Ergodic Average Dominance for Unimodular Amenable Groups

TL;DR

This paper demonstrates that ergodic averages for unimodular amenable groups can be controlled by Cesàro means of a Markov operator, enabling derivation of ergodic theorems from integer actions.

Contribution

It introduces a method to dominate ergodic averages of unimodular amenable groups by Markov operator means, extending ergodic theorems to broader group actions.

Findings

Ergodic averages can be dominated by Cesàro means of a Markov operator.

Maximal and pointwise ergodic theorems are derived for unimodular amenable groups.

The approach applies to both commutative and noncommutative ergodic theorems.

Abstract

In this paper we show that the ergodic averages of the action of any unimodular amenable group along certain F{\o}lner sequences can be dominated by the Ces\`aro means of a suitably constructed Markov operator, that is, the ergodic averages of an integer action. Moreover, the restriction on these F{\o}lner sequences are mild enough so that every two-sided F{\o}lner sequence has a subsequence satisfying these conditions. As a consequence of this inequality, we obtain the maximal and pointwise (individual) ergodic theorems for actions of unimodular amenable groups directly from the corresponding ergodic theorems for integer actions. This allows us to deal with the commutative and noncommutative ergodic theorems on an equal footing.