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 R^2 by translation. We show that this action is an almost 1:1 extension of a minimal R 2 R^2 action by rotations on T 4 T^4 , i.e., it is an R 2 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 T^4 . The local topological structure of the set of Penrose tilings is described, and some generalizations are discussed.
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E. Powell Robinson (1996) studied this question.