Eliminating Thurston obstructions and controlling dynamics on curves

Abstract

Every Thurston map f ∶ S 2 → S 2 on a 2-sphere S 2 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 f for which this obstruction is eliminated. We prove that no other obstruction arises and so f 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.