Measure-theoretic metric mean dimension

TL;DR

This paper introduces a new measure-theoretic metric mean dimension concept for infinite entropy systems, linking it to packing mean dimension of generic points, advancing understanding of complex dynamical systems.

Contribution

It defines measure-theoretic metric mean dimension for infinite entropy systems and establishes its equivalence with packing mean dimension of generic points.

Findings

Measure-theoretic metric mean dimensions coincide across different types of measure-theoretic ε-entropies.

The measure-theoretic metric mean dimension equals the packing metric mean dimension of generic points.

Provides a new framework for analyzing infinite measure-theoretic entropy systems.

Abstract

For infinite measure-theoretic entropy systems, we introduce the notion of measure-theoretic metric mean dimension of invariant measures for different types of measure-theoretic $\epsilon$-entropies, and show that measure-theoretic metric mean dimensions of different types of measure-theoretic $\epsilon$-entropies coincide with the packing metric mean dimension of the set of generic points of ergodic measures.