Linearly Repetitive Delone Sets
Résumé
The 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) < LR for all 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) < c(d)R for some 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 Euclidean
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