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

• Published in: Mathematical proceedings of the Cambridge Philosophical Society Vol. 170; no. 2; pp. 265 - 289
• Main Author: MARTÍ-PETE, DAVID
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.03.2021
• Subjects: Analytic functions; Domains; Entire functions; 37F10; 30D05
• ISSN: 0305-0041; 1469-8064
• DOI: 10.1017/S0305004119000409

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.