Topology of a class of p2-crystallographicreplication tiles

We study the topological properties of a class of planar crystallographic replication tiles. Let M∈Z2×2 be an expanding matrix with characteristic polynomial x2+Ax+B (A,B∈Z, B≥2) and v∈Z2 such that (v,Mv) are linearly independent. Then the equation MT+B−12v=T∪(T+v)∪(T+2v)∪⋯∪(T+(B−2)v)∪(−T)defines a unique nonempty compact set T satisfying To¯=T. Moreover, T tiles the plane by the crystallographic group p2 generated by the π-rotation and the translations by integer vectors. It was proved by Leung and Lau in the context of self-affine lattice tiles with collinear digit set that T∪(−T) is homeomorphic to a closed disk if and only if 2|A|