Linearly repetitive Delone sets are rectifiable
Daniel Coronel
Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Campus San Joaquín, Avenida Vicuña Mackenna 4860, Santiago, ChileJean-Marc Gambaudo
Institut Non Linéaire de Nice–Sophia Antipolis, UMR CNRS 7335, Université de Nice – Sophia Antipolis, 1361, Route des Lucioles, 06560 Valbonne, FranceJosé Aliste-Prieto
Centro de Modelamiento Matemático, Universidad de Chile, Blanco Encalada 2120 7to. piso, Santiago, Chile
![Linearly repetitive Delone sets are rectifiable cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-aihpc-volume-30-issue-2.png&w=3840&q=90)
Abstract
We show that every linearly repetitive Delone set in the Euclidean -space , with , is equivalent, up to a bi-Lipschitz homeomorphism, to the integer lattice . In the particular case when the Delone set in comes from a primitive substitution tiling of , we give a condition on the eigenvalues of the substitution matrix which ensures the existence of a homeomorphism with bounded displacement from to the lattice for some positive . This condition includes primitive Pisot substitution tilings but also concerns a much broader set of substitution tilings.
Cite this article
Daniel Coronel, Jean-Marc Gambaudo, José Aliste-Prieto, Linearly repetitive Delone sets are rectifiable. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 2, pp. 275–290
DOI 10.1016/J.ANIHPC.2012.07.006