Self-affine sets and graph-directed systems

A self-affine set in R-n is a compact set T with A(T) = U-dis an element ofD(T+d) where A is an expanding n x n matrix with integer entries and D = {d(1), d(2), . . ., d(N)} subset of Z(n) is an N-digit set. For the case N = \det(A)\ the set T has been studied in great detail in the context of self-affine tiles. Our main interest in this paper is to consider the case N > \det(A)\, but the theorems and proofs apply to all the N. The self-affine sets arise naturally in fractal geometry and, moreover, they are the support of the scaling functions in wavelet theory. The main difficulty in studying such sets is that the pieces T + d, d is an element of D, overlap and it is harder to trace the iteration. For this we construct a new graph-directed system to determine whether such a set T will have a nonvoid interior, and to use the system to calculate the dimension of T or its boundary (if T-0 not equal circle divide). By using this setup we also show that the Lebesgue measure of such T is a rational number, in contrast to the case where, for a self-affine tile, it is an integer.