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Shape convergence for aggregate tiles in conformal tilings


Authors: R. Kenyon and K. Stephenson
Journal: Proc. Amer. Math. Soc. 147 (2019), 4275-4287
MSC (2010): Primary 37F30; Secondary 05B45
DOI: https://doi.org/10.1090/proc/14453
Published electronically: June 27, 2019
MathSciNet review: 4002541
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Abstract: Given a polygonal substitution tiling $T$ of the plane with subdivision rule $\tau$, we study the conformal tilings $T_n$ associated with $\tau ^nT$. We prove, under the assumption that there is somewhere a non-real similarity mapping one tile to another, that aggregate tiles within $T_n$ converge in shape as $n\rightarrow \infty$ to their associated Euclidean tiles in $T$.


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Additional Information

R. Kenyon
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
MR Author ID: 307971
Email: rkenyon@math.brown.edu

K. Stephenson
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996
MR Author ID: 216579
Email: kstephe2@utk.edu

Received by editor(s): March 16, 2017
Received by editor(s) in revised form: October 30, 2017, and October 22, 2018
Published electronically: June 27, 2019
Additional Notes: Research of the first author was supported by NSF grants DMS-1612668, DMS-1713033 and the Simons Foundation award 327929.
Research of the second author was supported by Simons Foundation award 208523.
Communicated by: Ken Bromberg
Article copyright: © Copyright 2019 American Mathematical Society