Boundary parametrization of planar self-affine tiles with collinear digit set

Abstract

We consider a class of planar self-affine tiles $T = M^{ - 1} \cup _{a \in \mathcal{D}} (T + a)$ generated by an expanding integral matrix M and a collinear digit set $\mathcal{D}$ as follows:

$$ M = \left( \begin{gathered} 0 - B \hfill \\ 1 - A \hfill \\ \end{gathered} \right),\mathcal{D} = \left\{ {\left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ \end{gathered} \right),...,\left( \begin{gathered} |B| - 1 \hfill \\ 0 \hfill \\ \end{gathered} \right)} \right\} $$

. We give a parametrization $\mathbb{S}^1 \to \partial T$ of the boundary of T with the following standard properties. It is Hölder continuous and associated with a sequence of simple closed polygonal approximations whose vertices lie on ∂T and have algebraic preimages. We derive a new proof that T is homeomorphic to a disk if and only if 2|A| ⩽ |B + 2|.