Metric Mean Dimension and Mean Hausdorff Dimension Varying the Metric

Abstract

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