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

 

Generalized $ \beta$-expansions, substitution tilings, and local finiteness


Authors: Natalie Priebe Frank and E. Arthur Robinson Jr.
Journal: Trans. Amer. Math. Soc. 360 (2008), 1163-1177
MSC (2000): Primary 52C20; Secondary 37B50.
Published electronically: October 23, 2007
MathSciNet review: 2357692
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Abstract: For a fairly general class of two-dimensional tiling substitutions, we prove that if the length expansion $ \beta$ is a Pisot number, then the tilings defined by the substitution must be locally finite. We also give a simple example of a two-dimensional substitution on rectangular tiles, with a non-Pisot length expansion $ \beta$, such that no tiling admitted by the substitution is locally finite. The proofs of both results are effectively one-dimensional and involve the idea of a certain type of generalized $ \beta$-transformation.


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Additional Information

Natalie Priebe Frank
Affiliation: Department of Mathematics, Vassar College, Box 248, Poughkeepsie, New York 12604
Email: nafrank@vassar.edu

E. Arthur Robinson Jr.
Affiliation: Department of Mathematics, George Washington University, Washington, DC 20052
Email: robinson@gwu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04527-8
PII: S 0002-9947(07)04527-8
Keywords: Substitution sequence, self-similar tiling
Received by editor(s): June 6, 2005
Published electronically: October 23, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.