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Spectral study of automatic and substitutive sequences

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Beyond Quasicrystals

Part of the book series: Centre de Physique des Houches ((LHWINTER,volume 3))

Abstract

A natural question about a sequence taking its values in a finite alphabet, is the following: how far from a random sequence is it and which appropriate quantitative analysis can we imagine?

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References

  1. ALLOUCHE J.P., LIARDET P.: Generalized Rudin-Shapiro sequences, Acta Arith. 60 (1991) 1–27.

    MathSciNet  MATH  Google Scholar 

  2. ALLOUCHE J.P., MENDES FRANCE M.: Suite de Rudin-Shapiro et modèle d’Ising, Bull. S.M.F. 113 (1985) 273–283.

    MathSciNet  MATH  Google Scholar 

  3. ALLOUCHE J.P., MENDES FRANCE M.: Quasi-crystal Ising chain and automata theory, J. Stat. Phys. 42 (1986).

    Google Scholar 

  4. ALLOUCHE J.P., MENDES FRANCE M.: this volume.

    Google Scholar 

  5. ARNOLD V.I., AVEZ.A.: Problèmes ergodiques de la mécanique classique, Gauthier- Villars Paris 1967.

    Google Scholar 

  6. BERTHE V.: this volume.

    Google Scholar 

  7. BOMBIERI E., TAYLOR J.E.: Quasicrystals, tilings and algebraic number theory: some preliminary connections, Contemporary Math. 64 (1987) 241–264.

    Article  MathSciNet  Google Scholar 

  8. BRAUER A.: On algebraic equations with all but one root in the interior of the unit circle, Math. Nachr. 4 (1951) 250–257.

    Article  MathSciNet  MATH  Google Scholar 

  9. BRILLHART J., CARLITZ L.: Note on the Rudin-Shapiro polynomials, Proc. Amer. Math. Soc. 25 (1970) 114–118.

    Article  MathSciNet  MATH  Google Scholar 

  10. BROWN G., MORAN W.: A dichotomy for infinite convolutions of discrete measures, Proc. Camb. Phil. Soc. 73 (1973) 307–316.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. CHACON R.V.:Weakly mixing transformations which are not strongly mixing, Proc. Amer.Math.Soc. 22 (1969) 559–562.

    Google Scholar 

  12. CHRISTOL G., KAMAE T., MENDES FRANCE M., RAUZY G.: Suites algébriques, automates et substitutions, Bull. S.M.F. 108 (1980) 401–419.

    MathSciNet  Google Scholar 

  13. COBHAM A.: Uniform tag sequences, Math. systems theory, 6 (1972) 164–192.

    Article  MathSciNet  MATH  Google Scholar 

  14. COQUET J., KAMAE T., MENDES FRANCE M.: Sur la mesure spectrale de certaines suites arithmétiques, Bull. S.M.F. 105 (1977) 369–384.

    MathSciNet  MATH  Google Scholar 

  15. CORNFELD J.P., FOMIN S.V., SINAI Y.G.: Ergodic theory. Springer Verlag, 1982.

    Google Scholar 

  16. COVEN E.M., HEDLUND G.A.: Sequences with minimal block growth, Math. Systems Th. 7 (1973) 138–153

    Article  MathSciNet  MATH  Google Scholar 

  17. CRISP D., MORAN W., POLLINGTON A., SHIUE P.: Substitution invariant cutting sequences, J.theorie des nombres de Bordeaux 5 (1993) 123–143.

    Article  MathSciNet  Google Scholar 

  18. DEKKING F.M.: The spectrum of dynamical systems arising from substitutions of constant length, Zeit. Wahr., 41 (1978) 221–239.

    Article  MathSciNet  MATH  Google Scholar 

  19. DEKKING F.M.: On the Prouhet-Thue-Morse measure, Acta Univ. Carolinae, mathematica et physica, 33 (1992) 35–40.

    MathSciNet  MATH  Google Scholar 

  20. DEKKING F.M., KEANE M.: Mixing properties of substitutions, Zeit. Wahr., 42 (1978) 23–33.

    Article  MathSciNet  MATH  Google Scholar 

  21. DEKKING F.M., MENDES FRANCE M.: Uniform distribution modulo 1; a geometrical viewpoint, J.Reine Angew. Math. 329 (1981) 143–153.

    MathSciNet  MATH  Google Scholar 

  22. FERENCZI S.: Les transformations de Chacon: combinatoire, structure géométrique, lien avec les systèmes de complexité 2n + 1, to appear in BuIl.S.M.F. 123 (1995).

    Google Scholar 

  23. GARSIA A.: Arithmetic properties of Bernoulli convolutions, Trans. Amer. Math. Soc. 102 (1962) 409–432.

    Article  MathSciNet  MATH  Google Scholar 

  24. HELSON H., PARRY W.: Cocycles and spectra, Arkiv för Mat. 16 (1978) 195–206.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. HOF A.: Quasicrystals, Aperiodicity and Lattice Systems, Thesis Rijksuniversiteit Groningen (1992)

    Google Scholar 

  26. HOST B.: Valeurs propres des systèmes dynamiques définis par des substitutions de longueur variable, Erg. th. et dyn. Sys. 6 (1986) 529–540.

    MathSciNet  MATH  Google Scholar 

  27. HOST B., MELA J.F., PARREAU F.: Analyse harmonique des mesures, Astérisque (1986) 135–136.

    Google Scholar 

  28. HOST B., PARREAU F.: Private communication.

    Google Scholar 

  29. HOST B., PARREAU F.: The generalized purity law for ergodic maesures: a simple proof, Colloq. Math. 60 /61 (1990) 205–212.

    MathSciNet  Google Scholar 

  30. JACOBS K., KEANE M.: 0–1 sequences of Toeplitz type, Z. Wahrsch. Verw. Geb. 13 (1969) 123–131.

    Article  MathSciNet  MATH  Google Scholar 

  31. KAKUTANI S.: Strictly ergodic symbolic dynamical systems, Proceedings of the sixth Berkeley Symposium on Mathematical statistics and Probability (1967) 319–326.

    Google Scholar 

  32. KATZNELSON Y.: An introduction to Harmonic Analysis, Wiley (1968).

    Google Scholar 

  33. KEANE M.: Generalized Morse sequences, Zeit. Wahr. 10 (1968) 335–353.

    Article  MathSciNet  MATH  Google Scholar 

  34. KWIATKOVSKI J.: Isomorphism of regular Morse dynamical systems induced by arbitrary blocks, Stud. Math. 84 (1986) 219–246.

    Google Scholar 

  35. LEDRAPPIER F.: Des produits de Riesz comme mesures spectrales, Ann. I.H.P. section B, 4 (1970) 335–344.

    MathSciNet  Google Scholar 

  36. LESIGNE E.: Le théorème ergodique ponctuel le long de la suite de Morse, Preprint.

    Google Scholar 

  37. LESIGNE E., PETERSEN K.: Bounded exponential sums, Osaka J. Math. 27 (1990) 277–280.

    MathSciNet  MATH  Google Scholar 

  38. LUCK J.M., GODRECHE C., JANNER A., JANSSEN T.: The nature of the atomic surfaces of quasi-periodic self-similar structures, J. Phys. A: Math. Gen. 26 (1993) 1951–1999.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  39. MARTIN J.C.: Substitution minimal flows, Amer. J. Math. 93 (1971) 503–526.

    Article  MATH  Google Scholar 

  40. MARTIN J.: Minimal flows arising from substitutions of non-constant length, Math. Syst. Th. 7 (1973) 73–82.

    Article  MATH  Google Scholar 

  41. MARTIN J.C.: Generalized Morse sequences on n-symbols. Proc. A.M.S. 54 (1976) 379–383.

    MATH  Google Scholar 

  42. MAUDUIT C.: Automates finis et ensembles normaux. Ann. Ins. Fourier 36 (1986) 1–25.

    Article  MathSciNet  MATH  Google Scholar 

  43. MAUDUIT C.: Sur l’ensemble normal des substitutions de longueur quelconque, J. Number Theory, 29 (1988) 235–250.

    Article  MathSciNet  MATH  Google Scholar 

  44. MENDES FRANCE M., VAN DER POORTEN A.J.: Arithmetic and analytic properties of paper folding sequences, Bull. Austral. Math. Soc. 24 (1981) 123–131.

    Article  MathSciNet  Google Scholar 

  45. MEYER Y. Algebraic numbers and Harmonic analysis, North Holland (1972).

    Google Scholar 

  46. MEYER Y., WEISS B.: Les produits de Riesz sont des Bernoulli shifts, Séminaire de probabilités de Rennes, (1974).

    Google Scholar 

  47. MICHEL P.: Coincidence values and spectra of substitutions. Zeit. Wahr. 42 (1978) 205–227.

    Article  MATH  Google Scholar 

  48. MOSSE B.: Notions de reconnaissabilité pour les substitutions et complexité des suites automatiques, (submitted to Theor.Comput.Sci.)

    Google Scholar 

  49. PARRY W.: Topics in ergodic theory. Cambridge University Press, (1981)

    Google Scholar 

  50. PEYRIERE J.: Etude de quelques propriétés des produits de Riesz, Ann. Inst. Fourier 25 (1975) 127–169.

    Article  MathSciNet  MATH  Google Scholar 

  51. QUEFFELEC M.: Mesures spectrales associées a certaines suites arithmétiques, Bull. S.M.F. 107 (1979) 385–421.

    MathSciNet  MATH  Google Scholar 

  52. QUEFFELEC M.: Substitution dynamical systems - Spectral analysis. Lecture Note in Math. n°1294, Springer-Verlag (1987).

    Google Scholar 

  53. QUEFFELEC M.: Une nouvelle propriété des suites de Rudin-Shapiro, Ann. Inst. Fourier 37 (1987) 115–138.

    Article  MathSciNet  MATH  Google Scholar 

  54. RAUZY G.: Propriétés statistiques des suites arithmétiques, P.U.F. Coll. Sup (1976).

    Google Scholar 

  55. RAUZY G.: Suites à termes dans un alphabet fini, Sém. théorie des nombres, Bordeaux, exp. 25 (1982–1983).

    Google Scholar 

  56. RAUZY G.: Structure des systèmes dynamiques définis par substitutions (unpublished).

    Google Scholar 

  57. RIESZ M., NAGY B.SZ.: Lecons d’analyse fonctionnelle, Gauthier-Villars (1972).

    Google Scholar 

  58. RUDIN W.: Real and complex analysis, Mac Graw Hill (1968).

    Google Scholar 

  59. SENECHAL M., TAYLOR J.E.: Quasicrystals: the view from Les Houches, Math. Intelligencer, 12 (1990) 54–64.

    Article  MathSciNet  MATH  Google Scholar 

  60. SOLOMYAK B.: Substitutions, adic transformations and beta-expansions, Contemp. Math. 135 (1992) 361–372.

    MathSciNet  Google Scholar 

  61. SERIES C.: The geometry of Markoff numbers, Math. Intelligencer 7 (1985) 20–29.

    Article  MathSciNet  MATH  Google Scholar 

  62. SUTO A.: this volume

    Google Scholar 

  63. WEN Zhi-Ying: Substitutions, fractals et dimensions. Habilitation Univ. Paris-Sud (1990).

    Google Scholar 

  64. WEN Zhi-Ying: this volume.

    Google Scholar 

  65. WEN Zhi-Ying, WEN Zhi-Xiong: Des remarques sur les suites engendrées par des substitutions composées, Ann. Fac. Sci. Toulouse, 9 (1988) 56–65.

    Google Scholar 

  66. ZYGMUND A. Trigonometric series. Cambridge University Press, London, (1959).

    MATH  Google Scholar 

  67. GODRECHE C., LUCK J.M.: Multifractal Analysis in reciprocal space and the nature of the Fourier Transform of self-similar structure, J.Phys.A.Math.and Gen. 23 (1990) 37–6 9.

    Google Scholar 

  68. PEYRIERE J.: this volume.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Université des Sciences et Technologies de Lille, F-59655, Villeneuve d’Asq Cedex, France

    Martine Queffelec

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  1. Martine Queffelec
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Editor information

Editors and Affiliations

  1. Université Paris VII-Denis Diderot, France

    Françoise Axel

  2. CECN-CNRS, Vitry, France

    Denis Gratias

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© 1995 Springer-Verlag Berlin Heidelberg

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Queffelec, M. (1995). Spectral study of automatic and substitutive sequences. In: Axel, F., Gratias, D. (eds) Beyond Quasicrystals. Centre de Physique des Houches, vol 3. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03130-8_12

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  • DOI: https://doi.org/10.1007/978-3-662-03130-8_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-59251-8

  • Online ISBN: 978-3-662-03130-8

  • eBook Packages: Springer Book Archive

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