Genericity of continuous maps with positive metric mean dimension

TL;DR

This paper demonstrates that continuous maps with any specified metric mean dimension are dense in the space of all continuous maps on a compact manifold, with some sets being residual, highlighting the genericity of positive metric mean dimension.

Contribution

It establishes the density and residuality of continuous maps with prescribed metric mean dimension on compact manifolds and product spaces, extending the understanding of generic properties in dynamical systems.

Findings

Maps with any metric mean dimension are dense in C^0(N).

For a=n, the set of maps with metric mean dimension n is residual.

Existence and density results for maps with positive metric mean dimension on Cantor sets and product spaces.

Abstract

M. Gromov introduced the mean dimension for a continuous map in the late 1990's, which is an invariant under topological conjugacy. On the other hand, the notion of metric mean dimension for a dynamical system was introduced by Lindenstrauss and Weiss in 2000 and this refines the topological entropy for dynamical systems with infinite topological entropy. In this paper we will show if $N$ is a $n$ dimensional compact riemannian manifold then, for any $a\in [0,n]$, the set consisting of continuous maps with metric mean dimension equal to $a$ is dense in $C^{0}(N)$ and for $a=n$ this set is residual. Furthermore, we prove some results related to existence and density of continuous maps on Cantor sets with positive metric mean dimension and on product spaces with positive mean dimension.