The dynamical properties of Penrose tilings

Robinson, E., Jr.

doi:10.1090/S0002-9947-96-01640-6

The dynamical properties of Penrose tilings
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by E. Arthur Robinson Jr. PDF

Trans. Amer. Math. Soc. 348 (1996), 4447-4464 Request permission

Abstract:

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.

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