Linearly repetitive Delone sets are rectifiable

2011/03/28 by José Aliste‐Prieto, Aliste-Prieto, J., Daniel Coronel +3 · 1 citation
Materials Science · #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Metric Geometry (math.MG) #Quasicrystal Structures and Properties #Solidification and crystal growth phenomena

paper · pdf · doi:10.48550/arxiv.1103.5423

openalex publication_date 2011/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Metallic Phase with Long-Range Orientational Order and No Translational Symmetry

Cited by

• Fast repetitivity in non-rectifiable Delone sets

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