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 ∈ [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 ∈ [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 withupper metric mean dimension equal to n is residual in Hom(N).