Structure of Planar Integral Self-Affine Tilings

Document Type

Article

Publication Date

2012

Publication Title

Mathematische Nachrichten

DOI

10.1002/mana.201000061

Abstract

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.

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