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

Abstract

Consider an \(n\times n\) matrix A such that \(|\lambda_j|>1\) for all eigenvalues \(\lambda_j\) of A and a set of \(n\)-vectors \(\{d_1,\dots,d_q\}\) with integer elements. Define affine maps \(S_j(x)=A^{-1}(x+d_j)\). There exists a unique nonempty compact set \(T\) such that \(T=\bigcup S_j(T)\). This set \(T\) (called self-affine) is the attractor of the iterated function system \(\{S_j\}\). The author gives conditions under which the set \(T\) is homeomorphic to a closed disk.