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