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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Generalized $\beta$-expansions, substitution tilings, and local finiteness
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by Natalie Priebe Frank and E. Arthur Robinson Jr. PDF
Trans. Amer. Math. Soc. 360 (2008), 1163-1177 Request permission

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.
References
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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
  • Received by editor(s): June 6, 2005
  • Published electronically: October 23, 2007
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 1163-1177
  • MSC (2000): Primary 52C20; Secondary 37B50
  • DOI: https://doi.org/10.1090/S0002-9947-07-04527-8
  • MathSciNet review: 2357692