On the connectivity of Julia sets of transcendental entire functions

We have two main purposes in this paper. One is to give some sufficient conditions for the Julia set of a transcendental entire function to be connected or to be disconnected as a subset of the complex plane . The other is to investigate the boundary of an unbounded periodic Fatou component , which is known to be simply-connected. These are related as follows: let from a unit disk , then under some mild conditions we show that the set of all angles where is dense in is an attracting basin, a parabolic basin or a Siegel disk. If is not univalent, then or at least its closure contains a certain perfect set, which means the boundary has a very complicated structure. In all cases, this result leads to the disconnectivity of the Julia set . If is a Baker domain on which is univalent, however, we shall show by giving an example that , which has a rather simple structure, and, moreover, in the Riemann sphere is connected if and only if has no multiply-connected wandering domains.