Multi-Component Model Sets and Invariant Densities

Model sets (also called cut and project sets) are generalizations of lattices, and multi-component model sets are generalizations of lattices with colourings. In this paper, we study self-similarities of multi-component model sets. The main point may be simply summarized: whenever there is a self-similarity, there are also naturally related density functions. As in the case of ordinary model sets, we show that invariant densities exist and that they produce absolutely continuous invariant measures in internal space, these features now appearing in matrix form. We establish a close connection between the theory of invariant densities and the spectral theory of matrix continuous refinement operators.