H\"older 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 $H^{\alpha}(X)$ 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.