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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Functions of substitution tilings as a Jacobian
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by Yaar Solomon PDF
Proc. Amer. Math. Soc. 141 (2013), 3853-3863 Request permission

Abstract:

A tiling $\tau$ of the Euclidean space gives rise to a function $f_\tau$, which is constant $1/\left |{T}\right |$ on the interior of every tile $T$. In this paper we give a local condition to know when $f_\tau$, which is defined by a primitive substitution tiling of the Euclidean space, can be realized as a Jacobian of a biLipschitz homeomorphism of $\mathbb {R}^d$. As an example we show that this condition holds for any star-shaped substitution tiling of $\mathbb {R}^2$. In particular, the result holds for any Penrose tiling.
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Additional Information
  • Yaar Solomon
  • Affiliation: Department of Mathematics, Ben-Gurion University of The Negev, Beer-Sheva, Israel
  • Email: yaars@bgu.ac.il
  • Received by editor(s): August 17, 2010
  • Received by editor(s) in revised form: January 12, 2012
  • Published electronically: July 16, 2013
  • Additional Notes: This research was supported by the Israel Science Foundation, grant No. 190/08
  • Communicated by: Michael Wolf
  • © Copyright 2013 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 141 (2013), 3853-3863
  • MSC (2010): Primary 46-XX, 51-XX
  • DOI: https://doi.org/10.1090/S0002-9939-2013-11663-1
  • MathSciNet review: 3091774