Topological entropy of sets of generic points for actions of amenable groups (1602.08242v2)

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.