Hyperspatiality for isomorphisms of stabilized automorphism groups of shifts of finite type

TL;DR

This paper investigates the structure of stabilized automorphism groups of shifts of finite type, proving that isomorphisms between these groups are spatially induced by homeomorphisms, and explores their automorphism properties.

Contribution

It establishes the spatiality of automorphism group isomorphisms for shifts of finite type and analyzes the uncountability of their outer automorphism groups.

Findings

Isomorphisms are spatially induced by homeomorphisms.

Bijection between periodic points preserves shift powers.

Outer automorphism group is uncountable.

Abstract

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 $\Psi \colon \textrm{Aut}^{\infty}(\sigma_{m}) \to \textrm{Aut}^{\infty}(\sigma_{n})$ between stabilized automorphism groups of full shifts is spatially induced by a homeomorphism $\hat{\Psi}$ 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 $\textrm{Aut}^{\infty}(\sigma_{n})$ is uncountable, and deduce several other properties of $\textrm{Aut}^{\infty}(\sigma_{n})$ using the spatiality results.