Amenable equivalence relations and the construction of ergodic averages for group actions

TL;DR

This paper introduces a unified method for constructing pointwise ergodic sequences applicable to all countable groups, leveraging the theory of amenable equivalence relations to encompass both amenable and non-amenable cases.

Contribution

The paper develops a novel, general approach to ergodic averages that unifies the treatment of amenable and non-amenable groups through amenable equivalence relations.

Findings

Applicable to both amenable and non-amenable groups

Provides a new framework for ergodic averages

Unifies existing theories under a common approach

Abstract

We present a general new method for constructing pointwise ergodic sequences on countable groups, which is applicable to amenable as well as to non-amenable groups and treats both cases on an equal footing. The principle underlying the method is that both cases can be viewed as instances of the general ergodic theory of amenable equivalence relations.