Hostname: page-component-7dd5485656-wlg5v Total loading time: 0 Render date: 2025-10-31T01:51:13.827Z Has data issue: false hasContentIssue false
  • English
  • Français

Recognizability of morphisms

Part of: Topological dynamics

Published online by Cambridge University Press:  13 January 2023

MARIE-PIERRE BÉAL Open the ORCID record for MARIE-PIERRE BÉAL [Opens in a new window] ,
DOMINIQUE PERRIN  and
ANTONIO RESTIVO
MARIE-PIERRE BÉAL
Affiliation:
Université Gustave Eiffel, CNRS, LIGM, F-77454 Marne-la-Vallée, France (e-mail: marie-pierre.beal@univ-eiffel.fr)
DOMINIQUE PERRIN*
Affiliation:
Université Gustave Eiffel, CNRS, LIGM, F-77454 Marne-la-Vallée, France (e-mail: marie-pierre.beal@univ-eiffel.fr)
ANTONIO RESTIVO
Affiliation:
Dipartimento di Matematica e Informatica, Università di Palermo, Italy (e-mail: antonio.restivo@unipa.it)
*
e-mail: dominique.perrin@esiee.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate several questions related to the notion of recognizable morphism. The main result is a new proof of Mossé’s theorem and actually of a generalization to a more general class of morphisms due to Berthé et al [Recognizability for sequences of morphisms. Ergod. Th. & Dynam. Sys. 39(11) (2019), 2896–2931]. We actually prove the result of Berthé et al for the most general class of morphisms, including ones with erasable letters. Our result is derived from a result concerning elementary morphisms for which we also provide a new proof. We also prove some new results which allow us to formulate the property of recognizability in terms of finite automata. We use this characterization to show that for an injective morphism, the property of being recognizable on the full shift for aperiodic points is decidable.

Keywords

symbolic dynamicssubstitutionsautomata

MSC classification

Primary: 37B10: Symbolic dynamics

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 11 , November 2023 , pp. 3578 - 3602
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Aubrun, N. and Sablik, M.. Simulation of effective subshifts by two-dimensional subshifts of finite type. Acta Appl. Math. 126 (2013), 35–63.10.1007/s10440-013-9808-5CrossRefGoogle Scholar
Aubrun, N. and Sablik, M.. Multidimensional effective $S$ -adic systems are sofic. Unif. Distrib. Theory 9(2) (2014), 7–29.Google Scholar
Béal, M.-P., Perrin, D. and Restivo, A.. Unambiguously coded systems. European J. Combin., to appear, Preprint, 2021, arXiv:2103.01012.Google Scholar
Berstel, J., Perrin, D., Perrot, J.-F. and Restivo, A.. Sur le théorème du défaut. J. Algebra 60(1) (1979), 169–180.10.1016/0021-8693(79)90113-3CrossRefGoogle Scholar
Berstel, J., Perrin, D. and Reutenauer, C.. Codes and Automata. Cambridge University Press, Cambridge, 2009.10.1017/CBO9781139195768CrossRefGoogle Scholar
Berthé, V., Steiner, W., Thuswaldner, J. M. and Yassawi, R.. Recognizability for sequences of morphisms. Ergod. Th. & Dynam. Sys. 39(11) (2019), 2896–2931.10.1017/etds.2017.144CrossRefGoogle Scholar
Bezuglyi, S., Kwiatkowski, J. and Medynets, K.. Aperiodic substitution systems and their Bratteli diagrams. Ergod. Th. & Dynam. Sys. 29(1) (2009), 37–72.10.1017/S0143385708000230CrossRefGoogle Scholar
Donoso, S., Durand, F., Maass, A. and Petite, S.. Interplay between finite topological rank minimal Cantor systems, $S$ -adic subshifts and their complexity. Trans. Amer. Math. Soc. 374 (2021), 3453–3489.10.1090/tran/8315CrossRefGoogle Scholar
Durand, F. and Perrin, D.. Dimension Groups and Dynamical Systems. Cambridge University Press, Cambridge, 2022.10.1017/9781108976039CrossRefGoogle Scholar
Ehrenfeucht, A. and Rozenberg, G.. Elementary homomorphisms and a solution of the $\mathrm{DOL}$ sequence equivalence problem. Theoret. Comput. Sci. 7(2) (1978), 169–183.10.1016/0304-3975(78)90047-6CrossRefGoogle Scholar
Karhumäki, J., Maňuch, J. and Plandowski, W.. A defect theorem for bi-infinite words. Theoret. Comput. Sci. 292(1) (2003), 237–243. Selected Papers in honor of J. Berstel.10.1016/S0304-3975(01)00225-0CrossRefGoogle Scholar
Kurka, P.. Topological and Symbolic Dynamics (Cours Spécialisés [Specialized Courses], 11). Société Mathématique de France, Paris, 2003.Google Scholar
Kyriakoglou, R.. Recognizable substitutions. PhD Thesis, Université Paris Est, 2019.Google Scholar
Lind, D. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding (Cambridge Mathematical Library), 2nd edn. Cambridge University Press, 2021.10.1017/9781108899727CrossRefGoogle Scholar
Linna, M.. The decidability of the $\mathrm{DOL}$ prefix problem. Int. J. Comput. Math. 6(2) (1977), 127–142.10.1080/00207167708803132CrossRefGoogle Scholar
Lothaire, M.. Algebraic Combinatorics on Words (Encyclopedia of Mathematics and its Applications, 90). Cambridge University Press, Cambridge, 2002.10.1017/CBO9781107326019CrossRefGoogle Scholar
Martin, J. C.. Minimal flows arising from substitutions of non-constant length. Math. Syst. Theory 7 (1973), 72–82.10.1007/BF01824809CrossRefGoogle Scholar
Mossé, B.. Puissances de mots et reconnaissabilité des points fixes d’une substitution. Theoret. Comput. Sci. 99(2) (1992), 327–334.10.1016/0304-3975(92)90357-LCrossRefGoogle Scholar
Mossé, B.. Reconnaissabilité des substitutions et complexité des suites automatiques. Bull. Soc. Math. France 124(2) (1996), 329–346.10.24033/bsmf.2283CrossRefGoogle Scholar
Mozes, S.. Tilings, substitution systems and dynamical systems generated by them. J. Anal. Math. 53(1) (1989), 139–186.10.1007/BF02793412CrossRefGoogle Scholar
Perrin, D. and Pin, J.-E.. Infinite Words: Automata, Semigroups, Logic and Games. Elsevier, Amsterdam, 2004.Google Scholar
Perrin, D. and Rindone, G.. On syntactic groups. Bull. Belg. Math. Soc. Simon Stevin 10(suppl.) (2003), 749–759.CrossRefGoogle Scholar
Perrin, D. and Ryzhikov, A.. The degree of a finite set of words. Proc. 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2020, December 14–18, 2020, BITS Pilani, K K Birla Goa Campus, Goa, India (Virtual Conference) (LIPIcs, 182). Eds. Saxena, N. and Simon, S.. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2020, pp. 54:1–54:16.Google Scholar
Perrin, D. and Ryzhikov, A.. The degree of a finite set of words. Preprint, 2022, arXiv:2106.14471.Google Scholar
Queffélec, M.. Substitution Dynamical Systems—Spectral Analysis (Lecture Notes in Mathematics, 1294), 2nd edn. Springer-Verlag, Berlin, 2010.10.1007/978-3-642-11212-6CrossRefGoogle Scholar
Restivo, A.. On a question of McNaughton and Pappert. Inform. Control 25 (1974), 1.10.1016/S0019-9958(74)90821-3CrossRefGoogle Scholar
Schützenberger, M.-P.. A property of finitely generated submonoids of free monoids. Algebraic Theory of Semigroups. Ed. Pollak, G.. North-Holland, Amsterdam, 1979, pp. 545–576.Google Scholar
Solomyak, B.. Nonperiodicity implies unique composition for self-similar translationally finite tilings. Discrete Comput. Geom. 20(2) (1998), 265–279.10.1007/PL00009386CrossRefGoogle Scholar