# Fluctuation bounds for ergodic averages of amenable groups on uniformly convex Banach spaces

## TL;DR

This paper establishes uniform bounds on the fluctuations of ergodic averages for actions of amenable groups on uniformly convex Banach spaces, extending to $L^p$ spaces and providing quantitative convergence insights.

## Contribution

It introduces a highly uniform fluctuation bound for ergodic averages under amenable group actions on uniformly convex Banach spaces, generalizing previous ergodic theorems.

## Key findings

- Bound on the number of fluctuations for ergodic averages
- Uniform bounds over long distances for F{\

## 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 $p\in(1,\infty)$.