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

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

Bearbeitet von

Volker Mayer, Mariusz Urbanski, Bartlomiej Skorulski

1. Auflage 2011. Taschenbuch. x, 112 S. Paperback ISBN 978 3 642 23649 5

Format (B x L): 15,5 x 23,5 cm Gewicht: 199 g

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Chapter 2

Expanding Random Maps

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

2.1 Introductory Examples

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

The first one is a known random version of the Sierpi´nski gasket (see, for example [15]). Let D .A; B; C /be a triangle with vertexesA; B; C and choose a 2 .A; B/,b2.B; C /andc2.C; A/. Then we can associate toxD.a; b; c/a map

f_x W.A; a; c/[.a; B; b/[.b; C; a/!;

such that the restriction off_x to each one of the three subtriangles is a affine map onto. The mapf_x is nothing else than the generator of a deterministic Sierpi´nski gasket. Note that this map can be made continuous by identifying the vertices A; B; C(Fig.2.1).

Now, supposex₁ D .a₁; b₁; c₁/; x₂ D .a₂; b₂; c₂/; ::: are chosen randomly which, for example, may mean that they form sequences of three dimensional independent and identically distributed (i.i.d.) random variables. Then they generate compact sets

Jx₁;x₂;x₃;:::D \

n1

.f_xnı:::ıf_x1/¹./

called random Sierpi´nski gaskets having the invariance propertyf_x¹₁ .J_x2;x3;:::/D J_x1;x2;x3;:::. For a little bit simpler example of random Cantor sets we refer the reader to Sect.5.3. In that example we provide a more detailed analysis of such random sets.

V. Mayer et al., Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry, Lecture Notes in Mathematics 2036,

DOI 10.1007/978-3-642-23650-1 2, © Springer-Verlag Berlin Heidelberg 2011

5

Fig. 2.1 Two different generators of Sierpi´nski gaskets

Fig. 2.2 A generator of degree6

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

Another point is that the mapsf_x that generate the random Sierpi´nski gasket have degree3. In the sequel of this manuscript, we will allow the degreed_x of all maps to be different (see Fig.2.2) and only require that the functionx7!log.d_x/is measurable.

Finally, the above examples are all expanding with an expanding constant _x > 1 :

As already explained in the introduction, the present monograph concerns random maps for which the expanding constants _x can be arbitrarily close to one.

Furthermore, using an inducing procedure, we will even weaken this to the maps that are only expanding in the mean (see Chap.7).

2.1 Introductory Examples 7 The example of random Sierpi´nski gasket is not conformal. Random iterations of rational functions or of holomorphic repellers are typical examples of conformal random dynamical systems. Random iterations of the quadratic familyf_c.z/ D z² C chave been considered, for example, by Br¨uck and B¨uger among others (see [8] and [9]). In this case, one chooses randomly a sequence of bounded parameters cD.c₁; c₂; :::/and considers the dynamics of the family

F_c1;:::;cnDf_cnıf_cn1ı:::ıf_c1; n1:

This leads to the dynamical invariant sets

K_c D fz2CI F_c1;:::;cn.z/6! 1g and J_c D@K_c:

The setK_cis the filled in Julia set andJ_cthe Julia set associated to the sequencec.

The simplest case is certainly the one when we consider just two polynomials z7!z²C₁and z7!z²C₂and we build a random sequence out of them. Julia sets that come out of such a choice are presented in Fig.2.3. Such random Julia sets are different objects as compared to the Julia sets for deterministic iteration of quadratic polynomials. But not only the pictures are different and intriguing, we

Fig. 2.3 Some quadratic random Julia sets

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

2.2 Preliminaries

Suppose.X;B; m; / is a measure preserving dynamical system with invertible and ergodic map W X ! X which is referred to as the base map. Assume further that.Jx; _x/,x 2 X, are compact metric spaces normalized in size by diam_x.Jx/1. Let

J D [

x2X

fxg J_x: (2.1)

We will denote byB_x.z; r/the ball in the space.Jx; %_x/centered at z 2Jx and with radiusr. Frequently, for ease of notation, we will writeB.y; r/forB_x.z; r/, whereyD.x;z/. Let

T_x WJx !J.x/; x 2X;

be continuous mappings and letT W J ! J be the associated skew-product defined by

T .x;z/D..x/; T_x.z//: (2.2) For everyn0we denoteT_xⁿWD Tn1.x/ı:::ıT_x WJx !Jⁿ_.x/. With this notation one hasTⁿ.x; y/D.ⁿ.x/; T_xⁿ.y//. We will frequently use the notation

x_nDⁿ.x/; n2Z:

If it does not lead to misunderstanding we will identifyJxandfxg Jx.

2.3 Expanding Random Maps

A mapT W J ! J is called a expanding random map if the mappings T_x W Jx ! J_.x/ are continuous, open, and surjective, and if there exist a function W X !RC,x 7!_x, and a real number > 0such that following conditions hold.

Uniform Openness.T_x.B_x.z; _x//B_.x/

T_x.z/;

for every.x;z/2J. Measurably Expanding. There exists a measurable function W X ! .1;C1/, x7!_x such that, form-a.e.x2X,

%_.x/.T_x.z₁/; T_x.z₂//_x%_x.z₁;z₂/ whenever %.z₁;z₂/ < _x;z₁;z₂2Jx :

2.4 Uniformly Expanding Random Maps 9 Measurability of the Degree. The mapx 7!deg.T_x/WDsup_y2J.x/#T_x¹.fyg/is measurable.

Topological Exactness. There exists a measurable functionx7!n.x/such that T_x^n.x/.B_x.z; //DJ_n.x/.x/ for every z2J_x and a.e. x2X : (2.3) Note that the measurably expanding condition implies thatT_xj_B.z;x/is injective for every.x;z/ 2 J. Together with the compactness of the spacesJx it yields the numbers deg.T_x/ to be finite. Therefore the supremum in the condition of measurability of the degree is in fact a maximum.

In this work we consider two other classes of random maps. The first one consists of the uniform expanding maps defined below. These are expanding random maps with uniform control of measurable “constants”. The other class we consider is composed of maps that are only expanding in the mean. These maps are defined like the expanding random maps above excepted that the uniform openness and the measurable expanding conditions are replaced by the following weaker conditions (see Chap.7for detailed definition).

1. All local inverse branches do exist.

2. The functionin the measurable expanding condition is allowed to have values in.0;1/but subjects only the condition

Z

Xlog_x d m > 0:

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

2.4 Uniformly Expanding Random Maps

Most of this paper and, in particular, the whole thermodynamical formalism is devoted to measurable expanding systems. The study of fractal and geometric properties (which starts with Chap.5), somewhat against our general philosophy, but with agreement with the existing tradition (see for example [5,12,17]), we will work mostly with uniform and conformal systems (the later are introduced in Chap.5).

A expanding random mapT WJ !J is called uniformly expanding if – WDinf_x2Xx> 1,

– deg.T /WDsup_x2Xdeg.T_x/ <1, – nWDsup_x2Xn.x/ <1.

2.5 Remarks on Expanding Random Mappings

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

T_y¹WB_.x/.T .y/; /!B_x.y; _x/

ofT_xsendingT_x.z/to z. By the measurably expanding property we have

%.T_y¹.z₁/; T_y¹.z₂//_x¹%.z₁;z₂/ for z₁;z₂2B_.x/

T .y/;

(2.4) and

T_y¹.B_.x/.T .y/; //B_x.y; _x¹/B_x.y; /:

Hence, for everyn0, the composition

T_yⁿDT_y¹ıT_{T .y/}¹ ı: : :ıT_T¹_n1.y/WBn.x/.Tⁿ.y/; /!B_x.y; / (2.5) is well defined and has the following properties:

T_yⁿWBn.x/.Tⁿ.y/; /!B_x.y; /

is continuous,

TⁿıT_yⁿDIdj_Bn.x/.Tn.y/;/, T_yⁿ.T_xⁿ.z//Dz and, for every z₁;z₂ 2Bn.x/

Tⁿ.y/;

,

%.T_yⁿ.z₁/; T_yⁿ.z₂//._xⁿ/¹%.z₁;z₂/; (2.6) where_xⁿD_x.x/ n1.x/:Moreover,

T_xⁿ.Bn.x/.Tⁿ.y/; //B_x.y; ._xⁿ/¹/B_x.y; /: (2.7) Lemma 2.1 For everyr > 0, there exists a measurable functionx 7! n_r.x/such that a.e.

T_x^nr.x/.B_x.z; r//DJnr .x/.x/ for every z2J_x: (2.8) Moreover, there exists a measurable functionj WX !Nsuch that a.e. we have

T_x^j.x/_j.x/.B_xj.x/.z; //DJ_x for every z2J_xj.x/: (2.9) Proof. In order to prove the first statement, consider₀ > 1and let F be the set ofx 2 X such that _{x 0}. If₀ 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 setX_CF⁰ (see (2.10)). Then forx 2 X_CF⁰ , the limit lim_n!1._xⁿ/¹D0. Define

X_CF;kWD fx2X_CF⁰ W._x^k/¹ < rg:

ThenX_CF;k X_CF;kC1andS

k2NX_CF;k DX_CF⁰ . By measurability ofx7!_x, X_CF;kis a measurable set. Hence the function

X_CF⁰ 3x7!n_r.x/WDminfkWx2X_CF;kg Cn.x/

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

T_x^nr.x/.B_x.z; r//DJnr .x/.x/:

In order to prove the existence of a measurable functionj W X ! N define measurable sets

X_;nWD fx2X Wn.x/ng,X_;n⁰ WDⁿ.X_;n/andX⁰ D [

n2N

X_;n⁰ :

Then the map

X⁰ 3x7!j.x/WDminfn2NWx2X_;n⁰ g

satisfies (2.9) forx2X⁰. Sincem.ⁿ.X_;n//Dm.X_;n/%1asntends to1we

havem.X⁰/D1. ut

2.6 Visiting Sequences

LetF 2F be a set with a positive measure. Define the sets

V_CF.x/WD fn2NWⁿ.x/2Fg and V_F.x/WD fn2NWⁿ.x/2Fg:

The setV_CF.x/is called visiting sequence (ofF atx). Then the setV_F.x/is just a visiting sequence for¹and we also call it backward visiting sequence. Byn_j.x/

we denote thejth-visit inF atx. Sincem.F / > 0, by Birkhoff’s Ergodic Theorem we have that

m.X_CF⁰ /Dm.X_F⁰ /D1;

where

X_CF⁰ WDn

x2X WV_CF.x/is infinite and lim

j!1

n_jC1.x/

n_j.x/ D1 o

(2.10)

andX_F⁰ is defined analogously. The setsX_CF⁰ andX_F⁰ are respectively called forward and backward visiting forF.

Let .x; n/be a formula which depends on x 2 X andn 2 N. We say that .x; n/holds in a visiting way, if there existsFwithm.F / > 0such that, form-a.e.

x 2 X_CF⁰ and allj 2 N, the formula .^nj.x/; n_j.x//holds, where.n_j.x//¹_jD0 is the visiting sequence ofF atx. We also say that .x; n/holds in a exhaustively visiting way, if there exists a familyF_k 2 F with lim_k!1m.F_k/D 1such that, for allk,m-a.e.x 2 X_CF⁰

k, and allj 2 N, the formula .^nj.x/; n_j.x//holds, where.n_j.x//¹_jD0is the visiting sequence ofF_katx.

Now, letf_nWX!Rbe a sequence of measurable functions. We write that s-lim

n!1f_nDf;

if that there exists a familyF_k 2 F with lim_k!1m.F_n/D 1such that, for allk andm-a.e.x2X_CF⁰

kand allj 2N,

jlim!1f_nj.x/Df .x/;

where.n_j/¹_jD0is the visiting sequence ofF_katx.

Suppose thatg₁; : : : ; g_k WX !Ris a finite collection of measurable functions and letb₁; : : : ; b_nbe a collection of real numbers. Consider the set

F WD\^k

iD1

g_i¹..1; b_i/:

Ifm.F / > 0, thenF is called essential forg₁; : : : ; g_kwith constantsb₁; : : : ; b_n(or just essential, if we do not want explicitly specify functions and numbers). Note that by measurability of the functionsg₁; : : : ; g_k, for every" > 0we can always find finite numbersb₁; : : : ; b_nsuch that the essential setFforg₁; : : : ; g_kwith constants b₁; : : : ; b_nhas the measurem.F /1".

2.7 Spaces of Continuous and H¨older Functions

We denote byC.Jx/the space of continuous functionsg_x W Jx ! Rand by C.J/the space of functionsg WJ !Rsuch that, for a.e.x 2 X,x 7!g_x WD g_jJx 2 C.Jx/. The set C.J/contains the subspacesC⁰.J/of functions for which the functionx7! kg_xk₁is measurable, andC¹.J/for which the integral

kgk₁ WD Z

X

kg_xk₁d m.x/ <1:

2.8 Transfer Operator 13 Now, fix˛ 2 .0; 1. By H^˛.Jx/we denote the space of H¨older continuous functions onJx with an exponent˛. This means that'_x 2H^˛.Jx/if and only if'_x 2C.Jx/and v.'_x/ <1where

v_˛.'_x/WDinffH_x W j'.z₁/'.z₂/j H_x%^˛_x.z₁;z₂/g; (2.11) where the infimum is taken over all z₁;z₂2Jx with%.z₁;z₂/.

A function' 2C¹.J/is called H¨older continuous with an exponent˛provided that there exists a measurable functionH W X ! Œ1;C1/,x 7! H_x, such that logH 2L¹.m/and such that v_˛.'_x/ H_x for a.e.x 2 X. We denote the space of all H¨older functions with fixed˛andH byH^˛.J; H /and the space of all

˛-H¨older functions byH^˛.J/DS

H1H^˛.J; H /.

2.8 Transfer Operator

For every functiongWJ !Cand a.e.x2Xlet S_ng_x D

n1X

jD0

g_xıT_x^j; (2.12)

and, ifg WX !C, thenS_ng D P_n1

jD0gı^j. Let' be a function in the H¨older spaceH^˛.J/. For everyx2X, we consider the transfer operatorLx DL';xW C.Jx/!C.J_.x//given by the formula

L_xg_x.w/D X

Tx.z/Dw

g_x.z/e^'x.z/; w2J_.x/: (2.13)

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

kLxk₁deg.T_x/exp.k'k₁/: (2.14) This family of operators gives rise to the global operatorL W C.J/ ! C.J / defined as follows:

.Lg/_x.w/DL1.x/g1.x/.w/:

For everyn > 1and a.e.x2X, we denote

L_xⁿWDLn1.x/ı:::ıLx WC.Jx/!C.Jⁿ_.x//:

Note that

L_xⁿg_x.w/D X

z2Txⁿ.w/

g_x.z/e^Sn'x.z/, w2Jⁿ.x/; (2.15) whereS_n'_x.z/has been defined in (2.12). The dual operatorL_xmapsC.J_.x// intoC.Jx/.

2.9 Distortion Properties

Lemma 2.2 Let'2H^˛.J; H /, letn1and lety D.x;z/2J. Then jS_n'_x.T_yⁿ.w₁//S_n'_x.T_yⁿ.w₂//j %^˛.w₁;w₂/

n1X

jD0

Hj.x/.^nj

^j.x//^˛

for all w₁;w₂2B.T_xⁿ.z/; /.

Proof. We have by (2.6) and H¨older continuity of'that jSn'x.T_yⁿ.w1//Sn'x.T_yⁿ.w2//j

n1X

jD0

j'x.T_x^j.T_yⁿ.w1///'x.T_x^j.T_yⁿ.w2///j

D

n1

X

jD0

ˇˇˇ'x.T^{.nj /}

Tx^j.y/ .w1//'x.T^{.nj /}

Tx^j.y/ .w2//ˇˇˇ

n1

X

jD0

%^˛ T^{.nj /}

Tx^j.x/ .w₁/; T^{.nj /}

Tx^j.x/ .w₂/ Hj.x/;

hencejS_n'_x.T_yⁿ.w₁//S_n'_x.T_yⁿ.w₂//j %^˛.w₁;w₂/P_n1

jD0Hj.x/.^nj_j.x//^˛. u t Set

Q_x WDQ_x.H /D X¹

jD1

Hj.x/.^j_j.x//^˛: (2.16) Lemma 2.3 The functionx 7!Q_x is measurable andm-a.e. finite. Moreover, for every'2H^˛.J; H /,

jS_n'_x.T_yⁿ.w₁//S_n'_x.T_yⁿ.w₂//j Qn.x/%^˛.w₁;w₂/

for alln1, a.e.x2X, every z2Jxand w₁;w₂2B.Tⁿ.z/; /and where again yD.x;z/.

Proof. The measurability ofQ_xfollows directly from (2.16). Because of Lemma2.2 we are only left to show thatQ_x ism-a.e. finite. Letbe a positive real number less or equal toR

log_xd m.x/. Then, using Birkhoff’s Ergodic Theorem for¹, we get that

lim inf

j!1

1 j

j1

X

kD0

logj.x/

form-a.e.x2X. Therefore, there exists a measurable functionC WX !Œ1;C1/

m-a.e. finite such thatC¹.x/e^{j=2 j}

^jC1.x/for allj 0 and a.e.x 2 X.

2.9 Distortion Properties 15 Moreover, since logH 2L¹.m/it follows again from Birkhoff’s Ergodic Theorem that

jlim!1

1

j logHj.x/D0 m-a:e:

There thus exists a measurable functionC_H WX !Œ1;C1/such that

Hj.x/C_H.x/e^j˛=4 and Hj.x/C_H.x/e^j˛=4 (2.17) for allj 0and a.e.x2X. Then, for a.e.x2X, alln0and allaj n1, we have

Hj.x/DH.nj /.ⁿ.x//C_H.ⁿ.x//e^{.nj /˛=4}: Therefore, still withx_nDⁿ.x/,

Q_xn D

n1X

jD0

H_xj._x^nj

j /^˛

n1X

jD0

C_H.x_n/e^{.nj /˛=4}C^˛.x_n1/e^{˛.nj /=2}

C^˛.x_n1/C_H.x_n/

n1X

jD0

e^{˛.nj /=4}C^˛.x_n1/C_H.x_n/.1e^˛=4/¹:

Hence

Q_xC^˛.¹.x//C_H.x/.1e^˛=4/¹<C1:

u t