Sequence entropy and independence in free and minimal actions

TL;DR

The paper constructs minimal (and in some cases free minimal) dynamical actions of countable groups with prescribed topological sequence entropy values, extending the Z-action paradigm to broader group classes. By leveraging Toeplitz $G$-subshifts and their maximal equicontinuous factors (the $G$-odometer), the authors realize $h_{top}^* = \log(m)$ for groups with a $\mathbb{Z}$ quotient and establish zero entropy yet controlled sequence entropy ranges for groups with a finite-index subgroup isomorphic to $\mathbb{Z}^r$, including virtually $\mathbb{Z}^r$ groups. They connect entropy values to independence structures via IN/IT tuples, proving the existence of $n$-tuples for certain $n$ while ruling out larger independent sizes, and they quantify fiber bounds of the maximal equicontinuous factor, yielding precise entropy intervals and ergodic measure counts. The results broaden the scope of zero-entropy systems with positive sequence entropy, provide explicit constructions, and raise questions about uniquely ergodic versions, tameness, and broader group classes.

Abstract

For every countable infinite group that admits $\mathbb{Z}$ as a homomorphic image, we show that for each $m\in\mathbb{N}$, there exists a minimal action whose topological sequence entropy is $\log(m)$. Furthermore, for every countable infinite group $G$ that contains a finite index normal subgroup $G'$ isomorphic to $\mathbb{Z}^r$, and for every $m\in \mathbb{N}$, we found a free minimal action with topological sequence entropy $\log(n)$, where $m\leq n\leq m^{2^r[G:G']}$. In both cases, we also show that the aforementioned minimal actions admit non-trivial independence tuples of size $n$ but do not admit non-trivial independence tuples of size $n+1$ for some $n\geq m$.

Theorems & Definitions (82)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• proof
• Remark 2.3
• Proposition 2.4: KeLi07
• Proposition 2.5: HuYe09
• Proposition 2.6
• proof