SINGULARITIES AND LIMIT FUNCTIONS IN ITERATION OF MEROMORPHIC FUNCTIONS

Abstract

Let f(z) be a transcendental meromorphic function. The paper investigates, using the hyperbolic metric, the relation between the forward orbit P(f) of singularities of f−1 and limit functions of iterations of f in its Fatou components. It is mainly proved, among other things, that for a wandering domain U, all the limit functions of {fn|U} lie in the derived set of P(f) and that if fnp|V→ q(n→ +∞) for a Fatou component V, then either q is in the derived set of Sp (f) or fp(q) = q. As applications of main theorems, some sufficient conditions of the non-existence of wandering domains and Baker domains are given.