Metric mean dimension and mean Hausdorff dimension varying the metric

Let $f:M\to M$ 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 $f:M\to M$, the functions ${\text{mdim}}_{\text{M}}(M,f):M(\tau )\to \mathbb{R}\cup \{\mathrm{\infty }\}$ and ${\text{mdim}}_{\text{H}}(M,f):M(\tau )\to \mathbb{R}\cup \{\mathrm{\infty }\}$ 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.