Thurston’s pullback map, invariant covers, and the global dynamics on curves

We consider rational maps f on the Riemann sphere Ĉ with an f-invariant set P ⊂ Ĉ of four marked points containing the postcritical set of f. We show that the dynamics of the corresponding Thurston pullback map σf on the completion T̅p of the associated Teichmüller space Tp with respect to the Weil-Petersson metric is easy to understand when T̅p admits a cover by sets with good combinatorial and dynamical properties. In particular, the map f has a finite global curve attractor in this case. Using a result by Eremenko and Gabrielov, we also show that if P contains all critical points of f and each point in P is periodic, then such a cover of T̅p can be obtained from a σf-invariant tessellation by ideal hyperbolic triangles.