Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [ACG11] J. Aliste-Prieto, D. Coronel and J. M. Gambaudo, Linearly repetitive Delone sets are rectifiable, arXiv:1103.5423v3 (2011).
  • [BK98] D. Burago and B. Kleiner, Separated nets in Euclidean space and Jacobians of biLipschitz maps, Geom. Func. Anal. 8 (1998), no. 2, 273-282. MR 1616135 (99d:26018)
  • [BK02] D. Burago and B. Kleiner, Rectifying separated nets, Geom. Func. Anal. 12 (2002), 80-92. MR 1904558 (2003h:26020)
  • [McM98] C. T. McMullen, Lipschitz maps and nets in Euclidean space, Geom. Func. Anal. 8 (1998), no. 2, 304-314. MR 1616159 (99e:58017)
  • [Ro04] E. A. Robinson, Jr. Symbolic dynamics and tilings of $ \mathbb{R}^d$, Proc. Sympos. Appl. Math. 60 (2004), 81-119. MR 2078847 (2005h:37036)
  • [S11] Y. Solomon, Substitution tilings and separated nets with similarities to the integer lattice, Israel J. Math. 181 (2011), 445-460. MR 2773052 (2012c:37030)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46-XX, 51-XX

Retrieve articles in all journals with MSC (2010): 46-XX, 51-XX

Additional Information

Yaar Solomon
Affiliation: Department of Mathematics, Ben-Gurion University of The Negev, Beer-Sheva, Israel

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

American Mathematical Society