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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Email: robinson@math.gwu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01640-6
PII: S 0002-9947(96)01640-6
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