Direct singularities and completely invariant domains of entire functions

Walter Bergweiler; Alexandre Eremenko

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Direct singularities and completely invariant domains of entire functions

Walter Bergweiler·Alexandre Eremenko

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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.

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BibTeX RIS

Walter Bergweiler, Alexandre Eremenko. 2006-11-14. Direct singularities and completely invariant domains of entire functions. https://arxiv.org/abs/math/0608027

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