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Conformal Geometry and Dynamics

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Constructing rational maps from subdivision rules

Authors: J. W. Cannon, W. J. Floyd, R. Kenyon and W. R. Parry
Journal: Conform. Geom. Dyn. 7 (2003), 76-102
MSC (2000): Primary 37F10, 52C20; Secondary 57M12
Published electronically: July 28, 2003
MathSciNet review: 1992038
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Abstract: Suppose $\mathcal{R}$ is an orientation-preserving finite subdivision rule with an edge pairing. Then the subdivision map $\sigma_{\mathcal{R}}$ is either a homeomorphism, a covering of a torus, or a critically finite branched covering of a 2-sphere. If $\mathcal{R}$ has mesh approaching $0$ and $S_{\mathcal{R}}$ is a 2-sphere, it is proved in Theorem 3.1 that if $\mathcal{R}$ is conformal, then $\sigma_{\mathcal{R}}$ is realizable by a rational map. Furthermore, a general construction is given which, starting with a one-tile rotationally invariant finite subdivision rule, produces a finite subdivision rule $\mathcal{Q}$ with an edge pairing such that $\sigma_{\mathcal{Q}}$ is realizable by a rational map.

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Additional Information

J. W. Cannon
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602

W. J. Floyd
Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061

R. Kenyon
Affiliation: Laboratoire de Topologie, Université Paris-Sud, Bat. 425, 91405 Orsay Cedex-France

W. R. Parry
Affiliation: Department of Mathematics, Eastern Michigan University, Ypsilanti, Michigan 48197

Keywords: Finite subdivision rule, rational map, conformality
Received by editor(s): September 5, 2001
Received by editor(s) in revised form: April 4, 2003
Published electronically: July 28, 2003
Additional Notes: This research was supported in part by NSF grants DMS-9803868, DMS-9971783, and DMS-10104030.
Article copyright: © Copyright 2003 American Mathematical Society