Dynamics of functions meromorphic outside a small set

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.

Totally disconnected Julia set for different classes of meromorphic functions

“…In Section 3 we prove Theorem A which generalize a result given by Baker, Domínguez and Herring in [7].…”

“…For functions in class M ∩ S M there are neither wandering domains nor Baker domains [7]. Since the proofs remain valid for functions in class K ∩ S K we do not prove them.…”

“…We follow and verfy that the proof given in [7] for functions in class M ∩ S M works with some changes for functions in class A, where f is typical.…”

Transcendental Harmonic Mappings and Gravitational Lensing by Isothermal Galaxies

“…This falls short, since we are considering here a function g with infinitely many essential singularities and infinitely many critical points. However, the more updated version of the Fatou Theorem provided by the following Lemma found in [1] (we formulate a special case of Lemma 10) is perfectly suited to our situation. For an exposition of the extensions of the Fatou theorem leading up to this modern formulation, see the survey [3].…”

Geometric rigidity of transcendental meromorphic functions

“…Consequently f is a dynamically tame map. We would like to make it explicit that Bolsch maps form a special subclass of function meromorphic outside a small set; see [1] and [2] for its definition.…”