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The celebrated Thurston's theorem in complex dynamics provides a criterion when a Thurston map is realized by a rational map. However, checking this criterion for a given Thurston map turns out to be a non-trivial problem in itself.

We consider a family of Thurston maps that arises from Schwarz reflections on flapped pillows. We use Thurston's result and a counting argument to establish a necessary and sufficient condition for a map in this family to be realized by a rational map. This result generalizes to all Thurston maps with a hyperbolic orbifold and four postcritical points. Namely, if such a Thurston map has an obstruction, then one can “blow up” suitable arcs in the underlying 2-sphere and construct a new Thurston map that is realized. In particular, this allows for the combinatorial construction of a large class of rational Thurston maps with four postcritical points.

If time permits, I will also briefly talk about a subclass of our rational Thurston maps with four postcritical points for which we can give a positive answer to the global curve attractor problem. (All these results are joint with M.Bonk and M.Hlushchanka.)

COMD Research Seminar Series: Eliminating Thurston Obstructions And Controlling The Dynamics Of Curves