Bowen entropy for actions of amenable groups

TL;DR

This paper extends Bowen's entropy concept to amenable group actions, demonstrating that under certain conditions, Bowen entropy equals topological entropy, supported by a variational principle and local entropy formula.

Contribution

It introduces a variational principle and proves Bowen entropy equals topological entropy for amenable group actions under the tempered condition.

Findings

Bowen entropy equals topological entropy under the tempered condition.

Established a variational principle for Bowen entropy in amenable group actions.

Proved a Brin-Katok local entropy formula for these systems.

Abstract

Bowen introduced a definition of topological entropy of subset inspired by Hausdorff dimension in 1973 \cite{B}. In this paper we consider the Bowen's entropy for amenable group action dynamical systems and show that under the tempered condition, the Bowen entropy of the whole compact space for a given F{\o}lner sequence equals to the topological entropy. For the proof of this result, we establish a variational principle related to the Bowen entropy and the Brin-Katok's local entropy formula for dynamical systems with amenable group actions.