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