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Eventual Conjugacy of Free Inert $G$-SFTs

Jeremias Epperlein

TL;DR

The paper proves that any two free inert actions of a finite group $G$ on an SFT are eventually conjugate by an automorphism after taking a high enough power of the shift, and provides a parallel algebraic result: inert matrices over $igmathbb{Z}_+[G]$ with SE augmentations over $igmathbb{Z}_+$ are SE over $igmathbb{Z}_+[G]$. This is achieved by representing free $G$-SFTs as $G$-extensions defined by matrices in $igmathbb{Z}_+[G]$ and leveraging lift-of-SE techniques to transfer SE from augmentations to the full group-ring setting. Consequences include eventual $G$-conjugacy of free finite-order automorphisms of full shifts and corollaries for stabilized automorphism groups, as well as a equivariant flow-equivalence perspective generalizing prior results. The work connects inertness, shift equivalence, and equivariant flow invariants, enriching the algebraic toolkit for classifying $G$-SFTs and their automorphism structures.

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.

Eventual Conjugacy of Free Inert $G$-SFTs

TL;DR

The paper proves that any two free inert actions of a finite group on an SFT are eventually conjugate by an automorphism after taking a high enough power of the shift, and provides a parallel algebraic result: inert matrices over with SE augmentations over are SE over . This is achieved by representing free -SFTs as -extensions defined by matrices in and leveraging lift-of-SE techniques to transfer SE from augmentations to the full group-ring setting. Consequences include eventual -conjugacy of free finite-order automorphisms of full shifts and corollaries for stabilized automorphism groups, as well as a equivariant flow-equivalence perspective generalizing prior results. The work connects inertness, shift equivalence, and equivariant flow invariants, enriching the algebraic toolkit for classifying -SFTs and their automorphism structures.

Abstract

The action of a finite group on a subshift of finite type is called free, if every point has trivial stabilizer, and it is called inert, if the induced action on the dimension group of is trivial. We show that any two free inert actions of a finite group 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 -SFTs.
Paper Structure (8 sections, 24 theorems, 34 equations, 2 figures)

This paper contains 8 sections, 24 theorems, 34 equations, 2 figures.

Table of Contents

  1. Introduction and Main Results
  2. Preliminaries
  3. Finite Groups Acting Freely on SFTs
  4. Matrices Over The Integral Group Ring and $G$-Extensions of SFTs
  5. A Characterization of Free Inert $G$-SFTs
  6. Proof of the Main Theorem
  7. A Theorem of Kim and Roush
  8. Shift Equivalence Over $\mathbb{Z}_+[G]$ and Equivariant Flow Equivalence

Key Result

Theorem 1

Let $(Y_1,\sigma)$ and $(Y_2,\sigma)$ be SFTs such that $(Y_1,\sigma^\ell)$, $(Y_2,\sigma^\ell)$ are conjugate for all sufficiently large $\ell$. Let $\alpha_1$ and $\alpha_2$ be free inert actions of a finite group $G$ on $(Y_1,\sigma)$ and $(Y_2,\sigma)$, respectively. Then for every sufficiently

Figures (2)

  • Figure 1: The subshift $Y$ from \ref{['ex:000']}. The automorphism induced by $\tau$ corresponds to a point reflection of this graph across its center.
  • Figure 2: A $\mathbb{Z}/2\mathbb{Z}$ extension of the golden mean shift.

Theorems & Definitions (47)

  • Example 1.2: Obstruction to topological conjugacy
  • Theorem 1: \ref{['thm:main']} (Main theorem - dynamical version)
  • Theorem 2: \ref{['thm:main-algebraic-version']} (Main theorem - algebraic version)
  • Corollary 1: \ref{['thm:main-corr-classic-version']} (Eventual conjugacy for free finite order automorphisms of full shifts)
  • Corollary 2: \ref{['cor:main-corr-stabilized-version']} (Conjugacy of free finite order elements in the stabilized automorphism group)
  • Definition 3.1: $G$-SFT
  • Definition 3.2: $G$-conjugacy
  • Definition 3.3: Eventual $G$-conjugacy
  • Proposition 3.4: Representation of free $G$-SFTs by graph automorphisms
  • Definition 3.5: Inert free $G$-SFTs
  • ...and 37 more