Disk-like tiles and self-affine curves with noncollinear digits

Abstract

Let A ∈ M n (ℤ) be an expanding matrix, D C ℤ n a digit set and T = T(A, D) the associated self-affine set. It has been asked by Grochenig and Haas (1994) that given any expanding matrix A ∈ M 2 (Z), whether there exists a digit set such that T is a connected or disk-like (i.e., homeomorphic to the closed unit disk) tile. With regard to this question, collinear digit sets have been studied in the literature. In this paper, we consider noncollinear digit sets and show the existence of a noncollinear digit set corresponding to each expanding matrix such that T is a connected tile. Moreover, for such digit sets, we give necessary and sufficient conditions for T to be a disk-like tile.