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arXiv · 1103.5423

Linearly repetitive Delone sets are rectifiable

J. Aliste-Prieto·D. Coronel·J. -M. Gambaudo
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Abstract

In this paper we prove that, for any integer $d>0$, every linearly repetitive Delone set in the Euclidean $d$-space $\RR^d$ is equivalent, up to a bi-Lipschitz homeomorphism, to the integer lattice $\ZZ^d$. In the particular case when the Delone set $X$ in $\RR^d$ comes from a primitive substitution tiling of $\RR^d$, we give a condition on the eigenvalues of the substitution matrix which implies the existence of a homeomorphism with bounded displacement from $X$ to the lattice lattice $λ\ZZ^d$ for some positive $λ$. This condition includes primitive Pisot substitution tilings but also concerns a much broader set of substitution tilings.

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BibTeXRIS

J. Aliste-Prieto, D. Coronel, J. -M. Gambaudo. 2011-10-24. Linearly repetitive Delone sets are rectifiable. https://arxiv.org/abs/1103.5423

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