Linearly repetitive Delone sets are rectifiable

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

doi:10.1016/j.anihpc.2012.07.006

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

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Résumé

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 $ℤ^{d}$ . 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 $β ℤ^{d}$ for some positive β. This condition includes primitive Pisot substitution tilings but also concerns a much broader set of substitution tilings.

MR Zbl

DOI : 10.1016/j.anihpc.2012.07.006

@article{AIHPC_2013__30_2_275_0,
     author = {Aliste-Prieto, Jos\'e and Coronel, Daniel and Gambaudo, Jean-Marc},
     title = {Linearly repetitive {Delone} sets are rectifiable},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {275--290},
     publisher = {Elsevier},
     volume = {30},
     number = {2},
     year = {2013},
     doi = {10.1016/j.anihpc.2012.07.006},
     mrnumber = {3035977},
     zbl = {1288.52011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2012.07.006/}
}

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Aliste-Prieto, José; Coronel, Daniel; Gambaudo, Jean-Marc. Linearly repetitive Delone sets are rectifiable. Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 2, pp. 275-290. doi: 10.1016/j.anihpc.2012.07.006

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