Complex flows, escape to infinity and a question of Rubel

Abstract

Let $f$ be a transcendental entire function. It was shown in a previous paper (2017) 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 $\dot{z}=\overline{f}(z)$, 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.