Fluctuation bounds for ergodic averages of amenable groups

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 Følner 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 Følner 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 $p\in (1,\infty )$.

14 pages. Journal article version of results previously appearing in the thesis arXiv:1901.08538 (with some minor corrections). To appear in Bull. London Math. Soc