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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
Published electronically: June 27, 2019
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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

K. Stephenson
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996

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