Repetitive Delone Sets and Quasicrystals

This paper considers the problem of characterizing the simplest discrete point sets that are aperiodic, using invariants based on topological dynamics. A Delone set whose patch-counting function N(T), for radius T, is finite for all T is called repetitive if there is a function M(T) such that every ball of radius M(T)+T contains a copy of each kind of patch of radius T that occurs in the set. This is equivalent to the minimality of an associated topological dynamical system with R^n-action. There is a lower bound for M(T) in terms of N(T), namely N(T) = O(M(T)^n), but no general upper bound. The complexity of a repetitive Delone set can be measured by the growth rate of its repetitivity function M(T). For example, M(T) is bounded if and only if the set is a crystal. A set is called is linearly repetitive if M(T) = O(T) and densely repetitive if M(T) = O(N(T))^{1/n}). We show that linearly repetitive sets and densely repetitive sets have strict uniform patch frequencies, i.e. the associated topological dynamical system is strictly ergodic. It follows that such sets are diffractive. In the reverse direction, we construct a repetitive Delone set in R^n which has M(T) = O(T(log T)^{2/n}(log log log T)^{4/n}), but does not have uniform patch frequencies. Aperiodic linearly repetitive sets have many claims to be the simplest class of aperiodic sets, and we propose considering them as a notion of "perfectly ordered quasicrystal".Comment: To appear in "Ergodic Theory and Dynamical Systems" vol.23 (2003). 37 pages. Uses packages latexsym, ifthen, cite and files amssym.def, amssym.te

Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies

Delone sets of finite local complexity in Euclidean space are investigated. We show that such a set has patch counting and topological entropy 0 if it has uniform cluster frequencies and is pure point diffractive. We also note that the patch counting entropy is 0 whenever the repetitivity function satisfies a certain growth restriction.Comment: 16 pages; revised and slightly expanded versio

Linearly Repetitive Delone Sets

International audienceThe notion of linearly recurrent subshift has been introduced in [Du, DHS] to study the relations between the substitutive dynamical systems and the stationary dimension groups. In an independent way, the similar notion of linearly repetitive Delone sets of the Euclidean space R d appears in [LP1]. For a Delone set X of R d , the repetitivity function M X (R) is the least M (possibly infinite) such that every closed ball B of radius M intersected with X contains a translated copy of any patch with diameter smaller than 2R. A Delone set X is said linearly repetitive if there exists a constant L such that M X (R) 0. Observe that we can assume that the constant L is greater than 1. According to the following theorem, the slowest growth for the repetitivity function of an aperiodic Delone set is linear. Theorem 1 ([LP1] Theo. 2.3). Let d ≥ 1. There exists a constant c(d) > 0 such that for any Delone set X of R d such that M X (R) 0, then X has a non-zero period. Even more, if for some R, M X (R) < 4 3 R, then the Delone set X is a crystal i.e. has d independent periods (Theo. 2.2 [LP1]). The classical examples of aperiodic Delone systems (i.e. arising form substitution) are linearly repetitive. Lemma 2 ([So2] Lem. 2.3). A primitive self similar tiling is linearly repetitive. In many senses that we will not specify, the family of linearly repetitive Delone sets is small inside the family of all the Delone sets of the Euclidea

SPECTRAL TRIPLES AND APERIODIC ORDER

International audienceWe construct spectral triples for compact metric spaces (X, d). This provides us with a new metric ¯ ds on X. We study its relation with the original metric d. When X is a subshift space, or a discrete tiling space, and d satisfies certain bounds we advocate that the property of ¯ ds and d to be Lipschitz equivalent is a characterization of high order. For episturmian subshifts, we prove that ¯ ds and d are Lipschitz equivalent if and only if the subshift is repulsive (or power free). For Sturmian subshifts this is equivalent to linear recurrence. For repetitive tilings we show that if their patches have equi-distributed frequencies then the two metrics are Lipschitz equivalent. Moreover, we study the zeta-function of the spectral triple and relate its abscissa of convergence to the complexity exponent of the subshift or the tiling. Finally, we derive Laplace operators from the spectral triples and compare our construction with that of Pearson and Bellissard