Substitutions, abstract number systems and the space filling property
[Substitutions et propriété de remplissage de l’espace]
Annales de l'Institut Fourier, Tome 56 (2006) no. 7, pp. 2345-2389.

Dans cet article nous étudions des mots multidimensionnels engendrés par des points fixes de substitutions, et obtenus en projetant les points entiers sur la demi-droite brisée correspondante. Nous montrons que pour une grande classe de substitutions le mot correspondant est la restriction d’une fonction linéaire modulo 1 et qu’il est possible de décider si le mot résultant remplit l’espace. La preuve utilise des réseaux et le système de numération abstrait associé à la substitution.

In this paper we study multi-dimensional words generated by fixed points of substitutions by projecting the integer points on the corresponding broken halfline. We show for a large class of substitutions that the resulting word is the restriction of a linear function modulo 1 and that it can be decided whether the resulting word is space filling or not. The proof uses lattices and the abstract number system associated with the substitution.

DOI : 10.5802/aif.2243
Classification : 11A63, 68R15, 37B10, 05A05, 05B25, 52C07, 52C22, 68Q45, 11K16
Keywords: Substitutions, limit word, discretisation of the hyperplane, lattices, automata, abstract number systems
Mot clés : substitutions, mot limite, hyperplan discret, réseaux, automates, systèmes de numération abstraits
Fuchs, Clemens 1 ; Tijdeman, Robert 2

1 ETH Zürich Departement Mathematik Rämistrasse 101 8092 Zürich (Switzerland)
2 Universiteit Leiden Mathematisch Instituut Niels Bohrweg 1 2300 RA Leiden (The Netherlands)
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Fuchs, Clemens; Tijdeman, Robert. Substitutions, abstract number systems and the space filling property. Annales de l'Institut Fourier, Tome 56 (2006) no. 7, pp. 2345-2389. doi : 10.5802/aif.2243. http://www.numdam.org/articles/10.5802/aif.2243/

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