Absorbing sets and Baker domains for holomorphic maps

We consider holomorphic maps f : U → U for a hyperbolic domain U in the complex plane, such that the iterates of f converge to a boundary point ζ of U . By a previous result of the authors, for such maps there exist nice absorbing domains W ⊂ U . In this paper we show that W can be chosen to be simply connected, if f has parabolic I type in the sense of the Baker–Pommerenke–Cowen classification of its lift by a universal covering (and ζ is not an isolated boundary point of U). Moreover, we provide counterexamples for other types of the map f and give an exact characterization of parabolic I type in terms of the dynamical behaviour of f .