Direct singularities and completely invariant domains of entire functions

Abstract

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.