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 \(\dot z = \bar f(z)\), 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.