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 E c = C ˆ \\ E is equal to C ˆ . 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 ○ g and g ○ f . Denote by F ( f ) and J ( f ) the Fatou and Julia sets of f. Let U be a component of F ( f ○ g ) and V be a component of F ( g ○ 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.