On relative metric mean dimension with potential and variational principles

TL;DR

This paper introduces a new concept of relative mean metric dimension with potential in topological dynamical systems and establishes multiple variational principles linking it with various entropy notions.

Contribution

It defines relative mean metric dimension with potential and proves four variational principles connecting it with different entropy measures, addressing an open question.

Findings

Established four variational principles linking metric dimension with entropy.

Partially answered an open question by Shi for well-partitionable spaces.

Derived a variational inequality involving box dimension.

Abstract

In this article, we introduce a notion of relative mean metric dimension with potential for a factor map $\pi: (X,d, T)\to (Y, S)$ between two topological dynamical systems. To link it with ergodic theory, we establish four variational principles in terms of metric entropy of partitions, Shapira's entropy, Katok's entropy and Brin-Katok local entropy respectively. Some results on local entropy with respect to a fixed open cover are obtained in the relative case. We also answer an open question raised by Shi \cite{Shi} partially for a very well-partitionable compact metric space, and in general we obtain a variational inequality involving box dimension of the space. Corresponding inner variational principles given an invariant measure of $(Y,S)$ are also investigated.