On automorphism groups of low complexity minimal subshifts
Résumé
In this article we study the automorphism group Aut(X, σ) of a minimal subshift (X, σ) of low word complexity. In particular, we prove that Aut(X, σ) is virtually Z for aperiodic minimal subshifts with affine complexity on a subsequence, more precisely, the quotient of this group by the one gen-erated by the shift map is a finite group. In addition, we provide examples to show that any finite group can be obtained in this way. The class con-sidered includes minimal substitutions, linearly recurrent subshifts and even some minimal subshifts with polynomial complexity. In the case of polyno-mial complexity, first we prove that for minimal subshifts with polynomial recurrence any finitely generated subgroup of Aut(X, σ) is virtually nilpotent. Then, we describe several subshifts of polynomial complexity to illustrate that the group of automorphisms can still be virtually Z. This behavior seems to be very frequent for low complexity minimal subshifts. The main technique in this article relies on the study of classical relations among points used in topological dynamics, in particular asymptotic pairs.
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