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Escaping Fatou components of transcendental self-maps of the punctured plane

Part of: Complex dynamical systems Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  28 November 2019

DAVID MARTÍ-PETE
DAVID MARTÍ-PETE*
Affiliation:
Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan. e-mail: martipete@math.kyoto-u.ac.jp
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Abstract

We study the iteration of transcendental self-maps of $\mathcal{C}^*:\=\mathcal{C}\{0}$, that is, holomorphic functions $\fnof:\mathcal{C}^*:\rarr\mathcal{C}^*$ for which both zero and infinity are essential singularities. We use approximation theory to construct functions in this class with escaping Fatou components, both wandering domains and Baker domains, that accumulate to $\{0},\infin$ in any possible way under iteration. We also give the first explicit examples of transcendental self-maps of $\mathcal{C}^*$ with Baker domains and with wandering domains. In doing so, we developed a sufficient condition for a function to have a simply connected escaping wandering domain. Finally, we remark that our results also provide new examples of entire functions with escaping Fatou components.

MSC classification

Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions
Secondary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 2 , March 2021 , pp. 265 - 289
Copyright
© Cambridge Philosophical Society 2019

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