ON THE CONNECTIVITY OF JULIA SETS OF TRANSCENDENTAL ENTIRE-FUNCTIONS

We have two main purposes in this paper. One is to give some sufficien t conditions for the Julia set of a transcendental entire function f t o be connected or to be disconnected as a subset of the complex plane C. The other is to investigate the boundary of an unbounded periodic F atou component U, which is known to be simply-connected. These are rel ated as follows: let phi : D --> U be a Riemann map of U from a unit d isk D, then under some mild conditions we show that the set Theta(infi nity) of all angles where to admits the radial limit oo is dense in pa rtial derivative D if U is an attracting basin, a parabolic basin or a Siegel disk. If U is a Baker domain on which f is not univalent, then Theta(infinity), is dense in partial derivative D or at least its clo sure <(Theta(infinity))over bar>, contains a certain perfect set, whic h means the boundary partial derivative U has a very complicated struc ture. In all cases, this result leads to the disconnectivity of the Ju lia set J(f) in C. If U is a Baker domain on which f is univalent, how ever, we shall show by giving an example that partial derivative U can be a Jordan are in C, which has a rather simple structure, and, moreo ver, J(f) can be connected. We also consider the connectivity of the s et J(f) U {infinity}) in the Riemann sphere (C) over cap and show that J(f) U {infinity} is connected if and only if f has no multiply-conne cted wandering domains.