Finitary Ryan's and local $\mathcal{Q}$ entropy for $\mathbb{Z}^{d}$ subshifts

TL;DR

The work extends Ryan's centralizer results to shifts of finite type over general groups, proving finitary Ryan-type theorems for contractible $\mathbb{Z}^{d}$-SFTs with a fixed point and for full shifts on suitable infinite groups. It introduces local $\mathcal{Q}$ entropy to relate stabilized automorphism groups to the underlying topological entropy, and proves that for contractible $\mathbb{Z}^{d}$-SFTs this entropy recovers $h_{top}$ up to a rational multiple, enabling a complete classification of stabilized automorphism groups of full shifts over $\mathbb{Z}^{d}$. The paper develops gate lattices and ghost centers, and employs transitivity-based arguments to realize finitary Ryan phenomena in both general and full-shift settings. Consequently, stabilized automorphism groups become powerful invariants, with entropy-driven rational relations and precise isomorphism classifications, while also highlighting distinctions between contractible SFTs and full shifts. The results advance understanding of stabilized symmetries in symbolic dynamics and their connections to entropy and structure theory.

Abstract

For the action of a group $G$ by homeomorphisms on a space $X$, the automorphism group $\mathrm{Aut}(X,G)$ consists of all self-homeomorphisms of $X$ which commute with $x \mapsto g \cdot x$ for every $g \in G$. A theorem of Ryan shows that for an irreducible $\mathbb{Z}$-shift of finite type $(X,σ_{X})$, the center of $\mathrm{Aut}(X,σ_{X})$ is generated by the shift $σ_{X}$. A finitary version of this for $\mathbb{Z}$-shifts of finite type was proved by the second author for certain full shifts, and later generalized by Kopra to irreducible $\mathbb{Z}$-shifts of finite type. We generalize these finitary Ryan's theorems to shifts of finite type over more general groups. We prove that for contractible $\mathbb{Z}^{d}$-shifts of finite type with a fixed point, there is a finitely generated subgroup of the automorphism group whose centralizer in the group of homeomorphisms is the subgroup of shifts. We also prove versions of this for full shifts over any infinite, finitely generated group on sufficiently nice alphabet sizes. The stabilized automorphism group $\mathrm{Aut}^{(\infty)}(X,G)$ is the union of $\mathrm{Aut}(X,H)$ over all finite index subgroups $H \subset G$. Aimed at studying stabilized automorphism groups for shifts of finite type, we introduce an entropy-like quantity for pointed groups which we call local $\mathcal{Q}$ entropy, a generalization of a notion called local $\mathcal{P}$ entropy previously introduced by the first author. Using the finitary Ryan's theorems, we prove that the local $\mathcal{Q}$ entropy of the stabilized automorphism group of a contractible $\mathbb{Z}^{d}$-shift of finite type recovers the topological entropy of the underlying shift system up to a rational multiple. We then use this to give a complete classification up to isomorphism of the stabilized automorphism groups of full shifts over $\mathbb{Z}^{d}$.

Theorems & Definitions (133)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Definition 6
• Proposition 7
• Proposition 8
• proof
• Proposition 9