Unbounded fast escaping wandering domains

TL;DR

This paper develops a new approximation method in complex dynamics to construct transcendental entire functions with unbounded wandering domains, including those with fast escaping behavior, addressing longstanding open questions.

Contribution

It introduces a novel approximation technique enabling the construction of entire functions with unbounded wandering domains of various types, including those with specific orders.

Findings

Constructed entire functions with unbounded fast escaping wandering domains.

Provided examples of functions with orders between 1/2 and 1 having unbounded wandering domains.

Answered a long-standing question of Rippon and Stallard.

Abstract

We introduce a new approximation technique into the context of complex dynamics that allows us to construct examples of transcendental entire functions with unbounded wandering domains. We provide examples of entire functions with an orbit of unbounded fast escaping wandering domains, answering a long-standing question of Rippon and Stallard. Moreover, these examples cover all possible types of simply connected wandering domains in terms of convergence to the boundary. In relation to a conjecture of Baker, it was unknown whether functions of order less than one could have unbounded wandering domains. For any given order greater than $1/2$ and smaller than $1$, we provide an entire function of such order with an unbounded wandering domain.