Complex Analysis, Dynamics and Geometry Seminar

William Thurston proved that a post-critically finite branched cover of the plane is either equivalent to a polynomial or it has a topological obstruction. Each branched cover determines a combinatorial map on Teichmueller space called a lifting map. We prove that for those branched covers that are equivalent to polynomials, the lifting map converges to a compact set in Teichmueller space. We can then determine to which polynomial the branched cover is equivalent. For the obstructed branched covers, we give an algorithm to find an obstruction. Our algorithm also resolves Pilgrim's "Finite Global Attractor" conjecture for topological polynomials. This is joint work with Jim Belk, Justin Lanier, and Dan Margalit. Speaker(s): Becca Winarski (College of the Holy Cross)