Hölder continuous maps on the interval with 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 $\alpha$-Hölder continuous maps on $X$, for $\alpha \in (0,1].$ $H^1(X)$ is the space of Lipschitz continuous maps on $X$. We have $$H^1(X)\subset H^{\beta }(X)\subset H^{\alpha }(X)\subset C^0(X),\text{ where }0<\alpha <\beta <1.$$ It is well-known that if $\varphi \in H^1(X)$, then $\varphi$ 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 $\varphi \in H^{\alpha }([0,1])$ with positive metric mean dimension.

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