The dynamical properties of Penrose tilings

The set of Penrose tilings, when provided with a natural compact metric topology, becomes a strictly ergodic dynamical system under the action of by translation. We show that this action is an almost 1:1 extension of a minimal action by rotations on , i.e., it is an generalization of a Sturmian dynamical system. We also show that the inflation mapping is an almost 1:1 extension of a hyperbolic automorphism on . The local topological structure of the set of Penrose tilings is described, and some generalizations are discussed.