Structure of Planar Integral Self-affine Tilings

Da-Wen Deng, Xiangtan University
Tao Jiang, Xiangtan University
Sze-Man Ngai, Georgia Southern University


For a self-affine tile in generated by an expanding matrix and an integral consecutive collinear digit set , 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 . 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.