Illinois Journal of Mathematics

Direct singularities and completely invariant domains of entire functions

Walter Bergweiler and Alexandre Eremenko

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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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Illinois J. Math., Volume 52, Number 1 (2008), 243-259.

First available in Project Euclid: 15 May 2009

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Primary: 30D20: Entire functions, general theory


Bergweiler, Walter; Eremenko, Alexandre. Direct singularities and completely invariant domains of entire functions. Illinois J. Math. 52 (2008), no. 1, 243--259. doi:10.1215/ijm/1242414130.

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