The automorphism group of a shift of linear growth: beyond transitivity

TL;DR

This paper investigates the algebraic structure of automorphism groups of shifts with linear complexity growth, revealing they are virtually abelian, and classifies all finite automorphism groups for such shifts.

Contribution

It establishes the virtually abelian nature of automorphism groups for linearly complex shifts and classifies all finite automorphism groups in this context.

Findings

Finitely generated subgroups are virtually a0Z^d.

If the shift is transitive, the automorphism group is virtually a0Z.

Classified all finite groups that can be automorphism groups of shifts.

Abstract

For a finite alphabet $\mathcal{A}$ and shift $X\subseteq\mathcal{A}^{\mathbb{Z}}$ whose factor complexity function grows at most linearly, we study the algebraic properties of the automorphism group ${\rm Aut}(X)$. For such systems, we show that every finitely generated subgroup of ${\rm Aut}(X)$ is virtually ${\mathbb Z}^d$, in contrast to the behavior when the complexity function grows more quickly. With additional dynamical assumptions we show more: if $X$ is transitive, then ${\rm Aut}(X)$ is virtually $\mathbb Z$; if $X$ has dense aperiodic points, then ${\rm Aut}(X)$ is virtually ${\mathbb Z}^d$. We also classify all finite groups that arise as the automorphism group of a shift.