The combinatorics of rational maps

Talk - mathematics
In the 1980ies William Thurston raised the question about the topological behavior of rational maps on the Riemann sphere. In his famous characterization of rational maps, he proved that a postcritically finite branched covering of the two-sphere is equivalent to a rational map if and only if it does not admit a Thurston obstruction. The latter constitutes a purely topological criterion based on the behavior of curves under pullback through the map. There has since been a great effort in the dynamics community to develop methods that allow for hands-on applications and to prove combinatorial version of this theorem.
In this talk, I will present recent joint work with M.Bonk and M.Hlushchanka on eliminating obstructions for Thurston maps with four postcritical points, and, outline a current project in progress about the realization of dynamical portraits.