On the Rank of the Thurston Pullback Map

Abstract Under some mild assumptions, an orientation-preserving branched covering map of marked $2$-spheres induces a pullback between the corresponding Teichmüller spaces. By analyzing associated pushforward operator acting on integrable quadratic differentials, we obtain global lower bound rank derivative in terms action cover points. In dynamical context, two sets points target and source coincide with postcritical set. Investigating resulting is central part Thurston’s topological characterization postcritically finite rational maps. Postcritically maps constant have been studied by various authors. that direction, our approach provides upper bounds size set pullback, shows dynamics highly restricted.