Download PDF - A generalization of Sturmian sequences: Combinatorial structure and transcendenceAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

A generalization of Sturmian sequences: Combinatorial structure and transcendence

Authors 1, 1

Affiliations

  1. Department of Mathematics, University of North Texas, Denton, TX 76203-5116, U.S.A.

Bibliography

  1. J.-P. Allouche and L. Q. Zamboni, Algebraic irrational binary numbers cannot be fixed points of non-trivial constant length or primitive morphisms, J. Number Theory 69 (1998), 119-124.
  2. P. Arnoux and S. Ito, Pisot substitutions and Rauzy fractals, preprint, 1999.
  3. P. Arnoux et G. Rauzy, Représentation géométrique de suites de complexité 2n+1, Bull. Soc. Math. France 119 (1991), 199-215.
  4. J. Berstel, Mots de Fibonacci, Séminaire d'Informatique Théorique, LITP, Universités Paris 6-7 (1980-1981), 57-78.
  5. J. Berstel et P. Séébold, Morphismes de Sturm, Bull. Belg. Math. Soc. 1 (1994), 175-189.
  6. V. Berthé, Fréquences des facteurs des suites sturmiennes, Theoret. Comput. Sci. 165 (1996), 295-309.
  7. M. G. Castelli, F. Mignosi and A. Restivo, Fine and Wilf's theorem for three periods and a generalization of sturmian words, ibid. 218 (1999), 83-94.
  8. N. Chekhova, Les suites d'Arnoux-Rauzy : algorithme d'approximation et propriétés ergodiques, preprint, 1998.
  9. N. Chekhova, P. Hubert et A. Messaoudi, Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci, J. Théor. Nombres Bordeaux, to appear.
  10. E. M. Coven and G. A. Hedlund, Sequences with minimal block growth, Math. Systems Theory 7 (1973), 138-153.
  11. A. de Luca and F. Mignosi, Some combinatorial properties of Sturmian words, Theoret. Comput. Sci. 136 (1994), 361-385.
  12. F. Durand, A characterization of substitutive sequences using return words, Discrete Math. 179 (1998), 89-101.
  13. F. Durand, Linearly recurrent subshifts, Ergod. Theory Dynam. Systems, to appear.
  14. F. Durand and B. Host, private communication.
  15. F. Durand, B. Host and C. Skau, Substitution dynamical systems, Bratteli diagrams and dimension groups, Ergod. Theory Dynam. Systems 19 (1999), 953-993.
  16. S. Ferenczi, Les transformations de Chacon : combinatoire, structure géométrique, lien avec les systèmes de complexité 2n+1, Bull. Soc. Math. France 123 (1995), 271-292.
  17. S. Ferenczi and C. Mauduit, Transcendence of numbers with a low complexity expansion, J. Number Theory 67 (1997), 146-161.
  18. C. Holton and L. Q. Zamboni, Descendants of primitive substitutions, Theory Comput. Syst. 32 (1998), 133-157.
  19. C. Holton and L. Q. Zamboni, Directed graphs and substitutions, in: From Crystals to Chaos, P. Hubert, R. Lima and S. Vaienti (eds.), World Sci., 1999, to appear.
  20. J. H. Loxton and A. van der Poorten, Arithmetic properties of automata: regular sequences, J. Reine Angew. Math. 392 (1988), 57-69.
  21. K. Mahler, Lectures on Diophantine Approximations, Part I: g-adic Numbers and Roth's Theorem, Univ. of Notre Dame, 1961.
  22. K. Mahler, Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen, Proc. Konink. Nederl. Akad. Wetensch. Ser. A 40 (1937), 421-428.
  23. F. Mignosi, Infinite words with linear subword complexity, Theoret. Comput. Sci. 65 (1989), 221-242.
  24. F. Mignosi, On the number of factors of Sturmian words, ibid. 82 (1991), 71-84.
  25. F. Mignosi and G. Pirillo, Repetitions in the Fibonacci infinite word, RAIRO Inform. Théor. Appl. 26 (1992), 199-204.
  26. M. Morse and G. A. Hedlund, Symbolic dynamics, Amer. J. Math. 60 (1938), 815-866.
  27. M. Morse and G. A. Hedlund, Symbolic dynamics II: Sturmian sequences, ibid. 62 (1940), 1-42.
  28. M. Queffélec, Substitution Dynamical Systems-Spectral Analysis, Lecture Notes in Math. 1294, Springer, 1987.
  29. G. Rauzy, Mots infinis en arithmétique, in: Automata on Infinite Words, M. Nivat and D. Perrin (eds.), Lecture Notes in Comput. Sci. 192, Springer, Berlin, 1985, 165-171.
  30. G. Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France 110 (1982), 147-178.
  31. G. Rote, Sequences with subword complexity 2n, J. Number Theory 46 (1994), 196-213.
  32. J.-I. Tamura, A class of transcendental numbers having explicit g-adic and Jacobi-Perron expansions of arbitrary dimension, Acta Arith. 71 (1995), 301-329.
  33. N. Wozny and L. Q. Zamboni, Frequencies of factors in Arnoux-Rauzy sequences, Acta Arith., to appear.
  34. L. Q. Zamboni, Une généralisation du théorème de Lagrange sur le développement en fraction continue, C. R. Acad. Sci. Paris Sér. I, 327 (1998), 527-530.
Acta Arithmetica
2000/95/2
Export to PBN, Opens in new tab
Pages:
167-184
Main language of publication
English
Received
1998-12-23
Accepted
1999-05-24
Published
2000
Exact and natural sciences