On automorphism groups of low complexity subshifts

TL;DR

This paper investigates the automorphism groups of low complexity subshifts, showing they are often virtually D7, and explores their structure and examples across various complexity classes.

Contribution

It establishes that automorphism groups of certain low complexity subshifts are virtually D7 and provides methods to realize any finite group as such an automorphism group.

Findings

Automorphism groups are virtually D7 for aperiodic minimal subshifts.

Any finite group can be realized as an automorphism group of some low complexity subshift.

Automorphism groups of nilsystems are nilpotent of order d.

Abstract

In this article we study the automorphism group ${\rm Aut}(X,\sigma)$ of subshifts $(X,\sigma)$ of low word complexity. In particular, we prove that Aut$(X,\sigma)$ is virtually $\mathbb{Z}$ for aperiodic minimal subshifts and certain transitive subshifts with non-superlinear complexity. More precisely, the quotient of this group relative to the one generated by the shift map is a finite group. In addition, we show that any finite group can be obtained in this way. The class considered includes minimal subshifts induced by substitutions, linearly recurrent subshifts and even some subshifts which simultaneously exhibit non-superlinear and superpolynomial complexity along different subsequences. The main technique in this article relies on the study of classical relations among points used in topological dynamics, in particular, asymptotic pairs. Various examples that illustrate the technique developed in this article are provided. In particular, we prove that the group of automorphisms of a $d$-step nilsystem is nilpotent of order $d$ and from there we produce minimal subshifts of arbitrarily large polynomial complexity whose automorphism groups are also virtually $\mathbb{Z}$.