Baranski, KrzysztofFagella Rabionet, NúriaJarque i Ribera, XavierKarpinska, Boguslawa2020-06-032020-06-032015-06-100024-6107https://hdl.handle.net/2445/164104We consider holomorphic maps $f: U \rightarrow U$ for a hyperbolic domain $U$ in the complex plane, such that the iterates of $f$ converge to a boundary point $\zeta$ of $U$. By a previous result of the authors, for such maps there exist nice absorbing domains $W \subset U$. In this paper we show that $W$ can be chosen to be simply connected, if $f$ has doubly parabolic type in the sense of the Baker-Pommerenke-Cowen classification of its lift by a universal covering (and $\zeta$ is not an isolated boundary point of $U$). We also provide counterexamples for other types of the map $f$ and give an exact characterization of doubly parabolic type in terms of the dynamical behaviour of $f$.19 p.application/pdfeng(c) London Mathematical Society, 2015Funcions de variables complexesFuncions meromorfesSistemes dinàmics complexosFunctions of complex variablesMeromorphic functionsComplex dynamical systemsinfo:eu-repo/semantics/article6454702020-06-03info:eu-repo/semantics/openAccess