Generic homeomorphisms have full metric mean dimension

TL;DR

This paper proves that for generic homeomorphisms on high-dimensional manifolds, the metric mean dimension equals the manifold's dimension, and for interval maps, all metric mean dimension levels are dense.

Contribution

It establishes the full metric mean dimension for generic homeomorphisms and density of level sets for continuous interval maps.

Findings

Metric mean dimension of generic homeomorphisms equals manifold dimension

Level sets of metric mean dimension are dense for continuous interval maps

Results hold for manifolds of dimension greater than one

Abstract

We prove that the upper metric mean dimension of $C^0$-generic homeomorphisms, acting on a compact smooth boundaryless manifold with dimension greater than one, coincides with the dimension of the manifold. In the case of continuous interval maps we also show that each level set for the metric mean dimension is $C^0$-dense in the space of continuous endomorphisms of $[0,1]$ with the uniform topology.