Self-similar lattice tilings and subdivision schemes

Let M is an element of Z(8x8) be a dilation matrix and let D subset of Z(8) be a complete set of representatives of distinct cosets of Z(8)/MZ(8). The self-similar tiling associated with M and D is the subset of R-s given by T(M,D) = {Sigma (infinity)(j=1) M(-j)alpha (j) : alpha (j) is an element of D}. The purpose of this paper is to characterize self-similar lattice tili ngs, i.e., tilings T(M,D) which have Lebesgue measure one. In particular, i t is shown that T(M,D) is a lattice tiling of and only if there is no nonem pty finite set Lambda subset of Z(8)\(D-D) such that M-1 ((D-D) + Lambda) b oolean ANDZ(8) subset of Lambda. this set Lambda can be restricted to be co ntained in a finite set K depending only on M and D. We also give a new pro of for the fact that T(M,D) is a lattice tiling if and only if U-n=1(infini ty) (Sigma (n-1)(j=0) M-j (D-D)) = Z(8). two approaches are provided, one b ased on scrambling matrices and the other based on primitive matrices. Thes e will follow from the characterization of subdivision schemes associated w ith nonnegative masks in terms of finite powers of finite matrices, without computing eigenvalues or spectral radii. Our characterization shows that t he convergence of the subdivision scheme with a nonnegative mask depends on ly on location of its positive coefficients.