Mean dimension explosion of induced homeomorphisms

Given X a compact metric space and T: X → X a continuous map, the induced hyperspace map TK acts on the hyperspace K(X) of closed and nonempty subsets of X, and on the continuum hyperspace C(X) ⊂ K(X) of connected sets. This work studies the mean dimension explosion phenomenon: when the base system T has zero topological entropy, but the mean dimension of the induced map TK is infinite. In particular, this phenomenon is attained for Morse-Smale diffeomorphisms. Furthermore, for a circle homeomorphism H, the mean dimension explosion does not occur if, and only if, H is conjugated to a rotation. Finally, if the topological entropy of T is positive, then the metric mean dimension of TK is infinite.