Double variational principle for mean dimension

TL;DR

This paper establishes a new variational principle linking mean dimension and rate distortion theory, demonstrating their equality under certain conditions and combining multiple mathematical disciplines.

Contribution

It introduces a novel minimax framework connecting mean dimension with rate distortion dimension for systems with the marker property.

Findings

Minimax value equals mean dimension for systems with the marker property.

Existence of a metric where upper metric mean dimension equals mean dimension.

New integration of ergodic theory, rate distortion, and geometric measure theory.

Abstract

We develop a variational principle between mean dimension theory and rate distortion theory. We consider a minimax problem about the rate distortion dimension with respect to two variables (metrics and measures). We prove that the minimax value is equal to the mean dimension for a dynamical system with the marker property. The proof exhibits a new combination of ergodic theory, rate distortion theory and geometric measure theory. Along the way of the proof, we also show that if a dynamical system has the marker property then it has a metric for which the upper metric mean dimension is equal to the mean dimension.