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Limit Theorems for Ergodic Group Actions and

Random Walks

MICHAEL BJÖRKLUND

Doctoral Thesis

Stockholm, Sweden 2009

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TRITA-MAT-09-MA-06 ISSN 1401-2278

ISRN KTH/MAT/DA 09/04-SE ISBN 978-91-7415-282-1

Department of Mathematics Royal Institute of Technology SE-100 44 Stockholm, SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen i matema-tik tisdagen den 26 maj 2009 klockan 13.00 i Sal F3, Kungl Tekniska högskolan, Lindstedtsvägen 26, Stockholm.

© Michael Björklund, Maj 2009 Tryck: Universitetsservice US AB

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iii

Abstract

This thesis consists of an introduction, a summary and 7 papers. The papers are devoted to problems in ergodic theory, equidistribution on compact manifolds and random walks on groups.

In Papers A and B, we generalize two classical ergodic theorems for actions of abelian groups. The main result is a generalization of Kingman’s subadditive ergodic theorem to ergodic actions of the group Zd.

In Papers C,D and E, we consider equidistribution problems on nilmanifolds. In Paper C we study the asymptotic behavior of dilations of probability measures on nilmanifolds, supported on singular sets, and prove, under some technical assumptions, effective con-vergences to Haar measure. In Paper D, we give a new geometric proof of an old result by Koksma on almost sure equidistribution of expansive sequences. In paper E we give nec-essary and sufficient conditions on a probability measure on a homogeneous Riemannian manifold to be non–atomic.

Papers F and G are concerned with the asymptotic behavior of random walks on groups. In Paper F we consider homogeneous random walks on Gromov hyperbolic groups and establish a central limit theorem for random walks satisfying some technical moment conditions. Paper G is devoted to certain Bernoulli convolutions and the regularity of their value distributions.

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iv

Sammanfattning

Denna avhandling består av en inledning, en sammanfattning och 7 artiklar. Alla artiklar behandlar problem inom ergodteori, likformig fördelning på kompakta mångfalder och slumpvandringar på grupper.

I Artiklarna A och B generaliserar vi två klassiska ergodsatser för verkningar av abelska grupper. Huvudresultatet är en utvidgning av Kingmans subadditiva ergodsats till ergodiska verkningar av gruppen Zd.

I Artiklarna C, D och E behandlar vi problem om likformig fördelning på nilmångfal-der. I Artikel C studerar vi det asymptotiska beteendet av dilatationer av sannolikhetsmått på nilmångfalder, med stöd på singulära mängder, och vi etablerar, under vissa tekniska antaganden, effektiv konvergens mot Haarmåttet. I Artikel D ger vi ett nytt geometriskt bevis av ett gammalt resultat av Koksma om generisk likformig fördelning av expansiva sviter. I Paper E ger vi nödvändiga och tillräckliga villkor för existens av atomer för ett sannolikhetsmått på kompakta och homogena Riemannmångfalder.

Artiklar F och G behandlar det asymptotiska beteendet av slumpvandringar på grup-per. Artikel F rör fallet med Gromovhyperboliska grupper och vi etablerar en central gränsvärdessats för under vissa tekniska momentvillkor. Artikel G behandlar den generis-ka regulariteten av värdesfördelningarna av Bernoullifaltningar.

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Acknowledgements

First I want to express my deepest gratitude to my supervisor, Anders Karlsson. He introduced me to the wonderful interplay between metric geometry and probability theory; an interplay which I hope this thesis will reflect. I am also very grateful to my second advisor, Michael Benedicks, who has been extremely important to my mathematical development. I would also like to extend my gratitude to three very important teachers: Jan-Erik Björk, Per Sjölin and Lars Svensson.

I have enjoyed the hospitality of many universities, and I would especially like to thank the mathematics departments at Ohio State University, Yale University, Hebrew University and ETH in Zürich for their generosity. I have benefited enor-mously from discussions with Vitaly Bergelson, Francois Ledrappier and Manfred Einsiedler.

I had the great pleasure to collaborate with Alexander Engström, Alexander Fish, Tobias Hartnick, Richard Miles and Daniel Schnellmann, and I hope that I will have more opportunities to do so in the future as well. I am also grateful to my family and all my fellow graduate students at KTH for their support over the years.

During periods of inefficiency, I have found it helpful to spend some time on other activities. I would like to thank Oskar Wändell for his company and friendship during many hours of bird-watching.

Finally, I wish to thank Kathrin for being her.

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Contents

Acknowledgements v

Contents vi

I

Introduction and Summary

1

1 Introduction 3

1.1 Ergodic Theory . . . 3

1.2 Equidistribution . . . 5

1.3 Random Walks on Groups . . . 6

2 Summary 9 2.1 Summary of Paper A . . . 9 2.2 Summary of Paper B . . . 10 2.3 Summary of Paper C . . . 11 2.4 Summary of Paper D . . . 11 2.5 Summary of Paper E . . . 11 2.6 Summary of Paper F . . . 12 2.7 Summary of Paper G . . . 13 Bibliography 15

II Scientific Papers

17

Paper A:

The Asymptotic Shape Theorem for Generalized First Passage Percolation. Submitted.

Paper B:

Ergodic Theorems for Homogeneous Dilations. Submitted.

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vii

Paper C:

Equidistribution of Dilations of Polynomial Curves in Nilmanifolds. Joint with Alexander Fish.

Proc. Amer. Math. Soc. 137 (2009), 2111-2123.

Paper D:

Almost Sure Equidistribution in Expansive Families. Joint with Daniel Schnellmann.

Submitted.

Paper E:

Continuous Measures on Homogeneous Spaces. Joint with Alexander Fish.

To appear in Annales d’Institut Fourier.

Paper F:

Central Limit Theorems for Gromov Hyperbolic Groups. Submitted.

Paper G:

Almost Sure Absolute Continuity of Bernoulli Convolutions. Joint with Daniel Schnellmann.

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Part I

Introduction and Summary

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Chapter 1

Introduction

The aim of this chapter is to provide some background to the results presented in this thesis. In the first three sections we define basic concepts and frequent terminology used throughout the thesis. The main results are presented in Sections 2.1 to 2.6.

1.1

Ergodic Theory

Let (X, F , µ) be a σ–finite measure space. An endomorphism of the measure space (X, F , µ) is a measurable self-map of X with the property that µ(T−1(A)) = µ(A) for all A ∈ F . We say that T is an automorphism if it admits a measurable inverse map. We let Aut(X, F , µ) denote the group of all automorphisms of (X, F , µ). Let G be a locally compact and second countable group. A measurable homomorphism φ : G → Aut(X, F , µ) is called a measurable action of G on X. An element A in F is invariant under the action of G if φ(g)A = A for all g ∈ G. We say that the action is ergodic if every invariant set is either null or conull. Ergodic theory is the study of ergodic group actions.

Note that the definitions above are not quite standard. In general, automorphisms are only required to preserve a fixed measure class and not necessarily a fixed mea-sure. However, this weaker notion of an automorphism will only be used in one section of Paper A.

One of the most important theorems in ergodic theory is undoubtedly the individual ergodic theorem due to G. D. Birkhoff [1] in 1931. If T is an ergodic automorphism of a probability measure space (X, F , µ), and f is a µ–integrable function on X, then there is a conull subset X0 of X, which may depend on f , such that

lim n→∞ 1 2n + 1 n X k=−n f (Tkx) = Z X f dµ, 3

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4 CHAPTER 1. INTRODUCTION

for all x in X0. A weaker version of this result is due to J. von Neumann in 1931, who

proved that if f is square–integrable on X with respect to µ, then the convergence above holds in the norm of L2(X, F , µ). Extensions of the ergodic theorem to

ergodic actions by Zd are also possible to prove and were first established by N.

Wiener in [12]. These can be described as follows. Let T1, . . . , Td be commuting

automorphisms of (X, F , µ) and let Fndenote the ball of radius n in Zdwith respect

to the standard word metric. Then, if the action generated by T1, . . . , Td is ergodic

and if f is a µ–integrable function on X, we have lim n→∞ 1 |Fn| X k∈Fn f (Tkx) = Z X f dµ

for almost every x in X. Here Tk = Tk1 1 · · · T

kd

d , with k = (k1, . . . , kd) in Zd.

There is an analogue of this theorem for Rd–actions. Let t 7→ T

t be a measurable

ergodic action of Rdon a probability measure space (X, F , µ). If f is a µ–integrable

function on X, then lim λ→∞ 1 |Bλ| Z Bλ f (Ttx) dt = Z X f dµ,

for almost every x in X, where Bλ denotes the ball in Rd, centered at the point

0 with radius λ. Here, |Bλ| denoted the Lebesgue measure of Bλ. We will extend

this theorem to a more general situation in paper B.

Another generalization of Birkhoff’s ergodic theorem to Zd–actions was suggested

by S.M. Kozlov [8] and D. Boivin and Y. Derriennic [3]. If f1, . . . , fdare measurable

real–valued functions on a probability measure space (X, F , µ), we define a recursive sequence of functions on X by

Sn+m(x) = Sn(x) + Sm(Tnx), ∀ n, m ∈ Zd,

and Sek = fk for k = 1, . . . , d, with ek the k:th standard basis vector in Z d. A

sequence of this form is called a cocycle, with generators f1, . . . , fd. If the

ac-tion generated by T1, . . . , Td is ergodic and f1, . . . , fd belong to the Lorentz space

Ld,1(X, µ), D. Boivin and Y. Derriennic [3] proved that there are real numbers L1, . . . , Ld such that lim |n|→∞ Sn− (L1 R Xf1dµ + . . . + Ld R Xfddµ) |n| = 0

almost everywhere on X. Birkhoff’s ergodic theorem corresponds to the case d = 1, since L1,1(X, µ) = L1(X, µ).

There are also non–linear extensions of Birkhoff’s ergodic theorem. We say that a sequence a : Z × X → R is a subadditive cocycle if

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1.2. EQUIDISTRIBUTION 5

almost everywhere on X. Clearly, every cocycle is a subadditive cocycle. The first general study of these sequences was undertaken by J.F.C. Kingman [6] in 1968. He proved that ifR

X|a(n, x)| dµ(x) < +∞ for all n ∈ Z, then the limit

A(x) = lim

n→+∞

an(x)

n ,

exists almost everywhere on X and is invariant under T . Thus, if the action is ergodic, A is necessarily constant. One of the main applications of Kingman’s subadditive ergodic theorem arise in percolation theory. In paper A, we will describe this connection more thoroughly and extend Kingman’s theorem to measurable actions by Zd. We also establish a general representation theorem for subadditive

cocycles.

1.2

Equidistribution

Let Z be a compact and metrizable space, and let M1(Z) denote the convex set of

probability measures on Z. We say that a sequence νn in M1(Z) converges to ν in

the weak*–topology on M1(Z) if lim n→∞ Z Z φ dνn= Z Z φ dν

for every continuous function φ on Z. Recall that M1(Z) is sequentially weak*–

compact, i.e. every sequence of probability measures on Z has a weak*–convergent subsequence. We say that a sequence znof points in Z equidistribute with respect

to a measure ν if lim n→∞ 1 n n X k=1 δzk= ν,

where the limit is taken in the weak*–topology. More generally, we can consider sequences which depend on some parameter θ, and ask if equidistribution of the sequence holds for a large set of θ’s. For example, if θ belongs to some bounded subset of Rd, we may want to establish equidistribution for almost every θ with

respect to the Lebesgue measure. We will consider this situation in Paper D. If Z is a compact Riemannian homogenous space, i.e. if Z is a compact Riemannian manifold with a transitive action by isometries, the eigenfunctions of the Laplace operator on Z can be used to establish equidistribution of certain sequences of measures. More precisely, since the linear span of all eigenfunctions φk of the

Laplace operator on a compact Riemannian manifold is uniformly dense, to prove that a sequence νn converges to ν, it suffices to prove that

lim n→∞ Z Z φjdνn = Z Z φjdν,

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6 CHAPTER 1. INTRODUCTION

for all j. This is a useful technique when the measures are well adapted to the ho-mogeneous structure and the eigenfunctions are explicitly known. However, if this is not the case, other methods have to be employed. We will discuss an alternative moment approach to equidistribution in Paper D.

In the general situation it is interesting to find necessary and sufficient conditions to ensure that a given measure on Z is continuous, i.e. does not give positive mass to individual points of Z. We prove a general characterization in Paper E.

1.3

Random Walks on Groups

Let G be a locally compact and second countable group. Given two probability measures µ and ν on G we define the convolution of µ and ν as the push–forward of µ × ν under the multiplication map G × G → G. This construction yields a new probability measure on G, which we denote by µ ∗ ν. Given a sequence µ1, µ2, . . .

of probability measures on G, we define νn = µ1∗ . . . ∗ µn, which we will refer

to as the random walk induced by the sequence µ1, µ2, . . .. If µk = µ for all k,

we will refer to µ as the random walk, and write νn = µ∗n. It is of great

inter-est in probability theory to understand various asymptotic aspects of random walks. We first consider the homogeneous situation, i.e. µk = µ for all k. Let G be a

finitely generated group, and let S be a symmetric generating set. Let d be a G– invariant metric on G, not necessarily the word–metric. Define X = GZ with the

natural product σ–algebra and let P = µZ. Note that right–shift T on X preserves

P and is ergodic. Let g : X → G be the projection onto the zero coordinate, and define Z0= e and Z1= g, and recursively

Zn+m(x) = Zn(x)Zm(Tnx)

for all m, n ∈ Z. Let an(x) = d(Zn(x), e) for n ∈ Z. We are interested in the

generic asymptotic behavior of the sequence an. Suppose

Z

X

d(g(x), e) dµ(x) < ∞.

By Kingman’s subadditive ergodic theorem, there is a non–negative constant A such that

A = lim

n→∞

d(Zn(x), e)

n

almost everywhere on X with respect to P. It is in general very hard to compute the constant A, given a measure µ. Nevertheless, Y. Guivarc’h [5] was able to prove that, if d is the word–metric on G with respect to some finite generating set, then A > 0 if G is non–amenable and if the support of µ generates G as a group. This raises the question about the speed of convergence. In general this is a very hard problem. We address this question in Paper F, and give a partial answer in the

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1.3. RANDOM WALKS ON GROUPS 7

case of hyperbolic groups.

In Paper G we turn to the inhomogeneous situation. Suppose λk is a sequence of

real numbers and define the probability measures, µk = 1 2(δλk+ δ−λk) , k ≥ 1. SupposeP k≥1λ 2

k< +∞. The infinite convolution product

ν =

Y

k=1

µk

exists and is well–defined. We will refer to the measure ν as a Bernoulli convolution. A well-studied problem is to determine the measure class of ν for different choices of sequences λk. In Paper G we consider sequences of the form λk = λφ(k) for

0 < λ < 1 and functions φ : R+→ R+. We establish conditions on φ to ensure that

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Chapter 2

Summary

2.1

Summary of Paper A

First passage percolation is concerned with the generic large scale geometry of ran-dom semimetric spaces. We start by describing the classical situation. Let (X, F , µ) be a probability space, and suppose T is a measurable ergodic Zd–action by

au-tomorphisms of (X, F , µ). Suppose that f1, . . . , fd are non–negative measurable

functions on X, we define the weight of the edge between n and n + eiin the

stan-dard Cayley graph of Zd as f

i(Tnx). Thus, given x ∈ X, we define the semimetric

ρx(m, n) between two points m, n ∈ Zd as the infinum of all the weighted paths

between m and n in the standard Cayley graph of Zd. This produces a measurable

functions ρ into the space of semimetrics on Zd with the following equivariance

property:

ρTkx(m, n) = ρx(m + k, n + k), ∀ k, m, n ∈ Zd, ∀ x ∈ X.

We will refer ρ as a random semimetric on Zd. Assume that all fi belong to the

Lorentz space Ld,1(X, µ). The asymptotic shape theorem, due to D. Boivin [2], asserts the existence of a seminorm L on Rd such that

lim

|n|→∞

ρx(0, n) − L(n)

|n| = 0

almost everywhere on X, where | · | is the standard word–metric on Zd. Note that

the limits along the coordinate axes exist by Kingman’s subadditive ergodic the-orem, so Boivin’s result can be viewed as a weak multidimensional generalization of Kingman’s theorem. Note that the case d = 1 corresponds to Birkhoff’s ergodic theorem.

The main goal of paper A is to generalize this theorem to general random semi-metric, not necessarily induced from a randomly weighted standard Cayley graph of Zd. This requires new ideas since Boivin’s proof heavily uses the geometry of

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10 CHAPTER 2. SUMMARY

the standard Cayley graph. The key observation is that any random semimetric can be realized as the norm of an additive cocycle with values in L∞(Zd), and the

additivity can be used to transfer the proof of the necessary maximal inequalities to Boivin’s and Derriennic’s maximal inequality, proved in [3]. This observation allows us to generalize the concept of first passage percolation, by which we now mean the asymptotic behavior of Banach space–valued ergodic cocycles. If we as-sume that the Banach spaces are separable and reflexive, more refined versions of the asymptotic shape theorem can be derived.

2.2

Summary of Paper B

The ergodic theorem for Rd actions was formulated in section 1.1. A.P. Calderón [4] realized that this theorem is an easy consequence of the Hardy-Littlewood’s maximal inequality, which lead to a revolution in ergodic theory. The starting point in Paper B is the following straightforward paraphrase of the averages above, for fixed λ > 0: 1 |Bλ| Z Bλ f (Ttx) dt = Z Rd f (Tλtx)χB(t) |B| dt.

We now ask if the pointwise ergodic theorem continues to hold if the probability measure χB

|B| is replaced by a more general measure ν, not necessarily absolutely

continuous with respect to the Lebesgue measure. Note however that ν has to be non-atomic. We first establish the easy fact that the mean ergodic theorem only holds when ν is a Rajchman measure, i.e. when the Fourier transform of ν tends to 0 at infinity. This is a large set of measures on Rd, and include many measures which are supported on subsets of Hausdorff dimension 0. To address the question about almost everywhere convergence, we have to restrict the class of measures further. Previous works by J. Bourgain, M. Lacey and R.L. Jones have established pointwise results when ν is supported on spheres in Rd, d ≥ 2. We extend their work to measures whose Fourier transforms decay sufficiently fast at infinity. More precisely, recall that the Fourier dimension of a probability measure ν on Rd is

defined as the largest constant a for which

|ˆν(ξ)| ≤ C 1

|ξ|a/2, ∀ ξ ∈ R d\{0},

for some constant C. We prove that if ν has Fourier dimension a > 1, then for all function f ∈ Lp(X), with p > 1+aa ,

lim λ→∞ Z Rd f (Tλtx) dν(t) = Z X f dµ almost everywhere on X.

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2.3. SUMMARY OF PAPER C 11

2.3

Summary of Paper C

This is a joint paper with A. Fish. Let γ denote a continuous curve in Rn, and

assume that the domain of definition of γ is the unit interval [0, 1]. Let m denote the Lebesgue measure on [0, 1], and let ν = γ∗m denote the push–forward of m

under the map γ, i.e.

γ∗m(B) = m(γ−1(B)),

for all Borel sets B in Rn. We also define

γ.γ(t) = (λγ1(t), . . . , λγn(t)),

for λ > 0 and set νλ= (λ.γ)∗m. This paper is concerned with the asymptotic

be-havior of the push–forward of νλ under the canonical projection onto the compact

torus Tn= Rn/Zn. We address quantitative statements about weak*–convergence

of νλ to the Haar measure on Tn, and analogous questions for more complicated

nilmanifolds.

2.4

Summary of Paper D

This a joint paper with D. Schnellmann. We give a new proof of an old result by J.F. Koksma [7] on equidistribution of sequences in R/Z which are parameterized by points in R. One of the main results in Koksma paper from 1935, that for almost every θ > 1, the projection of the sequence xj= θj onto T equidistributes. Certain

higher dimensional analogues were later established . In this paper we are inspired by a technique used in one–dimensional dynamics to prove geometric criteria for the equidistribution of higher–dimensional versions of Koksma’s theorem. We consider sequences of maps ˜fj: Ω → Rn, where Ω is an open set in Rn, and give conditions

on the expansion and distortion of the sequence, which ensure that the projection of the sequence ˜fj(θ) equidistributes in Rn/Γ for almost every θ ∈ Ω and for a fixed

lattice Γ in Rn.

2.5

Summary of Paper E

This is a joint paper with A. Fish. An old lemma by N. Wiener characterizes probability measures on the unit circle which does not charge individual points in terms of the the Fourier transform. More precisely, it is known that if ν is a probability measure on Z, then

lim n→∞ 1 n n−1 X k=0 |ˆν(k)|2=X x∈T |µ({x})|2, where ˆν is defined by ˆ ν(k) = Z T e−2πikxdν(x).

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12 CHAPTER 2. SUMMARY

In particular, ν is continuous if and only if

lim n→∞ 1 n n−1 X k=0 |ˆν(k)|2= 0.

The aim of this paper is to generalize this result to more general compact mani-folds than T. The extension to higher dimensional tori is straightforward, and one generalization of Wiener’s lemma for compact semisimple Lie groups was proved by M. Anoussis and A. Bisbas, using the explicit harmonic analysis available in this setting. We consider a compact Riemannian manifold X, and assume that the isometry group of X acts transitively on X. It is known that the span of the eigenfunctions of the Laplace–Beltrami operator ∆ is uniformly dense in C(X), and by duality we know that a probability measure ν on X is characterized completely by the sequence

νk =

Z

X

ψkdν, k ≥ 0

where ψk is the k:th eigenfunction of ∆. The heat kernel associated to ∆ is defined

by Kt(x, y) = X k≥0 e−λktψ k(x)ψk(y),

where λk is the eigenvalue corresponding to ψk. This function has the important

property that for every smooth function φ, lim

t→0+Ktφ(x) = limt→0+

Z

X

Kt(x, y)φ(y) dσ(y) = φ(0),

where σ is the normalized volume form on X. We use this property and prove that for any probability measure on X,

lim t→0+ P k≥0e−λk t k|2 P k≥0e−λkt =X x∈X |ν({x})|2.

In the case X = Td, we recover Wiener’s theorem by a simple Tauberian argument. The above result is new for nilmanifolds.

2.6

Summary of Paper F

A metric space (X, d) is Gromov hyperbolic if there is a δ > 0 such that for all points x, y, z, w in X, we have the inequality

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2.7. SUMMARY OF PAPER G 13

where (·, ·)· denotes the Gromov product, defined by

(x, y)w=

1

2(d(x, w) + d(y, w) − d(x, y)).

Let G be a finitely generated group with finite generating set S, which we assume is symmetric, i.e. S−1 = S. We define the Cayley graph of (G, S) as the abstract graph with node set G and edge set consisting of pairs of the form (g, gs) with g ∈ G and s ∈ S. The graph structure induces a natural metric dS on G which is

invariant under left multiplication. We say that a group G is Gromov hyperbolic if the metric structure on the Cayley graph with respect to some finite generating set S is Gromov hyperbolic. This notion turns out to be independent of the given generating set. It is easy to see that virtually free groups are Gromov hyperbolic. Let µ be a probability measure on G and assume that the support of µ generates G as a group. Assume that G is non–amenable. A theorem by Guivarc’h [5] assures the almost sure positivity of the drift, i.e.

A = lim

n→∞

1

ndS(g1· · · gn, e) > 0,

where the gi’s are i.i.d. G–valued random variables with common distribution µ.

The positivity of the drift only depends on the quasi–isometry class of dS, i.e. if

we change dS to another invariant metric d which satisfies

1

CdS(x, y) − b ≤ d(x, y) ≤ CdS(x, y) + b

for some non-negative constants C and b, for all x, y in G, then the positivity of the drift, calculated with respect to d, will not change.

In this paper we address the asymptotic behavior of the sequence Yn=

d(g1. . . gn, e) − nA

n , n ≥ 1,

for invariant metrics which are quasi–isometric to dS. We prove that there is an

invariant metric d, which is quasi–isometric to dS, for which, under some technical

moment conditions on µ, there is a constant σ > 0, which depends on µ, such that Yn converges weakly to a centered Gaussian random variable with variance σ. We

also give a new description of this metric. This generalizes a theorem by Ledrappier [9] for free groups.

2.7

Summary of Paper G

This is a joint paper with D. Schnellmann. We extend a result by Peres and Solomyak [10] on Bernoulli convolutions. Let 0 < λ < 1, and define the random series

Yλ=

X

n≥0

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14 CHAPTER 2. SUMMARY

where the εnare independent Bernoulli variables with values in {−1, +1}. P. Erdös

asked whether the distribution of Yλ is absolutely continuous with respect to the

Lebesgue measure for λ > 1/2. This question was settled by Solomyak [11], and a simpler proof was later found by B. Solomyak and Y. Peres. We apply the methods developed in the second paper and prove almost sure absolute continuity for the distribution of random series of the form

Zλ=

X

n≥0

εnλαn,

where αnis a sequence of real numbers which satisfies some technical growth

condi-tions. The case αn = n was considered by Peres and Solomyak, and if the sequence

is O(√n), A. Wintner proved much stronger assertions using Fourier analysis. Our paper covers the intermediate situation.

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Bibliography

[1] Birkhoff, G. D. Proof of the ergodic theorem. Proceedings of the National Academy of Sciences 17, 656 - 660, or Collected Mathematical Papers, 3 vols., Providence: American Mathematical Society, 1950, 2: 404Ð408.

[2] Boivin, D. First passage percolation: the stationary case. Probab. Theory Re-lated Fields 86 (1990), no. 4, 491–499.

[3] Boivin, D. and Derriennic, Y. The ergodic theorem for additive cocycles of Zd

or Rd. Ergodic Theory Dynam. Systems 11 (1991), no. 1, 19–39.

[4] Calderón, A.-P. Ergodic theory and translation-invariant operators. Proc. Nat. Acad. Sci. U.S.A. 59 1968 349–353.

[5] Guivarc’h, Y. Sur la loi des grands nombres et le rayon spectral d’une marche aléatoire. Conference on Random Walks (Kleebach, 1979) (French), pp. 47–98, 3, Astérisque, 74, Soc. Math. France, Paris, 1980.

[6] Kingman, J. F. C. The ergodic theory of subadditive stochastic processes. J. Roy. Statist. Soc. Ser. B 30 1968 499–510.

[7] Koksma, J. F. Ein mengentheoretischer Satz über die Gleichverteilung modulo Eins. Compositio Math. 2 (1935), 250–258.

[8] Kozlov, S. M. The averaging method and walks in inhomogeneous environ-ments. (Russian) Uspekhi Mat. Nauk 40 (1985), no. 2(242), 61–120, 238. [9] Ledrappier, F. Some asymptotic properties of random walks on free groups.

Topics in probability and Lie groups: boundary theory, 117–152, CRM Proc. Lecture Notes, 28, Amer. Math. Soc., Providence, RI, 2001.

[10] Peres, Y and Solomyak, B. Absolute continuity of Bernoulli convolutions, a simple proof. Math. Res. Lett. 3 (1996), no. 2, 231–239.

[11] Solomyak, B. On the random seriesP ±λn(an Erdös problem). Ann. of Math.

(2) 142 (1995), no. 3, 611–625.

[12] Wiener, N. The ergodic theorem. Duke Math. J. 5 (1939), no. 1, 1–18.

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Part II

Scientific Papers

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References

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