The dynamical properties of Penrose tilings

Abstract

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