Barański, KrzysztofFagella Rabionet, NúriaJarque i Ribera, XavieKarpińska, Bogusława2024-11-112024-11-112024-08-170025-5831https://hdl.handle.net/2445/216337We 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 $f$.65 p.application/pdfengcc by (c) Krzysztof Barańskit et al., 2024http://creativecommons.org/licenses/by/3.0/es/Sistemes dinàmics complexosFuncions de variables complexesFuncions meromorfesComplex dynamical systemsFunctions of complex variablesMeromorphic functionsinfo:eu-repo/semantics/article7501802024-11-11info:eu-repo/semantics/openAccess