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arXiv · 1411.0180

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

Van Cyr·Bryna Kra
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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.

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Van Cyr, Bryna Kra. 2014-11-01. The automorphism group of a shift of linear growth: beyond transitivity. https://arxiv.org/abs/1411.0180

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