Wandering Domains in the Iteration Ofcompositions of Entire Functionsi

If p is entire, g(z) = a + b exp(2i=c), where a , b , c are non-zero constants and the normal set of g(p) has no wandering components, then the same is true for the normal set of p(g). Let f be a rational function of degree at least 2 or a nonlinear entire function. Let f n , for n 2 N denote the nth iterate of f. Denote the set of normality by N(f) and the Julia set by J(f). Thus N(f) = fz : (f n) is normal in some neighbourhood of zg, J(f) = C ? N(f). By deenition N(f) is open (and possibly empty) and it is well known (see for example 8], 9]) that J(f) is nonempty and perfect and J(f) is completely invariant under f , that is, z 2 J(f) implies f(z) 2 J(f) and z 0 2 J(f) for any z 0 such that f(z 0) = z. Consequently N(f) is completely invariant. If U is a component of N(f) then f(U) lies in some component V of N(f). In fact V nf(U) is either ; or a single point, by an unpublished result of M. Herring. By a slight abuse of language we write V = f(U) even when V nf(U) is a singleton. If all f n (U) with n 2 N are diierent components of N(f) then U is called a wandering domain. D. Sullivan 13] proved that the set of normality of a rational function has no wandering domain, thus solving a problem open since the papers of Fatou and Julia. On the other hand this is not so for transcendental entire functions. In 1] the rst author constructed an entire function f such that N(f) has wandering domains. Since then several entire functions which have wandering domains with various diierent properties have been constructed, see for instance 3], 7]. Also at the same time there has been a move to classify those entire functions which do not have wandering domains 2], 6], 11]. In particular this is the case for functions which have only a nite number of asymptotic or critical values. Such functions are denoted as having nite type. In this paper we shall identify a class of composite entire functions which have no wandering domains and a class of composite entire functions which have wandering domains. We shall prove 1991 Mathematics Subject Classiication: …