On automorphism groups of low complexity subshifts
Abstract
In this article, we study the automorphism group Aut.X; / of subshifts .X; /
of low word complexity. In particular, we prove that Aut.X; / 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.
General note
Artículo de publicación ISI
Patrocinador
Grants Basal-CMM & Fondap 15090007, CONICYT Doctoral
fellowship 21110300, ANR grants SubTile, DynA3S and FAN and the cooperation project
MathAmSud DYSTIL
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Ergodic Theory and Dynamical Systems / Volume 36 / Issue 01 / February 2016, pp 64 - 95
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