Baker's conjecture for functions with real zeros

TL;DR

This paper proves Baker's conjecture for real transcendental entire functions with only real zeros, showing such functions have no unbounded wandering domains if their order is less than 1, using new complex analysis techniques.

Contribution

It introduces novel methods to confirm Baker's conjecture for a specific class of functions and establishes the absence of unbounded wandering domains for functions of order less than 1.

Findings

Baker's conjecture holds for real entire functions with only real zeros and order less than 1.

New techniques involving extremal length and growth estimates are developed.

The paper raises the question of whether such wandering domains can exist for all functions with order less than 1.

Abstract

Baker's conjecture states that a transcendental entire function of order less than $1/2$ has no unbounded Fatou components. It is known that, for such functions, there are no unbounded periodic Fatou components and so it remains to show that they can also have no unbounded wandering domains. Here we introduce completely new techniques to show that the conjecture holds in the case that the transcendental entire function is real with only real zeros, and we prove the much stronger result that such a function has no orbits of unbounded wandering domains whenever the order is less than 1. This raises the question as to whether such wandering domains can exist for any transcendental entire function with order less than 1. Key ingredients of our proofs are new results in classical complex analysis with wider applications. These new results concern: the winding properties of the images of certain curves proved using extremal length arguments, growth estimates for entire functions, and the distribution of the zeros of entire functions of order less than 1.