Structure of planar integral self‐affine tilings

Abstract

AbstractFor a self‐affine tile in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {R}^2$\end{document} generated by an expanding matrix \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$A\in M_2(\mathbf {Z})$\end{document} and an integral consecutive collinear digit set \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal D}$\end{document}, 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 \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal D}$\end{document}. 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.