Open Access
Spring 2008 Direct singularities and completely invariant domains of entire functions
Walter Bergweiler, Alexandre Eremenko
Illinois J. Math. 52(1): 243-259 (Spring 2008). DOI: 10.1215/ijm/1242414130


Let $f$ be a transcendental entire function which omits a point $a∈ℂ$. We show that if $D$ is a simply connected domain which does not contain $a$, then the full preimage $f^{−1}(D)$ is disconnected. Thus, in dynamical context, if an entire function has a completely invariant domain and omits some value, then the omitted value belongs to the completely invariant domain. We conjecture that the same property holds if $a$ is a locally omitted value (i.e., the projection of a direct singularity of $f^{−1})$. We were able to prove this conjecture for entire functions of finite order. We include some auxiliary results on singularities of $f^{−1}$ for entire functions $f$, which can be of independent interest.


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Walter Bergweiler. Alexandre Eremenko. "Direct singularities and completely invariant domains of entire functions." Illinois J. Math. 52 (1) 243 - 259, Spring 2008.


Published: Spring 2008
First available in Project Euclid: 15 May 2009

zbMATH: 1171.30009
MathSciNet: MR2507243
Digital Object Identifier: 10.1215/ijm/1242414130

Primary: 30D20

Rights: Copyright © 2008 University of Illinois at Urbana-Champaign

Vol.52 • No. 1 • Spring 2008
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