On a Rice theorem for dynamical properties of SFTs on groups

TL;DR

The paper investigates Rice-style undecidability for dynamical properties of $G$-SFTs when the domino problem $DP(G)$ is undecidable (as for $G=\mathbb{Z}^2$). It introduces Berger properties and proves that every nontrivial Berger property preserved by factors or extensions is undecidable via reductions to $DP(G)$, and it establishes a Rice-like theorem for computable invariants monotone by disjoint unions and products, showing such invariants must be constant; it also extends these ideas to dynamical properties of sofic $G$-subshifts. The results imply noncomputability of topological entropy for amenable groups with undecidable $DP(G)$ and provide a unified framework linking Rice-type and Adian–Rabin-type undecidability phenomena in symbolic dynamics on groups. Additionally, a parallel undecidability result is proved for sofic subshifts, reinforcing the broad reach of these undecidability phenomena across dynamical systems on groups. Overall, the work delineates the pervasive nature of undecidability in a broad class of dynamical properties and invariants for group-based symbolic systems.

Abstract

Let $G$ be a group with undecidable domino problem, such as $\mathbb{Z}^2$. We prove that all nontrivial dynamical properties for sofic $G$-subshifts are undecidable, that this is not true for $G$-SFTs, and an undecidability result for dynamical properties of $G$-SFTs similar to the Adian-Rabin theorem. Furthermore we prove a Rice-like result for dynamical invariants asserting that every computable real-valued invariant for $G$-SFTs that is monotone by disjoint unions and products is constant.

Theorems & Definitions (27)

• Definition 2.1
• Proposition 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• Remark 3.4
• Definition 3.5
• Theorem 3.6
• proof