Amos Nevo

A generalization of Birkhoff's pointwise ergodic theorem

Given an arbitrary invertible measure preserving transformation T on a probability space X, Birkhoffs pointwise ergodic theorem asserts that for any fELl(X), the averages of f along an orbit of T, namely the expressions (f(T-nx)+...+f(Tnx))/(2n+ 1) converge, for almost all xEX, to the limit ](x), where ] is the conditional expectation of f with respect to the a-algebra of T-invariant sets. It is natural to wonder whether, given two arbitrary invertible measure preserving transformations T and S, there is a natural way to average a function f along the orbits of the group generated by T and S, so as to obtain the same conclusion. If T and S happen to commute, then, as is well known, e.g. [OWl, the expressions (2n+l ) -2 E-n<~nl,n2<<.n f(TnlSn2x) converge for almost all xeX, for any fELl(X), and again thel imit is the conditional expectation of f with respect to the a-algebra of sets invariant under T and S . In other words, the pointwise ergodic theorem holds for finite-measure-preserving actions of the free Abelian group on two generators, namely Z 2. To answer the question posed above, we need to prove a pointwise ergodic theorem for finite-measure-preserving actions of the free non-Abelian group on two generators. Note that such a result implies a corresponding one for factor groups, when the weights used are those induced by the canonical factor map. The problem is thus naturally part of the following general framework:

Counting lattice points

Abstract For a locally compact second countable group G and a lattice subgroup Γ, we give an explicit quantitative solution of the lattice point counting problem in general domains in G, provided that G has finite upper local dimension, and the domains satisfy a basic regularity condition, the mean ergodic theorem for the action of G on G/Γ holds, with a rate of convergence. The error term we establish matches the best current result for balls in symmetric spaces of simple higher-rank Lie groups, but holds in much greater generality. A significant advantage of the ergodic theoretic approach we use is that the solution to the lattice point counting problem is uniform over families of lattice subgroups provided they admit a uniform spectral gap. In particular, the uniformity property holds for families of finite index subgroups satisfying a quantitative variant of property τ. We discuss a number of applications, including: counting lattice points in general domains in semisimple S-algebraic groups, counting rational points on group varieties with respect to a height function, and quantitative angular (or conical) equidistribution of lattice points in symmetric spaces and in affine symmetric varieties. We note that the mean ergodic theorems which we establish are based on spectral methods, including the spectral transfer principle and the Kunze–Stein phenomenon. We formulate and prove appropriate analogues of both of these results in the set-up of adele groups, and they constitute a necessary ingredient in our proof of quantitative results for counting rational points.

Chapter 13 - Pointwise Ergodic Theorems for Actions of Groups

This chapter discusses pointwise ergodic theorems for general measure-preserving actions of locally compact second countable (lcsc) groups. Ergodic theorems for actions of connected Lie groups and, particularly, equidistribution theorems on homogeneous spaces and moduli spaces, have been developed and used in a rapidly expanding array of applications. Equidistribution for every orbit does not hold in many cases, such as for the case of geodesic flows on compact surfaces of constant negative curvature. Convergence for every orbit fails even if the function is assumed continuous or even smooth. Thus, the restriction to almost every starting point is essential in the pointwise ergodic theorem. Furthermore, Calderons original formulation of his pointwise ergodic theorem did not prove or assume strict polynomial volume growth, but instead noted that the doubling condition implies the following property for the volume of the balls. Similarly, subexponential growth (which is equivalent to polynomial volume growth in the connected Lie group case but not in general) implies that a subsequence of the sequence of balls is asymptotically invariant. Hence, particularly, the mean ergodic theorem is true for the subsequence.

A structure theorem for actions of semisimple Lie groups

We consider a connected semisimple Lie group G with finite center, an admissible probability measure , on G, and an ergodic (G, ,u)-space (X, v). We first note (Lemma 0.1) that (X, v) has a unique maximal projective factor of the form (G/Q, vo): where Q is a parabolic subgroup of G, and then prove: 1. Theorem 1. If every noncompact simple factor of G has real rank at least two, then the maximal projective factor is nontrivial, unless v is a G

Maximal and pointwise ergodic theorems for word-hyperbolic groups

Let denote a word-hyperbolic group, and let S = S 1 denote a nite symmetric set of generators. Let S n = fw : jwj = ng denote the sphere of radius n, where j j denotes the word length on induced by S. De ne n d = 1 #S n P w2S n w, and n = 1 n+1 P n k=0 k . Let (X;B;m) be a probability space on which acts ergodically by measure preserving transformations. We prove a strong maximal inequality in L 2 for the maximal operator f = sup n 0 j n f(x)j. The maximal inequality is applied to prove a pointwise ergodic theorem in L 2 for exponentially mixing actions of , of the following form : n f (x) ! R X fdm almost everywhere and in the L 2 -norm, for every f 2 L 2 (X). As a corollary, for a uniform lattice G, where G is a simple Lie group of real rank one, we obtain a pointwise ergodic theorem for the action of on an arbitrary ergodic G-space. In particular, this result holds when X = G= is a compact homogeneous space, and yields an equidistribution result for sets of lattice points of the form g, for almost every g 2 G. x1 Definitions and statements of results 1.1 De nition of ergodic sequences. Let be a countable group, and let ` 1 ( ) = f = P 2 ( ) : P 2 j ( )j < 1g denote the group algebra. Given any unitary representation of in a Hilbert space H, extend to the group algebra by: ( ) = P 2 ( ) ( ). Denote by H 1 the space of vectors invariant under every ( ), 2 , and by E 1 the orthogonal projection on H 1 . De nition 1.1. Given a unitary representation ( ;H) of , a sequence n 2 ` 1 ( ) is a mean ergodic sequence in H if k ( n )f E 1 fk ! n!1 0 for all f 2 H. 1991 Mathematics Subject Classi cation. 22D40, 28D15, 43A20, 43A62.

Rigidity of Furstenberg entropy for semisimple Lie group actions

Abstract We consider the action of a semi-simple Lie group G on a compact manifold (and more generally a Borel space) X , with a measure ν stationary under a probability measure μ on G . We first establish some properties of the fundamental invariant associated with a (G,μ) -space (X,ν) , namely the Furstenberg entropy [? ] , given by h μ (X,ν)= ∫ G ∫ X − log dg −1 ν dν (x) dν(x) dμ(g). We then prove that when (X,ν) is a P -mixing (G,μ) -space [? ] , and R -rank (G)=r≥2 , the value of the Furstenberg entropy must coincide with one of the 2r values hμ(G/Q,ν0) , where Q⊂G is a parabolic subgroup. We also construct counterexamples to show that this conclusion fails for both non- P -mixing actions and actions of groups with R -rank 1 . We also characterize amenable actions with a stationary measure as the actions having the maximal possible value of the Furstenberg entropy. We give applications to geometric rigidity for actions with low Furstenberg entropy, to orbit equivalence and to the cohomology of actions with stationary measure.

Geometric covering arguments and ergodic theorems for free groups

We present a new approach to the proof of ergodic theorems for actions of free groups based on geometric covering and asymptotic invariance arguments. Our approach can be viewed as a direct generalization of the classical geometric covering and asymptotic invariance arguments used in the ergodic theory of amenable groups. We use this approach to generalize the existing maximal and pointwise ergodic theorems for free group actions to a large class of geometric averages which were not accessible by previous techniques. Some applications of our approach to other groups and other problems in ergodic theory are also briefly discussed.

Pointwise ergodic theorems beyond amenable groups

We prove pointwise and maximal ergodic theorems for probability measure preserving (p.m.p.) actions of any countable group, provided it admits an essentially free, weakly mixing amenable action of stable type

The Poisson boundary of CAT(0) cube complex groups

III_1

Metric Diophantine approximation on homogeneous varieties

. We show that this class contains all irreducible lattices in connected semisimple Lie groups without compact factors. We also establish similar results when the stable type is