Metric Mean Dimension and Mean Hausdorff Dimension Varying the Metric

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 τ be the topology on M M induced by d . We denote by M(τ ) M ( τ ) the set consisting of all metrics on M M that are equivalent to d . Let mdimM(M,d, f) mdim M ( M , d , f ) and mdimH (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 mdimM(M,d, f) mdim M ( M , d , f ) and mdimH (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 mdimM (M, f):M(τ )→ R∪ \∞ \ mdim M ( M , f ) : M ( τ ) → R ∪ { ∞ } and mdimH(M, f): M(τ )→ R∪ \∞ \ mdim H ( M , f ) : M ( τ ) → R ∪ { ∞ } are not continuous, where mdimM(M, f) (ρ )= mdimM (M,ρ , f) mdim M ( M , f ) ( ρ ) = mdim M ( M , ρ , f ) and mdimH(M, f) (ρ )= mdimH (M,ρ , f) mdim H ( M , f ) ( ρ ) = mdim H ( M , ρ , f ) for any ρ ∈ M(τ ) ρ ∈ M ( τ ) . Furthermore, we will present examples of certain classes of metrics for which the metric mean dimension is a continuous function.