Generalized entropy of induced zero-entropy systems

TL;DR

The paper develops a unified framework based on generalized entropy $o(T)$ and metric mean dimension to analyze the induced dynamics of zero-entropy systems. It proves that many base maps with super-polynomial, but sub-exponential growth—captured by $o(T)$—drive infinite metric mean dimension for the induced hyperspace map $T_{\mathcal{K}}$, while establishing exact bounds for the measure-induced map $T_*$ and its relation to $T_{\mathcal{K}}$; notably, $o(T_*) \ge o(T)$ and $o(T_*)$ lower-bounds $o(T_{\mathcal{K}})$, with sharp results in the Morse-Smale circle case where $o(F_*)=\sup\mathbb{P}$. The work provides explicit formulas, such as $o(T_{\mathcal{K}}) = \sup\{ [2^{\text{Span}(T,n,\varepsilon)}] : \varepsilon>0\}$, linking generalized entropy to the growth of spanning sets, and demonstrates that $o(T)=[n]$ yields nuanced behavior for the induced systems. A key contribution is the construction of comparison tools between $T_*$ and $T_{\mathcal{K}}$, including a structured embedding via $\Psi_L$ that shows $o(T_*) \le o(T_{\mathcal{K}})$ in general. The results illuminate how low complexity in the base system can still yield rich, even chaotic, induced dynamics and identify open questions around linear growth orders and exact entropy behaviours in induced settings.

Abstract

Given a compact metric space $X$ and a continuous map $T: X \to X$, the induced hyperspace map $T_\mathcal{K}$ acts on the hyperspace $\mathcal{K}(X)$ of nonempty closed sets of $X$, and the measure-induced map $T_*$ acts on the space of probability measures $\mathcal{M}(X)$. It is proven that a large class of zero-entropy dynamical systems exhibits infinite metric mean dimension in its induced hyperspace map $T_\mathcal{K}$. This work also builds on the concept of generalized entropy, which is fundamental for studying the complexity of zero-entropy systems. Lower bounds of the generalized entropy of the measure-induced map $T_*$ are established, assuming that the base system $T$ has zero topological entropy. Moreover, upper bounds of the generalized entropy are explicitly computed for the measure-induced map of the Morse-Smale diffeomorphisms on the circle. Finally, it is shown that the generalized entropy of $T_*$ is a lower bound for the generalized entropy of $T_\mathcal{K}$.

Figures (1)

• Figure 1: The orders of growth depicted in blue are those that are strictly less than $[n]$, and those in red are strictly greater.

Theorems & Definitions (35)

• Theorem A
• Theorem B
• Corollary 1
• Corollary 2
• Theorem C
• Corollary 3
• Corollary 4
• Theorem D
• Theorem E
• Corollary 5