Some variational principles for the metric mean dimension of a semigroup action

TL;DR

This paper establishes three variational principles for the metric mean dimension of free semigroup actions, connecting it with Shapira's entropy, Katok's entropy, and local entropy functions, extending previous dynamical systems results.

Contribution

It introduces and proves three variational principles for metric mean dimension in the context of free semigroup actions, extending classical entropy concepts.

Findings

Metric mean dimension satisfies variational principles based on Shapira's entropy.

Metric mean dimension satisfies variational principles based on Katok's entropy.

Metric mean dimension satisfies variational principles using local entropy functions.

Abstract

In this manuscript we show that the metric mean dimension of a free semigroup action satisfies three variational principles: (a) the first one is based on a definition of Shapira's entropy, introduced in \cite{SH} for a singles dynamics and extended for a semigroup action in this note; (b) the second one treats about a definition of Katok's entropy for a free semigroup action introduced in \cite{CRV-IV}; (c) lastly we consider the local entropy function for a free semigroup action and show that the metric mean dimension satisfies a variational principle in terms of such function. Our results are inspired in the ones obtained by \cite{LT2019}, \cite{VV}, \cite{GS1} and \cite{RX}.