Fluctuations of ergodic averages for actions of groups of polynomial growth
Volume 240 / 2018
Abstract
It was shown by S. Kalikow and B. Weiss that, given a measure-preserving action of $\mathbb {Z}^d$ on a probability space $\mathrm {X}$ and a nonnegative measurable function $f$ on $\mathrm {X}$, the probability that the sequence of ergodic averages $$ \frac 1 {(2k+1)^d} \sum _{g \in [-k,\dots ,k]^d} f(g \cdot x) $$ 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 the effective Vitali covering theorem to groups of polynomial growth.
