Linearly repetitive Delone sets are rectifiable

Aliste-Prieto, José ; Coronel, Daniel ; Gambaudo, Jean-Marc

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013), p. 275-290 / Harvested from Numdam

Résumé

We show that every linearly repetitive Delone set in the Euclidean d-space $ {\mathbb{R}}^{d}$, with $ d⩾2$, is equivalent, up to a bi-Lipschitz homeomorphism, to the integer lattice $ {\mathbb{Z}}^{d}$. In the particular case when the Delone set X in $ {\mathbb{R}}^{d}$ comes from a primitive substitution tiling of $ {\mathbb{R}}^{d}$, we give a condition on the eigenvalues of the substitution matrix which ensures the existence of a homeomorphism with bounded displacement from X to the lattice $ \beta {\mathbb{Z}}^{d}$ for some positive β. This condition includes primitive Pisot substitution tilings but also concerns a much broader set of substitution tilings.

Détail

Publié le : 2013-01-01

MR 3035977 | Zbl 1288.52011

DOI : https://doi.org/10.1016/j.anihpc.2012.07.006

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