## 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 - DOI: https://doi.org/10.1090/S1088-4173-03-00082-1
- Published electronically: July 28, 2003
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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.## References

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## Bibliographic Information

**J. W. Cannon**- Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
- Email: cannon@math.byu.edu
**W. J. Floyd**- Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061
- MR Author ID: 67750
- Email: floyd@math.vt.edu
**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: walter.parry@emich.edu
- 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: https://doi.org/10.1090/S1088-4173-03-00082-1
- MathSciNet review: 1992038