JournalsaihpcVol. 30, No. 2pp. 275–290

Linearly repetitive Delone sets are rectifiable

  • Daniel Coronel

    Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Campus San Joaquín, Avenida Vicuña Mackenna 4860, Santiago, Chile
  • Jean-Marc Gambaudo

    Institut Non Linéaire de Nice–Sophia Antipolis, UMR CNRS 7335, Université de Nice – Sophia Antipolis, 1361, Route des Lucioles, 06560 Valbonne, France
  • José Aliste-Prieto

    Centro de Modelamiento Matemático, Universidad de Chile, Blanco Encalada 2120 7to. piso, Santiago, Chile
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Abstract

We show that every linearly repetitive Delone set in the Euclidean d-space Rd\mathbb{R}^{d}, with d2d⩾2, is equivalent, up to a bi-Lipschitz homeomorphism, to the integer lattice Zd\mathbb{Z}^{d}. In the particular case when the Delone set X in Rd\mathbb{R}^{d} comes from a primitive substitution tiling of Rd\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 βZd\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.

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