On holomorphic families of rational maps: finiteness, rigidity and stability

Abstract

We consider holomorphic families of rational maps from the viewpoint of complex dynamics.

First, we consider some classes of families of rational maps which satisfy a certain stability condition. We show a finiteness theorem for such holomorphic families of rational maps parameterized by a Riemann surface of finite type.

Next, we consider the monodromy of quasiconformally stable holomorphic families of rational maps over a punctured disk, and study the action of the monodromy on the Julia set.