Self-similar Discrete Rotation Configurations and Interlaced Sturmian Words

  • Bertrand Nouvel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4992)


Rotation configurations for quadratic angles exhibit self-similar dynamics. Visually, it may be considered as quite evident. However, no additional details have yet been published on the exact nature of the arithmetical reasons that support this fact. In this paper, to support the existence of self-similar dynamic in 2d-configuration, we will use the constructive 1-d substitution theory in order to iteratively build quadratic rotation configurations from substitutive Sturmian words. More specifically : the self-similar rotation configurations are first shown to be an interlacing of configurations that are direct product of superposition of Sturmian words.


  1. [Ad02]
    Adamczewski, B.: Approche dynamique et combinatoire de la notion de discrépance. PhD thesis, Institut de Mathématiques de Luminy (2002)Google Scholar
  2. [Al96]
    Alessandri, P.: Codages de rotations et basse complexité. PhD thesis, Université de la Méditerranée (1996)Google Scholar
  3. [Be02]
    Berthé, V.: Autour du systéme de numération d’Ostrowski. Source: Bull. Belg. Math. Soc. Simon Stevin, 8(2), 209–239 (preprint, 2001) (English version: About Ostrowski Numeration System)Google Scholar
  4. [BHZ]
    Berthé, V., Holton, C., Zamboni, L.Q.: Initial Power of Sturmian Sequences (preprint)Google Scholar
  5. [Fo02]
    Fogg, P.: Substitutions in Dynamics, Arithmetics and Combinatorics. In: Berthé, V., Ferenczi, S., Mauduit, C., Siegel, A.: Lecture Notes in Mathematics, vol. 1794, 402 pages, (2002), ISBN: 3-540-44141-7Google Scholar
  6. [LKV04]
    Lowenstein, J.H., Koupstov, K.L., Vivaldi, F.: Recursive tiling and geometry of piecewise rotations by pi/7. Nonlinearity 17, 371–395 (2004)MATHCrossRefMathSciNetGoogle Scholar
  7. [MH38]
    Morse, M., Hedlund, G.A.: Symbolic Dynamics. American Journal of Mathematics 60(4), 815–866 (1938)MATHCrossRefMathSciNetGoogle Scholar
  8. [No06]
    Nouvel, B.: Rotations Discrétes et Automates Cellulaires. PhD thesis, École Normale Supérieure de Lyon (2006)Google Scholar
  9. [NR95]
    Nehlig, P.W., Réveilles, J.-P.: Fractals and Quasi-Affine Transformations. Computer Graphics Forum 14(2), 147–157 (1995)CrossRefGoogle Scholar
  10. [NR05]
    Nouvel, B., Rémila, É.: Configurations Induced by Discrete Rotations: Periodicity and Quasiperiodicity Properties. Discrete Applied Mathematics 127(2-3), 325–343 (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Bertrand Nouvel
    • 1
  1. 1.Center for Frontier Medical EngineeringChiba UniversityChibaJapan

Personalised recommendations