Infinite topological entropy, positive mean dimension, and factors of subshifts

TL;DR

The paper investigates when all nontrivial factors of a dynamical system have infinite topological entropy (equivalently, positive mean dimension), focusing on subshifts of block type in the Hilbert cube. It proves an if-and-only-if criterion: for the subshift of block type associated with $(2,B)$, the properties that every nontrivial factor has infinite topological entropy, every nontrivial factor has positive mean dimension, and the diagonal intersection condition $B \cap \{(x,x): 0 \le x \le 1\} \neq \emptyset$ are all equivalent. The proof combines a decomposition $X = X_0 \cup X_1$, a two-point factor when $B$ avoids the diagonal, Li13’s results on full shifts with path-connected alphabets, and a general lemma that transfers mean-dimension properties from subsystems to the whole. Consequently, the paper yields a broad family of new concrete examples and clarifies the structural role of the diagonal-intersection condition, while also discussing variants and open problems in this domain.

Abstract

We study dynamical systems with the property that all the nontrivial factors have infinite topological entropy (or, positive mean dimension). We establish an ``if and only if'' condition for this property among a typical class of dynamical systems, the subshifts of block type in the Hilbert cube. This in particular leads to a large class of concrete (and new) examples of dynamical systems having this property.

Theorems & Definitions (9)

• Theorem 1.1: Main theorem
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Lemma 2.1
• proof
• Theorem 3.1
• Theorem 3.2
• Conjecture 3.3