Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry

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

For further volumes: http://www.springer.com/series/304

2036

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Volker Mayer  Bartlomiej Skorulski Mariusz Urbanski

Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry

123

Volker Mayer Universit´e Lille 1 D´epartement de Math´ematiques 59655 Villeneuve d’Ascq France [email protected]

Mariusz Urbanski University of North Texas Department of Mathematics Denton, TX 76203-1430 USA [email protected]

Bartlomiej Skorulski Universidad Catolica del Norte Departamento de Matematicas Avenida Angamos 0610 Antofagasta Chile [email protected]

ISBN 978-3-642-23649-5 e-ISBN 978-3-642-23650-1 DOI 10.1007/978-3-642-23650-1 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2011940286 Mathematics Subject Classification (2010): 37-XX c Springer-Verlag Berlin Heidelberg 2011  This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

In this book we introduce measurable expanding random systems, develop the thermodynamical formalism and establish, in particular, exponential decay of correlations and analyticity of the expected pressure although the spectral gap property does not hold. This theory is then used to investigate fractal properties of conformal random systems. We prove a Bowen’s formula and develop the multifractal formalism of the Gibbs states. Depending on the behavior of the Birkhoff sums of the pressure function we get a natural classifications of the systems into two classes: quasi-deterministic systems which share many properties of deterministic ones and essential random systems which are rather generic and never bi-Lipschitz equivalent to deterministic systems. We show in the essential case that the Hausdorff measure vanishes which refutes a conjecture of Bogensch¨utz and Ochs. We finally give applications of our results to various specific conformal random systems and positively answer a question of Br¨uck and B¨uger concerning the Hausdorff dimension of randomJulia sets.

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Acknowledgements

The second author was supported by FONDECYT Grant no. 11060538, Chile and Research Network on Low Dimensional Dynamics, PBCT ACT 17, CONICYT, Chile. The research of the third author is supported in part by the NSF Grant DMS 0700831. A part of his work has been done while visiting the Max Planck Institute in Bonn, Germany. He wishes to thank the institute for support.

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Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1

2 Expanding Random Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Introductory Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Expanding Random Maps . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Uniformly Expanding Random Maps . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Remarks on Expanding Random Mappings . . . . .. . . . . . . . . . . . . . . . . . . . 2.6 Visiting Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7 Spaces of Continuous and H¨older Functions . . . .. . . . . . . . . . . . . . . . . . . . 2.8 Transfer Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.9 Distortion Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

5 5 8 8 9 10 11 12 13 14

3 The RPF-Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Formulation of the Theorems . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Frequently used Auxiliary Measurable Functions . . . . . . . . . . . . . . . . . . . 3.3 Transfer Dual Operators . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 Invariant Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5 Levels of Positive Cones of H¨older Functions . . .. . . . . . . . . . . . . . . . . . . . 3.6 Exponential Convergence of Transfer Operators . . . . . . . . . . . . . . . . . . . . 3.7 Exponential Decay of Correlations . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.8 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.9 Pressure Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.10 Gibbs Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.11 Some Comments on Uniformly Expanding Random Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

17 17 19 19 22 24 27 31 32 33 35

4 Measurability, Pressure and Gibbs Condition . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Measurable Expanding Random Maps . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Measurability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 The Expected Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

39 39 41 42

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Contents

Ergodicity of  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Random Compact Subsets of Polish Spaces . . . . .. . . . . . . . . . . . . . . . . . . .

43 43

5 Fractal Structure of Conformal Expanding Random Repellers . . . . . . . . 5.1 Bowen’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Quasi-Deterministic and Essential Systems . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 Random Cantor Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

47 47 51 54

6 Multifractal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Concave Legendre Transform . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Multifractal Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Analyticity of the Multifractal Spectrum for Uniformly Expanding Random Maps . . . . . . .. . . . . . . . . . . . . . . . . . . .

57 57 59

7 Expanding in the Mean. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 Definition of Maps Expanding in the Mean . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 Associated Induced Map . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3 Back to the Original System . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

69 69 70 72 73

8 Classical Expanding Random Systems . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 Definition of Classical Expanding Random Systems . . . . . . . . . . . . . . . 8.2 Classical Conformal Expanding Random Systems . . . . . . . . . . . . . . . . . . 8.3 Complex Dynamics and Br¨uck and B¨uger Polynomial Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.4 Denker–Gordin Systems . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.5 Conformal DG*-Systems . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.6 Random Expanding Maps on Smooth Manifold . . . . . . . . . . . . . . . . . . . . 8.7 Topological Exactness . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.8 Stationary Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

75 75 80

4.4 4.5

67

81 84 87 89 89 90

9 Real Analyticity of Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 93 9.1 The Pressure as a Function of a Parameter . . . . . .. . . . . . . . . . . . . . . . . . . . 93 9.2 Real Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 97 9.3 Canonical Complexification . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 100 9.4 The Pressure is Real-Analytic . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 103 9.5 Derivative of the Pressure . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 106 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 109 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 111

Chapter 1

Introduction

In this monograph we develop the thermodynamical formalism for measurable expanding random mappings. This theory is then applied in the context of conformal expanding random mappings where we deal with the fractal geometry of fibers. Distance expanding maps have been introduced for the first time in Ruelle’s monograph [25]. A systematic account of the dynamics of such maps, including the thermodynamical formalism and the multifractal analysis, can be found in [24]. One of the main features of this class of maps is that their definition does not require any differentiability or smoothness condition. Distance expanding maps comprise symbol systems and expanding maps of smooth manifolds but go far beyond. This is also a characteristic feature of our approach. We first define measurable expanding random maps. The randomness is modeled by an invertible ergodic transformation  of a probability space .X; B; m/. We investigate the dynamics of compositions Txn D T n1 .x/ ı ::: ı Tx ;

n  1;

where the Tx W Jx ! J.x/ (x 2 X ) is a distance expanding mapping. These maps are only supposed to be measurably expanding in the sense that their R expanding constant is measurable and a.e. x > 1 or log x dm.x/ > 0. In so general setting we build the thermodynamical formalism for arbitrary H¨older continuous potentials 'x . We show, in particular, the existence, uniqueness and ergodicity of a family of Gibbs measures fx gx2X . Following ideas of Kifer [17], these measures are first produced in a pointwise manner and then we carefully check their measurability. Often in the literature all fibers are contained in one and the same compact metric space and symbolic dynamics plays a prominent role. Our approach does not require the fibers to be contained in one metric space neither we need any Markov partitions or, even auxiliary, symbol dynamics. Our results contain those in [5] and in [17] (see also the expository article [20]). Throughout the entire monograph where it is possible we avoid, in hypotheses, absolute constants. Our feeling is that in the context of random systems all (or at least V. Mayer et al., Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry, Lecture Notes in Mathematics 2036, DOI 10.1007/978-3-642-23650-1 1, © Springer-Verlag Berlin Heidelberg 2011

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1 Introduction

as many as possible) absolute constants appearing in deterministic systems should become measurable functions. With this respect the thermodynamical formalism developed in here represents also, up to our knowledge, new achievements in the theory of random symbol dynamics or smooth expanding random maps acting on Riemannian manifolds. Unlike recent trends aiming to employ the method of Hilbert metric (as for example in [12, 19, 26, 27]) our approach to the thermodynamical formalism stems primarily from the classical method presented by Bowen in [7] and undertaken by Kifer [17]. Developing it in the context of random dynamical systems we demonstrate that it works well and does not lead to too complicated (at least to our taste) technicalities. The measurability issue mentioned above results from convergence of the Perron–Frobenius operators. We show that this convergence is exponential, which implies exponential decay of correlations. These results precede investigations of a pressure function x 7! Px .'/ which satisfies the property Z e 'x dx ; .x/ .Tx .A// D e Px .'/ A

where A is any measurable set such that Tx jA is injective. The integral, against the measure m on the base X , of this function is a central parameter EP .'/ of random systems called the expected pressure. If the potential ' depends analytically on parameters, we show that the expected pressure also behaves real analytically. We would like to mention that, contrary to the deterministic case, the spectral gap methods do not work in the random setting. Our proof utilizes the concept of complex cones introduced by Rugh in [26], and this is the only place, where we use the projective metric. We then apply the above results mainly to investigate fractal properties of fibers of conformal random systems. They include Hausdorff dimension, Hausdorff and packing measures, as well as multifractal analysis. First, we establish a version of Bowen’s formula (obtained in a somewhat different context in [6]) showing that the Hausdorff dimension of almost every fiber Jx is equal to h, the only zero of the expected pressure EP .'t /, where 't D t log jf 0 j and t 2 R. Then we analyze the behavior of h-dimensional Hausdorff and packing measures. It turned out that the random dynamical systems split into two categories. Systems from the first category, rather exceptional, behave like deterministic systems. We call them, therefore, quasi-deterministic. For them the Hausdorff and packing measures are finite and positive. Other systems, called essentially random, are rather generic. For them the h-dimensional Hausdorff measure vanishes while the h-packing measure is infinite. This, in particular, refutes the conjecture stated by Bogensch¨utz and Ochs in [6] that the h-dimensional Hausdorff measure of fibers is always positive and finite. In fact, the distinction between the quasi-deterministic and the essentially random systems is determined by the behavior of the Birkhoff sums Pxn .'/ D P n1 .x/ .'/ C ::: C Px .'/

1 Introduction

3

of the pressure function for potential 'h D h log jf 0 j. If these sums stay bounded then we are in the quasi-deterministic case. On the other hand, if these sums are neither bounded below nor above, the system is called essentially random. The behavior of Pxn , being random variables defined on X , the base map for our skew product map, is often governed by stochastic theorems such as the law of the iterated logarithm whenever it holds. This is the case for our primary examples, namely conformal DG-systems and classical conformal random systems. We are then in position to state that the quasi-deterministic systems correspond to rather exceptional case where the asymptotic variance  2 D 0. Otherwise the system is essential. The fact that Hausdorff measures in the Hausdorff dimension vanish has further striking geometric consequences. Namely, almost all fibers of an essential conformal random system are not bi-Lipschitz equivalent to any fiber of any quasideterministic or deterministic conformal expanding system. In consequence almost every fiber of an essentially random system is not a geometric circle nor even a piecewise analytic curve. We then show that these results do hold for many explicit random dynamical systems, such as conformal DG-systems, classical conformal random systems, and, perhaps most importantly, Br¨uck and B¨uger polynomial systems. As a consequence of the techniques we have developed, we positively answer the question of Br¨uck and B¨uger (see [9] and Question 5.4 in [8]) of whether the Hausdorff dimension of almost all naturally defined random Julia set is strictly larger than 1. We also show that in this same setting the Hausdorff dimension of almost all Julia sets is strictly less than 2. Concerning the multifractal spectrum of Gibbs measures on fibers, we show that the multifractal formalism is valid, i.e. the multifractal spectrum is Legendre conjugated to a temperature function. As usual, the temperature function is implicitly given in terms of the expected pressure. Here, the most important, although perhaps not most strikingly visible, issue is to make sure that there exists a set Xma of full measure in the base such that the multifractal formalism works for all x 2 Xma . If the system is in addition uniformly expanding then we provide real analyticity of the pressure function. This part is based on work by Rugh [27] and it is the only place where we work with the Hilbert metric. As a consequence and via Legendre transformation we obtain real analyticity of the multifractal spectrum. Random transformations have already a long history and the present manuscript does, by no means, cover all its topics. Some of them can be found in Arnold’s book [1] and in Kifer and Liu’s chapter in [20]. Let us however mention some interesting results. Denote by Mm1 .T / the set of T -invariant measures from Mm1 .J /. Let  2 Mm1 .T /. The fiber entropy hr .T / of  is given as follows. If R D fR1 ; R2 ; : : : ; Rn g is a finite partition of J , then by Rx we denote the partition of J given by sets Rxk WD Rk \ Jx , k D 1; : : : ; n. Then hr .T /

1 WD sup lim Hx n!1 n R

n1 _ i D0

! R i .x/ :

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1 Introduction

In fairly general random setting one can prove that this limit m-almost surely exists (see, e.g. [4]). Moreover, there is a following relation between the fiber entropy and the topological pressure called Variational Principle (see, e.g. [2, 4, 14]) Z EP .'/ WD sup

2.T /

 'd C hr .T / :

It is also worth noting that in many cases the entropy and averaged positive Lyapunov exponents can satisfy so called Margulis–Ruelle inequality (see, e.g. [3]) or Pesin formula (see, e.g. [21]). We also refer the reader to [22]. We would like to thank Yuri I. Kifer for his remarks which improved the final version of this monograph.

Chapter 2

Expanding Random Maps

For the convenience of the reader, we first give some introductory examples. In the remaining part of this chapter we present the general framework of expanding random maps.

2.1 Introductory Examples Before giving the formal definitions of expanding random maps, let us now consider some typical examples. The first one is a known random version of the Sierpi´nski gasket (see, for example [15]). Let  D .A; B; C / be a triangle with vertexes A; B; C and choose a 2 .A; B/, b 2 .B; C / and c 2 .C; A/. Then we can associate to x D .a; b; c/ a map fx W .A; a; c/ [ .a; B; b/ [ .b; C; a/ ! ; such that the restriction of fx to each one of the three subtriangles is a affine map onto . The map fx is nothing else than the generator of a deterministic Sierpi´nski gasket. Note that this map can be made continuous by identifying the vertices A; B; C (Fig. 2.1). Now, suppose x1 D .a1 ; b1 ; c1 /; x2 D .a2 ; b2 ; c2 /; ::: are chosen randomly which, for example, may mean that they form sequences of three dimensional independent and identically distributed (i.i.d.) random variables. Then they generate compact sets \ Jx1 ;x2 ;x3 ;::: D .fxn ı ::: ı fx1 /1 ./ n1

.Jx2 ;x3 ;::: / D called random Sierpi´nski gaskets having the invariance property fx1 1 Jx1 ;x2 ;x3 ;::: . For a little bit simpler example of random Cantor sets we refer the reader to Sect. 5.3. In that example we provide a more detailed analysis of such random sets. V. Mayer et al., Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry, Lecture Notes in Mathematics 2036, DOI 10.1007/978-3-642-23650-1 2, © Springer-Verlag Berlin Heidelberg 2011

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Fig. 2.1 Two different generators of Sierpi´nski gaskets

Fig. 2.2 A generator of degree 6

Such examples admit far going generalizations. First of all, we will consider much more general random choices than i.i.d. ones. We model randomness by taking a probability space .X; B; m/ along with an invariant ergodic transformation  W X ! X . This point of view was up to our knowledge introduced by the Bremen group (see [1]). Another point is that the maps fx that generate the random Sierpi´nski gasket have degree 3. In the sequel of this manuscript, we will allow the degree dx of all maps to be different (see Fig. 2.2) and only require that the function x 7! log.dx / is measurable. Finally, the above examples are all expanding with an expanding constant x   > 1 : As already explained in the introduction, the present monograph concerns random maps for which the expanding constants x can be arbitrarily close to one. Furthermore, using an inducing procedure, we will even weaken this to the maps that are only expanding in the mean (see Chap. 7).

2.1 Introductory Examples

7

The example of random Sierpi´nski gasket is not conformal. Random iterations of rational functions or of holomorphic repellers are typical examples of conformal random dynamical systems. Random iterations of the quadratic family fc .z/ D z2 C c have been considered, for example, by Br¨uck and B¨uger among others (see [8] and [9]). In this case, one chooses randomly a sequence of bounded parameters c D .c1 ; c2 ; :::/ and considers the dynamics of the family Fc1 ;:::;cn D fcn ı fcn1 ı ::: ı fc1 ;

n  1:

This leads to the dynamical invariant sets Kc D fz 2 CI Fc1 ;:::;cn .z/ 6! 1g and Jc D @Kc : The set Kc is the filled in Julia set and Jc the Julia set associated to the sequence c. The simplest case is certainly the one when we consider just two polynomials z 7! z2 C 1 and z 7! z2 C 2 and we build a random sequence out of them. Julia sets that come out of such a choice are presented in Fig. 2.3. Such random Julia sets are different objects as compared to the Julia sets for deterministic iteration of quadratic polynomials. But not only the pictures are different and intriguing, we

Fig. 2.3 Some quadratic random Julia sets

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2 Expanding Random Maps

will see in Chap. 5 that also generically the fractal properties of such Julia sets are fairly different as compared with the deterministic case even if the dynamics are uniformly expanding. In Chap. 8 we present a more general class of examples and we explain their dynamical and fractal features.

2.2 Preliminaries Suppose .X; B; m; / is a measure preserving dynamical system with invertible and ergodic map  W X ! X which is referred to as the base map. Assume further that .Jx ; x /, x 2 X , are compact metric spaces normalized in size by diamx .Jx /  1. Let [ fxg  Jx : (2.1) J D x2X

We will denote by Bx .z; r/ the ball in the space .Jx ; %x / centered at z 2 Jx and with radius r. Frequently, for ease of notation, we will write B.y; r/ for Bx .z; r/, where y D .x; z/. Let Tx W Jx ! J.x/ ;

x 2 X;

be continuous mappings and let T W J ! J be the associated skew-product defined by T .x; z/ D ..x/; Tx .z//:

(2.2)

For every n  0 we denote Txn WD T n1 .x/ ı ::: ı Tx W Jx ! J n .x/ . With this notation one has T n .x; y/ D . n .x/; Txn .y//. We will frequently use the notation xn D  n .x/;

n 2 Z:

If it does not lead to misunderstanding we will identify Jx and fxg  Jx .

2.3 Expanding Random Maps A map T W J ! J is called a expanding random map if the mappings Tx W Jx ! J.x/ are continuous, open, and surjective, and if there exist a function  W X ! RC , x 7! x , and a real number  > 0 such that following conditions hold.   Uniform Openness. Tx .Bx .z; x //  B.x/ Tx .z/;  for every .x; z/ 2 J . Measurably Expanding. There exists a measurable function  W X ! .1; C1/, x 7! x such that, for m-a.e. x 2 X , %.x/ .Tx .z1 /; Tx .z2 //  x %x .z1 ; z2 /

whenever

%.z1 ; z2 / < x ; z1 ; z2 2 Jx :

2.4 Uniformly Expanding Random Maps

9

Measurability of the Degree. The map x 7! deg.Tx / WD supy2J.x/ # Tx1 .fyg/ is measurable. Topological Exactness. There exists a measurable function x 7! n .x/ such that n .x/

Tx 

.Bx .z; // D J n .x/ .x/

for every z 2 Jx and a.e. x 2 X :

(2.3)

Note that the measurably expanding condition implies that Tx jB.z;x / is injective for every .x; z/ 2 J . Together with the compactness of the spaces Jx it yields the numbers deg.Tx / to be finite. Therefore the supremum in the condition of measurability of the degree is in fact a maximum. In this work we consider two other classes of random maps. The first one consists of the uniform expanding maps defined below. These are expanding random maps with uniform control of measurable “constants”. The other class we consider is composed of maps that are only expanding in the mean. These maps are defined like the expanding random maps above excepted that the uniform openness and the measurable expanding conditions are replaced by the following weaker conditions (see Chap. 7 for detailed definition). 1. All local inverse branches do exist. 2. The function  in the measurable expanding condition is allowed to have values in .0; 1/ but subjects only the condition Z X

log x d m > 0:

We employ an inducing procedure to expanding in the mean random maps in order to reduce then to the case of random expanding maps. This is the content of Chap. 7 and the conclusion is that all the results of the present work valid for expanding random maps do also hold for expanding in the mean random maps.

2.4 Uniformly Expanding Random Maps Most of this paper and, in particular, the whole thermodynamical formalism is devoted to measurable expanding systems. The study of fractal and geometric properties (which starts with Chap. 5), somewhat against our general philosophy, but with agreement with the existing tradition (see for example [5,12,17]), we will work mostly with uniform and conformal systems (the later are introduced in Chap. 5). A expanding random map T W J ! J is called uniformly expanding if –  WD infx2X x > 1, – deg.T / WD supx2X deg.Tx / < 1, – n WD supx2X n .x/ < 1.

10

2 Expanding Random Maps

2.5 Remarks on Expanding Random Mappings The conditions of uniform openness and measurably expanding imply that, for every y D .x; z/ 2 J there exists a unique continuous inverse branch Ty1 W B.x/ .T .y/; / ! Bx .y; x / of Tx sending Tx .z/ to z. By the measurably expanding property we have %.Ty1 .z1 /; Ty1 .z2 //  x1 %.z1 ; z2 / and

  for z1 ; z2 2 B.x/ T .y/; 

(2.4)

Ty1 .B.x/ .T .y/; //  Bx .y; x1 /  Bx .y; /:

Hence, for every n  0, the composition 1 n Tyn D Ty1 ı TT1 .y/ ı : : : ı TT n1 .y/ W B n .x/ .T .y/; / ! Bx .y; /

(2.5)

is well defined and has the following properties: Tyn W B n .x/ .T n .y/; / ! Bx .y; / is continuous, T n ı Tyn D IdjB n .x/ .T n .y/;/ , Tyn .Txn .z// D z   and, for every z1 ; z2 2 B n .x/ T n .y/;  , %.Tyn .z1 /; Tyn .z2 //  .xn /1 %.z1 ; z2 /;

(2.6)

where xn D x .x/     n1 .x/ : Moreover, Txn .B n .x/ .T n .y/; //  Bx .y; .xn /1 /  Bx .y; /:

(2.7)

Lemma 2.1 For every r > 0, there exists a measurable function x 7! nr .x/ such that a.e. Txnr .x/ .Bx .z; r// D J nr .x/ .x/ for every z 2 Jx : (2.8) Moreover, there exists a measurable function j W X ! N such that a.e. we have .Bxj.x/ .z; // D Jx Txj.x/ j.x/

for every z 2 Jxj.x/ :

(2.9)

Proof. In order to prove the first statement, consider 0 > 1 and let F be the set of x 2 X such that x  0 . If 0 is sufficiently close to 1, then m.F / > 0. In the following section such a set will be called essential. In that section we also

2.6 Visiting Sequences

11

0 0 associate to such an essential set a set XCF (see (2.10)). Then for x 2 XCF , the n 1 limit limn!1 .x / D 0. Define 0 XCF ;k WD fx 2 XCF W .xk /1  < rg:

S 0 . By measurability of x 7! x , Then XCF ;k  XCF ;kC1 and k2N XCF ;k D XCF XCF ;k is a measurable set. Hence the function 0 3 x 7! nr .x/ WD minfk W x 2 XCF ;k g C n .x/ XCF

is finite and measurable. By (2.7) and (2.3), Txnr .x/ .Bx .z; r// D J nr .x/ .x/ : In order to prove the existence of a measurable function j W X ! N define measurable sets [ 0 0 X;n WD fx 2 X W n .x/  ng, X;n WD  n .X;n / and X0 D X;n : n2N

Then the map

0 X0 3 x 7! j.x/ WD minfn 2 N W x 2 X;n g

satisfies (2.9) for x 2 X0 . Since m. n .X;n // D m.X;n / % 1 as n tends to 1 we t u have m.X0 / D 1.

2.6 Visiting Sequences Let F 2 F be a set with a positive measure. Define the sets VCF .x/ WD fn 2 N W  n .x/ 2 F g

and VF .x/ WD fn 2 N W  n .x/ 2 F g:

The set VCF .x/ is called visiting sequence (of F at x). Then the set VF .x/ is just a visiting sequence for  1 and we also call it backward visiting sequence. By nj .x/ we denote the j th-visit in F at x. Since m.F / > 0, by Birkhoff’s Ergodic Theorem we have that 0 0 m.XCF / D m.XF / D 1; where o n nj C1 .x/ 0 XCF D1 WD x 2 X W VCF .x/ is infinite and lim j !1 nj .x/

(2.10)

12

2 Expanding Random Maps

0 0 0 and XF is defined analogously. The sets XCF and XF are respectively called forward and backward visiting for F . Let .x; n/ be a formula which depends on x 2 X and n 2 N. We say that .x; n/ holds in a visiting way, if there exists F with m.F / > 0 such that, for m-a.e. 0 x 2 XCF and all j 2 N, the formula . nj .x/; nj .x// holds, where .nj .x//1 j D0 is the visiting sequence of F at x. We also say that .x; n/ holds in a exhaustively visiting way, if there exists a family Fk 2 F with limk!1 m.Fk / D 1 such that, 0 for all k, m-a.e. x 2 XCF , and all j 2 N, the formula . nj .x/; nj .x// holds, k 1 where .nj .x//j D0 is the visiting sequence of Fk at x. Now, let fn W X ! R be a sequence of measurable functions. We write that

s-lim fn D f; n!1

if that there exists a family Fk 2 F with limk!1 m.Fn / D 1 such that, for all k 0 and m-a.e. x 2 XCF and all j 2 N, k lim fnj .x/ D f .x/;

j !1

where .nj /1 j D0 is the visiting sequence of Fk at x. Suppose that g1 ; : : : ; gk W X ! R is a finite collection of measurable functions and let b1 ; : : : ; bn be a collection of real numbers. Consider the set F WD

k \

gi1 ..1; bi /:

i D1

If m.F / > 0, then F is called essential for g1 ; : : : ; gk with constants b1 ; : : : ; bn (or just essential, if we do not want explicitly specify functions and numbers). Note that by measurability of the functions g1 ; : : : ; gk , for every " > 0 we can always find finite numbers b1 ; : : : ; bn such that the essential set F for g1 ; : : : ; gk with constants b1 ; : : : ; bn has the measure m.F /  1  ".

2.7 Spaces of Continuous and H¨older Functions We denote by C .Jx / the space of continuous functions gx W Jx ! R and by C .J / the space of functions g W J ! R such that, for a.e. x 2 X , x 7! gx WD gjJx 2 C .Jx /. The set C .J / contains the subspaces C 0 .J / of functions for which the function x 7! kgx k1 is measurable, and C 1 .J / for which the integral Z kgk1 WD

X

kgx k1 d m.x/ < 1:

2.8 Transfer Operator

13

Now, fix ˛ 2 .0; 1 . By H ˛ .Jx / we denote the space of H¨older continuous functions on Jx with an exponent ˛. This means that 'x 2 H ˛ .Jx / if and only if 'x 2 C .Jx / and v.'x / < 1 where v˛ .'x / WD inffHx W j'.z1 /  '.z2 /j  Hx %˛x .z1 ; z2 /g;

(2.11)

where the infimum is taken over all z1 ; z2 2 Jx with %.z1 ; z2 /  . A function ' 2 C 1 .J / is called H¨older continuous with an exponent ˛ provided that there exists a measurable function H W X ! Œ1; C1/, x 7! Hx , such that log H 2 L1 .m/ and such that v˛ .'x /  Hx for a.e. x 2 X . We denote the space of all H¨older functions with fixed ˛Sand H by H ˛ .J ; H / and the space of all ˛-H¨older functions by H ˛ .J / D H 1 H ˛ .J ; H /.

2.8 Transfer Operator For every function g W J ! C and a.e. x 2 X let Sn gx D

n1 X

gx ı Txj ;

(2.12)

j D0

P j older and, if g W X ! C, then Sn g D n1 j D0 g ı  . Let ' be a function in the H¨ ˛ space H .J /. For every x 2 X , we consider the transfer operator Lx D L';x W C .Jx / ! C .J.x/ / given by the formula Lx gx .w/ D

X

gx .z/e 'x .z/ ;

w 2 J.x/ :

(2.13)

Tx .z/Dw

It is obviously a positive linear operator and it is bounded with the norm bounded above by kLx k1  deg.Tx / exp.k'k1 /: (2.14) This family of operators gives rise to the global operator L W C .J / ! C .J / defined as follows: .L g/x .w/ D L 1 .x/ g 1 .x/ .w/: For every n > 1 and a.e. x 2 X , we denote Lxn WD L n1 .x/ ı ::: ı Lx W C .Jx / ! C .J n .x/ /: Note that Lxn gx .w/ D

X

gx .z/e Sn 'x .z/ ,

w 2 J n .x/ ;

(2.15)

z2Txn .w/

where Sn 'x .z/ has been defined in (2.12). The dual operator Lx maps C  .J.x/ / into C  .Jx /.

14

2 Expanding Random Maps

2.9 Distortion Properties Lemma 2.2 Let ' 2 H ˛ .J ; H /, let n  1 and let y D .x; z/ 2 J . Then jSn 'x .Tyn .w1 //  Sn 'x .Tyn .w2 //j  %˛ .w1 ; w2 /

n1 X j D0

nj

H j .x/ . j .x/ /˛

for all w1 ; w2 2 B.Txn .z/; /. Proof. We have by (2.6) and H¨older continuity of ' that n1 X

jSn 'x .Tyn .w1 //  Sn 'x .Tyn .w2 //j 

j'x .Txj .Tyn .w1 ///  'x .Txj .Tyn .w2 ///j

j D0

D

n1 X j D0



n1 X j D0

ˇ ˇ ˇ ˇ .nj / .nj / .w1 //  'x .T j .w2 //ˇ ˇ'x .T j Tx .y/

 .nj /  .nj / %˛ T j .w1 /; T j .w2 / H j .x/ ; Tx .x/

hence jSn 'x .Tyn .w1 //Sn 'x .Tyn .w2 //j  %˛ .w1 ; w2 / Set Qx WD Qx .H / D

1 X j D1

Tx .y/

Tx .x/

Pn1

nj ˛ . j D0 H j .x/ . j .x/ /

H j .x/ .jj .x/ /˛ :

t u

(2.16)

Lemma 2.3 The function x 7! Qx is measurable and m-a.e. finite. Moreover, for every ' 2 H ˛ .J ; H /, jSn 'x .Tyn .w1 //  Sn 'x .Tyn .w2 //j  Q n .x/ %˛ .w1 ; w2 / for all n  1, a.e. x 2 X , every z 2 Jx and w1 ; w2 2 B.T n .z/; / and where again y D .x; z/. Proof. The measurability of Qx follows directly from (2.16). Because of Lemma 2.2 we are only leftRto show that Qx is m-a.e. finite. Let  be a positive real number less or equal to log x d m.x/. Then, using Birkhoff’s Ergodic Theorem for  1 , we get that j 1 1X lim inf log  j .x/   j !1 j kD0

for m-a.e. x 2 X . Therefore, there exists a measurable function C W X ! Œ1; C1/ m-a.e. finite such that C1 .x/e j=2  jj C1 .x/ for all j  0 and a.e. x 2 X .

2.9 Distortion Properties

15

Moreover, since log H 2 L1 .m/ it follows again from Birkhoff’s Ergodic Theorem that 1 lim log H j .x/ D 0 m-a:e: j !1 j There thus exists a measurable function CH W X ! Œ1; C1/ such that H j .x/  CH .x/e j˛=4

and H j .x/  CH .x/e j˛=4

(2.17)

for all j  0 and a.e. x 2 X . Then, for a.e. x 2 X , all n  0 and all a  j  n  1, we have H j .x/ D H .nj / . n .x//  CH . n .x//e .nj /˛=4 : Therefore, still with xn D  n .x/, Qxn D

n1 X j D0

Hxj .xnj /˛ j

 C˛ .xn1 /CH .xn /



n1 X

CH .xn /e .nj /˛=4 C˛ .xn1 /e ˛.nj /=2

j D0 n1 X

e ˛.nj /=4  C˛ .xn1 /CH .xn /.1  e ˛=4 /1 :

j D0

Hence

Qx  C˛ . 1 .x//CH .x/.1  e ˛=4 /1 < C1: t u

•

Chapter 3

The RPF-Theorem

We now establish a version of Ruelle–Perron–Frobenius (RPF) Theorem along with a mixing property. Notice that this quite substantial fact is proved without any measurable structure on the space J . In particular, we do not address measurability issues of x and qx . In order to obtain this measurability we will need and we will impose a natural measurable structure on the space J . This will be done in the next chapter.

3.1 Formulation of the Theorems Let T W J ! J be a expanding random map. Denote by M 1 .Jx / the set of all Borel probability measures on Jx . A family of measures fx gx2X such that x 2 M 1 .Jx / is called T -invariant if x ı Tx1 D .x/ for a.e. x 2 X . This chapter is devoted to the thermodynamical formalism. The main results proved here are listed below. Theorem 3.1 Let ' 2 H ˛ .J / and let L D L' be the associated transfer operator. Then the following holds. 1. There exists a unique family of probability measures x 2 M .Jx / such that Lx .x/ D x x

where

x D .x/ .Lx 1/ m-a:e:

(3.1)

2. There exists a unique function q 2 C 0 .J / such that m-a.e. Lx qx D x q.x/

and

x .qx / D 1:

(3.2)

Moreover, qx 2 H ˛ .Jx / for a.e. x 2 X . 3. The family of measures fx WD qx x gx2X is T -invariant. V. Mayer et al., Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry, Lecture Notes in Mathematics 2036, DOI 10.1007/978-3-642-23650-1 3, © Springer-Verlag Berlin Heidelberg 2011

17

18

3 The RPF-Theorem

Theorem 3.2 1. Let 'Ox D 'x C log qx  log q.x/ ı T  log x : Denote LO WD L'O . Then, for a.e. x 2 X and all gx 2 C.Jx /, LOxn gx ! n!1

Z gx qx dx :

2. Let 'Q x D 'x  log x . Denote LQ WD L'Q . There exist a constant B < 1 and a measurable function A W X ! .0; 1/ such that for every function g 2 C 0 .J / with gx 2 H ˛ .Jx / there exists a measurable function Ag W X ! .0; 1/ for which Z  g n .x/ d n .x/ qx k1  Ag . n .x//A.x/B n k.LQ n g/x  for a.e. x 2 X and every n  1. 3. There exists B < 1 and a measurable function A0 W X ! .0; 1/ such that for every f n .x/ 2 L1 . n .x/ / and every gx 2 H ˛ .Jx /, ˇ  ˇ  ˇx .f n .x/ ı T n /gx   n .x/ .f n .x/ /x .gx /ˇ x Z v˛ .gx qx /  n 0 n jgx jdx C 4   n .x/ .jf n .x/ j/A . .x// B : Qx A collection of measures fx gx2X such that x 2 M 1 .Jx / is called a Gibbs family for ' 2 H ˛ .J / provided that there exists a measurable function D' W X ! Œ1; C1/ and a function x 7! Px , called the pseudo-pressure function, such that .D' .x/D' . n .x///1 

x .Tyn .B.T n .y/; /// exp.Sn '.y/  Sn Px /

 D' .x/D' . n .x//

(3.3)

for every n  0, a.e. x 2 X and every z 2 Jx and with y D .x; z/. Towards proving uniqueness type result for Gibbs families we introduce the following concept. Notice that in the case of random compact subsets of a Polish space (see Sect. 4.5) this condition always holds (see Lemma 4.11). Measurability of Cardinality of Covers. There exists a measurable function X 3 x 7! ax 2 N such that for almost every x 2 X there exists a finite sequence S w1x ; : : : ; waxx 2 Jx such that aj xD1 B.wjx ; / D Jx : Theorem 3.3 The collections fx gx2X and fx gx2X are Gibbs families. Moreover, if J satisfies the condition of measurability of cardinality of covers and if fx0 gx2X is a Gibbs family, then x0 and x are equivalent for almost every x 2 X .

3.3 Transfer Dual Operators

19

3.2 Frequently used Auxiliary Measurable Functions Some technical measurable functions appear throughout the paper so frequently that, for convenience of the reader, we decided to collect them in this section together. However, the reader may skip this part now without any harm and come back to it when it is appropriately needed. First, define 1  D .x/ WD deg Txn exp.2kSn 'x k1 / (3.4) with n D n .x/ being the index given by the topological exactness condition (cf. (2.3)). Then, let j D j.x/ be the number given by Lemma 2.1 and define ˚  C' .x/ WD e Qxj deg.Txjj / max exp.2kSk 'xk k1 / W 0  k  j  1:

(3.5)

Now let s > 1. Put

and

Cmin .x/ WD e sQx e kSj 'xj k1  1

(3.6)

  Cmax .x/ WD e sQx deg Txn exp.2kSn 'x k1 /;

(3.7)

where n WD n .x/. Then we define   1  exp  .s  1/Hx1 x˛ r˛ Cmin .x/ 1  inf : ˇx .s/ WD C' .x/ r2.0; 1  exp.2sQx r ˛ / Since by (2.16)

(3.8)

sQx D sQx1 x˛ C sHx1 x˛ ; 1 1

(3.9)

.sQx1 C Hx1 /x˛ D sQx  .s  1/Hx1 x˛ : 1 1

(3.10)

This, together with (3.5) and (3.6), gives us that 0 < ˇx .s/ D

Cmin .x/ .s  1/Hx1 x˛ Cmin .x/ 1 <  1: C' .x/ 2sQx C' .x/

3.3 Transfer Dual Operators In order to prove Theorem 3.1 we fix a point x0 2 X and, as the first step, we reduce the base space X to the orbit Ox0 D f n .x0 /; n 2 Zg:

20

3 The RPF-Theorem

The motivation for this is that then we can deal with a sequentially topological compact space on which the transfer (or related) operators act continuously. Our conformal measure then can be produced, for example, by the methods of the fixed point theory, similarly as in the deterministic case. Removing a set of measure zero, if necessary, we may assume that this orbit is chosen so that all the involved measurable functions are defined and finite on the points of Ox0 . For every x 2 Ox0 , let 'x 2 C .Jx / be the continuous potential of the transfer operator Lx W C.Jx / ! C.J.x/ / which has been defined in (2.13). Proposition 3.4 There exists probability measures x 2 M.Jx / such that Lx .x/ D x x where

for every x 2 Ox0 ;

x WD Lx ..x/ /.1/ D .x/ .Lx 1/:

(3.11)

Proof. Let C  .Jx / be the dual space of C .Jx / equipped with the weak topology. Consider the product space Y D.Ox0 / WD C  .Jx / x2Ox0

with the product topology. This is a locally convex topological space and the set Y

P.Ox0 / WD

M 1 .Jx /

x2Ox0

is a compact subset of D.Ox0 /. A simple observation is that the map x W M 1 .J.x/ / ! M 1 .Jx / defined by x ..x/ / D

Lx .x/ Lx .x/ .1/

is weakly continuous. Consider then the global map  W P.Ox0 / ! P.Ox0 / given by    D .x /x2Ox0 7!  ./ D x .x/ x2O : x0

Weak continuity of the x implies continuity of  with respect to the coordinate convergence. Since the space P.Ox0 / is a compact subset of a locally convex topological space, we can apply the Schauder–Tychonoff fixed point theorem to get  2 P.Ox0 / fixed point of  , i.e. Lx .x/ D x x for every x 2 Ox0 .

where x D Lx .x/ .1/ D .x/ .Lx .1// t u

3.3 Transfer Dual Operators

21

Remark 3.5 The relation (3.11) implies inf e 'x .y/  x  kLx 1k1 :

(3.12)

y2Jx

A straightforward adaptation of the proof of Proposition 2.2 in [13] leads to the following, to Proposition 3.4 equivalent, characterization of Gibbs states: if Txn jA is injective, then Z  n .x/ .Txn .A// D nx e Sn ' dx : (3.13) A

Here is one more useful estimate. Lemma 3.6 For every x 2 Ox0 and n  1,   inf exp Sn 'x .z/ 

z2Jx

  nx  sup exp Sn 'x .z/ : n deg.Tx / z2Jx

(3.14)

Moreover, for every z 2 Jx and every r > 0, x .B.z; r//  D.x; r/;

(3.15)

where  1  D.x; r/ WD deg.TxN / inf exp z2Jx

inf

a2B.z;r/

SN 'x .a/ 

 sup SN 'x .b/

b2B.z;r/

(3.16) with N D nr .x/ being the index given by Lemma 2.1. It follows that the set Jx is a topological support of x . In particular, with D .x/ defined in (3.4), x .B.z; //  D .x/:

(3.17)

Proof. The inequalities (3.14) immediately follow from  n .x/ .Lxn 1/ D ..Lxn /  n .x/ /.1/ D nx x .1/ D nx : Now fix an arbitrary z 2 Jx and r > 0. Put n D nr .x/ (see Lemma 2.1). Then, by (3.13), x .B.z; r//nx which implies (3.15).

sup e a2B.z;r/

Sn 'x .a/



nx

Z B.z;r/

e Sn 'x dx  1; t u

22

3 The RPF-Theorem

3.4 Invariant Density Consider now the normalized operator LQ given by LQx D 1 x Lx ;

x 2 X:

(3.18)

Proposition 3.7 For every x 2 Ox0 , there exists a function qx 2 H ˛ .Jx / such that Z Q Lx qx D q.x/ and qx dx D 1: Jx

In addition,

qx .z1 /  expfQx %˛ .z1 ; z2 /gqx .z2 /

for all z1 ; z2 2 Jx with %.z1 ; z2 /  , and 1=C' .x/  qx  C' .x/;

(3.19)

where C' was defined in (3.5). In order to prove this statement we first need a good uniform distortion estimate. Lemma 3.8 For all w1 ; w2 2 Jx and n  1 LQxnn 1.w1 / Lxn 1.w1 /  C' .x/; D nn Lxn 1.w2 / LQxn 1.w2 /

(3.20)

n

where C' is given by (3.5). If in addition %.w1 ; w2 /  , then LQxnn 1.w1 /  expfQx %˛ .w1 ; w2 /g: LQ n 1.w2 /

(3.21)

xn

Moreover, 1=C' .x/  LQxnn 1.w/  C' .x/

for every w 2 Jx and n  1:

(3.22)

Proof. First, (3.21) immediately follows from Lemma 2.3. Notice also that   exp Qx %˛ .w1 ; w2 /  exp Qx

(3.23)

since diam.Jx /  1. The global version of (3.20) can be proved as follows. If n D 0; : : : ; j.x/, then for every w1 ; w2 2 Jx , Lxnn 1.w1 / 

deg.Txnn / exp.kSn 'xn k1 / n Lxn 1.w2 /  C' .x/Lxnn 1.w2 /: exp.kSn 'xn k1 /

3.4 Invariant Density

23 j

Next, let n > j WD j.x/. Take w01 2 Txj .w1 / such that 0

1.w01 / D e Sj '.w1 / Lxnj n

sup j

y2Txj .w1 /

  e Sj '.y/ Lxnj 1.y/ n

.w2 / such that %xj .w01 ; w02 /  . Then, by (3.21) and (3.23), and w02 2 Txj j 0

1/.w1 /  deg.Txjj /e Sj '.w1 / Lxnj 1.w01 / Lxnn 1.w1 / D Lxjj .Lxnj n n 0

 deg.Txjj /e Sj '.w1 / e Qxj Lxnj 1.w02 /  C' .x/Lxnn 1.w2 /: n This shows (3.20). By Proposition 3.4 Z Jx

LQxnn .1/dx D

Z Jxn

1dxn D 1;

(3.24)

which implies the existence of w; w0 2 Jx such that LQxnn 1.w/  1 and LQxnn 1.w0 /  1. Therefore, by the already proved part of this lemma, we get (3.22). t u Proof. [Proof of Proposition 3.7] Let x 2 Ox0 . Then by Lemma 3.8, for every k  0 and all w1 ; w2 2 Jx with %.w1 ; w2 /  , we have that jLQxkk 1.w1 /  LQxkk 1.w2 /j  C' .x/2Qx %˛ .w1 ; w2 / and 1=C' .x/  LQxkk 1  C' .x/. It follows that the sequence qx;n WD

n1 1 X Qk Lxk 1; n

n1

kD0

is equicontinuous for every x 2 Ox0 . Therefore, there exists a sequence nj ! 1 such that qx;nj ! qx uniformly for every x of the countable set Ox0 . The functions qx have all the required properties. t u Let x WD qx x ;

(3.25)

and let LOx WD L';x O be the transfer operator with potential 'O x D 'x C log qx  log q.x/ ı Tx  log x : Then LOx gx D

1 q.x/

LQx .gx qx /

for every gx 2 L1 .x /:

(3.26)

24

3 The RPF-Theorem

Consequently

LOx 1x D 1.x/ :

(3.27)

Lemma 3.9 For all g.x/ 2 L1 ..x/ / D L1 ..x/ /, we have x .g.x/ ı Tx / D .x/ .g.x/ / :

(3.28)

Proof. From conformality of x (see Proposition 3.4) it follows that LOx ..x/ /.gx / D

Z J.x/

LOx .gx /d.x/ D 1 x

Z J.x/

.Lx gx qx /d.x/

O D 1 x Lx ..x/ /.gx qx / D x .gx qx / D x .gx /:

(3.29)

So, if fx  .g.x/ ı Tx / 2 L1 .x /, then     x .g.x/ ı Tx /fx D LOx ..x/ / .g.x/ ı Tx /fx      D .x/ LOx .g.x/ ı Tx /fx D .x/ g.x/ LOx .fx / (3.30) since

  LOx .g.x/ ı Tx /fx D g.x/ LOx .fx /:

Substituting in (3.30) 1x for fx and using (3.27), we get the lemma.

t u

Remark 3.10 In Chap. 4 we provide sufficient measurability conditions for these fiber measures x and x to be integrable to produce global measures projecting on X to m. The measure  defined by (4.2) is then T -invariant.

3.5 Levels of Positive Cones of H¨older Functions For s  1, set n ˛ sx D g 2 C .Jx / W g  0; x .g/ D 1 and g.w1 /  e sQx % .w1 ;w2 / g.w2 / o for all w1 ; w2 2 Jx with %.w1 ; w2 /   : (3.31) In fact all elements of sx belong to H ˛ .Jx /. This is proved in the following lemma. Lemma 3.11 If g  0 and if for all w1 ; w2 2 Jx with %.w1 ; w2 /   we have ˛ .w

g.w1 /  e sQx %

1 ;w2 /

g.w2 /;

3.5 Levels of Positive Cones of H¨older Functions

25

then v˛ .g/  sQx .exp.sQx  ˛ // ˛ jjgjj1 : Proof. Let w1 ; w2 2 Jx be such that %.w1 ; w2 /  . Without loss of generality we may assume that g.w1 / > g.w2 /. Then g.w1 / > 0 and therefore, because of our hypothesis, g.w2 / > 0. Hence, we get   jg.w1 /  g.w2 /j g.w1 / D  1  exp sQx %˛ .w1 ; w2 /  1: jg.z2 /j g.w2 / Then

jg.w1 /  g.w2 /j  sQx .exp.sQx  ˛ //%˛ .w1 ; w2 /jjgjj1 : t u

Hence the set

sx

is a level set of the cone defined in (9.13), that is sx D Cxs \ fg W x .g/ D 1g:

In addition, in the following lemma we show that this set is bounded in H ˛ .Jx /. Lemma 3.12 For a.e. x 2 X and every g 2 sx , we have kgk1  Cmax .x/, where Cmax is defined by (3.7). Proof. Let g 2 sx and let z 2 Jx . Since g  0 we get Z

Z g dx 

B.z;/

Jx

g dx D 1:

Therefore there exists b 2 B.z; / such that g.b/  1=x .B.z; //  1=D .x/; where the latter inequality is due to Lemma 3.6. Hence ˛ .b;z/

g.z/  e sQx %

g.b/ 

e sQx  Cmax .x/: D .x/

A kind of converse to Lemma 3.11 is given by the following. Lemma 3.13 If g 2 H ˛ .Jx / and g  0, then g C v˛ .g/=Qx 2 1x : x .g/ C v˛ .g/=Qx

t u

26

3 The RPF-Theorem

Proof. Consider the function h D g Cv˛ .g/=Qx . In order to get the inequality from the definition of sx , we take z1 ; z2 2 Jx . If h.z1 /  h.z2 / then this inequality is trivial. Otherwise h.z1 / > h.z2 /, and therefore h.z1 / jh.z1 /  h.z2 /j v˛ .g/%˛ .z1 ; z2 / D Qx %˛ .z1 ; z2 /: 1D  h.z2 / jh.z2 /j v˛ .g/=Qx

t u

An important property of the sets sx is their invariance with respect to the normalized operator LQx D 1 x Lx . Lemma 3.14 Let g 2 sx . Then, for every n  1,   LQxn g.w1 /  exp sQxn %˛ .w1 ; w2 / ; w1 ; w2 2 J n .x/ with %.w1 ; w2 /  : LQ n g.w2 / x

Consequently, LQxn . sx /  s n .x/ for a.e. x 2 X and all n  1. Notice that the constant function 1 2 sx for every s  1. For this particular function our distortion estimation was already proved in Lemma 3.8. Proof. [Proof of Lemma 3.14] Let g 2 sx , let % n .x/ .w1 ; w2 /  , and let z1 2 Txn .w1 /. For y D .x; z1 /, we put z2 D Tyn .w2 /. With this notation, we obtain from Lemma 2.2 and from the definition of sx that LQxn g.w1 /  LQ n g.w2 / x

  exp Sn 'x .z1 / g.z1 /   sup z1 2Txn .w1 / exp Sn 'x .z2 /g.z2 /

 n1   X ˛ ˛ H j .x/ .nj C sQx .xn /˛ :  exp % .w1 ; w2 / j .x/ / j D0

(3.32) Since Qx .xn /˛

C

n1 X j D0

˛ H j .x/ .nj D Q n .x/ ; j .x/ /

(3.33) t u

the lemma follows. Lemma 3.15 With Cmin the function given by (3.6) we have that LQxii g  Cmin .x/

for every i  j.x/ and g 2 sxi :

R Proof. First, let i D j.x/. Since Jx gdxi D 1 there exists a 2 Jxi such i that g.a/  1. By definition of j.x/, for any point w 2 Jx , there exists z 2 Txi .x/ \ B.a; /. Therefore i

3.6 Exponential Convergence of Transfer Operators

27

LQxii g.w/  e Si 'xi .z/ g.z/  e Si 'xi .z/ e sQx g.a/  Cmin .x/: The case i > j.x/ follows from the previous one, since LQxi

i j.x/

gxi 2 xj.x/ .

t u

3.6 Exponential Convergence of Transfer Operators Lemma 3.16 Let ˇx D ˇx .s/ (cf. (3.8)). Then for x 2 X , i  j.x/ and gxi 2 sxi , there exists hx 2 sx such that .LQ i g/x D LQxii gxi D ˇx qx C .1  ˇx /hx : Proof. By Lemma 3.15, we have LQxii gxi  Cmin .x/: Then by (3.19) for all w; z 2 Jx with %x .z; w/ < ,     ˇx exp sQx %˛x .z; w/ qx .z/  qx .w/        ˇx exp sQx %˛x .z; w/  exp  sQx %˛x .z; w/ qx .z/       ˇx exp sQx %˛x .z; w/  exp  sQx %˛x .z; w/ C' .x/      ˇx C' .x/ 1  exp.2sQx %˛x .z; w// exp sQx %˛x .z; w/      i ˛  exp sQx %˛x .z; w/  exp .sQx  Hx1 x˛ /% .z; w/ LQxi gxi .z/ x 1       exp sQx %˛ .z; w/  exp .sQx C Hx / ˛ %˛ .z; w/ LQ i gx .z/: x

1

1

x1 x

xi

i

Since by (3.32), for h 2 sx1 ,   LQx1 h.z/  exp .sQx1 C Hx1 /x˛ %˛ .z; w/ LQx1 h.w/; 1 x   %˛ .z; w/ LQxii gxi .w/: LQxii gxi .z/  exp .sQx1 C Hx1 /x˛ 1 x Then we have that     ˇx exp sQx %˛x .z; w/ qx .z/  qx .w/    exp sQx %˛x .z; w/ LQxii gxi .z/  LQxii gxi .w/ and then    LQxii gxi .w/  ˇx qx .w/  exp sQx %˛x .z; w/ LQxii gxi .z/  ˇx qx .z/ :

28

3 The RPF-Theorem

Moreover, ˇx qx  Cmin .x/  LQxii gxi . Hence the function hx WD

LQxii gxi  ˇx qx 1  ˇx

2 sx :

t u

We are now ready to establish the first result about exponential convergence. Proposition 3.17 Let s > 1. There exist B < 1 and a measurable function A W X ! .0; 1/ such that for a.e. x 2 X for every N  1 and gxN 2 sxN we have k.LQ N g/x  qx k1 D kLQ N gx  qx k1  A.x/B N : xN

N

Proof. Fix x 2 X . Put gn WD gxn , ˇn WD ˇxn , sn WD sxn , and .LQ n g/k WD 1 .LQ n g/xk . Let .i.n// Pn nD1 be a sequence of integers such that i.n C 1/  sj.xS.n/ /, where S.n/ D kD1 i.k/, n  1, and where S.0/ D 0. If gS.n/ 2 S.n/ , then Lemma 3.16 yields the existence of a function hn1 2 sS.n1/ such that   LQ i.n/ g

S.n1/

D ˇS.n1/ qS.n1/ C .1  ˇS.n1/ /hn1   D 1  .1  ˇS.n1/ / qS.n1/ C .1  ˇS.n1/ /hn1 :

Since 

LQi.n/Ci.n1/ g



   D LQi.n1/ LQi.n/ g

S.n2/



S.n2/

  D LQi.n1/ ˇS.n1/ qS.n1/ C .1  ˇS.n1/ /hn1 

S.n2/

 D ˇS.n1/ qS.n2/ C .1  ˇS.n1/ / LQi.n1/ .hn1 / S.n2/

it follows again from Lemma 3.16 that there is hn2 2 sS.n2/ such that 

LQ i.n/Ci.n1/ g

 S.n2/

D ˇS.n1/ qS.n2/   C.1  ˇS.n1/ / ˇS.n2/ qS.n2/ C.1  ˇS.n2/ /hn2   D 1  .1  ˇS.n2/ /.1  ˇS.n1/ / qS.n2/ C .1  ˇS.n2/ /.1  ˇS.n1/ /hn1 :

It follows now by induction that there exists h 2 sx such that 

LQ S.n/ g

 x

  D LQ i.n/C:::Ci.1/g D .1  ˘x.n/ /qx C ˘x.n/ h; x

3.6 Exponential Convergence of Transfer Operators .n/

where we set ˘x Therefore,

D

Qn1

kD0 .1

29

 ˇxS.k/ /: Since h 2 sx , we have jhj  Cmax .x/.

ˇ    ˇ ˇ ˇ Q S.n/ g  1  ˘x.n/ qx ˇ  Cmax .x/˘x.n/ ˇ L x

if gS.n/ 2 sS.n/ : (3.34)

By measurability of ˇ and j one can find M > 0 and J  1 such that the set G WD fx W ˇx  M and j.x/  J g

(3.35)

has a positive measure larger than or equal to 3=4. Now, we will show that for a.e. x 2 X there exists a sequence .nk /1 of non-negative integers such that n0 D 0, kD0 for k > 0, we have that xJ nk 2 G, and #fn W 0  n < nk and xJ n 2 Gg D k  1:

(3.36)

Indeed, applying Birkhoff’s Ergodic Theorem to the mapping  J we have that for almost every x 2 X , #f0  m  n  1 W  J m .x/ 2 Gg D E.1G jIJ /.x/; n!1 n lim

where E.1G jIJ / is the conditional expectation of 1G with the respect to the algebra IJ of  J -invariant sets. Note that if a measurable set A is  J -invariant, 1 j then set [jJD0  .A/ is  1 -invariant. If m.A/ > 0, then from ergodicity of  1 1 j we get that m.[jJD0  .A// D 1, and then by invariantness of the measure m, we conclude that m.A/  1=J . Hence we get that for almost every x the sequence nk is infinite and lim

k!1

3 k  : nk 4J

(3.37)

Fix N  0 and take l  0 so that J nl  N  J nlC1 . Define a finite sequence  l S.k/ kD1 by S.k/ WD J nk for k < l and S.l/ WD N , and observe that by (3.37), we have N  J nlC1  4J 2 l. Then (3.19) and (3.34) give ˇˇ  ˇˇ  ˇˇ ˇˇ jjLQxNN gxN  qx jj1  ˇˇLQxNN gxN  1  ˘x.l/ qx ˇˇ

1

C ˘x.l/ jjqx jj1

   .1  M /l C' .x/ C Cmax.x/ .

p

4J 2

  1  M /N C' .x/ C Cmax .x/ :

30

3 The RPF-Theorem

This establishes our proposition with B D

p

4J 2

1  M and



A.x/ WD maxf2Cmax.x/B J kx ; .C' .x/ C Cmax .x//g; where kx is a measurable function such that we have

k nk



1 2J

for all k  kx .

t u

From now onwards throughout this section, rather than the operator LQ , we consider the operator LOx defined previously in (3.26). Lemma 3.18 Let s > 1 and let g W J ! R be any function such that gx 2 H ˛ .Jx /. Then, with the notation of Proposition 3.17, we have ˇˇ  ˇˇ Z Z v˛ .gx qx /  ˇˇ ˇˇ O n n A. n .x//B n : gx dx 1ˇˇ  C' . .x// jgx jdx C4 ˇˇLx gx  1 Qx Proof. Fix s > 1. First suppose that gx  0. Consider the function hx D

gx C v˛ .gx /=Qx x

where x WD x .gx / C v˛ .gx /=Qx :

It follows from Lemma 3.13 that hx belongs to the set sx and from Proposition 3.17 we have ˇˇ ˇˇ Z  ˇˇ Qn ˇˇ gx dx q n .x/ ˇˇ ˇˇLx gx 

ˇˇ ˇˇ Z  v˛ .gx / n ˇˇ ˇˇ  ˇˇx LQxn hx  LQx 1x  gx dx q n .x/ ˇˇ 1 Qx ˇˇ ˇ  ˇ v˛ .gx / ˇˇ ˇˇ D ˇˇx LQxn hx  x q n .x/ C q n .x/  LQxn 1x ˇˇ 1 Qx  v˛ .gx /   x C A. n .x//B n : Qx

1

Then applying this inequality for gx qx and using (3.19) we get ˇˇ ˇˇ Z  ˇˇ O n ˇˇ gx dx 1 n .x/ ˇˇ ˇˇLx gx 

1

ˇˇ ˇˇ  ˇˇ

1 q n .x/

ˇˇ ˇˇ ˇˇ Z  ˇˇ ˇˇ Qn ˇˇ gx qx dx q n .x/ ˇˇ ˇˇ  ˇˇLx .gx qx / 

1

Z v˛ .gx qx /   C' . n .x// gx dx C 2 A. n .x//B n : Qx

So, we have the desired estimate for non-negative gx . In the general case we can use the standard trick and write gx D gxC  gx , where gxC ; gx  0. Then the lemma follows. t u The estimate obtained in Lemma 3.18 is a little bit inconvenient for it depends on the values of a measurable function, namely C' A, along the positive -orbit of x 2 X . In particular, it is not clear at all from this statement that the item 1 in Theorem 3.2 holds. In order to remedy this flaw, we prove the following proposition.

3.7 Exponential Decay of Correlations

31

Proposition 3.19 For m-a.e. x 2 X and every gx 2 C .Jx /, we have ˇˇ ˇˇ Z  ˇˇ O n ˇˇ gx dx 1 n .x/ ˇˇ ˇˇLx gx 

! 0:

1 n!1

Proof. First of all, we may assume without loss of generality that the function gx 2 H ˛ .Jx / since every continuous function is a limit of a uniformly convergent sequence of H¨older functions. Now, let A > 0 be sufficiently big such that the set XA D fx 2 X I A.x/  A g

(3.38)

has positive measure. Notice that, by ergodicity of m, some iterate of a.e. x 2 X is in the set XA . Then by Poincar´e recurrence theorem and ergodicity of m, for a.e. x 2 X , there exists a sequence nj ! 1 such that  nj .x/ 2 XA , j  1. Therefore we get, for such an x 2 XA , from Lemma 3.18 that   Z Z  v˛ .gx qx / 1  O nj  n gx dx 1 j .x/   A B nj jgx j dx C 4 Lx gx  1 Qx (3.39) for every j  1. Finally, to pass from the subsequence .nj / to the sequence of all natural numbers we employ the monotonicity argument that already appeared in Walters paper [29]. Since LOx 1x D 1.x/ , we have for every w 2 J.x/ that inf gx .z/ 

z2Jx

X

O gx .z/e '.z/  sup gx .z/: z2Jx

z2Tx1 .w/

Consequently, the sequence .Mn;x /1 nD0 D .

sup w2J n .x/

LOxn gx .w//1 nD0

is weakly decreasing. Similarly we have a weakly increasing sequence .mn;x /1 nD0 D .

inf

w2J n .x/

LOxn gx .w//1 nD0 :

The proposition follows since, by (3.39), both sequences converge on the subsequence .nj /. t u

3.7 Exponential Decay of Correlations The following proposition proves item 3 of Theorem 3.2. For a function fx 2 L1 .x / we denote its L1 -norm with respect to x by Z kfx k1 WD jfx jdx :

32

3 The RPF-Theorem

Proposition 3.20 There exists a -invariant set X 0  X of full m-measure such that, for every x 2 X 0 , every f n .x/ 2 L1 . n .x/ / and every gx 2 H ˛ .Jx /, ˇ ˇ   ˇx .f n .x/ ı T n /gx   n .x/ .f n .x/ /x .gx /ˇ  A .gx ;  n .x//B n jjf n .x/ jj1 ; x

where Z v˛ .gx qx /  jgx jdx C 4 A. n .x//: A .gx ;  .x// WD C' . .x// Qx n

n

Proof. Set hx D gx 

R

gx dx and note that by (3.30) and (3.27) we have that

    x .f n .x/ ı Txn /gx   n .x/ .f n .x/ /x .gx / D  n .x/ f n .x/ LOxn .gx /  n .x/ .f n .x/ /x .gx /   D  n .x/ f n .x/ LOxn .hx / : (3.40) Since Lemma 3.18 yields jjLOxn hx jj1  A .gx ;  n .x//B n it follows from (3.40) that ˇ  ˇ  ˇx .f n .x/ ı T n /gx   n .x/ .f n .x/ /x .gx /ˇ  x

Z

ˇ ˇ ˇf n .x/ LO n .hx /ˇd n .x/ x Z

 A .gx ;  .x//B n

n

ˇ ˇ ˇf n .x/ ˇd n .x/ :

t u Using similar arguments like in Proposition 3.19 we obtain the following. Corollary 3.21 Let f n .x/ 2 L1 . n .x/ / and gx 2 L1 .Jx /, where x 2 X 0 and X 0 is the set given by Lemma 3.20. If jjf n .x/ jj1 ¤ 0 for all n, then ˇ  ˇ  ˇx .f n .x/ ı T n /gx   n .x/ .f n .x/ /x .gx /ˇ x

jjf n .x/ jj1

! 0

as n ! 1:

Remark 3.22 Note that if jjf n .x/ jj1 grows subexponentially, then ˇ ˇ   ˇx .f n .x/ ı T n /gx   n .x/ .f n .x/ /x .gx /ˇ ! 0 x

as n ! 1:

(3.41)

This is for example the case if x 7! log jjfx jj1 is m-integrable since Birkhoff’s Ergodic Theorem implies that .1=n/ log jjf n .x/ jj1 ! 0 for a.e. x 2 X .

3.8 Uniqueness Lemma 3.23 The family of measures x 7! x is uniquely determined by condition (3.1).

3.9 Pressure Function

33

Proof. Let Q x be a family of probability measures satisfying (3.1). For x 2 X choose arbitrarily a sequence of points wn 2 J n .x/ and define x;n WD

.Lxn / ıwn : Lxn 1.wn /

Then, by Proposition 3.19, for a.e. x 2 X and all gx 2 C .Jx / we have Lxn gx .wn / LOxn .gx =qx /.wn / x .gx / D lim D x .gx /: D n!1 L n 1.wn / n!1 LO n .1=q /.w / x .1/ x x n x (3.42)

lim x;n .gx / D lim

n!1

In other words, x;n ! x : n!1

in the weak* topology. Uniqueness of the measures x follows.

(3.43) t u

Lemma 3.24 There exists a unique function q 2 C 0 .J / that satisfies (3.2). Proof. Follows from Proposition 3.17.

t u

3.9 Pressure Function The pressure function is defined by the formula x 7! Px .'/ WD log x : If it does not lead to misunderstanding, we will also denote the pressure function by Px . It is important to note that this function is generally non-constant, even for a.e. x 2 X . Actually, if the pressure function is a.e. constant, then the random map shares many properties with a deterministic system. This will be explained in detail in Sect. 5. Note that (3.42) and (3.11) imply an alternative definition of Px .'/, namely L nC1 1.wnC1 / ; (3.44) Px .'/ D log..x/ .Lx 1// D lim log xn n!1 L.x/ 1.wnC1 / where, for every n 2 N, wn is an arbitrary point from J n .x/ . Lemma 3.25 For m-a.e. x 2 X and every sequence .wn /n  Jx lim

n!1

1 1 Sn Pxn  log Lxnn 1xn .wn / D 0: n n

Proof. By (3.19) and Proposition 3.17, we have that

34

3 The RPF-Theorem

Lxnn 1xn .w/ 1  C' .x/ C A.x/B n  A.x/B n  C' .x/ nxn for every w 2 Jx and every n 2 N. Therefore  log

   1  A.x/  log Lxnn 1xn .w/  log nxn  log C' .x/ C A.x/ : C' .x/ t u

Lemma 3.26 For m-a.e. x 2 X and for every sequence yn 2 Jxn , n  0, s-lim

1

n!1

n

Sn P x 

 1 log Lxn 1x .yn / D 0: n

Proof. Using Egorov’s Theorem and Lemma 3.25 we have that for each ı > 0 there exists a set Fı such that m.X n Xı / < ı and 1 1 Sn Pxn  max log Lxnn 1xn .y/ ! 0 n!1 n n y2Jxn uniformly on Fı . The lemma follows now from Birkhoff’s Ergodic Theorem.

t u

Lemma 3.27 If there exist g 2 L1 .m/ such that log kLx 1k1  g.x/, then   1  1 n  Sn Px  log Lx 1x  lim  D 0: n!1  n n 1 0 and let Proof. Let F WD Fı be the set from the proof of Lemma 3.26, let x 2 XCF .nj / be the visiting sequence. Let j be such that nj < n  nj C1 . Then n

log Lxn 1.y/  log kLx j 1k C Snnj g. nj .x//

for every y 2 J n .x/ : (3.45) Now, let h.x/ WD k'x k1 . Since by (3.12)  log x  k'x k1 , n

nn

 log nx D  log xj  log xnj j  Snj Px C Snnj h. nj .x//: Then by (3.45) 1 1 1 1 1 nj Sn Px  log Lx 1x .yxnj / C Snnj .g C h/. nj .x//: Sn Px  log Lxn 1x .yxn /  n n nj j nj n

On the other hand, for y 2 J n .x/ , n

n

C1 log Lxn 1.y/  log Lx j C1 1.T nj.x/

n

.y//  Snj C1 n g. n .x//

3.10 Gibbs Property

35

and by (3.12), n

n

log nx D log xj C1  log xnj C1

n

n

 log kLx j C1 1k C Snj C1 n h. n .x//: t u

The lemma follows now by Birkhoff’s Ergodic Theorem.

3.10 Gibbs Property Lemma 3.28 Let w 2 Jx , set y D .x; w/ and let n  0. Then e Q n .x/ .D . n .x/// 

x .Tyn .B.T n .y/; /// exp.Sn '.y/  Sn Px .'//

 e Q n .x/ :

Proof. Fix an arbitrary z 2 Jx and set y D .x; z/. Then by Lemma 2.3 and (3.13) we have that x .Tyn .B.T n .y/; /// exp.Sn '.y/  Sn Px .'//

0



.nx /1  n .x/ .B.T n .y/; // supz0 2Tyn .B.T n .y/;// e Sn '.z / .nx /1 e Sn '.y/

 e QS n .x/ :

On the other hand x .Tyn .B.T n .y/; /// exp.Sn '.y/  Sn Px .'//

0



.nx /1  n .x/ .B.T n .y/; // infz0 2Tyn .B.T n .y/;// e Sn '.z / .nx /1 e Sn '.y/

  n .x/ .B.T n .y/; //e QS n .x/ :

The lemma follows by (3.17).

t u

Lemma 3.29 Let T W J ! J satisfy the condition of measurability of cardinality of covers and let fi;x g, where i D 1; 2, be two Gibbs families with pseudo-pressure functions x 7! Pi;x . Then, for a.e. x, the measures 1;x and 2;x are equivalent and 1 1 1 Snk P1;x D lim Snk P2;x D lim S n k Px ; k!1 nk k!1 nk k!1 nk lim

where .nk / D .nk .x// is the visiting sequence of an essential set. Proof. Let A be compact subset of Jx and let ı > 0. By regularity of 2;x we can find " > 0 such that 2;x .Bx .A; "//  2;x .A/ C ı:

(3.46)

36

3 The RPF-Theorem

Now, let Nx be a measurable function such that .xNx /1  "=2. Set Ajn WD fy 2 Txn .yxjn / W A \ Tyn .B.yxjn ; // ¤ ;g: Let Z be a L; N; D; D-essential set of ax ; Nx ; D1 ; D2 and let .nk / D .nk .x// be the visiting sequence of Z. Fix k 2 N and put n D nk .x/. Then we have axn

A

[ [

j D1 y2Aj

Tyn B.yxjn ; /  Bx .A; "/:

n

By (3.3) it follows that 1;x .A/ 

a xn X X

1;x .Tyn B.yxjn ; // D1 .x/D

j D1 y2Aj

L X X

exp.Sn '.y/  Sn P1;x .'//:

j D1 y2Aj

n

n

(3.47)

Then by (3.46) and again by (3.3) 1;x .A/  D1 .x/D exp.Sn P2;x  Sn P1;x /

axn X X

exp.Sn '.y/  Sn P2;x .'//

j D1 y2Aj

n

 D1 .x/D2 .x/D 2 exp.Sn P2;x  Sn P1;x /

axn X X j D1 y2Aj

2;x .Tyn B.yxjn ; //

n

2

 D1 .x/D2 .x/D L exp.Sn P2;x  Sn P1;x /2;x .B.A; "//  D1 .x/D2 .x/D 2 L exp.Sn P2;x  Sn P1;x /.2;x .A/ C ı/;

(3.48)

since for y ¤ y 0 such that y; y 0 2 Txn .yxjn /, we have that j Tyn B.yxjn ; / \ Tyn 0 B.yx ; / D ;: n

Hence the difference Snk P2;x  Snk P1;x is bounded from below by some constant, since otherwise taking A D Jx we would obtain that 1;x .Jx / D 0 on a subsequence of .nk / in (3.48). Similarly, exchanging 1;x with 2;x we obtain that Snk P1;x  Snk P2;x is bounded from above. Then, letting ı go to zero, we have that 1;x and 2;x are equivalent. Note that exp.Sn P1;x /Lxn 1x .yn / D

X

e Sn 'x .y/Sn P1;x

y2Txn .yn /

 D1 .x/D

X

y2Txn .yn /

D D1 .x/D:

1;x .Tyn B.yn ; //  D1 .x/D1;x .Jx /

3.11 Some Comments on Uniformly Expanding Random Maps

37

Then

1 1 1 log Lx 1x .yn /  log.D1 .x/D/  Sn P1;x : n n n On the other hand, by (3.47), on the same subsequence X

1 D x1 .Jx /  D1 .x/DL

e Sn 'x .y/Sn P1;x

y2Txn .yn / a

for some yn 2 fyx1n ; : : : ; yxnxn g. Therefore, using Lemma 3.26 and the Sandwich Theorem, we have that, for x 2 XZ0 \ XP0 , 1 1 Snk P1;x D lim Snk Px : k!1 nk k!1 nk lim

t u

Remark 3.30 We cannot expect that P1;x D Px .'/ m-almost surely since, for any measurable function x 7! gx , P1;x WD Px .'/Cgx g.x/ , is also a pseudo-pressure function (see Lemma 3.28).

3.11 Some Comments on Uniformly Expanding Random Maps By C1 .J / we denote the space of B-measurable mappings g W J ! R with gx W Jx ! R continuous such that supx2X kgx k1 < 1. For H0  0, by H˛ .J ; H0 / we denote the space of all functions ' in Hm˛ .J / \ Cm1 .J / such that all of Hx are bounded above by H0 . Let [

H˛ .J / D

H˛ .J ; H0 /:

H0 0

For ' 2 H ˛ .J ; H0 / we put Q WD H0

1 X

 ˛j D

j D1

H0  ˛ : 1   ˛

Then Lemma 2.3 takes on the following form. Lemma 3.31 For every ' 2 H˛ .J ; H0 /, jSn 'x .Tyn .w1 //  Sn 'x .Tyn .w2 //j  Q%˛ .w1 ; w2 / for all n  1, all x 2 X , every z 2 Jx and every w1 ; w2 2 B.T n .z/; / and where y D .x; z/.

38

3 The RPF-Theorem

In this paper, whenever we deal with uniformly expanding random maps, we always assume that potentials belong to H˛ .J /. Hence all the functions C' .x/, Cmax .x/, Cmin .x/ and ˇx defined, respectively, by (3.5)–(3.8) are uniformly bounded on X . Therefore, there exists A 2 R such that A.x/  A for all x 2 X , where A.x/ is the function from Proposition 3.17. In particular, we can prove the following. Lemma 3.32 There exists a constant A such that, for x 2 X and all y1 ; y2 2 Jxn ˇ L n 1.y / ˇ 1 ˇ x ˇ  x ˇ  A B n : ˇ n1 Lx1 1.y1 / Proof. It follows from Proposition 3.17 that jLQx1 .LQ 1/.y1 /  LQx1 1.y2 /j  2AB n1 : Then by Lemma 3.6 and (3.22) we have, for some x-independent constant A , that ˇ 2AB 1 B n  ˇ L n 1.y / ˇ 1 x ˇ x  A B n : x ˇ  ˇ n1 Lx1 1.y2 / LQxn1 .1/.y2 /

t u

Chapter 4

Measurability, Pressure and Gibbs Condition

We now study measurability of the objects produced in the previous section. Up to now we do not know, for example, whether the family of measures x represents the disintegration of a global Gibbs state  with marginal m on the fibered space J . Therefore, we define abstract measurable expanding random maps for which the above measurabilities of x , qx , x and x can be shown. Then, we can construct a Borel probability invariant ergodic measure on J for the skewproduct transformation T with Gibbs property and study the corresponding expected pressure. Our settings are related to those of smooth expanding random mappings of one fixed Riemannian manifold from [17] and those of random subshifts of finite type whose fibers are subsets of NN from [5]. One possible extension of these works is to consider expanding random transformations on subsets of a fixed Polish space. A general framework for this was, in fact, prepared by Crauel in [10]. In Chap. 4.5 we show how Crauel’s random compact subsets of Polish spaces fit into our general framework and, therefore, our settings comprise all these options and go beyond. The issue of measurability of x , qx , x and x does not seem to have been treated with care in the literature. As a matter of fact, it was not quite clear to us even for symbol dynamics or random expanding systems of smooth manifolds until, very recently, when Kifer’s paper [19] has appeared to take care of these issues.

4.1 Measurable Expanding Random Maps Let T W J ! J be a general expanding random map. Define X W J ! X by X .x; y/ D x. Let B WD BJ be a -algebra on J such that 1. X and T are measurable, 2. for every A 2 B, X .A/ 2 F , 3. BjJx is the Borel -algebra on Jx .

V. Mayer et al., Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry, Lecture Notes in Mathematics 2036, DOI 10.1007/978-3-642-23650-1 4, © Springer-Verlag Berlin Heidelberg 2011

39

40

4 Measurability, Pressure and Gibbs Condition

By L0m .J / we denote the set of all BJ -measurable functions and by Cm0 .J / the set of all BJ -measurable functions g such that gx 2 C .Jx /. Lemma 4.1 If g 2 Cm0 .J /, then x 7! kgx k1 is measurable. Proof. The proof is a consequence of item 2. Indeed, Plet .Gn / be an increasing approximation of jgj by step functions. So let Gn D m kD1 ak 1Ak ; where .ak / is an increasing sequence of non-negative real numbers, and Ak are BJ -measurable. Then, define Xm WD X .Am /

and Xk WD X .Ak / n [m j DkC1 X .Aj /;

where k D 1; : : : ; m  1. Let Hn .x/ WD

m X

ak 1Xk .x/ D sup Gn .x; y/: y2Jx

kD0

Then the sequence .Hn / is increasing and converges pointwise to the function x ! 7 kgx k1 . u t 1 0 R The space Lm .J / is, by definition, the set of all g 2 Lm .J /, such that kgx k1 dm.x/ < 1: We also define

Cm1 .J / WD Cm0 .J / \ L1m .J / and

Hm˛ .J / WD Cm1 .J / \ H ˛ .J /:

By M 1 .J / we denote the set of probability measures and by Mm1 .J / its subset consisting of measures  0 such that there exists a system of fiber R measures fx0 gx2X with the property that for every g 2 L1m .J /, the map x 7! Jx gx dx0 is measurable and Z Z Z 0 gd D gx dx0 d m.x/: J

Then

X

Jx

1 m D  0 ı X

.x0 /x2X

(4.1) 0

is the canonical system of conditional measures of  with and the family respect to the measurable partition fJx gx2X of J . It is also instructive to notice that in the case when J is a Lebesgue space then (4.1) implies that  0 2 Mm1 .J /. The measure 0 2 M 1 .J / is called T -invariant if 0 ı T 1 D 0 . If 0 2 Mm1 .J /, then, in terms of the fiber measures, clearly T -invariance equivalently means that the family f0x gx2X is T -invariant; see Chap. 3.1 for the definition of T -invariance of a family of measures. Fix ' 2 Hm1 .J /. Then the general expanding random map T W J ! J is called a measurable expanding random map if the following conditions are satisfied.

4.2 Measurability

41

Measurability of the Transfer Operator. The transfer operator is measurable, i.e. L g 2 Cm0 .J / for every g 2 Cm0 .J /. Integrability of the Logarithm of the Transfer Operator. The function X 3 x 7! log kLx 1x k1 belongs to L1 .m/. We shall now provide a simple, easy to verify, sufficient condition for integrability of the logarithm of the transfer operator. Lemma 4.2 If log.deg.Tx // 2 L1 .m/, then x 7! log kLx 1x k1 belongs to L1 .m/. Proof. Recall that X

e k'x k1 

e 'x .z/  deg.Tx /e k'x k1 :

Tx .z/Dw

Hence k'x k1  log kLx 1x k1  log.deg.Tx // C k'x k1 :

t u

4.2 Measurability Now, we assume that T W J ! J is a measurable expanding random map. In particular, the operator L is measurable. Armed with these assumptions, we come back to the families of Gibbs states fx gx2X and fx gx2X whose pointwise construction was given in Theorem 3.1. Since we have already established good convergence properties, especially the exponential decay of correlations, it will follow rather easily that these families form in fact conditional measures of some measures  and  from Mm1 .J /. As an immediate consequence of item 3 of Theorem 3.1, we get that the probability measure  is invariant under the action of the map T W J ! J . All of this is shown in the following lemmas. Lemma 4.3 For every g 2 L1m .J /, the map x 7! x .gx / is measurable. Proof. It follows from (3.42) that kLxn gx k1 D x .gx /: n!1 kL n 1k1 x lim

Then measurability of x 7! x .gx / is a direct consequence of measurability of the transfer operator. t u This lemma enables us to introduce the probability measure  on J given by the formula: Z Z gx dx d m.x/: .g/ D X

Jx

This measure, therefore, belongs to Mm1 .J /.

42

4 Measurability, Pressure and Gibbs Condition

Lemma 4.4 The map X 3 x 7! x 2 R is measurable and the function q W J 3 .x; y/ 7! qx .y/ belongs to L0m .J /. Proof. Since  2 Mm1 .J /, measurability of ’s follows from the formula (3.11) and measurability of the transfer operator. Then measurability of ’s and of the transfer operator together with limn!1 LQxnn 1 D qx (see Proposition 3.17) imply measurability of q. t u From this lemma and Lemma 4.3 it follows that we can define a measure  by the formula: Z Z qx gx dx d m.x/: (4.2) .g/ D X

Jx

4.3 The Expected Pressure The pressure function of a measurable expanding random map has the following important property. Lemma 4.5 The pressure function X 3 x 7! Px .'/ is integrable. Proof. It follows from the definition of the transfer operator, that k'x k1  log .x/ .Lx 1/  log kLx 1k1 :

(4.3)

Then, by (3.11) and integrability of the logarithm of the transfer operator, the function Px .'/ is bounded above and below by integrable functions, hence integrable. t u Therefore, the expected pressure of ' given by Z EP .'/ D

X

Px .'/d m.x/

is well defined. The equality (3.42) yields alternative formulas for the expected pressure. In order to establish them, observe that by Birkhoff’s Ergodic Theorem EP .'/ D lim

n!1

1 log nx n

for a:e: x 2 X:

In addition, by (3.11), nx D nx x .1/ D  n .x/ .Lxn .1//: Thus, it follows that 1 L kCn 1x .wkCn / 1 log nx D lim log k x : n k!1 n L n .x/ 1 n .x/ .wkCn / However, by Lemma 3.27 we can get even more interesting formula.

(4.4)

4.5 Random Compact Subsets of Polish Spaces

43

Lemma 4.6 For every ' 2 Hm˛ .J / and for almost every x 2 X 1 log Lxn 1.wn /; n!1 n

EP .'/ D lim

where the points wn 2 J n .x/ are arbitrarily chosen.

4.4 Ergodicity of  Proposition 4.7 The measure  is ergodic. Proof. Let B be a measurable set such that T 1 .B/ D B and, for x 2 X , denote by Bx the set fy 2 Jx W .x; y/ 2 Bg. Then we have that Tx1 .B.x/ / D Bx . Now let X0 WD fx 2 X W x .Bx / > 0g: This is clearly a -invariant subset of X . We will show that, if m.X0 / > 0, then x .Bx / D 1 for a.e. x 2 X0 . Since  is ergodic with respect to m, this implies ergodicity of T with respect to . Define a function f by fx WD 1Bx . Clearly fx 2 L1 .x / and f n .x/ ı Txn D fx 0 m-a.e. Let x 2 X 0 \X R 0 , where X is given by Proposition 3.20. Let gx be a function 1 from L .Jx / with gx dx D 0. Then using (3.41) we obtain that   lim x .f n .x/ ı Txn /gx ! 0:

n!1

Consequently

Z Bx

gx dx D 0:

Since this holds for every mean zero function gx 2 L1 .Jx / , we have that x .Bx / D 1 for every x 2 X 0 \ X0 . This finishes the proof of ergodicity of T with respect to the measure . t u A direct consequence of Lemma 3.29 and ergodicity of T is the following. Proposition 4.8 The measure  2 Mm1 .J / is a unique T -invariant measure satisfying (3.3).

4.5 Random Compact Subsets of Polish Spaces Suppose that .X; F ; m/ is a complete measure space. Suppose also that .Y; %/ is a Polish space which is normalized so that diam.Y / D 1. Let BY be the -algebra of Borel subsets of Y and let KY be the space of all compact subsets of Y topologized

44

4 Measurability, Pressure and Gibbs Condition

by the Hausdorff metric. Assume that a measurable mapping X 3 x 7! Jx 2 KY is given. Following Crauel [10, Chap. 2], we say that a map X 3 x 7! Yx  Y is measurable if for every y 2 Y; the map x 7! d.y; Yx / is measurable, where d.y; Yx / WD inffd.y; yx / W yx 2 Yx g: This map is also called a random set. If every Yx is closed (res. compact), it is called a closed (res. compact) random set. With this terminology X 3 x 7! Jx  Y is a compact random set (see [10, Remark 2.16, p. 16]). Closed random sets have the following important properties (cf. [10, Proposition 2.4 and Theorem 2.6]). Theorem 4.9 Suppose that X 3 x 7! Yx is a closed random set such that Yx ¤ ;. (a) For all open sets V  Y , the set fx 2 X W Yx \ V ¤ ;g is measurable. (b) The set J WD graph.x 7! Yx / WD f.x; yx / W x 2 X and yx 2 Yx g is a measurable subset of X  Y , i.e. J is a subset of F ˝ BY , the product algebra of F and BY . (c) For every n, there exists a measurable function X 3 x 7! yx;n 2 Yx such that Yx D clfyx;n W n 2 Ng: In particular, there exists a measurable map X 3 x 7! yx 2 Yx . Note that item (b) implies that J is a measurable subset of X Y . Let BJ WD F ˝ BY jJ . Then by Theorem 2.12 from [10] we get that for all A 2 BJ , X .A/ 2 F . Now, let X 3 x 7! Yx be a compact random set and let r > 0 be a real number. Then every set Yx can be covered by some finite number ax D ax .r/ 2 N of open balls with radii equal to r. Moreover, by Lebesgue’s Covering Lemma, there exits Rx D Rx .r/ > 0 such that every ball B.yx ; Rx / with yx 2 Yx is contained in a ball from this cover. As we prove below, we can actually choose ax and Rx in a measurable way. Hence for the compact random set x 7! Jx the measurability of cardinality of covers (see Chap. 3.1, just before Theorem 3.3) holds automatically. In the proof of Lemma 4.11 we will use the following Proposition 2.1 from [10, p. 15]. Proposition 4.10 For compact random set x 7! Yx and for every ", there exists a (non-random) compact set Y"  Y such that m.fx 2 X W Yx  Y" g/  1  ": Lemma 4.11 There exists a measurable set Xa0  X of full measure m such that, for every r > 0 and every positive integer k, there exists a measurable function Xa0 3 x 7! yx;k 2 Yx and there exist measurable functions Xa0 3 x 7! ax 2 N and Xa0 3 x 7! Rx 2 RC such that for every x 2 Xa0 ,

4.5 Random Compact Subsets of Polish Spaces ax [

45

Bx .yx;k ; r/  Yx ;

kD1

and for every yx 2 Yx , there exists k D 1; : : : ; ax for which Bx .yx ; Rx /  Bx .yx;k ; r/: Proof. For n 2 N let Y1=n  Y be a compact set given by Proposition 4.10. Then the set Xn WD fx 2 X W Yx  Y1=n g is measurable and has the measure m.Xn / greater or equal to 1  1=n. Define Xa0 WD

[

Xn :

n2N

Then m.Xa0 / D 1. Let fyn W n 2 NC g be a dense subset of Y . Since Y1=n is compact, there exists a positive integer a.n/ such that a.n/

[

B.yk ; r=2/  Y1=n :

(4.5)

kD1

Define a function Xa0 3 x 7! ax , by ax D a.n/ where n WD minfk W x 2 Xk g: The measurability of Xn gives us the required measurability of x 7! ax . Let fyk W k 2 Ng be a countable dense set of Y and m 2 N. For every k 2 N define a function x 7! Gx;k by  Gx;k D

B.yk ; r=2/ if Yx \ B.yk ; r=2/ ¤ ; otherwise. Yx

Since, by Theorem 4.9(a), the set fx 2 X W Yx \ B.yk ; r=2/ ¤ ;g is measurable, it follows that X 3 x 7! Gx;k is a closed random set. Hence, by Theorem 4.9(c), there exists a measurable selection X 3 x 7! yx:k 2 Gx;k . Note that, if yx:k 2 B.yk ; r=2/, then B.yk ; r=2/  B.yx:k ; r/. Therefore, by (4.5), Ux [

B.yx;k ; r/  Y1=n  Yx

for all x 2 Xn :

kD1

Finally, for x 2 Xn , let Rx > 0 be a real number such that, for y 2 Y1=n , there exists k D 1; : : : ; U.n/ for which B.y; Rx /  B.yk ; r=2/  B.yx;k ; r/: Then XU0 3 x 7! Rx 2 RC is also measurable. t u

•

Chapter 5

Fractal Structure of Conformal Expanding Random Repellers

We now deal with conformal expanding random maps. We prove an appropriate version of Bowen’s Formula, which asserts that the Hausdorff dimension of almost every fiber Jx , denoted throughout the paper by HD, is equal to a unique zero of the function t 7! EP .t/. We also show that typically Hausdorff and packing measures on fibers respectively vanish and are infinite. A simple example of such a phenomenon is a Random Cantor Set described. Later in this paper the reader will find more refined and general examples of Random Conformal Systems notably Classical Random Expanding Systems, Br¨uck and B¨uger Polynomial Systems and DG-Systems. In the following we suppose that all the fibers Jx are in an ambient space Y which is a smooth Riemannian manifold. We will deal with C 1C˛ -conformal mappings fx and denote then jfx0 .z/j the norm of the derivative of fx which, by conformality, is nothing else than the similarity factor of fx0 .z/. Finally, let jjfx0 jj1 be the supremum of jfx0 .z/j over z 2 Jx . Since we deal with expanding systems we have jfx0 j  x for a.e. x 2 X: (5.1) Definition 5.1 Let f W .x; z/ 7! ..x/; fx .z// be a measurable expanding random map having fibers Jx  Y and such that the mappings fx W Jx ! J.x/ can be extended to a neighborhood of Jx in Y to conformal C 1C˛ mappings. If in addition log jjfx0 jj1 2 L1 .m/ then we call f conformal expanding random map. A conformal random map f W J ! J which is uniformly expanding is called conformal uniformly expanding.

5.1 Bowen’s Formula For every t 2 R we consider the potential 't .x; z/ D t log jfx0 .z/j. The associated topological pressure P .'t / will be denoted P .t/. Let

V. Mayer et al., Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry, Lecture Notes in Mathematics 2036, DOI 10.1007/978-3-642-23650-1 5, © Springer-Verlag Berlin Heidelberg 2011

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48

5 Fractal Structure of Conformal Expanding Random Repellers

Z EP .t/ D

X

Px .t/dm.x/

be its expected value with respect to the measure m. In view of (5.1), it follows from Lemma 9.6 that the function t 7! EP .t/ has a unique zero. Denote it by h. The result of this subsection is the following version of Bowen’s formula, identifying the Hausdorff dimension of almost all fibers with the parameter h. Theorem 5.2 (Bowen’s Formula) Let f be a conformal expanding random map. The parameter h, i.e. the zero of the function t 7! EP .t/, is m-a.e. equal to the Hausdorff dimension HD.Jx / of the fiber Jx . Bowen’s formula has been obtained previously in various settings first by Kifer [18] and then by Crauel and Flandoli [11], Bogensch¨utz and Ochs [6], and Rugh [26]. Proof. Let .x;h /x2X be the measures produced in Theorem 3.1 for the potential 'h . Fix x 2 X and z 2 Jx and set again y D .x; z/. For every r 2 .0;  let k D k.z; r/ be the largest number n  0 such that B.z; r/  fyn .B.fxn .z/; //:

(5.2)

By the expanding property this inclusion holds for all 0  n  k and limr!0 k.z; r/ D C1. Fix such an n. By Lemma 3.28,   x;h .B.z; r//  x;h .fyn .B.fxn .z/; ///  exp hQ n .x/ j.fxn /0 .z/jh exp.Pxn .h//: (5.3) .sC1/

On the other hand, B.z; r/ 6 fy by Lemma 2.3,

.B.fxsC1 .z/; // for every s  k. But, since

B.z; exp.Q sC1 .x/  ˛ /j.fxsC1 /0 .z/j1 /  fy.sC1/ .B.fxsC1 .z/; //; we get and j.fxs /0 .z/j1

  exp Q sC1 .x/  ˛ j.fxsC1 /0 .z/j1   r (5.4)     1 exp Q sC1 .x/  ˛ r: Inserting this to (5.3) we obtain,

    x;h .B.z; r//   h exp hQ n .x/ exp hQ sC1 .x/  ˛ r h 0 n   exp.Pxn .h//j fsC1n .fx .z//jh n .x/

(5.5)

or, equivalently, ˛

hQ sC1 .x/  hQ n .x/ log x;h .B.z; r//  hC C C log r log r log r C

h log  C log r

Pxn .h/ log r

:

ˇ ˇ 0 n ˇ ˇ .fx .z//ˇ h log ˇ fsC1n n .x/ log r (5.6)

5.1 Bowen’s Formula

49

Our goal is to show that lim inf r!0

log x;h .B.z; r//  h for a.e. x 2 X and all z 2 Jx : log r

Since the function x 7! Qx is measurable and almost everywhere finite, there exists M > 0 such that m.A/ > 0, where A D fx 2 X W Qx  M g: Fix n D nk  0 to be the largest integer less than or equal to k such that  n .x/ 2 A and s D sk to be the least integer greater than or equal to k such that  sC1 .x/ 2 A. It follows from Birkhoff’s Ergodic Theorem that limk!1 sk =nk D 1: Of course if for k  1 we take any rk > 0 such that k.z; rk / D k, then limk!1 rk D 0. Now, note that by (5.2), the formula fyn .B.fxn .z/; //  B.z; exp.Q n .x/  ˛ /j.fxn /0 .z/j1 / yields r  exp.Q n .x/  ˛ /j.fxn /0 .z/j1 : Equivalently,  log r  log j.fxn /0 .z/j   ˛ Q n .x/  log : Since log j.fxn /0 .z/j  log xn and since the function x 7! log x is integrable and Z  D minf1;

log  d mg > 0

we get from Birkhoff’s Ergodic Theorem that for a.e. x 2 X and all r > 0 small enough (so k and nk and sk large enough too)  log r 

  n  s: 2 3

(5.7)

Remember that  n .x/ 2 A and  sC1 .x/ 2 A. We thus obtain from (5.6) that ˇ ˇ 0 n log x;h .B.z; r// 1 1 ˇ ˇ .fx .z//ˇ  2 Pxn .h/  h  3h lim sup log ˇ fsC1n n .x/ r!0 log r n k!1 s (5.8) R for a.e. x 2 X and all z 2 Jx . But as Px .h/d m.x/ D 0, we have by Birkhoff’s Ergodic Theorem that 1 lim P n .h/ D 0: (5.9) n!1 n x Also, since the measure h is f -invariant, it follows from Birkhoff’s Ergodic Theorem that there exists a measurable set X0  X such that for every x 2 X0 there exists at least one (in fact of full measure x;h ) zx 2 Jx such that lim inf

Z ˇ  ˇ 1 ˇ j 0 ˇ log ˇ fx .zx /ˇ D O WD log jfx0 .z/jdh .x; z/ 2 .0; C1/: lim j !1 j J

50

5 Fractal Structure of Conformal Expanding Random Repellers

Hence, remembering that  n .x/ and  sC1 .x/ belong to A, we get lim sup k!1

ˇ ˇ  ˇ ˇ 0 n 1 1  ˇˇ sC1 0 ˇˇ 0 ˇ ˇ ˇ ˇ log ˇ fsC1n log ˇ fx .fx .z//ˇ D lim sup .z/ˇ  log ˇ fxn .z/ˇ n .x/ s k!1 s ˇ ˇ  ˇ 1  ˇˇ sC1 0 0 ˇ ˇ ˇ .zx /ˇ  log ˇ fxn .zx /ˇ D lim sup log ˇ fx s k!1 ˇ ˇ 0 1 ˇ ˇ  lim sup log ˇ fxsC1 .zx /ˇ k!1 s ˇ  ˇ 1 0 ˇ ˇ  lim inf log ˇ fxn .zx /ˇ D O  O D 0: k!1 s

Inserting this and (5.9) to (5.8) we get that lim inf r!0

log x;h .B.z; r//  h: log r

(5.10)

Keep x 2 X , z 2 Jx and r 2 .0; . Now, let l D l.z; r/ be the least integer  0 such that fyl .B.fxl .z/; //  B.z; r/: (5.11) Then, by Lemma 3.28, x;h .B.z; r//  x;h .fyl .B.fxl .z/; ///    D1 . l .x// exp Q l .x/ j.fxl /0 .z/jl exp.Pxl .h//: .l1/

On the other hand, fy

(5.12)

.B.fxl1 .z/; // 6 B.z; r/: But, since

fy.l1/ .B.fxl1 .z/; //  B.y; exp.Q l1 .x/ ˛ /j.fxl1 /0 .z/j1 /; we get Thus j.fxl1 /0 .z/j1

r   exp.Q l1 .x/  ˛ /j.fxl1 /0 .y/j1 : (5.13)     1 exp Q l1 .x/  ˛ r: Inserting this to (5.12) we obtain,

Q x;h .B.z; r//   h D1 . l .x//e  l .x/ j.f l1 .x/ /0 .fxl1 .z//jh    exp hQ l1 .x/  ˛ r h exp.Pxl .h//:

(5.14) (5.15)

Now, given any integer j  1 large enough, take Rj > 0 to be the least radius r > 0 such that fyj .B.fxj .z/; //  B.z; r/: Then l.y; Rj / D j . Since the function Q is measurable and almost everywhere finite, and  is a measure-preserving transformation, there exist a set  X with

5.2 Quasi-Deterministic and Essential Systems

51

positive measure m and a constant E > 0 such that Qx  E, D1 .x/  E and Q 1 .x/  E for all x 2 . It follows from Birkhoff’s Ergodic Theorem and ergodicity of the map  W X ! X that there exists a measurable set X1  X with m.X1 / D 1 such that for every x 2 X1 there exists an unbounded increasing ji sequence .ji /1 i D1 such that  .x/ 2 for all i  1. Formula (5.13) then yields  log Rji  E ˛ C log  C log j.fxji 1 .z/j  E ˛ C log  C log xji 1 

 ji ; 2

where the last inequality was written because of the same argument as (5.7) was, intersecting also X1 with an appropriate measurable set of measure 1. Now we get from (5.14) that   log x;h B.z; Rji / 2 log E 2E 2h 1 2h ˛ E  hC   log jj.f ji 1 .x/ /0 jj1  log Rji ji ji  ji ji 

2h log  2 1 ji  P .h/: ji  ji x

R Noting that X Px .t/d m.x/ D 0 and applying Birkhoff’s Ergodic Theorem, we see that the last term in the above estimate converges to zero. Also 1 log jj.f ji 1 .x/ /0 jj1 converges to zero because of Birkhoff’s Ergodic Theorem ji and integrability of the function x 7! log jjfx0 jj1 . Since all the other terms obviously converge to zero, we thus get for a.e. x 2 X and all z 2 Jx , that   log x;h B.z; Rji / log x;h .B.z; r//  h: lim inf  lim inf r!0 i !1 log r log Rji Combining this with (5.10), we obtain that lim inf r!0

log x;h .B.z; r// Dh log r

for a.e. x 2 X and all z 2 Jx . This gives that HD.Jx / D h for a.e. x 2 X . We are done. t u

5.2 Quasi-Deterministic and Essential Systems We now investigate the fractal structure of the Julia sets and we will see that the random systems naturally split into two classes depending on the asymptotic behavior of Birkhoff’s sums of the topological pressure Pxn .h/. Definition 5.3 Let f be a conformal uniformly expanding random map. It is called essentially random if for m-a.e. x 2 X ,

52

5 Fractal Structure of Conformal Expanding Random Repellers

lim sup Pxn .h/ D C1 and lim inf Pxn .h/ D 1;

(5.16)

n!1

n!1

where h is the Bowen’s parameter coming from Theorem 5.2. The map f is called quasi-deterministic if for m-a.e. x 2 X there exists Lx > 0 such that Lx  Pxn .h/  Lx

for m-almost all x 2 X and all n  0:

(5.17)

Remark 5.4 Because of ergodicity of the transformation  W X ! X , for a uniformly conformal random map to be essential it suffices to know that the condition (5.16) is satisfied for a set of points x 2 X with a positive measure m. Remark 5.5 If the number 1 n!1 n

2 .P .h// D lim

Z 

2 Sn .P .h// d m > 0

and if the Law of Iterated Logarithm holds, i.e. if p p P n .h/ P n .h/  lim sup p x D 2 2 .P .h//  2 2 .P .h// D lim inf p x n!1 n!1 n log log n n log log n m-a.e., then our conformal random map is essential. It is essential even if only the Central Limit Theorem holds, i.e. if  m

Pxn .h/
 !

1 p

2

Z

r

e s

2 =2 2 .P .h//

ds:

1

Remark 5.6 If there exists a bounded everywhere defined measurable function u W X ! R such that Px .h/ D u.x/  u ı .x/ (i.e. if P .h/ is a coboundary) for all x 2 X , then our system is quasi-deterministic. For every ˛ > 0 let H ˛ refer to the ˛-dimensional Hausdorff measure and let P refer to the ˛-dimensional packing measure. Recall that a Borel probability measure  defined on a metric space M is geometric with an exponent ˛ if and only if there exist A  1 and R > 0 such that ˛

A1 r ˛  .B.z; r//  Ar ˛ for all z 2 M and all 0  r  R. The most significant basic properties of geometric measures are the following: (GM1) The measures , H ˛ , and P ˛ are all mutually equivalent with Radon– Nikodym derivatives separated away from zero and infinity. (GM2) 0 < H ˛ .M /; P ˛ .M / < C1. ( GM3) HD.M / D h. The main result of this section is the following.

5.2 Quasi-Deterministic and Essential Systems

53

Theorem 5.7 Suppose f W J ! J is a conformal uniformly expanding random map. (a) If the system f W J ! J is essential, then H h .Jx / D 0 and P h .Jx / D C1 for m-a.e. x 2 X . (b) If, on the other hand, the system f W J ! J is quasi-deterministic, then for every x 2 X xh is a geometric measure with exponent h and therefore ( GM1)– (GM3) hold. R Proof. Part (a). Remember that by its very definition EP .h/ D Px .h/d m.x/ D 0. By Definition 5.3 there exists a measurable set X1 with m.X1 / D 1 such that for every x 2 X1 there exists an increasing unbounded sequence .nj /1 j D1 (depending on x) of positive integers such that n

lim Px j .h/ D 1:

j !1

(5.18)

Since we are in the uniformly expanding case, the formula (5.12) from the proof of Theorem 5.2 (Bowen’s Formula) takes on the following simplified form:   x .B.z; r//  D 1 r h exp Pxl.z;r/ .h/

(5.19)

with some D  1 and all z 2 Jx . Since the map is uniformly expanding, for all j  1 large enough, there exists rj > 0 such that l.z; rj / D nj . So disregarding finitely many terms, we may assume without loss of generality, that this is true for all j  1. Clearly limj !1 rj D 0: It thus follows from (5.19) that   n x;h .B.z; rj //  D 1 rjh exp Px j .h/ for all x 2 X1 , all z 2 Jx and all j  1. Therefore, by (5.18), lim sup r!0

  x;h .B.z; r// x;h .B.z; rj // n  lim sup D 1 lim sup exp Px j .h/ D C1; h h r rj j !1 j !1

which implies that H h .Jx / D 0. The proof for packing measures is similar. By Definition 5.3 there exists a measurable set X2 with m.X2 / D 1 such that for every x 2 X2 there exists an increasing unbounded sequence .sj /1 j D1 (depending on x) of positive integers such that s lim Px j .h/ D C1: (5.20) j !1

Since we are in the expanding case, formula (5.5) from the proof of Theorem 5.2 (Bowen’s Formula), applied with s D k.z; r/, takes on the following simplified form:   x .B.z; r//  Dr h exp Pxk.z;r/ .h/ (5.21)

54

5 Fractal Structure of Conformal Expanding Random Repellers

with D  1 sufficiently large, all x 2 X2 and all z 2 Jx . By our uniform assumptions, for all j  1 large enough, there exists Rj > 0 such that k.z; Rj / D sj . Clearly limj !1 Rj D 0: It thus follows from (5.21) that   s x;h .B.z; rj //  DRjh exp Px j .h/ for all x 2 X2 , all z 2 Jx and all j  1. Therefore, using (5.20), we get lim inf r!0

  x;h .B.z; r// x;h .B.z; Rj // s  lim inf  D lim inf exp Px j .h/ D 0: h j !1 j !1 rh Rj

Thus P h .Jx / D C1. We are done with part (a). Suppose now that the map f W J ! J is quasi-deterministic. It then follows from Definition 5.3 and (5.19) along with (5.21), that for every x 2 X and for every r > 0 small enough independently of x 2 X , we have .Lx D/1 r h  x;h .B.y; r//  Lx Dr h ; x 2 X; z 2 Jx : This means that each x;h , x 2 X , is a geometric measure with exponent h and the theorem follows. t u As a straightforward consequence of this theorem we get a corollary transparently stating that essential conformal random systems are entirely new objects, drastically different from deterministic self-conformal sets. Corollary 5.8 Suppose that conformal random map f W J ! J is essential. Then for m-a.e. x 2 X the following hold. (1) The fiber Jx is not bi-Lipschitz equivalent to any deterministic nor quasideterministic self-conformal set. (2) Jx is not a geometric circle nor even a piecewise smooth curve. (3) If Jx has a non-degenerate connected component (for example if Jx is connected), then h D HD.Jx / > 1. (4) Let d be the dimension of the ambient Riemannian space Y . Then HD.Jx / 0 whenever W is connected. The proof of (4) is similar. Since (3) obviously implies (2), we are done. t u

5.3 Random Cantor Set Here is a first example of an essentially random system. Define f0 .x/ D 3x.mod 1/ for x 2 Œ0; 1=3 [ Œ2=3; 1

5.3 Random Cantor Set

55

and f1 .x/ D 4x.mod 1/ for x 2 Œ0; 1=4 [ Œ3=4; 1: Let X D f0; 1gZ,  be the shift transformation and m be the standard Bernoulli measure. For x D .: : : ; x1 ; x0 ; x1 ; : : :/ 2 X define fx D fx0 , fxn D f n1 .x/ ı f n2 .x/ ı : : : ı fx and 1 \ Jx D .fxn /1 .Œ0; 1/: nD0

The skew product map defined on

S

x2X

Jx by the formula

f .x; y/ D ..x/; fx .y// generates a conformal random expanding system. We shall show that this system is essential. To simplify the next calculation, we define recurrently:  x .1/ D

3 if x0 D 0 ; x .n/ D  n1 .x/ .1/x .n  1/: 4 if x0 D 1

Consider the potential ' t defined by the formula 'xt D t log x .1/. Then Sn 'xt D t log x .n/: Let Cn be a cylinder of the order n that is Cn is a subset of Jx of diameter .x .n//1 such that fxn jCn is one-to-one and onto J n .x/ . We can project the measure m on Jx and we call this measure x . In other words, x is such a measure that all cylinders of level n have the measure 1=2n. Then by Law of Large Numbers for m-almost every x lim

n!1

log 2 log 4 log x .Cn / D D DW h: log diam.Cn / .1=n/ log x .n/ log 12

Therefore the Hausdorff dimension of Jx is for m-almost every x constant and equal to h. Next note that x .Cn / D exp.Sn Px /; diam.Cn /h

(5.22)

where Px WD log 2  h log x .1/: This will give us the value of the Hausdorff and packing measure. So let Z0 ; Z1 ; : : : be independent random variables, each having the same distribution such that the probability of Zn D log 2h log 3 is equal to the probability of Zn D log 2h log 4

56

5 Fractal Structure of Conformal Expanding Random Repellers

and is equal to 1=2. The expected value of Zn , EP , is zero and its standard deviation

> 0. Then the Law of the Iterated Logarithm tells us that the following equalities: p Z1 C : : : C Zn lim inf p D  2

n!1 n log log n

and

p Z1 C : : : C Zn lim sup p D 2 (5.23) n!1 n log log n

hold with probability one. Then, by (5.22), lim sup n!1

x .Cn / D1 diam.Cn /h

and

lim inf n!1

x .Cn / D0 diam.Cn /h

for m-almost every x. In particular, the Hausdorff measure of almost every fiber Jx vanishes and the packing measure is infinite. Note also that the Hausdorff dimension of fibers is not constant as clearly HD.J01 / D log 2= log 3, whereas HD.J11 / D log 2= log 4 D 1=2.

Chapter 6

Multifractal Analysis

The second direction of our study of fractal properties of conformal random expanding maps is to investigate the multifractal spectrum of Gibbs measures on fibers. We show that the multifractal formalism is valid. It seems that it is impossible to do it with a method inspired by the proof of Bowen’s formula since one gets full measure sets for each real ˛ and not one full measure set Xma such that for all x 2 Xma , the multifractal spectrum of the Gibbs measure on the fiber over x is given by the Legendre transform of a temperature function which is independent of x 2 Xma . In order to overcome this problem we work out a different proof in which we minimize the use Birkhoff’s Ergodic Theorem and instead we base the proof on the definition of Gibbs measures and the behavior of the Perron–Frobenius operator. In this point we were partially motivated by the approach presented in Falconer’s book [15]. Another issue we would like to bring up here is real analyticity of the multifractal spectrum which we establish for uniformly expanding systems. The proof is based on real-analiticity results for the expected pressure which are treated separately in Chap. 9 since this part involves different methods.

6.1 Concave Legendre Transform Let ' 2 Hm .J / be such that EP .'/ D 0. Fix q 2 R. We will not use the function qx and therefore this will not cause any confusion. Define auxiliary potentials 'q;x;t .y/ WD q.'x .y/  Px .'//  t log jfx0 .y/j: By Lemma 9.5, the function .q; t/ 7! EP .q; t/ WD EP .'q;t / is convex. Moreover, since log jfx0 .y/j  log x > 0, it follows from Lemma 9.6 that for every q 2 R there exists a unique T .q/ 2 R such that

V. Mayer et al., Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry, Lecture Notes in Mathematics 2036, DOI 10.1007/978-3-642-23650-1 6, © Springer-Verlag Berlin Heidelberg 2011

57

58

6 Multifractal Analysis

EP .'q;T .q// D 0: The function q 7! T .q/ defined implicitly by this formula is referred to as the temperature function. Put 'q WD 'q;T .q/ : By DT we denote the set of differentiability points of the temperature function T . By convexity of EP , for  2 .0; 1/, EP .q1 C .1  /q2 ; T .q1 / C .1  /T .q2 //  EP .q1 ; T .q1 // C .1  /EP .q2 ; T .q2 // D 0: Since t 7! EP .q1 C .1  /q2 ; t/ is decreasing, T .q1 C .1  /q2 /  T .q1 / C .1  /T .q2 /: Hence the function q 7! T .q/ is convex and continuous. Furthermore, it follows from its convexity that the function T is differentiable everywhere but a countable set, where it is left and right differentiable. Define L.T /.˛/ WD where ˛ 2 Dom.L/ D



inf

1


 ˛q C T .q/ ;

 lim T 0 .q  /; lim T 0 .q C / :

q!1

q!1

We call L the concave Legendre transform. This transform is related to the (classical) Legendre transform L by the formula L.T /.˛/ D  L.T /.˛/. The transform L sends convex functions to concave ones and, if q 2 DT , then L.T /.T 0 .q// D T 0 .q/q C T .q/: Lemma 6.1 Let q 2 DT . Then for every " > 0 there exists ı" > 0, such that, for all ı 2 .0; ı" /, we have EP ..1 C ı/q; T .q/ C .qT 0 .q/ C "/ı/ < 0 and

EP ..1  ı/q; T .q/ C .qT 0 .q/ C "/ı/ < 0:

Proof. Since the temperature function T is differentiable at the point q, we may write T .q C ıq/ D T .q/ C T 0 .q/ıq C o.ı/

6.2 Multifractal Spectrum

59 .1/

for all ı > 0 sufficiently small, say ı 2 .0; ı" /. So, T .q/ C .qT 0 .q/ C "/ı  T ..1 C ı/q/ D "ı C o.ı/ > 0: Then, in virtue of Lemma 9.6, we get that EP ..1 C ı/q; T .q/ C .qT 0 .q/ C "/ı/ < EP ..1 C ı/q/; T ..1 C ı/q// D 0; meaning that the first assertion of our lemma is proved. The second one is proved similarly producing a positive number ı".2/ . Setting then ı" D minfı".1/ ; ı".2/ g completes the proof. t u

6.2 Multifractal Spectrum Let  be the invariant Gibbs measure for ' and let  be the '-conformal measure. For every ˛ 2 R define n o log x .B.y; r// Kx .˛/ WD y 2 Jx W dx .y/ WD lim D˛ r!0 log r and

n o log x .B.y; r// Kx0 WD y 2 Jx W the limit lim does not exist : r!0 log r

This gives us the multifractal decomposition Jx WD

]

Kx .˛/ ] Kx0 :

˛0

The multifractal spectrum is the family of functions fgx gx2X given by the formulas: gx .˛/ WD HD.Kx .˛//: The function dx .y/ is called the local dimension of the measure x at the point y. Since for m almost every x 2 X the measures x and x are equivalent with Radon– Nikodym derivatives uniformly separated from 0 and infinity (though the bounds may and usually do depend on x), we conclude that we get the same set Kx .˛/ if in its definition the measure x is replaced by x . Our goal now is to get a “smooth” formula for gx . Let q and q be the measures for the potential 'q given by Theorem 3.1. The main technical result of this section is this. Proposition 6.2 For every q 2 DT there exists a measurable set Xma  X with m.Xma / D 1 and such that, for every x 2 Xma , and all q 2 DT , we have

60

6 Multifractal Analysis

gx .T 0 .q// D qT 0 .q/ C T .q/ Proof. Firstly, by Lemma 9.4, for every 0 < R   there exists a measurable function DR W X ! .0; C1/ such that for all q 2 R, all x 2 X , all y 2 Jx , and all integers n  0, we have q;x .fyn .B.f n .y/; R/// q    DR . n .x//; n n 0 T .q/ exp q.Sn '.y/  Px .'// j.fx / .y/j (6.1) where q  WD .q; T .q// as defined in (9.1). In what follows we keep the notation from the proof of Theorem 5.2. The formulas (5.2) and (5.11) then give for every j  l and every 0  i  k, that 

q DR . n .x// 

   Dq . j .x///1 exp q.Sj '.y/  Pxj .'// j.fxj /0 .y/jT .q/  q;x .B.y; r//

(6.2)

   D . i .x// exp q.Si '.y/  Pxi .'// j.fxi /0 .y/jT .q/ : q

By Qx we denote the measurable function given by Lemma 2.3 for the function  log jf 0 j. Let X be an essential set for the functions X 3 x 7! Rx , X 3 x 7! O a, O and DO  . Let .nj /1 be a.x/, x 7! Qx , and X 3 x 7! D .x/ with constants R, O Q, 1 0 the positively visiting sequence for X at x. Let XE be the set given by Lemma 9.5 for potentials q;t , q; t 2 R2 . Let 0 0 XC WD XE0 \ XCX :  0 . Fix "1 > 0. For Let us first prove the upper bound on gx .T 0 .q//. Fix x 2 XC every j  1 let fwk .xnj / W 1  k  a.xnj /g be a  spanning set of Jxnj . As EP .q / D 0, it follows from Lemma 9.6 that  WD 12 EP .q;T .q/C"1 / < 0. So, in virtue of Lemma 9.5, there exists C  1 such that

Lq;T .q/C"1 ;x 1.wk .xnj //  C e  nj

(6.3)

O Now, fix an arbitrary "2 2 R for all j  1 and all k D 1; 2; : : : ; a. nj .x//  a. such that q"2  0. For every integer l  1 let  1 log x .B.y; r// 1  T 0 .q/ C j"2 j Kx ."2 ; l/ D y 2 Kx .T 0 .q// W  T 0 .q/  j"2 j  2 log r 2  for all 0 < r  1= l :

6.2 Multifractal Spectrum

61

Note that

1 [

Kx .T 0 .q// D

Kx ."2 ; l/:

(6.4)

lD1

Let 8 <

a.xnj /

nj .x/ D z 2 :

[

n fx j .wk .xnj //

W Kx ."2 ; l/ \

n fz j .B.f nj .z/; =2//

kD1

[

Then Kx ."2 ; l/ 

nj

fz

9 =

¤; : ;

.B.f nj .z/; =2//:

(6.5)

z2nj .x/ nj

For every z 2 nj .x/, say z 2 fx

.wk .xnj //, choose nj

zO 2 Kx ."2 ; l/ \ fz

.B.wk .xnj /; =2//:

  Then B wk .xnj /; =2  B.f nj .z/; /, and therefore nj

fz

nj 

.B.wk .xnj /; =2//  fzO

 B.f nj .Oz/; / :

It follows from this and (6.5) that [

Kx ."2 ; l/ 

nj

z2nj .x/

Put

fzO

.B.f nj .Oz/; //:

(6.6)

n O xnj /0 .Oz/j1 rj.1/ .Oz/ D QO 1 j.fx j /0 .Oz/j1 and rj.2/ .Oz/ D Qj.f

We then have  .2/   .1/  n B zO; rj .Oz/  fzO j .B.f nj .Oz/; //  B zO; rj .Oz/ : .1/

.1/

Therefore, assuming j  1 to be sufficiently large so that the radii rj .Oz/ and rj .Oz/ are sufficiently small, particularly  1= l, we get  n  log x fzO j .B.f nj .Oz/; // n

 log j.fx j /0 .Oz/j



  n log x B.Oz/; QO 1 j.fx j /0 .Oz/j1 n

 log j.fx j /0 .Oz/j   .1/ log x B.Oz/; rj .Oz//   T 0 .q/ C j"2 j .1/ log.rj .Oz// C log QO

62

6 Multifractal Analysis

and   n log x fzO j .B.f nj .Oz/; // n

 log j.fx j /0 .Oz/j



  O xnj /0 .Oz/j1 log x B.Oz/; Qj.f n

 log j.fx j /0 .Oz/j   .2/ log x B.Oz/; rj .Oz//  T 0 .q/  j"2 j:  log.rj.2/ .Oz//  log QO

Hence,    n  n jqj log x fzO j .B.f nj .Oz/; //  .T 0 .q/ C j"2 j/ log j.fx j /0 .Oz/j  0 and     n n jqj log x fzO j .B.f nj .Oz/; //  .T 0 .q/  j"2 j/ log j.fx j /0 .Oz/j  0: So, in either case (as "2 q > 0),     n n q log x fzO j .B.f nj .Oz/; //  .T 0 .q/  j"2 j/ log j.fx j /0 .Oz/j  0 or equivalently,   n 0 xq fzO j .B.f nj .Oz/; // j.f nj /0 .Oz/jqT .q/"2 q  1:

(6.7)

Put t D qT 0 .q/ C T .q/ C "1 C "2 q. Using (6.7) and (6.3) we can then estimate as follows: X

diamqT

0 .q/CT .q/C" C" q 1 2

z2nj .x/

D

X

 nj  fzO .B.f nj .Oz/; //

   n  n 0 diamT .q/C"1 fzO j .B.f nj .Oz/; // diamqT .q/C"2 q fzO j .B.f nj .Oz/; //

z2nj .x/



X

O 1 /t j.f nj /0 .z/j.T .q/C"1 / .Q/ O t j.f nj /0 .Oz/jqT 0 .q/"2 q .Q

z2nj .x/

O 1 /2t D .Q

X

  n n exp q.Snj '.z/  Px j .'//  .T .q/ C "1 / log j.fx j /0 .z/j

z2nj .x/

  0 n  exp q.Px j .'/  Snj '.z/ j.f nj /0 .Oz/jqT .q/"2 q X X  n O 1 /2t e q QO  O 1 /2t  .Q .Q exp q.Snj '.z/  Px j .'// z2nj .x/

z2nj .x/

   0 n n .T .q/ C "1 / log j.fx j /0 .z/j exp q.Px j .'/  Snj '.Oz// j.f nj /0 .Oz/jqT .q/"2 q

6.2 Multifractal Spectrum

O 1 /2t e q QO   .Q

63

X

X

O 1 /2t .Q

z2nj .x/

 n exp q.Snj '.z/  Px j .'//

z2nj .x/

 q  n  0 n .T .q/ C "1 / log j.fx j /0 .z/j x fzO j .B.f nj .Oz/; // j.f nj /0 .Oz/jqT .q/"2 q O 1 /2t e q QO   .Q

X

X

O 1 /2t .Q

z2nj .x/

 n exp q.Snj '.z/  Px j .'//

z2nj .x/

 n .T .q/ C "1 / log j.fx j /0 .z/j O

O 1 /2t e q Q  .Q

a.xnj /

X

Lq;T .q/C"

kD1

1

;x 1.wk .xnj //

O

O 1 /2t e q Q a.xn /e  nj  C.Q j

O

O 1 /2t e q Q ae  nj :  C.Q

Letting j !1 and looking also at (6.6), we thus conclude that H t .Kx ."2 ; l// D 0. In virtue of (6.4) this implies that H t .Kx .T 0 .q/// D 0. Since "1 > 0 and "2 q > 0 were arbitrary, it follows that gx .T 0 .q// D HD.Kx .T 0 .q///  qT 0 .q/ C T .q/:

(6.8)

Let us now prove the opposite inequality. For every s  1 let s be the largest integer in Œ0; s  1 such that  s .x/ 2 X and let sC be the least integer in Œs C 1; C1/ such that  sC .x/ 2 X . It follows from (6.2) applied with j D lC and i D k , that (5.4) is true with s C 1 replaced by kC , and (5.13) is true with l  1 replaced by l , that   l l q  log DO  C q SlC '.y/  PxC .'/  T .q/ log j.fx C /0 .y/j log q;x .B.y; r//  log r log  C  ˛ QO  log j.fxl /0 .y/j and   q  log DO  C q Sk '.y/  Pxk .'/  T .q/ log j.fxk /0 .y/j log q;x .B.y; r//  : k log r log    ˛ QO  log j.fx C /0 .y/j Hence,

lim sup r!0

log q;x .B.y; r// log r n

Px C .'/  SnC '.y/  lim sup q log j.fxn /0 .y/j n!1

!

n

log j.fx C /0 .y/j C T .q/lim sup n 0 n!1 log j.fx / .y/j

(6.9)

64

6 Multifractal Analysis

and log q;x .B.y; r// r!0 log r  Pxn .'/  Sn '.y/ log j.fxn /0 .y/j  lim inf q C T .q/lim inf : n n n!1 n!1 log j.f C /0 .y/j log j.fx C /0 .y/j x

lim inf

(6.10)

Now, given " > 0 and ı" > 0 ascribed to " according to Lemma 6.1, fix an arbitrary ı 2 .0; ı" . Set   .1/ D .1Cı/q;T .q/C.qT 0 .q/C"/ı exp .1 C ı/P .q /  .1/ D ";ı and

  .2/  .2/ D ";ı D .1ı/q;T .q/C.qT 0 .q/C"/ı exp .1 C ı/P .q / :

Since  EP . .1/ / D EP ..1Cı/q;T .q/C.qT 0 .q/C"/ı C .1 C ı/ D EP ..1Cı/q;T .q/C.qT 0 .q/C"/ı

Z P .q /d m



and EP .

.2/



/ D EP ..1ı/q;T .q/C.qT 0 .q/C"/ı C .1  ı/

Z P .q /d m

 D EP ..1ı/q;T .q/C.qT 0 .q/C"/ı ; it follows from Lemmas 6.1 and 9.5, there exists D .q; "; ı/ 2 .0; 1/ such that for all k D 1; 2, and all n  1 sufficiently large, we have n1 log L n.k/ .1/.w/  log  0 for all x 2 XC and all w 2 J n .x/ . Equivalently,

L n.k/ .1/.w/   n : x

x

(6.11) nj

0 , all j  1, all 1  k  a. nj .x/  a, O and all z 2 fx Now, for all x 2 XC .wk .xnj //, define

o n n A.z/ WD y 2 fz j .B.wk .xnj /; // W B.f nj .y/; R/  B.wk .xnj /; / : Note that

a.xnj /

[

[

kD1 z2f nj .w .x // x k nj

A.z/ D J .x/:

(6.12)

6.2 Multifractal Spectrum

65

Fix any q 2 DT and set n

" D sup

0<ıı"

maxf..1 C ı/q; T .q/ C .qT 0 .q/ C "/ı/ ; o ..1  ı/q; T .q/ C .qT 0 .q/ C "/ı/ g :

0 Let x 2 XC . Set

  0 O .q/ C T .q/  "/ : M WD exp Qı.qT

Then, using (6.12), Lemma 2.3 (for the potential .x; z/ 7! log jfx0 .z/j, (6.2), and (6.11), we obtain   n   0 n q;x fy 2 Jx W q;x fy j .B.f nj .y/; R//  j.fx j /0 .y/j.qT .q/CT .q//C"g   n  n  0 D q;x fy 2 Jx W q;x fy j .B.f nj .y/; R// j.fx j /0 .y/jqT .q/CT .q/"  1g   nj  n  0 ı fy .B.f nj .y/; R// j.fx j /0 .y/jı.qT .q/CT .q/"/  1g D q;x fy 2 Jx W q;x Z  n  nj 0 ı q;x fy .B.f nj .y/; R// j.fx j /0 .y/jı.qT .q/CT .q/"/dq;x .y/  Jx

a.xnj /



X

Z

X

kD1 z2f nj .w .x // A.z/ x k nj n

j.fx j /0 .y/jı.qT a.xnj /



X

X

 nj  ı fy .B.f nj .y/; R/// q;x

0 .q/CT .q/"

/dq;x .y/

  nj ı q;x fz .B.wk .xnj /; ///

kD1 z2f nj .w .x // x k nj n

j.fx j /0 .z/jı.qT a.xnj /

M

X

X

0 .q/CT .q/"/

M q;x .A.z//

 nj  n 0 ı fz .B.wk .xnj /; /// j.fx j /0 .z/jı.qT .q/CT .q/"/ q;x

kD1 z2f nj .w .x // x k nj

 n  q;x fz j .B.wk .xnj /; ///

a.xnj /

DM

X

X

 n  nj 0 1Cı q;x fz .B.wk .xnj /; /// j.fx j /0 .z/jı.qT .q/CT .q/"/

kD1 z2f nj .w .x // x k nj a.xnj /



MD "

X

X

   n n exp .1 C ı/q Snj .z/  Px j ..z/  .1 C ı/Px .q j /

kD1z2f nj .w .x // x k nj

66

6 Multifractal Analysis n

j.fx j /0 .z/j.T .q/.1Cı/Cı.qT a.xnj /

D

MD "

X

0 .q/T .q/C"//

n

exp..1 C ı/Pxj .q .z///

   n n exp .1 C ı/q Snj .z/  Px j ..z/  .1 C ı/Px .q j /

X

kD1z2f nj .w .x // x k nj n

j.fx j /0 .z/j.T .q/C.qT

0 .q/C"/ı/

n

exp..1 C ı/Px j .q .z///

a.xnj /

D MD "

X

kD1

L

nj .1/

x

.1/.wk .xnj //  MD " a nj :

(6.13)

Therefore, 1 X

  nj   0 nj q;x fy 2 Jx W q;x fy .B.f nj .y/; R//  j.fx /0 .y/j.qT .q/CT .q//C" g < C1:

j D1

q

Hence, by the Borel–Cantelli Lemma, there exists a measurable set J1;";x  Jx q such that q;x .J1;";x / D 1 and n    n # j  1 W q;x fy 2 Jx W q;x fy j .B.f nj .y/; R// n

 j.fx j /0 .y/j.qT

0 .q/CT .q//"

g

o

< 1: (6.14)

Arguing similarly, with the function  .1/ replaced by  .2/ , we produce a measurable q q set J2;";x  Jx such that q;x .J2;";x / D 1 and n   n  # j  1 W q;x fy 2 Jx W q;x fy j .B.f nj .y/; R// n

 j.fx j /0 .y/j.qT Set Jxq D

1 \

0 .q/CT .q//C"

g

o

< 1: (6.15)

q q J1;1=n;x \ J2;1=n;x :

nD1

Then have

q q;x .Jx /

q

D 1 and, it follows from (6.13) and (6.1), that for all y 2 Jx , we n

lim

j !1

Since limn!1

n nC

q.Px j .'/  Snj '.y// n

log j.fx j /0 .y/j

D qT 0 .q/:

D 1, it thus follows from (6.9) and (6.10) that dq;x .y/ D qT 0 .q/ C T .q/;

(6.16)

6.3 Analyticity of the Multifractal Spectrum for Uniformly Expanding Random Maps

67

and (recall that 1;x D x and T .1/ D 0) lim

r!0

log x .B.x; r// D T 0 .q/ log r

q

q

for all y 2 Jx . As the latter formula implies that Jx  K.T 0 .q//, and as q q;x .Jx / D 1, applying (6.16), we get that gx .T 0 .q// D HD.Kx .T 0 .q///  HD.Jxq // D qT 0 .q/ C T .q/: t u

Combining this formula with (6.8) completes the proof. As an immediate consequence of this proposition we get the following theorem.

Theorem 6.3 Suppose that f .x; z/ D ..x/; fx .z// is a conformal random expanding map. Then the Legendre conjugate, gW Range.T 0 / ! Œ0; C1/, to the temperature function R 3 q 7! T .q/ is differentiable everywhere except a countable set of points, call it DT , and there exists a measurable set Xma  X with m.Xma / D 1 such that for every ˛ 2 DT / and every x 2 Xma , we have gx .˛/ D g.˛/:

6.3 Analyticity of the Multifractal Spectrum for Uniformly Expanding Random Maps Now, as in Chap. 9.4, we assume that we deal with a conformal uniform random expanding map. In particular, the essential infimum of x is larger than some  > 1 and functions Hx , n .x/, j.x/ are finite. In addition, we have that there exist constants L and c > 0 such that Sn 'x .y/  nc C L

(6.17)

for every y 2 Jx and n and EP .'/ D 0. With these assumptions we can get the following property of the function T . Proposition 6.4 Suppose that f WJ ! J is a conformal uniformly random expanding map. Then the temperature function T is real-analytic and for every q, we have R J 'dq 0 < 0: (6.18) T .q/ D R 0 J log jf jdq Proof. The potentials 'q;x;t .y/ WD q.'x .y/  Px .'//  t log jfx0 .y/j

68

6 Multifractal Analysis

extend by the same formula to holomorphic functions C  C 3 .q; t/ 7! 'q;x;t .y/. Since these functions are in fact linear, we see that the assumptions of Theorem 9.17 are satisfied, and therefore the function R  R 3 .q; t/ 7! EP .q; t/ is real-analytic. Since jfx0 .y/j > 0, in virtue of Proposition 9.18 we obtain that @EP .q; t/ D dt

Z J

log jfx0 jdq;x;t d m.x/ < 0:

(6.19)

Hence, we can apply the Implicit Function Theorem to conclude that the temperature function R 3 q 7! T .q/ 2 R, satisfying the equation, EP .q; T .q// D 0; is real-analytic. Hence, 0D

@EP .q; t / ˇˇ @EP .q; t / ˇˇ d EP .'q / C T 0 .q/: D ˇ ˇ t DT .q/ t DT .q/ dq @q @t

Then ˇ

T 0 .q/ D 

@EP .q;t / ˇ @q t DT .q/ ˇ @EP .q;t / ˇ @t t DT .q/

R D

J

R

J .'x

D R

J

 Px /dq;x d m.x/

 log jfx0 jdq;x d m.x/

R R 'x dq;x d m.x/  X Px d m.x/ J 'dq R R D : 0 0 J log jfx jdq;x d m.x/ J log jf jdq

So, we obtain (6.18). It follows, in particular, that T 0 .q/ < 0; since by (6.17), the integral

R J

'dq is negative.

(6.20) t u

Combining this proposition with Proposition 6.2 we get the following result which concludes this section. Theorem 6.5 Suppose that f WJ ! J is a conformal uniformly random expanding map. Then the Legendre conjugate, g W Range.T 0 / ! Œ0; C1/, to the temperature function R 3 q 7! T .q/ is real-analytic, and there exists a measurable set Xma  X with m.Xma / D 1 such that for every ˛ 2 Range.T 0 / and every x 2 Xma , we have gx .˛/ D g.˛/:

Chapter 7

Expanding in the Mean

In this chapter we show that the main achievements of this manuscript, including thermodynamical formalism, Bowen’s formula and multifractal analysis, also hold for a class of random maps satisfying an allegedly weaker expanding condition Z log x dm.x/ > 0: We start with a precise definition of this class. Then we explain how this case can be reduced to random expanding maps by looking at an appropriate induced map. The picture is completed by providing and discussing a concrete map that is not expanding but expanding in the mean.

7.1 Definition of Maps Expanding in the Mean Let T W J ! J be a skew-product map as defined in Sect. 2.2 satisfying the properties of Measurability of the Degree and Topological Exactness. Such a random map is called expanding in the mean, if for some  > 0 and some measurable function X 3 x 7! x 2 RC with Z log x dm.x/ > 0; we have that all inverse branches of every Txn are well defined on balls of radii  and are .xn /1 -Lipschitz continuous. More precisely, for every y D .x; z/ 2 J and every n 2 N, there exists Tyn W B n .x/ .T n .y/; / ! Jx

V. Mayer et al., Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry, Lecture Notes in Mathematics 2036, DOI 10.1007/978-3-642-23650-1 7, © Springer-Verlag Berlin Heidelberg 2011

69

70

7 Expanding in the Mean

such that 1. T n ı Tyn D IdjB n .x/ .T n .y/;/ and Tyn .Txn .z// D z,   2. %.Tyn .z1 /; Tyn .z2 //  .xn /1 %.z1 ; z2 / for all z1 ; z2 2 B n .x/ T n .y/;  .

7.2 Associated Induced Map In this section we show how the expanding in the mean maps can be reduced to our setting from Sect. 2.3. Let T W J ! J be an expanding in the mean random map. To this map and to a set A  X of positive measure we associate an induced map T in the following way. Let A be the first return map to the set A, that is A .x/ D minfn  1 W  n .x/ 2 Ag: Define also A .x/ WD  A .x/ .x/

and A;x WD

AY .x/1

 j .x/ :

j D0

Then the induced map T is the random map over .A; B; mA / defined by T x D TxA .x/

for a.e. x 2 A:

The following lemma show that the set A can be chosen such that T is an expanding random map. Lemma 7.1 There exists a measurable set A  X with m.A/ > 0 such that A;x > 1

for all x 2 A :

Proof. First, define inductively A1 WD fx W log x > 0g and, for k  1,

AkC1 WD fx 2 Ak W log Ak ;x > 0g:

Since Z 0<

X

Z log x dm.x/ D

Z A1

log A1 ;x dm.x/ D

Ak

log Ak ;x dm.x/;

7.2 Associated Induced Map

71

we have that m.Ak / > 0 for all k  1. Obviously, the sequence .Ak /1 is kD1 decreasing. Let 1 \ AD Ak and E D X n A : kD1

Notice that the points x 2 E have the property that log xn  0 for some n  1. Claim: m.A/ > 0. If on the contrary m.A/ D limk!1 m.Ak / D 0, then m.E/ D 1. Since the measure m is -invariant, we have that m.E1 / D 1 where E1 D

1 \

 n .E/ :

nD0

For x 2 E1 we have that log xn  0 for infinitelyR many n  1. This contradicts Birkhoff’s Ergodic Theorem since, by hypothesis, log x > 0. Therefore the set A has positive measure. Since m.A/ > 0, A is almost surely finite. Now let x 2 A. Then, for every point  j .x/, j D 1; : : : ; A .x/  1, we can find k.j / such that  j .x/ 2 X n Ak.j / . Put K.x/ D maxfk.j / W j D 1; : : : ; A .x/  1g C 1: Hence x and A .x/ are in AK.x/ and  j .x/ … AK.x/ for j D 1; : : : ; A .x/  1. Hence A .x/ D AK.x/ .x/, and therefore A;x D AK.x/ ;x > 1:

t u

In the following A  X will be some set coming from Lemma 7.1 and T D T A the associated induced map. For this map, we have to consider the following appropriated class of H¨older potentials. First, to every y D .x; z/ we associate the neighborhood 1 [    U.z/ D Tyn B n .x/ T n .y/;   Jx : nD0

Fix ˛ 2 .0; 1. As in Sect. 2.7 a function ' 2 C 1 .J / is called H¨older continuous with an exponent ˛ provided that there exists a measurable function H W X ! Œ1; C1/, x 7! Hx , such that Z X

log Hx dm.x/ < 1

(7.1)

72

7 Expanding in the Mean

and v˛ .'x /  Hx

for a.e. x 2 X:

The subtlety here is that the infimum in the definition (2.11) of v˛ is now taken over all z1 ; z2 2 Jx with z1 ; z2 2 U.z/, z 2 Jx . For example, any function, which is Hx -H¨older on the entire set Jx is fine. Let T be an expanding in the mean random map and ' a H¨older potential according to the definition above. Having associated to T the induced map T , one naturally has to replace the potential ' by the induced potential A .x/1

' x .z/ D

X

' j .x/ .Txj .z//:

j D0

Although, it is not clear if the potential ' satisfies the condition (7.1), the choice of the neighborhoods U.z/ and the definition of H¨older potentials make that Lemma 2.3 still holds. This gives us an important control of the distortion which is what is needed in the rest of the paper rather than the condition (7.1) leading to it. The hypothesis (7.1) is only used in the proof of Lemma 2.3.

7.3 Back to the Original System In this section we explain how to get the Thermodynamic Formalism for the original system. With the preceding notations, for the expanding induced map T the Thermodynamical Formalism of Chap. 3 and, in particular, the Theorems 3.1 and 3.2 do apply. We denote by  x , x and q x , x 2 A, the resulting conformal and invariant measures and the invariant density, respectively, for T . We now explain how the corresponding objects can be recovered for the original map T . Notice that this is possible since we only induced in the base system. First, we consider the case of the conformal measures. Let  x , x 2 A be the measure such that  L x  A .x/ D x  x : If x 2 A we put x D  x . If x … A, then by ergodicity of , almost surely there exists k 2 N, such that  k .x/ 2 A and  j .x/ … A for j D 0; : : : ; k  1. Then we put .Lxk /  k .x/ x D : (7.2) Lxk .1/ Therefore, the family fx gx2X 0 is a family of probability measures well defined for X in a subset X 0 of X with full measure. This family of measures has the conformality property Lx  k .x/ D x x ; where x D .x/ .Lx 1/, x 2 X 0 . Notice also that EP .'/ D EP .'/.

7.4 An Example

73

Similarly, from the family fx gx2A of T -invariant measures one can recover a family fx gx2X of invariant measures for the original map T . Indeed, for x 2 A and j D 0; : : : ; A .x/  1 it suffices to put  j .x/ D x ı Txj : Then, with q j .x/ D Lxj .q x /, we have that d j .x/ D q j .x/ d j .x/ : Hence Theorems 3.1 and 3.2 among with all statistical consequences hold for the original map. Moreover, since EP .' t / D EP .'t / their zeros coincide and consequently Bowen’s Formula and the Multifractal Analysis are also true for conformal expanding in the mean random maps.

7.4 An Example Here is an example of an expanding in the mean random system. Define  f0 .x/ D

1 x 2

C 15 x 2 if x 2 Œ0; 1=3 2 16x  15 if x 2 Œ15=16; 1

and f1 .x/ D 16x.mod 1/ for x 2 Œ0; 1=16 [ Œ15=16; 1: Let X D f0; 1gZ,  be the shift transformation and m be the standard Bernoulli measure. For x D .: : : ; x1 ; x0 ; x1 ; : : :/ 2 X define fx D fx0 , fxn D f n1 .x/ ı f n2 .x/ ı : : : ı fx and Jx D

1 \

.fxn /1 .Œ0; 1/:

nD0

For this map, 0 D 1=2 and 1 D 16 are the best expanding constants that one can take. With these constants we have Z log x dm.x/ > 0: Therefore, the map is expanding in the mean but not expanding. Note that the size of each component of fxn .Œ0; 1/ is bounded by an D 16n1 .1=2/n0 ;

(7.3)

74

7 Expanding in the Mean

where ni WD #fj D 0; : : : ; n  1 W xj D i g, i D 0; 1. Since 1 n0 n1 D lim D n!1 n n!1 n 2 lim

almost surely, we have that limn!1 an D 0. Hence, for almost every x 2 X , Jx is a Cantor set. Moreover, by (7.3), almost surely we have, that, 1 log 2n 16n1 t .1=2/n0 t n!1 n     n1 n0 3  log 2  t lim log 16  log 2 D log 2 1  t : n!1 n n 2

EP .t/  lim

Therefore, by Bowen’s Formula, the Hausdorff dimension of almost every fiber Jx is smaller than or equal to 2=3. Notice however that for some choices of x 2 X the fiber Jx contains open intervals.

Chapter 8

Classical Expanding Random Systems

Having treated a very general situation up to here, we now focus on more concrete random repellers and, in the next section, random maps that have been considered by Denker and Gordin. The Cantor example of Chap. 5.3 and random perturbations of hyperbolic rational functions like the examples considered by Br¨uck and B¨uger are typical random maps that we consider now. We classify them into quasi-deterministic and essential systems and analyze then their fractal geometric properties. Here as a consequence of the techniques we have developed, we positively answer the question of Br¨uck and B¨uger (see [9] and Question 5.4 in [8]) of whether the Hausdorff dimension of almost all (most) naturally defined random Julia sets is strictly larger than 1. We also show that in this same setting the Hausdorff dimension of almost all Julia sets is strictly less than 2.

8.1 Definition of Classical Expanding Random Systems Let .Y; / be a compact metric space normalized by diam.Y / D 1 and let U  Y . A repeller over U will be a continuous open and surjective map T W VT ! U where VT , the closure of the domain of T , is a subset of U . Let  > 1 and consider R D R.U;  / D fT W VT ! U  -expanding repeller over U g : Concerning the randomness we will consider classical independently and identically distributed (i.i.d.) choices. More precisely, we suppose the repellers Tx0 ; Tx1 ; :::; Txn ; :::

(8.1)

are chosen i.i.d. with respect to some arbitrary probability space .I; F0 ; m0 /. This gives rise to a random repeller Txn0 D Txn1 ı:::ıTx0 , n  1. The natural associated Julia set is V. Mayer et al., Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry, Lecture Notes in Mathematics 2036, DOI 10.1007/978-3-642-23650-1 8, © Springer-Verlag Berlin Heidelberg 2011

75

76

8 Classical Expanding Random Systems

Jx D

\ n1

Txn .U / where x D .x0 ; x1 ; :::/ : 0

Notice that compactness of Y together with the expanding assumption, we recall that  -expanding means that the distance of all points z1 ; z2 with .z1 ; z2 /  T is expanded by the factor  , implies that Jx is compact and also that the maps T 2 R are of bounded degree. A random repeller is therefore the most classical form of a uniformly expanding random system. The link with the setting of the preceding sections goes via natural extension. Set X D I Z , take the Bernoulli measure m D mZ0 and let the ergodic invariant map  be the shift map  W I Z ! I Z . If  W X ! I is the projection on the 0th coordinate and if x 7! Tx is a map from I to R then the repeller (8.1) is given by the skew-product   T .x; z/ D .x/; T.x/ .z/ ;

.x; z/ 2 J D

[

fxg  Jx :

(8.2)

x2X

The particularity of such a map is that the mappings Tx do only depend on the 0th coordinate. It is natural to make the same assumption for the potentials, i.e. 'x D '.x/ . We furthermore consider the following continuity assumptions: (T0) I is a bounded metric space. (T1) .x; z/ 7! Tx1 .z/ is continuous from J to K .U /, the space of all non-empty compact subsets of U equipped with the Hausdorff distance. (T2) For every z 2 U , the map x 7! 'x .z/ is continuous. A classical expanding random system is a random repeller together with a potential depending only on the 0th-coordinate such that the conditions (T0), (T1) and (T2) hold. Example 8.1 Suppose V; U are open subsets of C with V compactly contained in U and consider the set R.V; U / of all holomorphic repellers T W VT ! U having uniformly bounded degree and a domain VT  V . This space has natural topologies, for example the one induced by the distance      T1 ; T2 D dH VT1 ; VT2 C k.T1  T2 /jVT1 \VT2 k1 ; where dH denotes the Hausdorff metric. Taking then geometric potentials t log jT 0 j we get one of the most natural example of classical expanding random system. Proposition 8.2 The pressure function x 7! Px .'/ of a classical expanding random system is continuous. 1.y/ Proof. We have to show that x 7! x is continuous and since Lxn 1.y/=Lxn1 1 converges uniformly to x for every y 2 U (see Lemma 3.32) it suffices to show that x 7! Lxn 1.y/ does depend continuously on x 2 X . In order to do so, we first show that condition (T1) implies continuity of the function .x; y/ 7! #Tx1 .y/.

8.1 Definition of Classical Expanding Random Systems

77

Let .x; y/ 2 X  U and fix 0 < 0 < such that B.w1 ; 0 / \ B.w2 ; 0 / D ; for all disjoint w1 ; w2 2 Tx1 .y/. From (T1) follows that there exists ı > 0 such that 0 dH .Tx1 .y/; Tx1 0 .y // 

; 2

  whenever % .x; y/; .x 0 ; y 0 /  ı :

But this implies that for every w 2 Tx1 .y/ there exists at least one preimage w0 2 0 0 1 0 1 Tx1 0 .y / \ B.w; /. Consequently, #Tx 0 .y /  #Tx .y/. Equality follows since Tx 0 is injective on every ball of radius 0 , a consequence of the expanding condition. Let x 2 X , let W be a neighborhood of x and let y 2 U . From what was proved before we have that for every w 2 Tx1 .y/, there exists a continuous function x 0 7! zw .x 0 / defined on W such that Tx 0 .zw .x 0 // D y, zw .x/ D w and 0 1 Tx1 0 .y/ D fzw .x / W w 2 Tx .y/g:

The proposition follows now from the continuity of 'x , i.e. from (T2).

t u

We say that a function g W I Z ! R is past independent if g.!/ D g. / for any 1 Z !; 2 I Z with !j1 0 D j0 . Fix  2 .0; 1/ and for every function g W I ! R set v .g/ D supfv;n .g/g; n0

where

v;n .g/ D  n supfjg.!/  g. /j W !jn0 D jn0 g:

Denote by H the space of all bounded Borel measurable functions g W I Z ! R for which v .g/ < C1. Note that all functions in H are past independent. Let Z be the set of negative integers. If I is a metrizable space and d is a bounded metric on I , then the formula 1 X 2n d.!n ; n / dC .!; / D nD0 Z

defines a pseudo-metric on I , and for every 2 I Z , the pseudo-metric dC restricted to f g  N, becomes a metric which induces the product (Tychonoff) topology on f g  N. Theorem 8.3 Suppose that T W J ! J and W J ! R form a classical expanding random system. Let  W I Z ! .0; C1/ be the corresponding function coming from Theorem 3.1. Then both functions  and P . / belong to H with some  2 .0; 1/, and both are continuous with respect to the pseudo-metric dC . Proof. Let y 2 U be any point. Fix n  0 and !; 2 I Z with !jn0 D jn0 . By Lemma 3.32, we have ˇ ˇ ˇ ˇ ˇ L nC1 1.y/ ˇ ˇ ˇ L nC1 1.y/ ˇ ! ˇ ˇ ˇ  n  ! ˇ  A   ˇ  A n and ˇ n ˇ n ˇ L.!/ 1.y/ ˇ ˇ ˇ L./ 1.y/

78

8 Classical Expanding Random Systems

with some constants A > 0 and  2 .0; 1/. Since, by our assumptions, L!nC1 1.y/ D n n LnC1 1.y/ and L.!/ 1.y/ D L./ 1.y/, we conclude that j!   j  2A n . So, v ./  2A: Since, by Proposition 8.2, the function  W I Z ! .0; C1/ is continuous, it is therefore bounded above and separated from zero. In conclusion, both functions  and P . / belong to H with some  2 .0; 1/, and both are continuous with respect to the pseudo-metric dC . t u Corollary 8.4 Suppose that T W J ! J and W I Z ! R form a classical expanding random system. Then the number (asymptotic variance of P . /) 1 n!1 n

 2 .P . // D lim

Z  2 Sn .P . //  nEP . / dm  0

exists, and the Law of Iterated Logarithm holds, i.e. m-a.e we have p p Pxn  nEP . / P n . /  nEP . /  lim sup xp D 2 2 .P . //:  2 2 .P . // D lim inf p n!1 n!1 n log log n n log log n

Proof. Let  W I Z ! I be the canonical projection onto the 0th coordinate and let G D  1 .B/, where B is the -algebra of Borel sets of I . We want to apply Theorem 1.11.1 from [24]. Condition (1.11.6) is satisfied with the function (object being here as in Theorem 1.11.1 and by no means our potential!) identically equal to zero since jm.A \ B/  m.A/m.B/j D 0 for every A 2 G0m WD T j G \  1 .G / \ : : :  m .G / and B 2 Gn1 D C1 .G /, whenever n > m. j Dn  R 2Cı d m is finite (for every ı > 0) since, by Theorem 8.3, The integral jP . /j the pressure function P . / is bounded. This then implies that for all n  1, jP . /.!/  E .P . /jG0n /.!/j  v .P . // n , where v .P . // < C1. Therefore, Z jP . /  E .P . /jG0n /jd m  v .P . // n ; whence condition (1.11.7) from [24] holds. Finally, P . / is G01 -measurable, since P . / belonging to H is past independent. We have thus checked all the assumptions of Theorem 1.11.1 from [24] and, its application yields the existence of the asymptotic variance of P . / and the required Law of Iterated Logarithm to hold. t u Proposition 8.5 Let g 2 H . Then  2 .g/ D 0 if and only if there exists u 2 C..supp.m0 //Z / such that g  m.g/ D u  u ı  holds throughout .supp.m0 //Z . Proof. Denote the topological support of m0 by S . The implication that the cohomology equation implies vanishing of  2 is obvious. In order to prove the other implication, assume without loss of generality that m.g/ D 0. Because of Theorem 2.51 from [16] there exists u 2 L2 .m/ independent of the past (as so is g) such that

8.1 Definition of Classical Expanding Random Systems

79

g D uuı

(8.3)

in the space L2 .m/. Our goal now is to show that u has a continuous version and (8.3) holds at all points of S Z . In view of Lusin’s Theorem there exists a compact set K  S Z such that m.K/ > 1=2 and the function ujK is continuous. So, in view of Birkhoff’s Ergodic Theorem there exists a Borel set B  S Z such that m.B/ D 1, for S every ! 2 B,  n .!/ 2 K with asymptotic frequency > 1=2, u is well defined S n n on C1 .B/, and (8.3) holds on C1 .B/. Let Z D f1; 2; : : :g nD1  nD1  and let fm g2I Z be the canonical system of conditional measures for the partition ff gI N g2I Z with respect to the measure m. Clearly, each measure m , projected to I N , coincides with mC . Since m.B/ D 1, there exists a Borel set F  S Z such that m .F / D 1 and m .B \ .f g  I N // D 1 for all 2 F , where m is the infinite product measure on S Z . Fix 2 F and set Z D pN .B \ .f g  I N //, where pN W I Z ! I N is the natural projection from I Z to I N . The property that m .B \ .f g  I N // D 1 implies that Z D S N . Now, it immediately follows from the definitions of Z and B that for all x; y 2 Z there exists an increasing sequence .nk /1 of positive integers such that  nk . x/;  nk . y/ 2 K for all k  1. For kD1 every 0 < q  nk we have from (8.3) that nk q

X 

 g. j . nk . y///  g. j . nk . x///

j D0 nk X

C

  g. j . nk . y///  g. j . nk . x///

j Dnk qC1

D .u. nk . y//  u. nk . x// C .u. x/  u. y//: Since g 2 H , we have nk q

X

g. j . nk . y///  g. j . nk . y///



j D0 nk q



X

jg. j . nk . y///  g. j . nk . y///j

j D0 nk q



X

v .g/ nk j  v .g/.1  /1  q :

j D0

Now, fix " > 0. Take q  1 so large that v .g/.1  /1  q < "=2: Since the function g W I Z ! R is uniformly continuous with respect to the pseudo-metric d , " there exists ı > 0 such that jg.b/  g.a/j < 2q whenever d.a; b/ < ı. Assume that i i d.x; y/ < ı (so d. . x/;  . y// < ı for all i  0). It follows now that for every k  1 we have

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8 Classical Expanding Random Systems

ju. x/  u. y/j  v .g/.1  /1  q C q

" C ju. nk . y//  u. nk . x//j 2q

" " C C ju. nk . y//  u. nk . x//j 2 2 " D " C C ju. nk . y//  u. nk . x//j: 2



Since  nk . x/;  nk . y/ 2 K for all k  1, since limk!1 d. nk . x/,  nk . y// D 0, and since the function u, restricted to K, is uniformly continuous, we conclude that lim ju. nk . y//  u. nk . x//j D 0 :

k!1

We therefore get that ju. x/  u. y/j < " and this shows that the function u is uniformly continuous (with respect to the metric d ) on the set W D

[

B \ .f g  I N /:

2F

Since W D S Z (as m.W / D 1) and since u is independent of the past, we conclude that u extends continuously to S Z . Since both sides of (8.3) are continuous functions, and the equality in (8.3) holds on the dense set W \  1 .W /, we are done. t u

8.2 Classical Conformal Expanding Random Systems If a classical system is conformal in the sense of Definition 5.1 and if the potential is of the form ' D t log jf 0 j for some t 2 R then we will call it classical conformal expanding random system Theorem 8.6 Suppose f W J ! J is a classical conformal expanding random system. Then the following hold. (a) The asymptotic variance  2 .P .h// exists. (b) If  2 .P .h// > 0, then the system f W J ! J is essential, H h .Jx / D 0 and P h .Jx / D C1 for m-a.e. x 2 I Z . (c) If, on the other hand,  2 .P .h// D 0, then the system f W J ! J , reduced in the base to the topological support of m (equal to supp.m0 /Z ), is quasideterministic, and then for every x 2 supp.m/, we have: (c1) xh is a geometric measure with exponent h. (c2) The measures xh , H h jJx , and P h jJx are all mutually equivalent with Radon–Nikodym derivatives separated away from zero and infinity independently of x 2 I Z and y 2 Jx . (c3) 0 < H h .Jx /; P h .Jx / < C1 and HD.Jx / D h.

8.3 Complex Dynamics and Br¨uck and B¨uger Polynomial Systems

81

Proof. It follows from Corollary 8.4 that the asymptotic variance  2 .P .h// exists. Combining this corollary (the Law of Iterated Logarithm) with Remark 5.5, we conclude that the system f W J ! J is essential. Hence, item (b) follows from Theorem 5.7(a). If, on the other hand,  2 .P .h// D 0, then the system f W J ! J , reduced in the base to the topological support of m (equal to supp.m0 /Z ), is quasi-deterministic because of Proposition 8.5, Theorem 8.3 (P .h/ 2 H ), and Remark 5.6. Items (c1)–(c4) follow now from Theorem 5.7(b1)–(b4). We are done. t u As a consequence of this theorem we get the following. Theorem 8.7 Suppose f W J ! J is a classical conformal expanding random system. Then the following hold: (a) Suppose that for every x 2 I Z , the fiber Jx is connected. If there exists at least one w 2 supp.m/ such that HD.Jw / > 1, then HD.Jx / > 1 for m-a.e. x 2 I Z. (b) Let d be the dimension of the ambient Riemannian space Y . If there exists at least one w 2 supp.m/ such that HD.Jw / < d , then HD.Jx / < d for m-a.e. x 2 I Z. Proof. Let us proof first item (a). By Theorem 8.6(a) the asymptotic variance  2 .P .h// exists. If  2 .P .h// > 0, then by Theorem 8.6(a) the system f W J ! J is essential. Thus the proof is concluded in exactly the same way as the proof of Theorem 5.8(3). If, on the other hand,  2 .P .h// D 0, then the assertion of (a) follows from Theorem 8.6(c4) and the fact that HD.Jw / > 1 and w 2 supp.m/. Let us now prove item (b). If  2 .P .h// > 0, then, as in the proof of item (a), the claim is proved in exactly the same way as the proof of Theorem 5.8(4). If, on the other hand,  2 .P .h// D 0, then the assertion of (b) follows from Theorem 8.6(c4) and the fact that HD.Jw / < d and w 2 supp.m/. We are done. t u

¨ and Buger ¨ 8.3 Complex Dynamics and Bruck Polynomial Systems We now want to describe some classes of examples coming from complex dynamics. They will be classical conformal expanding random systems as well as G-systems defined later in this section. Indeed, having a sequence of rational functions F D O O ffn g1 nD0 on the Riemann sphere C we say that a point z 2 C is a member of the Fatou set of this sequence if and only if there exists an open set Uz containing z such that the family of maps ffn jUz g1 nD0 is normal in the sense of Montel. The Julia O of the Fatou set of F . For every set J .F / is defined to be the complement (in C) 1 k  0 put Fk D ffkCn gnD0 and observe that J .FkC1 / D fk .J .Fk //:

(8.4)

82

8 Classical Expanding Random Systems

Now, consider the maps fc .z/ D fd;c .z/ D zd C c,

d  2:

Notice that for every " > 0 there exists ı" > 0 such that if jcj  ı" , then fc .B.0; "//  B.0; "/: Consequently, if ! 2 B.0; "/Z , then J .ff!n g1 nD0 /  fz 2 C W jzj  "g and jf!0 k .z/j  d "d 1

(8.5)

n o p d 1 /. Let ı.d / D sup ı W " > 1=d : Fix 0 < ı < ı.d /. for all z 2 J .ff!kCn g1 " nD0 p d 1 1=d such that ı < ı" . Therefore, by (8.5), Then there exists " > jf!0 k .z/j  d "d 1

(8.6)

for all ! 2 B.0; ı/Z , all k  0 and all z 2 J .ff!kCn g1 nD0 /. A straight calculation ([8], p. 349) shows that ı.2/ D 1=4. Keep 0 < ı < ı.d / fixed. Let Fd;ı D ffd;c W c 2 B.0; ı/g: Consider an arbitrary ergodic measure-preserving transformation  W X ! X . Let m be the corresponding invariant probability measure. Let also H W X ! Fd;ı be an arbitrary measurable function. Set fd;x D H.x/ for all x 2 X . For every S x2X let Jx be the Julia set of the sequence ff n .x/ g1 , and then J D x2X Jx . nD0 Note that, because of (8.4), fd;x .Jx / D J.x/ : Thus, the map fd;ı;;H .x; y/ D ..x/; fd;x .y//;

x 2 X; y 2 Jx ;

(8.7)

defines a skew product map in the sense of Chap. 2.2 of our paper. In view of (8.7), when  W X ! X is invertible, fd;ı;;H is a distance expanding random system, and, since all the maps fx are conformal, fd;ı;;H is a conformal measurably expanding system in the sense of Definition 5.1. As an immediate consequence of Theorem 5.2 we get the following. Theorem 8.8 Let  W X ! X be an invertible measurable map preserving a probability measure m. Fix an integer d  1 and 0 < ı < ı.d /. Let H W X ! Fd;ı be an arbitrary measurable function. Finally, let fd;ı;;H be the distance expanding random system defined by formula (8.7). Then for almost all x 2 X the Hausdorff dimension of the Julia set Jx is equal to the unique zero of the expected value of the pressure function. Theorem 8.9 For the conformal measurably expanding systems fd;ı;;H defined in Theorem 8.8 the multifractal theorem, Theorem 6.4 holds.

8.3 Complex Dynamics and Br¨uck and B¨uger Polynomial Systems

83

We now define and deal with Br¨uck and B¨uger polynomial systems. We still keep d  2 and 0 < ı < ı.d / fixed. Let X D B.0; ı/Z and let  W B.0; ı/Z ! B.0; ı/Z to be the shift map denoted in the sequel by . Consider any Borel probability measure m0 on B.0; ı/ which is different from ı0 , the Dirac ı measure supported at 0. Define H W X ! Fd;ı by the formula H.!/ D fd;!0 . The corresponding skew-product map fd;ı W J ! J is then given by the formula: fd;ı .!; z/ D ..!/; fd;!0 .z// D ..!/; zd C !0 /; and fd;ı;! .z/ D zd C !0 acts from J! to J.!/ , where J! D J ..fd;!n /1 nD0 /: Then f W J ! J is called Br¨uck and B¨uger polynomial systems. Clearly, f W J ! J is a classical conformal expanding random system. In [8] Br¨uck speculated on p. 365 that if ı < 1=4 and m0 is the normalized Lebesgue measure on B.0; ı/, then HD.J! / > 1 for mC -a.e. ! 2 B.0; ı/N with respect to the skew-product map .!; z/ 7! ..!/; z2 C !0 /: In [9] this problem was explicitly formulated by Br¨uck and B¨uger as Question 5.4. Below (Theorem 8.10) we prove a more general result (with regard the measure on B.0; ı/ and the integer d  2 being arbitrary), which contains the positive answer to the Br¨uck and B¨uger question as a special case. In [8] Br¨uck also proved that if ı < 1=4 and the above skew product is considered then 2 .J! / D 0 for all ! 2 B.0; ı/N , where 2 denotes the planar Lebesgue measure on C. As a special case of Theorem 8.10 below we get a partial strengthening of Br¨uck’s result saying that HD.J! / < 2 for mC -a.e. ! 2 B.0; ı/N . Our results are formulated for the product measure m on B.0; ı/Z , but as mC is the projection from B.0; ı/Z to B.0; ı/N and as the Julia sets J! , ! 2 B.0; ı/Z depend only on !jC1 , i.e. on the future of !, 0 the analogous results for mC and B.0; ı/N follow immediately. Proving what we have just announced, note that if !0 2 supp.m0 / n f0g, then HD.J!01 // D HD.J .f!0 // 2 .1; 2/ (the equality holds already on the level of sets: J!01 D J .f!0 /), and by [9], all the sets J! , ! 2 B.0; ı/Z , are Jordan curves. Hence, since f W J ! J is a classical conformal expanding random system, as an immediate application of Theorem 8.7 we get the following. Theorem 8.10 If d  2 is an integer, 0 < ı < ı.d /, the skew-product map fd;ı W J ! J is given by the formula fd;ı .!; z/ D ..!/; fd;!0 .z// D ..!/; zd C !0 /;

84

8 Classical Expanding Random Systems

and m0 is an arbitrary Borel probability measure on B.0; ı/, different from ı0 , the Dirac ı measure supported at 0, then for m-almost every ! 2 B.0; ı/Z we have 1 < HD.J! / < 2.

8.4 Denker–Gordin Systems We now want to discuss another class of expanding random maps. This is the setting from [12]. In order to describe this setting suppose that X0 and Z0 are compact metric spaces and that 0 W X0 ! X0 and T0 W Z0 ! Z0 are open topologically exact distance expanding maps in the sense as in [24]. We assume that T0 is a skewproduct over SZ0 , i.e. for every x 2 X0 there exists a compact metric space Jx such that Z0 D X2X0 fxg  Jx and the following diagram commutes: Z0

T0 Z0





? X0

0 - ? X0

where .x; y/ D x and the projection  W Z0 ! X0 is an open map. Additionally, we assume that there exists L such that dX0 .0 .x/; 0 .x 0 //  LdX .x; x 0 /

(8.8)

for all x 2 X and that there exists 1 > 0 such that, for all x; x 0 satisfying dX0 .x; x 0 / < 1 there exist y; y 0 such that   d .x; y/; .x 0 ; y 0 / < :

(8.9)

We then refer to T0 W Z0 ! Z0 and 0 W X0 ! X0 as a DG-system. Note that T0 .fxg  Jx /  f0 .x/g  J0 .x/ and this gives rise to the map Tx W Jx ! J0 .x/ . Since T0 is distance expanding, conditions uniform openness, measurably expanding measurability of the degree, topological exactness (see Chap. 2) hold with some constants x   > 1, deg.Tx /  N1 < C1 and the number nr D nr .x/ in fact independent of x. Scrutinizing the proof of Remark 2.9 in [12] one sees that Lipschitz continuity (Denker and Gordin assume differentiability) suffices for it to go through and Lipschitz continuity is incorporated in the definition of expanding maps in [24]. Now assume that W Z ! R is a H¨older continuous

8.4 Denker–Gordin Systems

85

map. Then the hypothesis of Theorems 2.10, 3.1, and 3.2 from [12] are satisfied. Their claims are summarized in the following. Theorem 8.11 Suppose that T0 W Z0 ! Z0 and 0 W X0 ! X0 form a DG-system and that W Z ! R is a H¨older continuous potential. Then there exists a H¨older continuous function P . / W X0 ! R, a measurable collection f x gx2X0 and a continuous function q W Z0 ! Œ0; C1/ such that  R (a) 0 .x/ .A/ D exp Px . / A e x d x for all x 2 X0 and all Borel sets A  Jx Rsuch that Tx jA is one-to-one. (b) Jx qx d x D 1 for all x 2 X0 . (c) Denoting for every x 2 X0 by x the measure qx x we have X w .Tw1 .A// D x .A/ for every Borel set A  Jx : w201 .x/

This would mean that we got all the objects produced in Chap. 3 of our paper. However, the map 0 W X0 ! X0 need not be, and apart from the case when X0 is finite, is not invertible. But to remedy this situation is easy. We consider the projective limit (Rokhlin’s natural extension)  W X ! X of 0 W X0 ! X0 . Precisely, X D f.xn /n0 W 0 .xn / D xnC1 8n  1g and

   .xn /n0 D .0 .xn //n0 :

Then  W X ! X becomes invertible and the diagram X

 X

p

p

(8.10)

? 0 - ? X0 X0   commutes, where p .xn /n  0 D x0 . If in addition, as we assume from now on, the space X is endowed with a Borel probability 0 -invariant ergodic measure m0 , then there exists a unique -invariant probability measure m such that m ı  1 D m0 . Let [ Z WD fxg  Jx0 : x2X

We define the map T W Z ! Z by the formula T .x; y/ D ..x/; Tx0 .y// and the potential X 3 x 7! .x0 / from X to R. We keep for it the same symbol . Clearly the quadruple .T; ; m; / is a H¨older fiber system as defined in Chap. 2 of our paper. It follows from Theorem 8.11 along with the definition of  a commutativity of the diagram (8.10) for x 2 X all the objects Px . / D Px0 . /, x D exp.Px . //, qx D qx0 , x D x0 , and x D x0 enjoy all the properties required in

86

8 Classical Expanding Random Systems

Theorems 3.1 and 3.2; in particular they are unique. From now on we assume that the measure m is a Gibbs state of a H¨older continuous potential on X (having nothing to do with or P . /; it is only needed for the Law of Iterated Logarithm to hold). We call the quadruple .T; ; m; / DG*-system. The following H¨older continuity theorem appeared in the paper [12]. We provide here an alternative proof under weaker assumptions. Theorem 8.12 If dX .x; x 0 / < , then jx  x 0 j  HdX˛ .x; x 0 /. Proof. Let n be such that dX . 2n1 .x/;  2n1 .x 0 // < 1

and dX . 2n .x/;  2n .x 0 //  1 :

(8.11)

Let z 2 T 2nC1 .y/ and z0 2 T 2nC1 .y 0 /. Then for all k D 0; : : : ; n  1 j'.T k .z//  '.T k .z0 //j  Cd ˛ .T k .z/; T k .z0 //  C  ˛ n  ˛.nk1/ : Then jSn '.z/  Sn '.z0 /j 

C  ˛ n : 1   ˛

Put C 0 WD C =.1   ˛ /. Then ˇ L n 1.w/ ˇˇ ˇ ˇ log xn ˇ  C 0  ˛ n Lx 0 1.w0 / Then

n1 ˇ 1.w/ ˇˇ L.x/ ˇ and ˇ log n1 ˇ  C 0  ˛ n : L.x 0 / 1.w0 /

ˇ Lxn0 1.w0 / ˇˇ Lxn 1.w/ ˇ  log n1 ˇ log n1 ˇ  2C 0  ˛ n : L.x/ 1.w/ L.x 0 / 1.w0 /

(8.12)

Let ˛ 0 WD .˛ log  /=.2 log L/. Then by (8.11) 0

0

 n˛ D L2n˛ 

0

.d. 2n .x/;  2n .x 0 ///˛ .d.x; x 0 //˛  : 0 0 1˛ L2n˛ 0 1˛ t u

Then (8.12) finishes the proof.

Since the map 0 W X0 ! X0 is expanding, since m is a Gibbs state, and since P . / W X0 ! R is H¨older continuous, it is well known (see [24] for example) that the following asymptotic variance exists: 1 n!1 n

 2 .P . // D lim

Z  2 Sn .P . //  nEP . / dm:

The following theorem of Livsic flavor is (by now) well known (see [24]).

8.5 Conformal DG*-Systems

87

Theorem 8.13 Suppose .T; ; m; / is a DG*-system. Then the following are equivalent. (a)  2 .P . // D 0. (b) The function P . / is cohomologous to a constant in the class of real-valued continuous functions on X (resp. X0 ), meaning that there exists a continuous function u W X ! R (resp. u W X0 ! R) such that P . /  .u  u ı / (resp. P . /  .u  u ı 0 /) is a constant. (c) The function P . / is cohomologous to a constant in the class of real-valued H¨older continuous functions on X (resp. X0 ), meaning that there exists a H¨older continuous function u W X ! R (resp. u W X ! R) such that P . /  .u  u ı / (resp. P . /  .u  u ı 0 /) is a constant. (d) There exists R 2 R such that Pxn . / D nR for all n  1 and all periodic points x 2 X (resp. X0 ). As a matter of fact such theorem is formulated in [24] for non-invertible (0 ) maps only but it also holds for the Rokhlin’s natural extension . The following theorem follows directly from [24] and Theorem 8.11 (H¨older continuity of P . /). Theorem 8.14 (The Law of Iterated Logarithm) If .T; ; m; / is a DG*-system and if  2 .P . // > 0, then m-a.e. we have q q P n . /  nEP . / P n . /  nEP . /  2 2 .P . // D lim inf xp  lim sup xp D 2 2 .P . //: n!1 n!1 n log log n n log log n

8.5 Conformal DG*-Systems Now we turn to geometry. This section dealing with, below defined, conformal DG*-systems is a continuation of the previous one in the setting of conformal systems. We shall show that these systems naturally split into essential and quasideterministic, and will establish their fractal and geometric properties. Suppose that .f0 ; 0 / is a DG-system endowed with a Gibbs measure m0 at the base. Suppose also that this system is a random conformal expanding repeller in the sense of Chap. 5 and that the function W Z ! R given by the formula .x; y/ D  log jfx0 .y/j; is H¨older continuous.

88

8 Classical Expanding Random Systems

Definition 8.15 The corresponding system .f; ; m/ D .f; ; m; / (with  the Rokhlin natural extension of 0 as described above) is called conformal DG*system. For every t 2 R the potential t D t , considered in Chap. 5, is also H¨older continuous. As in Chap. 5 denote its topological pressure by P .t/. Recall that h is a unique solution to the equation EP .t/ D 0. By Theorem 5.2 (Bowen’s Formula) HD.Jx / D h for m-a.e. x 2 X . As an immediate consequence of Theorems 5.7, 8.14, and Remark 5.6, we get the following. Theorem 8.16 Suppose .f; ; m/ D .f; ; m; / is a random conformal DG*system. (a) If  2 .P .h// > 0, then the system .f; ; m/ is essential, and then H h .Jx / D 0

and

P h .Jx / D C1:

(b) If, on the other hand,  2 .P .h// D 0, then .f; ; m/ D .f; ; m; / is quasideterministic, and then for every x 2 X , we have that xh is a geometric measure with exponent h and, consequently, the geometric properties (GM1)–(GM3) hold. Exactly as Corollary 5.8 is a consequence of Theorem 5.7, the following corollary is a consequence of Theorem 8.16. Corollary 8.17 Suppose .f; ; m/ D .f; ; m; / is a conformal DG*-system and  2 .P .h// > 0. Then the system .f; ; m/ is essential, and for m-a.e. x 2 X the following hold. 1. The fiber Jx is not bi-Lipschitz equivalent to any deterministic nor quasideterministic self-conformal set. 2. Jx is not a geometric circle nor even a piecewise smooth curve. 3. If Jx has a non-degenerate connected component (for example if Jx is connected), then h D HD.Jx / > 1: 4. Let d be the dimension of the ambient Riemannian space Y . Then HD.Jx / < d . Now, in the same way as Theorem 8.7 is a consequence of Theorem 8.6, Corollary 8.17 yields the following. Theorem 8.18 Suppose .f; ; m/ D .f; ; m; / is a conformal DG*-system. Then the following hold. (a) Suppose that for every x 2 X , the fiber Jx is connected. If there exists at least one w 2 supp.m/ such that HD.Jw / > 1, then HD.Jx / > 1

for m-a.e. x 2 I Z :

(b) Let d be the dimension of the ambient Riemannian space Y . If there exists at least one w 2 X such that HD.Jw / < d , then HD.Jx / < d for m-a.e. x 2 X .

8.7 Topological Exactness

89

We end this subsection and the entire section with a concrete example of a conformal DG*-system. In particular, the three above results apply to it. Let X WD Sı1d D fz 2 C W jzj D ıg: Fix an integer k  2. Define the map 0 W X ! X by the formula 0 .x/ D ı 1k x k : Then 00 .x/ D kı 1k x k1 and therefore j00 .x/j D k  2 for all x 2 X . The normalized Lebesgue measure 0 on X is invariant under 0 . Define the map H W X ! Fd by setting H.x/ D fx . Then f0 ;H;0 .x; y/ D .kı 1k x k1 ; gd C x/:  Note that f0 ;H;0 ; 0 ; 0 / is a uniformly conformal DG-system and let .f;H ; ; / be the corresponding random conformal G-system, both in the sense of Chap. 5. Theorems 8.16, 8.18, and Corollary 8.17 apply.

8.6 Random Expanding Maps on Smooth Manifold We now complete the previous examples with some remarks on random maps on smooth manifolds. Let .M; / be a smooth compact Riemannian manifold. We recall that a differentiable endomorphism f W M ! M is expanding if there exists  > 1 such that jjfx0 .v/jj  jjvjj

for all x 2 M and all v 2 Tx M :

The largest constant  > 1 enjoying this property is denoted by .f /. If  >1, we denote by E .M / the set of all expanding endomorphisms of M for which .f /   . We also set [ E.M / D E .M /;  >1

i.e. E.M / is the set of all expanding endomorphisms of M .

8.7 Topological Exactness We shall prove the following. Proposition 8.19 Suppose that M is a connected and compact manifold and that fn 2 E.M /, n  1, are endomorphisms such that limn!1

n Y j D1

.fj / D C1 :

90

8 Classical Expanding Random Systems

Denote Fk D fk ı fk1 ı : : : ı f1 , k  1. Then, for every r > 0 there exist k  1 such that Fk .B.x; r// D M for every x 2 M : In particular, if U is a non-empty open subset of M , then there exists k  1 such that Fk .U / D M . Proof. Let f 2 E.M /, set  D .f / and notice first of all that for such a map the implicit function theorem applies and yields that f is an open map. The manifold M being connected, it follows that f is surjective. Moreover, if ˇ is any path starting at a point y D f .x/, then there is a lift ˛ starting at x. The expanding property implies that length.ˇ/ D length.f ı ˛/   length.˛/ : In particular, if ˇ is a geodesic between y D f .x/ and a point y 0 2 M , then there is a point x 0 2 M such that f .x 0 / D y 0 and .y; y 0 /   .x; x 0 / : This shows that for every r > 0 and every x 2 M we have f .B.x; r//  B.f .x/; r/ : The proposition follows now from the compactness of M .

t u

8.8 Stationary Measures Let M be an n-dimensional compact Riemannian manifold and let I be a set equipped with a probabilistic measure m0 . To every a 2 I we associate a differentiable expanding transformation fa of M into itself. Put X D I Z and let m be the product measure induced by m0 . For x D : : : a1 a0 a1 : : : consider 'x WD  log j det fa00 j. We assume that all our assumptions are satisfied. Then the measure D volM (where volM is the normalized Riemannian volume on M ) is  the fixed point of the operator Lx;' with x D 1. Let qx be the function given by Theorem 3.1, and let x be the measure determined by dx =d x D qx . We write I Z D I N  I N where points from I N are denoted by  x D : : : a2 a1 and from I N by x C D a0 a1 : : :. Then x  x C means x D : : : a1 a0 a1 : : :. Note that qx does not depend on x C , since nor does Lxnn 1.y/. Then we can write qx  WD qx and x  WD x . Since x .g ı fa0 / D .x/ we have that x  .g ı fa / D x  a .g/ (8.13) for every a 2 I .

8.8 Stationary Measures

91

Define a measure  by d D dx  d m .x  / where m is the product measure on I N . Then by (8.13) Z

 .g ı fa /d m0 .a/ D

Z

x  .g ı fa /d m .x  /

Z Z D

x  a .g/d m .x  /d m0 .a/ D  .g/:

Therefore,  is a stationary measure (see for example [28]).

•

Chapter 9

Real Analyticity of Pressure

Here we provide, in particular, the real analyticity results that where used in the proof of the real analyticity of the multifractal spectrum (Chap. 6.3). We putted this part at the end of the manuscript since, as already mentioned, it is of different nature. It is heavily based on ideas of Rugh [26] and uses the Hilbert metric on appropriately chosen cones.

9.1 The Pressure as a Function of a Parameter Here, we will have a careful close look at the measurable bounds obtained in Chap. 3 from which we deduce that the theorems from that section can be proved to hold for every parameter and almost every x (common for all parameters). In this section we only assume that T W J ! J is a measurable expanding random map. Let ' .1/ ; ' .2/ 2 Hm .J / and let t D .t1 ; t2 / 2 R2 . Put jtj WD maxfjt1 j; jt2 jg

and t  WD maxf1; jtjg:

(9.1)

Set 't DW t1 ' .1/ C t2 ' .2/ and ' WD j' .1/ j C j' .2/ j:

(9.2)

Fix ˛ > 0 and a measurable log-integrable function H W X ! Œ0; C1/ such that ' .1/ ; ' .2/ 2 Hm˛ .J ; H /. Then for all x 2 X and all y1 ; y2 2 Jx , we have j't;x .y2 /  't;x .y1 /j  Hx jt1 jx˛ .y2 ; y1 / C Hx jt2 jx˛ .y2 ; y1 /  2jtjHx x˛ .y2 ; y1 /: Therefore 't 2 Hm˛ .J ; 2jtjH /  Hm˛ .J ; 2t  H /. Also, for all x 2 X and all y 2 Jx , we have

V. Mayer et al., Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry, Lecture Notes in Mathematics 2036, DOI 10.1007/978-3-642-23650-1 9, © Springer-Verlag Berlin Heidelberg 2011

93

94

9 Real Analyticity of Pressure

jSn 't;x .y/j  jt1 jjSn 'x.1/ .y/j C jt2 jjSn 'x.2/ .y/j  jtjjSn 'x .y/j  jtjjjSn 'x jj1 : This implies

jjSn 't;x jj1  jtjjjSn 'x jj1  t  jjSn 'x jj1 :

(9.3)

Concerning the potential ', we get ˇ ˇ ˇ ˇ j'x .y2 /  'x .y1 /j  ˇj'x.1/ .y2 /  'x.1/ .y1 /ˇ C ˇ'x.2/ .y2 /  'x.2/ .y1 /ˇ  2Hx x˛ .y2 ; y1 /:

Thus

' 2 Hm˛ .J ; 2H /:

(9.4)

Denote by Ct , Ct;max , Ct;min , D;t and ˇt .s/, the respective functions associated to the potential 't as in Chap. 3.2. If the index t is missing, these numbers, as usually, refer to the potential ' given by (9.2). Using (9.3) and (9.4), we then immediately get t D;t .x/  D;' ; (9.5)   ˚   Ct .x/  exp Qx .2t  H / max exp 2t  jjSk 'xk jj1 0kj

  ˚   t    D C't ;  exp Qx .2H / max exp 2jjSk 'xk jj1

(9.6)

0kj

     Ct;min .x/  exp Qx .2t  H / exp 2t  jjSn 'x jj1 D Cmin .x/t ;

(9.7)

     Ct;max .x/ D exp Qx .2t  H / deg.Txn / exp 2t  jjSn 'x jj1  Cmax .x/t ;

(9.8)

and therefore,     Cmin .x/ t .s  1/2t  Hx1 x˛ Cmin .x/ t .s  1/Hx1 x˛ 1 1 D ˇt;x .s/  C' .x/ 4t  sQx C' .x/ 2sQx   t   Cmin .x/ t .s  1/Hx1 x˛ 1  D ˇxt .s/: C' .x/ 2sQx Finally we are going to look at the function A.x/ and the constant B obtained in Proposition 3.17. We fix the set G WD fx W ˇx  M and j.x/  J g as defined by (3.35). Note that by (9.1), for x 2 G we have, ˇx;t  M t . Denote by G0 the corresponding visiting set for backward iterates of , and by 3 k .nk /1 1 the corresponding visiting sequence. In particular limk!1 nk  4J : Putting p 2 4J 1  M t and Bt D  J kx

t .x/Bt At .x/ WD maxf2Cmax

t ; C't .x/ C Cmax .x/g;

9.1 The Pressure as a Function of a Parameter

95

as an immediate consequence of Proposition 3.17 and its proof along with our estimates above, we obtain the following. Proposition 9.1 For every t 2 R2 , for every x 2 G0 , and every gx 2 st;x kLQxnn ;t gxn  qt;x k1  At .x/Btn : More generally, if gx 2 H ˛ .Jx /, then ˇˇ Z  ˇˇ ˇˇ O n ˇˇ gx dt;x 1ˇˇ ˇˇLt;x gx  

1

Z

C't . n .x//

jgx j dt;x C 4

v˛ .gx qt;x /  At . n .x//Btn : t Q x

In here and in the sequel, by qt;x , st;x and Lt;x we denote the respective objects for the potential 't . Remark 9.2 It follows from the estimates of all involved measurable functions, that, for R > 0 and t 2 R such that jtj  R, the functions At and Bt in Proposition 9.1 can be replaced by AmaxfR;1g and BmaxfR;1g , respectively. Now, let us look at Proposition 3.19. Similarly as with the set G, we consider the set XA defined by (3.38) with A.x/ generated by '. So, if x 2 XA , then At .x/  At 0 for some finite number At which depends on t. Denote by XA;C the corresponding 0 visiting set intersected with GC . Therefore, the following is a consequence of the proof of Proposition 3.19 and the formula (3.43). 0 , and every gx 2 C .Jx / we Proposition 9.3 For every R > 0, every x 2 XA;C have that ˇˇ ˇˇ

Z  ˇˇ ˇˇ O n lim sup ˇˇLt;x gx  gx dt;x 1 n .x/ ˇˇ D 0: n!1 jt jR

1

Moreover, we obtain the following consequence of Lemma 3.28 and (9.5). Lemma 9.4 There exist a set X 0  X of full measure, and a measurable function X 3 x 7! D1 .x/ with the following property. Let x 2 X 0 , let w 2 Jx , and let n  0. Put y D .x; w/. Then .D1 . n .x///t 

t;x .Tyn .B.T n .y/; /// exp.Sn 't .y/  Sn Px .'t //

for all t 2 R2 . For all t 2 R2 set

EP .t/ WD EP .'t /:

 .D1 . n .x///t

96

9 Real Analyticity of Pressure

We now shall prove the following. Lemma 9.5 The function EP W R2 ! R is convex, and therefore, continuous. There exists a measurable set XE0 such that m.XE0 / D 1 and for all x 2 XE0 and all t 2 R2 , the limit 1 n lim log Lt;x 1.wn / (9.9) n!1 n exists, and is equal to EP .t/. Proof. By Lemmas 4.6 and 3.27 we know that for every t 2 R2 there exists a measurable Xt0 with m.Xt0 / D 1 and such that lim

n!1

1 1 n 1.wn / D lim log nt;x D EP .t/ log Lt;x n!1 n n

(9.10)

for all x 2 Xt0 . Fix 2 Œ0; 1/ and let t D .t1 ; t2 / and t 0 D .t10 ; t20 / 2 R2 . H¨older’s n inequality implies that all the functions R2 3 t 7! n1 log Lt;x 1.wn /, n  1, are 2 convex. It thus follows from (9.10), that the function R 3 t 7! EP .t/ is convex, whence continuous. Let \ XE0 D Xt0 : t 2Q2

Since the set Q2 is countable, we have that m.XE0 / D 1. Along with (9.10), and n density of Q2 in R2 , the convexity of the functions R2 3 t 7! n1 log Lt;x 1.wn / 1 0 2 n 1.wn / implies that for all x 2 XE and all t 2 R , the limit limn!1 n log Lt;x exists and represents a convex function, whence continuous. Since for all t 2 Q2 this continuous function is equal to the continuous function EP , we conclude that for all x 2 XE0 and all t 2 R2 , we have 1 n 1.wn / D EP .t/: log Lt;x n!1 n lim

t u

We are done.

Lemma 9.6 Fix t2 2 R and assume that there exist measurable functions L W X 3 x 7! Lx 2 R and c W X 3 x 7! cx > 0 such that Sn 'x;1 .z/  ncx C Lx

for every z 2 Jx and n  1 :

(9.11)

Then the function R 3 t1 7! EP .t1 ; t2 / 2 R is strictly decreasing and lim EP .t1 ; t2 / D 1

t1 !C1

and

lim EP .t1 ; t2 / D C1

t1 !1

m-a:e:

(9.12)

9.2 Real Cones

97

Proof. Fix x 2 XE0 . Let t1 < t10 . Then by (9.11) X z2Txn .wn /

D

  exp Sn '.t1 ;t2 / .z/ X

z2Txn .wn /

D

X

z2Txn .wn /



X

z2Txn .wn /

D

X

z2Txn .wn /

    exp t1 Sn '1 .z/ exp t2 Sn '2 .z/       exp t10 Sn '1 .z/ exp t2 Sn '2 .z/ exp .t1  t10 /Sn '1 .z/       exp t10 Sn '2 .z/ exp t2 Sn '2 .z/ exp .t1  t10 /.Lx  ncx /     exp Sn '.t10 ;t2 / .z/ exp .t10  t1 /.ncx  Lx /

Therefore, 1 1 n log Lt;x 1.wn /  log L.tn0 ;t2 /;x 1.wn / C .t10  t1 /.cx  Lx =n/ : 1 n n Hence, letting n ! 1, we get from Lemma 9.5 that EP .t1 ; t2 /  EP .t10 ; t2 / C .t10  t1 /cx . It directly follows from this inequality that the function t1 7! EP .t1 ; t2 / is strictly decreasing, that limt1 !C1 EP .t1 ; t2 / D 1 and that limt1 !1 EP .t1 ; t2 / D C1: t u

9.2 Real Cones We adapt the approach of Rugh [26] based on complex cones and establish real analyticity of the pressure function. Via Legendre transformation, this completes the proof of real analyticity of the multifractal spectrum (see Chap. 6). Let Hx WD HR;x WD H ˛ .Jx / and let HC;x WD HR;x ˚ i HR;x its complexification. ˛ .w ;w / 1 2

s Cxs WD CR;x WD fg 2 Hx W g.w1 /  e sQx %

g.w2 / if %.w1 ; w2 /  g: (9.13) Whenever it is clear what we mean by s, we also denote this cone by Cx . By CxC we denote the subset of all non-zero functions from Cxs . For l 2 .Hx / , the dual space of Hx , we define K.Cxs ; l/ WD sup

g2CxC

jjljj˛ jjgjj˛ : jhl; gij

98

9 Real Analyticity of Pressure

Then the aperture of Cxs is K.Cxs / WD inffK.Cxs ; l/ W l 2 .Hx / ; l ¤ 0g: Lemma 9.7 K.Cxs / < 1. This property of a cone is called an outer regularity. S x Proof. Let wk 2 Jx , k D 0; : : : ; N be such that L kD1 B.wk ; / D Jx . Define l0 .g/ WD

Lx X

g.wk /:

(9.14)

kD1

Then by Lemma 3.11 we have   jjgjj˛  sQx .exp.sQx  ˛ // C 1 jjgjj1    sQx .exp.sQx  ˛ // C 1 exp.sQx  ˛ /l0 .g/: Note that kl0 k˛ D Lx , since l0 .g/  Lx jjgjj1  Lx jjgjj˛ and l0 .1/ D Lx D Lx jj1jj˛ . Hence   kl0 k˛ kgk˛  Kx0 WD Lx sQx .exp.sQx  ˛ // C 1 exp.sQx  ˛ /: hl0 ; gi

(9.15) t u

Let sx0 WD

sQx1 x˛ C Hx1 x˛ 1 1 Qx

:

By (3.33) for s > 1, sx0 < s. Moreover, like in (3.32) we have the following. Lemma 9.8 Let g 2 Cxs and let w1 ; w2 2 J.x/ with %.w1 ; w2 /  . Then, for y 2 Tx1 .w1 / e '.y/ e Consequently

'.Ty1 .w2 //

o n g.y/ 0 ˛ Q % .w ; w / :  exp s 1 2 .x/ .x/ g.Ty1 .w2 //

o n Lx g.w1 / 0 Q.x/ %˛ .w1 ; w2 / :  exp s.x/ Lx g.w2 /

Lemma 9.9 There is a measurable function CR W X ! .0; 1/ such that Lxii g.w/ Lxii g.z/

 CR .x/

for every i  j.x/ and g 2 Cxs :

(9.16)

9.2 Real Cones

99

Proof. First, let i D j.x/. Let a 2 Txi .z/ be such that i e Si '.a/ g.a/ D

sup

e Si '.y/ g.y/:

y2Txi .z/ i

.w/ \ B.a; /. By definition of j.x/, for any point w 2 Jx there exists b 2 Txi i Therefore Lxii g.w/  e Si 'xi .b/ g.z/  exp.Si 'xi .b/  Si 'xi .a//e Si 'xi .a/ e sQx g.a/  where

exp.2kSj.x/ 'xj.x/ k1  sQx / deg.Txjj /

Lxii g.z/  .CR .x//1 Lxii g.z/;

  11 exp  sQx  2kSj.x/ 'xj.x/ k1 A  1: CR .x/ WD @ deg.Txj.x/ / j 0

(9.17)

j.x/ The case i > j.x/ follows from the previous one, since Lxii gxi 2 Cxsj.x/ .

t u

0

Let s > 1 and s < s. Define

x WD

x;s;s 0

  1  exp  .s C s 0 /Qx r ˛ s C s0   WD sup :  0 ˛ s  s0 r2.0; 1  exp  .s  s /Qx r

(9.18)

0

Lemma 9.10 For gx ; fx 2 Cxs ,

x

supy2Jx jgx .y/j infy2Jx jfx .y/j

s : fx  gx 2 CR;x

Proof. For all w; z 2 Jx with %x .z; w/ < ,    

x kgx =fx k1 exp sQx %˛x .z; w/ fx .z/  fx .w/       x kgx =fx k1 exp sQx %˛x .z; w/  exp s 0 Qx %˛x .z; w/ fx .z/       exp sQx %˛x .z; w/  exp  s 0 Qx %˛x .z; w/ gx .z/    exp sQx %˛x .z; w/ gx .z/  gx .w/:    Then exp sQx %˛x .z; w/ x kg=f k1 fx .z/  gx .z/  x kg=f k1 fx .w/  gx .w/. t u

100

9 Real Analyticity of Pressure

We say that gx 2 Cxs is balanced if fx .y1 /  CR .x/ fx .y2 /

for all y1 ; y2 2 Jx :

(9.19)

Let gx ; fx 2 Cxs . Put ˇx;s .fx ; gx / WD inff > 0 W fx  gx 2 Cxs g and define the Hilbert projective distance Pdist W Cxs  Cxs ! R by the formula: Pdistx .fx ; gx / WD Pdistx;s .fx ; gx / WD log.ˇx;s .fx ; gx /  ˇx;s .gx ; fx //: Let

j s s .L x WD diamCx;R xj .Cxj ;R //;

s where diamCx;R is the diameter with respect to the projective distance and j D j.x/. Then by Lemmas 9.8, 9.9 and 9.10 we get the following. 0

Lemma 9.11 If gx ; fx 2 Cxs are balanced, then Pdistx .fx ; gx /  2 log and, consequently, x  2 log

 s C s0

 s C s0 s  s0

s  s0

 CR .x/



  CR .x/ :

9.3 Canonical Complexification Following the ideas of Rugh [26] we now extend real cones to complex ones. Define Cx WD fl 2 .Hx / W ljCx  0g and s CC;x WD fg 2 HC;x W 8l1 ;l2 2Cx Rehl1 ; gihl2 ; gi  0g: C s Denote also by CC;x the set of all g 2 CC;x such that g 6 0. There are other s equivalent definitions of CC;x . The first one is called polarization identity by Rugh in [26, Proposition 5.2].

Proposition 9.12 (Polarization identity) s C D fa.f  C ig  / W f  ˙ g 2 CR;x and a 2 Cg: CC;x s as follows. Let %.w; w0 / < . Define In our case we can also define CC;x ˛ .w;w0 /

lw;w0 .g/ WD g.w/  e sQx % and

g.w0 /

9.3 Canonical Complexification

101

Fx WD flw;w0 W %.w; w0 / < g  Cx : Then

Cxs D fg 2 Hx W 8l2Fx l.g/  0g:

Later in this section we use the following two facts about geometry of complex numbers. The first one is obvious and the second is Lemma 9.3 from [26]. Lemma 9.13 Given c1 ; c2 > 0 there exist p1 ; p2 > 0 such that if s0 WD c1 p2 and Z 2 freiu W 1  1 C s02 , juj  2p1 C 2s0 g; then there exist ˛; ˇ;  > 0 such that Re Z  ˛, Re Z  ˇ, Im Z   and c2 < ˛. Lemma 9.14 Let z1 ; z2 2 C be such that Re z1 > Re z2 and define u 2 C though e i Im z1 u 

e z1  e z2 : e Re z1  e Re z2

Then j Arg uj 

 Im.z  z / 2 j Im.z1  z2 /j 1 2 : and 1  ju2 j  1 C Re.z1  z2 / Re.z1  z2 /

Let ' D Re ' C i Im ' be such that Re '; Im ' 2 H ˛ .J /. We now consider the corresponding complex Perron–Frobenius operators Lx;' defined by Lx;' gx .w/ D

X

e 'x .z/ gx .z/;

w 2 J.x/ :

Tx .z/Dw

Lemma 9.15 Let w; w0 ; z; z0 2 Jx such that %.w; w0 / <  and %.z; z0 / < . Then, s for all g1 ; g2 2 Cx;R , lw;w0 .Lx;' g1 /lz;z0 .Lx;' g2 / D Z; lw;w0 .Lx;Re ' g1 /lz;z0 .Lx;Re ' g2 / where Z 2 Ax WD freiu W 1  r  1 C s02 ; juj  2jj Im 'jj1 C 2s0 g and s0 WD

v˛ .Im '/x˛ : 0 .s  s.x/ /Q.x/

(9.20)

(9.21)

102

9 Real Analyticity of Pressure

Proof. For y 2 Tx1 .w/, by y 0 we denote Ty1 .w0 /. Then for g 2 Cx ˛

0

lw;w0 .Lx;' g/ WD Lx;' g.w/  e sQx % .w;w / Lx;' g.w0 / X X ˛ 0 0 D e '.y/ g.y/  e sQx % .w;w / e '.y / g.y 0 / D ny .'; g/; y2Tx1 .w/

where

y2Tx1 .w/

˛ .w;w0 /

ny .'; g/ WD e '.y/ g.y/  e sQx %

0

e '.y / g.y 0 /:

Define implicitly uy so that ny .Re '; g/e i Im '.y/ uy D ny .'; g/: Put z1 WD '.y/ C log g.y/ and z2 WD sQx %˛ .w; w0 / C '.y 0 / C log g.y 0 /. Then e i Im z1 uy D

e z1  e z2 : e Re z1  e Re z2

By (9.16) 0 Re '.y/  log g.y/  .Re '.y 0 / C log g.y 0 //  s.x/ Q.x/ %˛ .w1 ; w2 /:

Hence

0 Re.z1  z2 /  .s  s.x/ /Q.x/ %˛ .w1 ; w2 /:

We also have that j Im.z1  z2 /j  v˛ .Im '/x˛ %˛ .w1 ; w2 /; since Im.z1  z2 / D Im '.y/  Im '.y 0 /. Therefore, by Lemma 9.14 j Arg uy j  s0 WD

v˛ .Im '/x˛ 0 .s  s.x/ /Q.x/

and 1  juy j2  1 C s02 :

Since lw;w0 .Lx;' g/ D

X

ny .'; g/ D

y2Tx1 .w/

X

e i Im '.y/ uy ny .Re '; g/;

y2Tx1 .w/

lw;w0 .Lx;' g/ D Z; lw;w0 .Lx;Re ' g/ where Z 2 Ax WD freiu W 1  r  1 C s02 ; juj  2jj Im 'jj1 C 2s0 g: Similarly

9.4 The Pressure is Real-Analytic

103

lw;w0 .Lx;' g1 /lz;z0 .Lx;' g2 / DZ lw;w0 .Lx;Re ' g1 /lz;z0 .Lx;Re ' g2 / for possibly another Z 2 Ax .

t u

Let p1 ; p2 be the real numbers given by Lemma 9.13 with c1 D

x˛ x : and c2 D cosh 0 .s  sx /Qx 2

Having Lemmas 9.15, 9.13 and 9.11 the following proposition is a consequence of the proof of Theorem 6.3 in [26]. Proposition 9.16 Let j D j.x/. If k Im Sj 'xj k1  p1 then

and

v˛ .Im Sj 'xj /  p2 ;

(9.22)

s s Lxjj .CC;x /  CC;x : j

Let l0 (the functional defined by (9.14)). Then by Lemma 5.3 in [26] we get s ; l0 / WD sup K WD K.CC;x

C g2CC;x

p jjl0 jj˛ jjgjj˛  Kx WD 2 2Kx0 ; jhl0 ; gij

where Kx0 is defined by (9.15). By l we denote the functional which is a normalized s version of .1=Lx /l0 . So jjljj˛ D 1. Then, for every g 2 CC;x , 1

jjgjj˛  Kx : hl; gi

(9.23)

9.4 The Pressure is Real-Analytic We are now in position to prove the main result of this chapter. Here, we assume that T W J ! J is uniformly expanding random map. Then there exists j 2 N such that j.x/ D j for all x 2 X . Without loss of generality we assume that j D 1. Theorem 9.17 Let t0 D .t1 ; : : : ; tn / 2 Rn , R > 0 and let D.t0 ; R/ WD fz D .z1 ; : : : ; zn / 2 Cn W 8k jzk  tk j < Rg: Assume that the following conditions are satisfied. (a) For every x 2 X and every w 2 Jx , z 7! 'z;x .w/ is holomorphic on D.t0 ; R/. (b) For z 2 Rn \ D.t0 ; R/, 'z;x 2 HR;x .

104

9 Real Analyticity of Pressure

(c) For all z 2 D.t0 ; R/ and all x 2 X , there exists H such that k'z;x k˛  H . (d) For every " > 0 there exists ı > 0 such that for all z 2 D.t0 ; ı/ and all x 2 X , k Im 'z;x k˛  ": Then the function D.t0 ; R/ \ Rn 3 z 7! EP .'z / is real-analytic. Proof. Since we assume that the measurable constants are uniform for x 2 X we get that from Proposition 9.16 and condition (d) that there exists r > 0 such that, for all z 2 D.t0 ; r/ and all x 2 X , s s /  CC;x : Lz;x1 .CC;x 1

Then by (9.23),

n jjLz;x .1/jj˛ n

n lx .Lz;x .1// n

 K:

Ln

.1/.w/

z;xn Therefore, by Montel Theorem, the family lx .L is normal. Since, for all n z;xn .1// n z 2 R \ D.t0 ; r/ and all x 2 X we have that

n Lz;x .1/.w/ n

!

n lx .Lz;x .1// n!1 n

qz;x .w/ ; lx .qz;x /

we conclude that there exists an analytic function z 7! gz;x .w/ such that n Lz;x .1/.w/ n

n lx .Lz;x .1// n

! gz;x .w/: n!1

(9.24)

Since, in addition, Lx

Ln

z;xn .1/.w/ n lx .Lz;x .1// n



D

  L n .1/.w/  z;xn ; L  l x z;x 1 nC1 n l .L lx1 .Lz;xn .1// x z;xn .1// nC1 Lz;x .1/.w/ n

we therefore get that Lx

Ln

z;xn .1/.w/ n lx .Lz;x .1// n



! lx1 .Lx .gz;x //gx1 ;z : n!1

Thus, using again (9.24), we obtain Lz;x .gz;x / D lx1 .Lz;x .gz;x //gx1 ;z : As for all z 2 D.t0 ; r/ \ Rn , gz;x D

Lx qz;x qz;x D PN ; lx .qz;x / kD0 qz;x .wk /

9.4 The Pressure is Real-Analytic

105

we conclude that, lx1 .Lz;x gz;x / D lx1 .Lz;x

qz;x lx .qx ;z / / D z;x 1 1 : lx .qz;x / lx .qz;x /

(9.25)

By the very definitions lx1 .Lz;x gz;x / D .1=Lx /

Lx X

Lz;x gz;x .wk /

kD1

and Lz;x gz;x .w/ D

X

e 'z;x .y/ gz;x .y/:

y2Tx1 .w/

Denote gz;x .w/ by F .z/ and 'z;x .w/ by G.z/. Then, for z D .z1 ; : : : ; zn / 2 D.t0 ; r=2/, and .u/ D z C ..r=2/e 2 i u1 ; : : : ; .r=2/e 2 i un /, where u D .u1 ; : : : ; un / 2 Œ0; 2 n , by the Cauchy Integral Formula, ˇ ˇ @F ˇ ˇ 1 Z F ./ ˇ ˇ ˇ ˇ d  .z/ˇ D ˇ ˇ  2K=r ˇ @zk .2 i /2  .1  z1 / : : : .k  zk /2 : : : .2  z2 / for k D 1; : : : ; n. Similarly we obtain that ˇ @G ˇ ˇ ˇ .z/ˇ  2H=r ˇ @zk for k D 1; : : : ; n. Then, for k D 1; : : : ; n, ˇ @e 'z;x .y/ g .y/ ˇ ˇ @' .w/ @gz;x .y/ ˇˇ z;x ˇ ˇ ˇ z;t e 'z;x .y/ gz;x .y/ C e 'z;x .y/ ˇ ˇDˇ ˇ @zk @zk @zk  .2H=r/e H K C e H .2K=r/: It follows that there exists Cg such that for all x 2 X , ˇ @l .L g / ˇ ˇ x1 z;x z;x ˇ ˇ ˇ  Cg : @zk Using (3.19) we obtain that C'1  qt0 ;x .y/  C' ; and then

C'1  lx .qt0 ;x .y//  C'

(9.26)

106

9 Real Analyticity of Pressure

for all x 2 X . Moreover, it follows from Lemma 3.6 that t0 ;x  exp.k't0 ;x k1 /: Then z0 WD lx1 .Lt0 ;x gt0 ;x / D t;x

lx1 .qt;x1 /  exp. sup k'x k1 /C'2 > 0: lx .qt;x / x2X

Hence, by (9.26), there exists r1 > 0 so small that lx1 .Lz;x gz;x / 2 D.z0 ; z0 =2/ for all z 2 D.t0 ; r1 /. Therefore, for all x 2 X we can define the function D.t0 ; r1 / 3 z 7! log lx1 .Lz;x gz;x / 2 C: Now consider the holomorphic function Z z 7!

log lx1 .Lz;x gz;x /dm.x/:

Since the measure m is -invariant, by (9.25) Z

Z

lx1 .qz;x1 / dm.x/ lx .qz;x / Z Z Z D log z;x dm C lx1 .qz;x1 /dm  lx .qz;x /dm.x/

log lx1 .Lz;x gz;x /dm.x/ D

log z;x

Z D

log z;x dm D EP .'t /

for z 2 D.t0 ; r1 / \ Rn . Therefore the function D.t0 ; r1 / \ Rn 3 z 7! EP .'z / is real-analytic. t u

9.5 Derivative of the Pressure Now, let T W J ! J be uniformly expanding random map. Throughout the section, we assume that ' 2 Hm .J / is a potential such that there exist measurable functions L W X 3 x 7! Lx 2 R and c W X 3 x 7! cx > 0 such that Sn 'x .z/  ncx C Lx for every z 2 Jx and n and

2 Hm .J /. For t 2 R, define 't WD t' C :

(9.27)

9.5 Derivative of the Pressure

107

Let R > 0 and let jt0 j  R=2. Since we are in the uniform case, it follows from Remark 9.2 that there exist constants AR and BR such that, for t 2 ŒR; R, ˇˇ LQ n gx  Z  ˇˇ v.gx /  ˇˇ ˇˇ t;x n  gx dt;x ˇˇ  kgx k1 C 2 : A R BR ˇˇ 1 q n .x/ Q

(9.28)

Proposition 9.18 d EP .t/ D dt

Z

'x dtx dm.x/

Z D

'dt :

Proof. Assume without loss of generality that jtj  R=2 for some R > 0: Let x 7! y.x/ 2 Yx be a measurable function and let Z

1 n log Lt;x 1x .y.xn //dm.x/: n

EP .t; n/ WD

Then limn!1 EP .t; n/ D EP .t/ by Lemma 4.6. Fix x 2 X and put yn WD y.xn /. Observe that n 1x .yn / d Lt;x D dt

D

X

t

e Sn .'x /.y/ Sn 'x .y/

y2Txn .yn / n1 X

X

j D0

y2Txn .yn /

e

t /.y/ Sn .'x

'xj .Txj y/

n1 X

D

n Lt;x .'xj ı Txj /.yn /:

j D0

Since Sn .'xt /.y/ D Sj .'xt /.y/ C Snj .'xt j /.Txj y/ we have that nj

j

n .'xj ı Txj /.y.xn // D Lt;xj .'xj Lt;x 1x /.y.xn //: Lt;x

Then by a version of Leibniz integral rule (see for example [23], Proposition 7.8.4, p. 40) d EP .t; n/ D dt Since and we have that

Z

1 n

Pn1

nj j j D0 Lxj ;t .'xj Lt;x 1x /.y.xn // dm.x/: n Lt;x 1x .yn /

  nj j nj j Lt;xj .'xj Lt;x 1/.yn / D nx LQt;xj 't;xj LQt;x 1x .yn / n n Lt;x 1x .yn / D nx LQt;x 1x .yn /;

108

9 Real Analyticity of Pressure j n Lt;x .'xj ı Tx /.yn / n Lt;x 1x .yn /

D

  nj j LQt;xj 'xj LQt;x 1x .yn / n 1x .yn / LQt;x

:

(9.29)

j 1x is uniformly bounded. So does its H¨older variation. The function 'xj LQt;x Therefore it follows from (9.28), that there exists a constant AR and BR such that  Z     Q nj  j j 'xj LQt;x 1x dxt j   AR BRnj Lt;xj 'xj LQt;x 1x .yn /=qxn  1

   Qn  Lt;x .1x /.yn /=qxn  1xn 

and

1

n  AR BR ;

From this by (9.29) it follows that R

'xj LQt;x 1x dxt j  AR BR j

nj

1 C AR BRn

R

j



n Lt;x .'xj ı Tx /.yn /

Lxn 1Yx .yn /



'xj LQt;x 1x dxt j C AR BR j

nj

1  AR BRn

:

Since m is -invariant, we have that Z Z Z Z j 'xj LQt;x y j 1x dxt j dm.x/ D 'x LQxj ;t 1xj dxt dm.x/: Hence, for large n, RR

'x

 P n1 1

j D0

n

RR  Therefore

 LQxjj ;t 1xj dxt dm.x/ 

1 n

Pn1

nj j D0 .AR BR /

n AR BR

1C  P  t Qj 'x n1 n1 j D0 Lxj ;t 1xj dx dm.x/  n 1  AR BR

d EP .t; n/ D n!1 dt lim

uniformly for t 2 ŒR; R.

Z

1 n



d EP .' t ; n/ dt

Pn1

nj j D0 .AR BR /

:

'x dtx dm.x/ t u

References

1. Arnold, L.: Random Dynamical Systems. Springer Monographs in Mathematics. Springer, Berlin (1998) 2. Arnold, L., Evstigneev, I.V., Gundlach, V.M.: Convex-valued random dynamical systems: a variational principle for equilibrium states. Random Oper. Stochast. Equat. 7(1), 23–38 (1999). doi: 10.1515/rose.1999.7.1.23. http://dx.doi.org/10.1515/rose.1999.7.1.23 3. Bahnm¨uller, J., Bogensch¨utz, T.: A Margulis–Ruelle inequality for random dynamical systems. Arch. Math. (Basel) 64(3), 246–253 (1995). doi: 10.1007/BF01188575. http://dx.doi.org/10.1007/BF01188575 4. Bogensch¨utz, T.: Entropy, pressure, and a variational principle for random dynamical systems. Random Comput. Dyn. 1(1), 99–116 (1992/93) 5. Bogensch¨utz, T., Gundlach, V.M.: Ruelle’s transfer operator for random subshifts of finite type. Ergod. Theor. Dyn. Syst. 15(3), 413–447 (1995) 6. Bogensch¨utz, T., Ochs, G.: The Hausdorff dimension of conformal repellers under random perturbation. Nonlinearity 12(5), 1323–1338 (1999) 7. Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Mathematics, vol. 470. Springer, Berlin (1975) 8. Br¨uck, R.: Geometric properties of Julia sets of the composition of polynomials of the form z2 C cn . Pac. J. Math. 198(2), 347–372 (2001) 9. Br¨uck, R., B¨uger, M.: Generalized iteration. Comput. Meth. Funct. Theor. 3(1–2), 201–252 (2003) 10. Crauel, H.: Random Probability Measures on Polish Spaces. Stochastics Monographs, vol. 11. Taylor & Francis, London (2002) 11. Crauel, H., Flandoli, F.: Hausdorff dimension of invariant sets for random dynamical systems. J. Dyn. Differ. Equat. 10(3), 449–474 (1998). doi: 10.1023/A:1022605313961. http://dx.doi.org/10.1023/A:1022605313961 12. Denker, M., Gordin, M.: Gibbs measures for fibred systems. Adv. Math. 148(2), 161–192 (1999) 13. Denker, M., Urba´nski, M.: On the existence of conformal measures. Trans. Am. Math. Soc. 328(2), 563–587 (1991) 14. Deschamps, V.M.: Equilibrium states for non-H¨olderian random dynamical systems. Random Comput. Dyn. 5(4), 319–335 (1997) 15. Falconer, K.: Techniques in Fractal Geometry. Wiley, Chichester (1997) 16. Ibragimov, I.A., Linnik, Y.V.: Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff Publishing, Groningen (1971). With a supplementary chapter by I. A. Ibragimov and V. V. Petrov, Translation from the Russian edited by J. F. C. Kingman

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17. Kifer, Y.: Equilibrium states for random expanding transformations. Random Comput. Dyn. 1(1), 1–31 (1992/93) 18. Kifer, Y.: Fractal dimensions and random transformations. Trans. Am. Math. Soc. 348(5), 2003–2038 (1996) 19. Kifer, Y.: Thermodynamic formalism for random transformations revisited. Stochast. Dyn. 8, 77–102 (2008) 20. Kifer, Y., Liu, P.D.: Random dynamics. In: Handbook of Dynamical Systems, vol. 1B, pp. 379–499. Elsevier B.V., Amsterdam (2006) 21. Liu, P.D.: Entropy formula of Pesin type for noninvertible random dynamical systems. Math. Z. 230(2), 201–239 (1999). doi: 10.1007/PL00004694. http://dx.doi.org/10.1007/PL00004694 22. Liu, P.D., Qian, M.: Smooth Ergodic Theory of Random Dynamical Systems. Lecture Notes in Mathematics, vol. 1606. Springer, Berlin (1995) 23. Malliavin, P.: Integration and Probability, Graduate Texts in Mathematics, vol. 157. Springer, New York (1995). With the collaboration of H´el`ene Airault, Leslie Kay and G´erard Letac, Edited and translated from the French by Kay, With a foreword by Mark Pinsky 24. Przytycki, F., Urba´nski, M.: Conformal Fractals: Ergodic Theory Methods. London Mathematical Society, Lecture Note Series 371. Cambridge University Press, Cambridge (2010) 25. Ruelle, D.: Thermodynamic Formalism. Encyclopedia of Mathematics and its Applications, vol. 5. Addison-Wesley Publishing Co., Reading, Mass (1978). The mathematical structures of classical equilibrium statistical mechanics, With a foreword by Giovanni Gallavotti and GianCarlo Rota 26. Rugh, H.H.: Cones and gauges in complex spaces: spectral gaps and complex Perron– Frobenius theory. Ann. Math. 171(3), 1707–1752 (2010) 27. Rugh, H.H.: On the dimension of conformal repellors. randomness and parameter dependency. Preprint 2005, Ann. Math. 168(3), 695–748 (2008) 28. Viana, M.: Stochastic dynamics of deterministic systems. Lecture Notes XXI Brazilian Mathematics Colloquium, IMPA, Rio de Janeiro (1997) 29. Walters, P.: Invariant measures and equilibrium states for some mappings which expand distances. Trans. Am. Math. Soc. 236, 121–153 (1978)

Index

aperture, 98

Hilbert projective distance, 100

backward visiting sequence, 11 balanced, 100 base map, 8 Bowen’s Formula, 48 Br¨uck and B¨uger polynomial systems, 83

induced map, 70 induced potential, 72 integrability of the logarithm of the transfer operator, 41 invariant density, 23

classical conformal expanding random systems, 80 classical expanding random system, 76 concave Legendre transform, 58 conformal DG*-systems, 87 expanding random map, 47 uniformly expanding map, 47

measurability of the degree, 9 of cardinality of covers, 18 of the transfer operator, 41 measurable expanding random map, 40 measurably expanding, 8

outer regularity, 98 DG*-system, 86 DG-system, 84

essential, 12 essentially random, 51 exhaustively visiting way, 12 expanding in the mean, 69 expanding random map, 8 expected pressure, 42

Gibbs family, 18 Gibbs property, 35 H¨older continuous with an exponent ˛, 13, 71

polarization identity, 100 pressure function, 33 projective distance, 100 pseudo-pressure function, 18

random cantor set, 54 random compact subsets of Polish spaces, 43 random repeller, 75 random Sierpi´nski gasket, 5 repeller over U , 75 RPF-theorem, 17

T-invariance of a family of measures, 40

V. Mayer et al., Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry, Lecture Notes in Mathematics 2036, DOI 10.1007/978-3-642-23650-1, © Springer-Verlag Berlin Heidelberg 2011

111

112 of a family of measures, 17 of a measure, 40 temperature function, 58 topological exactness, 9 transfer dual operators, 19 operator, 13

Index uniform openness, 8 uniformly expanding random map, 9

visiting sequence, 11 visiting way, 12

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