Self-Affine Sets and Graph-Directed Systems

Abstract

Abstract. A self-affine set in R n is a compact set T with A(T ) = d∈D (T +d) whereA is an expanding n × n matrix with integer entries andis 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 ∈ 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 o = ∅). 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.

Self-affine measures and vector-valued representations

“…Hence we can determine whether K • = ∅ in at most 2 N steps. It is known that the Lebesgue measure of such a K is a rational number [14], and is an integer if K is a tile [19]. We prove Theorem 1.3.…”

“…This section is devoted to the calculation of the Lebesgue measure and Hausdorff dimension of integral selfaffine sets. These problems have been investigated in [37] and [14]. We will make use of the matrix representation of Section 3 to give an alternative approach, which unifies the considerations with the measures and functions and seems to be simpler.…”

“…This question has been studied in some detail in [14] and an algorithm was given there (see also [30], [37] for self-affine tiles). We make use of the main theorem to give a unified and more satisfactory criterion as follows:…”

On the boundaries of self-similar tiles in $\mathbb {R}^1$

“…Let e be the (m + 1)-vector with all entries equal to 1 and let e i be an (m + 1)-vector with the ith entry 1 and zero otherwise. It is not difficult to prove that #S n = e t 0 B n e where S n is used in (3.1) [HLR,Proposition 4.1], which satisfies…”

Integral self-affine sets with positive Lebesgue measures

“…From now on we omit "integral" in those notations for simplicity. When the Lebesgue measure of T , denoted by L(T ), is larger than zero, we call T a self-affine region ([6], [7]), which Partially supported by NNSF of China.…”