21734 articles – 15570 Notices  [english version]
 HAL : hal-00407999, version 2
 arXiv : 0907.4872
 Advances in Mathematics 226, 1 (2011) 139-175
 Versions disponibles : v1 (28-07-2009) v2 (01-09-2010)
 Fractal tiles associated with shift radix systems
 Valerie Berthe 1, 2, Anne Siegel 3
 (15/01/2011)
 Shift radix systems form a collection of dynamical systems depending on a parameter $\mathbf{r}$ which varies in the $d$-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. Beta-numeration and canonical number systems are known to be intimately related to fractal shapes, such as the classical Rauzy fractal and the twin dragon. These fractals turned out to be important for studying properties of expansions in several settings. In the present paper we associate a collection of fractal tiles with shift radix systems. We show that for certain classes of parameters $\mathbf{r}$ these tiles coincide with affine copies of the well-known tiles associated with beta-expansions and canonical number systems. On the other hand, these tiles provide natural families of tiles for beta-expansions with (non-unit) Pisot numbers as well as canonical number systems with (non-monic) expanding polynomials. We also prove basic properties for tiles associated with shift radix systems. Indeed, we prove that under some algebraic conditions on the parameter $\mathbf{r}$ of the shift radix system, these tiles provide multiple tilings and even tilings of the $d$-dimensional real vector space. These tilings turn out to have a more complicated structure than the tilings arising from the known number systems mentioned above. Such a tiling may consist of tiles having infinitely many different shapes. Moreover, the tiles need not be self-affine (or graph directed self-affine).
 1 : Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier (LIRMM) CNRS : UMR5506 – Université Montpellier II - Sciences et techniques 2 : Laboratoire d'informatique Algorithmique : Fondements et Applications (LIAFA) CNRS : UMR7089 – Université Paris VII - Paris Diderot 3 : SYMBIOSE (INRIA - IRISA) CNRS : UMR6074 – INRIA – Institut National des Sciences Appliquées (INSA) - Rennes – Université de Rennes 1 4 : Montan Universität Leoben Montanuniversität Leoben
 Équipe de recherche : INFO/ARITH
 Domaine : Mathématiques/Systèmes dynamiquesMathématiques/Théorie des nombres
 Mots Clés : beta expansion – canonical number system – shift radix system – tiling
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 hal-00407999, version 2 http://hal.archives-ouvertes.fr/hal-00407999 oai:hal.archives-ouvertes.fr:hal-00407999 Contributeur : Wolfgang Steiner <> Soumis le : Mardi 31 Août 2010, 16:44:30 Dernière modification le : Vendredi 5 Novembre 2010, 17:23:22