Structure of Planar Integral Self-Affine Tilings

For a self-affine tile in R{double-struck} ² generated by an expanding matrix M ₂ (Z) and an integral consecutive collinear digit set D, Leung and Lau [Trans. Amer. Math. Soc. 359, 3337-3355 (2007).] provided a necessary and sufficient algebraic condition for it to be disklike. They also characterized the neighborhood structure of all disklike tiles in terms of the algebraic data A and D. In this paper, we completely characterize the neighborhood structure of those non-disklike tiles. While disklike tiles can only have either six or eight edge or vertex neighbors, non-disklike tiles have much richer neighborhood structure. In particular, other than a finite set, a Cantor set, or a set containing a nontrivial continuum, neighbors can intersect in a union of a Cantor set and a countable set.