Metric mean dimension of free semigroup actions for non-compact sets

TL;DR

This paper introduces new notions of metric mean dimension for free semigroup actions on non-compact sets, establishes estimations and a variational principle, and explores the properties of irregular sets with the gluing orbit property.

Contribution

It generalizes existing results by defining novel metric mean dimensions for free semigroup actions and establishing a variational principle relating these dimensions to skew product transformations.

Findings

Derived bounds for upper metric mean dimension using local dimensions.

Proved a variational principle linking mean dimension with skew product transformations.

Showed that $ ext{varphi}$-irregular sets have full upper metric mean dimension under the gluing orbit property.

Abstract

In this paper, we introduce the notions of upper metric mean dimension, $u$-upper metric mean dimension, $l$-upper metric mean dimension of free semigroup actions for non-compact sets via Carath\'{e}odory-Pesin structure. Firstly, the lower and upper estimations of the upper metric mean dimension of free semigroup actions are obtained by local metric mean dimensions. Secondly, one proves a variational principle that relates the $u$-upper metric mean dimension of free semigroup actions for non-compact sets with the corresponding skew product transformation. Furthermore, using the variational principle above, $\varphi$-irregular set acting on free semigroup actions shows full upper metric mean dimension in the system with the gluing orbit property. Our analysis generalizes the results obtained by Carvalho et al. \cite{MR4348410}, Lima and Varandas \cite{MR4308163}.