Constructing rational maps from subdivision rules

Cannon, J.; Floyd, W.; Kenyon, R.; Parry, W.

doi:10.1090/S1088-4173-03-00082-1

Constructing rational maps from subdivision rules
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by J. W. Cannon, W. J. Floyd, R. Kenyon and W. R. Parry PDF

Conform. Geom. Dyn. 7 (2003), 76-102 Request permission

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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Lecture notes