Residual julia sets of meromorphic functions

Abstract

In this paper, we study the residual Julia sets of meromorphic functions. In fact, we prove that if a meromorphic function f belongs to the class S and its Julia set is locally connected, then the residual Julia set of f is empty if and only if its Fatou set F(f) has a completely invariant component or consists of only two components. We also show that if f is a meromorphic function which is not of the form α + (z - α)-keg(z), where k is a natural number, α is a complex number and g is an entire function, then f has buried components provided that f has no completely invariant components and its Julia set J(f) is disconnected. Moreover, if F(f) has an infinitely connected component, then the singleton buried components are dense in J(f). This generalizes a result of Baker and Domínguez. Finally, we give some examples of meromorphic functions with buried points but without any buried components. © 2006 Cambridge Philosophical Society.