Double variational principle for mean dimension with potential

TL;DR

This paper introduces a new concept called mean dimension with potential, establishing a variational principle that links it to rate distortion and geometric measure theory for dynamical systems with the marker property.

Contribution

It develops the theory of mean dimension with potential and proves a variational principle analogous to topological pressure for systems with the marker property.

Findings

Established a variational principle for mean dimension with potential.

Linked mean dimension with potential to rate distortion dimension and geometric measure theory.

Proved the minimax value equals mean dimension with potential for systems with the marker property.

Abstract

This paper contributes to the mean dimension theory of dynamical systems. We introduce a new concept called mean dimension with potential and develop a variational principle for it. This is a mean dimension analogue of the theory of topological pressure. We consider a minimax problem for the sum of rate distortion dimension and the integral of a potential function. We prove that the minimax value is equal to the mean dimension with potential for a dynamical system having the marker property. The basic idea of the proof is a dynamicalization of geometric measure theory.