  Model sets (also called cut and project sets) are generalizations of
lattices. Here we show how the self-similarities of model sets are a natural
replacement for the group of translations of a lattice. This leads us to the
concept of averaging operators and invariant densities on model sets. We prove
that invariant densities exist and that they produce absolutely continuous
invariant measures in internal space. We study the invariant densities and
their relationships to diffraction, continuous refinement operators, and
Hutchinson measures.
