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We have two main purposes in this paper. One is to give some sufficient conditions for the Julia set of a transcendental entire function f to 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 Fatou component U , which is known to be simply-connected. These are related as follows: let ϕ : D −→ U be a Riemann map of U from a unit disk D, then under some mild conditions we show that the set ∞ of all angles where ϕ admits the radial limit ∞ is dense in ∂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 ∞ is dense in ∂D or at least its closure ∞ contains a certain perfect set, which means the boundary ∂U has a very complicated structure. In all cases, this result leads to the disconnectivity of the Julia set J f in C. If U is a Baker domain on which f is univalent, however, we shall show by giving an example that ∂U can be a Jordan arc in C, which has a rather simple structure, and, moreover, J f can be connected. We also consider the connectivity of the set J f ∪ {∞} in the Riemann sphere C and show that J f ∪ {∞} is connected if and only if f has no multiply-connected wandering domains.

Let f n and f beentire functions and suppose that fn converges'to f locally uniformly on C. Then if the Fatou set o f f consists only of basins of attracting cycles or is empty, the Julia set of f n converges to that of f in the Hausdorff metric. We also show that expandingnesr of f implies the above assumption. Next we show that for each singular value c o f f there exists a singular value c(") off, (for each sufficiently large n) converging to c. As an application, we propose B criterion which dete@nes by the approximating sequence I fn),"=,, whether the Julia set o f f is the whole sphere C for a certain class of entire functions.

Prefecture (Communicated by Kiyosi IT0, M. J. A., Feb. 3, 1995) 1. Introduction. Throughout this paper let C U {co} be the Riemann sphere and R" --0 be a rational map of degree d_> 2.Periodic orbits are one of the most important objects to study in the theory of dynamical systems.By definition a point z 0 is an n-periodic point if R(zo) "= -R R(zo) Zo and R (Zo) :/: Zo for k n