The geometry of non-unit Pisot substitutions

Minervino, Milton; Thuswaldner, Jörg

Annales de l'Institut Fourier, Tome 64 (2014) no. 4, pp. 1373-1417.

On peut associer à une substitution de type Pisot non unimodulaire $\sigma$ 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 $\sigma$ 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 $\sigma$, des transformations adiques et un échange de morceaux.

It is known that with a non-unit Pisot substitution $\sigma$ 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 $\sigma$, 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 $\sigma$, to adic transformations, and a domain exchange.

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