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 R(2) by translation. We show that this action is an almost 1:1 extension of a minimal R(2) action by rotations on T-4, i.e., it is an R(2) generalization of a Sturmian dynamical system. We also show that the inflation mapping is an almost 1:1 extension of a hyperbolic automorphism on T-4. The local topological structure of the set of Penrose tilings is described, and some generalizations are discussed.