A note on dimensional entropy for amenable group actions

TL;DR

This paper provides topological proofs relating dimensional entropy and Hausdorff dimension to topological entropy for amenable group actions, answering previously posed questions in the field.

Contribution

It offers new topological proofs establishing the equality of Bowen topological entropy and topological entropy, as well as Hausdorff dimension and topological entropy, for amenable group actions.

Findings

Bowen topological entropy equals usual topological entropy along tempered F{}lner sequences.

Hausdorff dimension of certain amenable subshifts equals their topological entropy.

Answers questions posed by Zheng, Chen, and Simpson.

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 (Israel Journal of Mathematics 212 (2016), 895-911) and Simpson (Theory Comput. Syst. 56 (2015), 527-543).