Complex flows, escape to infinity and a question of Rubel
Annales Fennici Mathematici, Tome 47 (2022) no. 2, pp. 885-894
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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.
Keywords:
Holomorphic flows, antiholomorphic flows, trajectories
Affiliations des auteurs :
James K. Langley 1
James K. Langley. Complex flows, escape to infinity and a question of Rubel. Annales Fennici Mathematici, Tome 47 (2022) no. 2, pp. 885-894. doi: 10.54330/afm.120214
@article{AFM_2022_47_2_a12,
author = {James K. Langley},
title = {Complex flows, escape to infinity and a question of {Rubel}},
journal = {Annales Fennici Mathematici},
pages = {885--894},
year = {2022},
volume = {47},
number = {2},
doi = {10.54330/afm.120214},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.54330/afm.120214/}
}
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