2015/01/02 by Sebastián Donoso, Fabien Durand, Donoso, Sebastián +5 · 3 citations
Computer Science · Mathematics · #37B10 #54H20 #Cellular Automata and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1501.00510
openalex publication_date 2015/01/02 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
In this article we study the automorphism group rm Aut(X,\σ) of\nsubshifts (X,\σ) of low word complexity. In particular, we prove that\nAut(X,\σ) is virtually \ℤ for aperiodic minimal subshifts and\ncertain transitive subshifts with non-superlinear complexity. More precisely,\nthe quotient of this group relative to the one generated by the shift map is a\nfinite group. In addition, we show that any finite group can be obtained in\nthis way. The class considered includes minimal subshifts induced by\nsubstitutions, linearly recurrent subshifts and even some subshifts which\nsimultaneously exhibit non-superlinear and superpolynomial complexity along\ndifferent subsequences. The main technique in this article relies on the study\nof classical relations among points used in topological dynamics, in\nparticular, asymptotic pairs. Various examples that illustrate the technique\ndeveloped in this article are provided. In particular, we prove that the group\nof automorphisms of a d-step nilsystem is nilpotent of order d and from\nthere we produce minimal subshifts of arbitrarily large polynomial complexity\nwhose automorphism groups are also virtually \ℤ.\n