Ergodic Theorems for coset spaces

TL;DR

This paper investigates the conditions under which the mean ergodic theorem holds or fails along F ext{"o}lner sequences in countable amenable groups, revealing both positive results and counterexamples.

Contribution

It establishes new criteria for the validity of the mean ergodic theorem in amenable groups and constructs examples where the theorem fails under certain conditions.

Findings

Mean ergodic theorem always holds along any F ext{"o}lner sequence in virtually nilpotent groups.

Counterexamples show failure of the mean ergodic theorem in some amenable groups.

Embedding of groups with specific ergodic properties into larger groups with controlled behavior.

Abstract

We study in this paper the validity of the mean ergodic theorem along \emph{left} F\o lner sequences in a countable amenable group $G$. Although the \emph{weak} ergodic theorem always holds along \emph{any} left F\o lner sequence in $G$, we provide examples where the \emph{mean} ergodic theorem fails in quite dramatic ways. On the other hand, if $G$ does not admit any ICC quotients, e.g. if $G$ is virtually nilpotent, then we prove that the mean ergodic theorem does indeed hold along \emph{any} left F\o lner sequence. In the case when a unitary representation of a countable amenable group is induced from a unitary representation of a "sufficiently thin" subgroup, we prove that the mean ergodic theorem holds along any left F\o lner sequence for this representation. Furthermore, we show that every countable (infinite) amenable group $L$ embeds into a countable group $G$ which admits a unitary representation with the property that for any left F\o lner sequence $(F_n)$ in $L$, there exists a sequence $(s_n)$ in $G$ such that the mean (but \emph{not} the weak) ergodic theorem fails for this representation along the sequence $(F_n s_n)$. Finally, we provide examples of countable (not necessarily amenable) groups $G$ with proper, infinite-index subgroups $H$, so that the \emph{pointwise} ergodic theorem holds for averages along \emph{any} strictly increasing and nested sequence of finite subsets of the coset $G/H$.