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

期刊论文
出版模式：

开放获取
发表日期：

2024年2月10日

查看全文

Jeovanny M. Acevedo,
Sergio Romaña,
Raibel Arias

摘要

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