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Jewels are Forever pp 287–294Cite as

On the Index of Sturmian Words

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Summary

An infinite word x has finite index if the exponents of the powers of primitive words that are factors of x are bounded. F. Mignosi has proved that a Sturmian word has finite index if and only if the coefficients of the continued fraction development of its slope are bounded. Mignosi’s proof relies on a delicate analysis of the approximation of the slope by rational numbers. We give here a proof based on combinatorial properties of words, and give some additional relations between the exponents and the slope.

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References

  1. J. Berstel, Recent results on Sturmian words, in: J. Dassow, G. Rozenberg, A. Salomaa (eds.) Developments in Language Theory II, World Scientific, 1996.

    Google Scholar 

  2. J. Berstel, P. Séébold, Sturmian Words, in: M. Lothaire (ed.) Algebraic Combinatorics on Words, in preparation.

    Google Scholar 

  3. J. Berstel et P. Séébold, A remark on morphic Sturmian words, Informatique théorique et applications 28 (1994), 255–263.

    MathSciNet  MATH  Google Scholar 

  4. E. Bombieri, J. E. Taylor, Which distributions of matter diffract? An initial investigation, J. Phys. 47 (1986), Colloque C3, 19–28.

    MathSciNet  Google Scholar 

  5. J. E. Bresenham, Algorithm for computer control of a digital plotter, IBM Systems J. 4 (1965), 25–30.

    Article  Google Scholar 

  6. T. C. Brown, A characterization of the quadratic irrationals, Canad. Math. Bull. 34 (1991), 36–41.

    Article  MathSciNet  MATH  Google Scholar 

  7. D. Crisp, W. Moran, A. Pollington, P. Shiue, Substitution invariant cutting sequences, J. Théorie des Nombres de Bordeaux 5 (1933), 123–137.

    MathSciNet  Google Scholar 

  8. E. Coven, G. Hedlund, Sequences with minimal block growth, Math. Systems Theory 7 (1973), 138–153.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. De Luca, Sturmian words: structure, combinatorics, and their arithmetics, Theoret. Comput. Sci. 183 (1997), 45–82.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. De Luca, Combinatorics of standard Sturmian words, in: J. Mycielski, G. Rozenberg, A. Salomaa (eds.) Structures in Logic and Computer Science, Lect. Notes Comp. Sci. Vol. 1261, Springer-Verlag, 1997, pp 249–267.

    Google Scholar 

  11. A. De Luca, Standard Sturmian morphisms, Theoret. Comput. Sci. 178 (1997), 205–224.

    Article  MathSciNet  Google Scholar 

  12. A. De Luca et F. Mignosi, Some combinatorial properties of Sturmian words, Theoret. Comput. Sci. 136 (1994), 361–385.

    Article  MathSciNet  MATH  Google Scholar 

  13. S. Dulucq, D. Gouyou-Beauchamps, Sur les facteurs des suites de Sturm, Theoret. Comput. Sci. 71 (1990), 381–400.

    Article  MathSciNet  MATH  Google Scholar 

  14. A. S. Fraenkel, M. Mushkin, U. Tassa, Determination of ⌊nθ⌋ by its sequence of differences, Canad. Math. Bull. 21h (1978), 441–446.

    Article  MathSciNet  MATH  Google Scholar 

  15. G.A. Hedlund, Sturmian minimal sets, Amer. J. Math 66 (1944), 605–620.

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Morse, G.A. Hedlund, Symbolic dynamics, Amer. J. Math 60 (1938), 815–866.

    Article  MathSciNet  Google Scholar 

  17. M. Morse, G.A. Hedlund, Sturmian sequences, Amer. J. Math 61 (1940), 1–42.

    Article  MathSciNet  Google Scholar 

  18. S. Ito, S. Yasutomi, On continued fractions, substitutions and characteristic sequences, Japan. J. Math. 16 (1990), 287–306.

    MathSciNet  MATH  Google Scholar 

  19. J. Karhumäki, On strongly cube-free w-words generated by binary morphisms, in FCT’81, pp. 182–189, Lect. Notes Comp. Sci. Vol. 117, Springer-Verlag, 1981.

    Google Scholar 

  20. J. Karhumäki, On cube-free w-words generated by binary morphisms, Discr. Appl. Math. 5 (1983), 279–297.

    Article  MATH  Google Scholar 

  21. F. Mignosi, On the number of factors of Sturmian words, Theoret. Comput. Sci. 82 (1991), 71–84.

    Article  MathSciNet  MATH  Google Scholar 

  22. F. Mignosi et G. Pirillo, Repetitions in the Fibonacci infinite word, Theoret. Inform. Appl. 26,3 (1992), 199–204.

    MathSciNet  MATH  Google Scholar 

  23. F. Mignosi, R Séébold, Morphismes sturmiens et regies de Rauzy, J. Théorie des Nombres de Bordeaux 5 (1993), 221–233.

    MathSciNet  MATH  Google Scholar 

  24. M. Queffélec, Substitution Dynamical Systems - Spectral Analysis, Lecture Notes Math., vol. 1294, Springer-Verlag, 1987.

    Google Scholar 

  25. G. Rauzy, Suites à termes dans un alphabet fini, Sémin. Théorie des Nombres (1982–1983), 25–01,25–16, Bordeaux.

    Google Scholar 

  26. G. Rauzy, Mots infinis en arithmétique, in: Automata on infinite words (D. Perrin ed.), Lect. Notes Comp. Sci. 192 (1985), 165–171.

    Google Scholar 

  27. R Séébold, Fibonacci morphisms and Sturmian words, Theoret. Comput. Sci. 88 (1991), 367–384.

    Article  Google Scholar 

  28. C. Series, The geometry of Markoff numbers, The Mathematical Intelligencer 7 (1985), 20–29.

    Article  MathSciNet  MATH  Google Scholar 

  29. K. B. Stolarsky, Beatty sequences, continued fractions, and certain shift operators, Cand. Math. Bull. 19 (1976), 473–482.

    Article  MathSciNet  MATH  Google Scholar 

  30. B. A. Venkov, Elementary Number Theory, Wolters-Noordhoff, Groningen, 1970.

    MATH  Google Scholar 

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Authors
  1. Jean Berstel
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Editor information

Editors and Affiliations

  1. Department of Mathematics and Turku Centre for Computer Science, University of Turku, FIN-20014, Turku, Finland

    Juhani Karhumäki

  2. Institutes for Information Processing and Computer Supported Media (IICM), Technical University of Graz, Schiessstattgasse 4a, A-8010, Graz, Austria

    Hermann Maurer

  3. Institute of Mathematics, Romanian Academy of Sciences, PO Box l-764, RO-70700, Bucharest, Romania

    Gheorghe Păun

  4. Department of Computer Science, Leiden University, PO Box 9512, 2300 RA, Leiden, The Netherlands

    Grzegorz Rozenberg

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© 1999 Springer-Verlag Berlin Heidelberg

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Berstel, J. (1999). On the Index of Sturmian Words. In: Karhumäki, J., Maurer, H., Păun, G., Rozenberg, G. (eds) Jewels are Forever. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60207-8_25

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  • DOI: https://doi.org/10.1007/978-3-642-60207-8_25

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-64304-0

  • Online ISBN: 978-3-642-60207-8

  • eBook Packages: Springer Book Archive

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