THE AUTOMORPHISM GROUP OF A SHIFT OF LINEAR GROWTH: BEYOND TRANSITIVITY
Forum of Mathematics, Sigma, Tome 3 (2015)
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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 $\text{Aut}(X)$. For such systems, we show that every finitely generated subgroup of $\text{Aut}(X)$ 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 $\text{Aut}(X)$ is virtually $\mathbb{Z}$; if $X$ has dense aperiodic points, then $\text{Aut}(X)$ is virtually $\mathbb{Z}^{d}$. We also classify all finite groups that arise as the automorphism group of a shift.
@article{10_1017_fms_2015_3,
author = {VAN CYR and BRYNA KRA},
title = {THE {AUTOMORPHISM} {GROUP} {OF} {A} {SHIFT} {OF} {LINEAR} {GROWTH:} {BEYOND} {TRANSITIVITY}},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {3},
year = {2015},
doi = {10.1017/fms.2015.3},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2015.3/}
}
VAN CYR; BRYNA KRA. THE AUTOMORPHISM GROUP OF A SHIFT OF LINEAR GROWTH: BEYOND TRANSITIVITY. Forum of Mathematics, Sigma, Tome 3 (2015). doi: 10.1017/fms.2015.3
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