Fluctuations of ergodic averages for amenable group actions

Abstract

We show that for any countable amenable group action, along certain Følner sequences (those that have for any $c>1$ a two-sided $c$-tempered tail), one has a universal estimate for the number of fluctuations in the ergodic averages of $L^{\infty }$ functions. This estimate gives exponential decay in the number of fluctuations. Any two-sided Følner sequence can be thinned out to satisfy the above property. In particular, any countable amenable group admits such a sequence. This extends results of S. Kalikow and B. Weiss [1] for ${\mathbb{Z}}^d$ actions and of N. Moriakov [3] for actions of groups with polynomial growth.