Topology of planar self-affine tiles with collinear digit set

Abstract

We consider the self-affine tiles with collinear digit set defined as follows. Let $A,B\in \mathbb{Z}$ satisfy $|A|\le B\ge 2$ and $M\in {\mathbb{Z}}^{2\times 2}$ be an integral matrix with characteristic polynomial $x^2+Ax+B$. Moreover, let $\mathcal{D}=\{0,v,2v,\dots ,(B-1)v\}$ for some $v\in {\mathbb{Z}}^2$ such that $v,Mv$ are linearly independent. We are interested in the topological properties of the self-affine tile $\mathcal{T}$ defined by $M\mathcal{T}={\bigcup }_{d\in \mathcal{D}}(\mathcal{T}+d)$. Lau and Leung proved that $\mathcal{T}$ is homeomorphic to a closed disk if and only if $2|A|\le B+2$. In particular, $\mathcal{T}$ has no cut point. We prove here that $\mathcal{T}$ has a cut point if and only if $2|A|\ge B+5$. For $2|A|-B\in \{3,4\}$, the interior of $\mathcal{T}$ is disconnected and the closure of each connected component of the interior of $\mathcal{T}$ is homeomorphic to a closed disk.