  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.
