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

TL;DR

This paper establishes a variational principle linking Bowen topological entropy of generic points to measure-theoretic entropy for actions of amenable groups, generalizing Bowen's classical result.

Contribution

It extends Bowen's variational principle to actions of countable discrete amenable groups with tempered Følner sequences.

Findings

Proves the variational principle for Bowen entropy and measure-theoretic entropy.

Generalizes Bowen's classical result from $b{Z}$-actions to amenable group actions.

Shows the equality of Bowen topological entropy and measure-theoretic entropy for generic points.

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.