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Conformal Geometry and Dynamics

Published by the American Mathematical Society since 1997, the purpose of this electronic-only journal is to provide a forum for mathematical work in related fields broadly described as conformal geometry and dynamics. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.49.

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Constructing rational maps from subdivision rules
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by J. W. Cannon, W. J. Floyd, R. Kenyon and W. R. Parry
Conform. Geom. Dyn. 7 (2003), 76-102
Published electronically: July 28, 2003


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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Bibliographic Information
  • J. W. Cannon
  • Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
  • Email:
  • W. J. Floyd
  • Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061
  • MR Author ID: 67750
  • Email:
  • R. Kenyon
  • Affiliation: Laboratoire de Topologie, Université Paris-Sud, Bat. 425, 91405 Orsay Cedex-France
  • MR Author ID: 307971
  • W. R. Parry
  • Affiliation: Department of Mathematics, Eastern Michigan University, Ypsilanti, Michigan 48197
  • MR Author ID: 136390
  • Email:
  • 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.
  • © Copyright 2003 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 7 (2003), 76-102
  • MSC (2000): Primary 37F10, 52C20; Secondary 57M12
  • DOI:
  • MathSciNet review: 1992038