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Linearly repetitive Delone sets are rectifiable

By J. Aliste-Prieto, D. Coronel and J. -M. Gambaudo


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 $\lambda\ZZ^d$ for some positive $\lambda$. This condition includes primitive Pisot substitution tilings but also concerns a much broader set of substitution tilings

Topics: Mathematics - Dynamical Systems, Mathematical Physics, Mathematics - Metric Geometry
Year: 2011
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