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

Let N be an n -dimensional compact riemannian manifold, with n≥ 2 n≥2 . In this paper, we prove that for any α ∈ [0,n] α∈[0,n] , the set consisting of homeomorphisms on N with lower and upper metric mean dimensions equal to α α is dense in Hom(N) Hom(N) . More generally, given α ,β ∈ [0,n] α,β∈[0,n] , with α ≤ β α≤β , we show the set consisting of homeomorphisms on N with lower metric mean dimension equal to α α and upper metric mean dimension equal to β β is dense in 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 Hom(N) Hom(N) .