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On automorphism groups of low complexity subshifts

  • SEBASTIÁN DONOSO (a1) (a2), FABIEN DURAND (a3), ALEJANDRO MAASS (a4) and SAMUEL PETITE (a3)

Abstract

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 $\mathbb{Z}$ .

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[1]Arnoux, P. and Mauduit, C.. Complexité de suites engendrées par des récurrences unipotentes. Acta Arith. 76 (1996), 8597.
[2]Auslander, J.. Endomorphisms of minimal sets. Duke Math. J. 30 (1963), 605614.
[3]Auslander, J.. Minimal Flows and Their Extensions (North-Holland Mathematics Studies, 153). North-Holland, Amsterdam, 1988, 265pp.
[4]Auslander, L., Green, L. and Hahn, F.. Flows on homogeneous spaces. Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1963, 107pp; with the assistance of L. Markus and W. Massey, and an appendix by L. Greenberg.
[5]Barge, M. and Diamond, B.. A complete invariant for the topology of one-dimensional substitution tiling spaces. Ergod. Th. & Dynam. Sys. 21 (2001), 13331358.
[6]Boyle, M., Lind, D. and Rudolph, D.. The automorphism group of a shift of finite type. Trans. Amer. Math. Soc. 306 (1988), 71114.
[7]Cassaigne, J.. Complexité et facteurs spéciaux. Bull. Belg. Math. Soc. 4 (1997), 6788.
[8]Cassaigne, J.. Sequences with grouped factors. Developments in Language Theory III (DLT’97). Aristotle University of Thessaloniki, 1998, pp. 211222.
[9]Coven, E. M.. Endomorphisms of substitution minimal sets. Z. Wahrschein. Verw. Gebiete 20 (1971/72), 129133.
[10]Coven, E., Quas, A. and Yassawi, R.. Automorphisms of some Toeplitz and other minimal shifts with sublinear complexity. Preprint, 2015, arXiv:1505.02482.
[11]Cyr, V. and Kra, B.. The automorphism group of a shift of subquadratic growth. Proc. Amer. Math. Soc., to appear, arXiv:1403.0238.
[12]Cyr, V. and Kra, B.. The automorphism group of a shift of linear growth. Forum of Mathematics, Sigma 3 (2015), e5.
[13]del Junco, A.. A simple measure-preserving transformation with trivial centralizer. Pacific J. Math. 79 (1978), 357362.
[14]Donoso, S. and Sun, W.. Dynamical cubes and a criteria for systems having products extensions. Preprint, 2014, arXiv:1406.1220.
[15]Downarowicz, T.. Survey of odometers and Toeplitz flows. Algebraic and Topological Dynamics (Contemporary Mathematics, 385). American Mathematical Society, Providence, RI, 2005, pp. 737.
[16]Durand, F.. Linearly recurrent subshifts have a finite number of nonperiodic factors. Ergod. Th. & Dynam. Sys. 20 (2000), 10611078.
[17]Ferenczi, S.. Systems of finite rank. Colloq. Math. 73 (1997), 3565.
[18]Fiebig, D. and Fiebig, U.. The automorphism group of a coded system. Trans. Amer. Math. Soc. 348 (1996), 31733191.
[19]Hedlund, G. A.. Endomorphisms and automorphisms of the shift dynamical system. Math. System. Theory 3 (1969), 320375.
[20]Hedlund, G. A. and Morse, M.. Symbolic dynamics II. Sturmian trajectories. Amer. J. Math. 62 (1940), 142.
[21]Hochman, M.. On the automorphism groups of multidimensional shifts of finite type. Ergod. Th. & Dynam. Sys. 30 (2010), 809840.
[22]Host, B., Kra, B. and Maass, A.. Nilsequences and a structure theorem for topological dynamical systems. Adv. Math. 224 (2010), 103129.
[23]Host, B. and Parreau, F.. Homomorphismes entre systèmes dynamiques définies par substitutions. Ergod. Th. & Dynam. Sys. 9 (1989), 469477.
[24]Kim, K. H. and Roush, F. W.. On the automorphism groups of subshifts. Pure Math. Appl. Ser. B 1 (1990), 203230.
[25]King, J. and Thouvenot, J.-P.. A canonical structure theorem for finite joining-rank maps. J. Anal. Math. 56 (1991), 211230.
[26]Kůrka, P.. Topological and symbolic dynamics. Cours Spécialisés 11. Société Mathématique de France, Paris, 2003.
[27]Leibman, A.. Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold. Ergod. Th. & Dynam. Sys. 25 (2005), 201213.
[28]Lemańczyk, M. and Mentzen, M.. On metric properties of substitutions. Compos. Math. 65 (1988), 241263.
[29]Mossé, B.. Puissances de mots et reconnaissabilité des points fixes d’une substitution. Theoret. Comput. Sci. 99 (1992), 327334.
[30]Mossé, B.. Reconnaissabilité des substitutions et complexité des suites automatiques. Bull. Soc. Math. France 124 (1996), 329346.
[31]Olli, J.. Endomorphisms of Sturmian systems and the discrete chair substitution tiling system. Discrete Contin. Dyn. Syst. 33 (2013), 41734186.
[32]Ornstein, D.. On the root problem in ergodic theory. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (University of California, Berkeley, CA, 1970/1971) (Vol. II: Probability Theory). University of California Press, Berkeley, CA, 1972, pp. 347356.
[33]Pansiot, J.-J.. Complexité des facteurs des mots infinis engendrés par morphismes itérés. Automata, Languages and Programming (Lecture Notes in Computational Science, 172). Springer, Berlin, 1984, pp. 380389.
[34]Parry, W.. Ergodic properties of affine transformations and flows on nilmanifolds. Amer. J. Math. 91 (1969), 757771.
[35]Quas, A. and Zamboni, L.. Periodicity and local complexity. Theoret. Comput. Sci. 319 (2004), 229240.
[36]Queffélec, M.. Substitution Dynamical Systems–Spectral Analysis (Lecture Notes in Mathematics, 1294). Springer, Berlin, 1987.
[37]Salo, V. and Törmä, I.. Block maps between primitive uniform and Pisot substitutions. Ergod. Th. & Dynam. Sys., to appear. doi:10.1017/etds.2014.29. Published online 4 August 2014.
[38]Shalom, Y. and Tao, T.. A finitary version of Gromov’s polynomial growth theorem. Geom. Funct. Anal. 20 (2010), 15021547.
[39]Shao, S. and Ye, X.. Regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence. Adv. Math. 231 (2012), 17861817.
[40]Walters, P.. Topological conjugacy of affine transformations of tori. Trans. Amer. Math. Soc. 131 (1968), 4050.
[41]Ward, T.. Automorphisms of ℤd-subshifts of finite type. Indag. Math. (N.S.) 5 (1994), 495504.

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On automorphism groups of low complexity subshifts

  • SEBASTIÁN DONOSO (a1) (a2), FABIEN DURAND (a3), ALEJANDRO MAASS (a4) and SAMUEL PETITE (a3)

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