On Thurston's pullback map

Abstract

Let f: P^1 \to P^1 be a rational map with finite postcritical set P_f. Thurston showed that f induces a holomorphic map \sigma_f of the Teichmueller space T modelled on P_f to itself fixing the basepoint corresponding to the identity map (P^1, P_f) \to (P^1, P_f). We give explicit examples of such maps f showing that the following cases may occur: (1) the basepoint is an attracting fixed point, the image of \sigma_f is open and dense, and the map \sigma_f is a covering map onto its image; (2) the basepoint is a superattracting fixed point, \sigma is surjective, and \sigma is a ramified Galois covering, (3) \sigma_f is constant.