Combinatorial and Computational aspects of Tilings

After asking (and answering in the affirmative) the fundamental question whether a given substitution tiling is a model set, one might like to explore this connection further. For example, the (expansive) substitution on the direct space G corresponds to a (contractive) iterated function system on the internal space H, and we can even regard the substitution as a hyperbolic toral automorphism on the torus (G \times H)/L (where L is the lattice in the cut and project scheme). In fact, one can even construct a Markov partition of the torus which then yields geometrical insight into the dynamical system given by the action of the substitution.
We will give examples of these notions for (one-dimensional) non-unimodular Pisot substitutions, but also for two-dimensional examples like the Penrose tiling for which we will present several Markov partititons of the corresponding torus.