On the metric mean dimensions of saturated sets

TL;DR

This paper investigates the metric mean dimensions of saturated sets and generic points in infinite entropy systems, establishing variational principles and full dimension results for systems with the specification property.

Contribution

It introduces variational principles for metric mean dimensions of saturated sets and shows they have full metric mean dimension in systems with the specification property.

Findings

Variational principles for Bowen and packing metric mean dimensions.

Full metric mean dimension of saturated sets.

Characterization of dimensions of generic points and level sets.

Abstract

From a geometric perspective, we employ metric mean dimension to investigate the set of generic points of invariant measures and saturated sets in infinite entropy systems. For systems with the specification property, we establish certain variational principles for the Bowen and packing metric mean dimensions of saturated sets in terms of Kolmogorov-Sinai $\epsilon$-entropy, and prove that the upper capacity metric mean dimension of saturated sets has full metric mean dimension. Consequently, the Bowen and packing metric mean dimensions of the set of generic points of invariant measures coincide with the mean R\'enyi information dimension, and the upper capacity metric mean dimension of the set of generic points of invariant measures also has full metric mean dimension. As applications, for systems with the specification property, we present the qualitative characterization of the metric mean dimensions of level sets, the set of mean Li-Yorke pairs in infinite-entropy systems, and the set of generic points of invariant measures in full shifts over compact metric spaces.