## Constructing rational maps from subdivision rules

### 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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