Direct singularities and completely invariant domains of entire functions
arXiv:math/0608027
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.
This is the second, corrected version. There were gaps in the proof of the main theorem in the first version