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Abstract Numeration Systems and Tilings

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Mathematical Foundations of Computer Science 2005 (MFCS 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,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] .

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Authors and Affiliations

  1. LIRMM, CNRS-UMR 5506, Univ. Montpellier II, 161 rue Ada, 34392, Montpellier Cedex 5, France

    Valérie Berthé

  2. Institut de Mathématiques, Université de Liège, Grande Traverse 12 (B 37), B-4000, Liège, Belgium

    Michel Rigo

Authors
  1. Valérie Berthé
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  2. Michel Rigo
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Editors and Affiliations

  1. Institute of Mathematics, Gdansk University, Wita Stwosza 57, 80 952, Gdansk, Poland

    Joanna Jȩdrzejowicz  & Andrzej Szepietowski  & 

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Berthé, V., Rigo, M. (2005). Abstract Numeration Systems and Tilings. In: Jȩdrzejowicz, J., Szepietowski, A. (eds) Mathematical Foundations of Computer Science 2005. MFCS 2005. Lecture Notes in Computer Science, vol 3618. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11549345_13

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  • DOI: https://doi.org/10.1007/11549345_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-28702-5

  • Online ISBN: 978-3-540-31867-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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