Hyperspatiality for isomorphisms of stabilized automorphism groups of shifts of finite type
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909213590028288 |
|---|---|
| author | Epperlein, Jeremias Schmieding, Scott |
| author_facet | Epperlein, Jeremias Schmieding, Scott |
| contents | Given a homeomorphism $T \colon X \to X$ of a compact metric space $X$, the stabilized automorphism group $\textrm{Aut}^{\infty}(T)$ of the system $(X,T)$ is the group of self-homeomorphisms of $X$ which commute with some power of $T$. We study the question of spatiality for stabilized automorphism groups of shifts of finite type. We prove that any isomorphism $Ψ\colon \textrm{Aut}^{\infty}(σ_{m}) \to \textrm{Aut}^{\infty}(σ_{n})$ between stabilized automorphism groups of full shifts is spatially induced by a homeomorphism $\hatΨ$ between respective stabilized spaces of chain recurrent subshifts. This spatialization in particular gives a bijection between the sets of periodic points which intertwines some powers of the shifts, and this bijection recovers the isomorphism at the level of the faithful actions on the sets of periodic points. We also prove that the outer automorphism group of $\textrm{Aut}^{\infty}(σ_{n})$ is uncountable, and deduce several other properties of $\textrm{Aut}^{\infty}(σ_{n})$ using the spatiality results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_20463 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hyperspatiality for isomorphisms of stabilized automorphism groups of shifts of finite type Epperlein, Jeremias Schmieding, Scott Dynamical Systems 37B10 Given a homeomorphism $T \colon X \to X$ of a compact metric space $X$, the stabilized automorphism group $\textrm{Aut}^{\infty}(T)$ of the system $(X,T)$ is the group of self-homeomorphisms of $X$ which commute with some power of $T$. We study the question of spatiality for stabilized automorphism groups of shifts of finite type. We prove that any isomorphism $Ψ\colon \textrm{Aut}^{\infty}(σ_{m}) \to \textrm{Aut}^{\infty}(σ_{n})$ between stabilized automorphism groups of full shifts is spatially induced by a homeomorphism $\hatΨ$ between respective stabilized spaces of chain recurrent subshifts. This spatialization in particular gives a bijection between the sets of periodic points which intertwines some powers of the shifts, and this bijection recovers the isomorphism at the level of the faithful actions on the sets of periodic points. We also prove that the outer automorphism group of $\textrm{Aut}^{\infty}(σ_{n})$ is uncountable, and deduce several other properties of $\textrm{Aut}^{\infty}(σ_{n})$ using the spatiality results. |
| title | Hyperspatiality for isomorphisms of stabilized automorphism groups of shifts of finite type |
| topic | Dynamical Systems 37B10 |
| url | https://arxiv.org/abs/2405.20463 |