Topological Properties of a Class of Cubic Rauzy Fractals

with a b 1. The shift dynamical system induced by a,b is measure theoretically isomorphic to an exchange of three domains on a compact tile Ta,b with fractal boundary. We prove that Ta,b is homeomorphic to the closed disk iff 2b a 3. This solves a conjecture of Shigeki Akiyama posed in 1997. To this effect, we construct a Hölder continuous parametrization Ca,b W S1 ! Ta,b of the boundary of Ta,b. As a by-product, this parametrization gives rise to an increasing sequence of polygonal approximations of Ta,b, whose vertices lye on Ta,b and have algebraic pre-images in the parametrization.