Generalized thue-morse sequences

  • Patrice Séébold
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 199)


In a recent paper (8), Christol and al. introduce the following generalized Thue-Morse sequences over two letters a and b. Given a finite word u over {0,1}, the infinite word u has its i-th letter equal to a or b according to the number of occurrences of u in the binary expansion of i be even or odd.

Černý (7) has shown that these words do not contain any factor of the form (xy)nx, with n=2|u|.

We considerably strengthen this result, and prove that these words contain no cube exepted ap and bp, p≤n.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. (1).
    S. ADJAN: The Burnside problem and identities in groups, Math. Grenzgeb. vol.95 — Springer (1979).Google Scholar
  2. (2).
    S. ARSHON: Démonstration de l'existence de suites asymétriques infinies, Mat. Sb. 44 (1937), p. 769–777.Google Scholar
  3. (3).
    J.M. AUTEBERT — J. BEAUQUIER — L. BOASSON — M. NIVAT: Quelques problèmes ouverts en théorie des langages algébriques, RAIRO — Inf. Th. 13 (1979), p. 363–379.Google Scholar
  4. (4).
    J. BERSTEL: Mots sans carrés et morphismes itérés, Discrete Math. 29 (1980), p. 235–244.Google Scholar
  5. (5).
    J. BERSTEL: Some recent results on square-free words, Symposium for Theoretical Aspects of Computer Sciences Paris (1984).Google Scholar
  6. (6).
    J. BRZOZOWSKI — K. CULIK II — A. GABRIELIAN: Classification of noncounting events, J. Comp. Syst. Science 5 (1971), p. 41–53.Google Scholar
  7. (7).
    A. ČERNÝ: On generalized words of Thue-Morse, Technical Report LITP 83-44 (1983) To appear in RAIRO — Inf. Th.Google Scholar
  8. (8).
    G. CHRISTOL — T. KAMAE — M. MENDÈS-FRANCE — G. RAUZY: Suites algébriques, automates et substitutions, Bull. Soc. Math. France 108 (1980), p. 401–419.Google Scholar
  9. (9).
    A. COBHAM: Uniform tag sequences, Math. Systems Theory 6 (1972), p. 164–192.Google Scholar
  10. (10).
    K. CULIK II — I. FRIS: The decidability of the equivalence problem for DOL-systems, Inform. and Control 35 (1977), p. 20–39.Google Scholar
  11. (11).
    K. CULIK II — J. KARHUMÄKI: On the Ehrenfeucht conjecture for DOL-languages, RAIRO — Inf. Th. 17 (1983), p. 205–230.Google Scholar
  12. (12).
    A. EHRENFEUCHT — G. ROZENBERG: Elementary homomorphisms and a solution to the DOL-sequence equivalence problem, Theor. Comp. Sci. 7 (1978), p. 169–183.Google Scholar
  13. (13).
    G. HEDLUND: Remarks on the work of Axel Thue on sequences, Nord. Mat. Tidskr. 16 (1967), p. 148–150.Google Scholar
  14. (14).
    J. KARHUMÄKI: On cube-free w-words generated by binary morphisms, Discr. Appl. Math. 5 (1983), p. 279–297.Google Scholar
  15. (15).
    J. KARHUMÄKI: On the equivalence problem for binary DOL-systems, Inform. and Control 50 (1981), p. 276–284.Google Scholar
  16. (16).
    M. LOTHAIRE: Combinatorics on words, Addison-Wesley (1983).Google Scholar
  17. (17).
    M. MORSE: Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc. 22 (1921), p. 84–100.Google Scholar
  18. (18).
    M. MORSE — G. HEDLUND: Unending chess, symbolic dynamics and a problem in semi-groups, Duke Math. J. 11 (1944), p. 1–7.Google Scholar
  19. (19).
    J.J. PANSIOT: The Morse sequence and iterated morphisms, Inf. Proc. Letters 12 (1981), p. 68–70.Google Scholar
  20. (20).
    C. REUTENAUER: A new characterization of the regular languages, 8-th ICALP, Springer Lect. Notes in Comp. Sci. 115 (1981), p. 175–183.Google Scholar
  21. (21).
    W. RUDIN: Some theorems on Fourier coefficients, Proc. Amer. Math. Soc. 10 (1959), p. 855–859.Google Scholar
  22. (22).
    A. SALOMAA: Jewels of formal language theory, Pitman (1981).Google Scholar
  23. (23).
    P. SÉÉBOLD: Morphismes itérés, mot de Morse et mot de Fibonacci, C. R. Acad. Sci. Paris 295 (1982), p. 439–441.Google Scholar
  24. (24).
    P. SÉÉBOLD: Suites de Thue-Morse généralisées, to appear in RAIRO.Google Scholar
  25. (25).
    E.S. SHAPIRO: Extremal problems for polynomials and power series, Thesis MIT (1951).Google Scholar
  26. (26).
    A. THUE: Uber unendliche Zeichenreihen, Norske Vid. Selsk. Skr. I. Math. Nat. Kl. Christiania (1906), p. 1–22.Google Scholar
  27. (27).
    A. THUE: Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, Ibid. (1912), p. 1–67.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • Patrice Séébold
    • 1
    • 2
  1. 1.UER de MathématiquesUniversité Paris VIIFrance
  2. 2.Laboratoire d'Informatique Théorique et Programmation-LA 248 du CNRSParis Cedex 05France

Personalised recommendations