Title: Quasiconformal nonequivalence of Julia set and limit set

Abstract: The Julia set of a rational map and the limit set of a Kleinian group share many common features. In many cases, the two fractal sets can be homeomorphic. However, these homeomorphisms are usually not quasiconformal. In fact, we do not know any non-trivial examples where a Julia set of a rational map is quasiconformally homeomorphic to a limit set of a Kleinian group.

In this talk, I will discuss some non-equivalent phenomenon for some classes of Julia set and limit set. In particular, I will explain why a Julia set can never be quasiconformally homeomorphic to the Apollonian gasket.

The talk is based on a joint work with Yongquan Zhang.