Dynamics of composite functions meromorphic outside a small set

Abstract

Let M denote the class of functions f meromorphic outside some compact totally disconnected set E = E(f) and the cluster set of f at any a ∈ E with respect to Ec= ℂ̂\E is equal to ℂ̂. It is known that class M is closed under composition. Let f and g be two functions in class M, we study relationship between dynamics of f o g and g o f. Denote by F(f) and J(f) the Fatou and Julia sets of f. Let U be a component of F(f o g) and V be a component of F(g o f) which contains g (U). We show that under certain conditions U is a wandering domain if and only if V is a wandering domain; if U is periodic, then so is V and moreover, V is of the same type according to the classification of periodic components as U unless U is a Siegel disk or Herman ring. © 2004 Elsevier Inc. All rights reserved.