hal-03620380
https://u-picardie.hal.science/hal-03620380
arxiv:1501.00510
doi:10.1017/etds.2015.70
[CNRS] CNRS - Centre national de la recherche scientifique
[UNIV-PICARDIE] Université de Picardie Jules Verne
[INSMI] CNRS-INSMI - INstitut des Sciences Mathématiques et de leurs Interactions
[U-PICARDIE] Université de Picardie Jules Verne
[LAMFA] Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352
On automorphism groups of low complexity subshifts
Donoso, Sebastian
Durand, Fabien
Maass, Alejandro
Petite, Samuel
[MATH] Mathematics [math]
ART
In this article, we study the automorphism group Aut(X, sigma) of subshifts (X, sigma) of low word complexity. In particular, we prove that Aut(X, sigma) is virtually 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 Z.
2016
en
Ergodic Theory and Dynamical Systems
Cambridge University Press (CUP)