SINGULARITIES AND STRICTLY WANDERING DOMAINS OF TRANSCENDENTAL SEMIGROUPS

Abstract

In this paper, the dynamics on a transcendental entire semi- group G is investigated. We show the possible values of any limit function of G in strictly wandering domains and Fatou components, respectively. Moreover, if G is of class B, for any z in a Fatou domain, there does not exist a sequence {gk} of G such that gk(z) ! 1 as k ! 1. 1. Introduction and main results In a series of papers, Hinkkanen and Martin extended the classical theory of dynamics associated with the iteration of a single rational function to the more general setting of semigroups of rational functions, see (8, 9). In 1998, Poon (11, 12) extended the study to transcendental semigroups and obtained some basic results. The dynamics of transcendental semigroups actually has some rather different properties than the dynamics of rational semigroups or the iteration of a single function. Suppose ffj : j = 1,2,...,mg is a family of transcendental entire functions. We call the semigroup G = hf1,f2,...,fmigenerated by ffjg under functional composition a transcendental semigroup. Define the Fatou set of the semigroup G by F(G) = fz 2 C : G is normal in some neighbourhood of zg and the Julia set of G by J(G) = C nF(G). If G is generated by only one function f, then F(G) and J(G) are the Fatou set and Julia set respectively in the classical iteration theory of Fatou and Julia. We say that a set S is completely invariant under f if S is forward and backward invariant under f. It is well known that the Fatou set and Julia set of a single function f are both completely invariant. But F(G) and J(G) need