Some connectedness properties of julia sets

Abstract

In general the Julia set of an entire or meromorphic function is either connected or has an uncountable set of components. Conditions are found which ensure the second alternative. If a transcendental entire function has a completely invariant Fatou component then its Julia set is locally connected at no point in the plane. If the Julia set of a transcendental entire function is locally connected at some point, then the Julia set is connected. An entire function is constructed for which every periodic point is repelling, is a singleton component of the Julia set and lies on the boundary of no Fatou component.