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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

Functions of substitution tilings as a Jacobian


Author: Yaar Solomon
Journal: Proc. Amer. Math. Soc. 141 (2013), 3853-3863
MSC (2010): Primary 46-XX, 51-XX
Published electronically: July 16, 2013
MathSciNet review: 3091774
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Abstract: A tiling $ \tau $ of the Euclidean space gives rise to a function $ f_\tau $, which is constant $ 1/\left \vert{T}\right \vert$ 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

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11663-1
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
Article copyright: © Copyright 2013 American Mathematical Society