A characterization of Sturmian morphisms

  • Jean Berstel
  • Patrice Séébold
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 711)


A morphism is called Sturmian if it preserves all Sturmian (infinite) words. It is weakly Sturmian if it preserves at least one Sturmian word. We prove that a morphism is Sturmian if and only if it keeps the word ba2ba2baba2bab balanced. As a consequence, weakly Sturmian morphisms are Sturmian. An application to infinite words associated to irrational numbers is given.


Characteristic Sequence Irrational Number Symbolic Dynamic Empty Word Infinite Word 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E. Bombieri, J. E. Taylor, Which distributions of matter diffract? An initial investigation, J. Phys. 47 (1986), Colloque C3, 19–28.Google Scholar
  2. 2.
    J. E. Bresenham, Algorithm for computer control of a digital plotter, IBM Systems J.4 (1965), 25–30.Google Scholar
  3. 3.
    T. C. Brown, A characterization of the quadratic irrationals, Canad. Math. Bull.34 (1991), 36–41.Google Scholar
  4. 4.
    T. C. Brown, Descriptions of the characteristic sequence of an irrational, Canad. Math. Bull.36 (1993), 15–21.Google Scholar
  5. 5.
    D. Crisp, W. Moran, A. Pollington, P. Shiue, Substitution invariant cutting sequences, Sémin. Théorie des Nombres, Bordeaux, 1993, to appear.Google Scholar
  6. 6.
    E. Coven, G. Hedlund, Sequences with minimal block growth, Math. Systems Theory7 (1973), 138–153.Google Scholar
  7. 7.
    S. Dulucq, D. Gouyou-Beauchamps, Sur les facteurs des suites de Sturm, Theoret. Comput. Sci.71 (1990), 381–400.Google Scholar
  8. 8.
    A. S. Fraenkel, M. Mushkin, U. Tassa, Determination of ⌊⌋ by its sequence of differences, Canad. Math. Bull.21 (1978), 441–446.Google Scholar
  9. 9.
    G.A. Hedlund, Sturmian minimal sets, Amer. J. Math66 (1944), 605–620.Google Scholar
  10. 10.
    G.A. Hedlund, M. Morse, Symbolic dynamics, Amer. J. Math60 (1938), 815–866.Google Scholar
  11. 11.
    G.A. Hedlund, M. Morse, Sturmian sequences, Amer. J. Math61 (1940), 1–42.Google Scholar
  12. 12.
    S. Ito, S. Yasutomi, On continued fractions, substitutions and characteristic sequences, Japan. J. Math.16 (1990), 287–306.Google Scholar
  13. 13.
    F. Mignosi, On the number of factors of Sturmian words, Theoret. Comput. Sci.82 (1991), 71–84.Google Scholar
  14. 14.
    F. Mignosi, P. Séébold, Morphismes sturmiens et règles de Rauzy, Techn. Rep. LITP-91-74, Paris, France.Google Scholar
  15. 15.
    M. Queffélec, Substitution Dynamical Systems — Spectral Analysis, Lecture Notes Math.,vol. 1294, Springer-Verlag, 1987.Google Scholar
  16. 16.
    G. Rauzy, Suites à termes dans un alphabet fini, Sémin. Théorie des Nombres (1982–1983), 25–01,25–16, Bordeaux.Google Scholar
  17. 17.
    G. Rauzy, Mots infinis en arithmétique, in: Automata on infinite words (D. Perrin ed.), Lect. Notes Comp. Sci. 192 (1985), 165–171.Google Scholar
  18. 18.
    G. Rauzy, Sequences defined by iterated morphisms, in: Workshop on Sequences (R. Capocelli ed.), Lecture Notes Comput. Sci., to appear.Google Scholar
  19. 19.
    P. Séébold, Fibonacci morphisms and Sturmian words, Theoret. Comput. Sci.88 (1991), 367–384.Google Scholar
  20. 20.
    C. Series, The geometry of Markoff numbers, The Mathematical Intelligencer7 (1985), 20–29.Google Scholar
  21. 21.
    K. B. Stolarsky, Beatty sequences, continued fractions, and certain shift operators, Cand. Math. Bull.19 (1976), 473–482.Google Scholar
  22. 22.
    B. A. Venkov, Elementary Number Theory, Wolters-Noordhoff, Groningen, 1970.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Jean Berstel
    • 1
  • Patrice Séébold
    • 2
  1. 1.Institut Blaise PascalLITPParis
  2. 2.LAMIFAAmiensFrance

Personalised recommendations