Abstract:

In 1986, Thurston asked whether all degree d branched covering maps of the sphere were "essentially" rational maps. The answer is "usually" and this is proved using a mix of dynamical systems and complex analysis. It turns out that there can be a topological obstruction to the existence of a rational map which is quite subtle. In this talk we will look at this question for a larger class of branched coverings where again we use dynamics and complex analysis to understand the subtleties involved, but this time from a geometric perspective.