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On automorphism groups of low complexity subshifts

Published online by Cambridge University Press:  27 November 2015

Departamento de Ingeniería Matemática, Universidad de Chile, Beauchef 851, Santiago, Chile Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées, 5 bd Descartes, 77454 Marne la Vallée Cedex 2, France email,
Laboratoire Amiénois de Mathématiques Fondamentales et Appliquées, CNRS-UMR 7352, Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens Cedex 1, France email,
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, CNRS-UMI 2807, Universidad de Chile, Beauchef 851, Santiago, Chile email
Laboratoire Amiénois de Mathématiques Fondamentales et Appliquées, CNRS-UMR 7352, Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens Cedex 1, France email,


In this article, we study the automorphism group $\text{Aut}(X,{\it\sigma})$ of subshifts $(X,{\it\sigma})$ of low word complexity. In particular, we prove that $\text{Aut}(X,{\it\sigma})$ 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 $\mathbb{Z}$.

Research Article
© Cambridge University Press, 2015 

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