Density of the Level Sets of the Metric Mean Dimension for Homeomorphisms

Abstract

AbstractLet N be an n-dimensional compact riemannian manifold, with $$n\ge 2$$ n ≥ 2 . In this paper, we prove that for any $$\alpha \in [0,n]$$ α ∈ [ 0 , n ] , the set consisting of homeomorphisms on N with lower and upper metric mean dimensions equal to $$\alpha $$ α is dense in $$\text {Hom}(N)$$ Hom ( N ) . More generally, given $$\alpha ,\beta \in [0,n]$$ α , β ∈ [ 0 , n ] , with $$\alpha \le \beta $$ α ≤ β , we show the set consisting of homeomorphisms on N with lower metric mean dimension equal to $$\alpha $$ α and upper metric mean dimension equal to $$\beta $$ β is dense in $$\text {Hom}(N)$$ Hom ( N ) . Furthermore, we also give a proof that the set of homeomorphisms with upper metric mean dimension equal to n is residual in $$\text {Hom}(N)$$ Hom ( N ) .