Return times theorem for amenable groups

We extend Bourgain’s return times theorem to arbitrary locally compact second countable amenable groups. The proof is based on a version of the Bourgain-Furstenberg-Katznelson-Ornstein orthogonality criterion.     

A Banach space-valued ergodic theorem for amenable groups and applications

In this paper, we study unimodular amenable groups. The first part of the paper is devoted to results on the existence of uniform families of ε-quasi tilings for these groups. First we extend constructions of Ornstein and Weiss by quantitative estimates for the covering properties of the corresponding decompositions. Then we apply the methods developed to obtain an abstract ergodic theorem for a class of functions mapping subsets of a countable amenable group into some Banach space. This result extends significantly and complements related results found in the literature. Further, using the Lindenstrauss ergodic theorem, we link our results to classical ergodic theory. We conclude with two important applications: uniform approximation of the integrated density of states on amenable Cayley graphs and almost-sure convergence of cluster densities in an amenable bond percolation model.     

Group structure and the pointwise ergodic theorem for connected amenable groups

Let G be a connected amenable group (thus, an extension of a connected normal solvable subgroup R by a connected compact group $\text{K =}\text{G}\text{R}$). We show how to explicitly construct sequences {U_n} of compacta in G in terms of the structural features of G which have the following property: For any “reasonable” action G × L^p(X, μ) ↓ L^p(X, μ) on an L^p space, 1 <p < ∞, and any f ∈ L^p(X, μ), the averages

$\text{A}{}_{\mathrm{n}}\text{f=}\text{1}\text{|U}{}_{\mathrm{n}}\text{|}\text{∫}\text{U}{}_{\mathrm{n}}\text{T}{}_{\mathrm{g}}{}^{\mathrm{?1f}}\text{dg (|E|=}\text{left Haar measure in}\text{G)}$

converge in L^p norm, and pointwise μ-a.e. on X, to G-invariant functions $\text{f1}$ in L^p(X, μ). A single sequence {U_n} in G works for all L^p actions of G. This result applies to many nonunimodular groups, which are not handled by previous attempts to produce noncommutative generalizations of the pointwise ergodic theorem.     

Euler class groups and a theorem of Roitman

We define the Euler class group of a polynomial algebra in lower codimension and prove an analogue of a result of Roitman (on projective modules) for the Euler class groups.     

Pointwise theorems for amenable groups

In this paper we prove the pointwise ergodic theorem for general locally compact amenable groups along F?lner sequences that obey some restrictions. These restrictions are mild enough so that such sequences exist for all amenable groups. We also prove a generalization of the Shannon-McMillan-Breiman theorem to all discrete amenable groups. --> Oblatum 10-I-2000 & 9-V-2001?Published online: 13 August 2001     

Very weak Bernoulli for amenable groups

We strengthen Følner Independence (defined in an earlier paper) to a criterion (which we call Very Weak Bernoulli) and prove that this new condition is equivalent to Finitely Determined.     

Subsequence ergodic theorems for amenable groups

For amenable groups that have a Følner sequence {A _n} satisfying $\overline {lim} \left| {A_n^{ - 1} A_n } \right|/\left| {A_n } \right|< + \infty $ we show that a subsequence ergodic theorem is valid for the visit times to a set of positive measure.     

Bowen entropy for actions of amenable groups

Bowen introduced a definition of topological entropy of subset inspired by Hausdorff dimension in 1973 [1]. In this paper we consider the Bowen entropy for amenable group action dynamical systems and show that, under the tempered condition, the Bowen entropy of the whole compact space for a given Følner sequence equals the topological entropy. For the proof of this result, we establish a variational principle related to the Bowen entropy and the Brin–Katok local entropy formula for dynamical systems with amenable group actions.     

Rigid actions of amenable groups

Given a countable discrete amenable group G, does there exist a free action of G on a Lebesgue probability space which is both rigid and weakly mixing? The answer to this question is positive if G is abelian. An affirmative answer is given in this paper, in the case that G is solvable or residually finite. For a locally finite group, the question is reduced to an algebraic one. It is exemplified how the algebraic question can be positively resolved for some groups, whereas for others the algebraic viewpoint suggests the answer may be negative.     

Sub-additive ergodic theorems for countable amenable groups

In this paper we generalize Kingman's sub-additive ergodic theorem to a large class of infinite countable discrete amenable group actions.     

The Myhill property for strongly irreducible subshifts over amenable groups

Let G be an amenable group and let A be a finite set. We prove that if X ? A ^G is a strongly irreducible subshift then X has the Myhill property, that is, every pre-injective cellular automaton ?? : X ?? X is surjective.     

From acyclic groups to the Bass conjecture for amenable groups

We prove that the Bost Conjecture on the ¹-assembly map for countable discrete groups implies the Bass Conjecture. It follows that all amenable groups satisfy the Bass Conjecture.     

Reflection theorem for divisor class groups of relative quadratic function fields

We present the reflection theorem for divisor class groups of relative quadratic function fields. Let K be a global function field with constant field Fq. Let L₁ be a quadratic geometric extension of K and let L₂ be its twist by the quadratic constant field extension of K. We show that for every odd integer m that divides q+1 the divisor class groups of L₁ and L₂ have the same m-rank.     

Characteristic factors for commuting actions of amenable groups

We describe characteristic factors for certain averages arising from commuting actions of locally compact second-countable amenable groups. Under some ergodicity assumptions, we use these factors to prove a form of multiple recurrence for three such actions.     

On amenable and compact groups

In the sequel ofB. E. Johnson's work on amenable Banach algebras we characterize amenable and compact groups.