## Constructing rational maps from subdivision rules

###
J. W. Cannon, W. J. Floyd, R. Kenyon, and W. R. Parry

March 28, 2003

### Abstract

Suppose $\mathcal{R}$ is an orientation-preserving finite subdivision
rule with an edge pairing. Then the subdivision map
$\subm_{\mathcal{R}}$ 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~\ref{thm:conffsr} that if $\cR$ is
conformal then $\subm_{\mathcal{R}}$ 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
$\subm_{\mathcal{Q}}$ is realizable by a rational map.
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