Eliminating Thurston obstructions and controlling dynamics on curves

Every Thurston map f : S² → S² on a 2-sphere S² induces a pull-back operation on Jordan curves α ⊂ S^{2 \} P_f, where P_f is the postcritical set of f. Here the isotopy class [f⁻¹(α)] (relative to P_f) only depends on the isotopy class [α]. We study this operation for Thurston maps with four postcritical points. In this case, a Thurston obstruction for the map f can be seen as a fixed point of the pull-back operation. We show that if a Thurston map f with a hyperbolic orbifold and four postcritical points has a Thurston obstruction, then one can ‘blow up’ suitable arcs in the underlying 2-sphere and construct a new Thurston map f̂ for which this obstruction is eliminated. We prove that no other obstruction arises and so f̂ is realized by a rational map. In particular, this allows for the combinatorial construction of a large class of rational Thurston maps with four postcritical points. We also study the dynamics of the pull-back operation under iteration. We exhibit a subclass of our rational Thurston maps with four postcritical points for which we can give positive answer to the global curve attractor problem.