Sturmian and Episturmian Words

(A Survey of Some Recent Results)
  • Jean Berstel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4728)


This survey paper contains a description of some recent results concerning Sturmian and episturmian words, with particular emphasis on central words. We list fourteen characterizations of central words. We give the characterizations of Sturmian and episturmian words by lexicographic ordering, we show how the Burrows-Wheeler transform behaves on Sturmian words. We mention results on balanced episturmian words. We give a description of the compact suffix automaton of central Sturmian words.


Characteristic Word Arithmetical Complexity Binary Word Standard Word Block Complexity 
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.


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  1. 1.
    Klette, R., Rosenfeld, A.: Digital straightness—a review. Discrete Appl. Math. 139, 197–230 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Morse, M., Hedlund, G.A.: Symbolic dynamics II. Sturmian trajectories. Amer. J. Math. 62, 1–42 (1940)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Berstel, J., Boasson, L., Carton, O., Fagnot, I.: A first investigation of Sturmian trees. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 73–84. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  4. 4.
    de Luca, A., De Luca, A.: Pseudopalindrome closure operators in free monoids. Theoret. Comput. Sci. 362(1-3), 282–300 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Droubay, X., Justin, J., Pirillo, G.: Epi-Sturmian words and some constructions of de Luca and Rauzy. Theoret. Comput. Sci. 255(1-2), 539–553 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Justin, J., Pirillo, G.: On a characteristic property of Arnoux-Rauzy sequences. Theor. Inform. Appl. 36(4), 385–388 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Justin, J., Pirillo, G.: Episturmian words and episturmian morphisms. Theoret. Comput. Sci. 276(1-2), 281–313 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Justin, J.: Episturmian words and morphisms (results and conjectures). In: Crapo, H., Senato, D. (eds.) Algebraic Combinatorics and Computer Science, pp. 533–539. Springer, Heidelberg (2001)Google Scholar
  9. 9.
    Arnoux, P., Rauzy, G.: Représentation géométrique de suites de complexité 2n + 1. Bull. Soc. Math. France 119, 199–215 (1991)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Coven, E.M., Hedlund, G.A.: Sequences with minimal block growth. Math. Systems Theory 7, 138–153 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Allouche, J.P., Baake, M., Cassaigne, J., Damanik, D.: Palindrome complexity. Theoret. Comput. Sci. 292(1), 9–31 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Baláži, P., Masáková, Z., Pelantová, E.: Factor versus palindromic complexity of uniformly recurrent infinite words. Theoret. Comput. Sci. 380, 266–275 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Droubay, X., Pirillo, G.: Palindromes and Sturmian words. Theoret. Comput. Sci. 223(1-2), 73–85 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Damanik, D., Zamboni, L.Q.: Combinatorial properties of Arnoux-Rauzy subshifts and applications to Schrödinger operators. Rev. Math. Phys. 15(7), 745–763 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Avgustinovich, S.V., Fon-Der-Flaas, D.G., Frid, A.E.: Arithmetical complexity of infinite words. In: Words, Languages and Combinatorics. Proc. 3rd Conf. Words, Languages and Combinatorics, Kyoto, March 2000, vol. III, pp. 51–62. World Scientific, Singapore (2003)CrossRefGoogle Scholar
  16. 16.
    Cassaigne, J., Frid, A.E.: On the arithmetical complexity of Sturmian words. Theoret. Comput. Sci. 380, 304–316 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Avgustinovich, S.V., Cassaigne, J., Frid, A.E.: Sequences of low arithmetical complexity. Theor. Inform. Appl. 40(4), 569–582 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kamae, T., Zamboni, L.Q.: Maximal pattern complexity for discrete systems. Ergodic Theory Dynam. Systems 22(4), 1201–1214 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Kamae, T., Rao, H., Tan, B., Xue, Y.M.: Language structure of pattern Sturmian words. Discrete Math. 306(15), 1651–1668 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Kamae, T., Rao, H.: Maximal pattern complexity of words over l letters. European J. Combin. 27(1), 125–137 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Nakashima, I., Tamura, J.I., Yasutomi, S.I.: Modified complexity and ∗-Sturmian word. Proc. Japan Acad. Ser. A Math. Sci. 75(3), 26–28 (1999)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Nakashima, I., Tamura, J.I., Yasutomi, S.I.: *-Sturmian words and complexity. J. Theor. Nombres Bordeaux 15(3), 767–804 (2003)zbMATHMathSciNetGoogle Scholar
  23. 23.
    de Luca, A.: Sturmian words: structure, combinatorics, and their arithmetics. Theoret. Comput. Sci. 183(1), 45–82 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    de Luca, A.: Combinatorics of standard Sturmian words. In: Mycielski, J., Rozenberg, G., Salomaa, A. (eds.) Structures in Logic and Computer Science. LNCS, vol. 1261, pp. 249–267. Springer, Heidelberg (1997)Google Scholar
  25. 25.
    Risley, R., Zamboni, L.Q.: A generalization of Sturmian sequences: combinatorial structure and transcendence. Acta Arith. 95, 167–184 (2000)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Glen, A.: Powers in a class of \({\cal a}\)-strict standard episturmian words. Theoret. Comput. Sci. 380, 330–354 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Tan, B., Wen, Z.Y.: Some properties of the Tribonacci sequence. European J. Combin. (2007)Google Scholar
  28. 28.
    Chekhova, N., Hubert, P., Messaoudi, A.: Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci. J. Theor. Nombres Bordeaux 13(2), 371–394 (2001)zbMATHMathSciNetGoogle Scholar
  29. 29.
    Lothaire, M.: Applied Combinatorics on Words. Encyclopedia of Mathematics and its Applications, vol. 105. Cambridge University Press, Cambridge (2005)zbMATHGoogle Scholar
  30. 30.
    Series, C.: The geometry of Markoff numbers. Math. Intelligencer 7(3), 20–29 (1985)zbMATHMathSciNetGoogle Scholar
  31. 31.
    Berthé, V., Ei, H., Ito, S., Rao, H.: Invertible substitutions and Sturmian words: an application to Rauzy fractals. Theor. Inform. Appl. (to appear, 2007)Google Scholar
  32. 32.
    Lothaire, M.: Algebraic Combinatorics on Words. Encyclopedia of Mathematics and its Applications, vol. 90. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  33. 33.
    Pytheas Fogg, N.: Substitutions in dynamics, arithmetics and combinatorics. In: Berthé, V., Ferenczi, S., Mauduit, C., Siegel, A. (eds.) Lecture Notes in Mathematics, vol. 1794, Springer, Heidelberg (2002)Google Scholar
  34. 34.
    Christoffel, E.B.: Observatio arithmetica. Annali di Mathematica 6, 145–152 (1875)Google Scholar
  35. 35.
    Pirillo, G.: A curious characteristic property of standard Sturmian words. In: Crapo, H., Senato, D. (eds.) Algebraic Combinatorics and Computer Science. A tribute to Gian-Carlo Rota, pp. 541–546. Springer, Heidelberg (2001)Google Scholar
  36. 36.
    de Luca, A., Mignosi, F.: Some combinatorial properties of Sturmian words. Theoret. Comput. Sci. 136(2), 361–385 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Borel, J.P., Reutenauer, C.: Palindromic factors of billiard words. Theoret. Comput. Sci. 340(2), 334–348 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    de Luca, A., De Luca, A.: Combinatorial properties of Sturmian palindromes. Internat. J. Found. Comput. Sci. 17(3), 557–573 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Carpi, A., de Luca, A.: Codes of central Sturmian words. Theoret. Comput. Sci. 340(2), 220–239 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Berstel, J., de Luca, A.: Sturmian words, Lyndon words and trees. Theoret. Comput. Sci. 178(1-2), 171–203 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Berthé, V., de Luca, A., Reutenauer, C.: On an involution of Christoffel words and Sturmian morphisms. In: European J. Combinatorics (in press, 2007)Google Scholar
  42. 42.
    Chuan, W.F.: Moments of conjugacy classes of binary words. Theoret. Comput. Sci. 310(1-3), 273–285 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Jenkinson, O., Zamboni, L.Q.: Characterisations of balanced words via orderings. Theoret. Comput. Sci. 310(1-3), 247–271 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    de Luca, A., De Luca, A.: Some characterizations of finite Sturmian words. Theoret. Comput. Sci. 356(1-2), 118–125 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Fagnot, I., Vuillon, L.: Generalized balances in Sturmian words. Discrete Appl. Math. 121(1-3), 83–101 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Cassaigne, J., Ferenczi, S., Zamboni, L.Q.: Imbalances in Arnoux-Rauzy sequences. Ann. Inst. Fourier (Grenoble) 50(4), 1265–1276 (2000)zbMATHMathSciNetGoogle Scholar
  47. 47.
    Lipatov, E.P.: A classification of binary collections and properties of homogeneity classes. Problemy Kibernet 39, 67–84 (1982)MathSciNetGoogle Scholar
  48. 48.
    Mignosi, F.: On the number of factors of Sturmian words. Theoret. Comput. Sci. 82(1), 71–84 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    Berstel, J., Pocchiola, M.: A geometric proof of the enumeration formula for Sturmian words. Internat. J. Algebra Comput. 3(3), 349–355 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Berstel, J., Pocchiola, M.: Random generation of finite Sturmian words. In: Proceedings of the 5th Conference on Formal Power Series and Algebraic Combinatorics (Florence, 1993), vol. 153, pp. 29–39 (1996)Google Scholar
  51. 51.
    Berenstein, C.A., Lavine, D.: On the number of digital straight line segments. IEEE Trans. Pattern Anal. Mach. Intell. 10(6), 880–887 (1988)zbMATHCrossRefGoogle Scholar
  52. 52.
    Koplowitz, J., Lindenbaum, M., Bruckstein, A.M.: The number of digital straight lines on an n×n grid. IEEE Transactions on Information Theory 36(1), 192–197 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    Heinis, A.: On low-complexity bi-infinite words and their factors. J. Theor. Nombres Bordeaux 13(2), 421–442 (2001)zbMATHMathSciNetGoogle Scholar
  54. 54.
    Tarannikov, Y.: On the bounds for the number of ℓ-balanced words. Technical report, Mech. & Math. Department, Moscow State University (2007)Google Scholar
  55. 55.
    Mignosi, F., Zamboni, L.Q.: On the number of Arnoux-Rauzy words. Boolean Calculus of Differences 101(2), 121–129 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  56. 56.
    Paquin, G., Vuillon, L.: A characterization of balanced episturmian sequences. Electronic J. Combinatorics 14(1) R33, pages 12 (2007)Google Scholar
  57. 57.
    Glen, A., Justin, J., Pirillo, G.: Characterizations of finite and infinite episturmian words via lexicographic orderings. European Journal of Combinatorics (2007)Google Scholar
  58. 58.
    Pirillo, G.: Morse and Hedlund’s skew Sturmian words revisited. Annals Combinatorics  (to appear, 2007)Google Scholar
  59. 59.
    Gan, S.: Sturmian sequences and the lexicographic world. Proc. Amer. Math. Soc. (electronic) 129, 1445–1451 (2001)Google Scholar
  60. 60.
    Glen, A.: A characterization of fine words over a finite alphabet. Theoret. Comput. Sci. CANT conference, Liege, Belgium, May 8-19, 2007, 8–19 (to appear, 2007)Google Scholar
  61. 61.
    Pirillo, G.: Inequalities characterizing standard Sturmian and episturmian words. Theoret. Comput. Sci. 341, 276–292 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  62. 62.
    Burrows, M., Wheeler, D.J.: A block sorting data compression algorithm. Technical report, Digital System Research Center (1994)Google Scholar
  63. 63.
    Gessel, I., Reutenauer, C.: Counting permutations with given cycle structure and descent set. J. Comb. Theory A 64, 189–215 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  64. 64.
    Crochemore, M., Désarménien, J., Perrin, D.: A note on the Burrows-Wheeler transformation. Theoret. Comput. Sci. 332, 567–572 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  65. 65.
    Mantaci, S., Restivo, A., Sciortino, M.: Burrows Wheeler transform and Sturmian words. Inform. Proc. Letters 86, 241–246 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  66. 66.
    Mantaci, S., Restivo, A., Rosone, G., Sciortino, M.: An extension of the Burrows Wheeler transform. Theoret. Comput. Sci. (2007)Google Scholar
  67. 67.
    Rytter, W.: The structure of subword graphs and suffix trees of Fibonacci words. Theoret. Comput. Sci. 363(2), 211–223 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  68. 68.
    Epifanio, C., Mignosi, F., Shallit, J., Venturini, I.: On Sturmian graphs. Discrete Appl. Math 155, 1014–1030 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  69. 69.
    Blumer, A., Blumer, J.A., Haussler, D., Ehrenfeucht, A., Chen, M.T., Seiferas, J.I.: The smallest automaton recognizing the subwords of a text (Special issue: Eleventh international colloquium on automata, languages and programming (Antwerp, 1984)). Theoret. Comput. Sci. 40(1), 31–55 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  70. 70.
    Crochemore, M., Rytter, W.: Jewels of stringology. World Scientific Publishing Co. Inc, River Edge, NJ (2003)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Jean Berstel
    • 1
  1. 1.Institut Gaspard Monge, Université Paris-Est, Marne-la-ValléeFrance

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