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arXiv · 1401.7292

Absorbing sets and Baker domains for holomorphic maps

Krzysztof Barański·Núria Fagella·Xavier Jarque·Bogusława Karpińska
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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 $ζ$ 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 $ζ$ 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 $f$.

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Krzysztof Barański, Núria Fagella, Xavier Jarque, Bogusława Karpińska. 2014-01-28. Absorbing sets and Baker domains for holomorphic maps. https://doi.org/10.1112/jlms%2Fjdv016

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