Direct singularities and completely invariant domains of entire functions
Complex Variables
2009-06-30 v2 Dynamical Systems
Abstract
Let f be a transcendental entire function that omits a complex value a. We show that for every simply connected region D that does not contain a the full preimage of D is disconnected. We conjecture that the same holds if one only assumes that a is omitted locally. We were able to prove this conjecture under the additional assumption that f is of finite order. We include some auxilliary results on the singularities of the inverses of entire functions which are of independent interest.
Cite
@article{arxiv.math/0608027,
title = {Direct singularities and completely invariant domains of entire functions},
author = {Walter Bergweiler and Alexandre Eremenko},
journal= {arXiv preprint arXiv:math/0608027},
year = {2009}
}
Comments
This is the second, corrected version. There were gaps in the proof of the main theorem in the first version