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Journal of Dynamical and Control Systems

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23-05-2021

An Alternative Definition of Topological Entropy for Amenable Group Actions

Authors: Haiyan Wu, Zhiming Li

Published in: Journal of Dynamical and Control Systems | Issue 2/2022

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Abstract

We extend the definition of topological entropy for any (not necessarily continuous) amenable groups acting on a compact space by defining entropy of arbitrary subsets of a product space. We investigate how this new notion of topological entropy for amenable group actions behaves and some of its basic properties; among them are the behavior of the entropy with respect to disjoint union, Cartesian product, and some continuity properties with respect to Vietoris topology. As a special case for \(1\leq p\leq \infty \), the Bowen p-entropy of sets is introduced. It is shown that the notions of generalized topological entropy and Bowen \(\infty \)-entropy for compact metric spaces coincide.

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Title: An Alternative Definition of Topological Entropy for Amenable Group Actions
Authors: Haiyan Wu
Zhiming Li
Publication date: 23-05-2021
Publisher: Springer US
Published in: Journal of Dynamical and Control Systems / Issue 2/2022
Print ISSN: 1079-2724
Electronic ISSN: 1573-8698
DOI: https://doi.org/10.1007/s10883-021-09547-0

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