Substitution Delone Sets

Details Substitution Delone Sets Substitution Delone Sets

2001-10-19

arxiv.org/abs/math/0110222v2

Discrete & Computational Geometry 29, No. 2 (2003), 175--209.

Mathematics

Metric Geometry

34 pages, latex file; some results in Sect 5 rearranged and theorems reformulated

Scientific paper

This paper addresses the problem of describing aperiodic discrete structures that have a self-similar or self-affine structure. Substitution Delone set families are families of Delone sets (X_1, ..., X_n) in R^d that satisfy an inflation functional equation under the action of an expanding integer matrix in R^d. This paper studies such functional equation in which each X_i is a discrete multiset (a set whose elements are counted with a finite multiplicity). It gives necessary conditions on the coefficients of the functional equation for discrete solutions to exist. It treats the case where the equation has Delone set solutions. Finally, it gives a large set of examples showing limits to the results obtained.

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