Substitution Delone sets

Abstract

Substitution Delone set families are families of Delone sets χ = (X<inf>1</inf>, . . . , X<inf>n</inf>) which satisfy the inflation functional equation X<inf>i</inf> = V<inf>j=1</inf>^m(A(X<inf>j</inf>) + D<inf>ij</inf>), 1 ≤ i ≤ m, in which A is an expanding matrix, i.e., all of the eigenvalues of A fall outside the unit circle. Here the D<inf>ij</inf> are finite sets of vectors in ℝ^d and V denotes union that counts multiplicity. This paper characterizes families χ = (X<inf>1</inf>, . . . , X<inf>n</inf>) that satisfy an inflation functional equation, in which each X<inf>i</inf> 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 χ 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.