Metric Mean Dimension and Mean Hausdorff Dimension Varying the Metric

Abstract Let f: M M f: M → M be a continuous map on a compact metric space M M equipped with a fixed metric d, and let τ be the topology on M M induced by d. We denote by M () M (τ) the set consisting of all metrics on M M that are equivalent to d. Let mdim ₌ (M, d, f) mdim M (M, d, f) and mdim ₇ (M, d, f) mdim H (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 mdim ₌ (M, d, f) mdim M (M, d, f) and mdim ₇ (M, d, f) mdim H (M, d, f) depend on the metric d chosen for M M. In this work, we will prove that, for a fixed dynamical system f: M M f: M → M, the functions mdim ₌ (M, f): M () R \ \ mdim M (M, f): M (τ) → R ∪ ∞ and mdim ₇ (M, f): M () R \ \ mdim H (M, f): M (τ) → R ∪ ∞ are not continuous, where mdim ₌ (M, f) () = mdim ₌ (M, , f) mdim M (M, f) (ρ) = mdim M (M, ρ, f) and mdim ₇ (M, f) () = mdim ₇ (M, , f) mdim H (M, f) (ρ) = mdim H (M, ρ, f) for any M () ρ ∈ M (τ). Furthermore, we will present examples of certain classes of metrics for which the metric mean dimension is a continuous function.