Ergodic Theory and Dynamical Systems



Dynamics of functions meromorphic outside a small set


I. N. BAKER a1, P. DOMÍNGUEZ a2 and M. E. HERRING a1
a1 Department of Mathematics, Imperial College of Science, Technology and Medicine, London SW7 2BZ, UK
a2 F.C. Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Avenida San Claudio, Col. San Manuel, C.U., Puebla, Pue. C.P. 72570, México

Abstract

The theory of Fatou and Julia is extended to include the dynamics of functions f which are meromorphic in \widehat{\mathbb{C}} outside a totally disconnected compact set E(f) at whose points the cluster set of f is \widehat{\mathbb{C}}. The Julia set is defined not only by the standard approach but is also characterized in terms of the set of points whose orbits approach a point of E(f). For the subclass where E(f) has a complement of class OAD and the inverse of f has a finite set of singular points it is shown that neither wandering components nor Baker domains occur in F(f)>. As an application, functions of a certain general class are shown to have a totally disconnected Julia set.

(Received August 24 1999)
(Accepted July 7 2000)