Abstract:

Quasiconformal surgery in holomorphic dynamics is possible due to the celebrated theorem of Morrey, Ahlfors and Bers, which states that almost complex structures with bounded dilatation can be integrated. Recently, Guy David has proven a generalization of this theorem, which says that one can also integrate unbounded structures, as long as the regions where the dilatation is large, have sufficiently small area. This theorem opens the door to many new surgery constructions which where not possible before. In the talk we will present a construction due to C. Petersen and S. Zakeri which proves that the boundary of a Siegel disk of a quadratic polynomial, with rotation number in a certain full measure class of irrational numbers is a Jordan curve containing the critical point.