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

We consider a class of planar self-affine tiles \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ T = M^{ - 1} \cup _{a \in \mathcal{D}} (T + a) $$\end{document} generated by an expanding integral matrix M and a collinear digit set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{D} $$\end{document} as follows: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ M = \left( \begin{gathered} 0 - B \hfill \\ 1 - A \hfill \\ \end{gathered} \right),\mathcal{D} = \left\{ {\left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ \end{gathered} \right),...,\left( \begin{gathered} |B| - 1 \hfill \\ 0 \hfill \\ \end{gathered} \right)} \right\} $$\end{document}. We give a parametrization \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb{S}^1 \to \partial T $$\end{document} 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|.