2018•Topological Methods in Nonlinear AnalysisOpen access

A note on dimensional entropy for amenable group actions

Dou Dou, Ruifeng Zhang

Abstract

In this short note, for countably infinite amenable group actions, we provide topological proofs for the following results: Bowen topological entropy (dimensional entropy) of the whole space equals the usual topological entropy along tempered F{\o}lner sequences; the Hausdorff dimension of an amenable subshift (for certain metric associated to some F{\o}lner sequence) equals its topological entropy. This answers questions by Zheng and Chen \cite{ZC} and Simpson \cite{S}.

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Key concepts: Topological entropy, Mathematics, Amenable group, Countable set, Metric space, Entropy (arrow of time), Hausdorff dimension, Combinatorics