Advertisement

Abstract Numeration Systems and Tilings

  • Valérie Berthé
  • Michel Rigo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3618)

Abstract

An abstract numeration system is a triple S = (L,Σ,<) where (Σ,<) is a totally ordered alphabet and L a regular language over Σ; the associated numeration is defined as follows: by enumerating the words of the regular language L over Σ with respect to the induced genealogical ordering, one obtains a one-to-one correspondence between ℕ and L. Furthermore, when the language L is assumed to be exponential, real numbers can also be expanded. The aim of the present paper is to associate with S a self-replicating multiple tiling of ăthe space, under the following assumption: the adjacency matrix of the trimmed minimal automaton recognizing L is primitive with a dominant eigenvalue being a Pisot unit. This construction generalizes the classical constructions performed for Rauzy fractals associated with Pisot substitutions [16], and for central tiles associated with a Pisot beta-numeration [23] .

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Akiyama, S.: Self affine tiling and Pisot numeration system. In: Number theory and its applications (Kyoto 1997). Dev. Math, vol. 2, pp. 7–17. Kluwer Acad. Publ, Dordrecht (1999)Google Scholar
  2. 2.
    Akiyama, S.: On the boundary of self affine tilings generated by Pisot numbers. J. Math. Soc. Japan 54, 283–308 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Berthé, V., Rigo, M.: Odometers on regular languages. Theory Comput. Syst (to appear)Google Scholar
  4. 4.
    Berthé, V., Siegel, A.: Purely periodic expansions in the non-unit case (2004) (preprint)Google Scholar
  5. 5.
    Berthé, V., Siegel, A.: Tilings associated with beta-numeration and substitutions. Integers: Electronic Journal of Combinatorial Number Theory (to appear)Google Scholar
  6. 6.
    Dumont, J.-M., Thomas, A.: Systèmes de numération et fonctions fractales relatifs aux substitutions. Theoret. Comput. Sci. 65, 153–169 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dumont, J.-M., Thomas, A.: Digital sum moments and substitutions. Acta Arith. 64, 205–225 (1993)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Frougny, C., Solomyak, B.: Finite beta-expansions. Ergodic Theory Dynam. Systems 12, 713–723 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Grabner, P.J., Rigo, M.: Additive functions with respect to numeration systems on regular languages. Monatsh. Math. 139, 205–219 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Lecomte, P.B.A., Rigo, M.: Numeration systems on a regular language. Theory Comput. Syst. 34, 27–44 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Lecomte, P., Rigo, M.: On the representation of real numbers using regular languages. Theory Comput. Syst. 35, 13–38 (2002)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Lecomte, P., Rigo, M.: Real numbers having ultimately periodic representations in abstract numeration systems. Inform. and Comput. 192, 57–83 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lothaire, M.: Algebraic Combinatorics on words. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  14. 14.
    Mauldin, R.D., Williams, S.C.: Hausdorff dimension in graph directed constructions. Trans. Amer. Math. Soc. 309, 811–829 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Pytheas Fogg, N.: Substitutions in Dynamics, Arithmetics and Combinatorics. Lect. Notes in Math, vol. 1794. Springer, Berlin (2002)zbMATHCrossRefGoogle Scholar
  16. 16.
    Rauzy, G.: Nombres algébriques et substitutions. Bull. Soc. Math. France 110, 147–178 (1982)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Rigo, M.: Numeration systems on a regular language: arithmetic operations, recognizability and formal power series. Theoret. Comput. Sci. 269, 469–498 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Rigo, M.: Construction of regular languages and recognizability of polynomials. Discrete Math. 254, 485–496 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Rigo, M., Steiner, W.: Abstract β-expansions and ultimately periodic representations. J. Théor. Nombres Bordeaux 17, 283–299 (2005)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Siegel, A.: Représentation des systèmes dynamiques substitutifs non unimodulaires. Ergodic Theory Dynam. Systems 23, 1247–1273 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Sirvent, V., Wang, Y.: Self-affine tiling via substitution dynamical systems and Rauzy fractals. Pacific J. Math. 206, 465–485 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Thomas, W.: Automata on infinite objects. In: Handbook of theoret. comput. sci., vol. B, pp. 133–191. Elsevier, Amsterdam (1990)Google Scholar
  23. 23.
    Thurston, W.P.: Groups, tilings and finite state automata, Lectures notes distributed in conjunction with the Colloquium Series. AMS Colloquium lectures (1989)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Valérie Berthé
    • 1
  • Michel Rigo
    • 2
  1. 1.LIRMM, CNRS-UMR 5506Univ. Montpellier IIMontpellier Cedex 5France
  2. 2.Institut de MathématiquesUniversité de LiègeLiègeBelgium

Personalised recommendations