Absence of wandering domains for some real entire functions with bounded singular sets

TL;DR

This paper proves that certain real entire functions with bounded singular sets and satisfying a sector condition have no wandering domains, extending results to specific non-real functions and providing new insights into Julia sets and Baker domains.

Contribution

It establishes the absence of wandering domains for a class of real entire functions with bounded singular sets under a sector condition, including specific maps like sinh-based functions.

Findings

No wandering domains for functions satisfying the sector condition.

Julia set of e^z is the entire complex plane.

Extension of Baker domain results to meromorphic functions.

Abstract

Let f be a real entire function whose set S(f) of singular values is real and bounded. We show that, if f satisfies a certain function-theoretic condition (the "sector condition"), then $f$ has no wandering domains. Our result includes all maps of the form f(z)=\lambda sinh(z)/z + a, where a is a real constant and {\lambda} is positive. We also show the absence of wandering domains for certain non-real entire functions for which S(f) is bounded and the iterates of f tend to infinity uniformly on S(f). As a special case of our theorem, we give a short, elementary and non-technical proof that the Julia set of the complex exponential map f(z)=e^z is the entire complex plane. Furthermore, we apply similar methods to extend a result of Bergweiler, concerning Baker domains of entire functions and their relation to the postsingular set, to the case of meromorphic functions.