Baker's conjecture and Eremenko's conjecture for functions with negative real zeros

TL;DR

This paper advances understanding in transcendental dynamics by proving Baker's and Eremenko's conjectures for entire functions with zeros on the negative real axis, especially those of order less than 1/2.

Contribution

Introduces a new technique combining distortion theorems and curve winding properties to prove long-standing conjectures for specific classes of entire functions.

Findings

Baker's conjecture holds for functions with zeros on negative real axis of order less than 1/2.

Eremenko's conjecture is confirmed for these functions, showing all escaping set components are unbounded.

New methods involve hyperbolic metric contraction and curve winding analysis.

Abstract

We introduce a new technique that allows us to make progress on two long standing conjectures in transcendental dynamics: Baker's conjecture that a transcendental entire function of order less than 1/2 has no unbounded Fatou components, and Eremenko's conjecture that all the components of the escaping set of an entire function are unbounded. We show that both conjectures hold for many transcendental entire functions whose zeros all lie on the negative real axis, in particular those of order less than 1/2. Our proofs use a classical distortion theorem based on contraction of the hyperbolic metric, together with new results which show that the images of certain curves must wind many times round the origin.