Tilings Associated with Beta-numeration and Substitutions

Valerie Berthe 1 Anne Siegel 2
1 ARITH - Arithmétique informatique
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
2 Dyliss - Dynamics, Logics and Inference for biological Systems and Sequences
Inria Rennes – Bretagne Atlantique , IRISA-D7 - GESTION DES DONNÉES ET DE LA CONNAISSANCE
Abstract : This paper surveys different constructions and properties of some multiple tilings (that is, finite-to-one coverings) of the space that can be associated with beta-numeration and substitutions. It is indeed possible, generalizing Rauzy’s and Thurston’s constructions, to associate in a natural way either with a Pisot number β (of degree d) or with a Pisot substitution σ (on d letters) some compact basic tiles that are the closure of their interior, that have non-zero measure and a fractal boundary; they are attractors of some graph- directed Iterated Function System. We know that some translates of these prototiles under a Delone set Γ (provided by β or σ) cover Rd−1; it is conjectured that this multiple tiling is indeed a tiling (which might be either periodic or self-replicating according to the translation set Γ). This conjecture is known as the Pisot conjecture and can also be reformulated in spectral terms: the associated dynamical systems have pure discrete spectrum. We detail here the known constructions for these tilings, their main properties, some applications, and focus on some equivalent formulations of the Pisot conjecture, in the theory of quasicrystals for instance. We state in particular for Pisot substitutions a finiteness property analogous to the well-known (F) property in beta-numeration, which is a sufficient condition to get a tiling.
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Article dans une revue
Integers : Electronic Journal of Combinatorial Number Theory, State University of West Georgia, Charles University, and DIMATIA, 2005, Proceedings of the 2004 Number Theoretic Algorithms and Related Topics Workshop, 5 (3), pp.A2
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https://hal-lirmm.ccsd.cnrs.fr/lirmm-00105299
Contributeur : Christine Carvalho de Matos <>
Soumis le : mercredi 11 octobre 2006 - 07:51:14
Dernière modification le : jeudi 24 mai 2018 - 15:59:21
Document(s) archivé(s) le : mardi 6 avril 2010 - 19:15:56

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  • HAL Id : lirmm-00105299, version 1

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Valerie Berthe, Anne Siegel. Tilings Associated with Beta-numeration and Substitutions. Integers : Electronic Journal of Combinatorial Number Theory, State University of West Georgia, Charles University, and DIMATIA, 2005, Proceedings of the 2004 Number Theoretic Algorithms and Related Topics Workshop, 5 (3), pp.A2. 〈lirmm-00105299〉

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