Direct singularities and completely invariant domains of entire functions

Abstract

Let f be a transcendental entire function which omits a point a ε 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. © 2009 University of Illinois.