Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T12:24:17.330Z Has data issue: false hasContentIssue false
  • English
  • Français

Eventually dendric shift spaces

Part of: Theory of computing Topological dynamics

Published online by Cambridge University Press:  26 May 2020

FRANCESCO DOLCE Open the ORCID record for FRANCESCO DOLCE [Opens in a new window]  and
DOMINIQUE PERRIN
FRANCESCO DOLCE
Affiliation:
FNSPE, Czech Technical University in Prague—Trojanova 13, 120 00Praha 2, Czech Republic email [email protected]
DOMINIQUE PERRIN
Affiliation:
LIGM, Université Gustave Eiffel, France email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We define a new class of shift spaces which contains a number of classes of interest, like Sturmian shifts used in discrete geometry. We show that this class is closed under two natural transformations. The first one is called conjugacy and is obtained by sliding block coding. The second one is called the complete bifix decoding, and typically includes codings by non-overlapping blocks of fixed length.

Keywords

symbolic dynamicsconjugacybifix decoding

MSC classification

Primary: 37B10: Symbolic dynamics 68Q45: Formal languages and automata
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 7 , July 2021 , pp. 2023 - 2048
Check for updates
Copyright
© The Author(s) 2020. 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.)

References

Arnoux, P. and Rauzy, G.. Représentation géométrique de suites de complexité 2n + 1. Bull. Soc. Math. France 119(2) (1991), 199–215.CrossRefGoogle Scholar
Balková, L., Pelantová, E. and Steiner, W.. Sequences with constant number of return words. Monatsh. Math. 155(3–4) (2008), 251–263.CrossRefGoogle Scholar
Berstel, J., De Felice, C., Perrin, D., Reutenauer, C. and Rindone, G.. Bifix codes and Sturmian words. J. Algebra 369 (2012), 146–202.CrossRefGoogle Scholar
Berthé, V., De Felice, C., Delecroix, V., Dolce, F., Leroy, J., Perrin, D., Reutenauer, C. and Rindone, G.. Specular sets. Theoret. Comput. Sci. 684 (2017), 3–28.CrossRefGoogle Scholar
Berthé, V., De Felice, C., Dolce, F., Leroy, J., Perrin, D., Reutenauer, C. and Rindone, G.. Acyclic, connected and tree sets. Monatsh. Math. 176(4) (2015), 521–550.CrossRefGoogle Scholar
Berthé, V., De Felice, C., Dolce, F., Leroy, J., Perrin, D., Reutenauer, C. and Rindone, G.. Bifix codes and interval exchanges. J. Pure Appl. Algebra 219(7) (2015), 2781–2798.CrossRefGoogle Scholar
Berthé, V., De Felice, C., Dolce, F., Leroy, J., Perrin, D., Reutenauer, C. and Rindone, G.. Maximal bifix decoding. Discrete Math. 338 (2015), 725–742.CrossRefGoogle Scholar
Cassaigne, J.. Complexité et facteurs spéciaux. Bull. Belg. Math. Soc. Simon Stevin 4(1) (1997), 67–88. Journées Montoises (Mons, 1994).10.36045/bbms/1105730624CrossRefGoogle Scholar
Cassaigne, J. and Nicolas, F.. Factor complexity. Combinatorics, Automata and Number Theory (Encyclopedia of Mathematics and its Applications, 135) . Cambridge University Press, Cambridge, 2010, pp. 163–247.CrossRefGoogle Scholar
Cornfeld, I. P., Fomin, S. V. and Sinai, Y. G.. Ergodic theory (Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 245) . Springer, New York, 1982, translated from the Russian by A. B. Sosinskii.CrossRefGoogle Scholar
Damron, M. and Fickenscher, J.. The number of ergodic measures for transitive subshifts under the regular bispecial condition. Preprint, 2019, arXiv:1902.04619.CrossRefGoogle Scholar
Dolce, F., Kyriakoglou, R. and Leroy, J.. Decidable properties of extension graphs for substitutive languages. 15èmes Journées Montoises d’informatique théorique (Liège, 2016), https://orbi.uliege.be/bitstream/2268/205115/1/jm16_new.pdf.Google Scholar
Dolce, F. and Perrin, D.. Neutral and tree sets of arbitrary characteristic. Theoret. Comput. Sci. 658(part A) (2017), 159–174.Google Scholar
Dolce, F. and Perrin, D.. Eventually dendric shifts. Computer Science – Theory and Applications (Proc. 14th International Computer Science Symposium in Russia, CSR 2019, Novosibirsk (Russia), July 1–5, 2019) (Lecture Notes in Computer Science, 11532). Ed. R. van Bevern and G. Kucherov. Springer, Berlin, 2019, pp. 106–118.CrossRefGoogle Scholar
Dolce, F. and Perrin, D.. Return words and bifix codes in eventually dendric sets. Combinatorics on Words. WORDS 2019 (Lecture Notes in Computer Science, 11682). Ed. R. Mercaş and D. Reidenbach. Springer, Berlin, 2019, pp. 167–179.CrossRefGoogle Scholar
Donoso, S., Durand, F., Maass, A. and Petite, S.. On automorphism groups of low complexity subshifts. Ergod. Th. & Dynam. Sys. 36 (2016), 64–95.CrossRefGoogle Scholar
Durand, F. and Leroy, J.. Decidability of the isomorphism and the factorization between minimal substitution subshifts, Preprint, 2018, arXiv:1806.04891.Google Scholar
Durand, F., Leroy, J. and Richomme, G.. Do the properties of an S-adic representation determine factor complexity? J. Integer Seq. 16(2) (2013), Article 13.2.6, 30.Google Scholar
Fogg, N. P.. Substitutions in Dynamics, Arithmetics and Combinatorics (Lecture Notes in Mathematics, 1794). Ed. V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel. Springer, Berlin, 2002.Google Scholar
Lind, D. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.Google Scholar
Queffélec, M.. Substitution Dynamical Systems – Spectral Analysis (Lecture Notes in Mathematics, 1294) , 2nd edn. Springer, Berlin, 2010.10.1007/978-3-642-11212-6CrossRefGoogle Scholar
Vesely, V.. Properties of morphic images of $S$ -adic sequences. Mgr. Thesis, Czech Technical University, 2018.Google Scholar