Metric mean dimension and mean Hausdorff dimension varying the metric

TL;DR

This paper investigates how metric mean dimension and mean Hausdorff dimension depend on the choice of metric in a compact dynamical system, proving their discontinuity in general and identifying cases of continuity.

Contribution

It establishes the non-continuity of metric mean and Hausdorff dimensions with respect to metric variations and provides examples where metric mean dimension remains continuous.

Findings

Metric mean dimension and mean Hausdorff dimension are generally discontinuous functions of the metric.

Examples of metrics are provided where metric mean dimension is continuous.

Fundamental properties of mean Hausdorff dimension are established.

Abstract

Let $f: M\rightarrow M$ be a continuous map on a compact metric space $M$ equipped with a fixed metric $d$, and let $\tau$ be the topology on $M$ induced by $d$. First, we will establish some fundamental properties of the mean Hausdorff dimension. Furthermore, it is important to note that the metric mean dimension and mean Hausdorff dimension depend on the metric chosen for $M$. In this work, we will prove that, for a fixed dynamical system $f:M\rightarrow M$, the functions $\text{mdim}_{\text{M}}(M, f):M(\tau)\rightarrow \mathbb{R}\cup \{\infty\}$ and $\text{mdim}_{\text{H}}(M, f):M(\tau)\rightarrow \mathbb{R}\cup \{\infty\}$ are not continuous. Here, $ \text{mdim}_{\text{M}}(M, f)(\rho)= \text{mdim}_{\text{M}}(M,\rho, f)$ and $ \text{mdim}_{\text{H}}(M, f)(\rho)= \text{mdim}_{\text{H}}(M,\rho, f)$ represent, respectively, the metric mean dimension and the mean Hausdorff dimension of $f$ with respect to $\rho\in M(\tau)$ and $M(\tau)$ is the set consisting of all equivalent metrics to $d$ on $M$. Furthermore, we will present examples of certain classes of metrics for which the metric mean dimension is a continuous function.