Variational principles for Feldman-Katok metric mean dimension

TL;DR

This paper introduces Feldman-Katok metric mean dimensions, establishes conditions for their equivalence under certain growth constraints, and develops variational principles linking these dimensions to FK entropy measures.

Contribution

It presents the concept of Feldman-Katok metric mean dimensions and derives variational principles relating them to FK entropy, advancing the theoretical understanding of metric mean dimensions.

Findings

Metric mean dimensions coincide under weak tame growth.

Variational principles connect Feldman-Katok metric mean dimensions to FK entropy.

Established conditions for the equivalence of different metric mean dimensions.

Abstract

We introduce the notion of Feldman-Katok metric mean dimensions in this note. We show metric mean dimensions defined by different metrics coincide under weak tame growth of covering numbers, and establish variational principles for Feldman-Katok metric mean dimensions in terms of FK Katok $\epsilon$-entropy and FK local $\epsilon$-entropy function.