Journal of Mathematical Sciences

, Volume 156, Issue 2, pp 351–358 | Cite as

Balanced words and dynamical systems

  • A. L. Chernyatiev


This article is devoted to the description of all nonperiodic balanced words with n different letters. A superword W is called balanced if the numbers of equal letters in any two of its factors (subwords) u 1 and u 2 of equal length differ by at most 1. Balanced words are one of the possible generalizations of Sturmian words. We give a geometric interpretation of nonperiodic balanced sequences over an n-letter alphabet.


Blue Color Regular Polygon Symbolic Dynamic Equivalence Theorem Antipodal Point 
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  1. 1.
    A. Ya. Belov and G. V. Kondakov, “Reverse problems of symbolic dynamics,” Fundam. Prikl. Mat., 1, No. 1, 71–79 (1995).MATHMathSciNetGoogle Scholar
  2. 2.
    Ya. G. Sinai, Introduction to Ergodic Theory [in Russian], FAZIS, Moscow (1996).Google Scholar
  3. 3.
    J. Berstel, “Recent results on Sturmian words,” in: Developments in Language Theory. II, World Scientific (1996), pp. 13–24.Google Scholar
  4. 4.
    R. L. Graham, “Covering the positive integers by disjoints sets of the form {[ + β]: n = 1, 2,...},” J. Combin. Theory. Ser. A, 15, 354–358 (1973).MATHCrossRefGoogle Scholar
  5. 5.
    P. Hubert, “Well-balanced sequences,” Theoret. Comput. Sci., 242, 91–108 (2000).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    M. Lothaire, “Combinatorics on words,” in: Encyclopedia of Mathematics and Its Applications, Vol. 17, Addison-Wesley, Reading (1983).Google Scholar
  7. 7.
    A. de Luca, “Sturmian words: Structure, combinatorics and their arithmetics,” Theoret. Comput. Sci., 183, 45–82 (1997).MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    A. de Luca and S. Varricchio, “Combinatorial properties of uniformly recurrent words and an application to semigroups,” Internat. J. Algebra Comput., 1, No. 2, 227–246 (1991).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    M. Morse and G. A. Hedlund, “Symbolic dynamics. II. Sturmian trajectories,” Amer. J. Math., 62, 1–42 (1940).CrossRefMathSciNetGoogle Scholar
  10. 10.
    M. Newman, “Roots of unity and covering sets,” Math. Ann., 191, 279–282 (1971).CrossRefMathSciNetGoogle Scholar
  11. 11.
    R. Tijdeman, “Decomposition of the integers as a direct sum of two subsets,” in: S. David, ed., Number Theory. Number Theory Seminar Paris 1992–93, Cambridge Univ. Press (1995), pp. 261–276.Google Scholar
  12. 12.
    R. Tijdeman, “Fraenkel’s conjecture for six sequences,” Discrete Math., 222, Issue 1–3, 223–234 (2000).MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    L. Vuillon, “Balanced words,” Bull. Belg. Math. Soc. Simon Stevin, 10, 787–805 (2003).MATHMathSciNetGoogle Scholar
  14. 14.
    H. Weyl, “Über der Gleichverteilung von Zahlen mod 1,” Math. Ann., 77, 313–352 (1916).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  1. 1.Center for Supplementary Education “Distant Learning”MoscowRussia

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