Bowen's equations for upper metric mean dimension with potential

TL;DR

This paper introduces new notions of metric mean dimension with potential, establishes Bowen's equations and variational principles for these dimensions, and explores their applications to subsets and generic points of ergodic measures.

Contribution

It generalizes existing definitions of upper metric mean dimension with potential and establishes new Bowen's equations and variational principles for these generalized notions.

Findings

Established Bowen's equations for induced upper metric mean dimension with potential.

Proved variational principles for BS metric mean dimension and Packing BS metric mean dimension.

Analyzed Bowen upper metric mean dimension of generic points of ergodic measures.

Abstract

Firstly, we introduce a new notion called induced upper metric mean dimension with potential, which naturally generalizes the definition of upper metric mean dimension with potential given by Tsukamoto to more general cases, then we establish variational principles for it in terms of upper and lower rate distortion dimensions and show there exists a Bowen's equation between induced upper metric mean dimension with potential and upper metric mean dimension with potential. Secondly, we continue to introduce two new notions, called BS metric mean dimension and Packing BS metric mean dimension on arbitrary subsets, to establish Bowen's equations for Bowen upper metric mean dimension and Packing upper metric mean dimension with potential on subsets. Besides, we also obtain two variational principles for BS metric mean dimension and Packing BS metric mean dimension on subsets. Finally, the special interest about the Bowen upper metric mean dimension of the set of generic points of ergodic measures are also involved.