Abstract
We refine Douady and Hubbard's proof of Thurston's topological characterization of rational functions by proving the following theorem. Let f:S2→S2 be a branched covering with finite postcritical set Pf and hyperbolic orbifold. Let Γc denote the set of all homotopy classes γ of nonperipheral, simple closed curves in S2−Pf such that the length of the unique geodesic homotopic to γ tends to zero under iteration of the Thurston map induced by f on Teichmüller space. Then either Γc is empty, and f is equivalent to a rational function, or else Γc is a Thurston obstruction.
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