Canonical Thurston obstructions

We refine Douady and Hubbard's proof of Thurston's topological characteriza tion of rational functions by proving the following theorem. Let f:S-2 --> S-2 be a branched covering with finite postcritical set P-f and hyperbolic orbifold. Let Gamma (c) denote the set of all homotopy classes gamma of non peripheral, simple closed curves in S-2- P-f such that the length of the un ique geodesic homotopic to gamma tends to zero under iteration of the Thurs ton map induced by f on Teichmuller space. Then either Gamma (c) is empty, and f is equivalent to a rational function, or else Gamma (c) is a Thurston obstruction.