The limit set of recognizable substitution systems

  • Philippe Narbel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 665)


This paper introduces the global limit set of a morphism language. This set is generated by a process called “reversed morphism” which is a systematic way to embed words in each other. If the morphism is recognizable, i.e. locally invertible, then the limit set is shown in bijection with a rational language. This allows us to deduce that the limit set can be uncountable, can be structured by a natural dynamics and can contain only strictly quasiperiodic words.


Cauchy Sequence Substitution System Formal Language Theory Infinite Word Limit Word 
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  1. [Bea86]
    Beauquier (D.).-Automates de mots bi-infinis.-Paris, 1986. Thèse d'Etat.Google Scholar
  2. [BP85]
    Berstel (J.) and Perrin (D.).-Theory of Codes.-Academic Press, 1985.Google Scholar
  3. [CIK92]
    Culik II (K.) and Karhumäki (J.).-Iterative devices generating infinite words. In: Proceedings of STACS'92, pp. 531–543.Google Scholar
  4. [CIS82]
    Culik II (K.) and Salomaa (A.).-On infinite words obtained by iterating morphisms. Theoretical Computer Science, vol. 19, 1982, pp. 29–38.Google Scholar
  5. [DB89]
    De Bruijn (N.G.).-Updown generation of Beatty sequences. Indag. Math., vol. 51, 1989, pp. 385–407.Google Scholar
  6. [DB90]
    De Bruijn (N.G.).-Updown generation of Penrose patterns. Indag. Mathem., New Series., vol. 1, 1990, pp. 201–219.Google Scholar
  7. [DGS76]
    Denker (M.), Grillenberger (C.) and Sigmund (K.).-Ergodic theory on compact spaces.-Springer-Verlag, 1976, Lecture Notes in Mathematics, volume 527.Google Scholar
  8. [DL91]
    Devolder (J.) and Litovsky (I.).-Finitely generated biω-languages. Theoretical Computer Science, vol. 85, 1991, pp. 33–52.Google Scholar
  9. [GS87]
    Grunbaum (B.) and Shephard (G.C.).-Tilings and Patterns.-Freeman and co., 1987.Google Scholar
  10. [HL86]
    Harju (T.) and Linna (T.).-On the periodicity of morphisms on free monoids. RAIRO, Theoretical Informatics and Applications, vol. 20, number 1, 1986, pp. 47–54.Google Scholar
  11. [LP87]
    Lunnon (W.F.) and Pleasants (P.A.B.).-Quasicrystallographic tilings. J. Math. Pures et Appl., vol. 66, 1987, pp. 217–263.Google Scholar
  12. [MH38]
    Morse (M.) and Hedlund (G.A.).-Symbolic dynamics. American Journal of Mathematics, vol. 60, 1938, pp. 815–866.Google Scholar
  13. [Mos90]
    Mossé (B.).-Puissances de mots et reconnaissabilité des points fixes d'une substitution.-Technical report, PRO Mathématique Informatique, Université Aix-Marseille, 1990.Google Scholar
  14. [NP82]
    Nivat (N.) and Perrin (D.).-Ensembles reconnaissables de mots bi-infinis. In: 14th ACM Symp. on Theory of Computing, pp. 47–59.Google Scholar
  15. [PP92]
    Perrin (D.) and Pin (J.P.).-Mots infinis.-Technical Report number 92.17, LITP, april 1992.Google Scholar
  16. [Que87]
    Queffelec (M.).-Substitution Dynamical Systems — Spectral Analysis.-Springer-Verlag, 1987, Lecture Notes in Mathematics, volume 1294.Google Scholar
  17. [Sal81]
    Salomaa (A.).-Jewels of Formal Language Theory.-Rockville, MD, Computer Science Press, 1981.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Philippe Narbel
    • 1
  1. 1.L.I.T.PInstitut Blaise PascalParis 7

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