Groups, Geometry, and Dynamics

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Published online first: 2021-08-03
DOI: 10.4171/GGD/622

Fluctuations of ergodic averages for amenable group actions

Uri Gabor[1]

(1) Hebrew University of Jerusalem, Israel

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.

Keywords: Ergodic theorems, upcrossing inequalities, amenable group actions

Gabor Uri: Fluctuations of ergodic averages for amenable group actions. Groups Geom. Dyn. Electronically published on August 3, 2021. doi: 10.4171/GGD/622 (to appear in print)