Constructing rational maps from subdivision rules

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.