Fluctuation bounds for ergodic averages of amenable groups

TL;DR

This paper establishes uniform bounds on the fluctuations of ergodic averages for actions of amenable groups, extending classical ergodic theorems to a broader setting with quantitative fluctuation control.

Contribution

It introduces a new uniform fluctuation bound for ergodic averages of amenable group actions on uniformly convex Banach spaces, generalizing previous results.

Findings

Bound on the number of fluctuations for F{4}lner sequences satisfying temperedness.

Uniform fluctuation bounds over long distances for arbitrary F{4}lner sequences.

Implications for continuous group actions on $L^{p}$ spaces with $p eq 1, \

Abstract

We study fluctuations of ergodic averages generated by actions of amenable groups. In the setting of an abstract ergodic theorem for locally compact second countable amenable groups acting on uniformly convex Banach spaces, we deduce a highly uniform bound on the number of fluctuations of the ergodic average for a class of F{\o}lner sequences satisfying an analogue of Lindenstrauss's temperedness condition. Equivalently, we deduce a uniform bound on the number of fluctuations over long distances for arbitrary F{\o}lner sequences. As a corollary, these results imply associated bounds for a continuous action of an amenable group on a $\sigma$-finite $L^{p}$ space with $p\in(1,\infty)$.