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. In this paper, we prove that for any α ϵ 2 [0, n], the set consisting of homeomorphisms on N with lower and upper metric mean dimensions equal to α is dense in Hom(N). More generally, given α, β ϵ [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). Furthermore, we also give a proof that the set of homeomorphisms with upper metric mean dimension equal to n is residual in Hom(N).