Title:
Conformal Dynamics: Julia sets, Kleinian groups, and Schwarz reflections.
Abstract:
We will discuss the interplay between three branches of Conformal Dynamics: iteration of (anti-)rational maps, actions of Kleinian groups, and dynamics generated by Schwarz reflections in quadrature domains. We will present examples of matings between anti-quadratic maps and the triangle modular group, and of the Julia reasization for the classical Apollonian gasket. We will also describe the group of quasisymmetries for some classes of Julia sets, including the Apollonian one, certain Sierpinski carperts, and the Basilica. The Thompson group will naturally emerge in the latter case.