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) no. 2, pp. 275-290

We show that every linearly repetitive Delone set in the Euclidean d-space ${ℝ}^d$, 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 ${ℝ}^d$ comes from a primitive substitution tiling of ${ℝ}^d$, 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.

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