On automorphism groups of low complexity subshifts
By Sebastian Donoso, Fabien Durand, Alejandro Maass, S. Petite. Ergodic Theory and Dynamical Systems, 2015
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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 $\mathbb{Z}$.
Subfields: Artificial Intelligence, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/bbbb67cd4316208cb24eb54acf41c3529e1b769c
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Citations in the corpus
- The automorphism group of a minimal shift of stretched exponential growth (2015, methodology): “Further examples of minimal shifts with polynomial complexity and highly constrained automorphism groups were constructed by Donoso, Durand, Maass, and Petite [7], and an example of a minimal shift with subquadratic growth whose automorphism group is not finitely generated was given by Salo [14].”