Topological properties of a class of cubic Rauzy fractals

Abstract

We consider the substitution $\sigma_{a,b}$ defined by $$\begin{array}{rlcl} \sigma_{a,b}: & 1 & \mapsto & \underbrace{1\ldots 1}_{a}2 \\ & 2 & \mapsto & \underbrace{1\ldots 1}_{b}3 \\ & 3 & \mapsto & 1 \end{array} $$ 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 H\"older continuous parametrization $C_{a,b}:\mathbb{S}^1\to\partial \mathcal{T}_{a,b}$ 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.