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Automorphisms of automatic shifts

Part of: Sequences and sets Topological dynamics

Published online by Cambridge University Press:  20 February 2020

CLEMENS MÜLLNER Open the ORCID record for CLEMENS MÜLLNER [Opens in a new window]  and
REEM YASSAWI
CLEMENS MÜLLNER
Affiliation:
Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 Boulevard du 11 novembre 1918, 69100 Villeurbanne, France Discrete Mathematics and Geometry, TU Wien, Wiedner Hauptstr. 8, 1040 Wien, Austria
REEM YASSAWI
Affiliation:
Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 Boulevard du 11 novembre 1918, 69100 Villeurbanne, France School of Mathematics and Statistics, Walton Hall, Kents Hill, Milton Keynes, MK76AA, UK
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Abstract

In this paper we continue the study of automorphism groups of constant-length substitution shifts and also their topological factors. We show that, up to conjugacy, all roots of the identity map are letter-exchanging maps, and all other non-trivial automorphisms arise from twisted compressions of another constant-length substitution. We characterize the group of roots of the identity in both the measurable and topological setting. Finally, we show that any topological factor of a constant-length substitution shift is topologically conjugate to a constant-length substitution shift via a letter-to-letter code.

Keywords

substitution dynamical systemsautomatic sequencesautomorphisms

MSC classification

Primary: 37B10: Symbolic dynamics 11B85: Automata sequences
Secondary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.) 37B15: Cellular automata
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 5 , May 2021 , pp. 1530 - 1559
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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References

Baake, M. and Lenz, D.. Spectral notions of aperiodic order. Discrete Contin. Dyn. Syst. Ser. S 10(2) (2017), 161–190.Google Scholar
Baake, M., Roberts, J. A. G. and Yassawi, R.. Reversing and extended symmetries of shift spaces. Discrete Contin. Dyn. Syst. 38(2) (2018), 835–866.Google Scholar
Blanchard, F., Durand, F. and Maass, A.. Constant-length substitutions and countable scrambled sets. Nonlinearity 17(3) (2004), 817–833.Google Scholar
Cobham, A.. Uniform tag sequences. Math. Systems Theory 6 (1972), 164–192.CrossRefGoogle Scholar
Coven, E. M., Dekking, F. M. and Keane, M. S.. Topological conjugacy of constant length substitution dynamical systems. Indag. Math. (N.S.) 28(1) (2017), 91–107.CrossRefGoogle Scholar
Coven, E. M., Quas, A. and Yassawi, R.. Computing automorphism groups of shifts using atypical equivalence classes. Discrete Anal. (2016), Paper No. 3, 28.Google Scholar
Dekking, F. M.. The spectrum of dynamical systems arising from substitutions of constant length. Z. Wahrsch. Verw. Gebiete 41(3) (1977/78), 221–239.Google Scholar
Downarowicz, T. and Lemańczyk, M.. Private communication, 2019.Google Scholar
Durand, F. and Leroy, J.. Decidability of the isomorphism and the factorization between minimal substitution subshifts. Preprint, 2018, arXiv:1806.04891.Google Scholar
Ellis, R. and Gottschalk, W. H.. Homomorphisms of transformation groups. Trans. Amer. Math. Soc. 94 (1960), 258–271.Google Scholar
Gottschalk, W. H. and Hedlund, G. A.. Topological Dynamics (American Mathematical Society Colloquium Publications, 36) . American Mathematical Society, Providence, RI, 1955.CrossRefGoogle Scholar
Herning, J. L.. Spectrum and factors of substitution dynamical systems. PhD Thesis, George Washington University, ProQuest LLC, Ann Arbor, MI, 2013.Google Scholar
Host, B.. Valeurs propres des systèmes dynamiques définis par des substitutions de longueur variable. Ergod. Th. & Dynam. Sys. 6(4) (1986), 529–540.CrossRefGoogle Scholar
Host, B. and Parreau, F.. Homomorphismes entre systèmes dynamiques définis par substitutions. Ergod. Th. & Dynam. Sys. 9(3) (1989), 469–477.CrossRefGoogle Scholar
Kamae, T.. A topological invariant of substitution minimal sets. J. Math. Soc. Japan 24 (1972), 285–306.CrossRefGoogle Scholar
Lemańczyk, M. and Mentzen, M. K.. On metric properties of substitutions. Compos. Math. 65(3) (1988), 241–263.Google Scholar
Lemańczyk, M. and Müllner, C.. Automatic sequences are orthogonal to aperiodic multiplicative functions. Preprint, 2018, arXiv:1811.00594.Google Scholar
Martin, J. C.. Substitution minimal flows. Amer. J. Math. 93 (1971), 503–526.CrossRefGoogle Scholar
Mossé, B.. Puissances de mots et reconnaissabilité des points fixes d’une substitution. Theoret. Comput. Sci. 99(2) (1992), 327–334.Google Scholar
Müllner, C.. Automatic sequences fulfill the Sarnak conjecture. Duke Math. J. 166(17) (2017), 3219–3290.CrossRefGoogle Scholar
Queffélec, M.. Substitution Dynamical Systems—Spectral Analysis (Lecture Notes in Mathematics, 1294) , 2nd edn. Springer, Berlin, 2010.Google Scholar
Staynova, P.. The Ellis semigroup of a generalised morse system. Preprint, 2017, arXiv:1711.10484. Ergod. Th. & Dynam. Sys. accepted.Google Scholar