Self-affine tiling via substitution dynamical systems and Rauzy fractals

Abstract

In this paper we show that a class of sets known as the Rauzy fractals, which are constructed via substitution dynamical systems, give rise to self-affine multi-tiles and self-affine tilings. This provides an efficient and unconventional way for constructing aperiodic self-affine tilings. Our result also leads to a proof that a Rauzy fractal R associated with a primitive and unimodular Pisot substitution has nonempty interior.