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 $\{f^n\vert_U\}$ lie in the derived set of $P(f)$ and that if $f^{np}\vert_V\rightarrow q(n\rightarrow +\infty)$ for a Fatou component $V$ , then either $q$ is in the derived set of $S_p(f)$ or $f^p(q) = q$ . As applications of main theorems, some sufficient conditions of the non-existence of wandering domains and Baker domains are given.