Topological properties of a class of cubic Rauzy fractals

Abstract

We consider the substitution $\sigma_{a, b}$ defined by \begin{align*} \sigma_{a, b}\colon & 1 \mapsto \underbrace{1\ldots 1}_{a}2{,} \\ & 2 \mapsto \underbrace{1\ldots 1}_{b}3{,} \\ & 3 \mapsto 1 \end{align*} with $a \geq b \geq 1$. The shift dynamical system induced by $\sigma_{a, b}$ 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 $2b-a\leq 3$. This solves a conjecture of Shigeki Akiyama posed in 1997. To this effect, we construct a Hölder continuous parametrization $C_{a, b}\colon \mathbb{S}^{1} \to \partial\mathcal{T}_{a, b}$ of the boundary of $\mathcal{T}_{a, b}$. As a by-product, this parametrization gives rise to an increasing sequence of polygonal approximations of $\partial\mathcal{T}_{a, b}$, whose vertices lye on $\partial\mathcal{T}_{a, b}$ and have algebraic pre-images in the parametrization.