Some notes on variational principle for metric mean dimension

TL;DR

This paper advances the understanding of metric mean dimension by solving an open problem, establishing a double variational principle involving Rényi information dimension, and introducing the concept of maximal metric mean dimension measure.

Contribution

It provides a solution to an open problem, formulates a double variational principle for metric mean dimension, and introduces the notion of maximal metric mean dimension measure.

Findings

Solved an open problem posed by Gutman and Spiewak.

Established a double variational principle involving Rényi information dimension.

Introduced the concept of maximal metric mean dimension measure.

Abstract

Firstly, we answer the problem 1 asked by Gutman and $\rm \acute{\ S}$piewak in \cite{gs20}, then we establish a double variational principle for mean dimension in terms of R$\bar{e}$nyi information dimension and show the order of $\sup$ and $\limsup$ (or $\liminf$) of the variational principle for the metric mean dimension in terms of R$\bar{e}$nyi information dimension obtained by Gutman and $\rm \acute{\ S}$piewak can be changed under the marker property. Finally, we attempt to introduce the notion of maximal metric mean dimension measure, which is an analogue of the concept called classical maximal entropy measure related to the topological entropy.