NDLI: Substitution Delone Sets

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Abstract	. Substitution Delone set families are families of Delone sets X =(X1, . . ., Xn) which satisfy the inflation functional equation X_i = \bigvee_{j=1}^m ( \mbox{\phvr A} (X_j) + {\cal D}_ $ij$ ) , \qquad 1 \le i \le m, in which A is an expanding matrix, i.e., all of the eigenvalues of A fall outside the unit circle. Here the D ij are finite sets of vectors in R d and V denotes union that counts multiplicity. This paper characterizes families X =(X 1 , . . ., X n ) that satisfy an inflation functional equation, in which each X i is a multiset (set with multiplicity) whose underlying set is discrete. It then studies the subclass of Delone set solutions, and gives necessary conditions on the coefficients of the inflation functional equation for such solutions X to exist. It relates Delone set solutions to a narrower subclass of solutions, called self-replicating multi-tiling sets, which arise as tiling sets for self-replicating multi-tilings.
Starting Page	175
Ending Page	209
Page Count	35
File Format	PDF
ISSN	01795376
Journal	Discrete & Computational Geometry
Volume Number	29
Issue Number	2
e-ISSN	14320444
Language	English
Publisher	Springer-Verlag
Publisher Date	2003-01-22
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Theoretical Computer Science Computational Theory and Mathematics Geometry and Topology

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Substitution Delone Sets