H\"older continuous maps on the interval with positive metric mean dimension

TL;DR

This paper demonstrates that for any H"older exponent less than one, there exist continuous maps on the interval with positive metric mean dimension, contrasting with Lipschitz maps which always have zero metric mean dimension.

Contribution

It proves the existence of H"older continuous maps with positive metric mean dimension on the interval, expanding understanding beyond Lipschitz maps.

Findings

H"older maps with positive metric mean dimension exist for all exponents in (0,1)

Lipschitz maps on compact spaces have zero metric mean dimension

Residual sets of continuous maps can have positive metric mean dimension

Abstract

Fix a compact metric space $X$ with finite topological dimension. Let $C^{0}(X)$ be the space of continuous maps on $X$ and $ H^{\alpha}(X)$ the space of $\alpha$-H\"older continuous maps on $X$, for $\alpha\in (0,1].$ $H^{1}(X)$ is the space of Lipschitz continuous maps on $X$. We have $$H^{1}(X)\subset H^{\beta}(X) \subset H^{\alpha}(X) \subset C^{0}(X),\quad\text{ where }0<\alpha<\beta<1.$$ It is well-known that if $\phi\in H^{1}(X)$, then $\phi$ has metric mean dimension equal to zero. On the other hand, if $X$ is a finite dimensional compact manifold, then $C^{0}(X)$ contains a residual subset whose elements have positive metric mean dimension. In this work we will prove that, for any $\alpha\in (0,1)$, there exists $\phi\in H^{\alpha}([0,1]) $ with positive metric mean dimension.