Topological entropy of sets of generic points for actions of amenable groups

Abstract

Let $G$ be a countable discrete amenable group which acts continuously on a compact metric space $X$ and let $\mu$ be an ergodic $G-$invariant Borel probability measure on $X$. For a fixed tempered F{\o}lner sequence $\{F_n\}$ in $G$ with $\lim\limits_{n\rightarrow+\infty}\frac{|F_n|}{\log n}=\infty$, we prove the following variational principle: $$h^B(G_{\mu},\{F_n\})=h_{\mu}(X,G),$$ where $G_{\mu}$ is the set of generic points for $\mu$ with respect to $\{F_n\}$ and $h^B(G_{\mu},\{F_n\})$ is the Bowen topological entropy (along $\{F_n\}$) on $G_{\mu}$. This generalizes the classical result of Bowen in 1973.