Topological structure of a class of planar self-affine sets

Let T=T(A,D⁎)T=T(A,D⁎) be a disk-like Z2Z2-tile generated by an expanding 2×22×2 matrix A   and a digit set D⁎⊂Z2D⁎⊂Z2. We study the subset F of T   defined by AF=F+DAF=F+D, where D⊊D⁎D⊊D⁎ is a sub-digit set. By studying a periodic extension H=F+Z2H=F+Z2, we classify F into three types according to their topological properties, which generalizes a result of Lau et al. [13]. We also provide some simple criteria for such classification.