Hölder continuous maps on the interval with positive metric mean dimension

Abstract

Fix a compact metric space X with finite topological dimension. Let C0(X) be the space of continuous maps on X and Hα(X) the space of α-Hölder continuous maps on X, for α ∈ (0, 1]. Let H1(X) be the space of Lipschitz continuous maps on X. We have H1(X) ⊂ Hβ(X) ⊂ Hα(X) ⊂ C0(X), where 0 < α < β < 1. It is well-known that if φ ∈ H1(X), then φ has metric mean dimension equal to zero. On the other hand, if X is a manifold, then C0(X) contains a residual subset whose elements have positive metric mean dimension. In this work we will prove that, for any α ∈ (0, 1), there exists φ ∈ Hα([0, 1]) with positive metric mean dimension.