Are Devaney hairs fast escaping?

TL;DR

This paper investigates the properties of Devaney hairs in transcendental entire functions, showing that most points on these curves are fast escaping, but providing an example where they are not.

Contribution

It demonstrates that typically points on Devaney hairs are fast escaping, and constructs an example where a Devaney hair contains no fast escaping points.

Findings

Most points on Devaney hairs are in the fast escaping set.

An example of a Devaney hair with no fast escaping points is provided.

Devaney hairs can exist in logarithmic tracts without fast escaping points.

Abstract

Beginning with Devaney, several authors have studied transcendental entire functions for which every point in the escaping set can be connected to infinity by a curve in the escaping set. Such curves are often called Devaney hairs. We show that, in many cases, every point in such a curve, apart from possibly a finite endpoint of the curve, belongs to the fast escaping set. We also give an example of a Devaney hair which lies in a logarithmic tract of a transcendental entire function and contains no fast escaping points.