Metric Mean Dimension and Mean Hausdorff Dimension Varying the Metric (2024)
Let f : M → M be a continuous map on a compact metric space M equipped with a fixed metric d, and let τ be the topology on M induced by d. We denote by M(τ) the set consisting of all metrics on M that are equivalent to d. Let mdimM(M,d,f) and mdimH(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 mdimM(M,d,f) and mdimH(M,d,f) depend on the metric d chosen for M. In this work, we will prove that, for a fixed dynamical system f : M → M, the functions mdimM(M, f ) : M(τ) → R∪ {∞} and mdimH(M,f) : M(τ ) → R∪ {∞} are not continuous, where mdimM(M,f)(ρ) = mdimM(M,ρ,f) and mdimH(M,f)(ρ) = mdimH(M,ρ,f) for any ρ ∈ M(τ). Furthermore, we will present examples of certain classes of metrics for which the metric mean dimension is a continuous function.