Topological properties of a class of cubic Rauzy fractals

We consider the substitution ${\sigma }_{a,b}$ defined by $$\begin{array}{rlcl}{\sigma }_{a,b}:& 1& \mapsto & \underset{a}{\underbrace{1\dots 1}}2\\ & 2& \mapsto & \underset{b}{\underbrace{1\dots 1}}3\\ & 3& \mapsto & 1\end{array}$$ with $a\geq b\geq 1$\mathrm{.}Theshiftdynamicalsysteminducedby$\sigma_{a,b}$ is measure theoretically isomorphic to an exchange of three domains on a compact tile ${\mathcal{T}}_{a,b}$ with fractal boundary. We prove that ${\mathcal{T}}_{a,b}$ is homeomorphic to the closed disk iff $2b-a\le 3$. This solves a conjecture of Shigeki Akiyama posed in 1997. To this effect, we construct a H"older continuous parametrization $C_{a,b}:{\mathbb{S}}^1\to \mathrm{\partial }{\mathcal{T}}_{a,b}$ of the boundary of ${\mathcal{T}}_{a,b}$. As a by-product, this parametrization gives rise to an increasing sequence of polygonal approximations of $\partial $\mathcal{T}${a,b}$,whoseverticeslyeon$\partial $\mathcal{T}${a,b}$ and have algebraic pre-images in the parametrization.