On variational principle for upper metric mean dimension with potential

TL;DR

This paper develops a variational principle linking upper metric mean dimension with potential to measure-theoretical concepts in dynamical systems, introducing equilibrium states to identify measures that maximize this relationship.

Contribution

It establishes a variational principle for upper metric mean dimension with potential and introduces equilibrium states in this context.

Findings

Proves a variational principle connecting upper metric mean dimension and measure-theoretical dimensions.

Defines and characterizes equilibrium states for the variational principle.

Provides a framework for analyzing dynamical systems via these new concepts.

Abstract

Borrowing the idea of topological pressure determining measure-theoretical entropy in topological dynamical systems, we establish a variational principle for upper metric mean dimension with potential in terms of upper measure-theoretical metric mean dimension of invariant measures. Moreover, the notion of equilibrium state is introduced to characterize these measures that attain the supremum of the variational principle.