Abstract Numeration Systems and Tilings

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


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] .


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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

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