Around the variational principle for metric mean dimension

TL;DR

This paper advances the understanding of metric mean dimension by establishing new variational principles, simplifying existing conditions, and deriving novel formulas for information dimension rates and local entropy bounds.

Contribution

It proves that ergodic measures suffice in the variational principle, introduces a general variational formula involving entropy growth, and provides new expressions for information dimension rate.

Findings

Ergodic measures suffice in the variational principle.

A general variational principle involving entropy growth is established.

New formulas for information dimension rate and local entropy bounds are derived.

Abstract

We study variational principles for metric mean dimension. First we prove that in the variational principle of Lindenstrauss and Tsukamoto it suffices to take supremum over ergodic measures. Second we derive a variational principle for metric mean dimension involving growth rates of measure-theoretic entropy of partitions decreasing in diameter which holds in full generality and in particular does not necessitate the assumption of tame growth of covering numbers. The expressions involved are a dynamical version of Renyi information dimension. Third we derive a new expression for Geiger-Koch information dimension rate for ergodic shift-invariant measures. Finally we develop a lower bound for metric mean dimension in terms of Brin-Katok local entropy.