Metric vs topological receptive entropy of semigroup actions

TL;DR

This paper introduces and compares receptive metric and topological entropies for semigroup actions, establishing properties, relations, and variational principles, and extends local entropy concepts with partial formulas.

Contribution

It develops a framework for receptive entropy notions in semigroup actions, including new characterizations and partial variational principles, bridging metric and topological perspectives.

Findings

Receptive metric entropy analyzed and related to classical metric entropy.

Comparison between receptive metric and topological entropy with variational principles.

Introduction of receptive local metric entropy with partial Brin-Katok and Variational results.

Abstract

We study the receptive metric entropy for semigroup actions on probability spaces, inspired by a similar notion of topological entropy introduced by Hofmann and Stoyanov. We analyze its basic properties and its relation with the classical metric entropy. In the case of semigroup actions on compact metric spaces we compare the receptive metric entropy with the receptive topological entropy looking for a Variational Principle. With this aim we propose several characterizations of the receptive topological entropy. Finally we introduce a receptive local metric entropy inspired by a notion by Bowen generalized in the classical setting of amenable group actions by Zheng and Chen, and we prove partial versions of the Brin-Katok Formula and the local Variational Principle.