫ E(φ|𝒜n ) dμ = ∫ φ dμ = ∫ E(φ|𝒜∞ ) dμ. A

A

A

A

Letting n → ∞ yields ̂ dμ = ∫ E(φ|𝒜∞ ) dμ, ∫φ A

∀A ∈ 𝒜k .

A

As k is arbitrary, ̂ dμ = ∫ E(φ|𝒜∞ ) dμ, ∫φ B

B

∞

∀B ∈ ⋃ 𝒜k . k=1

̂ and E(φ|𝒜∞ ) are Since ⋃∞ k=1 𝒜k is a π-system generating 𝒜∞ and since both φ ̂ 𝒜∞ -measurable, Corollary A.1.53 affirms that φ = E(φ|𝒜∞ ) μ-a. e. There is also a counterpart of this theorem for descending sequences of σ-algebras.

A.1 Measure theory | 885

Theorem A.1.69 (Reversed martingale convergence theorem for conditional expectations). Let (X, 𝒜, μ) be a probability space and φ ∈ L1 (X, 𝒜, μ). If (𝒜n )∞ n=1 is a descending sequence of sub-σ-algebras of 𝒜, then 󵄩󵄩 󵄨󵄨 ∞ 󵄩󵄩 󵄩 󵄨 󵄩 lim 󵄩󵄩󵄩E(φ|𝒜n ) − E(φ 󵄨󵄨󵄨 ⋂ 𝒜n )󵄩󵄩󵄩 = 0 n→∞󵄩 󵄩󵄩1 󵄨 󵄨 n=1 󵄩

and

󵄨󵄨 ∞ 󵄨 lim E(φ|𝒜n ) = E(φ 󵄨󵄨󵄨 ⋂ 𝒜n ) μ-a. e. n→∞ 󵄨󵄨 n=1

Proof. See Theorem 35.9 of Billingsley [8]. Theorems A.1.68/A.1.69 are especially useful for the calculation of the conditional expectation of a function with respect to a sub-σ-algebra generated by an uncountable measurable partition which can be approached by an ascending/descending sequence of sub-σ-algebras generated by countable measurable partitions. See Exercise 8.5.22. They can also be used to approximate a measurable set by one from a generating sequence of sub-σ-algebras. Corollary A.1.70. Let (X, 𝒜, μ) be a probability space. Let (𝒜n )∞ n=1 be an ascending sequence of sub-σ-algebras of 𝒜 and set 𝒜∞ = σ(⋃∞ 𝒜 ). Let B ∈ 𝒜∞ . For every ε > 0, n=1 n ∞ there exists A ∈ ⋃n=1 𝒜n such that μ(A △ B) < ε. Proof. Let B ∈ 𝒜∞ . It ensues from Theorem A.1.68 that lim E(1B |𝒜n ) = E(1B |𝒜∞ ) = 1B

n→∞

μ-a. e. on X.

By Theorem A.1.48, we deduce that 󵄨 󵄨 1 lim μ({x ∈ X : 󵄨󵄨󵄨E(1B |𝒜n )(x) − 1B (x)󵄨󵄨󵄨 ≥ }) = 0. 8

n→∞

For every n ∈ ℕ, let 󵄨 󵄨 1 Bn := {x ∈ X : 󵄨󵄨󵄨E(1B |𝒜n )(x) − 1B (x)󵄨󵄨󵄨 ≥ }. 8 Then there exists N = N(ε) ∈ ℕ such that μ(Bn ) ≤ ε/2,

∀n ≥ N.

For every n ∈ ℕ, let 󵄨 󵄨 1 An := {x ∈ X : 󵄨󵄨󵄨E(1B |𝒜n )(x) − 1󵄨󵄨󵄨 ≤ } ∈ 𝒜n . 4 On one hand, 󵄨 󵄨 1 x ∈ B \ An 󳨐⇒ 󵄨󵄨󵄨E(1B |𝒜n )(x) − 1B (x)󵄨󵄨󵄨 > 󳨐⇒ x ∈ Bn . 4

886 | Appendix A – A selection of classical results This means that B \ An ⊆ Bn . On the other hand, 󵄨 3 󵄨 󵄨 󵄨 x ∈ An \ B 󳨐⇒ 󵄨󵄨󵄨E(1B |𝒜n )(x) − 1B (x)󵄨󵄨󵄨 = 󵄨󵄨󵄨E(1B |𝒜n )(x)󵄨󵄨󵄨 ≥ 󳨐⇒ x ∈ Bn . 4 This means that An \ B ⊆ Bn . Therefore, μ(An △ B) = μ(An \ B) + μ(B \ An ) ≤ 2μ(Bn ) ≤ ε,

∀n ≥ N.

As An ∈ 𝒜n , we have found some A ∈ ⋃∞ n=1 𝒜n with μ(A △ B) < ε. We will now give a proof of Lemma A.1.32 in the case where the algebra is countable. Proof. Let 𝒜 = {An }∞ n=1 be a countable algebra on a set X and let μ be a probability measure on (X, σ(𝒜)). Set 𝒜′n = {Ak }nk=1 . Since 𝒜′n is a finite set, the algebra 𝒜(𝒜′n ) it generates is also finite, and thus σ(𝒜′n ) = 𝒜(𝒜′n ). Moreover, since 𝒜 is an algebra, σ(𝒜′n ) = 𝒜(𝒜′n ) ⊆ 𝒜. It follows that ∞

σ(𝒜) = σ( ⋃ σ(𝒜′n )). n=1

Let B ∈ σ(𝒜) and ε > 0. Since (σ(𝒜′n ))∞ n=1 is an ascending sequence of sub-σ-algebras of ′ σ(𝒜) such that σ(𝒜) = σ(⋃∞ σ( 𝒜 )), it ensues from Corollary A.1.70 that there exists n=1 n ′ ′ A ∈ ⋃∞ σ( 𝒜 ) with μ(A △ B) < ε. Since ⋃∞ n=1 n n=1 σ(𝒜n ) ⊆ 𝒜, we have found some A ∈ 𝒜 such that μ(A △ B) < ε. Finally, we introduce the concept of conditional measure and relate it to the concept of expected value. Definition A.1.71. Let (X, 𝒜, μ) be a probability space and let B ∈ 𝒜 be such that μ(B) > 0. The set function μB : 𝒜 → [0, 1] defined by setting μB (A) :=

μ(A ∩ B) , μ(B)

∀A ∈ 𝒜

is a probability measure on (X, 𝒜) called the conditional measure of μ on B.

A.1 Measure theory |

887

Note that for every φ ∈ L1 (X, 𝒜, μ), 1 ∫ φ dμ + 0 = E(φ|B). μ(B)

∫ φ dμB = ∫ φ dμB + ∫ φ dμB = X

B

B

X\B

A.1.10 Characteristic functions and distributions Definition A.1.72. If μ is a Borel probability measure on ℝ, then the function ̂ (t) := ∫ eits dμ(s) ℝ ∋ t 󳨃󳨀→ μ ℝ

is called the characteristic function (commonly in probability theory) or Fourier transform (in analysis) of the measure μ. The first classical result about this notion is the following. Theorem A.1.73. Two Borel probability measures μ and ν on ℝ are equal if and only if ̂ = ν̂. their characteristic functions are equal, i. e., μ = ν if and only if μ The second classical result goes as follows. ∗ Theorem A.1.74. A sequence (μn )∞ n=1 of Borel probability measures on ℝ weak con∞ ̂ n )n=1 of their verges to a Borel probability measure μ on ℝ if and only if the sequence (μ ̂ , the characteristic function of μ, i. e., if characteristic functions converges pointwise to μ and only if

̂ (t) lim μ n→∞ n

= μ(t),

∀t ∈ ℝ.

Definition A.1.75. Let (X, 𝒜, μ) be a probability space and let g : X → ℝ be a function in L1 (μ). The function ℝ ∋ t 󳨃󳨀→ μ? ∘ g −1 (t) = ∫ eits dμ ∘ g −1 (s) = ∫ eitg(x) dμ(x) ℝ

(A.8)

X

is called the characteristic function (in probability theory) or Fourier transform (in analysis) of g. The Borel probability measure μ ∘ g −1 on ℝ is called the probability distribution of the function (random variable) g with respect to the measure μ. Note that it follows from (A.8) that the characteristic function of g depends only on the distribution of g. As a direct consequence of Theorem A.1.73, we obtain the following.

888 | Appendix A – A selection of classical results Theorem A.1.76. Any two random variables (not necessarily defined on the same probability space) have the same distribution if and only if they have the same characteristic function. Definition A.1.77. A sequence (gn )∞ n=1 of random variables (not necessarily defined on the same probability space) is said to converge in distribution to a random variable g if ∗ −1 the sequence (μ ∘ gn−1 )∞ n=1 of their probability distributions weak converges to μ ∘ g , the probability distribution of g. As an immediate repercussion of Theorem A.1.74, we get the following. Theorem A.1.78. A sequence (gn )∞ n=1 of random variables (not necessarily defined on the same probability space) converges in distribution to a random variable g if and only if the sequence (μ? ∘ g −1 )∞ of their characteristic functions converges pointwise to the n

n=1

characteristic function μ? ∘ g −1 of g.

A.2 Functional analysis Arzelà–Ascoli’s theorem (cf. Theorem IV.6.7 in Dunford and Schwartz [26]) is a fundamental theorem in functional analysis and topology. Before stating it, let us first recall the following two definitions. Definition A.2.1. Let X be a set and ℝX be the set of real-valued functions on X. A subset F of ℝX is said to be pointwise bounded if sup{|f (x)| : f ∈ F} < ∞,

∀x ∈ X.

Definition A.2.2. Let X be a topological space, and let C(X) be the space of real-valued continuous functions on X. A subset F of C(X) is said to be equicontinuous if for every x ∈ X and every ε > 0, there is a neighborhood U(x, ε) of x such that (y, f ) ∈ U(x, ε) × F

󳨐⇒

󵄨󵄨 󵄨 󵄨󵄨f (y) − f (x)󵄨󵄨󵄨 < ε.

The Arzelà–Ascoli theorem gives a characterization of the relative compactness of sets of continuous functions. Theorem A.2.3 (Arzelà–Ascoli theorem). Let X be a compact Hausdorff space. A subset F of C(X) is relatively compact (i. e., has compact closure in the topology of uniform convergence induced by the supremum norm ‖ ⋅ ‖∞ ) if and only if F is equicontinuous and pointwise bounded. Theorem A.2.4 (Inverse function theorem). Let F and G be Banach spaces and u0 ∈ F. If f : F → G is a C 1 (i. e., continuously differentiable) function on some neighborhood of

A.2 Functional analysis | 889

u0 such that f ′ (u0 ) is invertible, then there exists an open neighborhood U of u0 such that f (U) is open in G and f |U : U → f (U) is a bijection. Furthermore, the inverse function 1 f |−1 U : f (U) → U is C . Let F be a complex Banach space and L(F) be the Banach space of bounded (i. e., continuous) ℂ-linear operators from F to itself. For any operator L ∈ L(F), recall that the operator norm of L is defined by ‖L‖ := inf{c ≥ 0 : ‖L(υ)‖ ≤ c‖υ‖, ∀υ ∈ F} = sup{

‖L(υ)‖ : υ ∈ F \ {0}}. ‖υ‖

If L0 ∈ L(F) and δ > 0, then BL(F) (L0 , δ) = {L ∈ L(F) : ‖L − L0 ‖ < δ}. The following is a result on perturbations of linear operators. Theorem A.2.5 (Kato–Rellich perturbation theorem). Let F be a complex Banach space. If L0 ∈ L(F) has a simple isolated eigenvalue α(L0 ), then for all ε > 0 there exists δ > 0 such that every L ∈ BL(F) (L0 , δ) has a simple isolated eigenvalue α(L) and there are two operators QL , SL ∈ L(F) such that: (a) {α(L)} = σ(L) ∩ Bℂ (α(L0 ), ε) for all L ∈ BL(F) (L0 , δ), where σ(L) is the spectrum of L. (b) The mapping BL(F) (L0 , δ) ∋ L 󳨃→ α(L) ∈ ℂ is holomorphic. (c) L = α(L)QL + SL , QL SL = SL QL = 0, LQL = QL L, LSL = SL L, Q2L = QL , and dim(QL (F)) = 1. (d) The mappings BL(F) (L0 , δ) ∋ L 󳨃→ QL , SL ∈ L(F) are holomorphic. (e) Consequently, decreasing δ > 0 if necessary, for every vector υ0 ∈ F \ Ker(L0 ), the vector QL (υ0 ) is an eigenvector of the operator L corresponding to the eigenvalue α(L) and the function BL(F) (L0 , δ) ∋ L 󳨃→ QL (υ0 ) ∈ F is holomorphic. There is another Riesz’s Theorem that we will use in this book. This one is about the extension of a linear functional. Theorem A.2.6. Let E be a real vector space, F be a vector subspace of E, and K be a convex cone in E. A linear functional L : F → ℝ is called K-positive if L|F∩K ≥ 0. A linear functional L : E → ℝ is said to be a K-positive extension of L if L|F = L and L|K ≥ 0. If E ⊆ F + K, i. e. if for every y ∈ E there exists x ∈ F such that y − x ∈ K, then every K-positive linear functional on F can be extended to a K-positive linear functional on E. Note that, in general, a K-positive linear functional on F cannot be extended to a K-positive linear functional on E. Already in two dimensions there is a counterexample. Let E = ℝ2 , F = ℝ × {0}, and K = {(x, y) : y > 0} ∪ {(x, 0) : x > 0}. The K-positive functional L(x, 0) = x can not be extended to a K-positive functional on E.

890 | Appendix A – A selection of classical results

A.3 Complex analysis in one variable Theorem A.3.1 (Cauchy’s theorem). Let D ⊆ ℂ be a connected open set with piecewise smooth boundary 𝜕D. If f is an analytic function on D that extends smoothly to 𝜕D, then ∫ f (z) dz = 0. 𝜕D

Theorem A.3.2 (Cauchy’s integral formula). Let D ⊆ ℂ be a connected open set with piecewise smooth boundary 𝜕D. If f is an analytic function on D that extends smoothly to 𝜕D, then f has derivatives of all orders on D, which are given by f (n) (z) =

n! f (w) dw, ∀z ∈ D, ∀n ≥ 0. ∫ 2πi (w − z)n+1 𝜕D

Theorem A.3.3 (Morera’s theorem). Let D ⊆ ℂ be a connected open set. If a continuous function f : D → ℂ is such that ∫𝜕R f (z) dz = 0 for every closed rectangle R ⊆ D with sides parallel to the coordinate axes, then f is analytic on D. Theorem A.3.4 (Monodromy theorem). Let D ⊆ ℂ be a connected open set. Let f : D → ℂ be holomorphic at a point z0 ∈ D, and let γ0 , γ1 : [0, 1] → D be two homotopic paths from z0 to a point w. If f can be continued holomorphically along any path in D, then the holomorphic continuations of f along γ0 and γ1 coincide at w. Lemma A.3.5 (Schwarz lemma). Let 𝔻 = {z ∈ ℂ : |z| < 1}. If f : 𝔻 → ℂ is an holomorphic function such that f (0) = 0 and f (𝔻) ⊆ 𝔻, then |f ′ (0)| ≤ 1

and |f (z)| ≤ |z|,

∀z ∈ 𝔻.

If |f ′ (0)| = 1 or |f (z)| = |z| for some z ≠ 0 then f (z) = az for some a ∈ ℂ with |a| = 1. Theorem A.3.6 (Schwarz reflection principle). Let D be a connected open subset of ℂ that is symmetric with respect to the real axis, and let D+ := D ∩ {Im(z) > 0}. Let f be an holomorphic function on D+ such that Im(f (z)) → 0 as z ∈ D+ tends to D ∩ ℝ. Then f has an analytic extension to D, and that extension satisfies f (z) = f (z),

∀z ∈ D.

Theorem A.3.7 (Koebe quarter theorem). Let 𝔻 = {z ∈ ℂ : |z| < 1}. If f : 𝔻 → ℂ is a univalent (i. e., one-to-one) holomorphic function, then 1 f (𝔻) ⊇ D(f (0), |f ′ (0)|), 4 where D(w, r) = Bℂ (w, r) is the open disc/ball centered at z and of radius r.

A.3 Complex analysis in one variable

| 891

It is worth pointing out that this result is sharp, i. e., the constant 1/4 cannot be improved (read increased). A repercussion of this theorem is the following well-known distortion theorem. Theorem A.3.8 (Koebe distortion theorem). Let 𝔻 = {z ∈ ℂ : |z| < 1}. If f : 𝔻 → ℂ is a univalent holomorphic function such that f (0) = 0 and f ′ (0) = 1, then |z| |z| ≤ |f (z)| ≤ , (1 + |z|)2 (1 − |z|)2 1 + |z| 1 − |z| ≤ |f ′ (z)| ≤ , (1 + |z|)3 (1 − |z|)3 and 1 − |z| 󵄨󵄨󵄨󵄨 f ′ (z) 󵄨󵄨󵄨󵄨 1 + |z| ≤ 󵄨z , 󵄨≤ 1 + |z| 󵄨󵄨󵄨 f (z) 󵄨󵄨󵄨 1 − |z| with equality if and only if f is a Koebe function, i. e., for some θ > 0 f (z) =

z . (1 − eiθ z)2

The following is a variation of the part of the Koebe distortion theorem about the derivative of univalent functions. Theorem A.3.9. Let w ∈ ℂ and φ : D(w, R) → ℂ be a univalent holomorphic function. Then for all z ∈ D(w, R), 1− (1 +

|z−w| R |z−w| 3 ) R

≤

1 + |z−w| |φ′ (z)| R ≤ . |φ′ (w)| (1 − |z−w| )3 R

In this book, we will overwhelmingly use Koebe’s distortion theorem in the following form. Theorem A.3.10. Let w ∈ ℂ and φ : D(w, R) → ℂ be a univalent holomorphic function. There exists a function K : [0, 1) → [1, ∞) such that sup{|φ′ (z)| : z ∈ D(w, tR)} ≤ K(t) inf{|φ′ (z)| : z ∈ D(z, tR)}. We finish with a Koebe distortion theorem for the argument of univalent functions. Theorem A.3.11 (Koebe distortion theorem for argument). Let 𝔻 = {z ∈ ℂ : |z| < 1}. If f : 𝔻 → ℂ is a univalent holomorphic function such that f (0) = 0 and f ′ (0) = 1, then 1+r 󵄨󵄨 󵄨 ′ , 󵄨󵄨arg(f (z))󵄨󵄨󵄨 ≤ 2 log 1−r

∀|z| ≤ r < 1.

For more information on the Koebe distortion theorem, see [22, 53].

892 | Appendix A – A selection of classical results The next two results are exhibits of the general fact that the number of zeros of an analytic function does not change under small analytic perturbations. Theorem A.3.12 (Rouché’s theorem). Let D ⊆ ℂ be a connected open set with piecewise smooth boundary 𝜕D. If f and g are analytic functions on D ∪ 𝜕D such that |g(z)| < |f (z)| for all z ∈ 𝜕D, then f and f +g have the same number of zeros in D, counting multiplicities. Theorem A.3.13 (Hurwitz’s theorem). Let D ⊆ ℂ be a connected open set. Suppose that (fn )∞ n=1 is a sequence of holomorphic functions on D that converges normally on D (i. e. converges uniformly on compact subsets of D) to a function f , and suppose that f has a zero of order N at z0 ∈ D. Then there exists δ > 0 such that for all sufficiently large n, the functions fn have each exactly N zeros in the disc D(z0 , δ), counting multiplicities, and these zeros converge to z0 as n → ∞. (Note: The normal convergence imposes that f be holomorphic.) Many theorems are attributed to Hurwitz, including the next one. Theorem A.3.14 (Hurwitz’s theorem II). Let D ⊆ ℂ be a connected open set. Suppose that (fn )∞ n=1 is a sequence of univalent (i. e., one-to-one) holomorphic functions on D that converges normally on D to a function f . Then f is either constant or univalent on D. A partial converse to this theorem is the following. Theorem A.3.15 (Converse to Hurwitz’s theorem). Let D ⊆ ℂ be a connected open set. Suppose that a sequence (fn )∞ n=1 of holomorphic functions on D converges normally on D to a univalent function f . Then for any compact set K ⊆ D, all but finitely many of the functions fn are univalent on K. Theorem A.3.16 (Implicit function theorem). Let F(z, w) be a continuous function of z and w that depends analytically on z for each fixed w. Suppose that F(z0 , w0 ) = 0 and 𝜕F (z , w0 ) ≠ 0. Let δ > 0 be such that F(z, w0 ) ≠ 0 for all z ∈ D(z0 , δ) \ {z0 }. Then there 𝜕z 0 exists ε > 0 such that for every w ∈ D(w0 , ε), there is a unique z = g(w) ∈ D(z0 , δ) such that F(z, w) = 0. In fact, g(w) =

1 2πi

∫ |ζ −z0 |=δ

ζ 𝜕F (ζ , w) 𝜕z F(ζ , w)

dζ ,

∀w ∈ D(w0 , ε),

and g : D(w0 , ε) → D(z0 , δ) is thus continuous. If, additionally, F(z, w) is analytic in w for each fixed z, then g is analytic and 𝜕F

g ′ (w) = − 𝜕w 𝜕F 𝜕z

(g(w), w)

(g(w), w)

.

The following theorem is a corollary to the previous one.

A.4 Complex analysis in several variables | 893

Theorem A.3.17 (Inverse function theorem). Suppose that an holomorphic function f : D(z0 , δ) → ℂ is such that f ′ (z0 ) ≠ 0 and f (z) ≠ f (z0 ) for all z ∈ D(z0 , δ) \ {z0 }. Let ε > 0 be such that f (𝜕D(z0 , δ)) ⊆ ℂ \ D(f (z0 ), ε). Then for every w ∈ D(f (z0 ), ε), there is a unique z ∈ D(z0 , δ) such that f (z) = w and, writing z = f −1 (w), the inverse function f −1 : D(f (z0 ), ε) → D(z0 , δ) satisfies f −1 (w) =

1 2πi

∫ |ζ −z0 |=δ

ζ f ′ (ζ ) dζ , f (ζ ) − w

∀w ∈ D(f (z0 ), ε),

and f −1 is thus holomorphic. The next theorem concerns extensions of holomorphic motions. Theorem A.3.18 (Slodkowski’s theorem). Let 𝔻 = {z ∈ ℂ : |z| < 1}. Every holomorphic motion f : 𝔻 × A → ℂ of any set A ⊆ ℂ can be extended to a holomorphic motion F : 𝔻 × ℂ → ℂ of ℂ.

A.4 Complex analysis in several variables Theorem A.4.1 (Hartogs’ theorem). Let Λ be an open subset of ℂn and F be a complex Banach space. If a function f : Λ → F is separately holomorphic on Λ, then it is holomorphic on Λ.

Appendix B – The Ionescu-Tulcea and Marinescu theorem In this Appendix, we present a functional-analytic result originally due to IonescuTulcea and Marinescu from their acclaimed paper [57]. We slightly extend the theorem to serve our purposes in Section 13.8. Let E be a complex linear (i. e. vector) space and B a linear subspace of E. Suppose that E and B are Banach spaces when equipped with the norms |⋅| and ‖⋅‖, respectively. Suppose also that: (a1) If a sequence (xn )∞ n=1 in B is such that there exists a constant K > 0 with lim |xn − x| = 0 for some x ∈ E

n→∞

and ‖xn ‖ ≤ K for all n ∈ ℕ,

then x∈B

and ‖x‖ ≤ K.

Let C(B, E) be the subclass of all bounded linear operators T : B → B satisfying the following conditions: (b1) There exists a constant H > 0 such that |T n | ≤ H for all n ∈ ℕ. (c1) There exist N ∈ ℕ and constants 0 < r < 1 and R > 0 such that 󵄩󵄩 N 󵄩󵄩 󵄩󵄩T (x)󵄩󵄩 ≤ r‖x‖ + R|x|,

∀x ∈ B.

(d1) Every bounded subset of B is mapped by T to a relatively compact subset of E. That is, the operator T : (B, ‖ ⋅ ‖) → (E, | ⋅ |) is compact. Given the above notation, definitions and assumptions, the remainder of this Appendix will be dedicated to proving the following theorem. Theorem B.1.1 (Ionescu-Tulcea and Marinescu theorem). Let T ∈ C(B, E). Then: (a2) There are only a finite number of eigenvalues of T : B → B with modulus equal to 1. We denote them by λ1 , . . . , λp ∈ 𝕊1 := {λ ∈ ℂ : |λ| = 1}. (b2) There exist operators T1 , . . . , Tp , S ∈ C(B, E) such that p

T n = ∑ λin Ti + Sn , i=1

∀n ∈ ℕ,

where dim(Ti (B)) < ∞ for every 1 ≤ i ≤ p. (c2) Ti2 = Ti , Ti ∘ Tj = 0 (i ≠ j) and Ti ∘ S = S ∘ Ti = 0 for each 1 ≤ i, j ≤ p. (d2) T ∘ Ti = Ti ∘ T = λi Ti for every 1 ≤ i ≤ p. (e2) r(S) < 1, where r(S) is the spectral radius of S : B → B. https://doi.org/10.1515/9783110702699-023

896 | Appendix B – The Ionescu-Tulcea and Marinescu theorem

(f2)

Addendum: σ(T) \ {λ1 , . . . , λp } ⊆ σ(S) ⊆ B(0, r(S)) ⊆ ℂ, where σ(T) denotes the spectrum of T. This property is called the spectral gap of T.

Remark B.1.2. Because T : B → B is a bounded operator, instead of condition (d1) it suffices that each bounded subset of (B, ‖ ⋅ ‖) be relatively compact in (E, | ⋅ |). We will not prove Theorem B.1.1 directly; rather, the proof will result from the combination of a series of lemmas. First of all, let us observe that we may replace the norms ‖ ⋅ ‖ and | ⋅ | by any equivalent norms to prove this theorem. Lemma B.1.3. The norm ‖ ⋅ ‖∗ on B defined by N−1

󵄩 󵄩 ‖x‖∗ := ∑ 󵄩󵄩󵄩T n (x)󵄩󵄩󵄩 n=0

is equivalent to ‖ ⋅ ‖ and there exists a constant 0 < ̃r < 1 such that 󵄩󵄩 󵄩 󵄩󵄩T(x)󵄩󵄩󵄩∗ ≤ ̃r ‖x‖∗ + R|x|,

∀x ∈ B.

Proof. Let x ∈ B. Notice that N−1

‖x‖ ≤ ‖x‖∗ ≤ ∑ ‖T‖n ⋅ ‖x‖ =: m‖x‖. n=0

Now, let ̃r = (m + r − 1)/m. Clearly, 0 < ̃r < 1. Moreover, N−1

N−1

n=0

n=1

󵄩󵄩 󵄩 󵄩 n+1 󵄩 󵄩 n 󵄩 󵄩 N 󵄩 󵄩󵄩T(x)󵄩󵄩󵄩∗ = ∑ 󵄩󵄩󵄩T (x)󵄩󵄩󵄩 = ∑ 󵄩󵄩󵄩T (x)󵄩󵄩󵄩 + 󵄩󵄩󵄩T (x)󵄩󵄩󵄩 N−1

󵄩 󵄩 ≤ ∑ 󵄩󵄩󵄩T n (x)󵄩󵄩󵄩 + r‖x‖ + R|x| n=1

N−1

N−1

n=0

n=0

by (c1)

󵄩 󵄩 󵄩 󵄩 = ̃r ∑ 󵄩󵄩󵄩T n (x)󵄩󵄩󵄩 + (1 − ̃r ) ∑ 󵄩󵄩󵄩T n (x)󵄩󵄩󵄩 + (r − 1)‖x‖ + R|x| ≤ ̃r ‖x‖∗ + (1 − ̃r )m‖x‖ + (r − 1)‖x‖ + R|x| = ̃r ‖x‖∗ + (m + r − 1 − ̃r m)‖x‖ + R|x| = ̃r ‖x‖∗ + R|x|.

Remark B.1.4. This lemma means that we may, without loss of generality, assume that condition (c1) is satisfied with N = 1. ̃ > 0 such that Lemma B.1.5. There exists a constant R 󵄩󵄩 n 󵄩󵄩 n ̃ 󵄩󵄩T (x)󵄩󵄩 ≤ r ‖x‖ + R|x|,

∀x ∈ B, ∀n ∈ ℕ.

Appendix B – The Ionescu-Tulcea and Marinescu theorem

| 897

Proof. Let x ∈ B and n ∈ ℕ. By condition (c1) with N = 1, 󵄨 n−1 󵄨 󵄩 n−1 󵄩 󵄩 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 n−1 󵄩󵄩T (x)󵄩󵄩 = 󵄩󵄩T(T (x))󵄩󵄩󵄩 ≤ r 󵄩󵄩󵄩T (x)󵄩󵄩󵄩 + R󵄨󵄨󵄨T (x)󵄨󵄨󵄨. From condition (b1), it follows that 󵄨󵄨 n−1 󵄨󵄨 󵄨󵄨 n−1 󵄨󵄨 󵄨󵄨T (x)󵄨󵄨 ≤ 󵄨󵄨T 󵄨󵄨 ⋅ |x| ≤ H|x|. Thus, 󵄩󵄩 n 󵄩󵄩 󵄩 n−1 󵄩 󵄩󵄩T (x)󵄩󵄩 ≤ r 󵄩󵄩󵄩T (x)󵄩󵄩󵄩 + A|x|, A A where A = RH. Observe that we can write A|x| = ( 1−r − r 1−r )|x|, so

A A 󵄩󵄩 n 󵄩󵄩 󵄩 󵄩 |x| ≤ r(󵄩󵄩󵄩T n−1 (x)󵄩󵄩󵄩 − |x|). 󵄩󵄩T (x)󵄩󵄩 − 1−r 1−r Repeating this n − 1 more times, we deduce that A A 󵄩󵄩 n 󵄩󵄩 |x| ≤ r n (‖x‖ − |x|). 󵄩󵄩T (x)󵄩󵄩 − 1−r 1−r Hence, A 1 − rn 󵄩󵄩 n 󵄩󵄩 n A|x| ≤ r n ‖x‖ + |x|. 󵄩󵄩T (x)󵄩󵄩 ≤ r ‖x‖ + 1−r 1−r ̃ = A/(1 − r). Set R Lemma B.1.6. The norms ‖T n ‖, n ∈ ℕ, are uniformly bounded. In other words, 󵄩 󵄩 M := sup{󵄩󵄩󵄩T n 󵄩󵄩󵄩 : n ∈ ℕ} < ∞. Proof. Let x ∈ B. Recalling that r < 1, Lemma B.1.5 implies that 󵄩 󵄩 ̃ < ∞. sup{󵄩󵄩󵄩T n (x)󵄩󵄩󵄩 : n ∈ ℕ} ≤ ‖x‖ + R|x| The desired result follows upon applying the Banach–Steinhaus theorem (also known as the uniform boundedness principle; see Theorem 5.8 in Rudin [109]). For every complex number λ, define B(λ) := Ker(T − λ ⋅ Id) ⊆ B. In other words, if B(λ) ≠ {0} then λ is an eigenvalue of the operator T and B(λ) is the eigenspace corresponding to λ. It is easy to see that for each λ ∈ ℂ, the set B(λ) is a linear subspace of B. Let us now proceed with the next lemma. Lemma B.1.7. If λ ∈ 𝕊1 , then B(λ) is finite dimensional.

898 | Appendix B – The Ionescu-Tulcea and Marinescu theorem Proof. Let X denote the closed (with respect to | ⋅ |) unit ball of B(λ), i. e., let X := {x ∈ B(λ) : |x| ≤ 1}. Clearly, X is a closed set in E. Let x ∈ X. Then x = λ−1 T(x) and, therefore, ‖x‖ = ‖T(x)‖ ≤ r‖x‖ + R|x| ≤ r‖x‖ + R by condition (c1). Hence, ‖x‖ ≤

R . 1−r

Thus, X is a bounded subset of B. This allows us to infer from condition (d1) that T(X) is compact in E. Since X = λ−1 T(X), we deduce that X = λ−1 T(X) is compact in E. Because X is closed in E, it follows that X = X is compact in E. As a Banach space is finite dimensional if and only if its unit ball is compact, the compactness in E of the unit ball X of B(λ) implies that dim(B(λ)) < ∞. Lemma B.1.8. Let T ∈ C(B, E). Then T has only a finite number of eigenvalues with modulus 1. Proof. Suppose, by way of contradiction, that there exists an infinite sequence of distinct eigenvalues (λi )∞ i=1 with |λi | = 1 for all i ∈ ℕ. For each λi choose any eigenvector xi ∈ B(λi ) \ {0}. Then the eigenvectors (xi )∞ i=1 are linearly independent. For every n ∈ ℕ, let X(n) be the linear span of (x1 , . . . , xn ), i. e., the set of all vectors of the form a1 x1 +⋅ ⋅ ⋅+an xn , where a1 , . . . , an ∈ ℂ. Notice that X(n) is a proper closed linear subspace of X(n + 1). A straightforward calculation shows that cT(X(n)) ⊆ X(n),

∀n ∈ ℕ, ∀c ∈ ℂ.

(B.1)

Indeed, if x ∈ X(n), n

n

i=1

i=1

cT(x) = c ∑ ai T(xi ) = ∑ cai λi xi ∈ X(n). The remainder of the proof of this lemma depends on the following two claims. Claim 1. If x ∈ X(n), then for every k ∈ ℕ we have that T k (x) − λnk x ∈ X(n − 1). Proof of Claim 1. We will prove this claim by induction. Let x ∈ X(n) and observe that n

n

n−1

i=1

i=1

i=1

T(x) − λn x = ∑ ai λi xi − λn ∑ ai xi = ∑ (ai λi − ai λn )xi ∈ X(n − 1).

Appendix B – The Ionescu-Tulcea and Marinescu theorem

| 899

Assume now that T k−1 (x) − λnk−1 x ∈ X(n − 1). Since T is a linear operator, it follows that T k (x) − λnk x = T(T k−1 (x) − λnk−1 x) + λnk−1 (T(x) − λn x) ∈ T(X(n − 1)) + X(n − 1)

⊆ X(n − 1) + X(n − 1) ⊆ X(n − 1). This completes the proof of the first claim.

◼

Claim 2. There exists a sequence (yn )∞ n=1 such that yn ∈ X(n), |yn | = 1 and |yn − x| ≥ for all x ∈ X(n − 1).

1 2

Proof of Claim 2. Choose z ∈ X(n) \ X(n − 1). As X(n − 1) is closed, we have β := dist(z, X(n − 1)) = inf{|z − x| : x ∈ X(n − 1)} > 0. So there exists some x̂ ∈ X(n−1) with the property that |z−x̂| ≤ 2β. Let yn := (z−x̂)/|z−x̂|. Clearly, yn ∈ X(n) and |yn | = 1. Let x ∈ X(n − 1). Then 󵄨󵄨 󵄨󵄨 z − x̂ 1 1 1 󵄨󵄨 󵄨 󵄨 󵄨 − x󵄨󵄨󵄨 = β= . |yn − x| = 󵄨󵄨󵄨 󵄨z − (x̂ + x|z − x̂|)󵄨󵄨󵄨 ≥ 󵄨󵄨 |z − x̂| 󵄨 󵄨󵄨 |z − x̂| 2β 2 This ends the proof of the second claim.

◼

In light of Lemma B.1.5, for all j ∈ ℕ we have that 󵄩󵄩 −n n 󵄩󵄩 󵄩󵄩 n 󵄩󵄩 n ̃ 󵄩󵄩λj T (yj )󵄩󵄩 = 󵄩󵄩T (yj )󵄩󵄩 ≤ r ‖yj ‖ + R for all n ∈ ℕ. Since r < 1, for each j ∈ ℕ there exists nj ∈ ℕ such that for all n ≥ nj − 1, 󵄩󵄩 −(n+1) n 󵄩󵄩 󵄩󵄩 −n n 󵄩󵄩 ̃ T (yj )󵄩󵄩 = 󵄩󵄩λj T (yj )󵄩󵄩 ≤ 1 + R. 󵄩󵄩λj Thus, the set {λj j T nj −1 (yj ) : j ∈ ℕ} is bounded in B. By condition (d1), the closure of −n

the set {λj j T nj (yj ) : j ∈ ℕ} is compact in E. It ensues that the sequence (λj j T nj (yj ))∞ j=1 contains a subsequence that converges in E. To finish the proof of the lemma, consider −n

−n

−n −n 󵄨 −n 󵄨 󵄨 −n 󵄨 Dj := 󵄨󵄨󵄨λj+1j+1 T nj+1 (yj+1 ) − λj j T nj (yj )󵄨󵄨󵄨 = 󵄨󵄨󵄨λj+1j+1 T nj+1 (yj+1 ) − yj+1 + yj+1 − λj j T nj (yj )󵄨󵄨󵄨 = |yj+1 − xj |,

where xj := λj j T nj (yj ) − (λj+1j+1 T nj+1 (yj+1 ) − yj+1 ). According to Claim 1, −n

−n

λj+1j+1 T nj+1 (yj+1 ) − yj+1 ∈ X(j + 1 − 1) = X(j). −n

Also, λj j T nj (yj ) ∈ X(j) by (B.1). Therefore, xj ∈ X(j). Finally, by the choice of the sequence (yj )∞ j=1 , we obtain from Claim 2 that Dj ≥ 1/2 for all j ∈ ℕ. This contradicts the −n

fact that (λj j T nj (yj ))∞ j=1 contains a convergent subsequence. −n

900 | Appendix B – The Ionescu-Tulcea and Marinescu theorem In order to proceed further, we need the following theorem due to Kakutani and Yosida [60]. Here, we use the notation ℬ(X) for the set of bounded linear operators on a Banach space (X, | ⋅ |). Theorem B.1.9. Let T ∈ ℬ(X). Assume that: (a) There exists C ≥ 0 such that 󵄨󵄨 n 󵄨󵄨 󵄨󵄨T 󵄨󵄨 ≤ C,

∀n ≥ 0.

(b) For each x ∈ X, the sequence ( n1 ∑nj=1 T j (x))∞ n=1 contains a subsequence converging to a point denoted by T1 (x). Then the sequence ( n1 ∑nj=1 T j (x))∞ n=1 actually converges to T1 (x) for each x ∈ X. Furthermore, T1 : X → X is a bounded linear operator such that T ∘ T1 = T1 ∘ T = T12 = T1

and

|T1 | ≤ C.

Proof. Fix x ∈ X and, to shorten the notation, set xn :=

1 n j ∑ T (x). n j=1

By (b), let x be an accumulation point of the sequence (xn )∞ n=1 . We shall first prove that T(x) = x.

(B.2)

Indeed, using (a), 󵄨 󵄨󵄨 1 2C 󵄨󵄨 󵄨 󵄨󵄨 1 n+1 󵄨 󵄨 n+1 󵄨 󵄨 󵄨 |x|. 󵄨󵄨T(xn ) − xn 󵄨󵄨󵄨 = 󵄨󵄨󵄨 (T (x) − T(x))󵄨󵄨󵄨 ≤ (󵄨󵄨󵄨T (x)󵄨󵄨󵄨 + 󵄨󵄨󵄨T(x)󵄨󵄨󵄨) ≤ 󵄨󵄨 n 󵄨󵄨 n n So, if (xnk )∞ k=1 is a subsequence such that xnk → x, then 󵄨󵄨 󵄨 2C |x|. 󵄨󵄨T(xnk ) − xnk 󵄨󵄨󵄨 ≤ nk Upon letting k → ∞, we obtain that T(x) = x. Now we shall prove that the entire sequence (xn )∞ n=1 converges to x. Write x = x + (x − x). By (B.2), we have xn = x +

1 n j ∑ T (x − x). n j=1

Thus, in order to prove that (xn )∞ n=1 converges to x, it suffices to prove that 1 n j ∑ T (x − x) = 0. n→∞ n j=1 lim

(B.3)

Appendix B – The Ionescu-Tulcea and Marinescu theorem |

901

To that end, let the set R be defined by R := (Id −T)(X). We then have that n

(Id −

n −1 n −2 2 n − (nk − 1) nk −1 1 k j T+ k T + ⋅⋅⋅ + k T )(x) ∈ R ∑ T )(x) = (Id −T)(Id + k nk j=1 nk nk nk

for all k ∈ ℕ. Letting k → ∞, we deduce that x − x ∈ R. To obtain (B.3), it suffices to prove that for all y ∈ R, 1 n j ∑ T (y) = 0. n→∞ n j=1 lim

(B.4)

In order to show this, consider first y ∈ R. In that case, y = z − T(z) for some z ∈ X and so, using (a) again, 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 n+1 󵄨󵄨 1 󵄨 󵄨 󵄨 󵄨󵄨 ∑ T j (y)󵄨󵄨󵄨 = 󵄨󵄨󵄨 1 ∑ T j (z) − 1 ∑ T j (z)󵄨󵄨󵄨 = 1 󵄨󵄨󵄨T(z) − T n+1 (z)󵄨󵄨󵄨 ≤ 2C |z|. 󵄨 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 n 󵄨 n j=2 n 󵄨󵄨 j=1 󵄨󵄨 󵄨󵄨 j=1 󵄨󵄨 This last quantity tends to zero as n → ∞. On the other hand, if y ∈ R \ R then fix ε > 0 and pick y′ ∈ R such that |y − y′ | < ε. Using (a) once more, we obtain that 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 n 󵄨󵄨 1 󵄨 󵄨 󵄨 󵄨󵄨 ∑ T j (y)󵄨󵄨󵄨 = 󵄨󵄨󵄨 1 ∑ T j (y′ ) + 1 ∑ T j (y − y′ )󵄨󵄨󵄨 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 n j=1 󵄨󵄨 j=1 󵄨󵄨 󵄨󵄨 j=1 󵄨󵄨 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 1 n 󵄨 󵄨󵄨 1 󵄨 ≤ 󵄨󵄨󵄨 ∑ T j (y′ )󵄨󵄨󵄨 + ∑󵄨󵄨󵄨T j (y − y′ )󵄨󵄨󵄨 󵄨󵄨 n 󵄨󵄨 n j=1 j=1 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨 1 󵄨󵄨 1 ≤ 󵄨󵄨󵄨 ∑ T j (y′ )󵄨󵄨󵄨 + C 󵄨󵄨󵄨y − y′ 󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨 ∑ T j (y′ )󵄨󵄨󵄨 + Cε. 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨󵄨 n j=1 󵄨󵄨󵄨 n j=1 󵄨 Letting n → ∞, we deduce that 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 1 lim sup󵄨󵄨󵄨 ∑ T j (y)󵄨󵄨󵄨 ≤ Cε. 󵄨󵄨 󵄨 n n→∞ 󵄨 󵄨 j=1 󵄨 Letting ε → 0, we conclude that 1 n j ∑ T (y) = 0. n→∞ n j=1 lim

We have now established (B.4) for any y ∈ R. This completes the proof that the sequence (xn )∞ n=1 converges to x.

902 | Appendix B – The Ionescu-Tulcea and Marinescu theorem So it makes sense to define the operator T1 by setting T1 (x) := x. It is clear that T1 : X → X is linear and bounded with |T1 | ≤ C. Furthermore, (B.2) is equivalent to the statement T ∘ T1 = T1 . Thus, for each n ∈ ℕ, T n ∘ T1 = T1

and hence

1 n j ∑ T ∘ T1 = T1 . n j=1

Upon passing to the limit n → ∞, this implies that T12 = T1 . Finally, we have that 󵄨󵄨 n 󵄨󵄨 󵄨 n 󵄨󵄨 󵄨󵄨 1 󵄨 󵄨 󵄨󵄨 ∑ T j (T(x)) − 1 ∑ T j (x)󵄨󵄨󵄨 = 󵄨󵄨󵄨 1 (T n+1 (x) − T(x))󵄨󵄨󵄨 ≤ 2C |x|. 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 n j=1 n 󵄨󵄨 j=1 󵄨󵄨 󵄨 󵄨 Letting n → ∞, it follows that T1 (T(x)) = T1 (x) for every x ∈ X. This completes the proof of Theorem B.1.9. Theorem B.1.10. If T ∈ ℬ(X) satisfies the hypotheses of Theorem B.1.9, then so does λ−1 T for any λ ∈ 𝕊1 and there thus exists an operator Tλ ∈ ℬ(X) such that: (a) Tλ = limn→∞ n1 ∑nj=1 λ−j T j . (b) Tλ ∘ T = T ∘ Tλ = λTλ , Tλ2 = Tλ and |Tλ | ≤ C. (c) If λ, μ ∈ 𝕊1 with λ ≠ μ, then Tλ ∘ Tμ = 0. (d) T(x) = λx if and only if Tλ (x) = x. Furthermore, if S = T − ∑ki=1 λi Tλi for some λ1 , . . . , λk ∈ 𝕊1 , then: (e) Tλi ∘ S = S ∘ Tλi = 0 for all 1 ≤ i ≤ k. (f) T ∘ S = S ∘ T = S2 . (g) T n = ∑ki=1 λin Tλi + Sn for all n ∈ ℕ. (h) λ is an eigenvalue of S if and only if λ is an eigenvalue of T and λ ≠ λi for all 1 ≤ i ≤ k. Proof. Part (a), as well as Tλ2 = Tλ and |Tλ | ≤ C in part (b), follow directly from Theorem B.1.9 with Tλ being the operator T1 associated to λ−1 T. Hence, Tλ ∘ T = λTλ ∘ (λ−1 T) = λTλ = λ(λ−1 T) ∘ Tλ = T ∘ Tλ .

Appendix B – The Ionescu-Tulcea and Marinescu theorem |

903

Alternatively, 1 n −j j+1 1 n 1 n+1 −j j ∑ λ T = λ lim ∑ λ−(j+1) T j+1 = λ lim ∑ λ T = λTλ . n→∞ n n→∞ n n→∞ n + 1 j=1 j=1 j=1

Tλ ∘ T = T ∘ Tλ = lim

Part (b) is thus proved. Moreover, by induction, Tλ ∘ T n = T n ∘ Tλ = λn Tλ ,

(B.5)

∀n ∈ ℕ.

In order to prove part (c), applying Tμ to part (a) from the right-hand side and using (B.5) for Tμ , we see that μ n

1 1 − (λ) μ 1 n 1 n 1 T = 0. Tλ ∘ Tμ = lim ∑ λ−j T j ∘ Tμ = lim ∑ j μj Tμ = lim n→∞ n→∞ n n→∞ n n 1 − ( μλ ) λ μ j=1 j=1 λ Thus, part (c) is proved. For part (d), first notice that if λ−1 T(x) = x, then 1 n −j j 1 n ∑ λ T (x) = lim ∑ x = x. n→∞ n n→∞ n j=1 j=1

Tλ (x) = lim

Also, if Tλ (x) = x, it follows from (B.5) that T(x) = T ∘ Tλ (x) = λTλ (x) = λx. The proofs of parts (e) and (f) follow from the definition of S and parts (b) and (c): k

Tλi ∘ S = S ∘ Tλi = T ∘ Tλi − ∑ λj Tλj ∘ Tλi = λi Tλi − λi Tλ2i = 0. j=1

Therefore, k

k

i=1

i=1

S2 = (T − ∑ λi Tλi ) ∘ S = T ∘ S − ∑ λi Tλi ∘ S = T ∘ S = S ∘ T. The proof of part (g) follows by induction from parts (b) and (f): k

k

k

i=1

i=1

i=1

T n+1 = T ∘ T n = T ∘ (∑ λin Tλi + Sn ) = ∑ λin T ∘ Tλi + T ∘ S ∘ Sn−1 = ∑ λin+1 Tλi + Sn+1 . It only remains to demonstrate part (h). To that end, suppose that λ is an eigenvalue of T and that λ ≠ λi for all 1 ≤ i ≤ k. Then there exists x ≠ 0 such that T(x) = λx and, therefore, in light of part (b), λi Tλi (x) = Tλi (T(x)) = λTλi (x),

∀1 ≤ i ≤ k.

904 | Appendix B – The Ionescu-Tulcea and Marinescu theorem Since λ ≠ λi , it follows that Tλi (x) = 0 for all 1 ≤ i ≤ k. Hence, k

λx = T(x) = ∑ λi Tλi (x) + S(x) = S(x). i=1

In other words, λ is an eigenvalue of S. For the converse, suppose that λ is an eigenvalue of S. Then there exists x ≠ 0 such that S(x) = λx. Thus, 1 1 1 1 T(x) = T( S(x)) = T(S(x)) = S2 (x) = λ2 x = λx. λ λ λ λ So λ is also an eigenvalue of T. By way of contradiction, suppose that λ = λi for some 1 ≤ i ≤ k. Then 1 n −j j 1 n −j j ∑ λi T (x) = lim ∑ λi λi x = x ≠ 0. n→∞ n n→∞ n j=1 j=1

Tλi (x) = lim However,

1 1 Tλi (x) = Tλi ( S(x)) = Tλi ∘ S(x) = 0. λ λ This contradiction finishes the proof of part (h), which completes the proof of Theorem B.1.10. Let us now return to our particular situation. Lemma B.1.11. Let T ∈ C(B, E). For each λ ∈ 𝕊1 and each x ∈ B, there exists a unique x ∈ B such that 󵄨󵄨 1 n 󵄨󵄨 󵄨 󵄨 lim 󵄨󵄨󵄨 ∑ λ−j T j (x) − x 󵄨󵄨󵄨 = 0. n→∞󵄨󵄨 n 󵄨󵄨 j=1 Proof. Let λ ∈ 𝕊1 and x ∈ B. First, note that n

n

j=1

j=1

∑ λ−j T j (x) = T(∑ λ−j T j−1 (x)) and, by Lemma B.1.6, 󵄩󵄩 n 󵄩󵄩 n 󵄩󵄩 󵄩 󵄨 −j 󵄨 −j j−1 󵄩󵄩∑ λ T (x)󵄩󵄩󵄩 ≤ ∑󵄨󵄨󵄨λ 󵄨󵄨󵄨 ⋅ M‖x‖ = Mn‖x‖. 󵄩󵄩 󵄩󵄩 j=1 j=1 Consequently, by (d1), the sequence ∞

1 n 1 n ( ∑ λ−j T j (x) = T( ∑ λ−j T j−1 (x))) n j=1 n j=1 n=1

(B.6)

Appendix B – The Ionescu-Tulcea and Marinescu theorem |

905

contains a convergent subsequence in E. Let B denote the closure of B in E. Due to (b1), each map T n : B → B extends ̂n = T̂ n : B → B and continuously to a map T 󵄨󵄨 ̂ n 󵄨󵄨 󵄨󵄨T 󵄨󵄨 ≤ H,

(B.7)

∀n ∈ ℕ.

In order to shorten the notation, set R := λ−1 T̂ : B → B

Rn = λ−n T̂ n : B → B.

and so

Take x ∈ B. For all k ∈ ℕ there exists xk ∈ B such that 1 . k

|x − xk |
1,

∀n ∈ ℕ.

Fix an arbitrary y ∈ B. Then, for all n ∈ ℕ there exists xn ∈ B such that (ζn Id −S)(xn ) = y.

(B.11)

Claim. There exists a constant D ≥ 0 such that |xn | ≤ D,

∀n ∈ ℕ.

Proof of Claim. Suppose that the claim is not true. By passing to a subsequence if necessary, we may assume without loss of generality that each |xn | > 0 and lim |xn | = ∞.

n→∞

Set xn′ := xn /|xn |. Since (ζn Id −S)(xn′ ) =

y , |xn |

(B.12)

ζn xn′ = S(xn′ ) +

y . |xn |

(B.13)

we obtain that

Hence, using (B.10), ‖y‖ 󵄩󵄩 ′ 󵄩󵄩 󵄩󵄩 ′ 󵄩󵄩 󵄩󵄩 ′ 󵄩󵄩 ‖y‖ 󵄩 󵄩 󵄩 󵄩 󵄨 󵄨 ‖y‖ ≤ r 󵄩󵄩󵄩xn′ 󵄩󵄩󵄩 + (R + A)󵄨󵄨󵄨xn′ 󵄨󵄨󵄨 + ≤ r 󵄩󵄩󵄩xn′ 󵄩󵄩󵄩 + R + A + . 󵄩󵄩xn 󵄩󵄩 < 󵄩󵄩ζn xn 󵄩󵄩 ≤ 󵄩󵄩S(xn )󵄩󵄩 + |xn | |xn | |xn | Thus, ‖y‖ 󵄩󵄩 ′ 󵄩󵄩 R + A + |xn | . 󵄩󵄩xn 󵄩󵄩 < 1−r

(B.14)

As limn→∞ |xn | = ∞, we deduce that the sequence (‖xn′ ‖)∞ n=1 is bounded. By Lemma B.1.13, the sequence (S(xn′ ))∞ contains a convergent subsequence in the | ⋅ | norm. n=1 ′ ∞ By (B.13), so does the sequence (ζn xn )n=1 . In turn, this implies that the sequence (xn′ )∞ n=1 contains a subsequence (xn′ j )∞ j=1 convergent in E. Set x′ := lim xn′ j ∈ E j→∞

(i. e., in the | ⋅ | norm).

Appendix B – The Ionescu-Tulcea and Marinescu theorem |

909

′ Using (a1) and the fact that (‖xn′ ‖)∞ n=1 is a bounded sequence, we know that x ∈ B. ′ ′ We also have that |x | = 1, so x ≠ 0. According to Lemma B.1.12(h), the operator S : (B, | ⋅ |) → (B, | ⋅ |) is bounded. Since every bounded operator on a Banach space is continuous, S : (B, | ⋅ |) → (B, | ⋅ |) is continuous. We thus obtain from (B.12) that

(λ Id −S)(x′ ) = 0. But this means that λ is an eigenvalue of S of modulus 1, which contradicts Lemma B.1.12(g). This completes the proof of the claim. ◼ that

Continuing the proof of Lemma B.1.14, it follows from (B.14) and from the claim (R + A)|xn | + ‖y‖ (R + A)D + ‖y‖ 󵄩 󵄩 ‖xn ‖ = 󵄩󵄩󵄩xn′ 󵄩󵄩󵄩 ⋅ |xn | ≤ ≤ . 1−r 1−r

So the sequence (‖xn ‖)∞ n=1 is bounded. Thus, in light of Lemma B.1.13, the sequence ∞ (S(xn ))n=1 contains a convergent subsequence in | ⋅ |. It follows from (B.11) that (ζn xn )∞ n=1 also contains a convergent subsequence, as, therefore, does the sequence (xn )∞ . Den=1 note this latter convergent subsequence by (xnj )∞ j=1 and set x := lim xnj ∈ E. j→∞

Using (a1) and the fact that (‖xn ‖)∞ n=1 is bounded, we know that x ∈ B. Since the operator S : (B, | ⋅ |) → (B, | ⋅ |) is continuous, we deduce from (B.11) that (λ Id −S)(x) = y. Since y ∈ B is arbitrary, we conclude that (λ Id −S)(B) = B. We now turn our attention to the spectral radius of S. Lemma B.1.15. r(S) < 1. Proof. Lemma B.1.14 states that the operator (λ Id −S) : B → B is surjective for all λ ∈ 𝕊1 . It follows from the fact that S : B → B has no eigenvalue of modulus 1 that (λ Id −S) : B → B is also injective. Thus, the inverse (λ Id −S)−1 exists for all λ ∈ 𝕊1 . Since the set of invertible operators on a Banach space is open, we deduce that S cannot have eigenvalues with moduli arbitrarily close to 1. Finally, we describe a relation between the spectra of S and T. Lemma B.1.16. σ(T) \ {λ1 , . . . , λp } ⊆ σ(S) ⊆ B(0, r(S)) ⊆ ℂ. Proof. Recall that λ ∈ σ(T) if and only if the operator T − λ Id is not invertible. Notice that if a Banach space B can be written as the direct sum B1 ⨁ B2 of two Banach spaces

910 | Appendix B – The Ionescu-Tulcea and Marinescu theorem B1 and B2 and if T : B → B is a bounded linear operator under which B1 and B2 are invariant, i. e., T(B1 ) ⊆ B1 and T(B2 ) ⊆ B2 , then σ(T) = σ(T|B1 ) ∪ σ(T|B2 ). Indeed, in such circumstances, T − λ Id preserves B1 and B2 , and hence T − λ Id is noninvertible on B if and only if T − λ Id is noninvertible on one of B1 or B2 , i. e., if and only if λ ∈ σ(T|B1 ) ∪ σ(T|B2 ). Using Lemma B.1.12, split B into B1 := ⨁pi=1 Ti (B) and B2 := (Id − ∑pi=1 Ti )(B). Observe that T(Ti (B)) = Ti ∘ T(B) ⊆ Ti (B),

∀1 ≤ i ≤ p,

and p

p

p

p

i=1

i=1

i=1

i=1

T ∘ (Id − ∑ Ti ) = T − ∑ T ∘ Ti = T − ∑ Ti ∘ T = (Id − ∑ Ti ) ∘ T. So T preserves each Ti (B), as well as B1 and B2 . We must now show that B = B1 ⨁ B2 , i. e., that any x ∈ B can be uniquely written as x1 +x2 where x1 ∈ B1 and x2 ∈ B2 . Clearly, p

p

i=1

i=1

x = ∑ Ti (x) + (Id − ∑ Ti )(x). If this decomposition were not unique, then there would be x1 ≠ x2 ∈ B such that p

p

p

p

i=1

i=1

i=1

i=1

∑ Ti (x1 ) + (Id − ∑ Ti )(x1 ) = ∑ Ti (x2 ) + (Id − ∑ Ti )(x2 ). Equivalently, p

p

i=1

i=1

∑ Ti (x1 − x2 ) = (Id − ∑ Ti )(x2 − x1 ). Suppose that p

p

i=1

i=1

∑ Ti (z) = (Id − ∑ Ti )(y) for some y, z ∈ B. Then ∑pi=1 Ti (z + y) = y. Applying any Tj to both sides would result in p

Tj (y) = Tj ∘ ∑ Ti (z + y) = Tj ∘ Tj (z + y) = Tj (z + y). i=1

Appendix B – The Ionescu-Tulcea and Marinescu theorem

| 911

Hence, Tj (z) = 0 for all 1 ≤ j ≤ p and ∑pi=1 Ti (z) = 0 = (Id − ∑pi=1 Ti )(y). In particular, this would mean that Tj (x1 − x2 ) = 0 for all 1 ≤ j ≤ p, i. e. Tj (x1 ) = Tj (x2 ) for all 1 ≤ j ≤ p and the decomposition is unique. Thus, we have shown that p

B = B1 ⨁ B2 = ⨁ Ti (B) ⨁ B2 i=1

and that T preserves each subspace in these direct sums. Consequently, p

σ(T) = ⋃ σ(T|Ti (B) ) ⋃ σ(T|B2 ).

(B.15)

i=1

We now examine the pointwise action of T on the subspace Ti (B) for any fixed 1 ≤ i ≤ p. If x ∈ Ti (B), then x = Ti (y) for some y ∈ B. Therefore, p

T(x) = ∑ λj Tj (Ti (y)) + S(Ti (y)) = λi Ti2 (y) = λi Ti (y) = λi x. j=1

So T|Ti (B) = λi Id |Ti (B) , i. e., T acts on Ti (B) as the multiplication by λi , and hence σ(T|Ti (B) ) = {λi }. Let us now investigate the pointwise action of T on B2 . Let x ∈ B2 , i. e., x = (Id − ∑pi=1 Ti )(y) for some y ∈ B. Then p

p

p

p

j=1

i=1

j=1

i=1

T(x) = ∑ λj Tj ((Id − ∑ Ti )(y)) + S(x) = ∑ λj Tj (y) − ∑ λi Ti (y) + S(x) = S(x). Thus, σ(T|B2 ) = σ(S|B2 ). By (B.15), we know that σ(T) = {λ1 , . . . , λp } ∪ σ(S|B2 ). By Lemma B.1.15 and the fact that all λi ∈ 𝕊1 , we deduce that σ(T) \ {λ1 , . . . , λp } = σ(S|B2 ) ⊆ σ(S) ⊆ B(0, r(S)). Proof of Theorem B.1.1. Lemma B.1.8 establishes (a2). Properties (b2), (c2) and (d2) are proved in Lemma B.1.12, in combination with Lemma B.1.7. Property (e2) is the object of Lemma B.1.15 while property (f2) is demonstrated in Lemma B.1.16.

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Index δ-cover 568 λ-lemma 620 λ-system 857 π-system 855 σ-algebra 856 σ-stable 577 ε-independent 529 acceptable potential 671 algebra 856 almost periodic operator 472 almost uniform convergence 873 amalgamated function 773 Arzelà–Ascoli Theorem 888 asymptotic covariance 547 asymptotic variance 514 Banach–Alaoglu’s Theorem 878 Borel σ-algebra 857 Borel measure 862 Boundary Separation Condition (BSC) 763 Bounded Distortion Property (BDP) 744 bounded variation 557 boundedly sub/supermultiplicative 674, 703 Bowen parameter 753 Bowen’s Formula 615, 758 box dimension 580 Carathéodory’s Extension Theorem 864 Cauchy’s Integral Formula 890 Cauchy’s Theorem 890 Central Limit Theorem 527, 726 CGDMS 742 change of variables 448 characteristic function 887 coboundary 442 coding map 738 cofinitely regular 754 cohomologous 442, 705 comparable words 669 completeness of F -spectrum 653 complex continued fractions 796 complex-parametrized family 803 conditional expectation 879 conditional measure 886 cone condition 768 conformal 742 https://doi.org/10.1515/9783110702699-025

conformal expanding repeller 605 conformal map 604 conformal measure 464, 784 conformal-like 771 correlation 509 countable stability 577 covering theorem 587 critically regular CGDMS 754 distribution 887 dual space 876 Dynkin’s π-λ Theorem 858 Egorov’s Theorem 873 equilibrium state 446, 683 essential infimum/supremum 597 evenly varying 567 expected value 879 F -conformal 776 Fatou’s Lemma 869 finitely irreducible 670, 739 finitely primitive 671, 739 finitely supported measure 681 finiteness parameter 751 Fourier transform 887 gauge function 567 Gauss measure 796 GDMS 737 GDS 737 generator 606 Gibbs measure 438 graph directed Markov system 737 graph directed system 737 Halmos’ Monotone Class Theorem 859 Hartogs’ Theorem 893 Hausdorff dimension 577 – of a measure 595 Hentschel–Procaccia spectrum 649 Hölder continuous on cylinders 671 Hölder family 774 Hölder function 433 Hölder stable 821 holomorphic motion 620 holomorphic perturbation 619

920 | Index

HP-spectrum 649 Hurwitz’s Theorem 892 Hurwitz’s Theorem (Converse) 892 Hurwitz’s Theorem II 892 IFS 737 Implicit Function Theorem 892 incomparable words 669 information dimension 650 inner regularity 862 integrable function 867 Inverse Function Theorem 888, 892 irregular CGDMS 753 iterated function system 606, 737 Jacobian 449 Kato–Rellich Perturbation Theorem 889 Koebe Distortion Theorem 891 Koebe Quarter Theorem 890 Lasota–Yorke map 549 Law of the Iterated Logarithm 529, 727 Lebesgue’s Dominated Convergence Theorem 870 Legendre transform 652 Legendre transform pair 653 limit set 606, 738 Liouville’s Theorem 739 local dimension 648, 839 locally finite 798 lower and upper information dimensions 665 Lyapunov exponent 640, 789 Martingale Convergence Theorem 882 Martingale Convergence Theorem for Conditional Expectations 884 measurable sets 857 measurable space 857 measurable transformation 859 measure 861 – σ-finite 861 – absolutely continuous 875 – Borel 862 – complete 862 – dimensionally exact 599 – equivalent 875 – finite 861 – geometric 599

– Lebesgue 863 – mutually singular 875 – pointwise dimension 596 – probability 861 – regular 863 metric outer measure 565 mixing 509 Monodromy Theorem 890 monotone class 858 Monotone Convergence Theorem 868 Morera’s Theorem 890 multifractal analysis 648 multifractal decomposition 649 multifractal spectrum 649, 827, 840, 852 Open Set Condition (OSC) 606, 742 operator – transfer-type 452 operator norm topology 876 osc(f ), the sup of the variation of f over 1-cylinders 671 outer measure 564 – Hausdorff 569 outer regularity 863 packing 571 – r-packing 571 packing∗ dimension 578 packing dimension 579 – of a measure 596 packing measure 574 partition function 750 periodically separated 823 pointwise dimension 596, 648 – dimension spectrum 649 – regular part 648 – singular part 649 pointwise finite 738 Portmanteau Theorem 877 pressure gap 489 quasi-compact 503 quasisymmetric 619 Radon–Nikodym derivative 875 Radon–Nikodym Theorem 875 real continued fractions 795 regular CGDMS 753 regular measure 863

Index | 921

Rényi spectrum 649 Reversed Martingale Convergence Theorem for Conditional Expectations 885 Riesz Representation Theorem 876 Rouché’s Theorem 892 Scheffé’s Lemma 871 Schwarz Lemma 890 Schwarz Reflection Principle 890 semi-algebra 855 simple function 860 Slodkowski’s Theorem 893 strong open set condition (SOSC) 760 Strong Separation Condition (SSC) 606, 763 strongly regular CGDMS 754

strongly summable 801 sub/supermultiplicative 674, 703 summable 694, 774 temperature 833 temperature function 651 topological pressure 678, 751, 775 topology of uniform convergence 876 transfer(-type) operator 452 volume lemma 790, 793 weak∗ topology 877 weakly Bernoulli 529

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