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

We consider rational maps f f on the Riemann sphere C ^ {C} with an f f -invariant set P ⊂ C ^ P⊂ {C} of four marked points containing the postcritical set of f f . We show that the dynamics of the corresponding Thurston pullback map σ f σ _f on the completion T P ¯ T_P̄ of the associated Teichmüller space T P T_P with respect to the Weil–Petersson metric is easy to understand when T P ¯ T_P̄ admits a cover by sets with good combinatorial and dynamical properties. In particular, the map f f has a finite global curve attractor in this case. Using a result by Eremenko and Gabrielov, we also show that if P P contains all critical points of f f and each point in P P is periodic, then such a cover of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper T Subscript upper P Baseline