Structure of planar integral self-affine tilings

For a self-affine tile in R-2 generated by an expanding matrix A. 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. (C) 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim