Hölder continuous maps on the interval with positive metric mean dimension
Fix a compact metric space $X$ with finite topological dimension. Let $C^{0}(X)$ be the space of continuous maps on $X$ and $ H^α(X)$ the space of $α$-Hölder continuous maps on $X$, for $α\in (0,1].$ $H^{1}(X)$ is the space of Lipschitz continuous maps on $X$. We have $$H^{1}(X)\subset H^β(X) \subset H^α(X) \subset C^{0}(X),\quad\text{ where }0<α<β<1.$$ It is well-known that if $ϕ\in H^{1}(X)$, then $ϕ$ 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 $α\in (0,1)$, there exists $ϕ\in H^α([0,1]) $ with positive metric mean dimension.
math.DS↗