Fluctuations of Ergodic Averages for Actions of Groups of Polynomial Growth

TL;DR

This paper extends the understanding of fluctuations in ergodic averages from actions of $\\mathbb{Z}^d$ to more general groups of polynomial growth, using a generalized Vitali covering theorem.

Contribution

It generalizes fluctuation bounds for ergodic averages from $\\mathbb{Z}^d$ actions to groups of polynomial growth, introducing a new Vitali covering theorem adaptation.

Findings

Bound on the probability of multiple fluctuations in ergodic averages

Extension of fluctuation results to polynomial growth groups

Development of a generalized Vitali covering theorem

Abstract

It was shown by S. Kalikow and B. Weiss that, given a measure-preserving action of $\mathbb{Z}^d$ on a probability space $X$ and a nonnegative measurable function $f$ on $X$, the probability that the sequence of ergodic averages $$ \frac 1 {(2k+1)^d} \sum\limits_{g \in [-k,\dots,k]^d} f(g \cdot x) $$ has at least $n$ fluctuations across an interval $(\alpha,\beta)$ can be bounded from above by $c_1 c_2^n$ for some universal constants $c_1 \in \mathbb{R}$ and $c_2 \in (0,1)$, which depend only on $d,\alpha,\beta$. The purpose of this article is to generalize this result to measure-preserving actions of groups of polynomial growth. As the main tool we develop a generalization of effective Vitali covering theorem for groups of polynomial growth.