Eventual Conjugacy of Free Inert $G$-SFTs

TL;DR

This paper proves that free inert actions of a finite group on a subshift of finite type are conjugate in high powers, advancing understanding of automorphisms and flow equivalence in symbolic dynamics.

Contribution

It demonstrates conjugacy of free inert G-actions on SFTs in high powers and generalizes flow equivalence results for G-SFTs.

Findings

Any two free inert G-actions on an SFT are conjugate in high powers.

Every two free elements of the stabilized automorphism group are conjugate.

Generalizes flow equivalence classification for G-SFTs.

Abstract

The action of a finite group $G$ on a subshift of finite type $X$ is called free, if every point has trivial stabilizer, and it is called inert, if the induced action on the dimension group of $X$ is trivial. We show that any two free inert actions of a finite group $G$ on an SFT are conjugate by an automorphism of any sufficiently high power of the shift space. This partially answers a question posed by Fiebig. As a consequence we obtain that every two free elements of the stabilized automorphism group of a full shift are conjugate in this group. In addition, we generalize a result of Boyle, Carlsen and Eilers concerning the flow equivalence of $G$-SFTs.