Complex flows, escape to infinity and a question of Rubel

TL;DR

This paper investigates the behavior of trajectories tending to infinity in complex flows generated by transcendental entire functions, revealing their rarity, contrasting holomorphic and antiholomorphic cases, and confirming Rubel's conjecture for a specific function class.

Contribution

It proves the rarity of trajectories tending to infinity in holomorphic flows, contrasts with antiholomorphic flows, and confirms Rubel's conjecture for functions in the Eremenko-Lyubich class.

Findings

Holomorphic flows have infinitely many trajectories to infinity, but they are rare.

Antiholomorphic flows may lack such trajectories unless the function is in class B.

For functions in class B, there exists a path to infinity where the function and derivatives tend to infinity.

Abstract

Let $f$ be a transcendental entire function. It was shown in a previous paper that the holomorphic flow $\dot z = f(z)$ always has infinitely many trajectories tending to infinity in finite time. It will be proved here that such trajectories are in a certain sense rare, although an example will be given to show that there can be uncountably many. In contrast, for the classical antiholomorphic flow $\dot z = \bar f(z)$, such trajectories need not exist at all, although they must if $f$ belongs to the Eremenko-Lyubich class $\mathcal{B}$. It is also shown that for transcendental entire $f$ in $\mathcal{B}$ there exists a path tending to infinity on which $f$ and all its derivatives tend to infinity, thus affirming a conjecture of Rubel for this class.