It is known that with a non-unit Pisot substitution one can associate certain fractal tiles, so-called Rauzy fractals. In our setting, these fractals are subsets of a certain open subring of the adèle ring of the associated Pisot number field. We present several approaches on how to define Rauzy fractals and discuss the relations between them. In particular, we consider Rauzy fractals as the natural geometric objects of certain numeration systems, in terms of the dual of the one-dimensional realization of , and in the context of model sets for particular cut and project schemes. We also define stepped surfaces suited for non-unit Pisot substitutions. We provide basic topological and geometric properties of the Rauzy fractals, prove some tiling results for them, and provide relations to subshifts defined in terms of the periodic points of , to adic transformations, and a domain exchange.
On peut associer à une substitution de type Pisot non unimodulaire certaines tuiles fractales, appelés fractals de Rauzy. Dans ce contexte, ces fractals sont des sous-ensembles d’un certain sous-anneau ouvert de l’anneau des adèles du corps de nombres associé. On présente plusieurs approches sur la façon de définir les fractals de Rauzy. En particulier, on considère les fractals de Rauzy comme des objets géométriques naturels associés à certains systèmes de numération, en termes du dual de la réalisation unidimensionnelle de et comme des ensembles définis par coupe et projection. On définit également des surfaces discrètes adaptées aux substitutions de type Pisot non unimodulaires. On établit des propriétés topologiques et géométriques basiques des fractals de Rauzy, ainsi que des résultats de pavage. Finalement on fournit des relations entre des sous-décalages définis en termes de points périodiques de , des transformations adiques et un échange de morceaux.
Keywords: Rauzy fractal, tiling, $p$-adic completion, beta-numeration
Mot clés : fractals de Rauzy, pavage, complété $p-$-adique, beta-numération
@article{AIF_2014__64_4_1373_0, author = {Minervino, Milton and Thuswaldner, J\"org}, title = {The geometry of non-unit {Pisot} substitutions}, journal = {Annales de l'Institut Fourier}, pages = {1373--1417}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {64}, number = {4}, year = {2014}, doi = {10.5802/aif.2884}, zbl = {06387311}, mrnumber = {3329667}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2884/} }
TY - JOUR AU - Minervino, Milton AU - Thuswaldner, Jörg TI - The geometry of non-unit Pisot substitutions JO - Annales de l'Institut Fourier PY - 2014 SP - 1373 EP - 1417 VL - 64 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2884/ DO - 10.5802/aif.2884 LA - en ID - AIF_2014__64_4_1373_0 ER -
%0 Journal Article %A Minervino, Milton %A Thuswaldner, Jörg %T The geometry of non-unit Pisot substitutions %J Annales de l'Institut Fourier %D 2014 %P 1373-1417 %V 64 %N 4 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2884/ %R 10.5802/aif.2884 %G en %F AIF_2014__64_4_1373_0
Minervino, Milton; Thuswaldner, Jörg. The geometry of non-unit Pisot substitutions. Annales de l'Institut Fourier, Volume 64 (2014) no. 4, pp. 1373-1417. doi : 10.5802/aif.2884. http://archive.numdam.org/articles/10.5802/aif.2884/
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