Duality of Model Sets Generated by Substitutions

The nature of this paper is twofold: On one hand, we will give a short introduction and overview of the theory of model sets in connection with nonperiodic substitution tilings and generalized Rauzy fractals. On the other hand, we will construct certain Rauzy fractals and a certain substitution tiling with interesting properties, and we will use a new approach to prove rigorously that the latter one arises from a model set. The proof will use a duality principle which will be described in detail for this example. This duality is mentioned as early as 1997 by Gelbrich in the context of iterated function systems, but it seems to appear nowhere else in connection with model sets.