On Disk-like Self-affine Tiles Arising from Polyominoes

Abstract

In this paper we study a class of plane self-affine lattice tiles that are defined using polyominoes. In particular, we characterize which of these tiles are homeomorphic to a closed disk. It turns out that their topological structure depends very sensitively on their defining parameters.

In order to achieve our results we use an algorithm of Scheicher and the second author which allows to determine neighbors of tiles in a systematic way as well as a criterion of Bandt and Wang, with that we can check disk-likeness of a self-affine tile by analyzing the set of its neighbors.