RANDOM WALKS ON GROUPS AND RANDOM TRANSFORMATIONS

ALEX FURMAN

To appear in

Handbook of Dynamical Systems, Vol 1

Editors: Boris Hasselblatt and Anatole Katok.

Publisher: Elsevier.

Date: October 16, 2000.

This work was supported in part by NSF grant DMS-9803607 and by CNRS.

Contents

Introduction 3 Acknowledgements 4 1. Products of random matrices 5 1.1. General Overview 5 1.2. Preliminaries on Markov processes 7 1.3. A formula for the top Lyapunov exponent 9 1.4. Non-random filtration associated to μ 12 1.5. Furstenberg’s condition for positive growth 13 1.6. Unitary representation approach I 15 1.7. Unitary representation approach II 16 1.8. Simplicity of the spectrum 18 1.9. Quasi-Projective transformations and Flag Varieties 20 1.10. Some Auxiliary Results 23 1.11. From contractions to the simplicity of the spectrum 25 1.12. Zariski closures and the contraction properties 27 1.13. Regularity of the Lyapunov spectrum 28 1.14. Further Limit Theorems 30 1.15. Additional Remarks 33 2. Random Walks on General Groups 34 2.1. Recurrence of random walks 34 2.2. Harmonic functions and (G, μ)-spaces 36 2.3. (G, μ)-boundaries 38 2.4. The Poisson boundary 40 2.5. Semi-simple Lie groups and their lattices 44 2.6. The Poisson boundary for discrete linear groups 46 2.7. Boundary entropy 48 2.8. Random Walk Entropy 51 2.9. Triviality of the Poisson boundary 55 2.10. Identification of the Poisson boundary 56 2.11. Towards a Structure Theory for (G, μ)-spaces 58 3. Random Transformations 60 3.1. The Random Ergodic Theorem 60 3.2. Strong ergodicity and Rate of convergence 62 3.3. Entropy for random transformations 65 3.4. Lyapunov exponents and non-random filtration 66 3.5. Entropy for random diffeomorphisms 68 3.6. Measure-preserving K^{μ}-property 69 3.7. Cocycle growth along μ-random walks 70 4. Selected Proofs 75 References 81

Introduction

The study of asymptotic behavior of sums S_{n} = X_{n} + ⋯ + X_{1} of independent identically distributed real valued random variables is a well established part of the classical probability theory. Under suitable boundness conditions the phenomena of Recurrence, Law of Large Numbers, Functional Central Limit Theorem, Iterated Logarithm Law and further limit theorems have been well understood already in the first half of 20-th century. In the recent decades the study of more general noncommutative situation became an active area of research. The general setup that will be discussed in this paper consists of a group G with a probability measure μ on it, and the questions concern the statistic behavior of the products S_{n} = X_{n} ⋯ X_{1} of independent G-valued random variables {X_{n}} which have a common distribution μ. A sequence {S_{n}} of such products is referred to as a path of the μ-random walk on G.

In this paper we discuss three aspects of random walks. Chapter 1 is devoted to random walks on matrix groups G = SL_{k}(ℝ). This theory is mainly concerned with Laws of Large Numbers and other limit theorems. The techniques involved include Markov processes, the dynamics of the projective G-action on flag varieties, elements of the structure theory of SL_{k}(ℝ) as an algebraic group and some considerations with unitary representations. Although the presentation is restricted to SL_{k}(ℝ), most of the results can be formulated and proved in the more general context of semisimple real Lie groups.

In chapter 2 random walks on general (locally compact or discrete) groups are discussed. The focus is on the connections between the properties of the group (such as nilpotency, amenaibility, growth etc.) and the behavior of the random walks on it. After a discussion of recurrence properties of random walks, we discuss bounded μ-harmonic functions, the concept of the Poisson boundary and related notions of boundary entropy and random walk entropy.

Chapter 3 is about random walks on groups of transformations of measure spaces and manifolds. The chapter starts with Random Ergodic Theorem and related results, and then turns to the random-walk-based notion of entropy. This notion is discussed in the context of diffeomorphisms of manifolds and in the general measurable setting. The discussion closes with a connection between entropy of random volume-preserving diffeomorphisms and the starting topic of the paper - random walks on matrix groups.

Obviously, the choice of the presented material was dictated by time, space and expertise limitations of the author as well as the personal taste. In particular, several important topics are not represented in this paper at all, most notably: random walks on graphs, Martin boundary, random perturbations in smooth dynamics and applications to stochastic differential equations. References to these topics in the present context should include: Woess [69], Guivarc’h, Ji, Taylor [25] and Kaimanovich [33], and works of Y. Kifer [41], [43].

The results in the paper are often stated not in the most general form known. For general statements and detailed proofs the reader is referred to the original papers, references to which are included. Obviously, any mistakes in the statements or proofs are in the full responsibility of the present author. Proofs of the stated results are included in the paper only if they are relatively short and exhibit important ideas. In many cases the proofs are just outlined with some details left out, however an effort was made to achieve a consistent and “relatively self contained” presentation. Included proofs appear either in the main text, or in the last sections 4, in which case the statements are marked by a star.

Acknowledgements. Being a survey, this paper would be impossible without the works of all the cited authors. Specifically I would like to express my deep appreciation and gratitude to Hillel Furstenberg, Yves Guivarc’h, Vadim Kaimanovich, Yuri Kifer, Francois Ledrappier, Gregory Margulis, Amos Nevo, Yuval Peres, Yehuda Shalom, Benjamin Weiss and Robert Zimmer for sharing their knowledge, ideas and insights, and for their comments and corrections to the first draft of this paper. I would also like to thank the editors - Anatole Katok and Boris Hasselblatt - for the invitation to write the survey. Special thanks to Ola for the encouragement and support. Part of this work was done while I enjoying the warm hospitality of Ecole Normale Supérieure de Lyon (CNRS), during the most pleasant summer of 1999. This work was also supported in part by the NSF grant DMS-9803607.

1. Products of random matrices

1.1. General Overview. One of the most natural and important examples of transformation groups are linear transformations of finite dimensional real linear spaces, i.e. subgroups of GL_{k}(ℝ). In this chapter we shall focus on the behavior of random products of such transformations, namely the products of random matrices. If {X_{n}, n ≥ 1} is a sequence in GL_{k}(ℝ) consider the real numbers x_{n} = k^{−1} log | det X_{n}| and the sequence X′_{n} = e^{−x_{n}}X_{n} of k × k-matrices, for which det X′_{n} = ±1, and

X_{n} ⋯ X_{1} = e^{x_{n}+···+x_{1}} X′_{n} ⋯ X′_{1}

The study of sums of i.i.d. real variables x_{n} + ⋯ + x_{1} belongs to the classical probability theory, while the new aspects of the matrix setup arise from the absence of commutativity in matrix multiplication of the X′_{n} part. Therefore, it is customary to discuss random products of matrices in the group

SL′_{k}(ℝ) = {g ∈ GL_{k}(ℝ) | det g = ±1}

or in SL_{k}(ℝ) which is a subgroup of index two in SL′_{k}(ℝ).

Let μ be a Borel probability measure on G = SL′_{k}(ℝ), and let S_{n}(ω) = X_{n}(ω) ⋯ X_{1}(ω) be a sequence of random products, where {X_{n}(ω), n ≥ 1} are independent μ-distributed random variables. X_{n} can be viewed as coordinate projections of the product probability space

(Ω, P) = (G^{ℕ}, μ^{ℕ})

related by X_{n} = X_{k} ∘ θ^{n−k} for n > k, where θ : Ω → Ω is the shift (θω)_{i} = ω_{i+1} on (Ω, P). Often we shall omit the point ω ∈ Ω from the notation, and denote the random variables by just X_{1}, X_{2}, …, X_{n}, … and S_{n} = X_{n} ⋯ X_{1}.

We shall be interested in the qualitative and quantitative properties of the random products S_{n} ∈ G and in their actions on the vector space ℝ^{k}, the projective space ℙ^{k−1}, Grassmannians and the flag varieties. In the context of products of random matrices one can study a non-commutative analogue of the Law of Large Numbers which describes the limit

lim_{n→∞} \frac{1}{n} log ∥X_{n} ⋯ X_{1}∥

where ∥ · ∥ is some norm on the matrix algebra M_{n×n}(ℝ). Since any two such norms ∥ · ∥, ∥ · ∥′ are equivalent (in the sense that ∥ · ∥/∥ · ∥′ is bounded from 0 and ∞) the existence and the value of the above limit does not depend on the choice of the norm. Hereafter it will be convenient to use the operator norm ∥A∥ = sup_{∥v∥=1} ∥Av∥ with respect to the l^{2}-norm ∥v∥ = (v_{1}^{2} + ⋯ + v_{k}^{2})^{1/2} on ℝ^{k}.

Laws of large numbers require some, typically L^{1}, boundness condition on the variables. We shall say that μ has finite first moment if

∫_{G} log ∥g∥ dμ(g) < ∞  (1.1)

Proposition 1.1. Let μ on SL′_{k}(ℝ) be a probability measure with finite first moment, and let {X_{n}, n ≥ 1} denote the sequence of independent μ-distributed random variables. Then with P-probability one the limit

lim_{n→∞} \frac{1}{n} log ∥X_{n} ⋯ X_{1}∥

exists, its value λ_{1}(μ) is P-a.e. constant and can also be expressed as

λ_{1}(μ) = lim_{n→∞} \frac{1}{n} ∫_{G} log ∥g∥ dμ^{n}(g) = inf_{n>0} \frac{1}{n} ∫_{G} log ∥g∥ dμ^{n}(g)

where μ^{n} denotes the n-th convolution power of μ.

Proof. This fact was originally proved by Furstenberg and Kesten [18]. Now it can be easily deduced from the subadditive ergodic theorem of Kingman as follows: the functions h_{n}(ω) = log ∥S_{n}(ω)∥ on (Ω, P, θ) form a subadditive cocycle

h_{n+m}(ω) = log ∥S_{n+m}(ω)∥ = log ∥S_{n}(θ^{m}ω)S_{m}(ω)∥
≤ log ∥S_{n}(θ^{m}ω)∥ + log ∥S_{m}(ω)∥ = h_{n}(θ^{m}ω) + h_{m}(ω)

and condition (1.1) is h_{1} ∈ L^{1}(Ω, P). Since the the shift θ is ergodic on the space (Ω, P), there exists a P-a.e. constant limit λ_{1}(μ) = lim_{n→∞} n^{−1} · h_{n}(ω) which can also be expressed as

λ_{1}(μ) = lim_{n→∞} \frac{1}{n} ∫ h_{n} dP = inf_{n} \frac{1}{n} ∫ h_{n} dP

Since all matrices in G = SL′_{k}(ℝ) have norm of at least one, h_{n} ≥ 0 and therefore λ_{1} ≥ 0.

The number λ_{1}(μ) is the top Lyapunov exponent of the random matrices. It is the first in a sequence

λ_{1}(μ) ≥ λ_{2}(μ) ≥ ⋯ ≥ λ_{k}(μ)

of k constants referred to as the Lyapunov exponents or the Lyapunov spectrum of random matrices with the law μ. The exponents λ_{p}(μ), p > 1, are defined inductively, via the top Lyapunov exponents of the exterior powers ∧^{p}S_{n} = ∧^{p}X_{n} ⋯ ∧^{p} X_{1} by the relations

λ_{1}(μ) + ⋯ + λ_{p}(μ) = lim_{n→∞} \frac{1}{n} log ∥ ∧^{p} S_{n}∥ p = 1, 2, …, k.

For any A ∈ SL′_{k}(ℝ) one has ∧^{k}A = det A = ±1 and consequently λ_{1} + ⋯ + λ_{k} = 0.

This formal definition of λ_{p}, 1 ≤ p ≤ k has a more transparent interpretation. Recall that any matrix M ∈ SL′_{k}(ℝ) can be written in the polar form M = V D U, where D = diag [e^{a_{1}}, …, e^{a_{k}}] denotes the diagonal matrix with a_{1} ≥ a_{2} ≥ ⋯ ≥ a_{k} and U, V are orthogonal matrices. Due to our choice of the norm one has

∥M∥ = e^{a_{1}},  ∥ ∧^{p} M∥ = e^{a_{1}+···+a_{p}}  for 1 ≤ p ≤ k.  (1.2)

which in particular shows that a_{1} ≥ ⋯ ≥ a_{k} are uniquely determined by M. Proposition 1.1 asserts that writing the random products S_{n} in the polar form

S_{n} = X_{n} ⋯ X_{1} = V_{n} diag [e^{a_{1}(n)}, …, e^{a_{k}(n)}] U_{n}  (1.3)

with a_{1}(n) ≥ ⋯ ≥ a_{k}(n) and U_{n}, V_{n} ∈ O(k), one has with P-probability one: lim_{n→∞} a_{1}(n)/n = λ_{1}, and (1.2) implies that lim_{n→∞} a_{p}(n)/n = λ_{p} for all p = 1, …, k.

This argument constitutes a part of the Oseledec theorem, which applies to general (ergodic) stationary processes {X_{n}, n ≥ 1} in G. In what follows we shall specialize to the case of independent random variables, obtaining much more detailed information.

1.2. Preliminaries on Markov processes. Consider the following general setup: let M be a compact metric space and denote by P(M) the space of all probability measures on M, which is a convex compact metric space in the weak-* topology induced by C(M). Given a continuous map M → P(M), x ↦ μ_{x}, one can define a Markov operator P acting on C(M) by

P f(x) = ∫_{M} f(y) dμ_{x}(y)

The measures μ_{x}, x ∈ X, are called transition probabilities of P. The dual operator P^{∗} acts on the space of measures on M preserving the convex compact set P(M), and the set of P^{∗}-fixed measures in P(M) is a convex compact subset. By a standard fixed point theorem it is non-empty - indeed for any η_{0} ∈ P(M) any accumulation point of n^{−1} · ∑_{1}^{n}(P^{∗})^{j}η_{0} is fixed by P^{∗}. A probability measure η ∈ P(M) with P^{∗}η = η is called P -stationary.

A Markov operator P as above and an arbitrary measure θ ∈ P(M) gives rise to the corresponding Markov process {Z_{n}, n ≥ 0} on M, where Z_{0} is taken with distribution θ on M, and the conditional distribution E(Z_{n+1} | Z_{n}, …, Z_{0}) is μ_{Z_{n}}. Such a Markov process {Z_{n}, n ≥ 0} is stationary if and only if the initial distribution θ of Z_{0} is P -stationary. In this general context we shall use the following fact:

Theorem 1.2 (Furstenberg [13], see also [19]). Let P be a Markov operator on a compact metric space M and σ be a continuous function on M. Take an arbitrary (not necessarily P -stationary) measure θ ∈ P(M), and let {Z_{n}, n ≥ 0} denote the corresponding Markov process on M. Then with probability one

lim sup_{n→∞} \frac{1}{n} ∑_{k=1}^{n} σ(Z_{k}) ≤ sup \{∫_{M} σ dη | η ∈ P(M) s.t. P^{∗}η = η\}

If, moreover, ∫ σ dη takes the same value α for all P -stationary probability measures η on M, then with probability one n^{−1} ∑_{k=1}^{n} σ(Z_{k}) → α.

Next consider the situation where a locally compact group (or a semigroup) G acts on a compact metric space M so that the action map G × M → M, (g, x) ↦ g · x, is continuous. Hereafter such M with the G-action is called a G-space. Given a probability measure μ on G and a probability measure ν on M one can define the convolution probability measure μ ∗ ν on M as the image of μ × ν under the action map G × M → M. In other words

∫_{M} f d(μ ∗ ν) = ∫_{G} ∫_{M} f(g · x) dν(x) dμ(g) for f ∈ C(M)

In particular, taking Dirac measures ν = δ_{x}, x ∈ M, one obtains a continuous assignment x ↦ μ_{x} = μ ∗ δ_{x}, which defines a Markov operator P on C(M) by

(P f)(x) = ∫_{G} f(g · x) dμ(g) for f ∈ C(M), x ∈ M

Probability measures ν ∈ P(M) satisfying μ∗ν = ν (equivalently P^{∗}ν = ν) are called μ-stationary measures. As before, one notes that the set of all μ-stationary measures in P(M) is non-empty convex compact set, and it is easy to see that each μ-stationary measure ν gives rise to a stationary M-valued Markov process {U_{n}^{ν}, n ≥ 0}. It is a general fact about Markov processes that {U_{n}^{ν}, n ≥ 0} fails to be ergodic iff there exists a P -invariant measurable set E ⊂ M with 0 < ν(E) < 1 (see Theorem 3.1 below). In the present context of a compact G-space M one can check that a μstationary measure ν admits P -invariant measurable E ⊂ M with 0 < ν(E) < 1 iff ν is not an extremal point of the set of all μ-stationary measures. This discussion can be summarized in terms of ergodic theory as follows:

Proposition 1.3. Let G be a locally compact group, μ a probability measure on G, M a compact G-space, and ν - a probability measure on M. Consider the one-sided Bernoulli shift (Ω, P, θ) where (Ω, P) = (G^{ℕ}, μ^{ℕ}), (θω)_{i} = ω_{i+1}, i ∈ ℕ, and the transformation T of the product space Ω × M defined by T (ω, x) = (θω, ω_{1} · x). Then (a) The product measure P × ν is T -invariant if and only if ν is a μ-stationary

measure.

(b) The product measure P×ν is an ergodic T -invariant measure iff ν is an extremal point of the compact convex set of all μ-stationary measures on M.

Given a probability measure μ on G and a G-space M, one can also consider the product space M̃ = G × M with the Markov operator P̃ defined by

P̃f(g, x) = ∫_{G} f(g′, g′ · x) dμ(g′)  (1.4)

The corresponding M̃-valued Markov process Z_{n}^{ν} can be described as follows: let U_{0} be a M-valued random variable with distribution ν, {X_{n}, n ≥ 0} a sequence of Gvalued random variables with distribution μ which are independent from each other and from U_{0}; and let

Z_{n}^{ν} = (X_{n}, U_{n}) where U_{n} = X_{n} ⋯ X_{1} · U_{0} = S_{n} · U_{0}  (1.5)

Notice that this construction does not quite fit into the framework of Markov processes on compact spaces if G is not compact. If μ has a compact support S ⊂ G, then one may simply replace the non-compact space G × M by the compact one M̃ = S × M. In general, one can consider the compact space M̃ = Ĝ × M where Ĝ = G ∪ {∗} is a one point compactification of G, taking special care while applying Theorem 1.2 to functions σ on G which do not extend continuously to Ĝ.

One can easily check that for any μ-stationary measure ν ∈ P(M), the measure μ × ν on M̃ is P̃-stationary, and so is the Markov process {Z_{n}^{ν}, n ≥ 0} given by (1.5). Moreover

Lemma 1.4 (cf. [14]). The set of P̃-stationary probability measures on M̃ is

{μ × ν | ν ∈ P(M) s.t. μ ∗ ν = ν}

Hence every stationary Markov processes on M̃ has the form (1.5). Moreover, the stationary Markov processes {Z_{n}^{ν}, n ≥ 0} is ergodic if and only if the μ-stationary measure ν is an extremal point of the convex compact consisting of all μ-stationary probability measures on M.

1.3. A formula for the top Lyapunov exponent. Consider the natural projective action of G = SL′_{k}(ℝ) on the projective space M = ℙ^{k−1}, take M̃ = G × M and let σ_{1} : G × ℙ^{k−1} → ℝ be defined by

σ_{1}(g, ū) = log \frac{∥gu∥}{∥u∥}  (1.6)

where u ∈ ℝ^{k} is a non-zero vector in the line ū. Note that σ_{1} is a cocycle with respect to the G-action on ℙ^{k−1}, i.e. σ_{1}(g′g, ū) = σ_{1}(g′, g · ū) + σ_{1}(g, ū), and for any non-zero vector u ∈ ℝ^{k} and any g_{0}, g_{1}, …, g_{n} ∈ G one has

log ∥g_{n} ⋯ g_{0}u∥ − log ∥u∥ = σ_{1}(g_{n}, ū_{n}) + ⋯ + σ_{1}(g_{0}, ū_{0})  (1.7)

where ū_{0} is the line spanned by u and ū_{j+1} = g_{j} · ū_{j}, j = 0, …, n − 1.

Let μ be a probability measure on G, and let us assume for the moment that μ has compact support S ⊂ G. Taking M̃ = S × ℙ^{k−1} one obtains a continuous function σ_{1} : M̃ = S × ℙ^{k−1} → ℝ on a compact space, so that Theorem 1.2 applies to any Markov process Z_{n}^{ν} = (X_{n}, U_{n}) = (X_{n}, X_{n} ⋯ X_{1}U_{0}) where the distribution ν of U_{0} is arbitrary (not necessarily μ-stationary), while {X_{n}, n ≥ 0}, are independent μdistributed random variables. In particular, taking ν = δū where u is any non-zero vector u ∈ ℝ^{k}, one has with P-probability one

lim sup_{n→∞} \frac{1}{n} log ∥X_{n} ⋯ X_{1}u∥ = sup \{α_{μ,ν} | ν ∈ P(ℙ^{k−1}) s.t. μ ∗ ν = ν\} (1.8)

where α_{μ,ν} denotes the integral

α_{μ,ν} = ∫_{G} ∫_{ℙ_{k−1}} σ_{1} dμ × ν = ∫_{G} ∫_{ℙ_{k−1}} log \frac{∥gu∥}{∥u∥} dν(ū) dμ(g)  (1.9)

The equality (1.8) holds not only for compactly supported μ, but for any μ with finite first moment. To show this, one needs to consider the space M̃ = Ĝ × ℙ^{k−1} and a sequence of continuous functions σ_{1}^{(T)} : M̃ → ℝ, where σ_{1}^{(T)} are certain truncations of σ_{1}. We refer to [19] for the details of this argument. Let us also point out that often there is only one μ-stationary measure ν on ℙ^{k−1} (see Theorem 1.34).

Remark 1.5. The definition of σ_{1} involved the choice of a norm ∥ · ∥ on ℝ^{k}, which has no influence on the limit in the left hand side of (1.9). In order to see that the right hand side is not sensitive to such a choice either, consider a cocycle σ′_{1} and α′_{μ,ν} which are defined similarly to σ_{1} and α_{μ,ν}, with ∥ · ∥ been replaced by another norm ∥ · ∥′. Then the cocycles σ_{1} and σ′_{1} are cohomologous in the sense that

σ′_{1}(g, ū) − σ_{1}(g, ū) = ϕ(g · ū) − ϕ(ū)

where ϕ(ū) = log(∥u∥′/∥u∥) is a bounded function of ū. Hence

α′_{μ,ν} − α_{μ,ν} = ∫_{G} ∫_{ℙ_{k−1}} ϕ(g · ū) dμ(g) dν(ū) − ∫_{ℙ_{k−1}} ϕ(ū) dν(ū)
= ∫_{ℙ_{k−1}} ϕ d(μ ∗ ν) − ∫_{ℙ_{k−1}} ϕ dν = 0

Definitions 1.6. Hereafter the following definitions are frequently used

(a) Given a probability measure μ on G = SL′_{k}(ℝ), or more generally on an arbitrary locally compact second countable group, denote by grp(μ) (resp. sgr(μ)) the smallest closed subgroup (resp. semi-group) of G with full μ measure. Equivalently, grp(μ) (resp. sgr(μ)) is the closed subgroup (resp. semi-group) generated by supp(μ). If G = grp(μ) we shall say that μ is generating.

(b) A closed (semi)group T of G = SL′_{k}(ℝ) is said to be strongly irreducible, if there does not exist a finite union W = V_{1} ∪ ⋯ ∪ V_{r} of proper linear subspaces of ℝ^{k} with T W ⊆ W. If T is a group strong irreducibility is equivalent to the condition that all finite index subgroups of T act irreducibly on ℝ^{k}.

(c) The notion of strong irreducibility of a (semi)group T ⊆ G = SL′_{k}(ℝ) extends in an obvious way to any linear representation of G. For p ∈ {1, …, k} we shall say that a (semi)group T is strongly p-irreducible if it is strongly irreducible in the natural G-action on the exterior power ∧^{p}ℝ^{k}. A (semi)group T which is strongly p-irreducible for all p = 1, …, k, will be called totally irreducible.

(d) A probability measure μ on G = SL′_{k}(ℝ) is called strongly irreducible/strongly p-irreducible/ totally irreducible if G_{μ} = grp(μ) (equivalently T_{μ} = sgr(μ)) has the corresponding property.

(e) A probability measure ν on the projective space ℙ^{k−1} is called proper, if ν(L̄) = 0 for any proper linear subspace L ⊂ ℝ^{k}.

The key property of strongly irreducible measures is the following

Lemma (*) 1.7 (Furstenberg, [13]). A probability measure μ on G = SL′_{k}(ℝ) is strongly irreducible if and only if all μ-stationary measures ν on ℙ^{k−1} are proper.

Let μ be a fixed strongly irreducible probability measure μ on G = SL′_{k}(ℝ) with finite first moment. Choose an extremal μ-stationary measure ν on ℙ^{k−1}. Since such ν is necessarily proper (Lemma 1.7), taking k unit vectors u_{1}, …, u_{k} with the directions ū_{1}, …, ū_{k} ∈ ℙ^{k−1} being chosen independently with distribution ν, one obtains with probability one a basis for ℝ^{k}. For any such choice u_{1}, …, u_{k} there exists a constant C = C(u_{1}, …, u_{k}) such that for any matrix X ∈ SL′_{k}(ℝ) the norm ∥X∥ can be estimated by

max_{1≤i≤k} ∥Xu_{i}∥ ≤ ∥X∥ ≤ C · max_{1≤i≤k} ∥Xu_{i}∥

Since ν was chosen to be extremal, the process Z_{n}^{ν} is ergodic, so that for a.e. choice of ū_{i}, i = 1, …, k, Birkhoff’s Ergodic Theorem gives

\frac{1}{n} log ∥X_{n} ⋯ X_{1}u_{i}∥ = \frac{1}{n} ∑_{1}^{n} σ(Z_{j}^{ν}) → ∫_{G} ∫_{ℙ_{k−1}} σ_{1} dμ dν = α_{μ,ν}

while

\frac{1}{n} log ∥X_{n} ⋯ X_{1}u_{i}∥ ≤ \frac{1}{n} log ∥X_{n} ⋯ X_{1}∥ ≤ max_{1≤i≤m} \frac{1}{n} log ∥X_{n} ⋯ X_{1}u_{i}∥ + \frac{log C}{n}

Therefore, for strongly irreducible μ with finite first moment one has α_{μ,ν} = λ_{1}(μ) for all extremal μ-stationary measures ν. Since α_{μ,ν} is affine in ν (as well as in μ), it takes the same value λ_{1}(μ) for all μ-stationary measures ν - which form a convex closure of the extremal (i.e. ergodic) ones. Applying the second part of Theorem 1.2, (and taking special care of non-compactness of G, by considering appropriately truncated functions ±σ^{(T)}), one deduces the following result:

Theorem 1.8 (Furstenberg, [13]). Let μ on G = SL′_{k}(ℝ) be a strongly irreducible probability measure with finite first moment. Then

λ_{1}(μ) = α_{μ,ν} = ∫_{G} ∫_{ℙ_{k−1}} log \frac{∥gu∥}{∥u∥} dν(ū) dμ(g)

for any probability measure ν ∈ P(ℙ^{k−1}) satisfying μ ∗ ν = ν. Moreover, for every non-zero vector u ∈ ℝ^{k} with P-probability one the random products X_{n} ⋯ X_{1} satisfy

lim_{n→∞} \frac{1}{n} log ∥X_{n} ⋯ X_{1}u∥ = lim_{n→∞} \frac{1}{n} log ∥X_{n} ⋯ X_{1}∥ = λ_{1}(μ)

Remark 1.9. Note that the statement about vector growth n^{−1} · log ∥S_{n}u∥ → λ_{1}(μ) in the theorem differs, in general, from a similar assertion in Oseledec theorem. For example, if μ is strongly irreducible with λ_{2}(μ) < λ_{1}(μ), which as we shall see is a typical situation, Oseledec theorem states that there is a measurable family of random (i.e. depending on ω) codimension one subspaces E_{2}(ω) such that for P-a.e. ω

∀ u ∈ ℝ^{k} \ E_{2}(ω) :  lim_{n→∞} n^{−1} · log ∥S_{n}(ω)u∥ = λ_{1}(μ)
∀ u ∈ E_{2}(ω) \ {0} :  lim_{n→∞} n^{−1} · log ∥S_{n}(ω)u∥ ≤ λ_{2}(μ) < λ_{1}(ω)

Yet Theorem 1.8 asserts that for every fixed no-zero vector u, for P-a.e. ω one has

lim_{n→∞} n^{−1} · log ∥S_{n}(ω)u∥ → λ_{1}(μ)

This just means that for every non-zero vector u the event {ω | u ∈ E_{2}(ω)} has P-probability zero.

The formula of Theorem 1.8 for λ_{1}(μ) has many important applications, but in general it does not allow explicit computations of λ_{1}(μ) in terms of μ, for it involves auxiliary μ-stationary measures ν on the boundary, and these are generally hard to identify. However, an explicit computation of λ_{1}(μ) is available for some special family of measures μ on G:

Theorem 1.10 (Furstenberg, [13] 7.3). Let μ be a probability measure on G = SL_{k}(ℝ) of the form μ = μ_{1} ∗ m_{K} ∗ μ_{2}, where m_{K} denotes the Haar measure on the maximal compact K = SO(k). Then μ has a unique stationary measure ν = μ_{1} ∗ ν_{0} where ν_{0} is the unique K-invariant probability measure on ℙ^{k−1}, and if μ has finite first moment then

λ_{1}(μ) = ∫_{K} ∫_{K} ∫_{G} ∫_{G} ∫_{G} log \frac{∥g_{1}kg_{2}g′_{1}k′u∥}{∥g′_{1}k′u∥} dμ_{1}(g_{1}) dμ_{1}(g′_{1}) dμ_{2}(g_{2}) dk dk′ (1.10)

for any non zero vector u ∈ ℝ^{k}. For bi-K-invariant measures μ = m_{K} ∗ μ_{0} ∗ m_{K} with finite first moment (1.10) reduces to

λ_{1}(μ) = ∫_{G} ∫_{ℙ_{k−1}} log ∥gu∥ dν_{0}(u) dμ_{0}(g)

Proof. Observe that for any probability measure η on ℙ^{k−1} one has m_{K} ∗ η = ν_{0}, and therefore (μ_{1} ∗ m_{K} ∗ μ_{2}) ∗ η = μ_{1} ∗ ν_{0}. Thus μ_{1} ∗ ν_{0} is the unique μ-stationary measure for μ = μ_{1} ∗ m_{K} ∗ μ_{2}, and (1.10) follows from Theorem 1.8. If μ is bi-K-invariant, then ν_{0} is the unique μ-stationary measure and the second formula follows.

1.4. Non-random filtration associated to μ. The probabilistic approach based on Markov processes on G × ℙ^{k−1}, briefly described above, allows further analysis of vector growth under products of random matrices. In particular, strong irreducibility condition in Theorem 1.8 can be relaxed to just irreducibility, and furthermore, in the general case (of possibly reducible grp(μ)) one has the following:

Theorem 1.11 (Furstenberg-Kifer [19]). Let μ on G = SL′_{k}(ℝ) be a probability measure with finite first moment, and let G_{μ} = grp(μ). Then there exists an integer r with 1 ≤ r ≤ k, a sequence of G_{μ}-invariant subspaces

{0} = L_{r+1} ⊂ L_{r} ⊂ ⋯ ⊂ L_{2} ⊂ L_{1} = ℝ^{k}

and a sequence of real numbers λ̃_{1}(μ) > λ̃_{2}(μ) > ⋯ > λ̃_{r}(μ), with λ̃_{1}(μ) = λ_{1}(μ), such that for every vector u ∈ L_{i} \ L_{i+1} with P-probability one

lim_{n→∞} \frac{1}{n} log ∥X_{n} ⋯ X_{1}u∥ = λ̃_{i}(μ)

Moreover the set Λ̃ = {λ̃_{i} | 1 ≤ i ≤ r} coincides with the set of values of of α_{μ,ν}, where ν ranges over all extremal μ-stationary probability measures on ℙ^{k−1}. For an extremal μ-stationary measure ν, α_{μ,ν} = λ̃_{i}(μ) if and only if ν(\overline{L}_{i} \ \overline{L}_{i+1}) = 1.

Corollary 1.12. There exists a μ-stationary measure ν ∈ P(ℙ^{k−1}) with ν(\overline{L}_{2}) = 0. For any such measure ν, one has α_{μ,ν} = λ̃_{1}(μ) = λ_{1}(μ). If G_{μ} is irreducible then α_{μ,ν} = λ_{1}(μ) for all μ-stationary ν.

Remark 1.13. Note that the theorem describes a non-random filtration L_{r} ⊂ ⋯ ⊂ L_{2} ⊂ L_{1} = ℝ^{k} which differs, in general, from the random (i.e. depending on ω ∈ Ω) filtration provided by the Oseledec theorem. The set of the corresponding exponents Λ̃ = {λ̃_{i}(μ) | 1 ≤ i ≤ r} is contained in the Lyapunov spectrum Λ = {λ_{j}(μ) | 1 ≤ j ≤ k}, but is, typically, smaller. In particular if G_{μ} is irreducible then the filtration is trivial: {0} = L_{2} ⊂ L_{1} = ℝ^{k} and Λ̃ = {λ̃_{1}(μ) = λ_{1}(μ)}.

The key ingredients of the proof of Theorem 1.11 are the following observations:

(i) For a fixed non-zero vector v ∈ ℝ^{k} the upper limit

a(v) = lim sup_{n→∞} \frac{1}{n} log ∥X_{n} ⋯ X_{1}v∥

is P-a.e. constant. This is basically a 0 − 1 law. (ii) For any real a the collection of vectors E(a) = {v ∈ ℝ^{k} | v = 0 or a(v) ≤ a}

is a linear subspace of ℝ^{k}, which is invariant under μ-a.e. g. Hence E(a) is a G_{μ}-invariant subspace.

(iii) Inspecting the discrete drops in dim E(a) as a varies from λ_{1} to −∞, one recognizes the exponents λ̃_{i} and the spaces L_{i}.

1.5. Furstenberg’s condition for positive growth. The mere existence of the top Lyapunov exponent λ_{1}(μ) ≥ 0, or even the formulae in Theorems 1.8 and 1.11, do not give a clear indication whether the growth of the random products S_{n} = X_{n} ⋯ X_{1} is actually exponential (λ_{1}(μ) > 0) or sub-exponential (λ_{1}(μ) = 0). If μ happens to be supported by a compact subgroup of G = SL′_{k}(ℝ), then the random products X_{n} ⋯ X_{1} are bounded and clearly λ_{1}(μ) = 0. In his fundamental work [13] Furstenberg proved that for strongly irreducible μ, this obvious obstacle to exponential growth is the only one. More precisely

Theorem 1.14 (Furstenberg, [13]). Let μ on G = SL′_{k}(ℝ) be a probability measure with finite first moment. Then λ_{1}(μ) > 0 unless, G_{μ} = grp(μ) is not strongly irreducible or compact.

We postpone the proof of Theorem 1.14 to the next section.

Remark 1.15. A remarkable feature of Furstenberg’s condition for λ_{1}(μ) > 0 is that it is not given in terms of μ itself, but rather in terms of the closed subgroup grp(μ) generated by μ. In fact, since both (strong) irreducibility and compactness are algebraic properties (i.e. can be described by polynomial equations in the entries of the matrices) Furstenberg’s condition can be formulated in terms of the Zariski closure H_{μ} = \overline{grp(μ)}^{Z} of grp(μ) (compare this with Theorem 1.25 below).

To illustrate the (strong) irreducibility condition, consider a measure μ with finite first moment on G = SL_{k}(ℝ) such that G_{μ} is reducible. Then G_{μ} is conjugate into

( \begin{matrix}A′_{11} & A′_{12} \\ 0 & A′_{22} \end{matrix} ) , A′_{11} ∈ GL_{k_{1}}(ℝ), A′_{22} ∈ GL_{k_{2}}(ℝ), A′_{12} ∈ M_{k_{1}×k_{2}}(ℝ)

where k = k_{1} + k_{2} and 0 < k_{1}, k_{2} < k. Let μ′_{i} denote the distribution of the A′_{ii} part on GL_{k_{i}}(ℝ). It is not hard to show (cf. [59] or [19]) that in this case

λ_{1}(μ) = max{λ_{1}(μ′_{1}), λ_{1}(μ′_{2})}

that is to say that the growth of the A′_{12}-part is dominated by the maximum of the growths of the A′_{11} and A′_{22} parts. Separate the scalar parts a_{i} = | det A′_{ii}| from the truly non-commutative components

A_{ii} = a_{i}^{−1/k_{i}} · A′_{ii} ∈ SL′_{k_{i}}(ℝ)

and let μ_{i} denote the distribution of A_{ii} on SL′_{k_{i}}(ℝ), i = 1, 2. Since log a_{1} +log a_{2} = 0 one obtains

λ_{1}(μ) = |∫ log a_{1} dμ′_{1}| + max{λ_{1}(μ_{1}), λ_{1}(μ_{2})}

If G_{μ} is contained in (a conjugate of) the upper triangular group P then applying the above argument inductively one concludes that

λ_{1}(μ) = max_{1≤i≤k} |∫ log |g_{ii}| dμ(g)|

where g_{ij} denote the ij-entry of a matrix g ∈ SL_{k}(ℝ). Observe that the map g ↦ (log |g_{11}|, …, log |g_{kk}|) is a homomorphism

ρ : P → ℝ_{0}^{k} = {x ∈ ℝ^{k} | x_{1} + ⋯ + x_{k} = 0}

and λ_{1}(μ) is the ∥ · ∥_{∞} norm of the barycenter of the push-forward measure ρ_{∗}μ on ℝ_{0}^{k}.

1.6. Unitary representation approach I. Positivity of the top Lyapunov exponent λ_{1}(μ) > 0 describes the exponential growth of products of μ-distributed independent random matrices. In the projective action on ℙ^{k−1} such growth corresponds to certain (exponential) contraction, which can further be related to a “spectral gap” of a Markov operator acting on certain space of functions on ℙ^{k−1}. This idea was made precise by Virtzer, who deduced Theorem 1.8 from a spectral gap in the quasiregular unitary representation on L^{2}(ℙ^{k−1}). In fact, the result of Virtzer [67] is more general, in the sense that it gives a sufficient condition for λ_{1}(μ) > 0 not only for products of independent μ-distributed random matrices, but also for products of random matrices satisfying certain condition on their correlations. Here we shall discuss the independent case only:

Theorem 1.16 (Virtzer [67]). Let μ be a probability measure on G = SL′_{k}(ℝ), such that G_{μ} = grp(μ) does not have an invariant probability measure on ℙ^{k−1}. Then there exists a positive constant γ(μ), so that with P-probability one

lim inf_{n→∞} \frac{1}{n} log ∥X_{n} ⋯ X_{1}∥ ≥ γ(μ) > 0  (1.11)

In particular, if μ has a finite first moment then λ_{1}(μ) ≥ γ(μ) > 0.

Proof. Denote by ν_{0} the SO(k)-invariant probability measure on ℙ^{k−1}. One can verify by a direct computation that for g ∈ G = SL′_{k}(ℝ) and every non-zero u ∈ ℝ^{k}

\frac{dg^{−1}ν_{0}}{dν_{0}} (ū) = (\frac{∥gu∥}{∥u∥})^{−k}  (1.12)

This implies the following estimate

∥g∥ = max_{ū} \frac{∥gu∥}{∥u∥} = (min_{ū} \sqrt{\frac{dg^{−1}ν_{0}}{dν_{0}} (ū)})^{−2/k} ≥ (∫_{ℙ_{k−1}} \sqrt{\frac{dg^{−1}ν_{0}}{dν_{0}} (ū)} dν_{0}(ū))^{−2/k} (1.13)

We shall also make use of the following general facts.

Proposition (*) 1.17. Let H be a locally compact group, μ a generating probability measure on H and let X be a compact metric space with a continuous H-action on it, which has no invariant probability measures. Then there exists a positive constant ϵ = ϵ(X, H, μ) > 0, so that for any quasi-invariant probability measure ν on X, the average operator π_{ν}(μ) = ∫ π_{ν}(h) dμ(h) on L^{2}(X, ν) corresponding to the quasi-regular unitary H-representation π_{ν}

π_{ν}(h)f(x) = \sqrt{\frac{dhν}{dν} (x)} · f(h^{−1} · x) (f ∈ L^{2}(X, ν))

has a spectral gap ∥π_{ν}(μ)∥_{sp} ≤ 1 − ϵ, where ∥T ∥_{sp} = lim ∥T^{n}∥^{1/n} denotes the spectral radius of a bounded operator T.

Lemma (*) 1.18 ([11]). Let π be a unitary representation of some group H on a Hilbert space H, μ be a probability measure on H, and {X_{n}, n ≥ 1} be a sequence of independent μ-distributed H-valued random variables. Assume that f_{1}, f_{2} ∈ H are two vectors with the property that ⟨π(h)f_{1}, f_{2}⟩ ≥ 0 for every h ∈ H. Then with P-probability one

lim inf_{n→∞} −\frac{1}{n} log ⟨π(S_{n}(ω))f_{1}, f_{2}⟩ ≥ log \frac{1}{∥π(μ)∥_{sp}} ≥ 0

Now consider the projective action of G_{μ} = grp(μ) on ℙ^{k−1}. Since G_{μ} has no invariant measures, by Proposition 1.17, the quasi-regular G_{μ}-representation π_{ν_{0}} on L^{2}(ℙ^{k−1}, ν_{0}) has a spectral gap ∥π_{ν_{0}}(μ)∥_{sp} < 1. Using the estimate (1.13) and Lemma 1.18, applied to the constant functions f_{1} = f_{2} = 1 ∈ L^{2}(ℙ^{k−1}, ν_{0}), one has for P-a.e. ω ∈ Ω the sequence S_{n} = S_{n}(ω) satisfies

lim inf_{n→∞} \frac{1}{n} log ∥S_{n}∥ ≥ lim inf_{n→∞} −\frac{2}{nk} log ∫_{ℙ_{k−1}} \sqrt{\frac{dS_{n}^{−1}ν_{0}}{dν_{0}} (ū)} dν_{0}(ū)
=  lim inf_{n→∞} −\frac{2}{nk} log ⟨1, π_{ν_{0}}(S_{n}^{−1})1⟩ = lim inf_{n→∞} −\frac{2}{nk} log ⟨π_{ν_{0}}(S_{n})1, 1⟩
≥  \frac{2}{k} log \frac{1}{∥π_{ν_{0}}(μ)∥_{sp}} > 0

Observe that Theorem 1.14 is a corollary of Theorem 1.16: if G_{μ} = grp(μ) is strongly irreducible and there exists a G_{μ}-invariant probability measure ν on ℙ^{k−1}, then ν is proper (Lemma 1.7), while the stabilizers of proper measures in PSL_{k}(ℝ)- action on ℙ^{k−1} are compact (see 1.27 below).

1.7. Unitary representation approach II. In Theorem 1.16 the positivity of the top Lyapunov exponent λ_{1}(μ) > 0 was deduced from a spectral gap in the quasiregular representation of G_{μ}, which followed from the absence of G_{μ}-invariant measures on ℙ^{k−1}. The latter is a manifestation of non-amenability of G_{μ}. It is natural therefore to try to deduce λ_{1}(μ) > 0 directly from non-amenability of G_{μ} which can be characterized by a spectral gap in the regular representation π_{reg} of G_{μ}. This approach, taken up in a joint work with Y. Shalom [11], leads to a lower estimate of λ_{1}(μ) in terms of an intrinsic spectral gap of G_{μ}, which might be easier to compute.

Consider the following general setup: let G be locally compact group with a left invariant Haar measure m_{G}, and assume that G has a left invariant metric d with finite growth δ(G, d) < ∞, where

δ(G, d) = lim sup_{R→∞} \frac{1}{R} log m_{G} \{g ∈ G | d(g, e) < R\}  (1.14)

A probability measure μ on G is said to have a finite first moment (with respect to a metric d) if

∫_{G} d(g, e) dμ(g) < ∞

It follows from Kingman’s subadditive ergodic theorem that for μ with finite first moment there exists a P-a.e. constant finite limit

λ^{(d)}(μ) = lim_{n→∞} \frac{1}{n}d(S_{n}, e)  (1.15)

often called the escape rate of the μ-random walk.

Theorem 1.19 (Furman-Shalom, [11], [63]). Let d be a left invariant metric on a locally compact group G with finite growth δ(G, d) < ∞, and let μ ∈ P(G) be a probability measure on G. Assume that the closed group H = grp(μ) ⊆ G is nonamenable. Then for P-a.e. ω one has

lim inf_{n→∞} \frac{d(S_{n}(ω), e)}{n} ≥ \frac{2}{δ(G, d)} log \frac{1}{∥π_{reg}(μ)∥_{sp}} > 0

where π_{reg}(μ) = ∫ π_{reg}(h) dμ(h) denotes the average operator associated to the left regular H-representation π_{reg} on L^{2}(H, m_{H}). If μ has finite first moment then the escape rate λ^{(d)}(μ) is bounded below by the positive constant on the RHS above.

Example 1.20. Let F_{r} = ⟨a_{1}, …, a_{r}⟩ be a free group of rank r > 1, and G be a locally compact group with a left invariant metric d and growth δ(G, d) < ∞. Then for any discrete embedding j : F_{r} → G and the measure μ on G given by μ(j(a_{i})^{±1}) = 1/2r one has

λ^{(d)}(μ) ≥ \frac{2}{δ(G, d)} log \frac{r}{\sqrt{2r − 1}}

Proof of Theorem 1.19. Consider two unitary H-representations π_{reg} and π′ on L^{2}(H, m_{H}) and L^{2}(G, m_{G}) respectively, defined by

(π_{reg}^{H}(h)f)(h′) = f(h^{−1}h′) (f ∈ L^{2}(H, m_{H})),
(π′(h)f)(g) = f(h^{−1}g) (f ∈ L^{2}(G, m_{G}))

and the associated operators π_{reg}(μ), π′(μ), respectively. The left H-action on G is free and admits a measurable cross section G = H · X with some σ-finite measure η on X, so that m_{G} = m_{H} × η. Writing f ∈ L^{2}(G, m_{G}) as f(h′, x) with h′ ∈ H, x ∈ X, one has π′(h)f(h′, x) = f(h^{−1}h′, x), so that the representation π′ of H is just an integral of π_{reg}-representations π′ = ∫_{X} π_{reg} dη(x), which implies ∥π′(μ)∥ = ∥π_{reg}(μ)∥ and ∥π′(μ)∥_{sp} = ∥π_{reg}(μ)∥_{sp}.

Non-amenability of H implies (basically is equivalent to) the spectral gap condition ∥π_{reg}(μ)∥_{sp} < 1 (see Theorem 2.3 of Derriennic and Guivarc’h). Hence we have

∥π′(μ)∥_{sp} = ∥π_{reg}(μ)∥_{sp} < 1

Let U be some bounded subset of G of positive Haar measure and let f_{U} denote the function m_{G}(U)^{−1} · 1_{U} where 1_{U} is the characteristic function of U. Fix some δ > δ(G, d) and let

f(g) = e^{−δ/2 d(g,e)}

Then both f_{U} and f are in L^{2}(G, m_{G}). Observe that for any g ∈ G one has

− log ⟨π′(g)f, f_{U}⟩ = − log (\frac{1}{m_{G}(U)} ∫_{U} e^{−δ/2 d(g′,g)} dm_{G}(g′))
≤ \frac{1}{m_{G}(U)} ∫_{U} \frac{δ}{2} d(g′, g) dm_{G}(g′) ≤ \frac{δ}{2} (d(g, e) − R)

where R = sup{(.e, g′) | g′ ∈ U}. Using this inequality for g = S_{n}(ω) and applying Lemma 1.18, one has for P-a.e. ω

lim inf_{n→∞} \frac{1}{n} d(S_{n}(ω), e) ≥ \frac{2}{δ} log \frac{1}{∥π′(μ)∥_{sp}} = \frac{2}{δ} log \frac{1}{∥π_{reg}(μ)∥_{sp}}

Since δ can be chosen arbitrarily close to δ(G, d), the theorem is proved.

1.8. Simplicity of the spectrum. Let μ be a probability measure on G = SL′_{k}(ℝ) with a finite first moment. Note that since λ_{1}(μ)+⋯+λ_{k}(μ) = 0 condition λ_{1}(μ) > 0 means that not all of λ_{p}(μ), p = 1, …, k, vanish (i.e. the Lyapunov spectrum is not trivial). An important problem is to obtain conditions on μ which guarantee the simplicity of the top Lyapunov exponent, i.e. λ_{1}(μ) > λ_{2}(μ), and more generally conditions for the simplicity of the whole Lyapunov spectrum:

λ_{1}(μ) > λ_{2}(μ) > ⋯ > λ_{k}(μ)

To state the main results in this direction we shall need the following definitions.

Definitions 1.21. Consider the group G = SL′_{k}(ℝ) and its projective action on ℙ^{k−1} and on ℙ(∧^{p}ℝ^{k}) for p = 1, 2, …, k − 1. A sequence {g_{n}} in G is called

(a) contracting (on ℙ^{k−1}) to ξ ∈ ℙ^{k−1} if the K = SO(k) invariant measure ν_{0} on ℙ^{k−1} is contracted by g_{n} to the Dirac measure δ_{ξ}, i.e. if g_{n}ν_{0} → δ_{ξ} weakly.

(b) p-contracting (to an ξ ∈ ℙ(∧^{p}ℝ^{k})) for some p ∈ {1, …, k−1} if ∧^{p}g_{n} is contracting on ℙ(∧^{p}ℝ^{k}), i.e. if g_{n}ν_{0} → δ_{ξ} weakly, where ν_{0} is the K-invariant measure on ℙ(∧^{p}ℝ^{k}).

A (semi)group T ⊆ G is called

(c) contracting (resp. p-contracting) if T contains a contracting (resp. p-contracting) sequence.

(d) totally contracting if T is p-contracting for all p = 1, 2, …, k − 1.

The measure ν_{0} used in the above definition can be replaced by an arbitrary proper measure (see Lemma 1.30).

Remark 1.22. Let T be a topological semigroup acting continuously on a compact metric space Q.

(a) The T -action on compact metric Q is said to be proximal if for any x, y ∈ Q there exists a sequence g_{n} in T and a point z ∈ Q so that lim_{n→∞} g_{n} · x = lim_{n→∞} g_{n} · y = z. In view of the compactness, the following condition is equivalent to proximality: for any two point set F in Q there exists a sequence g_{n} in T so that lim_{n→∞} diam(g_{n}F) = 0. In fact, it can be shown that in this definition “two point sets F ” can be replaced by “finite sets F ”.

(b) The T -action on a compact metric Q is said to be strongly proximal if for any probability measure ν on Q there exists a sequence g_{n} ∈ T and a point z ∈ Q so that the measures g_{n} · ν converge weakly (with respect to C(Q)^{∗}-topology) to the Dirac measure δ_{z}. Strong proximality implies proximality (consider ν = (δ_{x} + δ_{y})/2), but in general this is a stronger notion.

(c) Let T be a semigroup in SL′_{k}(ℝ) which acts (strongly) irreducibly on ℝ^{k}. Then it can be shown (for example using Lemma 1.30 below) that for the T -action on Q = ℙ^{k−1} the three conditions proximality, strong proximality and contraction are equivalent. In fact, under strong irreducibility condition, these properties are also equivalent to an existence of g ∈ T with a dominant eigenvalue.

Let us point out an important instance of the contraction phenomenon. Let μ be a measure on SL′_{k}(ℝ) with finite first moment and λ_{p}(μ) > λ_{p+1}(μ). Then for P-a.e. μrandom walk S_{n} = X_{n} ⋯ X_{1}, the transposed sequence S_{n}^{t} = X_{1}^{t} ⋯ X_{n}^{t} is contracting on ℙ(∧^{p}ℝ^{k}); while the sequence S_{n} itself is not necessarily contracting, yet always contains contracting subsequences. Indeed, write

S_{n} = U_{n} D_{n} V_{n} S_{n}^{t} = V_{n}^{t}D_{n} U_{n}^{t}

with D_{n} = diag [e^{a_{1}(n)}, …, e^{a_{k}(n)}] and U_{n}, V_{n} ∈ K = SO(k). For P-a.e. ω one has a_{p}(n)/a_{p+1}(n) → ∞ which forces the measure D_{n}ν_{0} = D_{n}U_{n}^{t}ν_{0} = D_{n}V_{n}ν_{0} to converge to the Dirac measure δ_{ξ_{0}} at ξ_{0} = e_{1} ∧ ⋯ ∧ e_{p}. Oseledec Theorem also implies that V_{n}^{t} · ξ_{0} = V_{n}^{−1} · ξ_{0} converges to some subspace ξ(ω) (which is actually E_{p+1}(ω)^{⊥} where ℝ^{k} = E_{1} ⊃ E_{2}(ω) ⊃ … is the Oseledec filtration). Thus S_{n}^{t} is contracting: S_{n}^{t}·ν_{0} → δ_{ξ(ω)}. On the other hand, U_{n}ξ_{0} converges only after passing to a subsequence, which explains why S_{n} itself is not necessarily contracting.

The transposed sequence S_{n}^{t} will indeed play a role in the proof of the following remarkable result of Guivarc’h and Raugi.

Theorem 1.23 (Guivarc’h - Raugi [27]). Let μ be a probability measure on G = SL′_{k}(ℝ) with a finite first moment, Assume that the semigroup T_{μ} = sgr(μ) is strongly p-irreducible. Then T_{μ} is p-contracting iff λ_{p}(μ) > λ_{p+1}(μ). In particular, if T_{μ} is totally irreducible then T_{μ} is totally contracting iff the Lyapunov spectrum of μ is simple:

λ_{1}(μ) > λ_{2}(μ) > ⋯ > λ_{k}(μ)

Remark 1.24. Note the following phenomena emphasized by this result: under the (strong) irreducibility condition a typical random walk S_{n}(ω) (i) necessarily “detects” the contraction property of the ambient semi-group T_{μ}, and (ii) translates the “qualitative” contraction into the quantitative exponential contraction.

Unfortunately it is not easy to verify the condition of Theorem 1.23 because it is given in terms of the semi-group T_{μ}, which is often hard to identify. The next conceptual step was made by Goldsheid and Margulis who proved that in the conditions of Theorem 1.23 one can replace the semi-group T_{μ} by a larger and much more convenient object:

Theorem 1.25 (Goldsheid - Margulis, [23]). Let μ be a probability measure on G = SL′_{k}(ℝ) with finite first moment, and let H_{μ} denote the smallest real algebraic subgroup of G containing supp(μ) (i.e. H_{μ} = \overline{grp(μ)}^{Z}). If H_{μ} is strongly p-irreducible and p-contracting so is T_{μ} = sgr(μ), and therefore λ_{p}(μ) > λ_{p+1}(μ). Hence, if grp(μ) is Zariski dense in G then the Lyapunov spectrum of μ is simple:

λ_{1}(μ) > λ_{2}(μ) > ⋯ > λ_{k}(μ)

Remark 1.26. In a subsequent work [28] Guivarc’h and Raugi gave a general description of the multiplicities of the Lyapunov exponents of μ in terms of the algebraic closure H_{μ} of grp(μ).

Observe that Zariski density condition H_{μ} = G is a relatively weak one. For a typical choice (in the sense of Haar measure, Baire category etc.) of a pair (A, B) ∈ SL_{k}(ℝ) × SL_{k}(ℝ) the group generated by A and B is Zariski dense in SL_{k}(ℝ), and therefore every measure μ with finite first moment and A, B ∈ supp(μ) has a simple Lyapunov spectrum.

We shall outline the proofs of Theorems 1.23 and 1.25 in sections 1.11 and 1.12. The following sections contain some important notions and auxiliary facts needed for these proofs.

1.9. Quasi-Projective transformations and Flag Varieties. The contraction property, which is central to Theorems 1.23 and 1.25, describes limit behavior of a sequence of projective transformations of ℙ^{k−1}. In [13] Furstenberg has introduced a very useful notion of quasi-projective transformations (QP-transformations), which provides a convenient framework for the analysis of the contraction properties.

A transformation b of the projective space ℙ^{k−1} is called quasi-projective (QP) if there exists a sequence of projective transformations Ā_{n} of ℙ^{k−1}, given by some matrices A_{n} ∈ SL′_{k}(ℝ), so that for each ξ ∈ ℙ^{k−1} one has

Ā_{n} · ξ → b · ξ

Let A_{n} ∈ SL′_{k}(ℝ) be an arbitrary sequence. Consider the matrices A′_{n} = ∥A_{n}∥^{−1} · A_{n} which belong to the compact set M_{k×k}^{1} = {A′ ∈ M_{k×k}(ℝ) | ∥A′∥ = 1}. Being scalar multiples of A_{n}, the matrices A′_{n} define the same projective transformations \overline{A}_{n} = \overline{A′}_{n}.

Passing to a convergent subsequence, one can assume that A′_{n} → B_{0}, where B_{0} is a k × k matrix with ∥B_{0}∥ = 1 and det B_{0} = lim_{n→∞} ∥A_{n}∥^{−k}. If A_{n} are bounded, then Ā_{n} → B̄_{0} ∈ PSL_{k}(ℝ) and the process terminates with b = B̄_{0}. Otherwise B_{0} is a singular matrix with ∥B_{0}∥ = 1. Denote L_{0} = ℝ^{k}, L_{1} = Ker B_{0}, k_{1} = dim L_{1}. Note that for ξ ∈ L_{0} \ L_{1} one has

lim_{n→∞} \overline{A}_{n} · ξ = lim_{n→∞} \overline{A′}_{n} · ξ = \overline{B}_{0} · ξ

Next consider the restricted transformations A_{n}^{1} = A_{n}|_{L_{1}} : L_{1} → ℝ^{k}. Let B_{1} : L_{1} → ℝ^{k} be a limit of some convergent subsequence of the transformations ∥A_{n}^{1}∥^{−1} · A_{n}^{1} : L_{1} → ℝ^{k} which, as before, belong to a compact set M_{k_{1}×k}^{1} of k_{1} × k-matrices of norm one. Set L_{2} = Ker B_{1} ⊂ L_{1}. Continuing this procedure, one obtains a sequence of subspaces

ℝ^{k} = L_{0} ⊃ L_{1} ⊃ ⋯ ⊃ L_{s−1} ⊃ L_{r} = {0}

and a sequence of linear transformations {B_{i} : L_{i} → ℝ^{k}}_{i=0}^{r} with L_{i+1} = Ker B_{i} such that for a suitable subsequence A_{n_{j}} of A_{n} one has

lim_{j→∞} Ā_{n_{j}} · ξ = \overline{B_{i}} · ξ for ξ ∈ \overline{L}_{i} \ \overline{L}_{i+1}

Hence, the above limit defines a QP-transformation b = lim_{j→∞} Ā_{n_{j}} of the projective space ℙ^{k−1}.

Let us summarize some important facts which follow from these arguments:

(i) Any sequence of PT-s {Ā_{n}} contains a subsequence converging to a QP-transformation b. Such limit transformations b are called quasi-projective limits (QP-limits) of the sequence {A_{n}}.

(ii) A sequence {A_{n}} in G = SL′_{k}(ℝ) is bounded iff all QP-limits (of its subse-

quences) are actually projective transformations. (iii) Any QP-transformation b of ℙ^{k−1} admits a description by {B_{i} : L_{i} → ℝ^{k}}_{i=0}^{r},

where ℝ^{k} = L_{0} ⊃ L_{1} ⊃ ⋯ ⊃ L_{r} = {0} are nested subspaces, B_{i} : L_{i} → ℝ^{k} are linear transformations with L_{i+1} = Ker B_{i} and ∥B_{i}∥ = 1. In this description b : ℙ^{k−1} → ℙ^{k−1} is given by

b · ξ = B̄_{i} · ξ  for ξ ∈ L_{i} \ L_{i+1}  (1.16)

(It should be pointed out that such a description need not be unique). Conversely, one can show that any system {B_{i} : L_{i} → ℝ^{k}}_{i=0}^{r} with the above properties defines a QP-transformation by (1.16).

Observe that if a QP-limit of b of a sequence A_{n} ∈ SL′_{k}(ℝ) then for any ν ∈ P(ℙ^{k−1}) one has

lim_{n→∞} Ā_{n} · ν = b · ν  (1.17)

Already the first layer B_{0} of b, described by {B_{i} : L_{i} → ℝ^{k}}_{i=0}^{r}, contains important information on {A_{n}}. If A_{n} are unbounded then b is a genuine QP-transformation with L_{1} = Ker B_{0} and R_{1} = Im B_{0} being proper subspaces of ℝ^{k}. Writing ν = ν_{0} + ν_{1} where ν_{0} = ν|L̄_{0} and ν_{1} = ν − ν_{0}, the measure b · ν_{1} is supported on R̄_{1}, while b · ν_{0} = lim_{n→∞} Ā_{n} · ν_{0} is supported on the limit of projective spaces \overline{A_{n}(L_{0})}, which is also a proper projective subspace. This consideration leads to the following:

Lemma 1.27 (Furstenberg [12]). Let A_{n} ∈ G = SL′_{k}(ℝ), and ν, ν′ ∈ P(ℙ^{k−1}) be probability measures so that Ā_{n} · ν → ν′. Then either A_{n} are bounded, or there exist proper subspaces V, W ⊂ ℝ^{k} such that ν′ is supported on \overline{V} ∪ \overline{W}. In particular, the stabilizer Stab_{G}(ν) = {g ∈ G | gν = ν} of any proper measure ν is a compact subgroup of G.

Remark 1.28. Further analysis of QP-limits along similar lines was used by Zimmer (see [72] 3.4.2) to prove that the stabilizer Stab_{PSL_{k}(ℝ)}(ν) of any measure ν ∈ P(ℙ^{k−1}) has a normal subgroup of finite index which is algebraic (recall that compact subgroups of real algebraic groups are algebraic).

Typically, QP-transformations are not continuous. Following Goldsheid-Margulis [23], we shall denote by M_{1}(b) ⊆ ℙ^{k−1} the closure of the set of the discontinuity points of a QP-transformation b, and by M_{0}(b) the b-image of the set of its continuity points. With these definitions, M_{0}(b) is always a projective subspace of ℙ^{k−1}, more precisely

Lemma 1.29 (see [23] 2.8). If M_{0}(b) contains more than one point, then M_{0}(b) = \overline{Im B_{0}} and M_{1}(b) = \overline{L_{1}}. If M_{0}(b) is a single point, then either M_{1}(b) = \overline{L_{i}} for some 1 ≤ i ≤ r, or M_{1}(b) is empty.

The contraction property (Definition 1.21) has the following characterization:

Lemma 1.30 ([27]). Let {A_{n}} be a sequence in G = SL′_{k}(ℝ), and ν be an arbitrary proper probability measure on ℙ^{k−1}. Write A_{n} = U_{n} diag [e^{a_{1}(n)}, …, e^{a_{k}(n)}] V_{n} with U_{n}, V_{n} ∈ O(k) and a_{1}(n) ≥ a_{2}(n) ≥ ⋯ ≥ a_{k}(n). The following conditions are equivalent

(a) {A_{n}} is a contracting sequence with Ā_{n} · ν_{0} → δ_{ξ} where ξ ∈ ℙ^{k−1}.

(b) Ā_{n} · ν → δ_{ξ}.

(c) lim_{n→∞} a_{2}(n)/a_{1}(n) = 0 and lim_{n→∞} Ū_{n} · ē_{1} = ξ

(d) All QP-limits b of {A_{n}} have M_{0}(b) = {ξ}.

Remark 1.31. Note that a semigroup T ⊆ SL′_{k}(ℝ) is p-strongly irreducible and p-contracting iff the transposed semigroup T^{t} = {g^{t} | g ∈ T } has these properties. Indeed strong irreducibility is preserved by the transpose operation, as well as the existence of contracting sequences (use part (c) of the Lemma).

The linear action of G = SL′_{k}(ℝ) on ℝ^{k} defines the corresponding projective actions not only on the projective space, but also on more general flag varieties. Let us recall the basic definitions of the latter. A flag of type τ = (τ_{1}, …, τ_{r}), where 1 ≤ τ_{1} < τ_{2} < ⋯ < τ_{r} ≤ k are integers, is an r-tuple of nested linear subspaces V_{1} ⊂ V_{2} ⊂ ⋯ ⊂ V_{r} of ℝ^{k} with dim V_{i} = τ_{i} for i = 1, …, r. The flag variety F_{τ} of type τ = (τ_{1}, …, τ_{r}) is the collection of all flags of type τ. Typical examples of flag varieties are: F_{(1)} - the projective space ℙ^{k−1}, F_{(p)} - the Grassmannian Gr_{p}, and the full flags F_{(1,2,...,k)}. We shall also use F_{(1,2)} consisting of projective line elements (i.e. a point and a direction through the point) on ℙ^{k−1}. A flag variety F_{τ} forms a closed subset of the compact projective space of the linear space ⊕_{i=1}^{r} ∧^{τ_{i}} ℝ^{k}. This provides it with the natural topology, projective G = SL′_{k}(ℝ)-action and with a notion of QP-transformations. Any fixed Euclidean norm ∥ · ∥ on ℝ^{k} defines the corresponding norms on the exterior products ∧^{p}ℝ^{k}, which allow to define cocycles σ_{1}, …, σ_{k} for the G = SL′_{k}(ℝ)-action on the full flag variety F = F_{(1,...,k)} as follows: for g ∈ G and ξ ∈ F set

σ_{p}(g, ξ) = log \frac{∥gu_{1} ∧ ⋯ ∧ gu_{p}∥}{∥u_{1} ∧ ⋯ ∧ u_{p}∥}  (1.18)

where ξ is the flag (\overline{u_{1}}, \overline{u_{1} ∧ u_{2}}, …, \overline{u_{1} ∧ ⋯ ∧ u_{p}}). One easily checks that:

(i) the cocycles σ_{p} are well defined; (ii) each σ_{p} can be defined on the corresponding Grassmannian σ_{p} : G × F_{(p)} in a

way which is consistent with the natural G-equivariant quotient map F → F_{(p)}; (iii) the definition of σ_{1} coincides with the one given in (1.6).

Next consider the cocycle σ = σ_{2} − 2σ_{1} : G × F_{(1,2)} → ℝ, which can be explicitly defined as

σ(g, ξ) = log \frac{∥gu ∧ gv∥}{∥gu∥^{2}} − log \frac{∥u ∧ v∥}{∥u∥^{2}}  (1.19)

where ξ = (ū, \overline{u ∧ v}) ∈ F_{(1,2)}. One can verify that exp σ(g, ξ) is the dilatation coefficient of the projective action of g on ℙ^{k−1} in the direction of the projective line element ξ.

In the sequel we shall need the following property of contracting sequences: Lemma 1.32 (see [27]). Let {g_{n}} be a sequence in G = SL′_{k}(ℝ) contracting ℙ^{k−1} towards z̄ ∈ ℙ^{k−1}. Then for every line element ξ = (ū, \overline{u ∧ v}) ∈ F_{(1,2)} with u ∉ z^{⊥}, the transposed sequence {g_{n}^{t}} satisfies σ(g_{n}^{t}, ξ) → −∞. 1.10. Some Auxiliary Results. This section contain some basic ingredients of the proof of Theorem 1.23 (providing the grounds for the phenomena pointed out in Remark 1.24).

Th following fact plays a crucial role in Theorem 1.23 and in the notion of (G, μ)- boundaries discussed in the next chapter. Lemma (*) 1.33 (Furstenberg [13], Guivarc’h-Raugi [27]). Let G be a locally compact group, Q be a compact metric G-space, μ a probability measure on G, and let ν be a μ-stationary probability measure on Q. Denote by Y_{1}, Y_{2}, … a sequence of G-valued independent random variables with distribution μ. Then for P-a.e. sequence ω = (Y_{1}, Y_{2}, …) the measures Y_{1}Y_{2} ⋯ Y_{n} · ν converge in the weak topology to a limit probability measure ν_{ω} ∈ P(Q), while ν is the average of these random measures

ν = ∫ ν_{ω} dP(ω)  i.e.  ∫_{Q} f dν = ∫_{Ω} ∫_{Q} f dν_{ω} dP(ω) (f ∈ C(Q))

Moreover, with P-probability one the same limit ν_{ω} is obtained by

Y_{1}Y_{2} ⋯ Y_{n} g · ν → ν_{ω}

for μ_{∗}-a.e. g ∈ G, where μ_{∗} = ∑_{1}^{∞} 2^{−p}μ^{p}.

Note the order of applied transformations Y_{1}Y_{2} ⋯ Y_{n} g · ν, and the fact that the limit Y_{1} ⋯ Y_{n} g · ν = ν_{ω} depends only on ω = (Y_{1}, Y_{2}, …), but not on g. Applying this general fact to the projective action of G = SL′_{k}(ℝ) on Q = ℙ^{k−1} one obtains:

Theorem 1.34 ([27]). Let μ be a probability measure on G = SL′_{k}(ℝ) so that the semi-group T_{μ} = sgr(μ) (equivalently the algebraic group H_{μ} = \overline{grp(μ)}^{Z}) are contracting and strongly irreducible. Then there is a unique μ-stationary measure ν on ℙ^{k−1}; for P-a.e. ω ∈ Ω the limit measure

ν_{ω} = lim_{n→∞} Y_{1}(ω) ⋯ Y_{n}(ω) · ν

is a Dirac measure ν = δ_{z(ω)}, where the random point z(ω) ∈ ℙ^{k−1} has distribution ν. The set L = supp(ν) ⊆ ℙ^{k−1} is the unique minimal set for the T_{μ}-action on ℙ^{k−1}, and T_{μ} acts minimally and strongly proximally on L.

Proof. Let ν be some μ-stationary measure on ℙ^{k−1}. Lemma 1.33 and Fubini theorem ensure that for all ω from a subset Ω_{0} ⊆ Ω with P(Ω_{0}) = 1, one has

Y_{1}(ω) ⋯ Y_{n}(ω)g · ν → ν_{ω} as n → ∞

for μ^{∗}-a.e. g ∈ G. Fix an ω ∈ Ω_{0}, and let b be a QP-limit of some subsequence Z_{n_{i}} of Z_{n} = Y_{1}(ω) … Y_{n}(ω). Since μ is strongly irreducible, ν is a proper measure (Lemma 1.7) and so are g · ν, for every g ∈ G. Hence for μ^{∗}-a.e. g ∈ G one has

ν_{ω} = lim_{i→∞} Z_{n_{i}}g · ν = bg · ν

The relation bg · ν = ν_{ω} clearly holds for all g ∈ supp(μ^{∗}) = T_{μ}. Since T_{μ} contains contracting sequences g_{n} with g_{n} · ν → δw̄ one concludes that ν_{ω} = δ_{b·w̄} is a Dirac measure ν_{ω} = δ_{z(ω)} at some z(ω) ∈ ℙ^{k−1}. In particular b · ν = ν_{ω} = δ_{z(ω)}, and since ν is proper, one has M_{0}(b) = {z(ω)}. This argument applies to all QP-limits b of the sequence Z_{n} = Y_{1}(ω) ⋯ Y_{n}(ω), which means that {Z_{n}} is a contracting sequence with Z_{n} · ν_{0} → δ_{z(ω)}. In particular z(ω) is determined by ω ∈ Ω_{0} only (and not by the choice of the μ-stationary measure ν); and since ν satisfies

ν = ∫ ν_{ω} dP(ω) = ∫ δ_{z(ω)} dP(ω)

the measure ν is just the distribution of z(ω), and hence is unique.

Let L′ ⊆ ℙ^{k−1} be a non-empty closed T_{μ}-invariant set. Then L′ supports a μstationary probability measure ν′, which by uniqueness coincides with ν. Hence L ⊆ L′. This proves that L is the unique T_{μ}-minimal set. The strong proximality assertion for the projective action of T_{μ} on L follows from the contraction property.

We shall also use the following general fact from ergodic theory:

Lemma 1.35 (Kesten [40], also Guivarc’h-Raugi [26]). Let T be a measure preserving transformation of a probability space (X, m) and assume that a function f ∈ L^{1}(X, m) satisfies ∑_{k=0}^{n−1} f(T^{k}x) → −∞ for m-a.e. x ∈ X. Then ∫_{X} f dm < 0 and lim_{n→∞} \frac{1}{n} ∑_{k=0}^{n−1} f(T^{k}x) < 0, for m-a.e. x ∈ X.

This Lemma allows to translate a qualitative divergence to a divergence with a speed. In the context of Markov processes this gives the following

Proposition (*) 1.36 (Guivarc’h - Raugi [27]). Let M be a compact metric space, σ : M → ℝ a continuous function, and P a Markov operator on M. Let θ ∈ P(M) be a P -stationary measure, and let {Z_{n}^{θ}, n ≥ 0} denote the corresponding stationary Markov process on M. Suppose that with probability one

lim_{n→∞} ∑_{k=0}^{n} σ(Z_{k}^{θ}) = −∞  (1.20)

Then with probability one lim_{n→∞} n^{−1} ∑_{k=0}^{n} σ(Z_{k}^{θ}) < 0. Furthermore, if (1.20) holds for all P -stationary measures θ, then there exists a γ < 0 so that

lim_{n→∞} sup_{x∈M} \frac{1}{n} ∑_{k=0}^{n} P^{k}σ(x) = γ < 0

1.11. From contractions to the simplicity of the spectrum. With these preliminaries we can present the proof of Theorem 1.23, following Guivarc’h and Raugi [27].

If λ_{p+1}(μ) > λ_{p}(μ) then T_{μ} is p-contracting as follows from the discussion after Remark 1.22. The main content of the theorem is the sufficiency of p-contracting for λ_{p}(μ) > λ_{p+1}(μ). Clearly, the case of p > 1 can be reduced to the case p = 1 by passing to the G-action on the p-th exterior power ∧^{p}ℝ^{k}. Hence, it is enough to show that if μ has finite first moment and T_{μ} = sgr(μ) is contracting and strongly irreducible, then λ_{1}(μ) > λ_{2}(μ). To avoid estimates we shall assume that μ is compactly supported.

Denote Y_{n} = X_{n}^{t}. Then {Y_{n}} is a sequence of independent random variables with common distribution μ^{t}, where dμ^{t}(g) = dμ(g^{t}). Let ν and ν′ be stationary measures on F_{(1)} = ℙ^{k−1} for μ and μ^{t} respectively. Since T_{μ} is strongly irreducible and contracting the same applies to T_{μ^{t}} = T_{μ}^{t} (Remark 1.31) and Theorem 1.34 shows that ν and ν′ are uniquely determined and proper (Lemma 1.7).

For P-a.e. ω ∈ Ω random walk {S_{n} = X_{n} ⋯ X_{2}X_{1}} the transposed sequence is {S_{n}^{t} = (X_{n} ⋯ X_{1})^{t} = Y_{1} ⋯ Y_{n}}, and Theorem 1.34 implies that

S_{n}^{t} · ν′ = Y_{1} ⋯ Y_{n} · ν′ → δ_{z̄(ω)}

is a Dirac measure at z̄(ω) ∈ ℙ^{k−1}. By Lemma 1.30 the sequence {S_{n}^{t}} is contracting. Since the distribution ν′ of z̄(ω) is proper, for any fixed non-zero vector u ∈ ℝ^{k} with P-probability one u ∉ z(ω)^{⊥}, and therefore by Lemma 1.32 for any projective line element ξ ∈ F_{(1,2)}

σ(S_{n}, ξ) = σ((Y_{1} ⋯ Y_{n})^{t}, ξ) → −∞  (1.21)

with P-probability one, where σ = σ_{2} − 2σ_{1} was defined by (1.19).

Now consider the cocycle σ as a function on the compact (!) space M = supp(μ) × F_{(1,2)}, equipped with the Markov operator P̃

P̃f(g, ξ) = ∫ f(g′, g′ · ξ) dμ(g′)

All P̃-stationary measures on M are of the form μ × η where η is a μ-stationary measure on F_{(1,2)} (Lemma 1.4). Fix such η and denote by {Z_{n}^{η} = (X_{n}, ξ_{n}), n ≥ 0} the corresponding stationary Markov process on M. Then (1.21) implies that with probability one

∑_{j=0}^{n−1} σ(Z_{j}^{η})) = ∑_{j=0}^{n−1} σ(X_{j+1}, S_{j} · ξ) = σ(S_{n}, ξ) → −∞

and this property holds for all μ-stationary η. Proposition 1.36 gives

lim_{n→∞} sup_{(g,ξ)∈M} \frac{1}{n} ∑_{j=0}^{n} P̃^{j}σ(g, ξ) = γ < 0

which, in view of the cocycle property of σ, yields

lim_{n→∞} sup_{ξ∈F_{(1,2)}} \frac{1}{n} ∫_{G} σ(g, ξ) dμ^{n}(g) = γ < 0  (1.22)

Denoting by ν_{0} the K = SO(k)-invariant probability measure on F_{(1,2)} one can check that there exists a constant C, so that for any g ∈ G

log ∥g ∧ g∥ ≤ C + ∫ σ_{2}(g, ξ) dν_{0}(ξ)

Since obviously log ∥g∥ = sup_{ξ} σ_{1}(g, ξ) ≥ ∫ σ_{1}(g, ξ) dν_{0}, one has

∫_{G}(log ∥g ∧ g∥ − 2 log ∥g∥)dμ^{n}(g) ≤ C + ∫ ∫ σ(g, ξ) dν_{0}(ξ) dμ^{n}(g)
≤ C + sup_{ξ∈F_{(1,2)}} ∫_{G} σ(g, ξ) dμ^{n}(g)

Dividing by n and taking n → ∞, one finally obtains from (1.22) that

λ_{2}(μ) − λ_{1}(μ) = (λ_{1}(μ) + λ_{2}(μ)) − 2λ_{1}(μ) ≤ γ < 0

Thereby proving the theorem.

The contraction property λ_{1} > λ_{2}, established by the theorem, has the following quantitative form

Proposition 1.37 (LePage [46]). Under the assumption of Theorem 1.23 one has

lim_{n→∞} sup_{ū_{1}≠ū_{2}∈ℙ^{k−1}} \frac{1}{n} ∫_{G} log \frac{δ(g · ū_{1}, g · ū_{2})}{δ(ū_{1}, ū_{2})} dμ^{n}(g) < 0

where δ(·, ·) is the natural metric on ℙ^{k−1}, given by δ(ū_{1}, ū_{2}) = ∥u_{1} ∧ u_{2}∥/∥u_{1}∥ · ∥u_{2}∥.

Proof. The Proposition follows from (1.22) and the following identity, satisfied by all ū_{1} ≠ ū_{2} ∈ ℙ^{k−1} and all g ∈ G = SL′_{k}(ℝ)

log \frac{δ(g · ū_{1}, g · ū_{2})}{δ(ū_{1}, ū_{2})} = log \frac{∥gu_{1} ∧ gu_{2}∥ · ∥u_{1}∥ · ∥u_{2}∥}{∥gu_{1}∥ · ∥gu_{2}∥ · ∥u_{1} ∧ u_{2}∥} = \frac{σ(g, ξ_{1}) + σ(g, ξ_{2})}{2}

where ξ_{i} = (\overline{u_{i}}, \overline{u_{1} ∧ u_{2}}), i = 1, 2.

1.12. Zariski closures and the contraction properties. This section contains an outline of the proof of Theorem 1.25. For the details the reader is referred to the original paper [23] (see also [21]).

Real algebraic groups have the property that algebraic closure of a sub-semigroup forms an algebraic subgroup. Thus Theorem 1.25 can be deduced from

Theorem 1.38 (Goldsheid-Margulis, [23]). Let H be an algebraic closure of a semigroup T ⊆ SL_{k}(ℝ). If H is p-strongly irreducible and p-contracting, then the same applies to the semigroup T.

Passing to the p-th exterior power, the Theorem is reduced to the case p = 1. Since (strong) irreducibility is preserved by algebraic closures, it suffices to prove that if a semigroup T ⊆ SL_{k}(ℝ) is strongly irreducible and its algebraic closure H contains contracting sequences then so does T itself. The proof of this fact is based on the analysis of QP-limits (on ℙ^{k−1})) of sequences in T. In general, given a semigroup T ⊆ G = SL_{k}(ℝ) we shall denote by \overline{T}^{qp} the collection of all QP-limits (on ℙ^{k−1}) of sequences in T. The set \overline{T}^{qp} will be called the QP-closure of T.

Lemma 1.39 ([23] 2.7, 2.10). The QP-closure \overline{T}^{qp} of a semigroup T ⊆ SL_{k}(ℝ) forms a semigroup, which is closed with respect to pointwise convergence.

In order to show that T contains a contracting sequence, it is enough (Lemma 1.30) to find a QP-limit b ∈ \overline{T}^{qp} with M_{0}(b) ⊆ ℙ^{k−1} being a point. Assuming that this is not the case, let d > 0 denote the minimal (projective) dimension of the spaces M_{0}(b) as b varies over \overline{T}^{qp}; and let Q ⊆ \overline{T}^{qp} consist of all b ∈ \overline{T}^{qp} with M_{0}(b) achieving this minimal dimension d.

Claim (*) 1.40. For any b ∈ Q either M_{0}(b) ⊆ M_{1}(b) or M_{0}(b) ∩ M_{1}(b) = ∅.

Claim (*) 1.41. There exist b ∈ Q with M_{0}(b) ∩ M_{1}(b) = ∅.

QP-transformations b with M_{0}(b) ∩ M_{1}(b) = ∅ enjoy the following useful property

Lemma 1.42 ([23] 2.9). Let b be a QP-transformation with M_{0}(b) ∩ M_{1}(b) = ∅. Denote by V the linear subspace generated by the lines in M_{0}(b). Then the restriction of b to \overline{V} is an invertible projective transformation β(b) ∈ PGL(V).

Now fix a QP-transformation b ∈ Q with disjoint M_{0}(b) and M_{1}(b), and consider the set

H(b) = {h ∈ H | M_{0}(bh̄) ∩ M_{1}(bh̄) = ∅}

Let Φ and Φ_{0} denote the semi-groups of all QP-transformations generated by {bh̄ | h ∈ H(b)} and {bh̄ | h ∈ H(b) ∩ T } respectively, and denote by V ⊂ ℝ^{k} the subspace spanned by the directions of M_{0}(b), i.e. M_{0}(b) = \overline{V}. By Lemma 1.42, for each ϕ in the generating set of Φ, there is a uniquely defined element β(ϕ) ∈ PGL(V) such that the actions of ϕ and β(ϕ) agree on M_{0}(b) = \overline{V}. Extending β to the whole semigroup Φ one obtains a homomorphism β : Φ → PGL(V). The next crucial claim is:

Claim (*) 1.43. The (semigroup) homomorphism β : Φ → PGL(V) maps Φ_{0} ⊂ Φ to a relatively compact semigroup β(Φ_{0}) of PGL(V).

Hence the closure K of β(Φ_{0}) in PGL(V) forms a compact semigroup of the real algebraic group PGL(V). It is therefore a real algebraic subgroup of PGL(V), due to the following general facts: (i) any compact sub-semigroup of a topological group forms a (compact) subgroup, and (ii) any compact subgroup of a real algebraic group is a real algebraic subgroup.

The correspondence h ↦ β(bh̄) of H(b) to PGL(V) is a rational map. It maps T ∩ H(b) into the algebraic subgroup K ⊂ PGL(V). Since T is Zariski dense in H, one concludes that β(bh) ∈ K for all h ∈ H(b). This is used to reach a contradiction to the assumption that H is contracting, by showing that H(b) contains a sequence h_{n} with a QP-limit b′ = lim_{n→∞} h̄_{n} such that M_{0}(bb′) has strictly smaller dimension than the dimension d > 0 of M_{0}(b), contradicting the minimality of d. This completes the sketch of the proof of Theorem 1.25. For the proofs of the claims 1.40–1.43 see section 4.

1.13. Regularity of the Lyapunov spectrum. Consider the regularity properties of the maps

μ ↦ λ_{1}(μ) μ ↦ (λ_{1}(μ), λ_{2}(μ), …, λ_{k}(μ))

which are defined for all μ ∈ P(SL_{k}(ℝ)) with finite first moments. It is not difficult to see that there is a lower semi-continuity in the following sense Lemma 1.44. Let μ_{n} → μ be a weakly convergent sequence of probability measures on G = SL_{k}(ℝ), such that μ and μ_{n} have finite first moments. Then

lim sup_{n→∞} λ_{1}(μ_{n}) ≤ λ_{1}(μ)

Proof. By Kingman’s subadditive ergodic theorem, one has

λ_{1}(μ) = inf_{p∈ℕ} \frac{1}{p} L(μ^{p}) where L(μ) = ∫_{G} log ∥g∥ dμ(g)

For every fixed p ∈ ℕ, one has L(μ_{n}^{p}) → L(μ^{p}) as n → ∞, hence the lemma follows from the general inequality lim sup_{n→∞} inf_{p} L(μ_{n}^{p}) ≤ inf_{p} lim_{n→∞} L(μ_{n}^{p}).

However, in general, one should not expect the map μ ↦ λ_{1}(μ) to be continuous:

Example 1.45. Let G = SL_{2}(ℝ) and consider the matrices

A = ( \begin{matrix}2 & 0 \\ 0 & 1/2 \end{matrix} ) F = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )

For 0 < t ≤ 1 let μ_{t} be defined by μ_{t}({A}) = t and μ_{t}({F }) = 1 − t. Obviously λ_{1}(μ_{1}) = log 2, but one can check that λ_{1}(μ_{t}) = 0 for any t < 1. To see the latter fact intuitively, observe that while A expands and contracts the vectors e_{1}, e_{2} by the factor of 2, F flips their directions. Applying long random product S_{n} = X_{n} ⋯ X_{1}, which contains many A-s and some roughly equally spaces flips F, to vectors e_{1} and e_{2}, each of these vectors experiences long alternating periods of expansion and contraction resulting in a sub-exponential growth.

Note that in the above example the group G_{μ_{1}} is reducible.

Theorem 1.46 (Furstenberg - Kifer, [19]). Let μ_{n} → μ be a weakly convergent sequence of probability measures on G = SL_{k}(ℝ), such that μ_{n} have uniformly bounded first moments

sup_{n} ∫_{G} log ∥g∥ dμ_{n}(g) < ∞

Assume that G_{μ} = grp(μ) has at most one invariant subspace in its linear action on ℝ^{k}. Then λ_{1}(μ_{n}) → λ_{1}(μ).

Proof. Here we shall give the proof for the case of an irreducible G_{μ}, and refer to [19] for the case of a single invariant subspace and further results.

Theorem 1.11 in particular implies that for each n there exists a μ_{n}-stationary measure ν_{n} ∈ P(ℙ^{k−1}) such that λ_{1}(μ_{n}) = α_{μ_{n},ν_{n}}. Due to the compactness of P(ℙ^{k−1}) one can assume that some subsequence ν_{n_{i}} converges weakly to a probability measure ν on ℙ^{k−1}. This limit measure ν is necessarily μ-stationary, because

ν = lim_{i→∞} ν_{n_{i}} = lim_{i→∞} μ_{n_{i}} ∗ ν_{n_{i}} = μ ∗ ν

The uniform bound on the first moments of μ_{n} enables to apply Lebesgue dominated convergence theorem to deduce that

lim_{i→∞} λ_{1}(μ_{n_{i}}) = lim_{i→∞} α_{μ_{n_{i}},ν_{n_{i}}} = α_{μ,ν}

In general the right hand side gives just a lower bound for λ_{1}(μ) (cf. Lemma 1.44), but under the assumption of irreducibility of the G_{μ}-representation on ℝ^{k}, Theorem 1.11 states that λ_{1}(μ) = α_{μ,ν}, and therefore λ_{1}(μ_{n_{i}}) → λ_{1}(μ). This argument shows that any subsequence of {μ_{n}} contains a sub-subsequence with λ_{1}(μ_{n_{i_{j}}}) → λ_{1}(μ), which means that lim_{n→∞} λ_{1}(μ_{n}) = λ_{1}(μ).

There exists a variety of results on regularity of the Lyapunov spectrum under various assumptions on μ (cf. LePage [45], Ruelle [61] etc.) Let us mention the following regularity result, due to Y. Peres, for measures with a fixed finite support:

Theorem 1.47 (Peres, [57]). Let S = {A_{1}, …, A_{a}} be some fixed finite set of matrices in G = SL_{k}(ℝ). Consider all probability measures μw̄ with supp(μ) = S and weights μw̄(A_{i}) = w_{i} > 0 where w̄ = (w_{1}, …, w_{a}) is a probability vector. If λ_{p}(μw̄) is a simple Lyapunov exponent, i.e. λ_{p−1}(μw̄) > λ_{p}(μw̄) > λ_{p+1}(μw̄), then locally λ_{p}(μw̄) is a real analytic function of the weights (w_{1}, …, w_{a}). In particular, if S = {A_{1}, …, A_{a}} generates a Zariski dense subgroup of G, then all λ_{p}(μw̄), p = 1, …, k − 1, are locally real analytic functions of the weights w̄.

The proof of this theorem relies on Theorem 1.49 below.

1.14. Further Limit Theorems. So far the discussion of μ-random products S_{n} = X_{n} ⋯ X_{1} focused on forms of Laws of Large Numbers for matrix products. Other classical limit theorems such as (Functional) Central Limit Theorem, Iterated Logarithm Law, Large Deviation results etc. were also proved in the setting of matrix products. Main results in this direction were obtained by the works of LePage, Guivarc’h, Raugi, Bougerol, Goldsheid ([46], [27], [22]) after earlier results of Furstenberg-Kesten [18], Tutubalin [64] and Virtzer [68]. Here we shall briefly state some of these results and refer the reader to [27], [22], [4], [46] for details and further results.

Finer Limit theorems typically require stronger integrability assumption than Laws of Large Numbers. In the context of random matrices we shall impose the following condition: a probability measure μ on G = SL′_{k}(ℝ) is said to have finite exponential moment if for some ϵ > 0

∫_{G} ∥g∥^{ϵ} dμ(g) < ∞  (1.23)

Theorem 1.48 (LePage, [46]). Let μ be a probability measure on G = SL_{k}(ℝ). Assume that μ has finite exponential moment, is strongly irreducible and contracting on ℙ^{k−1}. Then there exists a constant σ > 0 so that for any fixed u ∈ ℝ^{k} \ {0} one has Central Limit Theorem: The random variables

\frac{log ∥S_{n}(ω)∥ − nλ_{1}}{σ\sqrt{n}}  and  \frac{log ∥S_{n}(ω)u∥ − nλ_{1}}{σ\sqrt{n}}

converge in distribution to the standard Gaussian distribution N(0, 1) on ℝ. Functional CLT: The linear interpolation functions Σ_{n}(ω, t), t ∈ [0, 1], determined

by the discrete values

Σ_{n}(ω, m/n) = \frac{log ∥S_{m}(ω)u∥ − mλ_{1}}{σ\sqrt{n}} (m = 0, …, n)

converge in distribution to the canonical Wiener measure on C([0, 1]). Joint Distribution: The ℝ × ℙ^{k−1}-random variables

(\frac{log ∥S_{n}u∥ − nλ_{1}}{σ\sqrt{n}} , S_{n} · ū)

converge in distribution to N(0, 1) × ν with an error estimate of C/\sqrt{n}. Iterated Logarithm Law: With P-probability one the sequences

\frac{log ∥S_{n}(ω)∥ − nλ_{1}}{σ\sqrt{n log log n}}  and  \frac{log ∥S_{n}(ω)u∥ − nλ_{1}}{σ\sqrt{n log log n}}

have the interval [−1, 1] as the set of cluster points.

Proofs of these results go far beyond the scope of this paper. Here let us just point out the main ideas linking these results on products of random matrices to, by now well established, techniques in the classical probability theory, and refer the reader to the above mentioned papers for the full exposition of these ideas. The main device used in the proof of Theorem 1.48 is the family

P_{s}ϕ(ū) = ∫_{G} e^{i s·σ_{1}(g,ū)} ϕ(g · ū) dμ(g)

of operators acting on a certain space L_{α} of Hölder functions ϕ on ℙ^{k−1}. (The relevance of these operators become clear from the fact that P_{s}^{n}1(ū) is the Fourier transform of the distribution of log ∥S_{n}(ω)u∥/∥u∥). The finite exponential moment assumption (1.23) is needed for the following key result

Theorem 1.49 (LePage, [46]). Let μ be a strongly irreducible and contracting probability measure on SL′_{k}(ℝ), satisfying (1.23). Then there exists an α_{0} > 0 such that for any α ∈ (0, α_{0}) one has

lim_{n→∞} ( sup_{ū_{1}≠ū_{2}∈ℙ^{k−1}} ∫_{G} (\frac{δ(g · ū_{1}, g · ū_{2})}{δ(ū_{1}, ū_{2})})^{α} dμ^{n}(g))^{1/n} = ρ < 1

This Theorem is a strengthening of Proposition 1.37, and it follows from the corresponding strengthening of (1.22) which can be proved under condition (1.23).

Second major ingredient is the following Theorem 1.50 (see Guivarc’h-Raugi [27] pp. 44–45, Goldsheid-Guivarc’h [22] section 4). Let μ satisfy the assumptions of Theorem 1.49. If a function ψ ∈ L_{α} satisfies the equation

e^{i s·σ_{1}(g,ū)} = e^{iθ}ψ(g · ū)/ψ(ū)

for some s, θ ∈ ℝ and all g ∈ supp(μ), then s = 0, e^{iθ} = 1 and ψ is a constant.

Theorems 1.49 and 1.50 together allow to prove certain spectral gap properties of the operators P_{s} acting on the Hölder space L_{α} (see [27] and [4] Chapter V), and this spectral gap property is used to derive the limit theorems.

Similar techniques allow to prove multidimensional Limit Theorems (replacing projective spaces by more general flag varieties). More precisely, if S_{n} = X_{n} ⋯ X_{1} is written in a polar form as S_{n} = U_{n}A_{n}V_{n} where U_{n}, V_{n} ∈ Ø(k) and A_{n} = diag[e^{a_{1}(n)}, … e^{a_{k}(n)}] with a_{1} ≥ ⋯ ≥ a_{k}, consider the full flags U_{n}, V_{n} ∈ F = F_{(1,2,...,k)} defined by

U_{n} = U_{n} · (\overline{e_{k}}, \overline{e_{k} ∧ e_{k−1}}, …, \overline{e_{k} ∧ ⋯ ∧ e_{1}})
V_{n} = V_{n}^{−1} · (\overline{e_{1}}, \overline{e_{1} ∧ e_{2}}, …, \overline{e_{1} ∧ ⋯ ∧ e_{k}})

These flags are well defined if a_{1}(n) > a_{2}(n) > ⋯ > a_{k}(n) as it will be the case in the theorem below, due to Theorem 1.25 which guarantees the simplicity of the Lyapunov spectrum

Λ = (λ_{1}(μ) > λ_{2}(μ) > ⋯ > λ_{k}(μ))

Theorem 1.51 (Goldsheid - Guivarc’h [22], see also Guivarc’h -Raugi [27] ). Let μ be a probability measure on SL_{k}(ℝ). Assume that μ has finite exponential moment (1.23) and supp(μ) generates a Zariski dense subgroup in SL_{k}(ℝ). Then

1. With probability one: lim_{n→∞} n^{−1} log A_{n} = Λ.

2. The random vectors ϵ_{n} = n^{−1/2}(log A_{n} − nΛ) converge in distribution to a Gaussian distribution N with the full support on ℝ_{0}^{k} = {x ∈ ℝ^{k} | x_{1} + ⋯ + x_{k} = 0}.

3. The sequence of flags {V_{n}} converges with probability one to a flag V in F. The distribution of the limit point V is the unique μ̌-stationary measure ν′ on F.

4. The sequence of flags {U_{n}} converges in law to the unique μ-stationary measure ν on F.

5. The three random variables V_{n}, ϵ_{n}, U_{n} are asymptotically independent, i.e.

lim_{n→∞} P{U_{n} ∈ E, ϵ_{n} ∈ F, V_{n} ∈ G} = ν(E) · N(F) · ν′(G).

Part (a) in the theorem is the conclusion of Theorem 1.25 which is stated here for completeness. Statements (c) and (d) are deduced from the Oseledec Theorem and the multidimensional analogues of Theorem 1.34. The main assertions of the Theorem are (b), stating that the limit Gaussian distribution is non degenerate, i.e. is not supported on a proper subspace of ℝ_{0}^{k}, and (d) which describes the limit joint distribution. In [22] Goldsheid and Guivarc’h further prove that if μ has finite exponential moment and the algebraic closure H_{μ} of G_{μ} is semisimple (but is possibly smaller than SL_{k}(ℝ)) then the dimension of the limit Gaussian distribution of the diagonal part A_{n} is the real rank of H_{μ}. A cohomological equation as in Theorem 1.50 plays a crucial role in the analysis of the dimension of the Gaussian distribution (see [22] section 4).

1.15. Additional Remarks. The phenomena discussed in this section depend heavily on the assumption that {X_{n}} is a sequence of independent identically distributed matrices, rather than a general stationary sequence of matrices. In a recent paper [44] Kifer studied a mixed model, where {μ_{n}} is some stationary process taking values in P(SL_{k}(ℝ)), and X_{n} are matrices chosen independently according to the “random” distribution μ_{n}. Kifer has generalized many of the results of this section to this more general setting.

Some of the techniques and ideas in the study of products of random matrices found applications in other areas unrelated to random transformations. Let us mention here some of these applications.

• The important phenomenon behind Lemma 1.27 led Furstenberg to a short and elegant proof [16] of Borel’s Density Theorem.

• Similar phenomena were exploited by Zimmer to prove the very important property of tameness of group actions on spaces of probability measures, arising from algebraic actions on projective varieties ([72] section 3.2).

• Non-vanishing of Lyapunov exponents, using a spectral approach similar to to the one described in section 1.6, was used in the original proof of Margulis’ superrigidity (see [54] V.4).

• There exist “random walk” approaches to the proofs of superrigidity theorems of Margulis and Zimmer (see Furstenberg [17] and Margulis [54] chapter VI).

• In [24] Guivarc’h used random walks on SL_{k}(ℝ) to investigate properties of linear groups, in particular giving a shorter proof of the Tits alternative.

• Recently, Abels, Margulis and Soifer [1] applied quasi-projective transformations (in a quantitative form) to an investigation of affine linear groups in the context of Auslander conjecture.

2. Random Walks on General Groups

We turn now to a discussion of random walks S_{n} = X_{n} ⋯ X_{1} on general countable discrete and locally compact groups G.

2.1. Recurrence of random walks. Let Γ be a countable discrete group with a generating symmetric probability measure μ. Let S_{n} = X_{n} … X_{1} denote a path of μ-generated random walk. If S_{n} = e for some n ≥ 1 with probability one then S_{n} returns to e infinitely often with probability one. In this case the μ-random walk is called recurrent. Otherwise, the probability of infinitely many returns to e is zero and the μ-random walk is called transient. It is well known that the simple random walk on ℤ^{k} is recurrent iff k = 1 or k = 2. The following is a far reaching generalization of this classical fact:

Theorem 2.1 (Varopoulos, [65]). Let μ be a symmetric finitely supported generating probability measure on a countable discrete group Γ. The μ-random walk is recurrent on Γ if and only if Γ contains a subgroup of finite index Γ_{0} which is isomorphic to either ℤ or to ℤ^{2}.

Hence for most infinite discrete groups symmetric random walks are transient. One may consider the quantitative characteristics of the transience behavior. One of such characteristics is the (exponential) rate of decay of the return probabilities

p_{n} = P{ω | S_{n}(ω) = e} = μ^{n}({e})

Clearly p_{n+m} ≥ p_{n} · p_{m} for all n, m ∈ ℕ, which allows to consider the limit of (p_{n})^{1/n} as n → ∞, provided p_{n} do not vanish. For a symmetric measure μ one always has p_{2n} > 0 and therefore the limit

R(Γ, μ) = lim_{n→∞}(p_{2n})^{1/2n}

is a well defined quantity R(Γ, μ) ∈ (0, 1].

Theorem 2.2 (Kesten, [38], [39]). Let μ be a symmetric generating probability measure on a countable discrete group Γ. Then R(Γ, μ) = ∥π_{reg}(μ)∥ = ∥π_{reg}(μ)∥_{sp}, where π_{reg} denotes the regular Γ-representation on l^{2}(Γ). In particular, R(Γ, μ) < 1 if and only if the group Γ is non-amenable.

Proof. Consider the semi-contraction P = π_{reg}(μ) on l^{2}(Γ). Since μ is symmetric P is self-adjoint and ∥P ∥ = ∥P ∥_{sp}. Note that for any γ ∈ Γ, the action of P on the Dirac function δ_{γ} ∈ l^{2}(Γ) satisfies

∥P^{n}δ_{γ}∥^{2} = ⟨P^{n}δ_{γ}, P^{n}δ_{γ}⟩ = ⟨P^{2n}δ_{γ}, δ_{γ}⟩ = ⟨P^{2n}δ_{e}, δ_{e}⟩ = p_{2n}

Hence

R(Γ, μ) = lim_{n→∞}(p_{2n})^{1/2n} = lim_{n→∞} ∥P^{n}δ_{e}∥^{1/n} ≤ ∥P ∥

On the other hand for any finitely supported f = ∑_{i=1}^{N} a_{i}δ_{γ_{i}} one has

lim_{n→∞} ∥P^{n}f∥^{1/n} ≤ lim_{n→∞} (∑_{i=1}^{N} |a_{i}|\sqrt{p_{2n}})^{1/n} = R(Γ, μ)

which implies ∥P ∥ ≤ R(Γ, μ) using the spectral theorem.

Kesten’s theorem can also be stated for locally compact groups, in which case one should consider the probabilities μ^{n}(V) of the returns to bounded neighborhoods V of the identity. In the statement of Kesten’s theorem we have used the fact that condition ∥π_{reg}(μ)∥_{sp} = 1 is equivalent to amenability of the group, generated by μ. The following is a more standard definition of amenability of a locally compact group G: there exists a (left) invariant mean (i.e. positive, normalized, finitely additive functional) on the space BC(G) of bounded continuous functions on G.

Theorem 2.3 (Derriennic-Guivarc’h [7]). Let G be a locally compact group and μ be a probability measure on G with grp(μ) = G. Then G is amenable if and only if ∥π_{reg}(μ)∥_{sp} = 1.

Note that in this result the generating measure μ is not assumed to be symmetric. The proof is similar to that of Proposition 1.17 and Theorem 3.17 below (actually, in the proofs of the latter facts we used an idea from [7] of Derriennic-Guivarc’h).

Returning to the context of discrete countable groups, let Γ be a finitely generated group and S be a generating set. The associated left invariant word metric on Γ is defined by

d_{S}(g_{1}, g_{2}) = min{n ∈ ℕ | g_{2}^{−1}g_{1} ∈ ({e} ∪ S ∪ S^{−1})^{n}}

The corresponding growth function is V_{S}(n) = |{g ∈ Γ | d_{S}(g, e) ≤ n}|. It is well known (exercise) that if S and T are two finite generating sets for Γ then the metrics d_{T} and d_{S} are bi-Lipschitz equivalent, i.e. there exists a constant 0 < c < ∞ so that c^{−1} · d_{S} ≤ d_{T} ≤ c · d_{S}. Let us say that two functions ϕ, ϕ′ : ℕ → ℝ_{+} are roughly equivalent (notation ϕ ≃ ϕ′) if there exists a constant 0 < c < ∞ so that

c^{−1} · ϕ ([n/c]) ≤ ϕ′(n) ≤ c · ϕ([cn])

The above bi-Lipschitz equivalence of word metrics implies that the growth functions V_{S}(n) and V_{T} (n) are roughly equivalent, and their rough equivalence class can be denoted by V_{Γ}(n).

In a recent paper Pittet and Saloff-Coste [58], in particular, proved that return probability functions ϕ_{μ}(n) = μ^{2n}({e}) are also independent of μ, up to rough equivalence. More precisely,

Theorem 2.4 (Pittet Saloff-Coste, [58]). For any finitely generated group Γ and any two symmetric finitely supported generating probability measures μ, μ′ on Γ the return probability functions ϕ_{μ}(n) and ϕ_{μ′}(n) are roughly equivalent: ϕ_{μ} ≃ ϕ_{μ′}.

Remarks 2.5. (a) Non amenable groups Γ have exponential growth V_{Γ}(n) ≃ e^{n} (but there exist many amenable, even polycyclic, groups of exponential growth). Kesten’s theorem states that Γ is non-amenable iff ϕ_{Γ}(n) ≃ e^{−n}.

(b) Polycyclic groups which are not virtually nilpotent have exponential growth V_{Γ}(n) ≃ e^{n} (Milnor - Wolf), and have ϕ_{Γ} ≃ e^{−n^{1/3}} (Varopoulos).

(c) Virtually nilpotent groups Γ are characterized (Gromov) by polynomial growth V_{Γ}(n) ≃ n^{d}. For these groups Varopoulos showed that ϕ_{Γ}(n) ≃ n^{−d/2}.

(d) Grigorchuck constructed groups with intermediate growth V_{Γ}(n) (which is slower than any exponential, but faster than any polynomial). For these groups the above result is particularly interesting since the type ϕ_{Γ}(n) is still unknown.

(e) Pittet and Saloff-Coste also prove that the rough equivalence class of the return probability function is a quasi-isometric invariant of groups.

2.2. Harmonic functions and (G, μ)-spaces. Let G be a locally compact group, and μ be a probability measure on G. It will be convenient, although not necessary, to impose the following regularity assumption on μ

Definition 2.6. We shall say that a probability measure μ on a locally compact second countable group G is admissible if sgr(μ) = G and μ is spread out, which means that some convolution power μ^{p} of μ is not singular with respect to the Haar measure on G.

Given a a locally compact group G with an admissible probability measure μ on it, one can study the Markov operator P

(P f)(g) = ∫_{G} f(gg′) dμ(g′)

on various spaces of functions on G (subject to an appropriate integrability condition). Functions f fixed by P are called μ-harmonic, and are characterized by the μ-mean value property:

f(g) = ∫_{G} f(gg′) dμ(g′),  (g ∈ G)  (2.1)

We shall be especially interested in bounded μ-harmonic functions on G. It turns out that due to the assumption that μ is admissible, any bounded μ-harmonic function is necessarily continuous on G. The space of all bounded μ-harmonic functions on G will be denoted by H^{∞}(G, μ).

We shall also have an occasion to consider the space B_{luc}(G) consisting of all bounded left uniformly continuous functions on G. Recall that a function f on G is l.u.c. if for any ϵ > 0 there is a neighborhood of the identity V = V (ϵ) in G s.t.

|f(gg′) − f(g′)| < ϵ for g ∈ V, g′ ∈ G

The subspace of l.u.c. bounded μ-harmonic functions on G will be denoted by H_{luc}^{∞}(G, μ). Any bounded μ-harmonic function is a pointwise limit of l.u.c. bounded μ-harmonic functions. Indeed one can verify that for any f ∈ H^{∞}(G, μ) and any non-negative compactly supported continuous h ∈ C_{c}(G), the convolution function h ∗ f, defined by

(h ∗ f)(g) = ∫_{G} h(g′)f(g′^{−1}g) dm_{G}(g′)

is in H_{luc}^{∞}(G, μ). Taking a sequence h_{n} ∈ C_{c}(G) with h_{n} ≥ 0, ∫ h_{n} = 1 and supp(h_{n}) shrinking to the identity, one obtains h_{n} ∗ f → f.

The following Lemma provides a connection between H^{∞}(G, μ) and the right μrandom walks on G:

Lemma 2.7. For every f ∈ H^{∞}(G, μ) the limits

f̃(g, ω) = lim_{n→∞} f(gω_{1}ω_{2} ⋯ ω_{n})  (2.2)

exist for all g ∈ G and P-a.e. sequence ω. We shall denote f̃(ω) = f̃(e, ω).

Proof. Fix f ∈ H^{∞}(G, μ) and g ∈ G and consider the random variables W_{n} = f(gω_{1} ⋯ ω_{n}) on (Ω, P). Observe that the sequence {W_{n}} forms a bounded martingale with respect to the increasing sequence A_{n} of the σ-algebras generated by ω_{1}, …, ω_{n}. Indeed

𝔼(W_{n+1} | A_{n}) = ∫ f(gω_{1} ⋯ ω_{n}h) dμ(h) = f(gω_{1} ⋯ ω_{n}) = W_{n}

By the martingale theorem W_{n} converge pointwise on (Ω, P) to some measurable bounded function W_{∞}, which we have denote by f̃(g, ω).

Remark 2.8. In view of the preceding lemma, we shall throughout this chapter focus on the right μ-random walks ω_{1} ⋯ ω_{n} rather than previously considered left random walks S_{n} = ω_{n} ⋯ ω_{1}. Clearly these two settings are equivalent by g ↦ g^{−1}. See also section 2.3 for a better perspective.

Proposition 2.9 below describes an important construction of bounded μ-harmonic functions on (G, μ), using (G, μ)-spaces. Let (M, ν) be a Lebesgue probability space with a measurable G-action such that ν is μ-stationary in the sense that μ ∗ ν = ν where the convolution measure μ ∗ ν is the image of μ × ν under the action map G × M → M. Such a space (M, ν) with the G-action will be called a (G, μ)-space.

A natural source for (G, μ)-spaces are compact (metric) G-spaces, i.e. compact (metric) spaces M with a continuous action G × M → M. Any such G-space M has at least one μ-stationary probability measure ν, which gives rise to a (G, μ)-space (M, ν). Since one can restrict the continuous G-action to the compact G-invariant set M_{0} = supp(ν), we shall always assume that the considered μ-stationary measure ν has full support on the compact G-space M. Such a pair (M, ν) will be called a compact (G, μ)-space.

Proposition 2.9 (Furstenberg). Given a measurable (G, μ)-space (M, ν) consider the transform F_{(M,ν)} : ϕ ↦ f_{ϕ} defined for all ϕ ∈ L^{∞}(M, ν) by

f_{ϕ}(g) = ∫_{M} ϕ(g · x) dν(x) = ∫_{M} ϕ dgν = ∫_{M} ϕ(x) · Π(g, x) dν(x)  (2.3)

where Π(g, x) = dgν/dν(x) is the Radon-Nikodym cocycle. Then

(a) f_{ϕ} ∈ H^{∞}(G, μ) for all ϕ ∈ L^{∞}(M, ν).

(b) f_{ϕ} are constant functions on G for all ϕ ∈ L^{∞}(M, ν) iff ν is G-invariant. If (M, ν) is a compact (G, μ)-space, then (a’) f_{ϕ} ∈ H_{luc}^{∞}(G, μ) for all ϕ ∈ C(M). (b’) f_{ϕ} are constant functions on G for all ϕ ∈ C(M) iff ν is G-invariant.

Proof. Given ϕ ∈ L^{∞}(M, ν) the function f_{ϕ} is bounded by ∥ϕ∥_{∞} and satisfies

∫_{G} f_{ϕ}(gg′) dμ(g′) = ∫_{G} ∫_{M} ϕ(g · x) dμ ∗ ν(x) = f_{ϕ}(g)

Hence f_{ϕ} ∈ H^{∞}(G, μ), proving statement (a). Statement (b) is obvious from the definition. If M is a compact G-space and ϕ ∈ C(M), then for any ϵ > 0 there is an open neighborhood V of the identity in G so that for all g ∈ V and x ∈ M |ϕ(gx) − ϕ(x)| < ϵ, and therefore for all g′ ∈ G

|f_{ϕ}(gg′x) − f_{ϕ}(g′)| ≤ ∫_{M} |ϕ(gy) − ϕ(y)| dg′ν(y) < ϵ

Hence f_{ϕ} is a l.u.c. function, which proves (a’), while (b’) is immediate.

2.3. (G, μ)-boundaries. Consider a compact (G, μ)-space (M, ν). Lemma 1.33 (the proof of which basically consists of Lemma 2.7 and Proposition 2.9), states that for P-a.e. path ω = (ω_{1}, ω_{2}, …) of a (right) μ-random walk on G there exists a limit probability measure ν_{ω} on M

ν_{ω} = lim_{n→∞} ω_{1} ⋯ ω_{n} · ν

In the notation of Lemma 2.7 and 2.9 one has

f̃_{ϕ}(g, ω) = ∫_{M} ϕ dgν_{ω}  and f̃_{ϕ}(ω) = ∫_{M} ϕ dν_{ω}  (2.4)

Definition 2.10. A compact (G, μ)-space (M, ν) is a compact (G, μ)-boundary if ν_{ω} is a Dirac measure ν_{ω} = δ_{z(ω)} for P-a.e. ω ∈ Ω. An abstract measurable (G, μ)- space (M, ν) is a (G, μ)-boundary, if the transform F_{(M,ν)} : L^{∞}(M, ν) → H^{∞}(G, μ), F (M, ν) : ϕ ↦ f_{ϕ} satisfies

f̃_{ϕ·ψ}(ω) = f̃_{ϕ}(ω) · f̃_{ψ}(ω)  (2.5)

for any ϕ, ψ ∈ L^{∞}(M, ν) and for P-a.e. ω.

It is straightforward to see that if (M, ν) is a (G, μ)-boundary then the transform F_{(M,ν)} : L^{∞}(M, ν) → H^{∞}(G, μ) is an isometric embedding, and if (M, ν) is a compact (G, μ)-boundary then also F_{(M,ν)} : C(M) → H_{luc}^{∞}(G, μ) is an (isometric) embedding.

Note that any (G, μ)-boundary (M, ν) gives rise to a measurable map of Lebesgue spaces bnd_{(M,ν)} : (Ω, P) → (M, ν), defined via its action on the commutative von Neumann algebra L^{∞}(M, ν)

ϕ(bnd_{(M,ν)}(ω)) = f̃_{ϕ}(ω) = lim_{n→∞} f_{ϕ}(ω_{1} ⋯ ω_{n})  (2.6)

In the setup of compact (G, μ)-boundaries one has bnd_{(M,ν)}(ω) = z(ω) ∈ M where ν_{ω} = δ_{z(ω)}.

To further clarify the above concepts consider the skew-product construction (Ω × M, P×ν, T), T : (ω, x) ↦ (θω, ω_{1}·x), discussed in Proposition 1.3. The transformation T of (Ω × M, P × ν) is measure-preserving but not invertible. Let θ̄ denote the twosided shift (θ̄ω̄)_{i} = ω̄_{i+1}, i ∈ ℤ, acting on the space (Ω̄, P̄) = (G^{ℤ}, μ^{ℤ}). This is the natural extension of the one-sided shift θ on (Ω, P). The natural extension of the skew-product (Ω × M, P × ν, T) can be realized in the form (Ω̄ × M, m̄, T̄) with

T̄ : (ω̄, x) ↦ (θ̄ω̄, ω̄_{1} · x)

and m̄ being the unique T̄-invariant measure which extends P × ν from B(Ω × M) to B(Ω̄ × M), where B denotes the Borel σ-algebra of the corresponding spaces. Note that the projection Ω̄ × M → Ω̄ maps m̄ to P̄, because P̄ is the only θ̄-invariant probability measure which extends P from B(Ω) to B(Ω̄). Let {mω̄ ∈ P(M)}_{ω̄∈Ω̄} be the disintegration of m̄ with respect to P̄. The T̄ invariance of m̄ translates into the identity ω_{1}mω̄ = mθ̄ω̄, and the fact that m̄ extends P × ν to the σ-algebra of Ω̄ × M means that

∫ mω̄ dP̄(ω̄) = ν

and that mω̄ depends only on (…, ω̄_{−1}, ω̄_{0}).

In fact, every T̄-invariant measure m̄ on Ω̄ × M which projects onto P̄ and has the property that the fiber measures mω̄ depend only on ω̄_{i}, i ≤ 0, defines a measure on Ω × M of the form P × ν, where ν is a μ-stationary measure.

Hence the conditional measure of m̄ with respect to {ω̄_{k} | k ≥ −n} is ω̄_{0}ω̄_{−1} ⋯ ω̄_{−n}ν. As n → ∞ this conditional measure converges to mω̄, but the same limit measure was also described in Lemma 2.7 as

ν_{(ω_{0},ω_{−1},...)} = lim_{n→∞} ω_{0}ω_{−1} ⋯ ω_{−n}ν

Using this description, one can say

Proposition 2.11. A (G, μ)-space (M, ν) forms a (G, μ)-boundary if and only if in the natural extension (Ω̄ × M, m̄, T̄) of the non-invertible measure-preserving skewproduct (Ω×M, P×ν, T) the measure m̄ disintegrates into Dirac measures with respect to P̄, or equivalently if m̄ is a lift of P̄ to a graph of a measurable function Ω̄ → M.

The following idea of Furstenberg allows to construct (G, μ)-boundaries from arbitrary compact metric G-spaces M.

Proposition 2.12 (Furstenberg [14]). Let M be an arbitrary compact metric G-space and μ be an (admissible) probability measure on G. Then the G-action on the compact space V = P(M) contains a (G, μ)-boundary.

Proof. Consider V = P(M) as a compact convex set in C(M)^{∗}. Consider also the space P(V) = P(P(M)), elements of which will be denoted by ν̃. Let δ : V → P(V) stand for the natural embedding δ : v ↦ δ_{v} which has the set of extremal points Ext(P(V)) of P(V) as its image δ(V). Let us denote by bar(ν) the natural barycenter map, which is a continuous extension of the affine map

bar(∑_{i=1}^{n} p_{i}δ_{v_{i}}) = ∑_{1}^{n} p_{i}v_{i}

Since M is a G-space, there exists a μ-stationary measure ν on M, and according to Lemma 1.33 there is a disintegration ν = ∫_{Ω} ν_{ω} dP. Let ν̃ ∈ P(V) be defined as

ν̃ = ∫_{Ω} δ_{ν_{ω}} dP  (2.7)

which implies that bar(ν̃) = ν and that ν̃ is a μ-stationary measure on V. Observe that for a.e. ω one has

bar(ω_{1} ⋯ ω_{n} · ν̃) = ω_{1} ⋯ ω_{n} · ν → ν_{ω}  (2.8)

Since δ_{ν_{ω}} are extremal points in P(V), the two facts (2.7) and (2.8) imply that ω_{1} ⋯ ω_{n} · ν̃ → δ_{ν_{ω}}, which means that supp(ν̃) ⊆ V = P(M) together with ν̃ form a (G, μ)-boundary.

2.4. The Poisson boundary. It is easy to see that any measurable G-equivariant quotient (M′, ν′) of a (G, μ)-boundary (M, ν) is a (G, μ)-boundary. It turns out that given a pair (G, μ) there exists a uniquely defined maximal and universal (G, μ)- boundary, associated to (G, μ).

Theorem 2.13 (Furstenberg). Given a locally compact group G with an admissible probability measure μ, there exists a uniquely defined maximal measurable (G, μ)- boundary (B, ν), called the Poisson boundary of (G, μ), which is uniquely characterized by each of the following properties:

Poisson Representation: The transform F_{(B,ν)} : L^{∞}(B, ν) → H^{∞}(G, μ) is an isometric bijection. In particular every bounded μ-harmonic function f can be presented as f_{ϕ} = F_{(B,ν)}(ϕ) for a unique ϕ ∈ L^{∞}(B, ν), which can be defined by

ϕ(bnd_{(B,ν)}(ω)) = f̃(ω) = lim_{n→∞} f(ω_{1}ω_{2} ⋯ ω_{n})

Universality: The Poisson boundary (B, ν) is the maximal measurable (G, μ)-boundary: any measurable (G, μ)-boundary (B′, ν′) is a G-equivariant measurable quotient p : (B, ν) → (B′, ν′) of the Poisson boundary (B, ν). The quotient map p is uniquely defined, up to sets of ν-measure zero.

Combined with Proposition 2.12, the universality of the Poisson boundary gives the following important

Corollary 2.14 (Furstenberg). Let (B, ν) be the Poisson boundary for a locally compact group G with respect to some admissible measure μ on G. Then for any compact metric G-space M, there exists a measurable, with respect to ν, G-equivariant map f : B → P(M).

Remark 2.15. This is one of the variants of the so called boundary maps which play an important role in rigidity results for group actions. In [70] Zimmer introduced the notion of amenability of an action of a group G on a measure space (X, ν) with a quasi-invariant measure ν. Amenability of a G-action on (X, ν) (which is basically characterized by the existence of a measurable G-equivariant maps f : X → P(M) for any G-space M) can often be verified by other more convenient means. For example, an action of a lattice Γ in a locally compact group G on a homogeneous space G/H is amenable iff H is an amenable group. Poisson boundaries are amenable G-spaces. Boundary maps play a crucial role in the superrigidity results (see Zimmer [72] and Margulis [54]).

There are several ways to construct the Poisson boundary (B, ν) for (G, μ). In the following construction, due to Furstenberg (the form described below is borrowed from Glasner’s [20]), one exhibits a compact G-space B̄ with a μ-stationary measure ν for which the transform F_{(B̄,ν)} gives isometric isomorphisms

C(B̄)⟶^{≅}H_{luc}^{∞}(G, μ) and L^{∞}(B̄, ν)⟶^{≅}H^{∞}(G, μ).

Denote by A the collection of all f ∈ B_{luc}(G) such that for all g ∈ G for P-a.e. ω ∈ Ω the limit f̃(g, ω) = lim_{n→∞} f(gω_{1} ⋯ ω_{n}) exists. A forms a commutative C^{∗}- algebra with respect to pointwise operations and the sup norm. Let Z denote the closed ideal in A consisting of those f ∈ A for which the f̃(g, ω) = 0, and by B̄ the Gelfund dual of the quotient commutative C^{∗}-algebra A/Z, so that C(B̄) becomes identified with A/Z. The natural G-action by left translations on A preserves Z and descends to A/Z ≅ C(B̄). This G-action preserves the C^{∗}-algebra operations and norms, and the fact that all functions involved are l.u.c. corresponds to continuity of the action. This defines a continuous G-action on B̄. Next note that for each g ∈ G the functional L_{g} : f ∈ A ↦ ∫_{Ω} f̃(g, ω) dP(ω) is a positive normalized functional on A, which vanishes on Z, and hence descends to a positive normalized functional on A/Z ≅ C(B̄). Thus L_{g} corresponds to a probability measure ν_{g} on B̄. It follows from the construction that: (i) ν_{g} = gν_{e} for every g ∈ G; (ii) the measure ν = ν_{e} is μ-stationary; (iii) (B̄, ν) is a (topological) (G, μ)-boundary (due to multiplicativity).

By Lemma 2.7, H = H^{∞}(G, μ) ⊆ B_{luc}(G) is a closed subspace of A, and we claim that in fact A = Z ⊕ H. Indeed, the martingale theorem shows that for f ∈ H one has

f(g) = ∫_{Ω} f̃(g, ω) dP(ω)

and therefore Z ∩ H = {0}. To show that H + Z = A, take an arbitrary h ∈ A and denote by ϕ ∈ C(B̄) its image in A → A/Z ≅ C(B̄). Then ϕ defines a function f_{ϕ} ∈ H, which has the same “boundary values” f̃_{ϕ}(ω) = lim_{n→∞} f_{ϕ}(ω_{1} ⋯ ω_{n}) = lim_{n→∞} h(ω_{1} ⋯ ω_{n}) as h, so that h − f_{ϕ} ∈ Z.

Hence we have a natural bijection H ≅ (H ⊕ Z)/Z ≅ A/Z ≅ C(B̄), i.e. F_{(B̄,ν)} : C(B̄) → H_{luc}^{∞}(G, μ) is an isomorphism.

Finally, observe that this fact implies the universality (and thereby the uniqueness) of (B̄, ν) in the topological sense. Indeed, if (B′, ν′) is a compact G-space and forms a (G, μ)-boundary, then the linear operator

C(B′) ⟶^{F_{(B′,ν′)}} H_{luc}^{∞}(G, μ) ⟶^{F_{(B̄,ν)}^{−1}}C(B̄)

preserves positivity, maps 1_{B′} to 1B̄, and respects multiplication (basically due to the fact that both (B̄, ν) and (B′, ν′) are (G, μ)-boundaries !). Hence it is induced by a uniquely defined continuous map p : B̄ → B′. It is not hard to deduce from the construction that p is G-equivariant and satisfies p_{∗}ν = ν′ (in particular p is onto, assuming ν′ has full support on B′).

The fact that every bounded μ-harmonic function is a limit of l.u.c. bounded μharmonic ones, mentioned at the beginning of the section, can be used to deduce that as a measurable (G, μ)-boundary (B̄, ν) defines a bijection

F_{(B̄,ν)} : L^{∞}(B̄, ν) → H^{∞}(G, μ)  (2.9)

This fact implies the universality property (and hence the uniqueness) in this measurable setup, due to the fundamental correspondence between subalgebras of L^{∞}(B, ν) and measurable factors of (B, ν).

Remarks 2.16. (i) Observe, that for a discrete group G any bounded function is automatically l.u.c., so that C(B̄) coincides with L^{∞}(B̄, ν). This means that for a discrete G with a non-trivial Poisson boundary the compact Hausdorff space B̄ described above is non metrizable. (We have ignored some technical difficulties caused by this fact in the above arguments). However, measure-theoretically the Poisson boundary is always a Lebesgue space.

(ii) This is one of the reasons to consider the Poisson boundary as a measuretheoretical rather than topological object. A more significant, but related, reason is the fact that in many cases as a measurable G-space the Poisson boundary can be realized on some nice familiar compact (metric) G-spaces naturally associated to G. Typically these spaces are much smaller than somewhat monstrous universal topological G-space B̄ (cf. sections 2.6, 2.10).

(iii) The pair (G, μ) gives rise yet to another topological G-space - the so called Martin boundary - which is the universal object responsible for the representation of all positive μ-harmonic functions f on G in terms of associated positive measures ν_{f} on Δ. The Poisson boundary for (G, μ) can be identified with (Δ, ν_{1}), where 1 is the constant function on G. Interested reader is referred to the paper [33] of Kaimanovich for a discussion of the Martin boundary in this context, and to the monograph [25] by Guivarc’h, Ji and Taylor for a detailed analysis of Martin boundaries for semisimple Lie groups and symmetric spaces.

The following is one of several purely measure-theoretical construction of the Poisson boundary, described by Kaimanovich and Vershik [36] (see also a more recent paper Kaimanovich [33] for more details and discussion). Given a pair (G, μ) consider an auxiliary probability measure ρ on G and let y = (y_{0}, y_{1}, …) be a path of a (right) random walk

y_{0},  y_{n} = y_{n−1}ω_{n} = y_{0}ω_{1} ⋯ ω_{n},  (n ≥ 1)

where y_{0} has distribution ρ and ω_{1}, … are chosen independently with distribution μ. Let Q_{ρ} denote the resulting distribution of the paths y = (y_{n})_{n≥0} on the path space Y = {y = (y_{n})}. If ρ is a Dirac measure at g ∈ G denote the corresponding Q_{ρ} by Q_{g}. The group G acts on Y by left multiplication coordinate-wise, so that g takes Q_{e} to Q_{g}, and one has Q_{ρ} = ∫ Q_{g} dρ(g) for every ρ. Let T denote the shift (T y)_{n} = y_{n+1}. It commutes with the G-action. If ρ is in the measure class of the Haar measure on G, then Q_{ρ} is quasi-invariant with respect to T and one can consider the space (X, ν_{ρ}) of T -ergodic components of (Y, Q_{ρ}), which comes equipped with the G-action induced by the G-action on Y. Denote by ν_{g} the image of Q_{g} under the factor map Y → X and set ν = ν_{e} (one can use the assumption that μ is admissible to verify that ν_{g} are indeed well defined). Then ν_{g} = gν and since T Q_{e} = ∫ gQ_{e} dμ(g) one has ν = ∫ gν dμ(g), i.e. ν is μ-stationary. It also follows from the construction that (X, ν) is a (G, μ)-boundary. In order to show that (X, ν) is the Poisson boundary, observe that for any f ∈ H^{∞}(G, μ) and Q_{e}-a.e. y = (y_{n}) there exists a finite limit lim_{n→∞} f(y_{n}) = lim_{n→∞} f(ω_{1} ⋯ ω_{n}), which is clearly T -invariant and therefore defines a unique function f̃ ∈ L^{∞}(X, ν), and one has f(e) = ∫_{X} f̃dν. The equivariant G-action gives

f(g) = ∫_{X} f̃dν_{g} = ∫_{X} f̃dgν

so that (X, ν) gives Poisson representation as described in Theorem 2.13, and is therefore the (unique up to measurable isomorphism) Poisson boundary for (G, μ).

In [70] Zimmer described the the Poisson boundary as (a version of) the Mackey range of of the measurable cocycle α : ℕ × (Ω, P) → G given by α(n, ω) = ω_{n} ⋯ ω_{1}. This is basically the same construction as the above one (the space of the ergodic components of the shift T) considered from a different point of view. In this paper Zimmer introduced the above mentioned important notion of amenable actions, in particular illustrated by the G-action on the Poisson boundary/ies.

2.5. Semi-simple Lie groups and their lattices. The following fundamental results of Furstenberg are naturally formulated for semisimple Lie groups G (which are always assumed to be connected, to have finite center and no non-trivial compact factors), rather than for the special case of SL_{k}(ℝ) which has been our focus so far. Semisimple Lie groups G admit Iwasawa decomposition G = KAN, where K is a maximal compact subgroup, N is nilpotent, A is Cartan subgroup. Let P = MAN, where M is the centralizer of A in K, and denote by B(G) = G/P. This is a compact homogeneous G-space. In the case of G = SL_{k}(ℝ), one can take K = SO(k), P - the subgroup of upper triangular matrices, and B(G) = F = F_{(1,...,k)} - the full flag variety.

Theorem 2.17 (Furstenberg, [12]). Let G be a semisimple Lie group as above, and let μ be an absolutely continuous probability measure on G, which contains the identity in the interior of its support. Then there is a unique μ-stationary probability measure ν on B(G), which belongs to the Lebesgue class, and (B(G), ν) is the Poisson boundary for (G, μ). Moreover, B(G) is the universal topological (G, μ)-boundary, i.e. the Poisson representation F_{(B(G),ν)} defines isometric isomorphisms

L^{∞}(B(G), ν)⟶^{≅}H^{∞}(G, μ) C(B(G))⟶^{≅}H_{luc}^{∞}(G, μ)

The G-equivariant factors G/Q of B(G) = G/P with P ⊆ Q, provide the complete list of all (G, μ)-boundaries.

Remark 2.18. The theorem in particular states that semisimple groups G act transitively on their Poisson boundaries (under mild assumptions on μ). One should not expect this to be the case for general groups. However Raugi [59] and more recently Jaworski [30], [31] proved that under certain conditions other classes of groups G and measures μ have the property that G acts transitively on the associated Poisson boundary, so that the latter can be presented as a homogeneous space G/H.

Consider the particular case of absolutely continuous measures μ which satisfy m_{K} ∗ μ = μ, where K ⊂ G is a maximal compact. For such μ one has kμ = μ for all k ∈ K, and this implies that the unique K-invariant probability measure ν = ν_{0} on B(G) = G/P is the only μ-stationary measure, because ν = μ∗ν = kμ∗ν = kν for all k ∈ K. Hence all K-invariant μ on G as in Theorem 2.17 give rise to the same space H^{∞}(G, μ) of bounded μ-harmonic functions. Such bounded μ-harmonic functions are also left K-invariant, and can be considered as lifts of bounded functions on the symmetric space K\G. In [12] Furstenberg showed that in these cases H^{∞}(G, μ) (resp. H_{luc}^{∞}(G, μ)) coincide with the spaces of bounded (resp. l.u.c. bounded) classical harmonic functions on the symmetric space K\G, i.e. solutions of the Laplace-Beltrami equation. Hence the Poisson representation in Theorem 2.17 describes all classical bounded (l.u.c. bounded) harmonic functions on symmetric spaces in terms of their boundary values on B(G). This generalizes the classical Poisson representation of bounded harmonic functions on the unit disc in terms of bounded functions on its boundary - the circle (this case corresponds to G = SL_{2}(ℝ)). The space B(G) is often referred to as the Furstenberg boundary, or Satake-Furstenberg boundary, of a semi-simple group G.

Remark 2.19. The space B(G) is one of the compactifications of the symmetric space K\G of G. The subject of the recent book [25] by Guivarc’h, Ji and Taylor is the connections between various compactifications of the symmetric spaces. Random walks in the form of Brownian motion on the symmetric space and Martin boundaries play a central role in their analysis.

An important theme in the theory of semisimple Lie groups is understanding the relationship between a semisimple group G and lattices Γ contained in it. Recall that a lattice Γ in G is a discrete subgroup of (generally locally compact group) G which has a measurable fundamental domain of finite Haar measure (standard example is Γ = SL_{k}(ℤ) in G = SL_{k}(ℝ)). Furstenberg introduced the use of random walks as a tool relating lattices to the ambient semisimple groups. One of such connections is given by the following:

Theorem 2.20 (Furstenberg, [15] 5.1). Let Γ ⊂ G be a lattice in a semisimple Lie group G, and let ν_{0} be the K-invariant probability measure on B(G). Then there exists a probability measure μ on on Γ with supp(μ) = Γ so that (B(G), ν_{0}) is the Poisson boundary for (Γ, μ).

One of the constructions of such a measure μ on Γ, discussed in [15], is based on a certain discretization procedure applied to the continuous time Markov process - the Brownian motion - on the symmetric space X = K\G. (Relevance of the Brownian motion is clear from the fact that (B(G), ν_{0}) defines all harmonic functions on X, which are closely related to the Brownian motion). This discretization, in particular, allows to associate a bounded μ-harmonic function f̄ on Γ to every bounded harmonic function h on X = K\G.

In [52] Lyons and Sullivan studied similar questions in a more general geometric situation. In particular they consider a regular cover X of a compact Riemannian manifold M = X/Γ where Γ is a discrete group of isometries of X. In this, and more general context (see [52]), a discretization of Brownian motion on X allows to construct a probability measure μ on Γ with supp(μ) = Γ, so that every bounded harmonic function h on X defines a bounded μ-harmonic function h̄ on Γ. On the other hand Lyons and Sullivan describe a natural projection p from L^{∞}(X) to the space H^{∞}(X) of bounded harmonic functions on X. Roughly speaking, the projection of f ∈ L^{∞}(X) is obtained by applying a fixed invariant mean ϕ on the abelian semigroup ℝ_{+} to the function P_{t}f(x) = ∫_{X} P (t, x, y)f(y) dy where P (t, x, y) denotes the transition density of the Brownian motion. In [52] it is shown that

ϕ[P_{t}f] = ϕ[P_{t+s}f] = ϕ[P_{s}P_{t}f] and P_{s}ϕ[P_{t}f]

are equal, which means that the bounded function f̂(x) = ϕ[P_{t}f(x)] is harmonic.

The combination of the discretization procedure and the projection gives a bijection between the spaces H^{∞}(X) and H^{∞}(Γ, μ). This in particular allowed Lyons and Sullivan to conclude that (i) a non-amenable cover X of a compact manifold M always has non-trivial bounded harmonic functions, while (ii) nilpotent covers X of compact manifolds (and more generally ω-nilpotent covers of recurrent Riemannian manifolds) admit only constant bounded harmonic functions (see 2.35 and 2.37.(b) below).

2.6. The Poisson boundary for discrete linear groups. Let us return to random walks on (subgroups of) the simple Lie group G = SL_{k}(ℝ). Let μ be a probability measure with finite first moment. Denote G_{μ} = grp(μ), and let λ_{1}(μ) ≥ λ_{2}(μ) ≥ ⋯ ≥ λ_{k}(μ) be the Lyapunov exponents of μ. Let 1 ≤ τ_{1} < τ_{2} < ⋯ < τ_{r} = k be the indices of strict inequalities: λ_{τ_{i}}(μ) > λ_{τ_{i}+1}(μ), and consider the flag variety F_{τ} of type τ = (τ_{1}, …, τ_{r}) consisting of the flags

{0} ⊂ E_{1} ⊂ E_{2} ⊂ ⋯ ⊂ E_{r} = ℝ^{k}  with  dim E_{i} = τ_{i}

Recall the cocycles σ_{τ_{i}} : G × F_{τ} → ℝ defined in (1.18) by

σ_{τ_{i}}(g, ξ) = log \frac{∥ ∧_{j=1}^{τ_{i}} gu_{j}∥}{∥ ∧_{j=1}^{τ_{i}} u_{j}∥}

where {u_{j}}_{1}^{k} are any vectors such that the i-th subspace of the flag ξ ∈ F_{τ} is spanned by u_{1}, …, u_{τ_{i}} for i = 1, …, r.

Theorem 2.21 (Ledrappier [47], Kaimanovich [32], [34]). There exists a unique probability measure ν on F_{τ} such that

∫_{G} ∫_{F_{τ}} σ_{τ_{i}}(g, ξ) dν(ξ) dμ(g) = ∑_{j=1}^{τ_{i}} λ_{j}(μ)  for i = 1, …, r.  (2.10)

Let B ⊆ F_{τ} be the support of ν. Then (B, ν) is a (G_{μ}, μ)-boundary. Moreover, if grp(μ) is a discrete subgroup Γ in SL_{k}(ℝ), then (B, ν) is the Poisson boundary for (Γ, μ).

Remarks 2.22. Let μ be supported on a discrete subgroup Γ = G_{μ} of SL_{k}(ℝ).

(a) Theorem 2.21 was proved independently by Kaimanovich [32] and Ledrappier [47]. Both proofs are based on the entropy criterion 2.31.(c), due to Kaimanovich and Vershik, for a (Γ, μ)-boundary to be the maximal, i.e. the Poisson boundary. Kaimanovich’ version of the result (the precise statement of which slightly differs from the above) concerns random walks on discrete groups in semisimple Lie groups with the generating measure μ satisfying finite logarithmic moment condition: ∫ log log ∥g∥ dμ(g) < ∞. On the other hand, Ledrappier’s proof, which is based on Oseledec theorem requiring finite first moment, provides additional important information linking the Hausdorff-type dimension of the stationary measure ν on F_{τ}, the Lyapunov exponents and the random walk entropy h(Γ, μ) (see Remark 2.34).

(b) Assuming that μ is symmetric and has finite first moment, amenability of the group G_{μ} is equivalent to the triviality of the Lyapunov spectrum: λ_{1}(μ) = ⋯ = λ_{k}(μ) = 0. By Theorem 2.21 this is also equivalent to the triviality of the Poisson boundary of (G_{μ}, μ). Compare this with Proposition 2.35, Theorem 2.36 and Remarks 2.37.

Here we shall prove only the first part of the Theorem, and will follow Ledrappier’s [47]. Section 2.10 indicates Kaimanovich’ argument showing that (B, ν) is the Poisson boundary for a discrete Γ.

Let (Ω̄, P̄) denote the bi-infinite product (G^{ℤ}, μ^{ℤ}) with the (right) shift transformation θ̄ and its inverse θ̄^{−1} i.e. (θ̄^{−1}ω̄)_{i} = ω̄_{i−1}, i ∈ ℤ. Applying Oseledec theorem to the matrix valued function A(ω̄) = ω̄_{−1}^{−1} on (Ω̄, P̄, θ̄^{−1}), which describes the asymptotic of the sequences of matrices

A_{n}(ω̄) = A(θ̄^{−n+1}ω̄) ⋯ A(θ̄^{−1}ω̄)A(ω̄) = ω̄_{−n}^{−1} ⋯ ω̄_{−1}^{−1} = (ω̄_{−1} ⋯ ω̄_{−n})^{−1}

one obtains

(a) Lyapunov exponents λ̌_{1} ≥ ⋯ ≥ λ̌_{k} defined by P̄-a.e. constant limits

λ̌_{1} + ⋯ + λ̌_{j} = lim_{n→∞} \frac{1}{n} · log ∥ ∧^{j} A_{n}(ω̄)∥

which are easily seen to satisfy λ̌_{j} = −λ_{k−j}(μ), j = 1, …, k.

(b) The measurable splitting ℝ^{k} = W_{1}(ω̄) ⊕ ⋯ ⊕ W_{r}(ω̄) into subspaces such that a non-zero vector v belongs to W_{i}(ω̄) iff

lim_{n→∞} \frac{1}{n} log ∥A_{n}(ω̄)∥ = lim_{n→∞} \frac{1}{n} log ∥ω̄_{−n}^{−1} ⋯ ω̄_{−1}^{−1}∥ = λ̌_{k−τ_{i}} = −λ_{τ_{i}}(μ)
lim_{n→∞} \frac{1}{n} log ∥A_{−n}(ω̄)∥ = lim_{n→∞} \frac{1}{n} log ∥ω̄_{n} ⋯ ω̄_{0}∥ = −λ̌_{k−τ_{i}} = λ_{τ_{i}}(μ)

These subspaces have dimensions dim W_{i}(ω̄) = τ_{i} − τ_{i−1} and satisfy W_{i}(θ̄^{−1}ω̄) = ω̄_{−1}^{−1}W_{i}(ω̄), or equivalently ω̄_{0}W_{i}(ω̄) = W_{i}(θ̄ω̄), for i = 1, …, r.

(c) Denote by ξ(ω̄) the flag in F_{τ} corresponding to the filtration

{0} ⊂ E_{1}(ω̄) ⊂ E_{2}(ω̄) ⊂ ⋯ ⊂ E_{r}(ω̄) = ℝ^{k}

where E_{j}(ω̄) = ⊕_{i=1}^{j}W_{i}(ω̄). The correspondence ω̄ ↦ ξ(ω̄) is in fact measurable with respect to negative coordinates {…, ω̄_{−2}, ω̄_{−1}}.

Consider the transformation T̄ of the space Ω̄ × F_{τ} defined by T̄(ω̄, ξ) = (θ̄ω̄, ω̄_{0}ξ) and denote by s_{i} : Ω̄ × F_{τ} → ℝ the functions s_{i}(ω̄, ξ) = σ_{τ_{i}}(ω̄_{0}, ξ). Let ν̄ denote the measure on Ω̄ × F_{τ} defined by

dν̄(ω̄, ξ) = ∫Ω̄ δω̄ × δ_{ξ(ω̄)} dP̄(ω̄)

that is ν̄ is the lifting of the measure P̄ to the graph of ω̄ ↦ ξ(ω̄). Using Oseledec theorem one can verify that ν̄ is the unique T̄-invariant probability measure on Ω̄×F_{τ} which projects onto P̄ on Ω̄ and satisfies

∫_{Ω̄×F_{τ}} s_{i} dν̄ = λ_{1}(μ) + ⋯ + λ_{τ_{i}}(μ) for i = 1, …, r.  (2.11)

The invertible system (Ω̄, P̄, θ̄) with the shift (θ̄ω̄)_{i} = ω̄_{i+1}, i ∈ ℤ, is the natural extension of the one-sided shift (θω)_{i} = ω_{i+1}, i ∈ ℕ, on (Ω, P). Consider the noninvertible transformation T (ω, ξ) = (θω, ω_{0}ξ) of Ω × F_{τ}, and the projection

π × Id : (Ω̄ × F_{τ}) → (Ω, F_{τ})

corresponding to the projection π : Ω̄ → Ω and the identity map in the F_{τ} coordinate. One clearly has (π × Id) ∘ T̄ = T ∘ (π × Id), and it is easy to see that the projection (π ×Id)_{∗}ν̄ of the above defined ν̄ is a product measure P×ν where ν is a μ-stationary measure on F_{τ}. The constructions are summarized by the following commutative diagram of ergodic transformations

(Ω̄ × F_{τ}, ν̄, T̄) ⟶^{π×Id} (Ω × F_{τ}, P × ν, T)
↓  ↓
(Ω̄, P̄, θ̄) ⟶^{π}  (Ω, P, θ)

where the systems on the left hand sides are the natural extensions of the noninvertible systems on the right hand sides.

It follows from the construction that ν satisfies the relation (2.10). On the other hand, any μ-stationary probability measure ν′ on F_{τ} gives rise to a T -invariant measure P × ν′ on (Ω × F_{τ}), the natural extension of which can be uniquely realized as (Ω̄×F_{τ}, T̄, ν̄′) where ν̄′ is a T̄-invariant probability measure ν̄′ with (π×Id)_{∗}ν̄′ = P×ν′. Since the measure ν̄ is the unique T̄-invariant probability measure satisfying (2.11), the uniqueness of ν as in the proposition is established.

Finally, the fact the disintegration of ν̄ with respect to P̄ consists of Dirac measures at ξ(ω̄) means precisely (Proposition 2.11) that (B, ν) is a (G_{μ}, μ)-boundary.

2.7. Boundary entropy. Let us return to the general setup of an admissible probability measure μ on a locally compact group G. Consider a measurable (G, μ)-space (M, ν), which is not necessarily assumed to be a (G, μ)-boundary.

As we have already mentioned, the assumption that μ is admissible implies that ν is G-quasi-invariant, which enables to define the following notion of boundary entropy:

h_{μ}(M, ν) = ∫_{G} ∫_{M} − log \frac{dg^{−1}ν}{dν} (x) dν(x) dμ(g)  (2.12)

provided the integral is defined. The boundary entropy (also known as Furstenberg entropy) was introduced by Furstenberg [13] as a quantity measuring the extent of non-invariance of a μ-stationary measure ν. Indeed, h_{μ}(M, ν) takes non-negative values

0 ≤ h_{μ}(M, ν) ≤ ∞

with h_{μ}(M, ν) = 0 iff ν is G-invariant. This follows from the strict convexity of − log x, because

h_{μ}(M, ν) ≥ ∫_{G} − log (∫_{M} \frac{dg^{−1}ν}{dν} (x) dν(x)) dμ(g) = ∫_{G} log 1 dμ(g) = 0

with the equality corresponding to ρ_{ν}(g, x) = − log dg^{−1}ν/dν(x) being ν-a.e. constant for μ-a.e. g ∈ G, which is equivalent to ν being G-invariant.

Remark 2.23. The quantity (2.12) can be defined for quasi-invariant probability measure ν, which are not necessarily μ-stationary. The computed value remains constant over all measures (with a suitable integrability condition) from a given measure class precisely if this measure class contains a μ-stationary measure (see Nevo-Zimmer [55]).

Proposition 2.24 ([36]). Let (M, ν) be a (G, μ)-space. Then for any convolution power μ^{p} of μ the measure ν is μ^{p}-stationary and h_{(μ^{p})}(M, ν) = p · h_{μ}(M, ν).

Proof. The function ρ_{ν} : G × M → ℝ is a cocycle, i.e. satisfies ρ_{ν}(gh, x) = ρ_{ν}(g, h · x) + ρ_{ν}(h, x). Thus for every p ∈ ℕ one has

h_{(μ^{p})}(M, ν) = ∫_{G} ∫_{M} ρ_{ν}(g, x) dν(x) dμ^{p}(g)
= ∫_{G} ∫_{M} ρ_{ν}(g, h · x) dν(x) dμ(h) dμ^{p−1}(g) + ∫_{G} ∫_{M} ρ_{ν}(h, x) dν(x) dμ(h)
= ∫_{G} ∫_{M} ρ_{ν}(g, y) dν(y) dμ^{p−1}(g) + h_{μ}(M, ν)
= h_{(μ^{p−1})}(M, ν) + h_{μ}(M, ν) = ⋯ = p · h_{μ}(M, ν)

Proposition 2.25 ([36]). Let p : (M, ν) → (M_{0}, ν_{0}) be a G-equivariant measurable map of (G, μ)-spaces, then h_{μ}(M, ν) ≥ h_{μ}(M_{0}, ν_{0}), with the equality achieved iff the quotient is relatively measure-preserving, i.e. gν_{x} = ν_{g·x} for ν_{0}-a.e. x ∈ M_{0} where ν = ∫_{M_{0}} ν_{x} dν_{0}(x) is the disintegration of ν with respect to ν_{0}.

This statement follows from the cocycle property of Radon-Nikodym derivatives, relations μ ∗ ν = ν and μ ∗ ν_{0} = ν_{0} and strong convexity of − log(·).

Given a (G, μ)-space (M, ν) one can consider the quasi-regular G-representation π_{ν} of G on L^{2}(M, ν). Then one has the estimate

h_{μ}(M, ν) = ∫_{G} ∫_{M} − log \frac{dg^{−1}ν}{dν} (x) dν(x) dμ(g)

≥ −2 log (∫_{G} ∫_{M} \sqrt{\frac{dg^{−1}ν}{dν} (x)} dν(x) dμ(g))

= −2 log ⟨1, π_{ν}(μ̌)1⟩ = −2 log ⟨π_{ν}(μ)1, 1⟩ ≥ 2 log \frac{1}{∥π_{ν}(μ)∥}

which can be further improved by replacing ∥π_{ν}(μ)∥ by the spectral radius ∥π_{ν}(μ)∥_{sp} = lim_{n→∞} ∥π_{ν}(μ^{p})∥^{1/p}. This estimate together with Proposition 1.17 gives

Proposition 2.26. Let M be a compact metric G-space which has no G-invariant probability measures. Then for any admissible measure μ on G

inf_{μ∗ν=ν} h_{μ}(M, ν) > 0

Remark 2.27. Recall Proposition 1.3 describing the measure-preserving skew-product system (Ω × M, P × M, T). Assume that the logarithmic Radon-Nikodym derivative ρ_{ν}(g, x) = − log dg^{−1}ν/dν(x) is in L^{1}(P × ν). Since T -invariant sets can be only of the form Ω × E, E ⊆ M, the Ergodic Theorem implies that for P-a.e. ω:

h_{μ}(M, ν) = ∫_{Ω×M} ρ_{ν} dP × ν = lim_{n→∞} \frac{1}{n} ∑_{k=1}^{n} ∫_{M} ρ_{ν}(θ^{k}ω, ω_{k} ⋯ ω_{1}x) dν(x)

= lim_{n→∞} \frac{1}{n} ∫_{M} ρ_{ν}(ω_{n} ⋯ ω_{1}, x) dν(x)
= lim_{n→∞} \frac{1}{n} ∫_{M} − log \frac{d(ω_{n} ⋯ ω_{1})^{−1}ν}{dν} (x) dν(x)

If the skew-product is ergodic then one has pointwise convergence

lim_{n→∞} −\frac{1}{n} log \frac{d(ω_{n} ⋯ ω_{1})^{−1}ν}{dν} (x) = h_{μ}(M, ν)

Observe the similarity between these expressions and the formulae for the Lyapunov exponents (Theorem 1.8 and (1.12)). In particular, if G is a strongly irreducible subgroup of SL_{k}(ℝ), μ - a probability measure on G with finite first moment, such that the Lebesgue measure ν_{0} on M = ℙ^{k−1} is μ-stationary, then

λ_{1}(μ) = ∫_{G} ∫_{ℙ_{k−1}} log \frac{∥gu∥}{∥u∥} dν_{0}(ū) dμ(g)
= k^{−1} · ∫_{G} ∫_{ℙ_{k−1}} − log \frac{dg^{−1}ν_{0}}{dν_{0}} (ū) dν_{0}(ū) dμ(g)
= k^{−1} · h_{μ}(ℙ^{k−1}, ν_{0})

One can check that these equalities remain true if there is a μ-stationary measure in the Lebesgue class [ν_{0}].

2.8. Random Walk Entropy. In this section it will be more convenient to restrict our attention to discrete groups which will be denoted by Γ. For μ to be admissible on Γ just means that Γ = sgr(μ), however in the current context this assumption can be relaxed to Γ = grp(μ). Furthermore, some of the statements and proofs can be reformulated for general locally compact groups.

Given an arbitrary probability measure η on Γ define the entropy of η to be

H(η) = ∑_{γ∈Γ} −η(γ) · log η(γ)

with the usual convention 0 · log 0 = 0. We shall say that η has finite entropy if H(η) < ∞. Given two probability measures η_{1}, η_{2} on Γ the convolution probability measure η_{1} ∗ η_{2} is the image of η_{1} × η_{2} ∈ P(Γ × Γ) under the map (g_{1}, g_{2}) ↦ g_{1}g_{2}. The fact that the function −x · log x is strictly convex, implies that

H(η_{1} ∗ η_{2}) ≤ H(η_{1}) + H(η_{2}) for η_{1}, η_{2} ∈ P(Γ)  (2.13)

and if both η_{1}, η_{2} have finite entropies then the equality in (2.13) is achieved iff every g ∈ supp(η_{1} ∗ η_{2}) has a unique factorization g = g_{1}g_{2} with g_{i} ∈ supp(η_{i}), i = 1, 2.

Now consider the sequence of convolutions μ^{n} of the given probability measure μ on Γ. The subadditivity property (2.13) allows to define (Avez [2], Kaimanovich-Vershik [36]) the following random walk entropy

h(Γ, μ) = lim_{n→∞} \frac{1}{n} H(μ^{n}) = inf_{n≥1} \frac{1}{n} H(μ^{n})

where μ^{n} is the n-th convolution power of μ. Note that if H(μ) < ∞ then h(Γ, μ) ≤ H(μ) and the the equality is achieved iff for each n ≥ 1 every g ∈ supp(μ^{n}) has a unique factorization g = g_{1} ⋯ g_{n} with g_{1}, …, g_{n} ∈ supp(μ). Thus h(Γ, μ) describes the average information about the steps X_{1}, …, X_{n} of a typical random walk, given its position of the product X_{1} ⋯ X_{n}.

The following analogue of Shannon-McMillan-Breiman theorem states that the μ^{n}- weights assigned to μ-random walks at time n are asymptotically equal, on the logarithmic scale. More precisely

Theorem 2.28 (Kaimanovich-Vershik [36], Derriennic [6]). Let Γ be a discrete countable group, and μ be a probability measure with H(μ) < ∞. Denote by y_{n}(ω) = ω_{1} ⋯ ω_{n}, ω ∈ Ω, the path of the (right) μ-random walk on Γ. Then

lim_{n→∞} −\frac{1}{n} log μ^{n}(y_{n}(ω)) = h(g, μ)

the convergence being both pointwise w.r.to P and in L^{1}(Ω, P).

Proof. Observe that for every n, m ∈ ℕ and every sequence x = (x_{i})_{i=1}^{∞} in Γ one has μ^{m+n}(x_{1} ⋯ x_{n+m}) ≥ μ^{n}(x_{1} ⋯ x_{n}) · μ^{m}(x_{n+1} ⋯ x_{n+m}). Therefore the sequence of functions f_{n}(ω) = − log μ^{n}(y_{n}(ω)) satisfies f_{n+m}(ω) ≤ f_{n}(ω) + f_{m}(θ^{n}ω), i.e. {f_{n}} is a subadditive sequence over the ergodic system (Ω, P, θ). The finite entropy condition gives f_{1} ∈ L^{1}(Ω, P) and Kingman’s subadditive ergodic theorem completes the proof.

Remark 2.29. One might view this result as the statement that the μ-random walk on Γ has the number h(G, μ) as its “large scale Hausdorff dimension”, by which we mean the following. Denoting h = h(Γ, μ)

(a) There exists a sequence of subsets A_{n} ⊂ Γ with log |A_{n}| ≤ (h + ϵ)n, such that with P-probability one g_{n}(ω) ∈ A_{n} for all, but finitely many, n-s.

(b) For any sequence of subsets B_{n} ⊂ Γ with log |B_{n}| ≤ (h − ϵ)n, with P-probability one g_{n}(ω) ∈ B_{n} for at most finitely many n-s.

Remark 2.30. In [49] (see also [50]) Ledrappier introduced the notion of entropy profile for a symmetric probability measure μ on Γ with finite entropy. Assume that μ^{n}(e) > 0 for all n (this is always true for μ′ = μ ∗ μ when μ is symmetric). Given such a μ consider the paths of μ-random walk y_{n}(ω) = ω_{1} ⋯ ω_{n}. Ledrappier proves that there exists a convex function β_{μ} : ℝ_{+} → ℝ_{+} such that for P-a.e. ω for every t ≥ 0 one has

lim_{n→∞} − log μ^{n}(y_{[tn]}(ω)) = β_{μ}(t)  (2.14)

Observe that the entropy profile connects the spectral radius to the random walk entropy in a very natural way:

β_{μ}(0) = − log ∥π_{reg}(μ)∥ and β_{μ}(1) = h(Γ, μ)

See Theorem 2.33 below for further properties.

The following fundamental facts relate the entropy of random walk to the boundary entropy:

Theorem 2.31 (Kaimanovich-Vershik, [36]). Let μ be a probability measure on a discrete countable group Γ with finite entropy. Then

(a) For any (Γ, μ)-space (M, ν) one has the inequality h_{μ}(M, ν) ≤ h(Γ, μ).

(b) For the Poisson boundary (B, ν) the equality holds h_{μ}(B, ν) = h(Γ, μ).

(c) If a (Γ, μ)-boundary (B′, ν′) satisfies h_{μ}(B′, ν′) = h(Γ, μ), then (B′, ν′) is (measurably isomorphic to) the Poisson boundary (B, ν).

(d) The Poisson boundary of (Γ, μ) is trivial if and only if h(Γ, μ) = 0.

Proof. Since ν = μ ∗ ν = ∑_{g∈Γ} μ(g) · gν one has dgν/dν(x) ≤ μ(g)^{−1} for ν-a.e. x ∈ M. Thus the cocycle ρ_{ν}(g, x) = − log dg^{−1}ν/dν(x) satisfies for every g ∈ Γ and ν-a.e. x ∈ M

ρ_{ν}(g, x) = −ρ_{ν}(g^{−1}, g · x) = log \frac{dgν}{dν} (g · x) ≤ − log μ(g)

so that h_{μ}(M, ν) ≤ ∑_{g∈Γ} −μ(g) log μ(g) = H(μ) and

h_{μ}(M, ν) = \frac{1}{n}h_{(μ^{n})}(M, ν) ≤ \frac{1}{n}H(μ^{n}) → h(Γ, μ)

which proves (a).

For the proof of (b) recall the Kaimanovich-Vershik construction of the Poisson boundary (B, ν) as the space of ergodic components of the shift T on (Y, Q_{e}) (section 2.4). One can also view (B, ν) as the factor (Y, Q_{e})/τ of (Y, Q_{e}) defined by the limit σ-algebra τ lim_{n→∞} τ_{n} of the decreasing sequence of measurable partitions τ_{n}, where y ∼ y′ mod (τ_{n}) iff y_{k} = y′_{k} for all k ≥ n, or equivalently if T^{n}y = T^{n}y′. Denote by α_{1} the partition of Y defined by the value of y_{1}. Let α_{1}(y), τ_{n}(y) denote the equivalence classes containing y with respect to the corresponding partitions. The computation of the conditional probabilities

Q_{e} {α_{1}(y) | τ_{n}(y)} = μ(y_{1}) · μ^{n−1}(y_{1}^{−1}y_{n})/μ^{n}(y_{n})

gives the following relation on conditional entropies

H(α_{1} | τ_{n}) = ∫_{Y} − log Q_{e} {α_{1}(y) | τ_{n}(y)} dQ_{e}(y) = H(μ) + H(μ^{n−1}) − H(μ^{n})

Since τ_{n+1} ≺ τ_{n} one has H(α_{1} | τ_{n}) ≤ H(α_{1} | τ_{n+1}) which means that the sequence H(μ^{n+1}) − H(μ^{n}) is decreasing. Therefore

h(Γ, μ) = lim_{n→∞} \frac{1}{n}H(μ^{n}) = lim_{n→∞} \frac{1}{n} ∑_{k=1}^{n} (H(μ^{k}) − H(μ^{k−1}))
= lim_{n→∞} (H(μ^{n}) − H(μ^{n−1}))

so that

H(α_{1} | τ) = lim_{n→∞} H(α_{1} | τ_{n}) = H(μ) − lim_{n→∞} (H(μ^{n−1}) − H(μ^{n})) = H(μ) − h(Γ, μ)

In the following computation we shall denote by b ∈ B the image of y ∈ Y under the projection (Y, Q_{e}) → (Y, Q_{e})/τ = (B, ν).

h(Γ, μ) = H(μ) − H(α_{1} | τ)
= ∑_{g∈Γ} −μ(g) log μ(g) + ∑_{g∈Γ} μ(g) · ∫_{Y} log Q_{e}(y_{1} = g | τ(y)) dQ_{e}(y)

= ∑_{g∈Γ} −μ(g) log μ(g) + ∑_{g∈Γ} μ(g) · ∫_{B} log (μ(g)\frac{dgν}{dν} (b)) dν(b)

= ∑_{g∈Γ} μ(g) ∫_{B} log \frac{dgν}{dν} (b) dν(b)

= ∑_{g∈Γ} μ(g) · ∫_{B} − log \frac{dg^{−1}gν}{dgν} (b) dν(b)

= ∑_{g∈Γ} μ(g) · ∫_{B} − log \frac{dg^{−1}ν}{dν} (b′) dν(b′) = h_{μ}(B, ν)

proving (b).

Claim (c). By the universality of the Poisson boundary (B, ν) there is a measurable Γ-equivariant map p : (B, ν) → (B′, ν′). Hence (a) and Proposition 2.25 imply

h(Γ, μ) ≥ h_{μ}(B, ν) ≥ h_{μ}(B′, ν′) ≥ h(Γ, μ)

i.e. all the quantities are equal. Thus p : (B, ν) → (B′, ν′) is relatively measurepreserving, and using the fact that (B, ν) is a boundary one concludes that the disintegration of ν with respect to ν′ consists of Dirac measures, i.e. p is an isomorphism.

Claim (d). If h(Γ, μ) = 0 then h_{μ}(B, ν) = 0 by (a) and hence (B, ν) is trivial. The opposite implication follows from (b).

We conclude with some general inequalities.

Proposition (*) 2.32 ([36]). Let Γ be a finitely generated with a left invariant metric d, and let μ be a probability measure on Γ with finite first moment. Then μ has finite entropy and

h(Γ, μ) ≤ δ(Γ, d) · λ^{(d)}(μ)

where δ(Γ, d) = lim_{n→∞} n^{−1} log |{g ∈ Γ | d(g, e) ≤ n}| denotes the growth exponent of Γ, and λ^{(d)}(μ) = lim_{n→∞} d(S_{n}, e)/n - the escape rate.

Theorem 2.33 (Ledrappier, [49] see also [50]). Let μ be a symmetric probability measure μ with finite entropy on Γ. Denote r_{μ} = ∥π_{reg}(μ)∥. Then

4(1 − r_{μ}) ≤ h(Γ, μ)

If, furthermore, there is a semigroup μ_{t}, t > 0, of symmetric measures so that μ_{1} = μ, then

4 log(1/r_{μ}) ≤ h(Γ, μ)

The entropy profile β_{μ}(t), defined in 2.30, is asymptotic to log(1/r_{μ}) · t + cnst as t → ∞.

Remark 2.34. Let Γ = grp(μ) be a discrete subgroup of SL_{k}(ℝ) where μ is a probability measure with finite first moment. In [47] Ledrappier established a relationship between Lyapunov exponents, the entropy h(Γ, μ) and Hausdorff-like dimension dim(ν) of the μ-stationary measure ν on the natural boundary F_{τ}. In the simplest case of k = 2 where F_{τ} = ℙ^{1} (unless λ_{1}(μ) = 0) the “dimension” of the stationary measure is

dim(ν) = lim_{δ→0} lim sup_{ϵ→0} \frac{log N(ϵ, δ, ν)}{log(1/ϵ)}

where N(ϵ, δ, ν) is the minimal number of ϵ-intervals needed to cover (1 − δ) mass of ν on ℙ^{1}. In this case Ledrappier proves the inequalities

h(Γ, μ) ≤ 2λ_{1}(μ) · dim(ν) ≤ h_{μ}(ℙ^{1}, ν)  (2.15)

where the discreteness assumption on Γ = grp(μ) ⊆ SL_{2}(ℝ) is used only in the proof of the first inequality. Since one always has h(Γ, μ) ≥ h_{μ}(ℙ^{1}, ν), there is an equality throughout in (2.15) which in particular shows that (B, ν) is the Poisson boundary for (Γ, μ) where B = supp(ν) ⊆ ℙ^{1}.

2.9. Triviality of the Poisson boundary. Let G be a non-amenable locally compact group, and let μ be an admissible probability measure on G. Non-amenability of G is equivalent to the existence of a compact (metric) G-space M which has no G-invariant probability measures on it. Any such space supports a μ-stationary measure ν, and by Proposition 2.9 the transform F_{(M,ν)} gives rise to a family of l.u.c. bounded μ-harmonic functions on G, not all of which are constant. This proves:

Proposition 2.35 (Furstenberg). For any admissible (or just generating) probability measure μ on a locally compact non-amenable group G, there exist non-constant bounded l.u.c. μ-harmonic functions.

There exist many examples of amenable groups (as small as two-step solvable) with probability measures μ which have non-constant μ-harmonic functions. Furstenberg conjectured that for any amenable group G there exists a probability measure μ with supp(μ) = G, so that G has no non-constant bounded μ-harmonic functions. This conjecture was proved independently by Kaimanovich-Vershik and Rosenblatt:

Theorem 2.36 (Kaimanovich-Vershik [36], Rosenblatt [60]). Let G be a locally compact amenable group. Then there exists a symmetric admissible probability measure μ with supp(μ) = G such that any bounded μ-harmonic function on G is constant.

In [36] Kaimanovich and Vershik give very interesting examples of finitely generated solvable groups (certain wreath products) for which any generating symmetric μ with finite support has h(Γ, μ) > 0, i.e. admits non constant bounded μ-harmonic functions.

Remarks 2.37. Let us point out some situations where a group Γ has no nonconstant bounded μ-harmonic functions, i.e. the Poisson boundary of (Γ, μ) is trivial:

(a) Groups Γ of sub-exponential growth (i.e. those with δ(Γ, d) = 0) and measures μ with a finite first moment (Proposition 2.32). Recall that besides finitely generated nilpotent groups, which have polynomial growth, there are many finitely generated groups, constructed by Grigorchuck, with so called intermediate which is faster than polynomial but slower than exponential.

(b) Groups of polynomial growth and any probability measure μ. By the famous theorem of Gromov such groups are virtually nilpotent. For nilpotent groups the Poisson boundary is known to be trivial (cf. Furstenberg [14] 11.2, see also Margulis [53] for the corresponding results on positive μ-harmonic functions).

(c) General finitely generated groups Γ and measures μ with finite entropy, finite fist moment and zero escape rate: λ^{(d)}(μ) = 0 (Proposition 2.32). In fact, Varopoulos proved in [66] that for finitely supported symmetric measures μ on a general discrete group Γ, the condition λ^{(d)}(μ) = 0 not only implies, but is actually equivalent to h(Γ, μ) = 0.

2.10. Identification of the Poisson boundary. As we have seen, the Poisson boundary (B, ν) associated to a pair (G, μ) allows to describe all bounded μ-harmonic functions on G. In order to make this description tangible one seeks to realize this abstract measurable G-space (B, ν) on some concrete (topological) G-spaces naturally associated to G. Theorems 2.17 and Theorem 2.21 are examples of such realizations. In these examples the Poisson boundary is realized on some boundary ∂G = Ḡ \ G associated to an appropriate compactification Ḡ of G, i.e. a compact G-space which has an open dense G-orbit homeomorphic to G.

In many examples a locally compact (or discrete) group G with, say an admissible, probability measure μ, has a natural compactification Ḡ such that ∂G = Ḡ \ G supports a unique stationary measure ν on ∂G, so that (∂G, ν) forms a (G, μ)-boundary. In such a situation a.e. path of μ-random walk ω_{1}ω_{2} ⋯ ω_{n} on G converges in Ḡ to a unique point ξ = ξ(ω) ∈ ∂G, where X_{1} ⋯ X_{n}ν → δ_{ξ(ω)} ∈ P(∂G). (Observe also that, by Lemma 1.33, the same limit point ξ = ξ(ω) is obtained as the limit ω_{1} ⋯ ω_{n}g → ξ_{ω} ∈ ∂G for every g ∈ G). In such a situation one would like to know whether (∂G, ν) is (measurably) the maximal (G, μ)-boundary, i.e. the Poisson boundary. In [34] and [33] Kaimanovich developed very powerful geometric criteria for maximality of such a (G, μ)-boundaries. Consider the discrete situation. Let Γ be a finitely generated group with a left invariant word metric d. For r ≥ 0 let B_{r} = {g ∈ Γ | d(g, e) ≤ r} denote the centered ball of radius r in Γ.

Theorem 2.38 (Strip Approximation, Kaimanovich [34] 14.4). Let μ be a finite entropy probability measure on Γ as above. Assume that (B_{+}, ν_{+}) and (B_{−}, ν_{−}) are boundaries for (Γ, μ) and (Γ, μ̌) respectively, where μ̌(g) = μ(g^{−1}). Assume that there exists a Γ-equivariant map S, assigning to ν_{−} × ν_{+}-a.e. pair (ξ_{−}, ξ_{+}) ∈ B_{−} × B_{+} a non-empty subset S(ξ_{−}, ξ_{+}) of Γ, such that for ν_{−} × ν_{+}-a.e. (ξ_{−}, ξ_{+})

\frac{1}{n} log |S(ξ_{−}, ξ_{+}) ∩ B_{d(g_{n}(ω),e)}| → 0

in probability with respect to P, where g_{n}(ω) = ω_{1} ⋯ ω_{n}. Then (B_{−}, ν_{−}) and (B_{+}, ν_{+}) are the Poisson boundaries for (Γ, μ) and (Γ, μ̌) respectively.

The proof of this criterion is based on Theorem 2.31.(c). Applicability of the criterion depends on the existence of the “strips” S(ξ_{−}, ξ_{+}) with sub-exponential intersection with the centered balls B_{d(g_{n}(ω,e))}. If μ has a finite first moment then d(g_{n}(ω), e) has a linear growth. Hence if one can find “strips” S(ξ_{−}, ξ_{+}) with sub-exponential growth the condition of the theorem would be satisfied. However, if one can find “strips” S(ξ_{−}, ξ_{+}) with a polynomial growth, the condition applies to a larger class of measures μ, namely those with a finite first logarithmic moment:

∑ log d(g, e) dμ(g) < ∞

In [34] Kaimanovich used this strip approximation criterion (and other related techniques) to realize the Poisson boundaries (B, ν) for several classes of groups Γ on certain boundaries ∂Γ associated to some natural compactifications Γ̄ of Γ. Here is a brief list of such results (see [34] for details):

1. Word hyperbolic groups Γ and μ with finite entropy, finite first logarithmic moment and non-elementary Γ_{μ} = grp(μ). The relevant “geometric boundary” ∂Γ is the hyperbolic (ideal) boundary.

2. Groups Γ with infinitely many ends, and measures μ with finite entropy, finite first logarithmic moment and non-elementary Γ_{μ} = grp(μ). The relevant “geometric boundary” ∂Γ is the the space of ends E(Γ).

3. Cocompact discrete groups Γ of isometries of rank one Cartan-Hadamard manifolds, and measures μ with finite first logarithmic moment, finite entropy and grp(μ) = Γ. The relevant “geometric boundary” ∂Γ is the the visual boundary, homeomorphic to a sphere (see also Ballmann-Ledrappier [4]).

4. Non elementary subgroups in the Mapping Class Groups (joint work of Kaimanovich and Masur [35]). The “geometric boundary” in this context is Thurston’s compactification of the Teichmüller space. This identification of the Poisson boundary allowed Kaimanovich and Masur to prove that a non-elementary subgroup of a mapping class group cannot be isomorphic to a higher rank lattice.

5. Discrete subgroups Γ of SL_{k}(ℝ) and measures μ with finite entropy, finite first logarithmic moments and grp(μ) = Γ. The relevant boundary is an appropriate flag variety (Stake-Furstenberg compactification) (see Theorem 2.21 and the following remarks).

In [34] one can also find descriptions of the Poisson boundary for several other classes of groups, including polycyclic groups, some solvable groups and wreath products.

2.11. Towards a Structure Theory for (G, μ)-spaces. Measures, especially invariant ones, have proved to be an important tool in the study of group actions on topological spaces, and manifolds in particular. Since actions of non-amenable groups G on compact spaces do not always have invariant probability measures, but always admit stationary measures (for any probability measure on the group), it is highly desirable to develop a structure theory for (G, μ)-spaces, at least in the measurable category. This goal is still beyond reach, but one can speculate that (G, μ)-boundaries and measure-preserving systems are the potential building blocks of such a structure theory.

Let (M, ν) be a (measurable) (G, μ)-space where μ is an admissible measure on a locally compact group G. Let B denote the Lebesgue (complete) σ-algebra of (M, ν) and let B_{RN} denote the smallest complete sub-σ-algebra B_{RN} of B with respect to which the Radon-Nikodym derivatives r_{ν}(g, ·) = dg^{−1}ν/dν(·), g ∈ G, are all measurable. This σ-algebra B_{RN} is G-invariant (this basically follows from the cocycle property of r_{ν}) and therefore gives rise to a measurable G-equivariant factor (G, μ)- space

p : (M, ν) → (M_{RN}, ν_{RN})

which is called the Radon-Nikodym factor. Radon-Nikodym factor was introduced by Kaimanovich and Vershik in [36] in the discrete groups setting. For the details of the construction in general case see Nevo and Zimmer in [55]. The Radon-Nikodym factor has the following nice properties with respect to boundary entropy:

Proposition 2.39 (Kaimanovich-Vershik [36], Nevo-Zimmer [55]). Let (M, ν) be a (G, μ)-space with finite boundary entropy, and let (M_{RN}, ν_{RN}) denote the Radon-Nikodym factor. Then

(i) h_{μ}(M, ν) = h_{μ}(M_{RN}, ν_{RN}) so that p : (M, ν) → (M_{RN}, ν_{RN}) is relatively measurepreserving extension of G-spaces.

(ii) Every non-trivial G-equivariant factor (M_{0}, ν_{0}) of the Radon-Nikodym factor (M_{RN}, ν_{RN}) has strictly smaller boundary entropy: h_{μ}(M_{0}, ν_{0}) < h_{μ}(M, ν).

In a sense the Radon-Nikodym factor filters out (the outer layer of) the measurepreserving part of the action.

Recently Nevo and Zimmer studied the structure of (G, μ)-spaces (M, ν) where G is a semisimple Lie group, μ is an admissible measure μ on G, and (M, ν) has finite positive boundary entropy, i.e. ν is not G-invariant. Let us just mention the following (the reader is referred to Nevo-Zimmer [55], [56] for further results):

(i) If G is a simple Lie group with rk_{ℝ}(G) = 1, there exists a (G, μ)-space (M, ν) with h_{μ}(M, ν) > 0 which does not have any non-trivial (G, μ)-boundary as a factor. In particular, for such spaces the Radon-Nikodym factor is not a (G, μ)- boundary (compare with [36] 3.6).

(ii) For (semi)simple Lie group G with rk_{ℝ}(G) ≥ 2 any (G, μ)-space (M, ν) with h_{μ}(M, ν) > 0 has a non-trivial measurable boundary factor (M, ν) → (G/Q, ν_{G/Q}); moreover under some necessary assumptions (M, ν) has a boundary factor G/Q with the full boundary entropy h_{μ}(M, ν) = h_{μ}(G/Q, ν_{G/Q}), in which case the G-action on (M, ν) is an induced G-action from G/Q by a measure-preserving Q-action on some space (Y, η).

These results turn out to be closely related to Margulis’ Normal Subgroup Theorem !

3. Random Transformations

Let Φ be a family of transformations of a space X, μ a probability measure on Φ, and {ϕ_{n}, n ≥ 0} a sequence of independent Φ-valued μ-distributed random variables defined on the product probability space (Ω, P) = (Φ^{ℕ}, μ^{ℕ}). Let

S_{n}(ω) = ϕ_{n}(ω) ∘ ⋯ ∘ ϕ_{2}(ω) ∘ ϕ_{1}(ω)

be the corresponding product of transformations. The subject of our discussion will be the typical (with respect to P) behavior of the sequence {S_{n}(ω), n ≥ 0} of transformations of the space X. (To make this setup precise, a measurable structure should be introduced on Φ and X so that the action Φ × X → X is measurable). If the transformations ϕ ∈ Φ are invertible, as we shall assume for convenience, one can talk about the group G of transformations of X, generated by the family Φ. Thus {S_{n}(ω)} become paths of μ-random walks on G. In the previous chapters we focused on the connections between the structure of the group G and properties of the random walks S_{n} on it, while here we shall be interested in understanding the action of G on X using the properties of random transformations S_{n}.

3.1. The Random Ergodic Theorem. Let (X, ν) be a probability space, Φ = {ϕ} a family of measure class preserving transformations of (X, ν), μ - a probability measure on Φ such that (ϕ, x) ↦ ϕ(x) is measurable map mapping μ × ν onto ν (in other words (X, ν) is a (G, μ)-space where G is the group generated by Φ). For 1 ≤ p ≤ ∞ consider the Markov operator P acting on L^{p}(X, ν) by

(P f)(x) = ∫_{Φ} f(ϕ(x)) dμ(ϕ)

Consider the product space (Ω, P) = (Φ^{ℕ}, μ^{ℕ}) with the shift θ, (θω)_{i} = ω_{i+1}, and the skew-product measure-preserving transformation T of (Ω × X, P × ν) defined by T (ω, x) = (θω, ω_{1}(x)).

Theorem 3.1 (Random Ergodic Theorem). With the notations as above, the following conditions are equivalent:

(a) Every measurable set E ⊆ X for which m(ϕ^{−1}E△E) = 0 for μ-a.e. ϕ ∈ Φ, is measurably trivial, i.e. ν(E) = 0 or ν(X \ E) = 0.

(b) The Markov operator P on L^{p}(X, ν), 1 ≤ p ≤ ∞, is ergodic, i.e. P f = f for f ∈ L^{p}(X, ν) iff f(x) = c constant ν-a.e.

(c) The skew-product (Ω × X, P × ν, T) is ergodic. If these conditions are satisfied then for every function f ∈ L^{1}(X, ν) with P-probability one the sequence of random products {S_{n}(ω) = ω_{n} ∘ ⋯ ∘ ω_{1}}_{n=1}^{∞} satisfies

lim_{n→∞} \frac{1}{N} ∑_{n=1}^{N} f(S_{n}(ω) · x) = ∫_{X} f dν

the convergence being pointwise w.r.to ν and in L^{1}(X, ν).

Remarks 3.2. (a) The case of ν being invariant, rather than μ-stationary measure, is known as Kakutani’s Random Ergodic Theorem [37]. The proof of the theorem in the general case below is borrowed from Kifer [41] pp. 19-21.

(b) If the skew-product (Ω×X, P×ν, T) is not ergodic, the T -invariant measure P×ν can be disintegrated into a family of T -ergodic probability measures, which can be shown to have a form P × ν_{t} where ν_{t} is a μ-stationary ergodic measure (see Kifer [41] A1). If X is a compact G-space (G-is a group generated by Φ) then this decomposition is precisely the presentation of the μ-stationary measure ν as a convex (integral) combination of the set of extremal μ-stationary measures ν_{t}.

(c) Let G be a locally compact group, M a compact metric G-space, μ - an admissible probability measure on G, then the Random Ergodic Theorem implies that for a μ-stationary measure ν on M the G-action on (M, ν) is ergodic iff ν is an extremal point in the convex compact set of μ-stationary measures on M.

(d) In the measure preserving case, where ν is an invariant measure, under fairly general conditions on μ (μ is symmetric or μ^{n} and μ^{n+1} are not mutually singular for some n ≥ 1) the skew-product (Ω × X, P × ν, T) is not merely ergodic but is exact, i.e. its natural extension is a K-automorphism ([10] Appendix B). However, in general the skew-product is not Bernoullian (Kalikow’s T, T^{−1}- theorem).

Proof of Theorem 3.1.

The implications (c) ⇒ (b) ⇒ (a) are evident. To show that (a) ⇒ (b) let f ∈ L^{p}(X, ν) be P -invariant. Then |f| = |P f| ≤ P |f| so that P |f| − |f| is a non-negative function in L^{p}(X, ν), and since P^{∗}ν = ν one has

∫_{X}(P |f| − |f|) dν = ∫_{X} |f| dP^{∗}ν − ∫_{X} |f| dν = 0

which means that ν-a.e. P |f|(x) = |f(x)|, i.e. |f| is P -invariant. Hence f^{+}(x) = max{f(x), 0} = (|f(x)| + |f(x)|)/2 is also a P -invariant function, so the set E = {x | f(x) ≥ 0} satisfies ν(ϕ^{−1}E△E) = 0 for μ-a.e. ϕ ∈ Φ and is therefore measurably trivial by (a). Repeating this argument for the P -invariant function f(x) − c, where c is a constant, one concludes that all sets {x | f(x) ≥ c} are measurably trivial, i.e. f is a constant function, proving (b).

Now assume (b), and let h(ω, x) ∈ L^{2}(Ω × X, P × ν) be a T -invariant function, so that h(θω, ω_{1} · x) = h(ω, x). For n ≥ 0 let B_{n} denote the σ-algebra generated by x, ω_{1}, …, ω_{n} and let h_{0}(x), h_{n}(ω_{1}, …, ω_{n}, x) denote the projections of h to H_{n} = L^{2}(B_{n}) ⊆ H = L^{2}(P × ν). In probabilistic terms

h_{n}(ω_{1}, …, ω_{n}, x) = E(h(ω, x) | ω_{1}, …, ω_{n})

are the conditional expectations of h with respect to ω_{1}, …, ω_{n}. Since h(ω, x) = h(θ^{n}ω, ω_{n} ⋯ ω_{1} · x) we have

h_{n}(ω_{1}, …, ω_{n}, x) = h_{n−1}(ω_{2}, …, ω_{n}, ω_{1} · x) = ⋯ = h_{0}(ω_{n} ⋯ ω_{1} · x) (3.1)

In particular h_{0}(x) = EE(h(ω, x) | ω_{1}) = Eh_{1}(ω_{1}, x) = Eh_{0}(ω_{1} · x) = P h_{0}(x), which by (b) means that h_{0}(x) = c is a constant ν-a.e. In view of (3.1) all functions h_{n}(ω_{1}, …, ω_{n}, x) = c are constant. Since H_{n} increase to the whole space H = L^{2}(P × ν) the limit function h(ω, x) has to be constant c, and assertion (c) follows.

The following is a topological analogue of the Random Ergodic Theorem. Recall, that a continuous action G × X → X of a locally compact group G on a metric compact X is minimal iff every orbit G · x, x ∈ X, is dense in X.

Theorem (*) 3.3. Let G be a locally compact group with a continuous minimal action G × X → X on a compact metric space X. Let μ be a probability measure on G so that G = sgr(μ), and let S_{n}(ω) = f_{n}(ω) ⋯ f_{1}(ω) denote the random products of the μ-generated random walk on G. Then there exists a set Ω_{0} ⊆ Ω with P(Ω_{0}) = 1 so that for all ω ∈ Ω_{0} and every x ∈ X the sequence {S_{n}(ω) · x, n ≥ 0} is dense in X.

3.2. Strong ergodicity and Rate of convergence. Consider a measure-preserving action (X, m, G) of a locally compact group G, i.e. a jointly measurable action G × X → X where each g ∈ G acts as a measure-preserving transformation of the standard probability space (X, m). As usual such an action is called ergodic if if there are no measurable sets E ⊆ X with 0 < m(E) < 1 such that m(g^{−1}E△E) = 0 for every g ∈ G. Random Ergodic Theorem 3.1 (in this case Kakutani’s Theorem) implies

Corollary 3.4 (Kakutani’ Random Ergodic Theorem). Let (X, m, G) be an ergodic measure-preserving system and μ a measure on G with grp(μ) = G. Then for every function f ∈ L^{1}(X, m) P-a.e. path {S_{n} = ω_{n} ⋯ ω_{1}} of the μ-random walk on G

lim_{N→∞} \frac{1}{N} ∑_{n=1}^{N} f(S_{n} · x) = ∫_{X} f dm

in L^{1}(X, m) and for m-a.e. x ∈ X.

Proof. It suffices to show that any measurable set E ⊆ X with m(g^{−1}E△E) = 0 for μ-a.e. g ∈ G has m(E) ∈ {0, 1}. The unitary G-representation π on L^{2}(X, m)

(π(g)f)(x) = f(g^{−1} · x) (f ∈ L^{2}(X, m), g ∈ G)

is continuous (in the weak topology), and the indicator function 1_{E} ∈ L^{2}(X, m) is a fixed vector for μ-a.e. g ∈ G. By continuity, it is fixed by all of grp(μ) = G. As the system (X, m, G) is ergodic, the only π(G)-invariant vectors are constant functions, which means that E is measurably trivial.

Note the following features of the corollary:

Representative behavior: The ergodicity of the G-action forces a.e. paths of the

μ-random walk on G to become equidistributed on (X, m). One dimensional behavior: The paths of a random walk are indexed by ℕ and there-

fore allow to obtain “random walk” analogues of the classical ergodic theorems for single transformations.

The dependence on μ: The random walks are defined using an auxiliary distribution μ on G, but the equidistribution phenomenon obtained is independent of μ, provided μ is generating G.

Let (X, m, G) be a measure-preserving system and π denote the unitary G-representation on L^{2}(X, m). As the constant functions form an invariant subspace, the orthogonal complement

L_{0}^{2}(X, m) = {f ∈ L^{2}(X, m) |∫_{X} f dm = 0}

is π(G)-invariant as well, and we shall denote by π_{0} the restriction of π to L_{0}^{2}(X, m). Given a probability measure μ on G form the average operator π_{0}(μ) on L_{0}^{2}(X, m). Then π_{0}(μ) is a semi-contractions, i.e. ∥π_{0}(μ)∥ ≤ 1, and if μ is symmetric then π_{0}(μ) is self adjoint. In any case, π_{0}(μ̌ ∗ μ) = π_{0}(μ)^{∗}π_{0}(μ) is a positive self-adjoint semi-contraction.

Suppose that μ is generating, i.e. grp(μ) = G. The following conditions are well known to be equivalent to ergodicity of the system (X, m, G):

(a) The G-representation π_{0} on L_{0}^{2}(C, m) has no non-trivial invariant vectors.

(b) The operator π_{0}(μ) has no non-trivial fixed vectors in L_{0}^{2}(X, m).

(c) m is the unique G-invariant mean on L^{p}(X, m) for 1 ≤ p < ∞. (Recall that a mean on L^{p}(X, m) for 1 ≤ p ≤ ∞ is a positive, finitely additive functional, normalized by m(1) = 1.)

We shall say that a probability measure μ on a locally compact group G is aperiodic if it is not supported on a coset of a proper closed subgroup of G, equivalently if grp(μ̌ ∗ μ) = G. Observe that if μ is aperiodic on G, then condition (b) above can be replaced by (b’) The semi-contraction π_{0}(μ) has no eigenvalues on the unit circle.

The following result describes natural strengthennings of conditions (a), (b’), (c). We shall formulate it for actions of discrete countable groups Γ.

Theorem 3.5 (Schmidt [62], Furman-Shalom [10]). Let (X, m, Γ) be a measure-preserving system, where Γ is a discrete countable group, and let μ be an aperiodic probability measure on Γ. Then the following conditions are equivalent:

(a) ∥π_{0}(μ)∥ < 1.

(b) The Γ-representation π_{0} on L_{0}^{2}(X, m) does not have almost invariant vectors.

(c) m is the unique Γ-invariant mean on L^{∞}(X, m).

If the above conditions are satisfied the system (X, m, Γ) will be called strongly ergodic.

Remarks 3.6. (i) Theorem 3.5 has a partial generalization to actions of locally compact groups, in which case the implications (c) ⇒ (b) ⇒ (a) hold assuming certain regularity of μ (see [10]), but the equivalence of these conditions ceases to hold in general.

(ii) Strong ergodicity is a feature of non-amenability. In fact, a group Γ is amenable iff Γ has no strongly ergodic actions. On the other hand a group Γ has Kazhdan’s property (T) iff all its ergodic actions are strongly ergodic (Connes-Weiss [5]).

(iii) There are many natural examples of strongly ergodic actions beyond those of Kazhdan groups. For example, any subgroup Γ ⊆ SL_{n}(ℤ), which is not virtually abelian and acts irreducibly on ℝ^{n}, acts strongly ergodically on the torus 𝕋^{n} = ℝ^{n}/ℤ^{n} (see Furman-Shalom [10] for details and further examples).

It is well known that in the classical ergodic theorems for ℤ or ℕ, von Neumann’s mean ergodic theorem or Birkhoff’s pointwise ergodic theorem, no uniform rate of convergence can be achieved for all L^{2} or even L^{∞} functions in any non-trivial action. However, the following result shows that strongly ergodic actions exhibit certain universal rate of convergence in the context of Kakutani’s Random Ergodic Theorem:

Theorem 3.7 (Furman-Shalom [10]). Let (X, m, G) be a measure-preserving ergodic system and μ a probability measure on G so that the unitary G-representation on L_{0}^{2}(X, m) has a spectral gap ∥π_{0}(μ)∥ < 1. Given an arbitrary measurable function f on (X, m) form the sequence of partial sums

F_{n}(f, ω, x) = ∑_{k=1}^{n} f(S_{k}(ω) · x) where S_{k}(ω) = ω_{k} ⋯ ω_{1}

Then, for any f ∈ L^{2}(X, m) and any ϵ > 0, for P-a.e. ω ∈ Ω one has Random Mean Ergodic Theorem:

∥\frac{1}{n}F_{n}(f, ω, ·) − ∫ f dm∥_{L^{2}(X,m)} = o (\frac{log^{1/2+ϵ} n}{\sqrt{n}})  (3.2)

Random Pointwise Ergodic Theorem: for m-a.e. x ∈ X

|\frac{1}{n}F_{n}(f, ω, x) − ∫ f dm| = o (\frac{log^{3/2+ϵ} n}{\sqrt{n}})

Functional CLT: If f ∈ L^{p}(X, m) for some p > 2, and f is not a constant function, then there exists a σ > 0 so that the linear approximation Σ_{n}(ω, x, t), t ∈ [0, 1], determined by the discrete values

Σ_{n}(ω, x, k/n) = \frac{1}{σ\sqrt{n}} (F_{k}(f, ω, x) − k ∫_{X} f dm) , (k = 0, …, n)

converge in distribution (w.r.to P × m) to the canonical Wiener measure on C([0, 1]).

Remarks 3.8. (i) The spectral gap assumption in the Theorem is essential: if π_{0} has almost invariant vectors in L_{0}^{2}(X, m) then for every sequence a_{n} → 0 one can find an f ∈ L^{2}(X, m) so that ∥F_{n}(f, ω, ·) − ∫ f dm∥ ≥ a_{n} infinitely often with P-probability one.

(ii) Recall that for a measure-preserving system (X, m, G) to be weakly mixing (i.e. not to have finite dimensional G-invariant subspaces in L^{2}(X, m)) is equivalent to ergodicity of the diagonal action (X ×X, m×m, G). If the latter diagonal action has a spectral gap ∥π_{0} ⊗ π_{0}(μ)∥ < 1 then random walks on G have exponentially fast mixing properties for the original G-action on (X, m) (see [10]).

3.3. Entropy for random transformations. Let Φ be a family of invertible measurable transformations of a probability measure space (X, ν), μ - a probability measure on Φ so that ν is a μ-stationary measure. As before we denote by G the group generated by Φ, (Ω, P) = (Φ^{ℕ}, μ^{ℕ}) and denote by S_{n} = ω_{n} ⋯ ω_{1} the product (actually composition) of μ-distributed independent Φ-valued variables, which we call μ-random transformations.

In [41] Kifer introduced the following notion of entropy for this setting of random transformations. Given a finite measurable partition P of (X, ν), define

h̃^{μ}(P, G) = lim_{n→∞} \frac{1}{n} ∫_{Ω} H(S_{n−1}^{−1}P ∨ ⋯ ∨ S_{1}^{−1}P ∨ P) dP

and let h̃^{μ}(X, G) = sup_{P} h̃^{μ}(P, G) over all finite measurable partitions P of (X, ν). One can verify that the limit exists. In fact, h̃^{μ}(X, Φ) is precisely the relative, or fiber, entropy introduced by Abramov and Rohlin for the computation of an entropy of a skew-product relative to the base system. In our setup the skew-product is (Ω × X, P × ν, T), T : (ω, x) ↦ (θω, ω_{1}(x)) and the base is (Ω, P, θ). So one has

h(Ω × X, T) = h(Ω, θ) + h̃^{μ}(X, G)

but h̃^{μ}(X, G) is well defined (finite or infinite) even when the other two entropies are infinite.

It is clear from the definition that h̃^{μ}(X, G) is an invariant with respect G-equivariant measurable isomorphisms, and it does not increase when (X, ν) is replaced by a measurable equivariant factor. One also has

h̃^{(μ^{p})}(X, G) = p · h̃^{μ}(X, G)  (3.3)

for every convolution power μ^{p} of μ.

If ν is not only stationary, but rather invariant measure for μ-a.e. g ∈ G, then for every finite partition P the functions

f_{n}(ω) = H(S_{n−1}^{−1}(ω)P ∨ ⋯ ∨ S_{1}(ω)^{−1}P ∨ P)

satisfy

f_{n+m}(ω) = H(∨_{i=n}^{n+m−1}S_{i}(ω)^{−1}P ∨ ∨_{j=0}^{n−1}S_{j}(ω)^{−1}P)
≤ H(∨_{i=n}^{n+m−1}S_{i}(ω)^{−1}P) + H(∨_{j=0}^{n−1}S_{j}(ω)^{−1}P)
≤ H(S_{n}(ω)^{−1} ∨_{i=0}^{m−1} S_{i}(θ^{n}ω)^{−1}P) + H(∨_{j=0}^{n−1}S_{j}(ω)^{−1}P)
= f_{m}(θ^{n}ω) + f_{n}(ω)

Thus {f_{n}(ω)} form a bounded (f_{n} ≤ H(P) < ∞) subadditive cocycle over the ergodic system (Ω, P, θ), and by Kingman’s subadditive ergodic theorem for P-a.e. ω

lim_{n→∞} \frac{1}{n}H(S_{n−1}(ω)^{−1}P ∨ ⋯ ∨ P) = h̃^{μ}(P, G)  (3.4)

In this case one also has h̃μ̌(X, G) = h̃^{μ}(X, G), where μ̌ denotes the reflected measure dμ̌(g) = dμ(g^{−1}).

In some cases h̃^{μ}(X, G) can be related to the usual entropy. For example, if μ is a distribution on powers {T^{n}, n ∈ ℤ} of a single measure-preserving transformation T (or on an ℝ-flow T^{t}) of a measures space (X, ν), then assuming that μ has finite first moment

∫ |x| dμ(x) < ∞

one can prove that

h̃^{μ}(X, ℤ) = |bar(μ)| · h(X, T)

where h(X, T) is the Kolmogorov-Sinai entropy and bar(μ) ∈ ℝ is the barycenter of of μ, defined by bar(μ) = ∫ x dμ(x). In particular, in the case of a measurable action of Γ = ℤ (or ℝ) the entropy of μ-random transformations vanishes for all symmetric μ with finite first moment.

3.4. Lyapunov exponents and non-random filtration. Let (X, ν) be a probability measure and G a group of measurable transformations endowed with a Borel structure and a probability measure μ on G so that ν is μ-stationary. Let D : G×X → GL_{k}(ℝ) be a measurable cocycle, i.e. D(gh, x) = D(g, h · x)D(h, x) for ν-a.e. x and μ-a.e. g, h ∈ G. Impose the following integrability condition on D

∫_{G} ∫_{X} (log^{+} ∥D(g, x)∥ + log^{+} ∥D(g, x)^{−1}∥) dν(x) dμ(g) < ∞  (3.5)

where log^{+} t = max{0, log t}.

Consider the skew-product (Ω × X, P × ν, T) and the matrix-valued measurable function A : Ω × X → GL_{k}(ℝ) defined by A(ω, x) = D(ω_{1}, x). For n ≥ 1 set

A_{n}(ω, x) = (A ∘ T^{n−1} ⋯ A ∘ T · A)(ω, x) = D(ω_{n} ⋯ ω_{2}ω_{1}, x) = D(S_{n}(ω), x)

Then A_{n+m} = A_{n} ∘T^{m}A_{m} for all m, n ∈ ℕ. Finite first moment condition (3.5) means that both log^{+} ∥A∥ and log^{+} ∥A^{−1}∥ are in L^{1}(P × ν).

Assume for convenience that the skew-product transformation T is ergodic (see Remark 3.2.(b)). Then the matrix-valued function A : Ω × X → GL_{k}(ℝ) gives rise to the Lyapunov exponents λ_{1} ≥ λ_{1} ≥ ⋯ ≥ λ_{k} defined inductively by

λ_{1} + ⋯ + λ_{p} = lim_{n→∞} \frac{1}{n} ∫_{Ω×X} log ∥ ∧^{p} A_{n}(ω, x)∥ dP × ν
= lim_{n→∞} \frac{1}{n} ∫_{G} ∫_{X} log ∥ ∧^{p} D(g, x)∥ dν(x) dμ^{n}(g)

for p = 1, 2, …, k. Since these exponents are associated to the cocycle D : G × X → GL_{k}(ℝ) and the measure μ, we shall denote them by

λ_{1}(D, μ) ≥ ⋯ ≥ λ_{k}(D, μ)

By Oseledec theorem there exists a measurable filtration

{0} = E_{s+1}(ω, x) ⊂ E_{s}(ω, x) ⊂ ⋯ ⊂ E_{2}(ω, x) ⊂ E_{1}(ω, x) = ℝ^{n}

of ℝ^{n} by nested subspaces of dimensions dim E_{i}(ω, x) = τ_{i}, i = 1, …, s, where τ_{i} are precisely the indices of strict inequalities λ_{τ_{i}−1}(D, μ) > λ_{τ_{i}}(D, μ), and E_{i}(ω, x) are characterized by the property that u ∈ E_{i}(ω, x) \ E_{i+1}(ω, x) iff

lim_{n→∞} \frac{1}{n} log ∥A_{n}(ω, x)u∥ = lim_{n→∞} \frac{1}{n} log ∥D(S_{n}(ω), x)u∥ = λ_{τ_{i}}(D, μ)

Moreover the subspaces E_{i}(ω, x) are related by A(ω, x)E_{i}(ω, x) = E_{i}(T (ω, x)), i.e.

D(ω_{1}, x)E_{i}(ω, x) = E_{i}(θω, ω_{1} · x)

However, besides the Oseledec filtration, which depends on both x ∈ X and ω ∈ Ω, there is a non-random filtration analogous to the one described in section 1.4, which describes the P-a.e. constant growth rate n^{−1} log ∥D(S_{m}(ω), x) u∥ in terms of x and u ∈ ℝ^{k} only. More precisely

Theorem 3.9 (Kifer, [41], chapter III). Let G, μ, (X, ν) be as above and let D : G × X → GL_{k}(ℝ) be a measurable cocycle satisfying (3.5). Assume that ν is an ergodic μ-stationary measure. Then there exist integers k = k_{1} > k_{2} > ⋯ > k_{r} > k_{r+1} = 0, constants

λ̃_{1}(D, μ) > λ̃_{2}(D, μ) > ⋯ > λ̃_{r}(D, μ) with λ̃_{1}(D, μ) = λ_{1}(D, μ)

and a measurable map X → F_{(k_{r},...,k_{1})} which assigns to ν-a.e. x ∈ X a flag

{0} = L_{r+1}(x) ⊂ ⋯ ⊂ L_{2}(x) ⊂ L_{1}(x) = ℝ^{k} dim E_{i}(x) = k_{i}

so that for each i = 1, …, r one has for ν-a.e. x ∈ X

D(g, x)E_{i}(x) = E_{i}(g · x) for μ − a.e. g

and u ∈ E_{i}(x) \ E_{i+1}(x) iff for P-a.e. ω

lim_{n→∞} \frac{1}{n} log ∥D(S_{n}(ω), x)u∥ = λ̃_{i}(D, μ)

Remarks 3.10. (a) This non-random filtration can also be described in terms of the supports of μ-stationary measures on X × ℙ^{k−1} which project onto ν, where G acts on X × ℙ^{k−1} by g : (x, ū) ↦ (g · x, D(g, x) · ū). See Kifer [41] for the precise statements which generalize the ones in Theorem 1.11, where the latter corresponds to the case of trivial X = {pt}.

(b) As in Theorem 1.11 let us point out that the exponents λ̃_{1}(D, μ), …, λ̃_{r}(D, μ) form a subset of the Lyapunov exponents λ_{1}(D, μ), …, λ_{k}(D, μ), and typically the former is a proper subset of the latter.

(c) If the cocycle D : G × X → GL_{k}(ℝ) is measurably irreducible in the sense that there does not exist a measurable B : X → GL_{k}(ℝ) (it is sufficient to consider B : X → SO(k)) so that the cocycle D_{1}(g, x) = B(g · x)^{−1}D(g, x) B(x) takes values in a reducible subgroup of GL_{k}(ℝ), then r = 1 and every u ≠ 0 satisfies

\frac{1}{n} log ∥D(S_{n}(ω), x)u∥ = λ̃_{1}(D, μ) = λ_{1}(D, μ)

for P-a.e. ω and ν-a.e. x ∈ X.

(d) One of the applications of Theorem 3.9 is in the study of regularity of λ_{1}(D, μ) as μ varies (compare Theorem 1.46). For example, one can show that if ν is an invariant measure and μ_{n} → μ are measures on a fixed, say locally compact group G, so that μ is generating, μ_{n} satisfy a uniform first bound condition (3.5) and D is measurably irreducible, then λ_{1}(D, μ_{n}) → λ_{1}(D, μ). See Kifer [41] for further results.

(e) If ν is μ-stationary but not ergodic, then one can apply the theorem to the ergodic components to obtain a non-random filtration where r, the dimensions k_{i} = dim E_{i}, and the corresponding exponents λ̃_{i} might depend on x but not on ω.

One of the natural situations where a cocycle G : G × X → GL_{k}(ℝ) appears naturally is when X is a (compact) Riemannian manifold and G is a group of diffeomorphisms of X, with D representing the derivative cocycle (under some measurable trivialization of the tangent bundle T X). The non-random filtration, described in Theorem 3.9 gives in this context a measurable family of G-invariant subbundles of T X, which describe the (non-random) growth of the tangent vectors to X under P-a.e. random product of diffeomorphisms from G.

3.5. Entropy for random diffeomorphisms. Let X be a compact Riemannian k-manifold and μ be a probability measure on G ⊂ Diff^{2}(X), considered with the Borel structure defined by the C^{2}-topology. Let ν be a μ-stationary measure on X. Choosing a measurable trivialization of the tangent bundle T X one can consider the derivative cocycle

D : G × X → GL_{k}(ℝ) D(gh, x) = D(g, h · x)D(h, x)

Assume that μ on G has finite first moment in the sense of (3.5), and consider the case where μ-stationary measure ν is in the Lebesgue class on the manifold X.

Theorem 3.11 (Kifer, [41] p. 160). With the notations as above, assume that the μ-stationary measure ν is in the Lebesgue class on X. Then

∑_{i=1}^{k} λ_{i}(D, μ) ≤ 0

and the equality holds if and only if ν is invariant, i.e. gν = g for μ-a.e. g ∈ G.

The next result requires a technical assumption of integrability of the C^{2}-norms |g|_{c^{2}} of the elements g ∈ G ⊂ Diff^{2}(X)

∫_{G} log^{+} |g|_{C^{2}} dμ(g) + log^{+} |g^{−1}|_{C^{2}} dμ(g) < ∞

Theorem 3.12 (Ledrappier-Young, [51]). Let G, μ, (X, ν) be as above where the μstationary measure ν on X is in Lebesgue class. Then

h̃^{μ}(X, G) = ∑_{λ_{i}(D,μ)≥0} λ_{i}(D, μ)

This is a random version of Pesin’s formula for a single diffeomorphisms preserving a smooth measure. In the case of ν being invariant it can be deduced from the relative version of Pesin’s formula on the skew-product. However, in Theorem 3.12 the individual diffeomorphisms are not assumed to preserve a smooth measure. Ledrappier and Young [51] further prove that if λ_{1}(D, μ) > 0 then the limit measures ν_{ω} = ω_{1} ⋯ ω_{n}ν are SRB measures with respect to a random (i.e. ω-dependent) stable foliation.

3.6. Measure-preserving K^{μ}-property. Let us return to a general measurable setting, focusing on measure-preserving actions. Consider, say locally compact, group G acting ergodically by measure-preserving transformations on a probability space (X, m) (we use m for the measure to emphasize that it is invariant). Take a generating probability measure μ on G. We shall consider μ and μ-random walks on G as an auxiliary tool for the study of the G-action on (X, m). The entropy h̃^{μ}(X, G) is an invariant of the action (X, m, G) which depends on μ.

The definition of entropy for μ-random transformations, is quite similar to the Kolmogorov-Sinai definition of the entropy of a single transformation. In particular, one has a natural random (i.e. path-dependent) analogues of the time direction, past and future, which allow to generalize many classical results to the random entropy setup. For example, the theory of K-automorphisms for actions of ℤ admits a generalization to actions of general groups G in the the random walk setting.

Theorem 3.13 ([9]). Let (X, m, G) be an ergodic measure preserving system, and let μ be a generating probability measure on G. The following conditions are equivalent:

(a) Any non-trivial measurable G-equivariant factor system (X_{0}, m_{0}, G) has positive μ-entropy h̃^{μ}(X_{0}, G) > 0.

(b) For any finite partition P of (X, m) the tail partition

P^{∞}(ω) = ∧_{n=1}^{∞} ∨_{k=n}^{∞} S_{k}(ω)^{−1}P

is measurably trivial for P-a.e. ω.

(c) For any non-trivial partition P for P-a.e. ω ∈ Ω one has

lim_{n→∞} sup_{A∈P, B∈∨_{k=n}^{∞}S_{k}(ω)^{−1}P} |m(A ∩ B) − m(A) · m(B)| = 0

A system (X, m, G) satisfying the above conditions can be called K^{μ}-system.

Remark 3.14. Condition (c) describes a phenomenon of strong (uniform) mixing along μ-random walks, taking place in μ-K-systems. However, this type of mixing, in general, does not imply that the Γ-action on (X, m) is mixing in the usual sense.

Example 3.15. Let Γ ⊆ SL_{k}(ℤ) act on the k-torus 𝕋^{k} = ℝ^{k}/ℤ^{k}, whose action on ℝ^{k} is strongly irreducible. Then (𝕋^{n}, Γ) is a K^{μ}-system for any generating probability measure μ on Γ with finite first moment.

3.7. Cocycle growth along μ-random walks. Let (X, m, G) be an ergodic measurepreserving system. The value of the numeric invariant h̃^{μ}(X, G) of the action typically depends on the auxiliary generating measure μ on G (observe the relation (3.3)). It is natural, however, to ask to what extent the positivity of h̃^{μ}(X, G) depends on μ ?

In the smooth setup, where X is a compact Riemannian k-manifold with the Riemannian volume m and G ⊂ Diff_{m}^{2}(X) is a group of volume preserving diffeomorphisms, positivity of the μ-entropy h̃^{μ}(X, G) > 0 is equivalent, by Theorem 3.12, to the condition

λ_{1}(D, μ) > 0

where D : G × X → SL′_{k}(ℝ) is the derivative cocycle (the target group is SL′_{k}(ℝ) rather than GL_{k}(ℝ) because we assume that elements of G preserves the volume m). So the question becomes: under what conditions on (X, m, G) and D : G × X → SL′_{k}(ℝ) the alternative λ_{1}(D, μ) > 0 or λ_{1}(D, μ) = 0 does not depend on an auxiliary measure μ ?

One can consider the following general setting: G is a group acting ergodically by measure-preserving transformations on an abstract probability space (X, m), and D : G × X → SL′_{k}(ℝ) is a measurable cocycle with

∫_{X} log ∥D(g, x)∥ dm(x) < ∞ ∀ g ∈ G

(this is automatically satisfied in the context of diffeomorphisms). Let μ be a generating probability measure on G and assume that μ has finite first moment in the following sense

∫_{G} ∫_{X} log ∥D(g, x)∥ dm(x) dμ(g) < ∞  (3.6)

Let λ_{1}(D, μ) ≥ ⋯ ≥ λ_{k}(D, μ) denote the corresponding Lyapunov exponents.

Observe that if (X, m) is the trivial space (single point) then any cocycle D : G×{pt} → SL′_{k}(ℝ) is just a homomorphism D : G → SL′_{k}(ℝ). Condition (3.6) in this case is equivalent to the finite first moment condition for the push-forward measure D_{∗}μ on SL′_{k}(ℝ), and λ_{p}(D, μ) = λ_{p}(D_{∗}μ), p = 1, 2, …, k. Recall that Furstenberg’s condition 1.14 for λ_{1}(D_{∗}μ) > 0 is stated in terms of the group grp(D_{∗}μ) ⊆ SL′_{k}(ℝ) rather than the measure D_{∗}μ itself. For every generating μ on G one has grp(D_{∗}μ) = \overline{D(G)} and assuming that the latter is not compact, and is strongly irreducible one has λ_{1}(D, μ) = λ_{1}(D_{∗}μ) > 0.

In the framework of measurable cocycles D : G × X → SL′_{k}(ℝ) we shall use the following:

Definition 3.16. Let G be a group acting by measure-preserving transformations on a probability space (X, m). Two cocycles C, D : G × X → L = SL′_{k}(ℝ) are said to be (measurably) cohomologous if there exists a measurable function B : X → L so that D(g, x) = B(g · x)^{−1}C(g, x)B(x) for every g ∈ G and m-a.e. x ∈ X. A cocycle D : G × X → L is said to be non-compact/strongly irreducible/Zariski dense respectively if it is not measurably cohomologous to a cocycle C : G × X → L_{0} ⊂ L, taking values in a compact/virtually reducible/proper algebraic subgroup L_{0} in L = SL′_{k}(ℝ), respectively.

With these definitions we can formulate a cocycle generalization of Furstenberg’s condition for non-triviality of the Lyapunov spectrum, for measurable cocycles over strongly ergodic systems. To avoid certain technicalities assume that G is a discrete countable group.

Theorem 3.17 ([9]). Let (X, m, G) be a strongly ergodic system and D : G × X → L = SL′_{k}(ℝ) be a measurable cocycle. Assume that the cocycle D is strongly irreducible and non-compact. Then for any generating measure μ on G with finite first moment (3.6) one has λ_{1}(D, μ) > 0.

We postpone the proof of this result to the end of the section.

Remarks 3.18. (a) The assumption that (X, m, G) is strongly ergodic is used in

the proof below, but it might be unnecessary for the result.

(b) It is conceivable that Theorems 1.23 and 1.25 can also be extended to the cocycle setting. In particular, it is plausible that the cocycle Lyapunov spectrum is simple, i.e. λ_{1}(D, μ) > λ_{2}(D, μ) > ⋯ > λ_{k}(D, μ), provided D : Γ × X → SL′_{k}(ℝ) is Zariski dense (and possibly assuming the action (X, m, Γ) is strongly ergodic).

Theorem 3.17 can be used in the analysis of not strongly-irreducible cocycles as well (for that one needs to consider certain finite extensions of the original G-action and reducible cocycles over these extensions). For a general cocycle D : G×X → SL′_{k}(ℝ) over a strongly ergodic system (X, m, G) it can be proved that for all symmetric generating measures μ on G satisfying finite first moment condition either λ_{1}(D, μ) = 0 or λ_{1}(D, μ) > 0 for all symmetric measures μ with finite first moment (the first case corresponds to the algebraic hull of the cocycle D being amenable). This gives the following

Corollary 3.19. Let X be a compact Riemannian manifold, Γ ⊂ Diff_{m}^{2}(X) a countable group of volume-preserving diffeomorphisms such that the Γ-action on (X, m) is strongly ergodic. Then the entropy h̃^{μ}(X, Γ) either vanishes or is strictly positive simultaneously for all symmetric generating measures μ on Γ with finite first moment (3.6).

In other words, under the above assumptions of smoothness and strong ergodicity the positivity of random entropies h̃^{μ}(X, Γ) is an invariant of the action (X, m, Γ) which is independent of the choice of the symmetric generating μ on Γ.

In the special case of G being a discrete group Γ with Kazhdan’s property T, a sharper dichotomy can be proved, namely

Theorem 3.20 ([9], [71]). Let Γ be a discrete group with Kazhdan property T acting by C^{∞} diffeomorphisms on a compact manifold X preserving a smooth measure m. Then either h̃^{μ}(X, Γ) > 0 for all aperiodic probability measures μ on Γ, or the system (X, m, Γ) has a discrete spectrum, i.e. the Γ-action on (X, m) is measure-theoretically isomorphic to an action of Γ on K/K_{0} given by γ : kK_{0} ↦ ρ(γ)kK_{0} where ρ : Γ → K is a homomorphism into a compact group K and K_{0} ⊆ K is a closed subgroup.

The last theorem combines two results. The first ([9]) states that for groups Γ with property T any measurable cocycle D : Γ × X → SL′_{k}(ℝ) over an ergodic system (X, m, Γ) either has λ_{1}(D, μ) ≥ ϵ(μ) > 0, where ϵ(μ) > 0 depends only on Γ, or D is a compact cocycle. The second result, due to Zimmer [71], states that for C^{∞}-smooth actions of groups Γ with Kazhdan’s property T if the derivative cocycle is compact (equivalently Γ preserves a measurable Riemannian structure on the manifold) then the underlying system (X, m, Γ) has discrete spectrum. Note that this last implication does not hold for ℤ-actions (see Gunesch and Katok [29]; the appendix to this paper grew out of only partially successful attempt of the author to generalize Zimmer’s result [71] to a group G ⊂ Diff^{1}(X) which acts strongly ergodically on (X, m)).

Let us also point out that for higher rank lattices Γ one can deduce the above results (and further information on entropies of individual elements of Γ) using superrigidity for measurable cocycles (Zimmer [72]). Sketch of the Proof of Theorem 3.17. We shall use a cocycle analogue of the unitary representation approach discussed in section 1.6. Consider the product measure space (X̃, m̃) = (X × ℙ^{k−1}, m × ν_{0}), where ν_{0} denotes the Lebesgue measure on the projective space ℙ^{k−1}. The measurable G-action defined by g : (x, ū) ↦ (g · x, D(g, x) · ū) leaves the measure m̃ quasi-invariant, and therefore defines a quasiregular Γ-representation π̃ on L^{2}(X̃, m̃) by

π̃(g)F (x, ū) = \sqrt{ρ(g, x, ū)} F (g^{−1} · x, D(g^{−1}, x) · ū)

where ρ is the multiplicative Radon-Nikodym cocycle

ρ(g, x, ū) = \frac{dgm̃}{dm̃}(x, ū) = \frac{d D(g, x)ν_{0}}{dν_{0}} (ū)

We claim that under the assumptions of the theorem there is a spectral gap ∥π̃(μ)∥_{sp} < 1.

Assume that ∥π̃(μ)∥_{sp} = 1. Then (see analogous argument in the proof of Proposition 1.17) for T = π̃(μ) or for T^{∗} = π̃(μ̌) there exists a complex number z with |z| = 1 which is either an eigenvalue for T, or an approximate eigenvalue for T, or z̄ is an eigenvalue for T^{∗}. In any case, taking μ′ = μ or μ̌, and z′ = z or z̄ one can find a sequence of unit vectors F_{n} ∈ L^{2}(X̃, m̃) so that

∥π̃(μ′)F_{n} − z′F_{n}∥ → 0  (3.7)

Consider the measures m̃_{n} on X̃ defined by dm̃_{n}(x, ū) = |F_{n}(x, ū)|^{2} dm(x) dν_{0}(ū) and their disintegrations m̃_{n} = ∫_{X} η_{n,x} dm(x) with respect to (X, m). The total masses of |η_{n,x}| = η_{n,x}(ℙ^{k−1}) of the positive measures η_{n,x} are given by the non-negative functions

f_{n}(x) = η_{n,x}(ℙ^{k−1}) = ∥F_{n}(x, ·)∥_{L^{2}(ℙ^{k−1},ν_{0})}^{2}

which are unit vectors in L^{1}(X, m) because ∥f_{n}∥_{1} = ∥F_{n}∥_{2}^{2} = 1. Passing to a subsequence, one can assume that f_{n} weakly converge to some ϕ - a positive normalized functional on L^{∞}(X, m), i.e. a mean on L^{∞}(X, m). Cauchy-Schwartz inequality gives for every g ∈ G

∥f_{n} ∘ g^{−1} − f_{n}∥_{1}
= ∫_{X} |∫_{ℙ_{k−1}} (|F_{n}(g^{−1} · x, ū)|^{2} \frac{dD(g, x)ν_{0}}{dν_{0}} (ū) − |F_{n}(x, ū)|^{2}) ν_{0}(ū)| dm(x)
≤ ∫_{X} ∫_{ℙ_{k−1}} |π̃(g)F_{n} − z′F_{n}| · |π̃(g)F_{n} + z′F_{n}| dν_{0} dm
≤ 2 · ∥π̃(g)F_{n} − z′F_{n}∥_{2}

which, in view of (3.7) converge μ′-a.e. to 0. Thus the mean ϕ is g-invariant for μ′-a.e. g ∈ G, so that ϕ is G by continuity and therefore ϕ = m. Thus f_{n}(x) = |η_{n,x}| → 1 for m-a.e. x.

Passing, if necessary to a further subsequence, one can assume that η_{n,x} converge weakly to probability measures: η_{n,x} → η_{x} ∈ P(ℙ^{k−1}), and (3.7) and Cauchy-Schwartz inequality can be used to show that that the measure ∫_{X} δ_{x} × η_{x} dm(x) on X × ℙ^{k−1} is G-invariant, which means that

D(g, x) · η_{x} = η_{g·x}  (3.8)

for m-a.e. x ∈ X.

The following argument is known as Zimmer’s “Cocycle Reduction Lemma”. The natural action of L = SL′_{k}(ℝ) on the space of probability measures P(ℙ^{k−1}) is tame, which (by definition) means that there exists a countable collection {M_{i} ⊂ P(ℙ^{k−1})} of L-invariant Borel sets, which separates points of P(ℙ^{k−1}). Let X_{i} = {x ∈ X | η_{x} ∈ M_{i}}. By (3.8) each X_{i} is a G-invariant set, and by ergodicity it has m-measure 0 or 1. Intersecting those X_{i} which have m(X_{i}) = 1 and the complements of the others, gives a measurable subset X′ ⊆ X for which {η_{x} | x ∈ X′} lie in a single L-orbit L·η_{0} on P(ℙ^{k−1}), i.e. η_{x} = A(x) · η_{0} for some measurable A : X′ → L and η_{0} ∈ P(ℙ^{k−1}). Let L_{0} denote the stabilizer of η_{0} in L, then the cocycle

C(g, x) = A(g, x)^{−1}D(g, x)A(x)

satisfies C(g, x) · η_{0} = η_{0}, i.e. takes values in L_{0} ⊂ L. If η_{0} is a proper measure, then L_{0} is a compact subgroup in L = SL′_{k}(ℝ); while if η_{0} is not proper then L_{0} is not strongly irreducible. Both cases contradict the assumption on D. Hence there is a spectral gap ∥π̃(μ)∥ < 1.

Recall that for A ∈ SL′_{k}(ℝ) one has

log ∥A∥ ≥ −\frac{2}{k} log (∫_{ℙ_{k−1}} \sqrt{\frac{dA^{−1}ν_{0}}{dν_{0}} (ū)} dν_{0}(ū))

which gives, using the convexity of − log() the following estimate

\frac{1}{n} ∫_{G} ∫_{X} log ∥D(g, x)∥ dm(x) dμ^{n}(g)
≥ \frac{1}{n} ∫_{G} ∫_{X} −\frac{2}{k} log (∫_{ℙ_{k−1}} \sqrt{ρ(g, x, ū)} dν_{0}(ū)) dm(x) dμ^{n}(g)
≥ −\frac{2}{nk} log (∫_{G} ∫_{X} ∫_{ℙ_{k−1}} \sqrt{ρ(g, x, ū)} dν_{0}(ū) dm(x) dμ^{n}(g))
= −\frac{2}{nk} log ⟨π̃(μ^{n})1, 1⟩ ≥ \frac{2}{k} log \frac{1}{∥π̃(μ^{n})∥^{1/n}}

so that λ_{1}(D, μ) ≥ 2/k log (1/ ∥π̃(μ)∥_{sp}) > 0.

4. Selected Proofs

This section contains the proofs of the statements marked by a star in the paper. We use the notations of the corresponding statements.

Proof of Lemma 1.7. Suppose that μ is a probability measure on G = SL′_{k}(ℝ) and ν a μ-stationary measure on ℙ^{k−1}. Assume that ν is not proper, and let {0} ⊂ W ⊂ ℝ^{k} be a proper subspace of minimal dimension, such that ν(\overline{W}) > 0. The function f : G → [0, 1] defined by f(g) = gν(W) = ν(\overline{g}^{−1}\overline{W}) satisfies

∫_{G} f(gh) dμ(h) = ∫_{G} ghν(\overline{W}) dμ(h)
= g μ ∗ ν(\overline{W}) = gν(\overline{W}) = f(g)

By minimality of dim W, one has ν(\overline{g}\overline{W} ∩ \overline{g}′\overline{W}) = 0, whenever gW ≠ g′W. Hence the set of gW with f(g) = ν(\overline{g}\overline{W}) > ϵ is finite for any ϵ > 0. Thus f reaches its maximum value f(g_{0}) = max{f(g) | g ∈ G}, while the relation above yields that f(g_{0}h) = f(g_{0}) for μ-a.e. h ∈ G. Thus there exists a finite collection E of proper subspaces, s.t. h^{−1}g_{0}^{−1}W belongs to E for μ-a.e. h, and therefore the whole group G_{μ} = grp(μ) permutes the finite collection E of proper subspaces. Consequently, μ is not strongly irreducible.

On the other hand, assuming that μ is not strongly irreducible there is a finite collection \overline{E} = \overline{W}_{1} ∪ ⋯ ∪ \overline{W}_{r} ⊂ ℙ^{k−1} of proper projective subspaces, which is invariant under μ-a.e. g ∈ G. Being a compact invariant subset of ℙ^{k−1} the set \overline{E} supports μ-stationary measures ν which are not proper.

Proof of Proposition 1.17. Let X be a compact metric space, H a locally compact group acting continuously on X, and μ - a generating probability measure on H. Let {ν_{n}} be a sequence of H-quasi-invariant probability measures on X, such that the unitary quasi-regular H-representations π_{ν_{n}} on L^{2}(X, ν_{n}) defined by

π_{n}(h)f(x) = \sqrt{\frac{dhν_{n}}{dν_{n}} (x)}f(h^{−1} · x)

satisfy ∥π_{ν_{n}}(μ)∥_{sp} → 1. Denote by T_{n} = π_{ν_{n}}(μ) the (semi)contractions on the Hilbert spaces H_{n} = L^{2}(X, ν_{n}).

For a bounded operator T on a Hilbert space H, ∥T ∥_{sp} = max_{ρ∈σ(T)} |ρ|, where σ(T) denotes the spectrum of T. Recall that ρ ∈ σ(T) if (i) ρ is an eigenvalue for T, or (ii) ρ is an approximate eigenvalue for T, i.e.

∃f_{k} ∈ H : ∥f_{k}∥ = 1 and ∥T f_{k} − ρf_{k}∥ → 0

or (iii) Im (T − ρ) is not dense in H, in which case Im (T − ρ)^{⊥} = Ker(T − ρ)^{∗} = Ker(T^{∗} − ρ̄) ≠ {0}, i.e. ρ̄ is an eigenvalue for T^{∗}.

Therefore there always exists a complex number ρ with |ρ| = ∥T ∥_{sp} and unit vectors f_{k} in H such that ∥T f_{k} − ρf_{k}∥ → 0 or ∥T^{∗}f_{k} − ρf_{k}∥ → 0.

Recall that if T_{n} = π_{ν_{n}}(μ) then T_{n}^{∗} = π_{ν_{n}}(μ̌). Using the above argument about the spectrum, and assuming ∥T_{n}∥_{sp} → 1 one concludes that for μ′ = μ or μ̌ and an appropriate subsequence of ν_{n}, one can find functions f_{n} ∈ L^{2}(X, ν_{n}), and numbers ρ_{n} with |ρ_{n}| = ∥{∥_{sp} T_{n}} → 1, so that

∥f_{n}∥_{2} = 1 and ∥π_{ν_{n}}(μ′)f_{n} − ρ_{n}f_{n}∥_{2} → 0

Passing to a further subsequence, one can assume that ∥π(h)_{ν_{n}}f_{n} − ρ_{n}f_{n}∥ → 0 for μ′-a.e. h ∈ H.

Now consider the measures η_{n} on X defined by dη_{n}(x) = |f_{n}(x)|^{2} dν_{n}(x). These are probability measures because ∥f_{n}∥_{2}^{2} = 1. Since P(X) is compact, passing to a subsequence one can assume η_{n} → η ∈ P(X) in weak-* topology. Let us verify that η is H-invariant. Fix ϕ ∈ C(X) and ϵ > 0, then for n ≥ n(ϕ, ϵ) for every h ∈ H one has

|∫_{X} ϕ dhη − ∫_{X} ϕ η| ≤ |∫_{X} ϕ dhη_{n} − ∫_{X} ϕ η_{n}| + ϵ
= |∫_{X} ϕ(x) · (|f_{n}(h^{−1}x)|^{2} \frac{dh^{−1}ν_{n}}{dν_{n}} (x) − |f_{n}(x)|^{2}) dν_{n}(x)| + ϵ
≤ ∥ϕ∥_{∞} · ∥π_{ν_{n}}(h)f_{n} − ρ_{n}f_{n}∥_{2} · ∥π_{ν_{n}}(h)f_{n} + ρ_{n}f_{n}∥_{2} + ϵ
≤ 2∥ϕ∥_{∞} · ∥π_{ν_{n}}(h)f_{n} − ρ_{n}f_{n}∥_{2} + ϵ

Hence for μ′-a.e. h ∈ H one has h · η = η, and by continuity, η is invariant under grp(μ′) = H.

Proof of Lemma 1.18. Take ϵ > 0 and consider the non-negative series h_{ϵ}(ω) = ∑_{n=0}^{∞}(∥π(μ)∥ + ϵ)^{−n}⟨π(S_{n}(ω))f_{1}, f_{2}⟩. Term-by-term integration gives

∫_{Ω} h_{ϵ} dP = ∑_{n=0}^{∞}(∥π(μ)∥ + ϵ)^{−n}⟨π(μ^{n})f_{1}, f_{2}⟩ ≤ ∥f_{1}∥ · ∥f_{2}∥ · ∑_{n=0}^{∞} \frac{∥π(μ)∥^{n}}{(∥π(μ)∥ + ϵ)^{n}} < ∞

Hence for P-a.e. ω

lim sup_{n∞} \frac{1}{n} log ⟨π(S_{n}(ω))f_{1}, f_{2}⟩ ≤ log(∥π(μ)∥ + ϵ)  (4.1)

and taking ϵ → 0 one obtains log ∥π(μ)∥ as an upper bound for the LHS for P-a.e. ω. Repeating this argument with the convolution power μ^{p} one has P-a.e.

lim sup_{n∞} \frac{1}{n} log ⟨π(S_{n}(ω))f_{1}, f_{2}⟩ ≥ \frac{1}{p} log ∥π(μ^{p})∥ = log ∥π(μ)^{p}∥^{1/p} → log ∥π(μ)∥_{sp}

as p → ∞. This completes the proof of the Lemma.

Proof of Lemma 1.33. Given any ϕ ∈ C(Q) consider the bounded continuous function F_{ϕ} on G defined by F_{ϕ}(g) = ∫ ϕ dgν. Note that F_{ϕ} satisfies the following μ-mean value property:

∫_{G} F_{ϕ}(gh) dμ(h) = ∫_{G} ∫_{Q} ϕ(gh · u) dν(u) dμ(h)
= ∫ ϕ(g · v) dμ ∗ ν(u) = ∫_{Q} ϕ dν = F_{ϕ}(g)

so that the sequence of random variables w_{n} = F_{ϕ}(Y_{1} ⋯ Y_{n}) on (Ω, P) forms a bounded martingale, i.e. the conditional expectations satisfy

E(w_{n+1} | Y_{1}, …, Y_{n}) = w_{n}

The martingale convergence theorem implies that with probability one w_{n} converge to some value

w(ϕ, ω) = lim_{n→∞} ∫_{Q} ϕ dY_{1} ⋯ Y_{n}ν

Repeating this argument for a countable dense family of functions ϕ_{i} ∈ C(Q), one obtains for P-a.e. ω a positive normalized functional ϕ_{i} ↦ w(ϕ_{i}, ω), which can be extended to all of C(Q), and is therefore given by a probability measure, which we denote by ν_{ω}. The fact that ν is the average of ν_{ω} follows from Lebesgue dominated convergence theorem.

Denote Z_{n} = Y_{1} ⋯ Y_{n}. We need to show Z_{n}g ν → ν_{ω} for μ_{∗}-a.e. g, and it suffices to show that for any ϕ ∈ C(Q) for all k ≥ 0 one has P × μ^{k}-a.e. |F_{ϕ}(Z_{n}g) − F_{ϕ}(Z_{n})| → 0. Note that

∫ |F_{ϕ}(Z_{n}g) − F_{ϕ}(Z_{n})|^{2} dμ^{k}(g) dP(ω)
= ∫ F_{ϕ}^{2} dμ^{n+k} + ∫ F_{ϕ}^{2} dμ^{n} − 2 ∫ F_{ϕ}(hg)F_{ϕ}(h) dμ^{n}(h) dμ^{k}(g)
= ∫ F_{ϕ}^{2} dμ^{n+k} − ∫ F_{ϕ}^{2} dμ^{n}

where the last equality uses the μ-mean value property of F_{ϕ}. Hence

∫_{G} ∫_{Ω} ∑_{n=0}^{p} |F_{ϕ}(Z_{n}g) − F_{ϕ}(Z_{n})|^{2} dP(ω) dμ^{k}(g) ≤ 2k∥ϕ∥_{∞}

for every k ≥ 0. Hence the series ∑_{n} |F_{ϕ}(Z_{n}g) − F_{ϕ}(Z_{n})|^{2} converge P × μ^{k}-a.e., and in particular |F_{ϕ}(Z_{n}g) − F_{ϕ}(Z_{n})| → 0, and the proof is completed.

Proof of Proposition 1.36.

The first statement follows directly from Lemma 1.35 by realizing the real-valued stationary process {σ(Z_{n}^{θ})} as values {f(T^{n}x)} of a measurable function f on a measure-preserving system (X, m, T) along a random orbit T^{n}x.

To prove the second statement consider the set S_{P} = {θ ∈ P(M) | P^{∗}θ = θ} and let γ = sup{∫_{M} σ dθ | θ ∈ S_{P} }. By compactness the sup is attained by some θ_{0} ∈ S_{P}, and the first assertion gives γ < 0. Consider the subadditive sequence

u_{n} = sup_{x∈M} ∑_{k=1}^{n} P^{k}σ(x)

and for each n denote by x_{n} ∈ M a point which attains this sup. Any cluster point θ of the sequence η_{n} = n^{−1} ∑_{k=1}^{n}(P^{∗})^{k}δ_{x_{n}} of probability measures is easily seen to be P -stationary. Since u_{n} are subadditive, u_{n}/n converges and one has

\frac{u_{n}}{n} = ∫_{M} σ dη_{n} → ∫_{M} σ dθ ≤ γ < 0

Proof of Claim 1.40. Let {B_{i} : L_{i} → ℝ^{k}}_{i=0}^{r} represent the QP-transformation b. By Lemma 1.29 one has M_{1}(b) = \overline{L_{1}} and M_{0}(b) = \overline{Im B_{0}}. The square b^{2} is also in \overline{T}^{qp} (Lemma 1.39). Assuming the claim is wrong, one has that Im B_{0} does not contain L_{1} = Ker B_{0}, but dim(Im B_{0} ∩ Ker B_{0}) > 0, so that B_{0}^{2} ≠ 0 and dim B_{0}^{2} < dim B_{0}. This would imply that dim M_{0}(b^{2}) = dim \overline{Im B_{0}^{2}} < dim M_{0}(b) = d, contrary to the assumption that b ∈ Q.

Proof of Claim 1.41. Observe that T Q = Q because T Q ⊆ \overline{T} and for every projective transformation ḡ one has M_{1}(ḡb) = M_{1}(b) while dim M_{0}(ḡb) = dim ḡM_{0}(b) = dim M_{0}(b). Since T is (strongly) irreducible, for every b ∈ Q with M_{0}(b) ⊆ M_{1}(b), one can find a g ∈ T so that

M_{0}(ḡb) ⊈ M_{1}(b) = M_{1}(ḡb)

so that claim (1) implies M_{0}(ḡb) ∩ M_{1}(ḡb) = ∅.

Proof of Claim 1.43. Given a sequence ϕ_{n} = bḡ_{n} in Φ_{0} one needs to find a convergent subsequence for the projective transformations β(ϕ_{n}) ⊆ PGL(V). Passing to a subsequence one can assume that ϕ_{n} converge to a QP-limit ϕ ∈ \overline{T}^{qp} on ℝ^{k−1}. Since M_{0}(bḡ_{n}) = M_{0}(b) the QP-limit ϕ is in Q. Repeating the argument of Claim 1.40 one shows that there is a g ∈ T such that ϕ′ = ϕḡ ∈ \overline{T}^{qp} satisfies M_{0}(ϕ′)∩M_{1}(ϕ′) = ∅.

Note that ϕ′_{n} = ϕ_{n}ḡ converge to ϕ′ pointwise in ℙ^{k−1} (because ϕ_{n} → ϕ), and since

\overline{V} = M_{0}(ϕ′) ⊆ ℙ^{k−1} \ M_{1}(ϕ′)

for every ū ∈ ℙ(V), one has ϕ′_{n}ū → ϕ′ū. In other words β(ϕ′_{n}) → β(ϕ′) ∈ PGL(V) is a convergent sequence with a projective (rather than quasi-projective) limit. Hence β(ϕ_{n}) = β(ϕ′_{n}g^{−1}) is also a convergent sequence in PGL(V), and the claim is proved.

Proof of Remark 2.29.

Define A_{n} = {g ∈ Γ | log μ^{n}(g) > (−h − ϵ)n}. Since μ^{n}(A_{n}) ≤ 1 we have the bound log |A_{n}| ≤ (h + ϵ)n, while pointwise convergence in Theorem 2.28 implies that for Pa.e. ω one has g_{n}(ω) ∈ A_{n} for all but finitely many n-s. This proves (a).

Now suppose that B_{n} ⊂ Γ are sets of size log |B_{n}| ≤ (h − ϵ)n. Let Ω_{n} = {w | log μ^{n}(g_{n}(ω)) < (h − ϵ/2)n}. The pointwise convergence in Theorem 2.28 implies that P-a.e. ω belongs to Ω_{n} for all but finitely many n-s. Observe that

P{ω ∈ Ω_{n} | g_{n}(ω) ∈ B_{n}} ≤ e^{(h−ϵ)n} · e^{(−h+ϵ/2)n} = e^{−ϵ/2n}

The easy direction of Borel-Cantelli lemma implies that for P-a.e. ω the event {ω ∈ Ω_{n} | g_{n}(ω) ∈ B_{n}} occurs only finitely many times, proving (b).

Proof of Proposition 2.32.

Let S_{k} = {g ∈ Γ | d(g, e) = k}, k ≥ 0, denote the k-sphere in Γ, p_{k} = μ(S_{k}) and μ_{k} = p_{k}^{−1} · μ|_{S_{k}} denote the normalized restrictions of μ to S_{k}. If α denotes the partition of (Γ, μ) into elements and β ≺ α the partition into spheres, then the standard formula H(α) = H(α) + H(α | β) gives

H(μ) = ∑_{k=0}^{∞} −p_{k} log p_{k} + ∑_{k=0}^{∞} p_{k} · H(μ_{k})

≤ ∑_{k=0}^{∞} p_{k} · max\{\sqrt{k}, − log p_{k}\} + ∑_{k=0}^{∞} p_{k} · log |S_{k}|

As the function −t log(t) is increasing on (0, e^{−1}), one has for k ≥ 1

∑_{1}^{∞} p_{k} · max\{\sqrt{k}, − log p_{k}\} ≤ ∑_{1}^{∞} \sqrt{k} p_{k} + \sqrt{k} e^{−\sqrt{k}}

Since ∑ \sqrt{k}e^{−\sqrt{k}} = C < ∞, log |S_{k}|^{1/k} → δ = δ(Γ, d) and

∑ \sqrt{k} p_{k} ≤ ∑ k p_{k} = ∑_{g} d(g, e) μ(g)

one concludes that H(μ) is finite, provided μ has finite first moment.

Next take an ϵ > 0 and choose K large enough so that for all k > K

log |S_{k}|^{1/k} ≤ (δ + ϵ) and \sqrt{k} < ϵk

Since the finite set B_{K} = {g | d(g, e) ≤ K} contributes at most log |B_{K}| to the entropy of any measure on Γ one has

h(Γ, μ) = lim_{n→∞} \frac{1}{n}H(μ^{n})
≤ (δ + 2ϵ) lim_{n→∞} (\frac{1}{n} ∑ d(g, e) μ^{n}(g) + \frac{C + log |B_{K}|}{n})
= (δ + 2ϵ)) · λ^{(d)}

and since ϵ > 0 can be taken arbitrarily small the proof is completed.

Proof of Theorem 3.3.

We need to show that there is a set Ω′ ⊆ Ω of full P-measure, so that for every ω ∈ Ω′, every x ∈ X and any non-empty open U ⊆ X, one has S_{n}(ω) · x ∈ U for some n ≥ 0, or equivalently, for every ω ∈ Ω′ and any open non-empty U

∪_{n=0}^{∞}S_{n}(ω)^{−1}U = X

Since X has a countable base of open sets, it suffices to show that for any fixed set U the event

E_{U} = {ω ∈ Ω | ∪_{n=0}^{∞}S_{n}(ω)^{−1}U = X}

has P-probability one. The minimality of the original G-action implies that X = ∪_{g∈G}gU, and since X is compact there exists a finite sub-cover g′_{0}U ∪ ⋯ ∪ g′_{N}U = X. By continuity of the G-action on X there exists a small neighborhood V of the identity in G, so that

g_{0}U ∪ … g_{N}U = X  (4.2)

for any g_{0}, …, g_{N} satisfying g_{i} ∈ V_{i}, where V_{i} = g′_{i}V, i = 0, …, N. In fact, for (4.2) it suffices to have g_{i} ∈ gV_{i}, 0 = 1, …, N, for any fixed g ∈ G; in particular, (4.2) is satisfied for an arbitrary g_{0} provided

g_{i}g_{i−1}^{−1} ∈ W_{i} = V_{i+1}V_{i}^{−1} for i = 1, 2, …, N  (4.3)

Therefore to show that P(E_{U}) = 1 it suffices to show that for P-a.e. ω the sequence S_{n}(ω) = ω_{n} ⋯ ω_{1} contains an N-long subsequence satisfying (4.3). Since G = sgr(μ) and W_{i} are open non-empty subsets of G, there are integers k_{i} > 0 so that μ^{k_{i}}(W_{i}) > 0. Denoting n_{i} = k_{1} + ⋯ + k_{i} one observes that g_{i} = S_{n_{i}}(ω), i = 1, …, N, satisfies (4.3) with positive probability, and therefore with P-probability one a sequence g_{i} = S_{t+n_{i}}(ω), i = 1, …, N, satisfies (4.3) for some t ∈ ℕ. This proves P(E_{U}) = 1 and the proof is complete.

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Department of Mathematics (m/c 249), University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL 60607-7045

E-mail address: furman@math.uic.edu