Metric Mean Dimension and Mean Hausdorff Dimension Varying the Metric

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