Lecture Notes in Mathematics

Lecture Notes in Mathematics 2036

Editors:J.-M. Morel, CachanB. Teissier, Paris

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

Volker Mayer � Bartlomiej SkorulskiMariusz Urbanski

Distance Expanding RandomMappings, ThermodynamicalFormalism, Gibbs Measuresand Fractal Geometry

123

Volker MayerUniversite Lille 1Departement de Mathematiques59655 Villeneuve d’[email protected]

Bartlomiej SkorulskiUniversidad Catolica del NorteDepartamento de MatematicasAvenida Angamos [email protected]

Mariusz UrbanskiUniversity of North TexasDepartment of MathematicsDenton, TX [email protected]

ISBN 978-3-642-23649-5 e-ISBN 978-3-642-23650-1DOI 10.1007/978-3-642-23650-1Springer Heidelberg Dordrecht London New York

Lecture Notes in Mathematics ISSN print edition: 0075-8434ISSN electronic edition: 1617-9692

Library of Congress Control Number: 2011940286

c� Springer-Verlag Berlin Heidelberg 2011This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, 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 publicationor 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. Violationsare liable to prosecution under the German Copyright Law.The use of general descriptive names, registered names, trademarks, etc. in this publication does notimply, even in the absence of a specific statement, that such names are exempt from the relevant protectivelaws and regulations and therefore free for general use.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

Mathematics Subject Classification (2010): 37-XX

Preface

In this book we introduce measurable expanding random systems, develop thethermodynamical formalism and establish, in particular, exponential decay ofcorrelations and analyticity of the expected pressure although the spectral gapproperty does not hold. This theory is then used to investigate fractal propertiesof conformal random systems. We prove a Bowen’s formula and develop the mul-tifractal formalism of the Gibbs states. Depending on the behavior of the Birkhoffsums of the pressure function we get a natural classifications of the systems into twoclasses: quasi-deterministic systems which share many properties of deterministicones and essential random systems which are rather generic and never bi-Lipschitzequivalent to deterministic systems. We show in the essential case that the Hausdorffmeasure vanishes which refutes a conjecture of Bogenschutz and Ochs. We finallygive applications of our results to various specific conformal random systemsand positively answer a question of Bruck and Buger concerning the Hausdorffdimension of randomJulia sets.

v

Acknowledgements

The second author was supported by FONDECYT Grant no. 11060538, Chile andResearch Network on Low Dimensional Dynamics, PBCT ACT 17, CONICYT,Chile. The research of the third author is supported in part by the NSF Grant DMS0700831. A part of his work has been done while visiting the Max Planck Institutein Bonn, Germany. He wishes to thank the institute for support.

vii

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Expanding Random Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Introductory Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Expanding Random Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Uniformly Expanding Random Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Remarks on Expanding Random Mappings . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 Visiting Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.7 Spaces of Continuous and Holder Functions . . . . . . . . . . . . . . . . . . . . . . . . 122.8 Transfer Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.9 Distortion Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 The RPF-Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1 Formulation of the Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Frequently used Auxiliary Measurable Functions . . . . . . . . . . . . . . . . . . . 193.3 Transfer Dual Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4 Invariant Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.5 Levels of Positive Cones of Holder Functions . . . . . . . . . . . . . . . . . . . . . . . 243.6 Exponential Convergence of Transfer Operators . . . . . . . . . . . . . . . . . . . . 273.7 Exponential Decay of Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.8 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.9 Pressure Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.10 Gibbs Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.11 Some Comments on Uniformly Expanding

Random Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Measurability, Pressure and Gibbs Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1 Measurable Expanding Random Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Measurability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.3 The Expected Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

ix

x Contents

4.4 Ergodicity of � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.5 Random Compact Subsets of Polish Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Fractal Structure of Conformal Expanding Random Repellers . . . . . . . . 475.1 Bowen’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.2 Quasi-Deterministic and Essential Systems . . . . . . . . . . . . . . . . . . . . . . . . . 515.3 Random Cantor Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6 Multifractal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.1 Concave Legendre Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.2 Multifractal Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.3 Analyticity of the Multifractal Spectrum

for Uniformly Expanding Random Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7 Expanding in the Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697.1 Definition of Maps Expanding in the Mean . . . . . . . . . . . . . . . . . . . . . . . . . 697.2 Associated Induced Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707.3 Back to the Original System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.4 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

8 Classical Expanding Random Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758.1 Definition of Classical Expanding Random Systems . . . . . . . . . . . . . . . 758.2 Classical Conformal Expanding Random Systems . . . . . . . . . . . . . . . . . . 808.3 Complex Dynamics and Bruck and Buger Polynomial

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818.4 Denker–Gordin Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 848.5 Conformal DG*-Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 878.6 Random Expanding Maps on Smooth Manifold . . . . . . . . . . . . . . . . . . . . 898.7 Topological Exactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 898.8 Stationary Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

9 Real Analyticity of Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 939.1 The Pressure as a Function of a Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . 939.2 Real Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 979.3 Canonical Complexification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1009.4 The Pressure is Real-Analytic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1039.5 Derivative of the Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Chapter 1Introduction

In this monograph we develop the thermodynamical formalism for measurableexpanding random mappings. This theory is then applied in the context of conformalexpanding random mappings where we deal with the fractal geometry of fibers.

Distance expanding maps have been introduced for the first time in Ruelle’smonograph [25]. A systematic account of the dynamics of such maps, includingthe 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 requireany differentiability or smoothness condition. Distance expanding maps comprisesymbol systems and expanding maps of smooth manifolds but go far beyond. Thisis also a characteristic feature of our approach.

We first define measurable expanding random maps. The randomness is modeledby an invertible ergodic transformation � of a probability space .X; B; m/. Weinvestigate the dynamics of compositions

T nx D T�n�1.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 theirexpanding constant is measurable and a.e. �x > 1 or

Rlog �x dm.x/ > 0.

In so general setting we build the thermodynamical formalism for arbitraryHolder continuous potentials 'x . We show, in particular, the existence, uniquenessand ergodicity of a family of Gibbs measures f�xgx2X . Following ideas of Kifer[17], these measures are first produced in a pointwise manner and then we carefullycheck their measurability. Often in the literature all fibers are contained in one andthe same compact metric space and symbolic dynamics plays a prominent role. Ourapproach does not require the fibers to be contained in one metric space neither weneed 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, abso-lute 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

1

2 1 Introduction

as many as possible) absolute constants appearing in deterministic systems shouldbecome measurable functions. With this respect the thermodynamical formalismdeveloped in here represents also, up to our knowledge, new achievements in thetheory of random symbol dynamics or smooth expanding random maps acting onRiemannian manifolds.

Unlike recent trends aiming to employ the method of Hilbert metric (as forexample in [12, 19, 26, 27]) our approach to the thermodynamical formalism stemsprimarily from the classical method presented by Bowen in [7] and undertakenby Kifer [17]. Developing it in the context of random dynamical systems wedemonstrate that it works well and does not lead to too complicated (at least toour taste) technicalities. The measurability issue mentioned above results fromconvergence of the Perron–Frobenius operators. We show that this convergence isexponential, which implies exponential decay of correlations. These results precedeinvestigations of a pressure function x 7! Px.'/ which satisfies the property

��.x/.Tx.A// D ePx.'/

Z

A

e�'x d�x ;

where A is any measurable set such that TxjA is injective. The integral, againstthe measure m on the base X , of this function is a central parameter EP.'/ ofrandom systems called the expected pressure. If the potential ' depends analyticallyon parameters, we show that the expected pressure also behaves real analytically.We would like to mention that, contrary to the deterministic case, the spectralgap methods do not work in the random setting. Our proof utilizes the concept ofcomplex cones introduced by Rugh in [26], and this is the only place, where we usethe projective metric.

We then apply the above results mainly to investigate fractal properties of fibersof conformal random systems. They include Hausdorff dimension, Hausdorff andpacking measures, as well as multifractal analysis. First, we establish a version ofBowen’s formula (obtained in a somewhat different context in [6]) showing thatthe Hausdorff dimension of almost every fiber Jx is equal to h, the only zeroof the expected pressure EP.'t /, where 't D �t log jf 0j and t 2 R. Then weanalyze the behavior of h-dimensional Hausdorff and packing measures. It turnedout that the random dynamical systems split into two categories. Systems from thefirst category, rather exceptional, behave like deterministic systems. We call them,therefore, quasi-deterministic. For them the Hausdorff and packing measures arefinite and positive. Other systems, called essentially random, are rather generic. Forthem the h-dimensional Hausdorff measure vanishes while the h-packing measure isinfinite. This, in particular, refutes the conjecture stated by Bogenschutz 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 randomsystems is determined by the behavior of the Birkhoff sums

P nx .'/ D P�n�1.x/.'/ C ::: C Px.'/

1 Introduction 3

of the pressure function for potential 'h D �h log jf 0j. If these sums stay boundedthen we are in the quasi-deterministic case. On the other hand, if these sums areneither bounded below nor above, the system is called essentially random. Thebehavior of P n

x , being random variables defined on X , the base map for our skewproduct map, is often governed by stochastic theorems such as the law of theiterated logarithm whenever it holds. This is the case for our primary examples,namely conformal DG-systems and classical conformal random systems. We arethen in position to state that the quasi-deterministic systems correspond to ratherexceptional case where the asymptotic variance �2 D 0. Otherwise the system isessential.

The fact that Hausdorff measures in the Hausdorff dimension vanish hasfurther striking geometric consequences. Namely, almost all fibers of an essentialconformal random system are not bi-Lipschitz equivalent to any fiber of any quasi-deterministic or deterministic conformal expanding system. In consequence almostevery fiber of an essentially random system is not a geometric circle nor even apiecewise analytic curve. We then show that these results do hold for many explicitrandom dynamical systems, such as conformal DG-systems, classical conformalrandom systems, and, perhaps most importantly, Bruck and Buger polynomialsystems. As a consequence of the techniques we have developed, we positivelyanswer the question of Bruck and Buger (see [9] and Question 5.4 in [8]) of whetherthe Hausdorff dimension of almost all naturally defined random Julia set is strictlylarger than 1. We also show that in this same setting the Hausdorff dimension ofalmost all Julia sets is strictly less than 2.

Concerning the multifractal spectrum of Gibbs measures on fibers, we show thatthe multifractal formalism is valid, i.e. the multifractal spectrum is Legendre con-jugated to a temperature function. As usual, the temperature function is implicitlygiven in terms of the expected pressure. Here, the most important, although perhapsnot most strikingly visible, issue is to make sure that there exists a set Xma of fullmeasure 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 analyticityof the pressure function. This part is based on work by Rugh [27] and it is the onlyplace where we work with the Hilbert metric. As a consequence and via Legendretransformation we obtain real analyticity of the multifractal spectrum.

Random transformations have already a long history and the present manuscriptdoes, 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 interestingresults. Denote by M 1

m.T / the set of T -invariant measures from M 1m.J /. Let

� 2 M 1m.T /. The fiber entropy hr

�.T / of � is given as follows. If R DfR1; R2; : : : ; Rng is a finite partition of J , then by Rx we denote the partitionof J given by sets Rk

x WD Rk \ Jx , k D 1; : : : ; n. Then

hr�.T / WD sup

Rlim

n!11

nH�x

n�1_

iD0

R� i .x/

!

:

4 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 andthe topological pressure called Variational Principle (see, e.g. [2, 4, 14])

EP.'/ WD sup�2.T /

�Z'd� C hr

�.T /

�

:

It is also worth noting that in many cases the entropy and averaged positiveLyapunov 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 finalversion of this monograph.

Chapter 2Expanding Random Maps

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

2.1 Introductory Examples

Before giving the formal definitions of expanding random maps, let us now considersome typical examples.

The first one is a known random version of the Sierpinski 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 maponto �. The map fx is nothing else than the generator of a deterministic Sierpinskigasket. Note that this map can be made continuous by identifying the verticesA; B; C (Fig. 2.1).

Now, suppose x1 D .a1; b1; c1/; x2 D .a2; b2; c2/; ::: are chosen randomlywhich, for example, may mean that they form sequences of three dimensionalindependent and identically distributed (i.i.d.) random variables. Then they generatecompact sets

Jx1;x2;x3;::: D\

n�1

.fxnı ::: ı fx1

/�1.�/

called random Sierpinski gaskets having the invariance property f �1x1

.Jx2;x3;:::/ DJx1;x2;x3;:::. For a little bit simpler example of random Cantor sets we refer thereader to Sect. 5.3. In that example we provide a more detailed analysis of suchrandom sets.


5

6 2 Expanding Random Maps

Fig. 2.1 Two different generators of Sierpinski gaskets

Fig. 2.2 A generator of degree 6

Such examples admit far going generalizations. First of all, we will considermuch more general random choices than i.i.d. ones. We model randomness by takinga probability space .X; B; m/ along with an invariant ergodic transformation � WX ! X . This point of view was up to our knowledge introduced by the Bremengroup (see [1]).

Another point is that the maps fx that generate the random Sierpinski gaskethave degree 3. In the sequel of this manuscript, we will allow the degree dx of allmaps to be different (see Fig. 2.2) and only require that the function x 7! log.dx/ ismeasurable.

Finally, the above examples are all expanding with an expanding constant

�x � � > 1 :

As already explained in the introduction, the present monograph concerns randommaps for which the expanding constants �x can be arbitrarily close to one.Furthermore, using an inducing procedure, we will even weaken this to the mapsthat are only expanding in the mean (see Chap. 7).

2.1 Introductory Examples 7

The example of random Sierpinski gasket is not conformal. Random iterations ofrational functions or of holomorphic repellers are typical examples of conformalrandom dynamical systems. Random iterations of the quadratic family fc.z/ Dz2 C c have been considered, for example, by Bruck and Buger among others (see[8] and [9]). In this case, one chooses randomly a sequence of bounded parametersc D .c1; c2; :::/ and considers the dynamics of the family

Fc1;:::;cnD fcn

ı fcn1ı ::: 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. Juliasets that come out of such a choice are presented in Fig. 2.3. Such random Juliasets are different objects as compared to the Julia sets for deterministic iteration ofquadratic polynomials. But not only the pictures are different and intriguing, we

Fig. 2.3 Some quadratic random Julia sets


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

2.2 Preliminaries

Suppose .X; B; m; �/ is a measure preserving dynamical system with invertibleand ergodic map � W X ! X which is referred to as the base map. Assumefurther that .Jx; �x/, x 2 X , are compact metric spaces normalized in size bydiam�x

.Jx/ � 1. Let

J D[

x2X

fxg � Jx: (2.1)

We will denote by Bx.z; r/ the ball in the space .Jx ; %x/ centered at z 2 Jx andwith 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-productdefined by

T .x; z/ D .�.x/; Tx.z//: (2.2)

For every n � 0 we denote T nx WD T�n�1.x/ ı ::: ı Tx W Jx ! J�n.x/. With this

notation one has T n.x; y/ D .�n.x/; T nx .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 WJx ! 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 conditionshold.

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/# T �1

x .fyg/ ismeasurable.

Topological Exactness. There exists a measurable function x 7! n� .x/ such that

Tn�.x/x .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 injectivefor every .x; z/ 2 J . Together with the compactness of the spaces Jx it yieldsthe numbers deg.Tx/ to be finite. Therefore the supremum in the condition ofmeasurability of the degree is in fact a maximum.

In this work we consider two other classes of random maps. The first one consistsof the uniform expanding maps defined below. These are expanding random mapswith uniform control of measurable “constants”. The other class we consider iscomposed of maps that are only expanding in the mean. These maps are definedlike the expanding random maps above excepted that the uniform openness and themeasurable 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 dm > 0:

We employ an inducing procedure to expanding in the mean random maps in orderto reduce then to the case of random expanding maps. This is the content of Chap. 7and the conclusion is that all the results of the present work valid for expandingrandom 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 isdevoted to measurable expanding systems. The study of fractal and geometricproperties (which starts with Chap. 5), somewhat against our general philosophy, butwith agreement with the existing tradition (see for example [5,12,17]), we will workmostly 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.


2.5 Remarks on Expanding Random Mappings

The conditions of uniform openness and measurably expanding imply that, for everyy D .x; z/ 2 J there exists a unique continuous inverse branch

T �1y W B�.x/.T .y/; �/ ! Bx.y; �x/

of Tx sending Tx.z/ to z. By the measurably expanding property we have

%.T �1y .z1/; T �1

y .z2// � ��1x %.z1; z2/ for z1; z2 2 B�.x/

�T .y/; �

�(2.4)

andT �1

y .B�.x/.T .y/; �// � Bx.y; ��1x �/ � Bx.y; �/:

Hence, for every n � 0, the composition

T �ny D T �1

y ı T �1T .y/ ı : : : ı T �1

T n�1.y/W B�n.x/.T

n.y/; �/ ! Bx.y; �/ (2.5)

is well defined and has the following properties:

T �ny W B�n.x/.T

n.y/; �/ ! Bx.y; �/

is continuous,

T n ı T �ny D IdjB�n.x/.T n.y/;�/, T �n

y .T nx .z// D z

and, for every z1; z2 2 B�n.x/

�T n.y/; �

�,

%.T �ny .z1/; T �n

y .z2// � .�nx /�1%.z1; z2/; (2.6)

where �nx D �x��.x/ � � � ��n�1.x/: Moreover,

T �nx .B�n.x/.T

n.y/; �// � Bx.y; .�nx /�1�/ � Bx.y; �/: (2.7)

Lemma 2.1 For every r > 0, there exists a measurable function x 7! nr .x/ suchthat a.e.

T nr .x/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

T j.x/x

�j.x/.Bx

�j.x/.z; �// D Jx for every z 2 Jx

�j.x/: (2.9)

Proof. In order to prove the first statement, consider �0 > 1 and let F be the setof 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

associate to such an essential set a set X 0CF (see (2.10)). Then for x 2 X 0CF , thelimit limn!1.�n

x /�1 D 0. Define

XCF;k WD fx 2 X 0CF W .�kx /�1� < rg:

Then XCF;k � XCF;kC1 andS

k2N XCF;k D X 0CF . By measurability of x 7! �x ,XCF;k is a measurable set. Hence the function

X 0CF 3 x 7! nr .x/ WD minfk W x 2 XCF;kg C n�.x/

is finite and measurable. By (2.7) and (2.3),

T nr .x/x .Bx.z; r// D J�nr .x/.x/:

In order to prove the existence of a measurable function j W X ! N definemeasurable sets

X�;n WD fx 2 X W n� .x/ � ng, X 0�;n WD �n.X�;n/ and X 0

� D[

n2NX 0

�;n:

Then the mapX 0

� 3 x 7! j.x/ WD minfn 2 N W x 2 X 0�;ng

satisfies (2.9) for x 2 X 0�. Since m.�n.X�;n// D m.X�;n/ % 1 as n tends to 1 we

have m.X 0�/ D 1. ut

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 V�F .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 V�F .x/ is just avisiting 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 Theoremwe have that

m.X 0CF / D m.X 0�F / D 1;

where

X 0CF WD

nx 2 X W VCF .x/ is infinite and lim

j !1nj C1.x/

nj .x/D 1

o(2.10)


and X 0�F is defined analogously. The sets X 0CF and X 0�F are respectively calledforward 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.x 2 X 0CF 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 exhaustivelyvisiting way, if there exists a family Fk 2 F with limk!1 m.Fk/ D 1 such that,for all k, m-a.e. x 2 X 0CFk

, and all j 2 N, the formula .�nj .x/; nj .x// holds,where .nj .x//1

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-limn!1 fn D f;

if that there exists a family Fk 2 F with limk!1 m.Fn/ D 1 such that, for all k

and m-a.e. x 2 X 0CFkand all j 2 N,

limj !1 fnj

.x/ D f .x/;

where .nj /1j D0 is the visiting sequence of Fk at x.

Suppose that g1; : : : ; gk W X ! R is a finite collection of measurable functionsand let b1; : : : ; bn be a collection of real numbers. Consider the set

F WDk\

iD1

g�1i ..�1; bi /:

If m.F / > 0, then F is called essential for g1; : : : ; gk with constants b1; : : : ; bn (orjust essential, if we do not want explicitly specify functions and numbers). Note thatby measurability of the functions g1; : : : ; gk , for every " > 0 we can always findfinite numbers b1; : : : ; bn such that the essential set F for g1; : : : ; gk with constantsb1; : : : ; bn has the measure m.F / � 1 � ".

2.7 Spaces of Continuous and Holder Functions

We denote by C .Jx/ the space of continuous functions gx W Jx ! R and byC .J / the space of functions g W J ! R such that, for a.e. x 2 X , x 7! gx WDgjJx

2 C .Jx/. The set C .J / contains the subspaces C 0.J / of functions forwhich the function x 7! kgxk1 is measurable, and C 1.J / for which the integral

kgk1 WDZ

X

kgxk1 dm.x/ < 1:

2.8 Transfer Operator 13

Now, fix ˛ 2 .0; 1. By H ˛.Jx/ we denote the space of Holder continuousfunctions on Jx with an exponent ˛. This means that 'x 2 H ˛.Jx/ if and onlyif '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 Holder continuous with an exponent ˛ provided

that there exists a measurable function H W X ! Œ1; C1/, x 7! Hx , such thatlog H 2 L1.m/ and such that v˛.'x/ � Hx for a.e. x 2 X . We denote the spaceof all Holder functions with fixed ˛ and H by H ˛.J ; H/ and the space of all˛-Holder functions by H ˛.J / D S

H�1 H ˛.J ; H/.

2.8 Transfer Operator

For every function g W J ! C and a.e. x 2 X let

Sngx Dn�1X

j D0

gx ı T jx ; (2.12)

and, if g W X ! C, then Sng D Pn�1j D0 g ı �j . Let ' be a function in the Holder

space H ˛.J /. For every x 2 X , we consider the transfer operator Lx D L';x WC .Jx/ ! C .J�.x// given by the formula

Lxgx.w/ DX

Tx.z/Dw

gx.z/e'x.z/; w 2 J�.x/: (2.13)

It is obviously a positive linear operator and it is bounded with the norm boundedabove by

kLxk1 � 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

L nx WD L�n�1.x/ ı ::: ı Lx W C .Jx/ ! C .J�n.x//:

Note that

L nx gx.w/ D

X

z2T �nx .w/

gx.z/eSn'x.z/, w 2 J�n.x/; (2.15)

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

into C �.Jx/.


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.T �ny .w1// � Sn'x.T �n

y .w2//j � %˛.w1; w2/

n�1X

j D0

H�j .x/.�n�j

�j .x//�˛

for all w1; w2 2 B.T nx .z/; �/.

Proof. We have by (2.6) and Holder continuity of ' that


y .w2//j �n�1X

j D0

j'x.T jx .T �n

y .w1/// � 'x.T jx .T �n

y .w2///j

Dn�1X

j D0

ˇˇˇ'x.T

�.n�j /

Tjx .y/

.w1// � 'x.T�.n�j /

Tjx .y/

.w2//ˇˇˇ

�n�1X

j D0

%˛�T

�.n�j /

Tjx .x/

.w1/; T�.n�j /

Tjx .x/

.w2/�H�j .x/;

hence jSn'x.T �ny .w1//�Sn'x.T �n

y .w2//j � %˛.w1; w2/Pn�1

j D0 H�j .x/.�n�j

�j .x//�˛ .

utSet

Qx WD Qx.H/ D1X

j D1

H��j .x/.�j

��j .x//�˛ : (2.16)

Lemma 2.3 The function x 7! Qx is measurable and m-a.e. finite. Moreover, forevery ' 2 H ˛.J ; H/,


y .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 againy D .x; z/.

Proof. The measurability of Qx follows directly from (2.16). Because of Lemma 2.2we are only left to show that Qx is m-a.e. finite. Let � be a positive real numberless or equal to

Rlog �xdm.x/. Then, using Birkhoff’s Ergodic Theorem for ��1,

we get that

lim infj !1

1

j

j �1X

kD0

log ��j .x/ � �

for m-a.e. x 2 X . Therefore, there exists a measurable function C� W X ! Œ1; C1/

m-a.e. finite such that C �1� .x/ej�=2 � �

j

��j 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 Theoremthat

limj !1

1

jlog H��j .x/ D 0 m-a:e:

There thus exists a measurable function CH W X ! Œ1; C1/ such that

H�j .x/ � CH .x/ej˛�=4 and H��j .x/ � CH .x/ej˛�=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��.n�j /.�n.x// � CH .�n.x//e.n�j /˛�=4:

Therefore, still with xn D �n.x/,

QxnD

n�1X

j D0

Hxj.�n�j

xj/�˛ �

n�1X

j D0

CH .xn/e.n�j /˛�=4C ˛� .xn�1/e�˛.n�j /�=2

� C ˛� .xn�1/CH .xn/

n�1X

j D0

e�˛.n�j /�=4 � C ˛� .xn�1/CH .xn/.1 � e�˛�=4/�1:

HenceQx � C ˛

� .��1.x//CH .x/.1 � e�˛�=4/�1 < C1:

ut

Chapter 3The RPF-Theorem

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

3.1 Formulation of the Theorems

Let T W J ! J be a expanding random map. Denote by M 1.Jx/ the set ofall Borel probability measures on Jx . A family of measures f�xgx2X such that�x 2 M 1.Jx/ is called T -invariant if �x ı T �1

x 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 transferoperator. Then the following holds.

1. There exists a unique family of probability measures �x 2 M .Jx/ such that

L �x ��.x/ D �x�x where �x D ��.x/.Lx1/ m-a:e: (3.1)

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

Lxqx D �xq�.x/ and �x.qx/ D 1: (3.2)

Moreover, qx 2 H ˛.Jx/ for a.e. x 2 X .

3. The family of measures f�x WD qx�xgx2X is T -invariant.


17

18 3 The RPF-Theorem

Theorem 3.2

1. LetO'x D 'x C log qx � logq�.x/ ı T � log�x :

Denote OL WD L O' . Then, for a.e. x 2 X and all gx 2 C.Jx/,

OL nx gx ��!

n!1

Zgxqxd�x:

2. Let Q'x D 'x � log�x . Denote QL WD L Q' . There exist a constant B < 1 and ameasurable 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/ forwhich

k. QL ng/x �� Z

g��n.x/d��n.x/

�qxk1 � Ag.�

�n.x//A.x/Bn

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 forevery f�n.x/ 2 L1.��n.x// and every gx 2 H ˛.Jx/,

ˇˇ�x

�.f�n.x/ ı T n

x /gx

� � ��n.x/.f�n.x//�x.gx/ˇˇ

� ��n.x/.jf�n.x/j/A0.�n.x//� Z

jgx jd�x C 4v˛.gxqx/

Qx

�Bn:

A collection of measures f�xgx2X such that �x 2 M 1.Jx/ is called a Gibbsfamily for ' 2 H ˛.J / provided that there exists a measurable functionD' W X !Œ1;C1/ and a function x 7! Px , called the pseudo-pressure function, such that

.D'.x/D'.�n.x///�1 � �x.T

�ny .B.T n.y/; �///

exp.Sn'.y/� SnPx/� 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 Polishspace (see Sect. 4.5) this condition always holds (see Lemma 4.11).

Measurability of Cardinality of Covers. There exists a measurable function X 3x 7! ax 2 N such that for almost every x 2 X there exists a finite sequencew1

x; : : : ;waxx 2 Jx such that

Sax

j D1 B.wjx ; �/ D Jx :

Theorem 3.3 The collections f�xgx2X and f�xgx2X are Gibbs families. Moreover,if J satisfies the condition of measurability of cardinality of covers and if f� 0

xgx2X

is a Gibbs family, then �0x 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 frequentlythat, for convenience of the reader, we decided to collect them in this sectiontogether. However, the reader may skip this part now without any harm and comeback to it when it is appropriately needed.

First, defineD� .x/ WD �

degT nx

��1exp.�2kSn'xk1/ (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 eQx

�j deg.T jx

�j/max

˚exp.2kSk'x

�kk1/ W 0 � k � j

� � 1: (3.5)

Now let s > 1. Put

Cmin.x/ WD e�sQxe�kSj 'x

�jk

1 � 1 (3.6)

andCmax.x/ WD esQx deg

�T n

x

�exp.2kSn'xk1/; (3.7)

where n WD n�.x/. Then we define

ˇx.s/ WD Cmin.x/

C'.x/� inf

r2.0;��

1 � exp�� .s � 1/Hx

�1��˛

x�1r˛�

1 � exp.�2sQxr˛/: (3.8)

Since by (2.16)sQx D sQx

�1��˛

x�1

C sHx�1��˛

x�1; (3.9)

.sQx�1

CHx�1/��˛

x�1

D sQx � .s � 1/Hx�1��˛

x�1: (3.10)

This, together with (3.5) and (3.6), gives us that

0 < ˇx.s/ D Cmin.x/

C'.x/

.s � 1/Hx�1��˛

x�1

2sQx

<Cmin.x/

C'.x/� 1:

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 reducethe base space X to the orbit

Ox0D f�n.x0/; n 2 Zg:


The motivation for this is that then we can deal with a sequentially topologicalcompact space on which the transfer (or related) operators act continuously. Ourconformal measure then can be produced, for example, by the methods of the fixedpoint theory, similarly as in the deterministic case.

Removing a set of measure zero, if necessary, we may assume that this orbit ischosen so that all the involved measurable functions are defined and finite on thepoints 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

L �x ��.x/ D �x�x for every x 2 Ox0

;

where�x WD L �

x .��.x//.1/ D ��.x/.Lx1/: (3.11)

Proof. Let C �.Jx/ be the dual space of C .Jx/ equipped with the weak�topology. Consider the product space

D.Ox0/ WD

Y

x2Ox0

C �.Jx/

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

P.Ox0/ WD

Y

x2Ox0

M 1.Jx/

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 L �x ��.x/

L �x ��.x/.1/

is weakly continuous. Consider then the global map � W P.Ox0/ ! P.Ox0

/

given by� D .�x/x2Ox0

7�! �.�/ D ��x��.x/

�x2Ox0

:

Weak continuity of the �x implies continuity of � with respect to the coordinateconvergence. Since the space P.Ox0

/ is a compact subset of a locally convextopological space, we can apply the Schauder–Tychonoff fixed point theorem toget � 2 P.Ox0

/ fixed point of � , i.e.

L �x ��.x/ D �x�x where �x D L �

x ��.x/.1/ D ��.x/.Lx.1//

for every x 2 Ox0. ut

3.3 Transfer Dual Operators 21

Remark 3.5 The relation (3.11) implies

infy2Jx

e'x .y/ � �x � kLx1k1: (3.12)

A straightforward adaptation of the proof of Proposition 2.2 in [13] leads to thefollowing, to Proposition 3.4 equivalent, characterization of Gibbs states: if T n

x jA isinjective, then

��n.x/.Tnx .A// D �n

x

Z

A

e�Sn'd�x : (3.13)

Here is one more useful estimate.

Lemma 3.6 For every x 2 Ox0and n � 1,

infz2Jx

exp�Sn'x.z/

� � �nx

deg.T nx /

� supz2Jx

exp�Sn'x.z/

�: (3.14)

Moreover, for every z 2 Jx and every r > 0,

�x.B.z; r// � D.x; r/; (3.15)

where

D.x; r/ WD�

deg.TNx /��1

infz2Jx

exp�

infa2B.z;r/

SN'x.a/ � supb2B.z;r/

SN'x.b/�

(3.16)

with N D nr .x/ being the index given by Lemma 2.1. It follows that the set Jx isa 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/.Lnx 1/ D ..L n

x /��n.x//.1/ D �n

x�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 sup

a2B.z;r/

e�Sn'x.a/ � �nx

Z

B.z;r/

e�Sn'xd�x � 1;

which implies (3.15). ut


3.4 Invariant Density

Consider now the normalized operator QL given by

QLx D ��1x Lx; x 2 X: (3.18)

Proposition 3.7 For every x 2 Ox0, there exists a function qx 2 H ˛.Jx/ such

thatQLxqx D q�.x/ and

Z

Jx

qxd�x D 1:

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

QL nx

�n1.w1/

QL nx

�n1.w2/

D L nx

�n1.w1/

L nx

�n1.w2/

� C'.x/; (3.20)

where C' is given by (3.5). If in addition %.w1;w2/ � �, then

QL nx

�n1.w1/

QL nx

�n1.w2/

� expfQx%˛.w1;w2/g: (3.21)

Moreover,

1=C'.x/ � QL nx

�n1.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/� � expQx (3.23)

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

L nx

�n1.w1/ � deg.T n

x�n/ exp.kSn'x

�nk1/

exp.�kSn'x�n

k1/L n

x�n

1.w2/ � C'.x/Lnx

�n1.w2/:

3.4 Invariant Density 23

Next, let n > j WD j.x/. Take w01 2 T �j

x�j.w1/ such that

eSj '.w0

1/L n�j

x�n

1.w01/ D sup

y2T�jx

�j.w1/

�eSj '.y/L n�j

x�n

1.y/�

and w02 2 T �j

x�j.w2/ such that %x

�j.w0

1;w02/ � �. Then, by (3.21) and (3.23),

L nx

�n1.w1/ D L j

x�j.L n�j

x�n

1/.w1/ � deg.T jx

�j/eSj '.w0

1/L n�jx

�n1.w0

1/

� deg.T jx

�j/eSj '.w0

1/eQx

�j L n�jx

�n1.w0

2/ � C'.x/Lnx

�n1.w2/:

This shows (3.20). By Proposition 3.4

Z

Jx

QL nx

�n.1/d�x D

Z

Jx�n

1d�x�n

D 1; (3.24)

which implies the existence of w;w0 2 Jx such that QL nx

�n1.w/ � 1 and

QL nx

�n1.w0/ � 1. Therefore, by the already proved part of this lemma, we get (3.22).

utProof. [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

j QL kx

�k1.w1/� QL k

x�k

1.w2/j � C'.x/2Qx%˛.w1;w2/

and 1=C'.x/ � QL kx

�k1 � C'.x/. It follows that the sequence

qx;n WD 1

n

n�1X

kD0

QL kx

�k1; n � 1

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 functionsqx have all the required properties. ut

Let�x WD qx�x ; (3.25)

and let OLx WD L O';x be the transfer operator with potential

O'x D 'x C log qx � log q�.x/ ı Tx � log�x :

ThenOLxgx D 1

q�.x/

QLx.gxqx/ for every gx 2 L1.�x/: (3.26)


ConsequentlyOLx1x 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

OL �x .��.x//.gx/ D

Z

J�.x/

OLx.gx/d��.x/ D ��1x

Z

J�.x/

.Lxgxqx/d��.x/

D ��1x

OL �x .��.x//.gxqx/ D �x.gxqx/ D �x.gx/: (3.29)

So, if fx � .g�.x/ ı Tx/ 2 L1.�x/, then

�x

�.g�.x/ ı Tx/fx

� D OL �x .��.x//


�

D ��.x/

� OLx


�� D ��.x/

�g�.x/

OLx.fx/�

(3.30)

sinceOLx


� D g�.x/OLx.fx/:

Substituting in (3.30) 1x for fx and using (3.27), we get the lemma. utRemark 3.10 In Chap. 4 we provide sufficient measurability conditions for thesefiber measures �x and �x to be integrable to produce global measures projectingon X to m. The measure � defined by (4.2) is then T -invariant.

3.5 Levels of Positive Cones of Holder Functions

For s � 1, set

sx D

ng 2 C .Jx/ W g � 0; �x.g/ D 1 and g.w1/ � esQx%˛.w1;w2/g.w2/

for all w1;w2 2 Jx with %.w1;w2/ � �o: (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

g.w1/ � esQx%˛.w1;w2/g.w2/;

3.5 Levels of Positive Cones of Holder Functions 25

then

v˛.g/ � sQx.exp.sQx�˛//�˛ jjgjj1:

Proof. Let w1;w2 2 Jx be such that %.w1;w2/ � �. Without loss of generality wemay assume that g.w1/ > g.w2/. Then g.w1/ > 0 and therefore, because of ourhypothesis, g.w2/ > 0. Hence, we get

jg.w1/ � g.w2/jjg.z2/j D g.w1/

g.w2/� 1 � exp

�sQx%

˛.w1;w2/� � 1:

Thenjg.w1/ � g.w2/j � sQx.exp.sQx�

˛//%˛.w1;w2/jjgjj1:ut

Hence the set sx is a level set of the cone defined in (9.13), that is

sx D C s

x \ 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

B.z;�/

g d�x �Z

Jx

g d�x 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

g.z/ � esQx%˛.b;z/g.b/ � esQx

D�.x/� Cmax.x/: ut

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

�x.g/C v˛.g/=Qx

2 1x :


Proof. Consider the function h D gCv˛.g/=Qx. In order to get the inequality fromthe definition of s

x , we take z1; z2 2 Jx . If h.z1/ � h.z2/ then this inequality istrivial. Otherwise h.z1/ > h.z2/, and therefore

h.z1/

h.z2/� 1 D jh.z1/� h.z2/j

jh.z2/j � v˛.g/%˛.z1; z2/

v˛.g/=Qx

D Qx%˛.z1; z2/: ut

An important property of the sets sx is their invariance with respect to the

normalized operator QLx D ��1x Lx .

Lemma 3.14 Let g 2 sx . Then, for every n � 1,

QL nx g.w1/

QL nx g.w2/

� exp�sQxn

%˛.w1;w2/�; w1;w2 2 J�n.x/ with %.w1;w2/ � �:

Consequently, QL nx .

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

T �nx .w1/. For y D .x; z1/, we put z2 D T �n

y .w2/. With this notation, we obtainfrom Lemma 2.2 and from the definition of s

x that

QL nx g.w1/

QL nx g.w2/

� supz12T �n

x .w1/

exp�Sn'x.z1/

�g.z1/

exp�Sn'x.z2/g.z2/

�

� exp�%˛.w1;w2/

� n�1X

j D0

H�j .x/.�n�j

�j .x//�˛ C sQx.�

nx /

�˛��:

(3.32)

Since

Qx.�nx /

�˛ Cn�1X

j D0

H�j .x/.�n�j

�j .x//�˛ D Q�n.x/; (3.33)

the lemma follows. utLemma 3.15 With Cmin the function given by (3.6) we have that

QL ix

�ig � Cmin.x/ for every i � j.x/ and g 2 s

x�i:

Proof. First, let i D j.x/. SinceRJx

�igd�x

�iD 1 there exists a 2 Jx

�isuch

that g.a/ � 1. By definition of j.x/, for any point w 2 Jx , there exists z 2T �i

x�i.x/ \ B.a; �/. Therefore

3.6 Exponential Convergence of Transfer Operators 27

QL ix

�ig.w/ � eSi 'x

�i.z/g.z/ � eSi 'x

�i.z/e�sQxg.a/ � Cmin.x/:

The case i > j.x/ follows from the previous one, since QL i�j.x/x

�igx

�i2 x

�j.x/.ut

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 gx�i

2s

x�i

, there exists hx 2 sx such that

. QL ig/x D QL ix

�igx

�iD ˇxqx C .1 � ˇx/hx :

Proof. By Lemma 3.15, we have QL ix

�igx

�i� 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/

� ˇxC'.x/�1 � exp.�2sQx%

˛x.z;w//

�exp

�sQx%

˛x.z;w/

�

��

exp�sQx%

˛x.z;w/

� � exp�.sQx �Hx

�1��˛

x�1/%˛

x.z;w/�� QL i

x�igx

�i.z/

��

exp�sQx%

˛x.z;w/

� � exp�.sQx

�1CHx

�1/��˛

x�1%˛

x.z;w/�� QL i

x�igx

�i.z/:

Since by (3.32), for h 2 sx

�1,

QLx�1h.z/ � exp

�.sQx

�1CHx

�1/��˛

x�1%˛

x.z;w/� QLx

�1h.w/;

QL ix

�igx

�i.z/ � exp

�.sQx

�1CHx

�1/��˛

x�1%˛

x.z;w/� QL i

x�igx

�i.w/:

Then we have that

ˇx

�exp

�sQx%

˛x.z;w/

�qx.z/ � qx.w/

�

� exp�sQx%

˛x.z;w/

� QL ix

�igx

�i.z/� QL i

x�igx

�i.w/

and then

QL ix

�igx

�i.w/ � ˇxqx.w/ � exp

�sQx%

˛x.z;w/

�� QL ix

�igx

�i.z/ � ˇxqx.z/

�:


Moreover, ˇxqx � Cmin.x/ � QL ix

�igx

�i. Hence the function

hx WDQL ix

�igx

�i� ˇxqx

1 � ˇx

2 sx : ut

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 WX ! .0;1/ such that for a.e. x 2 X for every N � 1 and gx

�N2 s

x�N

wehave

k. QL Ng/x � qxk1 D k QL Nx

�Ngx

�N� qxk1 � A.x/BN :

Proof. Fix x 2 X . Put gn WD gxn, ˇn WD ˇxn

, sn WD s

xn, and . QL ng/k WD

. QL ng/xk. Let .i.n//1nD1 be a sequence of integers such that i.nC 1/ � j.x�S.n//,

where S.n/ D PnkD1 i.k/, n � 1, and where S.0/ D 0. If g�S.n/ 2 s

�S.n/, then

Lemma 3.16 yields the existence of a function hn�1 2 s�S.n�1/

such that

� QL i.n/g�

�S.n�1/D ˇ�S.n�1/q�S.n�1/ C .1� ˇ�S.n�1//hn�1

D�1 � .1 � ˇ�S.n�1//

�q�S.n�1/ C .1 � ˇ�S.n�1//hn�1:

Since� QL i.n/Ci.n�1/g

�

�S.n�2/D� QL i.n�1/

� QL i.n/g��

�S.n�2/

D� QL i.n�1/

�ˇ

�S.n�1/q�S.n�1/ C .1� ˇ�S.n�1//hn�1

��

�S.n�2/

D ˇ�S.n�1/q�S.n�2/ C .1� ˇ

�S.n�1//� QL i.n�1/.hn�1/

��S.n�2/

it follows again from Lemma 3.16 that there is hn�2 2 s�S.n�2/

such that

� QL i.n/Ci.n�1/g�

�S.n�2/D ˇ

�S.n�1/q�S.n�2/

C.1 � ˇ�S.n�1//

�ˇ

�S.n�2/qS.n�2/C.1 � ˇ�S.n�2//hn�2

�

D�1 � .1 � ˇ

�S.n�2//.1 � ˇ�S.n�1//

�qS.n�2/

C .1 � ˇ�S.n�2//.1 � ˇ

�S.n�1//hn�1:

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

� QL S.n/g�

xD� QL i.n/C:::Ci.1/g

�

xD .1 �˘ .n/

x /qx C˘ .n/x h;

3.6 Exponential Convergence of Transfer Operators 29

where we set ˘ .n/x D Qn�1

kD0.1 � ˇx�S.k/

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

Therefore,

ˇˇˇ� QL S.n/g

�

x��1 �˘ .n/

x

�qx

ˇˇˇ � Cmax.x/˘

.n/x if g�S.n/ 2 s

�S.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/

1kD0

of non-negative integers such that n0 D 0,for k > 0, we have that x�J nk

2 G, and

#fn W 0 � n < nk and x�J n 2 Gg D k � 1: (3.36)

Indeed, applying Birkhoff’s Ergodic Theorem to the mapping ��J we have that foralmost every x 2 X ,

limn!1

#f0 � m � n � 1 W ��J m.x/ 2 Ggn

D E.1GjIJ /.x/;

where E.1GjIJ / 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,then set [J �1

j D0�j .A/ is ��1-invariant. If m.A/ > 0, then from ergodicity of ��1

we get that m.[J �1j D0�

j .A// D 1, and then by invariantness of the measure m, weconclude that m.A/ � 1=J . Hence we get that for almost every x the sequence nk

is infinite and

limk!1

k

nk

� 3

4J: (3.37)

Fix N � 0 and take l � 0 so that Jnl � N � JnlC1. Define a finite sequence�S.k/

�lkD1

by S.k/ WD Jnk for k < l and S.l/ WD N , and observe that by (3.37),we have N � JnlC1 � 4J 2l . Then (3.19) and (3.34) give

jj QL Nx

�Ngx

�N� qxjj1 �

ˇˇˇˇˇˇ QL N

x�Ngx

�N��1 �˘ .l/

x

�qx

ˇˇˇˇˇˇ1 C˘ .l/

x jjqxjj1

� .1 �M/l�C'.x/C Cmax.x/

�

� .4J 2p

1 �M/N�C'.x/C Cmax.x/

�:


This establishes our proposition with B D 4J 2p1 �M and

A.x/ WD maxf2Cmax.x/B�J k�

x ; .C'.x/C Cmax.x//g;

where k�x is a measurable function such that we have k

nk� 1

2Jfor all k � k�

x . utFrom now onwards throughout this section, rather than the operator QL , we

consider the operator OLx defined previously in (3.26).

Lemma 3.18 Let s > 1 and let g W J ! R be any function such that gx 2H ˛.Jx/. Then, with the notation of Proposition 3.17, we have

ˇˇˇˇˇˇ OL n

x gx�� Z

gxd�x

�1

ˇˇˇˇˇˇ1 �C'.�

n.x//� Z

jgx jd�xC4v˛.gxqx/

Qx

�A.�n.x//Bn:

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 setsx and from Proposition 3.17

we have

ˇˇˇˇˇˇ QL n

x gx �� Z

gx d�x

�q�n.x/

ˇˇˇˇˇˇ1

�ˇˇˇˇˇˇ�x QL n

x hx � v˛.gx/

Qx

QL nx 1x �

� Z

gx d�x

�q�n.x/

ˇˇˇˇˇˇ1

Dˇˇˇˇˇˇ�x QL n

x hx ��xq�n.x/ C v˛.gx/

Qx

�q�n.x/ � QL n

x 1x

�ˇˇˇˇˇˇ1

��x C v˛.gx/

Qx

�A.�n.x//Bn:

Then applying this inequality for gxqx and using (3.19) we get

ˇˇˇˇˇˇ OL n

x gx �� Z

gx d�x

�1�n.x/

ˇˇˇˇˇˇ1

�ˇˇˇˇˇˇ1

q�n.x/

ˇˇˇˇˇˇ �ˇˇˇˇˇˇ QL n

x .gxqx/�� Z

gxqx d�x

�q�n.x/

ˇˇˇˇˇˇ1

� C'.�n.x//

� Z

gx d�x C 2v˛.gxqx/

Qx

�A.�n.x//Bn:

So, we have the desired estimate for non-negativegx . In the general case we can usethe standard trick and write gx D gC

x � g�x , where gC

x ; g�x � 0. Then the lemma

follows. utThe 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 �-orbitof x 2 X . In particular, it is not clear at all from this statement that the item 1 inTheorem 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

ˇˇˇˇˇˇ OL n

x gx �� Z

gxd�x

�1�n.x/

ˇˇˇˇˇˇ1 ��!

n!1 0:

Proof. First of all, we may assume without loss of generality that the functiongx 2 H ˛.Jx/ since every continuous function is a limit of a uniformly convergentsequence of Holder 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 isin the set XA . Then by Poincare 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. Thereforewe get, for such an x 2 XA , from Lemma 3.18 that

�� OL

njx gx �

� Zgxd�x

�1�

nj .x/

��1

� Zjgx j d�x C 4

v˛.gxqx/

Qx

��1 � A Bnj

(3.39)for every j � 1. Finally, to pass from the subsequence .nj / to the sequence ofall natural numbers we employ the monotonicity argument that already appeared inWalters paper [29]. Since OLx1x D 1�.x/, we have for every w 2 J�.x/ that

infz2Jx

gx.z/ �X

z2T �1x .w/

gx.z/eO'.z/ � sup

z2Jx

gx.z/:

Consequently, the sequence

.Mn;x/1nD0 D . sup

w2J�n.x/

OL nx gx.w//

1nD0

is weakly decreasing. Similarly we have a weakly increasing sequence

.mn;x/1nD0 D . inf

w2J�n.x/

OL nx gx.w//

1nD0:

The proposition follows since, by (3.39), both sequences converge on the subse-quence .nj /. ut

3.7 Exponential Decay of Correlations

The following proposition proves item 3 of Theorem 3.2. For a function fx 2L1.�x/ we denote its L1-norm with respect to �x by

kfxk1 WDZ

jfxjd�x :


Proposition 3.20 There exists a �-invariant set X 0 � X of full m-measure suchthat, for every x 2 X 0, every f�n.x/ 2 L1.��n.x// and every gx 2 H ˛.Jx/,

ˇˇ�x

�.f�n.x/ ı T n

x /gx

� � ��n.x/.f�n.x//�x.gx/ˇˇ � A�.gx ; �

n.x//Bnjjf�n.x/jj1;

where

A�.gx ; �n.x// WD C'.�

n.x//� Z

jgxjd�x C 4v˛.gxqx/

Qx

�A.�n.x//:

Proof. Set hx D gx � Rgxd�x and note that by (3.30) and (3.27) we have that

�x

�.f�n.x/ ı T n

x /gx

� � ��n.x/.f�n.x//�x.gx/ D ��n.x/

�f�n.x/

OL nx .gx/

�

��n.x/.f�n.x//�x.gx/

D ��n.x/

�f�n.x/

OL nx .hx/

�: (3.40)

Since Lemma 3.18 yields jj OL nx hx jj1 � A�.gx ; �

n.x//Bn it follows from (3.40)thatˇˇ�x

�.f�n.x/ ı T nx /gx

� � ��n.x/.f�n.x//�x.gx/ˇˇ �

Z ˇˇf�n.x/ OL n

x .hx/ˇˇd��n.x/

� A�

.gx; �n.x//Bn

Z ˇˇf�n.x/

ˇˇd��n.x/:

utUsing 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 andX 0is the set given by Lemma 3.20. If jjf�n.x/jj1 ¤ 0 for all n, then

ˇˇ�x

�.f�n.x/ ı T n

x /gx

� � ��n.x/.f�n.x//�x.gx/ˇˇ

jjf�n.x/jj1 �! 0 as n ! 1:

Remark 3.22 Note that if jjf�n.x/jj1 grows subexponentially, then

ˇˇ�x

�.f�n.x/ ı T n

x /gx

� � ��n.x/.f�n.x//�x.gx/ˇˇ �! 0 as n ! 1: (3.41)

This is for example the case if x 7! log jjfxjj1 is m-integrable since Birkhoff’sErgodic 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 .L nx /

�ıwn

L nx 1.wn/

:

Then, by Proposition 3.19, for a.e. x 2 X and all gx 2 C .Jx/ we have

limn!1 �x;n.gx/ D lim

n!1L n

x gx.wn/

L nx 1.wn/

D limn!1

OL nx .gx=qx/.wn/

OL nx .1=qx/.wn/

D �x.gx/

�x.1/D �x.gx/:

(3.42)In other words,

�x;n ��!n!1 �x: (3.43)

in the weak* topology. Uniqueness of the measures �x follows. utLemma 3.24 There exists a unique function q 2 C 0.J / that satisfies (3.2).

Proof. Follows from Proposition 3.17. ut

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 functionby Px . It is important to note that this function is generally non-constant, even fora.e. x 2 X . Actually, if the pressure function is a.e. constant, then the randommap shares many properties with a deterministic system. This will be explained indetail in Sect. 5. Note that (3.42) and (3.11) imply an alternative definition of Px.'/,namely

Px.'/ D log.��.x/.Lx1// D limn!1 log

L nC1x 1.wnC1/

L n�.x/

1.wnC1/; (3.44)

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

limn!1

1

nSnPx

�n� 1

nlog L n

x�n

1x�n.wn/ D 0:

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


1

C'.x/�A.x/Bn � L n

x�n

1x�n.w/

�nx

�n

� C'.x/C A.x/Bn

for every w 2 Jx and every n 2 N. Therefore

log� 1

C'.x/�A.x/

�� log L n

x�n

1x�n.w/ � log�n

x�n

� log�C'.x/C A.x/

�:

utLemma 3.26 For m-a.e. x 2 X and for every sequence yn 2 Jxn

, n � 0,

s-limn!1

� 1

nSnPx � 1

nlog L n

x 1x.yn/�

D 0:

Proof. Using Egorov’s Theorem and Lemma 3.25 we have that for each ı > 0 thereexists a set Fı such that m.X n Xı/ < ı and

1

nSnPx

�n� 1

nmax

y2Jxn

log L nx

�n1x

�n.y/ ��!

n!1 0

uniformly on Fı . The lemma follows now from Birkhoff’s Ergodic Theorem. utLemma 3.27 If there exist g 2 L1.m/ such that log kLx1k1 � g.x/, then

limn!1

��1

nSnPx � 1

nlog L n

x 1x

��1

D 0:

Proof. Let F WD Fı be the set from the proof of Lemma 3.26, let x 2 X 0CF and let.nj / be the visiting sequence. Let j be such that nj < n � nj C1. Then

log L nx 1.y/ � log kL

njx 1k C Sn�nj

g.�nj .x// for every y 2 J�n.x/:

(3.45)Now, let h.x/ WD k'xk1. Since by (3.12) � log�x � k'xk1,

� log�nx D � log�

njx � log�

n�njxnj

� SnjPx C Sn�nj

h.�nj .x//:

Then by (3.45)

1

nSnPx � 1

nlog L n

x 1x.yxn / � 1

njSnj Px � 1

njlog L

njx 1x.yxnj /C 1

nSn�nj .g C h/.�nj .x//:

On the other hand, for y 2 J�n.x/,

log L nx 1.y/ � log L

nj C1x 1.T

nj C1�n

�n.x/.y// � Snj C1�ng.�

n.x//

3.10 Gibbs Property 35

and by (3.12),

log�nx D log�

nj C1x � log�

nj C1�nxn

� log kLnj C1x 1k C Snj C1�nh.�

n.x//:

The lemma follows now by Birkhoff’s Ergodic Theorem. ut

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.T

�ny .B.T n.y/; �///

exp.Sn'.y/ � SnPx.'//� eQ�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.T�ny .B.T n.y/; �///

exp.Sn'.y/ � SnPx.'// �.�nx/

�1��n.x/.B.Tn.y/; �// supz0

2T�ny .B.T n.y/;�// e

Sn'.z0/

.�nx/�1eSn'.y/

� eQSn.x/ :

On the other hand

�x.T�ny .B.T n.y/; �///

exp.Sn'.y/ � SnPx.'// � .�nx/�1��n.x/.B.T

n.y/; �// infz0

2T�ny .B.T n.y/;�// e

Sn'.z0/

.�nx/�1eSn'.y/

� ��n.x/.B.Tn.y/; �//e�QSn.x/ :

The lemma follows by (3.17). utLemma 3.29 Let T W J ! J satisfy the condition of measurability of cardinalityof covers and let f�i;xg, where i D 1; 2, be two Gibbs families with pseudo-pressurefunctions x 7! Pi;x. Then, for a.e. x, the measures �1;x and �2;x are equivalent and

limk!1

1

nk

SnkP1;x D lim

k!11

nk

SnkP2;x D lim

k!11

nk

SnkPx;

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 canfind " > 0 such that

�2;x.Bx.A; "// � �2;x.A/C ı: (3.46)


Now, let Nx be a measurable function such that �.�Nxx /�1 � "=2. Set

Ajn WD fy 2 T �n

x .yjxn/ W A \ T �n

y .B.yjxn; �// ¤ ;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

A �axn[

j D1

[

y2Ajn

T �ny B.yj

xn; �/ � Bx.A; "/:

By (3.3) it follows that

�1;x.A/ �axnX

jD1

X

y2Ajn

�1;x.T�ny B.yjxn ; �//� D1.x/D

LX

jD1

X

y2Ajn

exp.Sn'.y/ � SnP1;x.'//:

(3.47)Then by (3.46) and again by (3.3)

�1;x.A/ � D1.x/D exp.SnP2;x � SnP1;x/

axnX

j D1

X

y2Ajn

exp.Sn'.y/� SnP2;x.'//

� D1.x/D2.x/D2 exp.SnP2;x � SnP1;x/

axnX

j D1

X

y2Ajn

�2;x.T�ny B.yj

xn; �//

� D1.x/D2.x/D2L exp.SnP2;x � SnP1;x/�2;x.B.A; "//

� D1.x/D2.x/D2L exp.SnP2;x � SnP1;x/.�2;x.A/C ı/; (3.48)

since for y ¤ y 0 such that y; y0 2 T �nx .y

jxn/, we have that

T �ny B.yj

xn; �/ \ T �n

y0

B.yjxn; �/ D ;:

Hence the difference SnkP2;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 asubsequence of .nk/ in (3.48). Similarly, exchanging �1;x with �2;x we obtain thatSnk

P1;x �SnkP2;x is bounded from above. Then, letting ı go to zero, we have that

�1;x and �2;x are equivalent.Note that

exp.�SnP1;x/L nx 1x.yn/ D

X

y2T�nx .yn/

eSn'x.y/�SnP1;x

� D1.x/DX

y2T�nx .yn/

�1;x.T�ny B.yn; �// � D1.x/D�1;x.Jx/

D D1.x/D:

3.11 Some Comments on Uniformly Expanding Random Maps 37

Then1

nlog Lx1x.yn/ � 1

nlog.D1.x/D/ � 1

nSnP1;x :

On the other hand, by (3.47), on the same subsequence

1 D �1x.Jx/ � D1.x/DL

X

y2T �nx .yn/

eSn'x .y/�SnP1;x

for some yn 2 fy1xn; : : : ; y

axnxn

g. Therefore, using Lemma 3.26 and the SandwichTheorem, we have that, for x 2 X 0

Z \ X 0P ,

limk!1

1

nk

SnkP1;x D lim

k!11

nk

SnkPx: ut

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

3.11 Some Comments on Uniformly ExpandingRandom Maps

By C 1� .J /we denote the space of B-measurable mappings g W J ! R with gx WJx ! R continuous such that supx2X kgxk1 < 1. ForH0 � 0, by H ˛� .J ;H0/

we denote the space of all functions ' in H ˛m .J / \ C 1

m .J / such that all of Hx

are bounded above by H0. Let

H ˛� .J / D[

H0�0

H ˛� .J ;H0/:

For ' 2 H ˛.J ;H0/ we put

Q WD H0

1X

j D1

�� j D H0��˛

1 � ��˛:

Then Lemma 2.3 takes on the following form.

Lemma 3.31 For every ' 2 H ˛� .J ;H0/,

jSn'x.T�ny .w1// � Sn'x.T

�ny .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 wherey D .x; z/.


In this paper, whenever we deal with uniformly expanding random maps,we always assume that potentials belong to H ˛� .J /. Hence all the functionsC'.x/, Cmax.x/, Cmin.x/ and ˇx defined, respectively, by (3.5)–(3.8) are uniformlybounded 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 thefollowing.

Lemma 3.32 There exists a constantA� such that, for x 2 X and all y1; y2 2 Jxn

ˇˇˇ

L nx 1.y1/

L n�1x1

1.y1/� �x

ˇˇˇ � A�B

n:

Proof. It follows from Proposition 3.17 that

j QLx1. QL 1/.y1/ � QLx1

1.y2/j � 2ABn�1:

Then by Lemma 3.6 and (3.22) we have, for some x-independent constant A�, that

ˇˇˇ

L nx 1.y1/

L n�1x1

1.y2/��x

ˇˇˇ � 2AB�1Bn�x

QL nx1.1/.y2/

� A�Bn: ut

Chapter 4Measurability, 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 fiberedspace J . Therefore, we define abstract measurable expanding random maps forwhich the above measurabilities of �x , qx , �x and �x can be shown. Then, wecan construct a Borel probability invariant ergodic measure on J for the skew-product transformation T with Gibbs property and study the corresponding expectedpressure.

Our settings are related to those of smooth expanding random mappings of onefixed Riemannian manifold from [17] and those of random subshifts of finite typewhose fibers are subsets of NN from [5]. One possible extension of these works isto 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. 4.5we show how Crauel’s random compact subsets of Polish spaces fit into our generalframework 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 beentreated with care in the literature. As a matter of fact, it was not quite clear to useven 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 .


39

40 4 Measurability, Pressure and Gibbs Condition

By L0m.J / we denote the set of all BJ -measurable functions and by C 0

m.J / theset of all BJ -measurable functions g such that gx 2 C .Jx/.

Lemma 4.1 If g 2 C 0m.J /, then x 7! kgxk1 is measurable.

Proof. The proof is a consequence of item 2. Indeed, let .Gn/ be an increasingapproximation of jgj by step functions. So let Gn D Pm

kD1 ak1Ak; 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 [mj DkC1�X .Aj /;

where k D 1; : : : ; m � 1. Let

Hn.x/ WDmX

kD0

ak1Xk.x/ D sup

y2Jx

Gn.x; y/:

Then the sequence .Hn/ is increasing and converges pointwise to the function x 7!kgxk1. ut

The space L1m.J / is, by definition, the set of all g 2 L0

m.J /, such thatR kgxk1dm.x/ < 1: We also define

C 1m.J / WD C 0

m.J / \ L1m.J /

andH ˛

m .J / WD C 1m.J / \ H ˛.J /:

By M 1.J / we denote the set of probability measures and by M 1m.J / its

subset consisting of measures � 0 such that there exists a system of fiber measuresf�0

xgx2X with the property that for every g 2 L1m.J /, the map x 7! R

Jxgx d�0

x

is measurable and Z

Jgd�0 D

Z

X

Z

Jx

gxd�0xdm.x/:

Thenm D �0 ı ��1

X (4.1)

and the family .� 0x/x2X is the canonical system of conditional measures of �0 with

respect to the measurable partition fJxgx2X of J . It is also instructive to noticethat in the case when J is a Lebesgue space then (4.1) implies that �0 2 M 1

m.J /.The measure �0 2 M 1.J / is called T -invariant if �0 ı T �1 D �0. If �0 2

M 1m.J /, then, in terms of the fiber measures, clearly T -invariance equivalently

means that the family f�0xgx2X is T -invariant; see Chap. 3.1 for the definition of

T -invariance of a family of measures.Fix ' 2 H 1

m .J /. Then the general expanding random map T W J ! Jis called a measurable expanding random map if the following conditions aresatisfied.

4.2 Measurability 41

Measurability of the Transfer Operator. The transfer operator is measurable, i.e.L g 2 C 0

m.J / for every g 2 C 0m.J /.

Integrability of the Logarithm of the Transfer Operator. The function X 3 x 7!log kLx1xk1 belongs to L1.m/.

We shall now provide a simple, easy to verify, sufficient condition for integrabil-ity of the logarithm of the transfer operator.

Lemma 4.2 If log.deg.Tx// 2 L1.m/, then x 7! log kLx1xk1 belongs to L1.m/.

Proof. Recall that

e�k'xk1 �

X

Tx.z/Dw

e'x.z/ � deg.Tx/ek'xk1 :

Hence �k'xk1 � log kLx1xk1 � log.deg.Tx// C k'xk1: ut

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, wecome back to the families of Gibbs states f�xgx2X and f�xgx2X whose pointwiseconstruction was given in Theorem 3.1. Since we have already established goodconvergence properties, especially the exponential decay of correlations, it willfollow rather easily that these families form in fact conditional measures of somemeasures � and � from M 1

m.J /. As an immediate consequence of item 3 ofTheorem 3.1, we get that the probability measure � is invariant under the actionof 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

limn!1

kL nx gxk1

kL nx 1k1

D �x.gx/:

Then measurability of x 7! �x.gx/ is a direct consequence of measurability of thetransfer operator. utThis lemma enables us to introduce the probability measure � on J given by theformula:

�.g/ DZ

X

Z

Jx

gxd�xdm.x/:

This measure, therefore, belongs to M 1m.J /.


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 L0

m.J /.

Proof. Since � 2 M 1m.J /, measurability of �’s follows from the formula (3.11)

and measurability of the transfer operator. Then measurability of �’s and of thetransfer operator together with limn!1 QL n

x�n

1 D qx (see Proposition 3.17) implymeasurability of q. ut

From this lemma and Lemma 4.3 it follows that we can define a measure � bythe formula:

�.g/ DZ

X

Z

Jx

qxgxd�xdm.x/: (4.2)

4.3 The Expected Pressure

The pressure function of a measurable expanding random map has the followingimportant 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'xk1 � log ��.x/.Lx1/ � log kLx1k1: (4.3)

Then, by (3.11) and integrability of the logarithm of the transfer operator, the func-tion Px.'/ is bounded above and below by integrable functions, hence integrable.

utTherefore, the expected pressure of ' given by

EP.'/ DZ

X

Px.'/dm.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 limn!1

1

nlog �n

x for a:e: x 2 X: (4.4)

In addition, by (3.11), �nx D �n

x�x.1/ D ��n.x/.Lnx .1//: Thus, it follows that

1

nlog �n

x D limk!1

1

nlog

L kCnx 1x.wkCn/

L k�n.x/

1�n.x/.wkCn/:

However, by Lemma 3.27 we can get even more interesting formula.

4.5 Random Compact Subsets of Polish Spaces 43

Lemma 4.6 For every ' 2 H ˛m .J / and for almost every x 2 X

EP.'/ D limn!1

1

nlog L n

x 1.wn/;

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 byBx the set fy 2 Jx W .x; y/ 2 Bg. Then we have that T �1

x .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 impliesergodicity of T with respect to �.

Define a function f by fx WD 1Bx. Clearly fx 2 L1.�x/ and f�n.x/ ı T n

x D fx

m-a.e. Let x 2 X 0 \X0, where X 0 is given by Proposition 3.20. Let gx be a functionfrom L1.Jx/ with

Rgxd�x D 0. Then using (3.41) we obtain that

limn!1 �x

�.f�n.x/ ı T n

x /gx

� ! 0:

Consequently Z

Bx

gx d�x 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 �. utA direct consequence of Lemma 3.29 and ergodicity of T is the following.

Proposition 4.8 The measure � 2 M 1m.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 aPolish space which is normalized so that diam.Y / D 1. Let BY be the �-algebra ofBorel subsets of Y and let KY be the space of all compact subsets of Y topologized


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 Yxg:

This map is also called a random set. If every Yx is closed (res. compact), it is calleda closed (res. compact) random set. With this terminology X 3 x 7! Jx � Y is acompact random set (see [10, Remark 2.16, p. 16]).

Closed random sets have the following important properties (cf. [10, Proposition2.4 and Theorem 2.6]).

Theorem 4.9 Suppose that X 3 x 7! Yx is a closed random set such thatYx ¤ ;.

(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 Yxg 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 openballs with radii equal to r . Moreover, by Lebesgue’s Covering Lemma, there exitsRx D Rx.r/ > 0 such that every ball B.yx ; Rx/ with yx 2 Yx is contained in aball from this cover. As we prove below, we can actually choose ax and Rx in ameasurable way. Hence for the compact random set x 7! Jx the measurability ofcardinality 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 X 0a � X of full measure m such that,

for every r > 0 and every positive integer k, there exists a measurable functionX 0

a 3 x 7! yx;k 2 Yx and there exist measurable functions X 0a 3 x 7! ax 2 N and

X 0a 3 x 7! Rx 2 RC such that for every x 2 X 0

a,

4.5 Random Compact Subsets of Polish Spaces 45

ax[

kD1

Bx.yx;k; r/ � Yx ;

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. Thenthe set Xn WD fx 2 X W Yx � Y1=ng is measurable and has the measure m.Xn/

greater or equal to 1 � 1=n. Define

X 0a WD

[

n2NXn:

Then m.X 0a/ D 1.

Let fyn W n 2 NCg be a dense subset of Y . Since Y1=n is compact, there exists apositive integer a.n/ such that

a.n/[

kD1

B.yk ; r=2/ � Y1=n: (4.5)

Define a function X 0a 3 x 7! ax , by ax D a.n/ where n WD minfk W x 2 Xkg: 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/ ¤ ;Yx otherwise.

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 2B.yk; r=2/, then B.yk ; r=2/ � B.yx:k ; r/. Therefore, by (4.5),

Ux[

kD1

B.yx;k ; r/ � Y1=n � Yx for all x 2 Xn :

Finally, for x 2 Xn, let Rx > 0 be a real number such that, for y 2 Y1=n, thereexists k D 1; : : : ; U.n/ for which B.y; Rx/ � B.yk ; r=2/ � B.yx;k; r/: ThenX 0

U 3 x 7! Rx 2 RC is also measurable. ut

Chapter 5Fractal Structure of Conformal ExpandingRandom Repellers

We now deal with conformal expanding random maps. We prove an appropriateversion of Bowen’s Formula, which asserts that the Hausdorff dimension of almostevery fiber Jx , denoted throughout the paper by HD, is equal to a unique zeroof the function t 7! EP.t/. We also show that typically Hausdorff and packingmeasures on fibers respectively vanish and are infinite. A simple example of such aphenomenon is a Random Cantor Set described.

Later in this paper the reader will find more refined and general examples ofRandom Conformal Systems notably Classical Random Expanding Systems, Bruckand Buger Polynomial Systems and DG-Systems.

In the following we suppose that all the fibers Jx are in an ambient space Ywhich is a smooth Riemannian manifold. We will deal with C 1C˛-conformalmappings fx and denote then jf 0

x.z/j the norm of the derivative of fx which, byconformality, is nothing else than the similarity factor of f 0

x.z/. Finally, let jjf 0xjj1

be the supremum of jf 0x.z/j over z 2 Jx . Since we deal with expanding systems

we havejf 0

xj � �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 randommap having fibers Jx � Y and such that the mappings fx W Jx ! J�.x/ can beextended to a neighborhood of Jx in Y to conformalC 1C˛ mappings. If in additionlog jjf 0

xjj1 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 jf 0x.z/j. The associated

topological pressure P.'t / will be denoted P.t/. Let


47

48 5 Fractal Structure of Conformal Expanding Random Repellers

EP.t/ DZ

X

Px.t/dm.x/

be its expected value with respect to the measurem. In view of (5.1), it follows fromLemma 9.6 that the function t 7! EP.t/ has a unique zero. Denote it by h. Theresult of this subsection is the following version of Bowen’s formula, identifyingthe 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 theHausdorff dimension HD.Jx/ of the fiber Jx .

Bowen’s formula has been obtained previously in various settings first byKifer [18] and then by Crauel and Flandoli [11], Bogenschutz and Ochs [6], andRugh [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/ � f �ny .B.f n

x .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.f�ny .B.f n

x .z/; �/// � exp�hQ�n.x/

�j.f nx /

0.z/j�h exp.�P nx .h//:

(5.3)

On the other hand, B.z; r/ 6� f�.sC1/

y .B.f sC1x .z/; �// for every s � k. But, since

by Lemma 2.3,

B.z; exp.�Q�sC1.x/�˛/j.f sC1

x /0.z/j�1�/ � f �.sC1/y .B.f sC1

x .z/; �//;

we getexp

��Q�sC1.x/�˛�j.f sC1

x /0.z/j�1� � r (5.4)

and j.f sx /

0.z/j�1 � ��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/�

˛�rh

� exp.�P nx .h//j

�f sC1�n

�n.x/

�0.f n

x .z//jh (5.5)

or, equivalently,

log �x;h.B.z; r//

log r� hC hQ�n.x/

log rC hQ�sC1.x/�

˛

log rC

�h log�ˇˇˇ�f sC1�n�n.x/

�0

.f nx .z//

ˇˇˇ�

log r

C�h log �

log rC �P n

x .h/

log r: (5.6)

5.1 Bowen’s Formula 49

Our goal is to show that

lim infr!0

log �x;h.B.z; r//

log r� h for a.e. x 2 X and all z 2 Jx :

Since the function x 7! Qx is measurable and almost everywhere finite, there existsM > 0 such that m.A/ > 0, where A D fx 2 X W Qx � M g: Fix n D nk � 0 tobe the largest integer less than or equal to k such that �n.x/ 2 A and s D sk to bethe least integer greater than or equal to k such that �sC1.x/ 2 A. It follows fromBirkhoff’s Ergodic Theorem that limk!1 sk=nk D 1: Of course if for k � 1 wetake any rk > 0 such that k.z; rk/ D k, then limk!1 rk D 0.

Now, note that by (5.2), the formula

f �ny .B.f n

x .z/; �// � B.z; exp.Q�n.x/�˛/j.f n

x /0.z/j�1�/

yields r � exp.Q�n.x/�˛/j.f n

x /0.z/j�1�: Equivalently,

� log r � log j.f nx /

0.z/j � �˛Q�n.x/ � log �:

Since log j.f nx /

0.z/j � log �nx and since the function x 7! log �x is integrable and

� D minf1;Z

log � dmg > 0

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

� log r � �

2n � �

3s: (5.7)

Remember that �n.x/ 2 A and � sC1.x/ 2 A. We thus obtain from (5.6) that

lim infr!0

log �x;h.B.z; r//

log r� h�3h lim sup

k!11

slog

�ˇˇˇ�f sC1�n

�n.x/

�0.f n

x .z//ˇˇˇ�

�21nP n

x .h/

(5.8)for a.e. x 2 X and all z 2 Jx . But as

RPx.h/dm.x/ D 0, we have by Birkhoff’s

Ergodic Theorem that

limn!1

1

nP n

x .h/ D 0: (5.9)

Also, since the measure �h is f -invariant, it follows from Birkhoff’s ErgodicTheorem 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

limj !1

1

jlog

ˇˇˇ�f j

x

�0.zx/

ˇˇˇ D O� WD

Z

Jlog jf 0

x.z/jd�h.x; z/ 2 .0;C1/:


Hence, remembering that �n.x/ and �sC1.x/ belong to A, we get

lim supk!1

1

slog

�ˇˇˇ�f sC1�n�n.x/

�0

.f nx .z//

ˇˇˇ�

D lim supk!1

1

s

�log

ˇˇˇ�f sC1x

�0

.z/ˇˇˇ � log

ˇˇˇ�f nx

�0

.z/ˇˇˇ�

D lim supk!1

1

s

�log

ˇˇˇ�f sC1x

�0

.zx/ˇˇˇ� log

ˇˇˇ�f nx

�0

.zx/ˇˇˇ�

� lim supk!1

1

slog

ˇˇˇ�f sC1x

�0

.zx/ˇˇˇ

� lim infk!1

1

slog

ˇˇˇ�f nx

�0

.zx/ˇˇˇ D O�� O� D 0:

Inserting this and (5.9) to (5.8) we get that

lim infr!0

log �x;h.B.z; r//

log r� h: (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 thatf �l

y .B.f lx .z/; �// � B.z; r/: (5.11)

Then, by Lemma 3.28,

�x;h.B.z; r// � �x;h.f�l

y .B.f lx .z/; �///

� D1.�l.x// exp

��Q�l .x/

�j.f lx /

0.z/j�l exp.�P lx.h//:

(5.12)

On the other hand, f �.l�1/y .B.f l�1

x .z/; �// 6� B.z; r/: But, since

f �.l�1/y .B.f l�1

x .z/; �// � B.y; exp.Q�l�1.x/�˛/j.f l�1x /0.z/j�1�/;

we getr � � exp.Q�l�1.x/�

˛/j.f l�1x /0.y/j�1: (5.13)

Thus j.f l�1x /0.z/j�1 � ��1 exp

��Q� l�1.x/�˛�r: Inserting this to (5.12) we obtain,

�x;h.B.z; r// � ��hD1.�l.x//e

�Q�l .x/j.f� l�1.x//

0.f l�1x .z//j�h (5.14)

� exp��hQ�l�1.x/�

˛�rh exp.�P l

x.h//: (5.15)

Now, given any integer j � 1 large enough, takeRj > 0 to be the least radius r > 0such that

f �jy .B.f j

x .z/; �// � B.z; r/:

Then l.y;Rj / D j . Since the function Q is measurable and almost everywherefinite, 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 andQ��1.x/ � E for all x 2 . It follows from Birkhoff’s Ergodic Theorem andergodicity 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 increasingsequence .ji /

1iD1 such that � ji .x/ 2 for all i � 1. Formula (5.13) then yields

� logRji� �E�˛ C log � C log j.f ji �1

x .z/j � �E�˛ C log � C log �ji �1x � �

2ji ;

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 getfrom (5.14) that

log �x;h

�B.z; Rji

/�

logRji

� hC 2 logE

�ji

� 2E

�ji

� 2h

�

1

ji

log jj.f�ji �1.x//0jj1 � 2h�˛E

�ji

�2h log �

�ji

� 2

�

1

ji

P jix .h/:

Noting thatR

XPx.t/dm.x/ D 0 and applying Birkhoff’s Ergodic Theo-

rem, we see that the last term in the above estimate converges to zero. Also1ji

log jj.f�ji �1.x//0jj1 converges to zero because of Birkhoff’s Ergodic Theorem

and integrability of the function x 7! log jjf 0x 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

lim infr!0

log �x;h.B.z; r//

log r� lim inf

i!1log �x;h

�B.z; Rji

/�

logRji

� h:

Combining this with (5.10), we obtain that

lim infr!0

log �x;h.B.z; r//

log rD h

for a.e. x 2 X and all z 2 Jx . This gives that HD.Jx/ D h for a.e. x 2 X . Weare done. ut

5.2 Quasi-Deterministic and Essential Systems

We now investigate the fractal structure of the Julia sets and we will see thatthe random systems naturally split into two classes depending on the asymptoticbehavior of Birkhoff’s sums of the topological pressure P n

x .h/.

Definition 5.3 Let f be a conformal uniformly expanding random map. It is calledessentially random if for m-a.e. x 2 X ,


lim supn!1

P nx .h/ D C1 and lim inf

n!1 P nx .h/ D �1; (5.16)

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

�Lx � P nx .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 auniformly conformal random map to be essential it suffices to know that thecondition (5.16) is satisfied for a set of points x 2 X with a positive measure m.

Remark 5.5 If the number

2.P.h// D limn!1

1

n

Z �Sn.P.h//

�2

dm > 0

and if the Law of Iterated Logarithm holds, i.e. if

�p22.P.h// D lim inf

n!1P n

x .h/pn log logn

� lim supn!1

P nx .h/p

n log lognDp22.P.h//

m-a.e., then our conformal random map is essential. It is essential even if only theCentral Limit Theorem holds, i.e. if

m

��

x 2 X W Pnx .h/pn

< r

�

! 1

p2�

Z r

�1e�s2=2�2.P.h// ds:

Remark 5.6 If there exists a bounded everywhere defined measurable function u WX ! R such that Px.h/ D u.x/ � u ı �.x/ (i.e. if P.h/ is a coboundary) for allx 2 X , then our system is quasi-deterministic.

For every ˛ > 0 let H ˛ refer to the ˛-dimensional Hausdorff measure and letP˛ refer to the ˛-dimensional packing measure. Recall that a Borel probabilitymeasure � defined on a metric spaceM is geometric with an exponent ˛ if and onlyif there exist A � 1 and R > 0 such that

A�1r˛ � �.B.z; r// � Ar˛

for all z 2 M and all 0 � r � R. The most significant basic properties of geometricmeasures 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 randommap.

(a) If the system f W J ! J is essential, then H h.Jx/ D 0 and Ph.Jx/ DC1 form-a.e. x 2 X .

(b) If, on the other hand, the system f W J ! J is quasi-deterministic, then forevery x 2 X �h

x is a geometric measure with exponent h and therefore (GM1)–(GM3) hold.

Proof. Part (a). Remember that by its very definition EP.h/ D RPx.h/dm.x/ D 0.

By Definition 5.3 there exists a measurable set X1 with m.X1/ D 1 such that forevery x 2 X1 there exists an increasing unbounded sequence .nj /

1j D1 (depending

on x) of positive integers such that

limj !1P

njx .h/ D �1: (5.18)

Since we are in the uniformly expanding case, the formula (5.12) from the proof ofTheorem 5.2 (Bowen’s Formula) takes on the following simplified form:

�x.B.z; r// � D�1rh exp��P l.z;r/

x .h/�

(5.19)

with some D � 1 and all z 2 Jx . Since the map is uniformly expanding, for allj � 1 large enough, there exists rj > 0 such that l.z; rj / D nj . So disregardingfinitely many terms, we may assume without loss of generality, that this is true forall j � 1. Clearly limj !1 rj D 0: It thus follows from (5.19) that

�x;h.B.z; rj // � D�1rhj exp

��P njx .h/

�

for all x 2 X1, all z 2 Jx and all j � 1. Therefore, by (5.18),

lim supr!0

�x;h.B.z; r//

rh� lim sup

j !1�x;h.B.z; rj //

rhj

�D�1lim supj !1

exp��P nj

x .h/�D C1;

which implies that H h.Jx/ D 0.

The proof for packing measures is similar. By Definition 5.3 there exists ameasurable set X2 with m.X2/ D 1 such that for every x 2 X2 there exists anincreasing unbounded sequence .sj /1j D1 (depending on x) of positive integers suchthat

limj !1P

sjx .h/ D C1: (5.20)

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 simplifiedform:

�x.B.z; r// � Drh exp��P k.z;r/

x .h/�

(5.21)


with D � 1 sufficiently large, all x 2 X2 and all z 2 Jx . By our uniformassumptions, for all j � 1 large enough, there existsRj >0 such that k.z; Rj /D sj .Clearly limj !1Rj D 0: It thus follows from (5.21) that

�x;h.B.z; rj // � DRhj exp

��P sjx .h/

�

for all x 2 X2, all z 2 Jx and all j � 1. Therefore, using (5.20), we get

lim infr!0

�x;h.B.z; r//

rh� lim inf

j !1�x;h.B.z; Rj //

Rhj

� D lim infj !1 exp

��P sjx .h/

� D 0:

Thus Ph.Jx/ D C1. We are done with part (a).

Suppose now that the map f W J ! J is quasi-deterministic. It then followsfrom Definition 5.3 and (5.19) along with (5.21), that for every x 2 X and for everyr > 0 small enough independently of x 2 X , we have

.LxD/�1rh � �x;h.B.y; r// � LxDr

h; x 2 X; z 2 Jx :

This means that each �x;h, x 2 X , is a geometric measure with exponent h and thetheorem follows. ut

As a straightforward consequence of this theorem we get a corollary transpar-ently 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 form-a.e. x 2 X the following hold.

(1) The fiber Jx is not bi-Lipschitz equivalent to any deterministic nor quasi-deterministic 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 .

Proof. Item (1) follows immediately from Theorem 5.7(a) and (b3). Item (3) fromTheorem 5.7(a) and the observation that H 1.W / > 0 whenever W is connected.The proof of (4) is similar. Since (3) obviously implies (2), we are done. ut

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 Bernoullimeasure. For x D .: : : ; x�1; x0; x1; : : :/ 2 X define fx D fx0

, f nx D f�n�1.x/ ı

f�n�2.x/ ı : : : ı fx and

Jx D1\

nD0

.f nx /

�1.Œ0; 1�/:

The skew product map defined onS

x2X Jx by the formula

f .x; y/ D .�.x/; fx.y//

generates a conformal random expanding system. We shall show that this system isessential. To simplify the next calculation, we define recurrently:

�x.1/ D�3 if x0 D 0

4 if x0 D 1; �x.n/ D ��n�1.x/.1/�x.n � 1/:

Consider the potential 't defined by the formula 'tx D �t log �x.1/. Then

Sn'tx D �t log �x.n/:

LetCn be a cylinder of the order n that isCn is a subset of Jx of diameter .�x.n//�1

such that f nx jCn

is one-to-one and onto J�n.x/. We can project the measure m onJx and we call this measure �x . In other words, �x is such a measure that allcylinders of level n have the measure 1=2n. Then by Law of Large Numbers form-almost every x

limn!1

log�x.Cn/

log diam.Cn/D log 2

.1=n/ log �x.n/D log 4

log 12DW h:

Therefore the Hausdorff dimension of Jx is for m-almost every x constant andequal to h. Next note that

�x.Cn/

diam.Cn/hD exp.�SnPx/; (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 theprobability ofZn D log 2�h log 3 is equal to the probability ofZn D log 2�h log 4


and is equal to 1=2. The expected value ofZn, EP , is zero and its standard deviation > 0. Then the Law of the Iterated Logarithm tells us that the following equalities:

lim infn!1

Z1 C : : :CZnpn log logn

D �p2 and lim sup

n!1Z1 C : : :CZnpn log logn

D p2 (5.23)

hold with probability one. Then, by (5.22),

lim supn!1

�x.Cn/

diam.Cn/hD 1 and lim inf

n!1�x.Cn/

diam.Cn/hD 0

form-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 dimensionof fibers is not constant as clearly HD.J01/ D log 2= log3, whereas HD.J11/ Dlog 2= log 4 D 1=2.

Chapter 6Multifractal Analysis

The second direction of our study of fractal properties of conformal randomexpanding maps is to investigate the multifractal spectrum of Gibbs measures onfibers. We show that the multifractal formalism is valid. It seems that it is impossibleto do it with a method inspired by the proof of Bowen’s formula since one gets fullmeasure sets for each real ˛ and not one full measure set Xma such that for allx 2 Xma, the multifractal spectrum of the Gibbs measure on the fiber over x isgiven by the Legendre transform of a temperature function which is independentof x 2 Xma. In order to overcome this problem we work out a different proof inwhich we minimize the use Birkhoff’s Ergodic Theorem and instead we base theproof on the definition of Gibbs measures and the behavior of the Perron–Frobeniusoperator. In this point we were partially motivated by the approach presented inFalconer’s book [15].

Another issue we would like to bring up here is real analyticity of the multifractalspectrum which we establish for uniformly expanding systems. The proof is basedon real-analiticity results for the expected pressure which are treated separately inChap. 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 functionqx and therefore this will not cause any confusion. Define auxiliary potentials

'q;x;t.y/ WD q.'x.y/ � Px.'//� t log jf 0x.y/j:

By Lemma 9.5, the function .q; t/ 7! EP.q; t/ WD EP.'q;t / is convex. Moreover,since log jf 0

x.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


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 thetemperature function. Put

'q WD 'q;T .q/:

ByDT 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 followsfrom its convexity that the function T is differentiable everywhere but a countableset, where it is left and right differentiable. Define

L.T /.˛/ WD inf�1<q<1�˛q C T .q/

�;

where˛ 2 Dom.L/ D �

limq!�1 �T 0.q�/; lim

q!1 �T 0.qC/�:

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 /.�˛/. Thetransform 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, forall ı 2 .0; ı"/, we have

EP..1C ı/q; T .q/C .qT 0.q/C "/ı/ < 0

andEP..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 maywrite

T .q C ıq/ D T .q/C T 0.q/ıq C o.ı/

6.2 Multifractal Spectrum 59

for all ı > 0 sufficiently small, say ı 2 .0; ı.1/" /. So,

T .q/C .qT 0.q/C "/ı � T ..1C ı/q/ D "ı C o.ı/ > 0:

Then, in virtue of Lemma 9.6, we get that

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

meaning that the first assertion of our lemma is proved. The second one isproved similarly producing a positive number ı.2/

" . Setting then ı" D minfı.1/" ; ı

.2/" g

completes the proof. ut

6.2 Multifractal Spectrum

Let � be the invariant Gibbs measure for ' and let � be the '-conformal measure.For every ˛ 2 R define

Kx.˛/ WDny 2 Jx W d�x

.y/ WD limr!0

log�x.B.y; r//

log rD ˛

o

and

K 0x WD

ny 2 Jx W the limit lim

r!0

log�x.B.y; r//

log rdoes not exist

o:

This gives us the multifractal decomposition

Jx WD]

˛�0

Kx.˛/ ]K 0x:

The multifractal spectrum is the family of functions fg�xgx2X given by the

formulas:g�x

.˛/ WD HD.Kx.˛//:

The function d�x.y/ is called the local dimension of the measure �x at the point y.

Since form almost every x 2 X the measures�x and �x are equivalent with Radon–Nikodym derivatives uniformly separated from 0 and infinity (though the boundsmay and usually do depend on x), we conclude that we get the same set Kx.˛/ if inits definition the measure �x is replaced by �x . Our goal now is to get a “smooth”formula for g�x

.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 withm.Xma/ D 1 and such that, for every x 2 Xma, and all q 2 DT , we have


g�x.�T 0.q// D �qT 0.q/C T .q/

Proof. Firstly, by Lemma 9.4, for every 0<R� � there exists a measurablefunction 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

D�q�

R .�n.x// � �q;x.f�n

y .B.f n.y/; R///

exp�q.Sn'.y/ � P n

x .'//�j.f n

x /0.y/j�T .q/

� Dq�

R .�n.x//;

(6.1)where q� WD .q; T .q//� as defined in (9.1). In what follows we keep the notationfrom the proof of Theorem 5.2. The formulas (5.2) and (5.11) then give for everyj � l and every 0 � i � k, that

D�q�

�.�j .x///�1 exp

�q.Sj'.y/� P j

x .'//�j.f j

x /0.y/j�T .q/

� �q;x.B.y; r//

� Dq�

�.� i .x// exp

�q.Si'.y/ � P i

x.'//�j.f i

x /0.y/j�T .q/:

(6.2)

By Qx we denote the measurable function given by Lemma 2.3 for the function� log jf 0j. Let X� be an essential set for the functions X 3 x 7! Rx , X 3 x 7!a.x/, x 7! Qx , andX 3 x 7! D�.x/ with constants OR, Oa, OQ, and OD� . Let .nj /

11 be

the positively visiting sequence for X� at x. Let XE0 be the set given by Lemma 9.5

for potentials �q;t , q; t 2 R2. Let

X 0C WD XE0 \ X 0CX

�

:

Let us first prove the upper bound on g�x.�T 0.q//. Fix x 2 X 0C. Fix "1 > 0. For

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 12EP.�q;T .q/C"1

/ < 0. So, invirtue of Lemma 9.5, there exists C � 1 such that

L�q;T .q/C"1;x1.wk.xnj

// � Ce��nj (6.3)

for all j � 1 and all k D 1; 2; : : : ; a.�nj .x// � Oa. Now, fix an arbitrary "2 2 R

such that q"2 � 0. For every integer l � 1 let

Kx."2; l/ D�

y 2 Kx.�T 0.q// W � T 0.q/� 1

2j"2j � log �x.B.y; r//

log r� �T 0.q/C 1

2j"2j

for all 0 < r � 1=l

�

:


Note that

Kx.�T 0.q// D1[

lD1

Kx."2; l/: (6.4)

Let

nj.x/ D

8<

:z 2

a.xnj/

[

kD1

f�nj

x .wk.xnj// W Kx."2; l/ \ f �nj

z .B.f nj .z/; �=2// ¤ ;9=

;:

ThenKx."2; l/ �

[

z2�nj .x/

f�nj

z .B.f nj .z/; �=2//: (6.5)

For every z 2 nj.x/, say z 2 f �nj

x .wk.xnj//, choose

Oz 2 Kx."2; l/\ f�nj

z .B.wk.xnj/; �=2//:

Then B�wk.xnj

/; �=2� � B.f nj .z/; �/, and therefore

f�nj

z .B.wk.xnj/; �=2// � f

�nj

Oz�B.f nj .Oz/; �/�:

It follows from this and (6.5) that

Kx."2; l/ �[

z2�nj.x/

f�nj

Oz .B.f nj .Oz/; �//: (6.6)

Putr

.1/j .Oz/ D OQ�1j.f nj

x /0.Oz/j�1 and r.2/j .Oz/ D OQj.f nj

x /0.Oz/j�1

We then have

B�Oz; r .1/

j .Oz/� � f�nj

Oz .B.f nj .Oz/; �// � B�Oz; r.2/

j .Oz/�:

Therefore, assuming j � 1 to be sufficiently large so that the radii r.1/j .Oz/ and r.1/

j .Oz/are sufficiently small, particularly � 1=l , we get

log �x

�f

�nj

Oz .B.f nj .Oz/; �//�

� log j.f njx /0.Oz/j � log �x

�B.Oz/; OQ�1j.f nj

x /0.Oz/j�1�

� log j.f njx /0.Oz/j

� log �x

�B.Oz/; r.1/

j .Oz//�

log.r .1/j .Oz//C log OQ

� �T 0.q/C j"2j


and

log �x

�f

�nj

Oz .B.f nj .Oz/; �//�

� log j.f njx /0.Oz/j � log �x

�B.Oz/; OQj.f nj

x /0.Oz/j�1�

� log j.f njx /0.Oz/j

� log �x

�B.Oz/; r .2/

j .Oz//�

log.r .2/j .Oz// � log OQ

� �T 0.q/ � j"2j:

Hence,

jqj�log �x

�f

�nj

Oz .B.f nj .Oz/; �//� � .T 0.q/C j"2j/ log j.f njx /0.Oz/j� � 0

and

jqj�log �x

�f

�nj

Oz .B.f nj .Oz/; �//� � .T 0.q/� j"2j/ log j.f njx /0.Oz/j� � 0:

So, in either case (as "2q > 0),

�q�log �x

�f

�nj

Oz .B.f nj .Oz/; �//� � .T 0.q/ � j"2j/ log j.f njx /0.Oz/j� � 0

or equivalently,

��qx

�f

�nj

Oz .B.f nj .Oz/; �//�j.f nj /0.Oz/jqT 0.q/�"2q � 1: (6.7)

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

X

z2�nj.x/

diam�qT 0.q/CT .q/C"1C"2q�f

�nj

Oz .B.f nj .Oz/; �//�

DX

z2�nj.x/

diamT .q/C"1�f

�nj

Oz .B.f nj .Oz/; �//�diam�qT 0.q/C"2q�f

�nj

Oz .B.f nj .Oz/; �//�

�X

z2�nj.x/

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

D . OQ��1/2tX

z2�nj.x/

exp�q.Snj

'.z/� P njx .'// � .T .q/C "1/ log j.f nj

x /0.z/j�

� exp�q.P

njx .'/ � Snj

'.z/�j.f nj /0.Oz/jqT 0.q/�"2q

� . OQ��1/2teq OQ�X

z2�nj.x/

. OQ��1/2tX

z2�nj.x/

exp�q.Snj

'.z/ � P njx .'//

�.T .q/C "1/ log j.f njx /0.z/j� exp

�q.P

njx .'/ � Snj

'.Oz//�j.f nj /0.Oz/jqT 0.q/�"2q



z2�nj.x/

. OQ��1/2tX

z2�nj.x/

exp�q.Snj

'.z/ � P njx .'//

�.T .q/C "1/ log j.f njx /0.z/j��q

x

�f

�nj

Oz .B.f nj .Oz/; �//�j.f nj /0.Oz/jqT 0.q/�"2q


z2�nj .x/

. OQ��1/2tX

z2�nj.x/

exp�q.Snj

'.z/ � P njx .'//

�.T .q/C "1/ log j.f njx /0.z/j�

� . OQ��1/2teq OQ�

a.xnj/

X

kD1

L�q;T .q/C"1;x1.wk.xnj

// � C. OQ��1/2teq OQ�a.xnj/e��nj

� C. OQ��1/2teq OQ�ae��nj :

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 "2q > 0

were arbitrary, it follows that

g�x.�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 integerin 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

log �q;x.B.y; r//

log r� �q� log OD� C q

�Sl

C

'.y/� PlC

x .'/�� T .q/ log j.f l

C

x /0.y/jlog � C �˛ OQ � log j.f l

�

x /0.y/jand

log �q;x.B.y; r//

log r� q� log OD� C q

�Sk

�

'.y/ � P k�

x .'/�� T .q/ log j.f k

�

x /0.y/jlog � � �˛ OQ � log j.f k

C

x /0.y/j:

Hence,

lim supr!0

log �q;x.B.y; r//

log r

� lim supn!1

qP

nC

x .'/ � SnC

'.y/

log j.f n�

x /0.y/j

!

C T .q/lim supn!1

log j.f nC

x /0.y/jlog j.f n

�

x /0.y/j (6.9)


and

lim infr!0

log �q;x.B.y; r//

log r

� lim infn!1

�

qP

n�

x .'/� Sn�

'.y/

log j.f nC

x /0.y/j

C T .q/lim infn!1

log j.f n�

x /0.y/jlog j.f n

C

x /0.y/j : (6.10)

Now, given " > 0 and ı" > 0 ascribed to " according to Lemma 6.1, fix an arbitraryı 2 .0; ı". Set

�.1/ D �.1/

";ıD �.1Cı/q;T .q/C.qT 0.q/C"/ı exp

��.1C ı/P.�q/�

and�.2/ D �

.2/

";ıD �.1�ı/q;T .q/C.�qT 0.q/C"/ı exp

��.1C ı/P.�q/�:

Since

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

�C .1C ı/

ZP.�q/dm

D EP.�.1Cı/q;T .q/C.qT 0.q/C"/ı

�

and

EP.�.2// D EP.�.1�ı/q;T .q/C.�qT 0.q/C"/ı

�C .1 � ı/ZP.�q/dm

D EP.�.1�ı/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 forall k D 1; 2, and all n � 1 sufficiently large, we have 1

nlog L n

�.k/x

.1/.w/ � log �

for all x 2 X 0C and all w 2 J�n.x/. Equivalently,

L n

�.k/x

.1/.w/ � �n: (6.11)

Now, for all x 2 X 0C, all j � 1, all 1 � k � a.�nj .x/ � Oa, and all z 2 f�nj

x

.wk.xnj//, define

A.z/ WDny 2 f �nj

z .B.wk.xnj/; �// W B.f nj .y/; R/ � B.wk.xnj

/; �/o:

Note thata.xnj

/[

kD1

[

z2f�nj

x .wk.xnj//

A.z/ D J .x/: (6.12)


Fix any q 2 DT and set

�" D sup0<ı�ı"

nmaxf..1C ı/q; T .q/C .qT 0.q/C "/ı/�;

..1 � ı/q; T .q/C .�qT 0.q/C "/ı/�go:

Let x 2 X 0C. Set

M WD exp� OQı.�qT 0.q/C T .q/� "/�:

Then, using (6.12), Lemma 2.3 (for the potential .x; z/ 7! log jf 0x.z/j, (6.2), and

(6.11), we obtain

�q;x�fy 2 Jx W �q;x

�f

�njy .B.f nj .y/;R//

� � j.f njx /0.y/j�.�qT 0.q/CT .q//C"g�

D �q;x�fy 2 Jx W �q;x

�f


�j.f njx /0.y/j�qT 0.q/CT .q/�" � 1g�

D �q;x�fy 2 Jx W �ıq;x

�f


�j.f njx /0.y/jı.�qT 0.q/CT .q/�"/ � 1g�

�Z

Jx

�ıq;x�f


�j.f njx /0.y/jı.�qT 0.q/CT .q/�"/d�q;x.y/

�a.xnj /X

kD1

X

z2f�njx .wk.xnj //

Z

A.z/�ıq;x

�f

�njy .B.f nj .y/;R///

�

j.f njx /0.y/jı.�qT 0.q/CT .q/�"/d�q;x.y/

�a.xnj /X

kD1

X


�ıq;x�f

�njz .B.wk.xnj /; �///

�

j.f njx /0.z/jı.�qT 0.q/CT .q/�"/M�q;x.A.z//

� M

a.xnj /X

kD1

X


�ıq;x�f


�j.f njx /0.z/jı.�qT 0.q/CT .q/�"/

��q;x�f


�

D M

a.xnj /X

kD1

X


�1Cıq;x

�f


�j.f njx /0.z/jı.�qT 0.q/CT .q/�"/

� MD�"�

a.xnj /X

kD1

X


exp.1C ı/q

�Snj �.z/ � P

njx .�.z/

� � .1C ı/Px.�njq /�


j.f njx /0.z/j�.T .q/.1Cı/Cı.qT 0.q/�T .q/C"// exp.�.1C ı/P

njx .�q.z///

D MD�"�

a.xnj /X

kD1

X


exp.1C ı/q

�Snj �.z/ � P

njx .�.z/

� � .1C ı/Px.�njq /�

�j.f njx /0.z/j�.T .q/C.qT 0.q/C"/ı/ exp.�.1C ı/P

njx .�q.z///

D MD�"�

a.xnj /X

kD1

Lnj

�.1/x

.1/.wk.xnj // � MD�"� a�

nj : (6.13)

Therefore,

1X

jD1

�q;x�fy 2 Jx W �q;x

�f

�njy .B.f nj .y/; R//

� � j.f njx /0.y/j�.�qT 0.q/CT .q//C"g� < C1:

Hence, by the Borel–Cantelli Lemma, there exists a measurable set J q1;";x � Jx

such that �q;x.Jq1;";x/ D 1 and

#nj � 1 W �q;x

�fy 2 Jx W �q;x

�f


�

� j.f njx /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 measurableset J q

2;";x � Jx such that �q;x.Jq2;";x/ D 1 and

#nj � 1 W �q;x

�fy 2 Jx W �q;x

�f


�

� j.f njx /0.y/j�.�qT 0.q/CT .q//C"g�

o< 1: (6.15)

Set

J qx D

1\

nD1

J q

1;1=n;x\ J q

2;1=n;x:

Then �q;x.Jq

x / D 1 and, it follows from (6.13) and (6.1), that for all y 2 J qx , we

have

limj !1

q.Pnjx .'/� Snj

'.y//

log j.f njx /0.y/j D �qT 0.q/:

Since limn!1 n�

nC

D 1, it thus follows from (6.9) and (6.10) that

d�q;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)

limr!0

log �x.B.x; r//

log rD �T 0.q/

for all y 2 J qx . As the latter formula implies that J q

x � K.�T 0.q//, and as�q;x.J

qx / D 1, applying (6.16), we get that

g�x.�T 0.q// D HD.Kx.�T 0.q/// � HD.J q

x // D �qT 0.q/C T .q/:

Combining this formula with (6.8) completes the proof. utAs 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 randomexpanding map. Then the Legendre conjugate, gW Range.�T 0/ ! Œ0;C1/, tothe temperature function R 3 q 7! T .q/ is differentiable everywhere except acountable set of points, call itD�

T , and there exists a measurable set Xma � X withm.Xma/ D 1 such that for every ˛ 2 D�

T / and every x 2 Xma, we have

g�x.˛/ D g.˛/:

6.3 Analyticity of the Multifractal Spectrumfor Uniformly Expanding Random Maps

Now, as in Chap. 9.4, we assume that we deal with a conformal uniform randomexpanding 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 existconstants 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 thefollowing property of the function T .

Proposition 6.4 Suppose that f WJ ! J is a conformal uniformly randomexpanding map. Then the temperature function T is real-analytic and for every q,we have

T 0.q/ DRJ 'd�q

RJ log jf 0jd�q

< 0: (6.18)

Proof. The potentials

'q;x;t.y/ WD q.'x.y/� Px.'//� t log jf 0x.y/j


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 9.17are satisfied, and therefore the function R � R 3 .q; t/ 7! EP.q; t/ is real-analytic.Since jf 0

x.y/j > 0, in virtue of Proposition 9.18 we obtain that

@EP.q; t/

dtD �

Z

J

log jf 0x jd�q;x;tdm.x/ < 0: (6.19)

Hence, we can apply the Implicit Function Theorem to conclude that the tempera-ture function R 3 q 7! T .q/ 2 R, satisfying the equation,

EP.q; T .q// D 0;

is real-analytic. Hence,

0 D dEP.'q/

dqD @EP.q; t/

@q

ˇˇˇtDT .q/

C @EP.q; t/

@t

ˇˇˇtDT .q/

T 0.q/:

Then

T 0.q/ D �@EP.q;t/

@q

ˇˇtDT .q/

@EP.q;t/@t

ˇˇtDT .q/

D �RJ .'x � Px/d�q;xdm.x/RJ � log jf 0

x jd�q;xdm.x/

DRJ 'xd�q;xdm.x/� R

X Pxdm.x/RJ log jf 0

x jd�q;xdm.x/D

RJ 'd�q

RJ log jf 0jd�q

:

So, we obtain (6.18). It follows, in particular, that

T 0.q/ < 0; (6.20)

since by (6.17), the integralRJ 'd�q is negative. ut

Combining this proposition with Proposition 6.2 we get the following resultwhich concludes this section.

Theorem 6.5 Suppose that f WJ ! J is a conformal uniformly randomexpanding map. Then the Legendre conjugate, g W Range.�T 0/ ! Œ0;C1/, to thetemperature function R 3 q 7! T .q/ is real-analytic, and there exists a measurableset Xma � X with m.Xma/ D 1 such that for every ˛ 2 Range.�T 0/ and everyx 2 Xma, we have

g�x.˛/ D g.˛/:

Chapter 7Expanding in the Mean

In this chapter we show that the main achievements of this manuscript, includingthermodynamical formalism, Bowen’s formula and multifractal analysis, also holdfor a class of random maps satisfying an allegedly weaker expanding condition

Zlog �xdm.x/ > 0:

We start with a precise definition of this class. Then we explain how this case canbe 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 notexpanding 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 satisfyingthe properties of Measurability of the Degree and Topological Exactness. Sucha random map is called expanding in the mean, if for some � > 0 and somemeasurable function X 3 x 7! �x 2 RC with

Zlog �xdm.x/ > 0;

we have that all inverse branches of every T nx are well defined on balls of radii �

and are .�nx /�1-Lipschitz continuous. More precisely, for every y D .x; z/ 2 J

and every n 2 N, there exists

T �ny W B�n.x/.T

n.y/; �/ ! Jx


69

70 7 Expanding in the Mean

such that

1. T n ı T �ny D IdjB�n.x/.T n.y/;�/ and T �n

y .T nx .z// D z,

2. %.T �ny .z1/; T �n

y .z2// � .�nx /�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 oursetting from Sect. 2.3.

Let T W J ! J be an expanding in the mean random map. To this map and toa set A � X of positive measure we associate an induced map T in the followingway. 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�A.x/�1Y

j D0

��j .x/:

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

T x D T �A.x/x for a.e. x 2 A:

The following lemma show that the set A can be chosen such that T is anexpanding 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

0 <

Z

X

log �xdm.x/ DZ

A1

log �A1;xdm.x/ DZ

Ak

log �Ak ;xdm.x/;

7.2 Associated Induced Map 71

we have that m.Ak/ > 0 for all k � 1. Obviously, the sequence .Ak/1kD1

isdecreasing. Let

A D1\

kD1

Ak and E D X n A :

Notice that the points x 2 E have the property that log �nx � 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 themeasure m is �-invariant, we have that m.E1/ D 1 where

E1 D1\

nD0

��n.E/ :

For x 2 E1 we have that log �nx � 0 for infinitely many n � 1. This contradicts

Birkhoff’s Ergodic Theorem since, by hypothesis,R

log �x > 0. Therefore the setA 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: ut

In the following A � X will be some set coming from Lemma 7.1 and T DT �A the associated induced map. For this map, we have to consider the followingappropriated class of Holder potentials. First, to every y D .x; z/ we associate theneighborhood

U.z/ D1[

nD0

T �ny

�B�n.x/

�T n.y/; �

�� Jx:

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

Z

X

log Hxdm.x/ < 1 (7.1)


andv˛.'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 overall z1; z2 2 Jx with z1; z2 2 U.z/, z 2 Jx . For example, any function, which isHx-Holder on the entire set Jx is fine.

Let T be an expanding in the mean random map and ' a Holder potentialaccording to the definition above. Having associated to T the induced map T , onenaturally has to replace the potential ' by the induced potential

'x.z/ D�A.x/�1X

j D0

'�j .x/.Tjx .z//:

Although, it is not clear if the potential ' satisfies the condition (7.1), the choiceof the neighborhoods U.z/ and the definition of Holder potentials make thatLemma 2.3 still holds. This gives us an important control of the distortion whichis 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 originalsystem.

With the preceding notations, for the expanding induced map T the Thermo-dynamical Formalism of Chap. 3 and, in particular, the Theorems 3.1 and 3.2 doapply. We denote by �x , �x and qx , x 2 A, the resulting conformal and invariantmeasures and the invariant density, respectively, for T . We now explain how thecorresponding objects can be recovered for the original map T . Notice that this ispossible since we only induced in the base system.

First, we consider the case of the conformal measures. Let �x , x 2 A be themeasure 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 thereexists k 2 N, such that �k.x/ 2 A and �j .x/ … A for j D 0; : : : ; k � 1. Then weput

�x D .L kx /��k.x/

L kx .1/

: (7.2)

Therefore, the family f�xgx2X 0 is a family of probability measures well definedfor X in a subset X 0 of X with full measure. This family of measures has theconformality property

L �x ��k.x/ D x�x ;

where x D ��.x/.Lx1/, x 2 X 0. Notice also that EP.'/ D EP.'/.

7.4 An Example 73

Similarly, from the family f�xgx2A of T -invariant measures one can recover afamily f�xgx2X 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 T �jx :

Then, with q�j .x/ D L jx .qx/, 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 forthe original map. Moreover, since EP.'t / D EP.'t/ their zeros coincide andconsequently Bowen’s Formula and the Multifractal Analysis are also true forconformal 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�

12x C 15

2x2 if x 2 Œ0; 1=3�

16x � 15 if x 2 Œ15=16; 1�

andf1.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 Bernoullimeasure. For x D .: : : ; x�1; x0; x1; : : :/ 2 X define

fx D fx0, f n

x D f�n�1.x/ ı f�n�2.x/ ı : : : ı fx

and

Jx D1\

nD0

.f nx /�1.Œ0; 1�/:

For this map, �0 D 1=2 and �1 D 16 are the best expanding constants that onecan take. With these constants we have

Zlog �xdm.x/ > 0:

Therefore, the map is expanding in the mean but not expanding.Note that the size of each component of f �n

x .Œ0; 1�/ is bounded by

an D 16�n1.1=2/�n0; (7.3)


where ni WD #fj D 0; : : : ; n � 1 W xj D ig, i D 0; 1. Since

limn!1

n0

nD lim

n!1n1

nD 1

2

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,

EP.t/ � limn!1

1

nlog 2n16�n1t .1=2/�n0t

� log 2 � t�

limn!1

n1

nlog 16 � n0

nlog 2

� D log 2

�

1 � 3

2t

�

:

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 thefiber Jx contains open intervals.

Chapter 8Classical Expanding Random Systems

Having treated a very general situation up to here, we now focus on moreconcrete random repellers and, in the next section, random maps that have beenconsidered by Denker and Gordin. The Cantor example of Chap. 5.3 and randomperturbations of hyperbolic rational functions like the examples considered byBruck and Buger are typical random maps that we consider now. We classifythem into quasi-deterministic and essential systems and analyze then their fractalgeometric properties. Here as a consequence of the techniques we have developed,we positively answer the question of Bruck and Buger (see [9] and Question 5.4in [8]) of whether the Hausdorff dimension of almost all (most) naturally definedrandom Julia sets is strictly larger than 1. We also show that in this same setting theHausdorff 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 whereVT , 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 identicallydistributed (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/. Thisgives rise to a random repeller T n

x0D Txn�1

ı :::ıTx0, n � 1. The natural associated

Julia set is


75

76 8 Classical Expanding Random Systems

Jx D\

n�1

T �nx0.U / where x D .x0; x1; :::/ :

Notice that compactness of Y together with the expanding assumption, we recallthat � -expanding means that the distance of all points z1; z2 with �.z1; z2/ � �T isexpanded by the factor � , implies that Jx is compact and also that the maps T 2 Rare of bounded degree. A random repeller is therefore the most classical form of auniformly expanding random system.

The link with the setting of the preceding sections goes via natural extension.Set X D IZ, take the Bernoulli measure m D mZ

0 and let the ergodic invariantmap � be the shift map � W I Z ! IZ. If � W X ! I is the projection on the 0thcoordinate and if x 7! Tx is a map from I to R then the repeller (8.1) is given bythe skew-product

T .x; z/ D ��.x/; T�.x/.z/

�; .x; z/ 2 J D

[

x2X

fxg � Jx : (8.2)

The particularity of such a map is that the mappings Tx do only depend on the0th 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! T �1

x .z/ is continuous from J to K .U /, the space of all non-emptycompact 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 apotential 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 containedin 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 naturaltopologies, 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 0j we get one of the most natural example of classical expanding randomsystem.

Proposition 8.2 The pressure function x 7! Px.'/ of a classical expandingrandom system is continuous.

Proof. We have to show that x 7! �x is continuous and since L nx 1.y/=L n�1

x11.y/

converges uniformly to �x for every y 2 U (see Lemma 3.32) it suffices to showthat x 7! L n

x 1.y/ does depend continuously on x 2 X . In order to do so, we firstshow that condition (T1) implies continuity of the function .x; y/ 7! #T �1

x .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 ; forall disjoint w1;w2 2 T �1

x .y/. From (T1) follows that there exists ı > 0 such that

dH .T�1x .y/; T �1

x0

.y 0// �

2; whenever %

�.x; y/; .x0; y0/

� � ı :

But this implies that for every w 2 T �1x .y/ there exists at least one preimage w0 2

T �1x0

.y0/ \ B.w; 0/. Consequently, #T �1x0

.y0/ � #T �1x .y/. Equality follows since

Tx0 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 T �1x .y/, there exists a continuous function

x0 7! zw.x0/ defined onW such that Tx0.zw.x

0// D y, zw.x/ D w and

T �1x0

.y/ D fzw.x0/ W w 2 T �1

x .y/g:

The proposition follows now from the continuity of 'x , i.e. from (T2). utWe say that a function g W IZ ! R is past independent if g.!/ D g./ for any

!; 2 IZ with !j10 D j10 . Fix � 2 .0; 1/ and for every function g W IZ ! R set

v�.g/ D supn�0

fv�;n.g/g;

wherev�;n.g/ D ��n supfjg.!/ � g./j W !jn0 D jn0g:

Denote by H� the space of all bounded Borel measurable functions g W IZ ! R forwhich v�.g/ < C1. Note that all functions in H� are past independent. Let Z� bethe set of negative integers. If I is a metrizable space and d is a bounded metric onI , then the formula

dC.!; / D1X

nD0

2�nd.!n; n/

defines a pseudo-metric on IZ, and for every 2 IZ, the pseudo-metric dCrestricted to fg � N, becomes a metric which induces the product (Tychonoff)topology on fg � N.

Theorem 8.3 Suppose that T W J ! J and � W J ! R form a classicalexpanding random system. Let � W IZ ! .0;C1/ be the corresponding functioncoming 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 IZ with !jn0 D jn0 . ByLemma 3.32, we have

ˇˇˇˇˇL nC1

! 1.y/

L n�.!/

1.y/� �!

ˇˇˇˇˇ

� A�n and

ˇˇˇˇˇL nC1

� 1.y/

L n�.�/

1.y/� ��

ˇˇˇˇˇ

� A�n


with some constantsA>0 and � 2 .0; 1/. Since, by our assumptions, L nC1! 1.y/ D

L nC1� 1.y/ and L n

�.!/1.y/ D L n

�.�/1.y/, we conclude that j�! �� j � 2A�n. So,

v�.�/ � 2A:

Since, by Proposition 8.2, the function � W IZ ! .0;C1/ is continuous, it istherefore 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 respectto the pseudo-metric dC. utCorollary 8.4 Suppose that T W J ! J and � W IZ ! R form a classicalexpanding random system. Then the number (asymptotic variance of P.�/)

�2.P.�// D limn!1

1

n

Z �Sn.P.�// � nEP.�/

�2

dm � 0

exists, and the Law of Iterated Logarithm holds, i.e. m-a.e we have

�p2�2.P.�// D lim inf

n!1

P nx � nEP.�/pn log log n

� lim supn!1

P nx .�/ � nEP.�/pn log log n

Dp2�2.P.�//:

Proof. Let � W IZ ! I be the canonical projection onto the 0th coordinate andlet G D ��1.B/, where B is the �-algebra of Borel sets of I . We want toapply 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!) identicallyequal to zero since jm.A \ B/ � m.A/m.B/j D 0 for every A 2 G m

0 WDG \ ��1.G / \ : : : ��m.G / and B 2 G 1

n D TC1j Dn �

�j .G /, whenever n > m.

The integralR jP.�/j2Cıdm is finite (for every ı > 0) since, by Theorem 8.3,

the pressure function P.�/ is bounded. This then implies that for all n � 1,jP.�/.!/� E .P.�/jG n

0 /.!/j � v�.P.�//�n, where v�.P.�// < C1. Therefore,

ZjP.�/ � E .P.�/jG n

0 /jdm � v�.P.�//�n;

whence condition (1.11.7) from [24] holds. Finally, P.�/ is G 10 -measurable,

since P.�/ belonging to H� is past independent. We have thus checked all theassumptions of Theorem 1.11.1 from [24] and, its application yields the existenceof the asymptotic variance of P.�/ and the required Law of Iterated Logarithm tohold. utProposition 8.5 Let g 2 H� . Then �2.g/ D 0 if and only if there exists u 2C..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 thecohomology equation implies vanishing of �2 is obvious. In order to prove the otherimplication, assume without loss of generality that m.g/ D 0. Because of Theorem2.51 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 u � u ı � (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 SZ. In view of Lusin’s Theorem there exists a compact setK � SZ such that m.K/ > 1=2 and the function ujK is continuous. So, in view ofBirkhoff’s Ergodic Theorem there exists a Borel set B � SZ such that m.B/ D 1,for every ! 2 B , ��n.!/ 2 K with asymptotic frequency > 1=2, u is well definedonSC1

nD�1 ��n.B/, and (8.3) holds onSC1

nD�1 ��n.B/. Let Z� D f�1;�2; : : :gand let fm�g�2IZ

�

be the canonical system of conditional measures for the partitionffg�INg�2IZ

�

with respect to the measurem. Clearly, each measurem� , projectedto IN, coincides with mC. Since m.B/ D 1, there exists a Borel set F � SZ

� suchthat m�.F / D 1 and m� .B \ .fg � IN// D 1 for all 2 F , where m� is theinfinite product measure on SZ

� . Fix 2 F and set Z D pN.B \ .fg � IN//,where pN W IZ ! IN is the natural projection from IZ to IN. The property thatm� .B \ .fg � IN// D 1 implies that Z D SN. Now, it immediately follows fromthe definitions of Z and B that for all x; y 2 Z there exists an increasing sequence.nk/

1kD1

of positive integers such that ��nk .x/; ��nk .y/ 2 K for all k � 1. Forevery 0 < q � nk we have from (8.3) that

nk�qX

j D0

�g.�j .��nk .y/// � g.�j .��nk .x///

�

CnkX

j Dnk�qC1

�g.�j .��nk .y/// � g.�j .��nk .x///

�

D .u.��nk .y// � u.��nk .x//C .u.x/ � u.y//:

Since g 2 H� , we have

nk�qX

j D0

�g.�j .��nk .y/// � g.�j .��nk .y///

�

�nk�qX

j D0

jg.�j .��nk .y/// � g.�j .��nk .y///j

�nk�qX

j D0

v�.g/�nk�j � v�.g/.1 � �/�1�q:

Now, fix " > 0. Take q � 1 so large that v�.g/.1 � �/�1�q < "=2: Since thefunction g W IZ ! R is uniformly continuous with respect to the pseudo-metric d ,there exists ı > 0 such that jg.b/�g.a/j < "

2qwhenever d.a; b/ < ı. Assume that

d.x; y/ < ı (so d.��i .x/; ��i .y// < ı for all i � 0). It follows now that forevery k � 1 we have


ju.x/� u.y/j � v�.g/.1 � �/�1�q C q"

2qC ju.��nk .y// � u.��nk .x//j

� "

2C "

2C ju.��nk .y// � u.��nk .x//j

D "C "

2C ju.��nk .y// � u.��nk .x//j:

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

limk!1

ju.��nk .y// � u.��nk .x//j D 0 :

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

W D[

�2F

B \ .fg � IN/:

Since W D SZ (asm.W / D 1) and since u is independent of the past, we concludethat u extends continuously to SZ. 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. ut

8.2 Classical Conformal Expanding Random Systems

If a classical system is conformal in the sense of Definition 5.1 and if the potential isof the form ' D �t log jf 0j for some t 2 R then we will call it classical conformalexpanding random system

Theorem 8.6 Suppose f W J ! J is a classical conformal expanding randomsystem. 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 Ph.Jx/ D C1 for m-a.e. x 2 IZ.(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 quasi-

deterministic, and then for every x 2 supp.m/, we have:

(c1) hx is a geometric measure with exponent h.

(c2) The measures hx , H hjJx

, and PhjJxare all mutually equivalent

with Radon–Nikodym derivatives separated away from zero and infinityindependently of x 2 IZ and y 2 Jx .

(c3) 0 < H h.Jx/;Ph.Jx/ < C1 and HD.Jx/ D h.

8.3 Complex Dynamics and Bruck and Buger 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, weconclude that the system f W J ! J is essential. Hence, item (b) follows fromTheorem 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), isquasi-deterministic because of Proposition 8.5, Theorem 8.3 (P.h/ 2 H�), andRemark 5.6. Items (c1)–(c4) follow now from Theorem 5.7(b1)–(b4). We are done.

utAs a consequence of this theorem we get the following.

Theorem 8.7 Suppose f W J ! J is a classical conformal expanding randomsystem. Then the following hold:

(a) Suppose that for every x 2 IZ, the fiber Jx is connected. If there exists atleast one w 2 supp.m/ such that HD.Jw/ > 1, then HD.Jx/ > 1 for m-a.e.x 2 IZ.

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

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 WJ ! J is essential. Thus the proof is concluded in exactly the same way as theproof 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), theclaim is proved in exactly the same way as the proof of Theorem 5.8(4). If, on theother 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. ut

8.3 Complex Dynamics and Bruck and Buger PolynomialSystems

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-systemsdefined later in this section. Indeed, having a sequence of rational functions F Dffng1

nD0 on the Riemann sphere OC we say that a point z 2 OC is a member of theFatou set of this sequence if and only if there exists an open set Uz containing zsuch that the family of maps ffnjUzg1

nD0 is normal in the sense of Montel. The Juliaset J .F / is defined to be the complement (in OC) of the Fatou set of F . For everyk � 0 put Fk D ffkCng1

nD0 and observe that

J .FkC1/ D fk.J .Fk//: (8.4)


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!ng1

nD0/ � fz 2 C W jzj � "g and

jf 0!k.z/j � d"d�1 (8.5)

for all z 2 J .ff!kCng1

nD0/. Let ı.d/ D supnı" W " > d�1

p1=d

o: Fix 0< ı < ı.d/.

Then there exists " > d�1p1=d such that ı < ı". Therefore, by (8.5),

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!kCng1

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 . Letm be the corresponding invariant probability measure. Let also H W X ! Fd;ı bean arbitrary measurable function. Set fd;x D H.x/ for all x 2 X . For every x 2 Xlet Jx be the Julia set of the sequence ff�n.x/g1

nD0, and then J D Sx2X Jx .

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 measurablyexpanding system in the sense of Definition 5.1. As an immediate consequence ofTheorem 5.2 we get the following.

Theorem 8.8 Let � W X ! X be an invertible measurable map preserving aprobability measurem. Fix an integer d � 1 and 0 < ı < ı.d/. LetH W X ! Fd;ı

be an arbitrary measurable function. Finally, let fd;ı;�;H be the distance expandingrandom system defined by formula (8.7). Then for almost all x 2 X the Hausdorffdimension of the Julia set Jx is equal to the unique zero of the expected value ofthe pressure function.

Theorem 8.9 For the conformal measurably expanding systems fd;ı;�;H defined inTheorem 8.8 the multifractal theorem, Theorem 6.4 holds.

8.3 Complex Dynamics and Bruck and Buger Polynomial Systems 83

We now define and deal with Bruck and Buger polynomial systems. We still keepd � 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 probabilitymeasure m0 on B.0; ı/ which is different from ı0, the Dirac ı measure supportedat 0. Define H W X ! Fd;ı by the formula H.!/ D fd;!0

. The correspondingskew-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/1nD0/:

Then f W J ! J is called Bruck and Buger polynomial systems. Clearly, f WJ ! J is a classical conformal expanding random system.

In [8] Bruck speculated on p. 365 that if ı < 1=4 and m0 is the normalizedLebesgue measure on B.0; ı/, then HD.J!/ > 1 for mC-a.e. ! 2 B.0; ı/N withrespect to the skew-product map

.!; z/ 7! .�.!/; z2 C !0/:

In [9] this problem was explicitly formulated by Bruck and Buger as Question 5.4.Below (Theorem 8.10) we prove a more general result (with regard the measure onB.0; ı/ and the integer d � 2 being arbitrary), which contains the positive answerto the Bruck and Buger question as a special case. In [8] Bruck also proved that ifı < 1=4 and the above skew product is considered then �2.J!/ D 0 for all ! 2B.0; ı/N, where �2 denotes the planar Lebesgue measure on C. As a special caseof Theorem 8.10 below we get a partial strengthening of Bruck’s result saying thatHD.J!/ < 2 formC-a.e. ! 2 B.0; ı/N. Our results are formulated for the productmeasure m on B.0; ı/Z, but as mC is the projection from B.0; ı/Z to B.0; ı/N andas the Julia sets J! , ! 2 B.0; ı/Z depend only on !jC1

0 , i.e. on the future of !,the analogous results for mC and B.0; ı/N follow immediately. Proving what wehave just announced, note that if !0 2 supp.m0/ n f0g, then

HD.J!1

0// D HD.J .f!0

// 2 .1; 2/

(the equality holds already on the level of sets: J!1

0D J .f!0

/), and by [9],

all the sets J! , ! 2 B.0; ı/Z, are Jordan curves. Hence, since f W J ! Jis a classical conformal expanding random system, as an immediate application ofTheorem 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 WJ ! J is given by the formula

fd;ı.!; z/ D .�.!/; fd;!0.z// D .�.!/; zd C !0/;


and m0 is an arbitrary Borel probability measure on B.0; ı/, different from ı0, theDirac ı measure supported at 0, then for m-almost every ! 2 B.0; ı/Z we have1 < HD.J!/ < 2.

8.4 Denker–Gordin Systems

We now want to discuss another class of expanding random maps. This is the settingfrom [12]. In order to describe this setting suppose that X0 and Z0 are compactmetric spaces and that �0 W X0 ! X0 and T0 W Z0 ! Z0 are open topologicallyexact distance expanding maps in the sense as in [24]. We assume that T0 is a skew-product overZ0, i.e. for every x 2 X0 there exists a compact metric space Jx suchthat Z0 D S

X2X0fxg � Jx and the following diagram commutes:

Z0T0 � 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; x0/ (8.8)

for all x 2 X and that there exists 1 > 0 such that, for all x; x0 satisfyingdX0

.x; x0/ < 1 there exist y; y0 such that

d�.x; y/; .x0; y0/

�< : (8.9)

We then refer to T0 W Z0 ! Z0 and �0 W X0 ! X0 as a DG-system. Note that

T0.fxg � Jx/ � f�0.x/g � J�0.x/

and this gives rise to the map Tx W Jx ! J�0.x/.Since T0 is distance expanding, conditions uniform openness, measurably

expanding measurability of the degree, topological exactness (see Chap. 2) holdwith some constants �x � � > 1, deg.Tx/ � N1 < C1 and the numbernr 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 definitionof expanding maps in [24]. Now assume that � W Z ! R is a Holder 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-systemand that � W Z ! R is a Holder continuous potential. Then there exists a Holdercontinuous function P.�/ W X0 ! R, a measurable collection f xgx2X0

and acontinuous function q W Z0 ! Œ0;C1/ such that

(a) �0.x/.A/ D exp�Px.�/

� RA e

��xd x for all x 2 X0 and all Borel setsA � Jx

such that Tx jA is one-to-one.(b)

RJx

qxd x D 1 for all x 2 X0.(c) Denoting for every x 2 X0 by �x the measure qx x we have

X

w2��10

.x/

�w.T�1

w .A// D �x.A/ for every Borel set A � Jx :

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 whenX0 is finite, is not invertible. But to remedy this situation is easy. We consider theprojective limit (Rokhlin’s natural extension) � W X ! X of �0 W X0 ! X0.Precisely,

X D f.xn/n�0 W �0.xn/ D xnC1 8n � �1gand

��.xn/n�0

� D .�0.xn//n�0:

Then � W X ! X becomes invertible and the diagram

X� � X

X0

p

� �0 � X0

p

�

(8.10)

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 Dm0. Let

Z WD[

x2X

fxg � Jx0:

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 �. Clearlythe quadruple .T; �;m; �/ is a Holder fiber system as defined in Chap. 2 of our paper.It follows from Theorem 8.11 along with the definition of � a commutativity of thediagram (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


Theorems 3.1 and 3.2; in particular they are unique. From now on we assume that themeasure m is a Gibbs state of a Holder continuous potential on X (having nothingto 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 Holder continuity theorem appeared in the paper [12]. We providehere an alternative proof under weaker assumptions.

Theorem 8.12 If dX .x; x0/ < , then j�x � �x0 j � Hd˛

X .x; x0/.

Proof. Let n be such that

dX .�2n�1.x/; �2n�1.x0// < 1 and dX .�

2n.x/; �2n.x0// � 1: (8.11)

Let z 2 T �2nC1.y/ and z0 2 T �2nC1.y0/. 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��˛.n�k�1/:

Then

jSn'.z/� Sn'.z0/j � C��˛n

1� ��˛:

Put C 0 WD C=.1 � ��˛/. Then

ˇˇˇ log

L nx 1.w/

L nx0

1.w0/

ˇˇˇ � C 0��˛n and

ˇˇˇ log

L n�1�.x/

1.w/

L n�1�.x0/

1.w0/

ˇˇˇ � C 0��˛n:

Then ˇˇˇ log

L nx 1.w/

L n�1�.x/

1.w/� log

L nx0

1.w0/L n�1

�.x0/1.w0/

ˇˇˇ � 2C 0��˛n: (8.12)

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

��n˛ D L�2n˛0 � .d.�2n.x/; �2n.x0///˛0

˛0

1 L�2n˛0

� .d.x; x0//˛0

˛0

1

:

Then (8.12) finishes the proof. utSince the map �0 W X0 ! X0 is expanding, since m is a Gibbs state, and since

P.�/ W X0 ! R is Holder continuous, it is well known (see [24] for example) thatthe following asymptotic variance exists:

�2.P.�// D limn!1

1

n

Z �Sn.P.�//� nEP.�/

�2

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 areequivalent.

(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 continuousfunction 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

Holder continuous functions onX (resp.X0), meaning that there exists a Holdercontinuous 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 P n

x .�/ D nR for all n � 1 and all periodic pointsx 2 X (resp. X0).

As a matter of fact such theorem is formulated in [24] for non-invertible (�0) mapsonly but it also holds for the Rokhlin’s natural extension � . The following theoremfollows directly from [24] and Theorem 8.11 (Holder continuity of P.�/).

Theorem 8.14 (The Law of Iterated Logarithm) If .T; �;m; �/ is a DG*-system andif �2.P.�// > 0, then m-a.e. we have

�q2�2.P.�// D lim inf

n!1P n

x .�/ � nEP.�/pn log log n

� lim supn!1

P nx .�/ � nEP.�/pn log log n

Dq2�2.P.�//:

8.5 Conformal DG*-Systems

Now we turn to geometry. This section dealing with, below defined, conformalDG*-systems is a continuation of the previous one in the setting of conformalsystems. We shall show that these systems naturally split into essential and quasi-deterministic, and will establish their fractal and geometric properties. Suppose that.f0; �0/ is a DG-system endowed with a Gibbs measurem0 at the base. Suppose alsothat this system is a random conformal expanding repeller in the sense of Chap. 5and that the function � W Z ! R given by the formula

�.x; y/ D � log jf 0x.y/j;

is Holder continuous.


Definition 8.15 The corresponding system .f; �;m/ D .f; �;m; �/ (with � theRokhlin 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 Holdercontinuous. As in Chap. 5 denote its topological pressure by P.t/. Recall that h isa 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 Ph.Jx/ D C1:

(b) If, on the other hand, �2.P.h// D 0, then .f; �;m/ D .f; �;m; �/ is quasi-deterministic, and then for every x 2 X , we have that h

x is a geometric measurewith exponent h and, consequently, the geometric properties (GM1)–(GM3) hold.

Exactly as Corollary 5.8 is a consequence of Theorem 5.7, the followingcorollary 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 thefollowing hold.

1. The fiber Jx is not bi-Lipschitz equivalent to any deterministic nor quasi-deterministic 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), thenh 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. Thenthe following hold.

(a) Suppose that for every x 2 X , the fiber Jx is connected. If there exists at leastone w 2 supp.m/ such that HD.Jw/ > 1, then

HD.Jx/ > 1 for m-a.e. x 2 IZ :

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

8.7 Topological Exactness 89

We end this subsection and the entire section with a concrete example of aconformal DG*-system. In particular, the three above results apply to it. Let

X WD S1ıd

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 ı1�kxk :

Then � 00.x/ D kı1�kxk�1 and therefore j� 0

0.x/j D k � 2 for all x 2 X . Thenormalized Lebesgue measure �0 on X is invariant under �0. Define the map H WX ! Fd by setting H.x/ D fx . Then

f�0;H;0.x; y/ D .kı1�kxk�1; gd C x/:

Note that�f�0;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 onsmooth manifolds. Let .M; �/ be a smooth compact Riemannian manifold. We recallthat a differentiable endomorphism f W M ! M is expanding if there exists � > 1such that

jjf 0x.v/jj � � jjvjj for all x 2 M and all v 2 TxM :

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[

�>1

E� .M/;

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 thatfn 2 E.M/, n � 1, are endomorphisms such that

limn!1nY

j D1

�.fj / D C1 :


Denote Fk D fk ı fk�1 ı : : : ı f1, k � 1. Then, for every r > 0 there exist k � 1

such thatFk.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 suchthat Fk.U / D M .

Proof. Let f 2 E.M/, set � D �.f / and notice first of all that for such a map theimplicit function theorem applies and yields that f is an open map. The manifoldMbeing connected, it follows that f is surjective. Moreover, if ˇ is any path startingat a point y D f .x/, then there is a lift ˛ starting at x. The expanding propertyimplies that

length.ˇ/ D length.f ı ˛/ � � length.˛/ :

In particular, if ˇ is a geodesic between y D f .x/ and a point y0 2 M , then thereis a point x0 2 M such that f .x0/ D y0 and

�.y; y0/ � � �.x; x0/ :

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 . ut

8.8 Stationary Measures

Let M be an n-dimensional compact Riemannian manifold and let I be a setequipped with a probabilistic measure m0. To every a 2 I we associate adifferentiable expanding transformation fa of M into itself. Put X D IZ andlet m be the product measure induced by m0. For x D : : : a�1a0a1 : : : consider'x WD � log j detf 0

a0j. We assume that all our assumptions are satisfied. Then the

measure D volM (where volM is the normalized Riemannian volume on M ) isthe fixed point of the operator L �

x;' with �x D 1. Let qx be the function given byTheorem 3.1, and let �x be the measure determined by d�x=d x D qx .

We write IZ D I�N � IN where points from I�N are denoted byx� D : : : a�2a�1 and from IN by xC D a0a1 : : :. Then x�xC meansx D : : : a�1a0a1 : : :. Note that qx does not depend on xC, since nor doesL n

x�n

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 d�x�dm�.x�/ where m� is the productmeasure on I�N. Then by (8.13)

Z��.g ı fa/dm0.a/ D

Z�x�.g ı fa/dm

�.x�/

DZ Z

�x�a.g/dm�.x�/dm0.a/ D ��.g/:

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

Chapter 9Real Analyticity of Pressure

Here we provide, in particular, the real analyticity results that where used in theproof of the real analyticity of the multifractal spectrum (Chap. 6.3). We putted thispart 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 appropriatelychosen 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. 3from which we deduce that the theorems from that section can be proved to hold forevery parameter and almost every x (common for all parameters).

In this section we only assume that T W J ! J is a measurable expandingrandom map. Let '.1/; '.2/ 2 Hm.J / and let t D .t1; t2/ 2 R

2. Put

jt j WD maxfjt1j; jt2jg and t� WD maxf1; jt jg: (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 H ˛

m .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 jt1j�˛x.y2; y1/CHx jt2j�˛

x.y2; y1/� 2jt jHx�˛x.y2; y1/:

Therefore 't 2 H ˛m .J ; 2jt jH/ � H ˛

m .J ; 2t�H/. Also, for all x 2 X and ally 2 Jx , we have


93

94 9 Real Analyticity of Pressure

jSn't;x.y/j � jt1jjSn'.1/x .y/j C jt2jjSn'

.2/x .y/j � jt jjSn'x.y/j � jt jjjSn'x jj1:

This impliesjjSn't;xjj1 � jt jjjSn'xjj1 � t�jjSn'xjj1: (9.3)

Concerning the potential ', we get

j'x.y2/ � 'x.y1/j � ˇˇj'.1/x .y2/ � '.1/x .y1/

ˇˇC ˇ

ˇ'.2/x .y2/ � '.2/x .y1/ˇˇ � 2Hx�

˛x.y2; y1/:

Thus' 2 H ˛

m .J ; 2H/: (9.4)

Denote by Ct , Ct;max, Ct;min, D�;t and ˇt .s/, the respective functions associated tothe 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 immediatelyget

D�;t .x/ � Dt�

�;' ; (9.5)

Ct .x/ � exp�Qx.2t

�H/�

max0�k�j

˚exp

�2t�jjSk'x

�kjj1

��

��

exp�Qx.2H/

�max

0�k�j

˚exp

�2jjSk'x

�kjj1

��t�

D C t�

' ;

(9.6)

Ct;min.x/ � exp��Qx.2t

�H/�

exp��2t�jjSn'xjj1

� D Cmin.x/t�

; (9.7)

Ct;max.x/ D exp�Qx.2t

�H/�

deg.T nx / exp

�2t�jjSn'xjj1

� � Cmax.x/t�

; (9.8)

and therefore,

ˇt;x.s/ ��Cmin.x/

C'.x/

�t� .s � 1/2t�Hx

�1��˛

x�1

4t�sQx

D�Cmin.x/

C'.x/

�t� .s � 1/Hx

�1��˛

x�1

2sQx

��Cmin.x/

C'.x/

�t�

�.s � 1/Hx

�1��˛

x�1

2sQx

�t�

D ˇt�

x .s/:

Finally we are going to look at the function A.x/ and the constant B obtained inProposition 3.17. We fix the set

G WD fx W ˇx � M and j.x/ � J gas defined by (3.35). Note that by (9.1), for x 2 G we have, ˇx;t � M t

� .Denote by G 0� the corresponding visiting set for backward iterates of � , and by.nk/

11 the corresponding visiting sequence. In particular limk!1 k

nk� 3

4J: Putting

Bt D 4J 2p1 �M t

� and

At .x/ WD maxf2C t�

max.x/B�J k�

xt ; C t

�

' .x/C C t�

max.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 ourestimates above, we obtain the following.

Proposition 9.1 For every t 2 R2, for every x 2 G0�, and every gx 2 �s

t;x

k QL nx

�n;tgx�n

� qt;xk1 � At .x/Bnt :

More generally, if gx 2 H ˛.Jx/, then

ˇˇˇˇˇˇ OL n

t;xgx �� Z

gxd�t;x

�1

ˇˇˇˇˇˇ1

� C t�

' .�n.x//

� Zjgx j d�t;x C 4

v˛.gxqt;x/

t�Qx

�At

�

.�n.x//Bnt�

:

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 jt j � R, the functions At and Bt in Proposition 9.1can 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 thesetXA defined by (3.38) withA.x/ generated by '. So, if x 2 XA, thenAt .x/ � At

for some finite number At which depends on t . Denote by X 0A;C the corresponding

visiting set intersected with G0C. Therefore, the following is a consequence of theproof of Proposition 3.19 and the formula (3.43).

Proposition 9.3 For every R > 0, every x 2 X 0A;C, and every gx 2 C .Jx/ we

have that

limn!1 sup

jt j�R

ˇˇˇˇˇˇ OL n

t;xgx �� Z

gxd�t;x

�1�n.x/

ˇˇˇˇˇˇ1

D 0:

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 functionX 3 x 7! D1.x/ with the following property. Let x 2 X 0, let w 2 Jx , and letn � 0. Put y D .x;w/. Then

.D1.�n.x///�t

� � �t;x.T�ny .B.T n.y/; �///

exp.Sn't.y/ � SnPx.'t //� .D1.�

n.x///t�

for all t 2 R2.

For all t 2 R2 set

EP.t/ WD EP.'t/:


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 thatm.XE

0/ D 1 and for all x 2 XE0 and all

t 2 R2, the limit

limn!1

1

nlog L n

t;x1.wn/ (9.9)

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 X 0t with m.X 0

t/ D 1 and such that

limn!1

1

nlog L n

t;x1.wn/ D limn!1

1

nlogn

t;x D EP.t/ (9.10)

for all x 2 X 0t . Fix 2 Œ0; 1/ and let t D .t1; t2/ and t 0 D .t 01; t 02/ 2 R

2. Holder’sinequality implies that all the functions R

2 3 t 7! 1n

log L nt;x1.wn/, n � 1, are

convex. It thus follows from (9.10), that the function R2 3 t 7! EP.t/ is convex,

whence continuous. LetXE

0 D\

t2Q2

X 0t :

Since the set Q2 is countable, we have that m.XE0/ D 1. Along with (9.10), and

density of Q2 in R2, the convexity of the functions R

2 3 t 7! 1n

log L nt;x1.wn/

implies that for all x 2 XE0 and all t 2 R

2, the limit limn!1 1n

log L nt;x1.wn/

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 thatfor all x 2 XE

0 and all t 2 R2, we have

limn!1

1

nlog L n

t;x1.wn/ D EP.t/:

We are done. utLemma 9.6 Fix t2 2 R and assume that there exist measurable functions L W X 3x 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

limt1!C1 EP.t1; t2/ D �1 and lim

t1!�1 EP.t1; t2/ D C1 m-a:e: (9.12)

9.2 Real Cones 97

Proof. Fix x 2 XE0. Let t1 < t 01. Then by (9.11)

X

z2T �nx .wn/

exp�Sn'.t1;t2/.z/

�

DX

z2T �nx .wn/

exp�t1Sn'1.z/

�exp

�t2Sn'2.z/

�

DX

z2T �nx .wn/

exp�t 01Sn'1.z/

�exp

�t2Sn'2.z/

�exp

�.t1 � t 01/Sn'1.z/

�

�X

z2T �nx .wn/

exp�t 01Sn'2.z/

�exp

�t2Sn'2.z/

�exp

�.t1 � t 01/.Lx � ncx/

�

DX

z2T �nx .wn/

exp�Sn'.t 0

1;t2/.z/�

exp�.t 01 � t1/.ncx � Lx/

�

Therefore,

1

nlog L n

t;x1.wn/ � 1

nlog L n

.t 0

1;t2/;x1.wn/C .t 01 � t1/.cx � Lx=n/ :

Hence, letting n ! 1, we get from Lemma 9.5 that EP.t1; t2/ � EP.t 01; t2/ C.t 01 � 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 thatlimt1!�1 EP.t1; t2/ D C1: ut

9.2 Real Cones

We adapt the approach of Rugh [26] based on complex cones and establish realanalyticity of the pressure function. Via Legendre transformation, this completesthe 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 ˚ iHR;x itscomplexification.

C sx WD C s

R;x WD fg 2 Hx W g.w1/ � esQx%˛.w1;w2/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 C C

x we denote the subset of all non-zero functions from C sx . For l 2 .Hx/

�,the dual space of Hx , we define

K.C sx ; l/ WD sup

g2C C

x

jjl jj˛jjgjj˛jhl; gij :


Then the aperture of C sx is

K.C sx / WD inffK.C s

x ; l/ W l 2 .Hx/�; l ¤ 0g:

Lemma 9.7 K.C sx / < 1. This property of a cone is called an outer regularity.

Proof. Let wk 2 Jx , k D 0; : : : ; N be such thatSLx

kD1 B.wk ; �/ D Jx . Define

l0.g/ WDLxX

kD1

g.wk/: (9.14)

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 kl0k˛ D Lx , since l0.g/ � Lx jjgjj1 � Lxjjgjj˛ and l0.1/ D Lx DLxjj1jj˛. Hence

kl0k˛kgk˛

hl0; gi � K 0x WD Lx

�sQx.exp.sQx�

˛//C 1�

exp.sQx�˛/: (9.15)

utLet

s0x WD sQx

�1��˛

x�1

CHx�1��˛

x�1

Qx

:

By (3.33) for s > 1, s0x < s. Moreover, like in (3.32) we have the following.

Lemma 9.8 Let g 2 C sx and let w1;w2 2 J�.x/ with %.w1;w2/ � �. Then, for

y 2 T �1x .w1/

e'.y/

e'.T �1y .w2//

g.y/

g.T �1y .w2//

� expns0

�.x/Q�.x/%˛.w1;w2/

o: (9.16)

ConsequentlyLxg.w1/

Lxg.w2/� exp

ns0

�.x/Q�.x/%˛.w1;w2/

o:

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

L ix

�ig.w/

L ix

�ig.z/

� CR.x/ for every i � j.x/ and g 2 C sx :

9.2 Real Cones 99

Proof. First, let i D j.x/. Let a 2 T �ix

�i.z/ be such that

eSi '.a/g.a/ D supy2T �i

x�i

.z/

eSi '.y/g.y/:

By definition of j.x/, for any point w 2 Jx there exists b 2 T �ix

�i.w/ \ B.a; �/.

Therefore

L ix

�ig.w/ � eSi 'x

�i.b/g.z/ � exp.Si'x

�i.b/� Si'x

�i.a//eSi 'x

�i.a/e�sQxg.a/

� exp.�2kSj.x/'x�j.x/

k1 � sQx/

deg.T jx

�j/

L ix

�ig.z/ � .CR.x//

�1L ix

�ig.z/;

where

CR.x/ WD0

@exp

�� sQx � 2kSj.x/'x

�j.x/k1�

deg.T j.x/x

�j/

1

A

�1

� 1: (9.17)

The case i > j.x/ follows from the previous one, since L i�j.x/x

�igx

�i2 C s

x�j.x/

.ut

Let s > 1 and s0 < s. Define

x WD x;s;s0 WD supr2.0;��

1 � exp�� .s C s0/Qxr

˛�

1� exp� � .s � s0/Qxr˛

� � s C s0

s � s0 : (9.18)

Lemma 9.10 For gx ; fx 2 C s0

x ,

x

supy2Jxjgx.y/j

infy2Jxjfx.y/j fx � gx 2 C s

R;x :

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

xkgx=fxk1�

exp�sQx%

˛x.z;w/

�fx.z/� fx.w/

�

� xkgx=fxk1�

exp�sQx%

˛x.z;w/

� � exp�s0Qx%

˛x.z;w/

��fx.z/

��

exp�sQx%

˛x.z;w/

� � exp� � s0Qx%

˛x.z;w/

��gx.z/

� exp�sQx%

˛x.z;w/

�gx.z/� gx.w/:

Then exp�sQx%

˛x.z;w/

��xkg=f k1fx.z/� gx.z/

�� xkg=f k1fx.w/� gx.w/.

ut


We say that gx 2 C sx is balanced if

fx.y1/

fx.y2/� CR.x/ for all y1; y2 2 Jx : (9.19)

Let gx; fx 2 C sx . Put ˇx;s.fx ; gx/ WD inff > 0 W fx � gx 2 C s

x g and define theHilbert projective distance Pdist W C s

x � C sx ! R by the formula:

Pdistx.fx ; gx/ WD Pdistx;s.fx; gx/ WD log.ˇx;s.fx; gx/ � ˇx;s.gx ; fx//:

Let�x WD diamC s

x;R.L j

x�j.C s

x�j ;R//;

where diamC sx;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.

Lemma 9.11 If gx ; fx 2 C s0

x are balanced, then

Pdistx.fx ; gx/ � 2 log�s C s0

s � s0 � CR.x/�

and, consequently,

�x � 2 log� s C s0

s � s0 � CR.x/�:

9.3 Canonical Complexification

Following the ideas of Rugh [26] we now extend real cones to complex ones. DefineC �

x WD fl 2 .Hx/� W l jCx

� 0g and

C sC;x WD fg 2 HC;x W 8l1;l22C �

xRehl1; gihl2; gi � 0g:

Denote also by C CC;x the set of all g 2 C s

C;x such that g 6� 0. There are otherequivalent definitions of C s

C;x . The first one is called polarization identity by Rughin [26, Proposition 5.2].

Proposition 9.12 (Polarization identity)

C sC;x D fa.f � C ig�/ W f � ˙ g� 2 C C

R;x and a 2 Cg:

In our case we can also define C sC;x as follows. Let %.w;w0/ < �. Define

lw;w0.g/ WD g.w/ � e�sQx%˛.w;w0/g.w0/

and

9.3 Canonical Complexification 101

Fx WD flw;w0 W %.w;w0/ < �g � C �x :

ThenC s

x D fg 2 Hx W 8l2Fxl.g/ � 0g:

Later in this section we use the following two facts about geometry of complexnumbers. 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 c1p2 and

Z 2 freiu W 1 � 1C s20 , juj � 2p1 C 2s0g;

then there exist ˛; ˇ; � > 0 such that ReZ � ˛, ReZ � ˇ, ImZ � � and �c2 < ˛.

Lemma 9.14 Let z1; z2 2 C be such that Re z1 > Re z2 and define u 2 C though

ei Im z1u � ez1 � ez2

eRe z1 � eRe z2:

Then

j Arg uj � j Im.z1 � z2/jRe.z1 � z2/

and 1 � ju2j � 1C� Im.z1 � z2/

Re.z1 � z2/

�2

:

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

Lx;'gx.w/ DX

Tx.z/Dw

e'x .z/gx.z/; w 2 J�.x/:

Lemma 9.15 Let w;w0; z; z0 2 Jx such that %.w;w0/ < � and %.z; z0/ < �. Then,for all g1; g2 2 C s

x;R,

lw;w0.Lx;'g1/lz;z0.Lx;'g2/

lw;w0.Lx;Re 'g1/lz;z0.Lx;Re 'g2/D Z;

where

Z 2 Ax WD freiu W 1 � r � 1C s20 ; juj � 2jj Im'jj1 C 2s0g (9.20)

and

s0 WD v˛.Im'/��˛x

.s � s0�.x/

/Q�.x/

: (9.21)


Proof. For y 2 T �1x .w/, by y0 we denote T �1

y .w0/. Then for g 2 Cx

lw;w0.Lx;'g/ WD Lx;'g.w/ � e�sQx%˛.w;w0/Lx;'g.w0/

DX

y2T �1x .w/

e'.y/g.y/ � e�sQx%˛.w;w0/e'.y0/g.y0/ DX

y2T �1x .w/

ny.'; g/;

whereny.'; g/ WD e'.y/g.y/ � e�sQx%˛.w;w0/e'.y0/g.y0/:

Define implicitly uy so that ny.Re '; g/ei Im '.y/uy D ny.'; g/: Put z1 WD '.y/Clogg.y/ and z2 WD �sQx%

˛.w;w0/C '.y0/C logg.y0/. Then

ei Im z1uy D ez1 � ez2

eRe z1 � eRe z2:

By (9.16)

Re '.y/ � logg.y/ � .Re '.y0/C logg.y0// � �s0�.x/Q�.x/%

˛.w1;w2/:

HenceRe.z1 � z2/ � .s � s0

�.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'.y0/. Therefore, by Lemma 9.14

j Arg uy j � s0 WD v˛.Im'/��˛x

.s � s0�.x/

/Q�.x/

and 1 � juyj2 � 1C s20 :

Since

lw;w0.Lx;'g/ DX

y2T �1x .w/

ny.'; g/ DX

y2T �1x .w/

ei Im '.y/uyny.Re '; g/;

lw;w0.Lx;'g/

lw;w0.Lx;Re 'g/D Z;

where

Z 2 Ax WD freiu W 1 � r � 1C s20 ; juj � 2jj Im'jj1 C 2s0g:

Similarly

9.4 The Pressure is Real-Analytic 103

lw;w0.Lx;'g1/lz;z0.Lx;'g2/

lw;w0.Lx;Re 'g1/lz;z0.Lx;Re 'g2/D Z

for possibly another Z 2 Ax . utLet p1; p2 be the real numbers given by Lemma 9.13 with

c1 D ��˛x

.s � s0x/Qx

and c2 D cosh�x

2:

Having Lemmas 9.15, 9.13 and 9.11 the following proposition is a consequenceof the proof of Theorem 6.3 in [26].

Proposition 9.16 Let j D j.x/. If

k ImSj'x�j

k1 � p1 and v˛.ImSj'x�j/ � p2; (9.22)

thenL j

x�j.C s

C;x�j/ � C s

C;x :

Let l0 (the functional defined by (9.14)). Then by Lemma 5.3 in [26] we get

K WD K.C sC;x; l0/ WD sup

g2C C

C;x

jjl0jj˛jjgjj˛jhl0; gij � Kx WD 2

p2K 0

x;

whereK 0x is defined by (9.15). By l we denote the functional which is a normalized

version of .1=Lx/l0. So jjl jj˛ D 1. Then, for every g 2 C sC;x ,

1 � jjgjj˛hl; gi � Kx : (9.23)

9.4 The Pressure is Real-Analytic

We are now in position to prove the main result of this chapter. Here, we assumethat 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 onD.t0; R/.(b) For z 2 R

n \D.t0; R/, 'z;x 2 HR;x .


(c) For all z 2 D.t0; R/ and all x 2 X , there exists H such that k'z;xk˛ � 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;xk˛ � ":

Then the functionD.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 weget 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 ,

Lz;x�1.C s

C;x�1/ � C s

C;x :

Then by (9.23),jjL n

z;x�n.1/jj˛

lx.L nz;x

�n.1//

� K:

Therefore, by Montel Theorem, the familyL n

z;x�n

.1/.w/

lx.L nz;x

�n .1//is normal. Since, for all

z 2 Rn \D.t0; r/ and all x 2 X we have that

L nz;x

�n.1/.w/

lx.L nz;x

�n.1//

��!n!1

qz;x.w/

lx.qz;x/;

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

L nz;x

�n.1/.w/

lx.L nz;x

�n.1//

��!n!1 gz;x.w/: (9.24)

Since, in addition,

Lx

�L nz;x

�n.1/.w/

lx.L nz;x

�n.1//

�D L nC1

z;x�n.1/.w/

lx1.L nC1

z;x�n.1//

� lx1

�Lz;x

�L nz;x

�n.1/.w/

lx.L nz;x

�n.1//

��;

we therefore get that

Lx

�L nz;x

�n.1/.w/

lx.L nz;x

�n.1//

��!n!1 lx1

.Lx.gz;x//gx1;z:

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 qz;x

lx.qz;x/D Lxqz;xPN

kD0 qz;x.wk/;

9.4 The Pressure is Real-Analytic 105

we conclude that,

lx1.Lz;xgz;x/ D lx1

.Lz;xqz;x

lx.qz;x// D z;x

lx1.qx1;z/

lx.qz;x/: (9.25)

By the very definitions

lx1.Lz;xgz;x/ D .1=Lx/

LxX

kD1

Lz;xgz;x.wk/

andLz;xgz;x.w/ D

X

y2T �1x .w/

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

Denote gz;x.w/ by F.z/ and 'z;x.w/ by G.z/. Then, for z D .z1; : : : ; zn/ 2D.t0; r=2/, and � .u/ D z C ..r=2/e2�iu1; : : : ; .r=2/e2�iun/, where u D.u1; : : : ; un/ 2 Œ0; 2 �n, by the Cauchy Integral Formula,

ˇˇˇ@F

@zk

.z/ˇˇˇ D

ˇˇˇ

1

.2 i/2

Z

�

F.�/

.�1 � z1/ : : : .�k � zk/2 : : : .�2 � z2/d�ˇˇˇ � 2K=r

for k D 1; : : : ; n. Similarly we obtain that

ˇˇˇ@G

@zk

.z/ˇˇˇ � 2H=r

for k D 1; : : : ; n. Then, for k D 1; : : : ; n,

ˇˇˇ@e'z;x.y/gz;x.y/

@zk

ˇˇˇ D

ˇˇˇ@'z;t.w/

@zk

e'z;x.y/gz;x.y/C e'z;x.y/ @gz;x.y/

@zk

ˇˇˇ

� .2H=r/eHK C eH .2K=r/:

It follows that there exists Cg such that for all x 2 X ,

ˇˇˇ@lx1

.Lz;xgz;x/

@zk

ˇˇˇ � Cg : (9.26)

Using (3.19) we obtain that

C�1' � qt0;x.y/ � C' ;

and thenC�1

' � lx.qt0;x.y// � C'


for all x 2 X . Moreover, it follows from Lemma 3.6 that t0;x � exp.�k't0;xk1/:Then

z0 WD lx1.Lt0;xgt0;x/ D t;x

lx1.qt;x1

/

lx.qt;x/� exp.� sup

x2X

k'xk1/C�2' > 0:

Hence, by (9.26), there exists r1 > 0 so small that

lx1.Lz;xgz;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;xgz;x/ 2 C:

Now consider the holomorphic function

z 7!Z

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

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

Zlog lx1

.Lz;xgz;x/dm.x/ DZ

logz;xlx1.qz;x1

/

lx.qz;x/dm.x/

DZ

logz;xdm CZlx1.qz;x1

/dm �Zlx.qz;x/dm.x/

DZ

logz;xdm D EP.'t /

for z 2 D.t0; r1/ \ Rn. Therefore the function D.t0; r1/ \ R

n 3 z 7! EP.'z/ isreal-analytic. ut

9.5 Derivative of the Pressure

Now, let T W J ! J be uniformly expanding random map. Throughout thesection, we assume that ' 2 Hm.J / is a potential such that there exist measurablefunctions 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 (9.27)

for every z 2 Jx and n and 2 Hm.J /. For t 2 R, define

't WD t' C :

9.5 Derivative of the Pressure 107

Let R > 0 and let jt0j � R=2. Since we are in the uniform case, it follows fromRemark 9.2 that there exist constants AR and BR such that, for t 2 Œ�R;R�,

ˇˇˇˇˇˇ

QL nt;xgx

q�n.x/

�� Z

gxd�t;x

�ˇˇˇˇˇˇ1 �

�kgxk1 C 2

v.gx/

Q

�ARB

nR: (9.28)

Proposition 9.18

dEP.t/

dtDZ'xd�

txdm.x/ D

Z'd�t :

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

EP.t; n/ WDZ1

nlog L n

t;x1x.y.xn//dm.x/:

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

dL nt;x1x.yn/

dtD

X

y2T �nx .yn/

eSn.'tx/.y/Sn'x.y/

Dn�1X

j D0

X

y2T �nx .yn/

eSn.'tx/.y/'xj

.T jx y/ D

n�1X

j D0

L nt;x.'xj

ı T jx /.yn/:

Since Sn.'tx/.y/ D Sj .'

tx/.y/C Sn�j .'

txj/.T

jx y/ we have that

L nt;x.'xj

ı T jx /.y.xn// D L n�j

t;xj.'xj

L jt;x1x/.y.xn//:

Then by a version of Leibniz integral rule (see for example [23], Proposition 7.8.4,p. 40)

dEP.t; n/

dtDZ 1

n

Pn�1j D0 L n�j

xj ;t .'xjL j

t;x1x/.y.xn//

L nt;x1x.yn/

dm.x/:

SinceL n�j

t;xj.'xj

L jt;x1/.yn/ D n

xQL n�jt;xj

�'t;xj

QL jt;x1x

�.yn/

andL n

t;x1x.yn/ D nx

QL nt;x1x.yn/;

we have that


L nt;x.'xj

ı T jx /.yn/

L nt;x1x.yn/

DQL n�jt;xj

�'xj

QL jt;x1x

�.yn/

QL nt;x1x.yn/

: (9.29)

The function 'xjQL jt;x1x is uniformly bounded. So does its Holder variation.

Therefore it follows from (9.28), that there exists a constant AR and BR such that

�� QL n�j

t;xj

�'xj

QL jt;x1x

�.yn/=qxn

�� Z

'xjQL jt;x1xd�

txj

��1 � ARB

n�jR

and �� QL n

t;x.1x/.yn/=qxn� 1xn

��1 � ARB

nR;

From this by (9.29) it follows that

R'xj QL

jt;x1xd�

txj

� ARBn�jR

1C ARBnR

� L nt;x .'xj ı T jx /.yn/

L nx 1Yx .yn/

�R'xj QL

jt;x1xd�

txj

C ARBn�jR

1� ARBnR

:

Since m is �-invariant, we have that

Z Z'xj

QLt;xyj1xd�

txj

dm.x/ DZ Z

'xQL jx

�j ;t1x�jd�t

xdm.x/:

Hence, for large n,

R R'x

�1n

Pn�1j D0

QL jx

�j ;t1x�j

�d�t

xdm.x/ � 1n

Pn�1j D0.ARB

n�jR /

1C ARBnR

� dEP.'t ; n/

dt

�R R

'x

�1n

Pn�1j D0

QL jx

�j ;t1x�j

�d�t

xdm.x/ � 1n

Pn�1j D0.ARB

n�jR /

1 �ARBnR

:

Therefore

limn!1

dEP.t; n/

dtDZ'xd�

txdm.x/

uniformly for t 2 Œ�R;R�. ut

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: avariational 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. Bahnmuller, J., Bogenschutz, T.: A Margulis–Ruelle inequality for random dynami-cal systems. Arch. Math. (Basel) 64(3), 246–253 (1995). doi: 10.1007/BF01188575.http://dx.doi.org/10.1007/BF01188575

4. Bogenschutz, T.: Entropy, pressure, and a variational principle for random dynamical systems.Random Comput. Dyn. 1(1), 99–116 (1992/93)

5. Bogenschutz, T., Gundlach, V.M.: Ruelle’s transfer operator for random subshifts of finite type.Ergod. Theor. Dyn. Syst. 15(3), 413–447 (1995)

6. Bogenschutz, T., Ochs, G.: The Hausdorff dimension of conformal repellers under randomperturbation. Nonlinearity 12(5), 1323–1338 (1999)

7. Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. LectureNotes in Mathematics, vol. 470. Springer, Berlin (1975)

8. Bruck, R.: Geometric properties of Julia sets of the composition of polynomials of the formz2 C cn. Pac. J. Math. 198(2), 347–372 (2001)

9. Bruck, R., Buger, 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 dynamicalsystems. 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., Urbanski, M.: On the existence of conformal measures. Trans. Am. Math. Soc.328(2), 563–587 (1991)

14. Deschamps, V.M.: Equilibrium states for non-Holderian random dynamical systems. RandomComput. 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 Vari-

ables. Wolters-Noordhoff Publishing, Groningen (1971). With a supplementary chapter byI. A. Ibragimov and V. V. Petrov, Translation from the Russian edited by J. F. C. Kingman

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

109

110 References

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 inMathematics, 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 Helene Airault, Leslie Kay and Gerard Letac,Edited and translated from the French by Kay, With a foreword by Mark Pinsky

24. Przytycki, F., Urbanski, M.: Conformal Fractals: Ergodic Theory Methods. London Mathema-tical 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 ofclassical equilibrium statistical mechanics, With a foreword by Giovanni Gallavotti and Gian-Carlo 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 BrazilianMathematics Colloquium, IMPA, Rio de Janeiro (1997)

29. Walters, P.: Invariant measures and equilibrium states for some mappings which expanddistances. Trans. Am. Math. Soc. 236, 121–153 (1978)

Index

aperture, 98

backward visiting sequence, 11balanced, 100base map, 8Bowen’s Formula, 48Bruck and Buger polynomial systems, 83

classical conformal expanding randomsystems, 80

classical expanding random system, 76concave Legendre transform, 58conformal

DG*-systems, 87expanding random map, 47uniformly expanding map, 47

DG*-system, 86DG-system, 84

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

Gibbs family, 18Gibbs property, 35

Holder continuous with an exponent ˛, 13, 71

Hilbert projective distance, 100

induced map, 70induced potential, 72integrability of the logarithm of the transfer

operator, 41invariant density, 23

measurabilityof the degree, 9of cardinality of covers, 18of the transfer operator, 41

measurable expanding random map, 40measurably expanding, 8

outer regularity, 98

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

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

T-invarianceof 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 Index

of a family of measures, 17of a measure, 40

temperature function, 58topological exactness, 9transfer

dual operators, 19operator, 13

uniform openness, 8uniformly expanding random map, 9

visiting sequence, 11visiting way, 12

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