Summary: 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.