Fluctuations of Ergodic Averages for Actions of Groups of Polynomial Growth

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 $$\frac{1}{(2k+1{)}^d}\sum _{g\in [-k,\dots ,k{]}^d}f(g\cdot x)$$ 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.

15 pages; corrections in Ex. 2.1 and Lemma 2.4, comments are welcome