Mean dimension explosion of induced homeomorphisms

TL;DR

This work analyzes the mean dimension and metric mean dimension of induced hyperspace maps $T_\u210A$ acting on $\mathcal{K}(X)$ and $\mathcal{C}(X)$, uncovering mean dimension explosion where zero-entropy base dynamics yield infinite mean dimension for the induced system. It proves that for locally connected continua with $\Omega(T)\subsetneq X$, $\text{mdim}(\mathcal{K}(X),T_\u210A)=\infty$, with Morse-Smale diffeomorphisms providing natural examples, and establishes a sharp circle dichotomy: $\text{mdim}(\mathcal{K}(S^1),H_\u210A)=0$ iff $H$ is conjugate to a rotation. The paper further develops the metric mean dimension theory, giving sufficient conditions (e.g., $h_{pol}(T)<1$ or isometric base) for zero mdim_M, while constructing counterexamples showing that the converse can fail and that positive entropy does not always force infinite mdim_M. A Morse-Smale continuum-dynamics dichotomy complements these results, and the findings illuminate the intricate relationship between base dynamics and induced hyperspace complexity, with implications for embeddings and quasi-factors in infinite-dimensional settings.

Abstract

Given $X$ a compact metric space and $T: X \to X$ a continuous map, the induced hyperspace map $T_\mathcal{K}$ acts on the hyperspace $\mathcal{K}(X)$ of closed and nonempty subsets of $X$, and on the continuum hyperspace $\mathcal{C}(X) \subset \mathcal{K}(X)$ 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 $T_\mathcal{K}$ is infinite. In particular, this phenomenon occurs for Morse-Smale diffeomorphisms. Furthermore, for a circle homeomorphism $H$, the mean dimension explosion does not occur if and only if $H$ is conjugate to a rotation. For the metric mean dimension, a different result is obtained: we establish sufficient conditions for the induced hyperspace map to have zero or infinite metric mean dimension.

Figures (1)

• Figure 1: The connected set in blue is an example of $\Phi(\xi)$ for $k = 2$.

Theorems & Definitions (42)

• Theorem A
• Corollary 1
• Theorem B
• Theorem C
• Corollary 2
• Theorem D
• Theorem E
• Theorem F
• Proposition 1: lindenstrauss_weiss_2000coornaert_2015
• Definition 1