Metric mean dimension and the variational principle for actions of amenable groups

TL;DR

This paper extends the classical variational principle to metric mean dimension for actions of amenable groups, introducing new measure-theoretic concepts and establishing fundamental relations between topological and measure-theoretic complexities.

Contribution

It defines a new measure-theoretic metric mean dimension for invariant measures and proves a classical-type variational principle, extending prior results to a broader setting.

Findings

Established a variational principle for metric mean dimension in amenable group actions.

Introduced infinite entropy dimensions for zero metric mean dimension systems.

Connected local and global metric mean dimensions via variational principles.

Abstract

Metric mean dimension is a dynamical counterpart of the box dimension in fractal geometry to characterize the topological complexity of infinite entropy systems. The classical variational principle states that topological entropy equals the supremum of measure-theoretic entropy over the set of invariant measures. Lindenstrauss and Tsukamoto proved that this variational principle fails for metric mean dimension in terms of rate-distortion dimensions. For the actions of amenable groups, we define a new measure-theoretic metric mean dimension for invariant measures and establish a classical-type variational principle for metric mean dimension. In particular, we extend the Lindenstrauss-Tsukamoto variational principles to the classical variational principle by defining modified rate-distortion dimensions. As applications, for systems with zero metric mean dimension, we introduce infinite entropy dimensions in both topological and measure-theoretic settings, and relate them via a variational principle. For systems with positive metric mean dimension, we introduce local metric mean dimension from a local perspective, and relate it to the metric mean dimension of the whole phase space via variational principles.