Are Devaney hairs fast escaping?

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.

22 pages, 1 figure

References in corpus (2)

• Dynamics of meromorphic functions with direct or logarithmic singularities
• On a question of Eremenko concerning escaping components of entire functions

Cited by in corpus (6)

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• On the dimension of points which escape to infinity at given rate under exponential iteration