The automorphism group of a strongly irreducible subshift on a group
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866929746903826432 |
|---|---|
| author | Barbieri, Sebastián Carrasco-Vargas, Nicanor Rivera-Burgos, Paola |
| author_facet | Barbieri, Sebastián Carrasco-Vargas, Nicanor Rivera-Burgos, Paola |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_14463 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The automorphism group of a strongly irreducible subshift on a group Barbieri, Sebastián Carrasco-Vargas, Nicanor Rivera-Burgos, Paola Dynamical Systems Group Theory 37B10, 20B27 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. |
| title | The automorphism group of a strongly irreducible subshift on a group |
| topic | Dynamical Systems Group Theory 37B10, 20B27 |
| url | https://arxiv.org/abs/2501.14463 |