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

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