Abstract:

It is known that the disconnected Julia set of any polynomial does not contain buried Julia components. But such Julia components may arise for rational maps. The first example is due to Curtis McMullen who provided a family of rational maps for which the disconnected Julia set is a Cantor of Jordan curves. However all known examples of buried Julia components, up to now, are wandering Jordan curves and comes from rational maps of degree at least 5.

I will present a way to construct some rational maps with elaborated disconnected Julia sets whose exchanging dynamics of Julia components is encoded by some weighted Hubbard trees. In particular, I will introduce a family of Julia sets with buried components of several types: wandering Jordan curves, wandering points but also preperiodic points and preperiodic infinitely connected Julia components. Moreover this family contains some rational maps of degree 3 with explicit formula that answers a question McMullen raised.

The used tools will be quasi-conformal surgery and a thorough understanding of obstructions that occur in Thurston's topological characterization theorem.