Wandering domains for composition of entire functions

TL;DR

This paper analyzes the wandering domains of a specific entire function constructed by Bishop, showing it has exactly two such domains, and explores how wandering domains relate to compositions of entire functions.

Contribution

It proves Bishop's example has exactly two wandering domains and constructs examples of entire functions with wandering domains in their composition but not in individual functions.

Findings

Bishop's example has exactly two wandering domains.

Existence of entire maps with wandering domains in their composition but not in individual functions.

Results extend understanding of Julia sets in composed entire functions.

Abstract

C. Bishop has constructed an example of an entire function f in Eremenko-Lyubich class with at least two grand orbits of oscillating wandering domains. In this paper we show that his example has exactly two such orbits, that is, f has no unexpected wandering domains. We apply this result to the classical problem of relating the Julia sets of composite functions with the Julia set of its members. More precisely, we show the existence of two entire maps f and g in Eremenko-Lyubich class such that the Fatou set of f compose with g has a wandering domain, while all Fatou components of f or g are preperiodic. This complements a result of A. Singh and results of W. Bergweiler and A. Hinkkanen related to this problem.