Structure of Planar Integral Self-Affine Tilings
For a self-affine tile in R2 generated by an expanding matrix A∈M2(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.
Deng, Da-Wen, Tao Jiang, Sze-Man Ngai.
"Structure of Planar Integral Self-Affine Tilings."
Mathematische Nachrichten, 285 (4): 447-475.