# The geometry of non-unit Pisot substitutions

Abstract : Let $\sigma$ be a non-unit Pisot substitution and let $\alpha$ be the associated Pisot number. It is known that one can associate certain fractal tiles, so-called \emph{Rauzy fractals}, with $\sigma$. In our setting, these fractals are subsets of a certain open subring of the ad\`ele ring $\mathbb{A}_{\mathbb{Q}(\alpha)}$. 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, define them in terms of the one-dimensional realization of $\sigma$ and its dual (in the spirit of Arnoux and Ito), and view them as the dual of multi-component 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 Rauzy fractals associated with non-unit Pisot substitutions, prove some tiling results for them, and provide relations to subshifts defined in terms of the periodic points of $\sigma$, to adic transformations, and a domain exchange. We illustrate our results by examples on two and three letter substitutions.
Type de document :
Article dans une revue
Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2014, 64, pp.1373 - 1417. <10.5802/aif.2884>
Domaine :

https://hal.archives-ouvertes.fr/hal-01485516
Contributeur : Milton Minervino <>
Soumis le : mercredi 8 mars 2017 - 23:07:15
Dernière modification le : vendredi 10 mars 2017 - 01:03:09
Document(s) archivé(s) le : vendredi 9 juin 2017 - 14:29:47

### Fichier

nonunitPaper.pdf
Fichiers produits par l'(les) auteur(s)

### Citation

Milton Minervino, Jörg Thuswaldner. The geometry of non-unit Pisot substitutions. Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2014, 64, pp.1373 - 1417. <10.5802/aif.2884>. <hal-01485516>

Consultations de
la notice

## 59

Téléchargements du document