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Transactions of the American Mathematical Society

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The dynamical properties of Penrose tilings

Author: E. Arthur Robinson Jr.
Journal: Trans. Amer. Math. Soc. 348 (1996), 4447-4464
MSC (1991): Primary 28D05; Secondary 28D20
MathSciNet review: 1355301
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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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Additional Information

E. Arthur Robinson Jr.
Affiliation: Department of Mathematics, The George Washington University, Washington, D.C. 20052

Keywords: Tilings, topological dynamics, almost periodicity
Received by editor(s): May 13, 1995
Additional Notes: Partially supported by a George Washington University Committee on Research UFF grant and by NSF grant DMS-9303498
Article copyright: © Copyright 1996 American Mathematical Society