On complex analytic properties of limit sets and Julia sets

Abstract

It is known that Kleinian groups and complex dynamics have many properties in common. We shall show that the structure of the Martin boundaries of the region of discontinuity and the Fatou set are similar under a certain condition. We also show that any conformal mapping defined on the region of discontinuity (resp. the Fatou set) has a continuous extension on the conical limit set (resp. the conical Julia set in the sense of Lyubich and Minsky).