Density of the level sets of the metric mean dimension for homeomorphisms

Abstract

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