Absorbing sets and Baker domains for holomorphic maps

Abstract

We consider holomorphic maps $f:U\to U$ 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 parabolic I 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 $U$). 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 $f$.

Cited by in corpus (4)

• Fatou components and singularities of meromorphic functions
• Accesses to infinity from Fatou components
• Escaping points in the boundaries of Baker domains
• Singular orbits and Baker domains