Constructing rational maps from subdivision rules

Abstract

Suppose R \mathcal {R} is an orientation-preserving finite subdivision rule with an edge pairing. Then the subdivision map σ R \sigma _{\mathcal {R}} is either a homeomorphism, a covering of a torus, or a critically finite branched covering of a 2-sphere. If R \mathcal {R} has mesh approaching 0 0 and S R S_{\mathcal {R}} is a 2-sphere, it is proved in Theorem 3.1 that if R \mathcal {R} is conformal, then σ R \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 Q \mathcal {Q} with an edge pairing such that σ Q \sigma _{\mathcal {Q}} is realizable by a rational map.