Topology of planar self-affine tiles with collinear digit set

TL;DR

This paper investigates the topological structure of certain self-affine tiles with collinear digit sets, establishing conditions under which these tiles have cut points or disconnected interiors, thus advancing understanding of their geometric properties.

Contribution

It characterizes when self-affine tiles with collinear digit sets have cut points or disconnected interiors, extending previous results on their topological structure.

Findings

$ ext{T}$ has a cut point if and only if } 2|A| ext{ is at least } B+5

Interior of $ ext{T}$ is disconnected for $2|A|-B ext{ in } oxed{ ext{3,4}}$

Closure of each interior component is homeomorphic to a closed disk

Abstract

We consider the self-affine tiles with collinear digit set defined as follows. Let $A,B\in\mathbb{Z}$ satisfy $|A|\leq B\geq 2$ and $M\in\mathbb{Z}^{2\times2}$ be an integral matrix with characteristic polynomial $x^2+Ax+B$. Moreover, let $\mathcal{D}=\{0,v,2v,\ldots,(B-1)v\}$ for some $v\in\mathbb{Z}^2$ such that $v,M v$ 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 $2|A|\leq B+2$. In particular, $\mathcal{T}$ has no cut point. We prove here that $\mathcal{T}$ has a cut point if and only if $2|A|\geq B+5$. 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.