The automorphism group of a strongly irreducible subshift on a group

TL;DR

The paper advances the theory of automorphism groups of symbolic dynamical systems by extending Ryan’s center description and Kim–Roush-type embeddings to strongly irreducible subshifts on arbitrary infinite groups. A novel marker framework and the new concept of egg markers enable explicit, transferizable automorphisms from full shifts to any nontrivial SI subshift, even beyond SFTs, under mild hypotheses such as strong TMP. The authors show that the center of Aut$(X)$ is generated by shifts along the center of $G$ modulo the shift kernel, and they prove that Aut$(A^{ ext{Z}})$ and Aut$(A^{F_k})$ embed into Aut$(X)$ under nontrivial conditions (notably non-locally-finite and nonamenable $G$). These results apply broadly, including new instances on $ ext{G}= ext{Z}$, and yield rigidity-type consequences and universality aspects for subgroup realizations within Aut$(X)$. The framework relies on a comprehensive marker lemma, signifying a robust toolkit for studying automorphism groups across general groups and subshift classes, with several open questions about weakening hypotheses and extending to further group families.

Abstract

We study the automorphism group $\operatorname{Aut}(X)$ of a non-trivial strongly irreducible subshift $X$ on an arbitrary infinite group $G$ and generalize classical results of Ryan, Kim and Roush. We generalize Ryan's theorem by showing that the center of $\operatorname{Aut}(X)$ is generated by shifts by elements of the center of $G$ modded out by the kernel of the shift action. We generalize Kim and Roush's theorem by showing that if the free group $F_k$ of rank $k\geq 1$ embeds into $G$, then the automorphism group of any full $F_k$-shift embeds into $\operatorname{Aut}(X)$. If $X$ is an SFT, or more generally, if $X$ satisfies the strong topological Markov property, then we can weaken the conditions on $G$. In this case we show that the automorphism group of any full $\mathbb{Z}$-shift embeds into $\operatorname{Aut}(X)$ provided $G$ is not locally finite, and that the automorphism group of any full $F_k$-shift embeds into $\operatorname{Aut}(X)$ whenever $G$ is nonamenable. Our results rely on a new marker lemma which is valid for any nonempty strongly irreducible subshift on an infinite group. We remark that our results are new even for $G=\mathbb{Z}$ as they do not require the subshift to be an SFT.

Figures (5)

• Figure 1: Three cones are fixed by the symbol at the origin.
• Figure 2: A $( \{2,3\}^2, \{0,\dots,5\}^2)$-marker in $\{0,1\}^{\mathbb Z^2}$.
• Figure 3: A collection of egg markers in $\{\texttt{0},\texttt{1}\}^{\mathbb Z^2}$.
• Figure 4: A portion of a configuration that is forward and backward consistent at $g$. The bottom row represents elements $g, gt$ and $gs$ in $G$ and the top row the symbols at those positions.
• Figure 5: Path types according to $f_x$.

Theorems & Definitions (96)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5