On the configuration of Herman rings of meromorphic functions
type: Item
instance of: scholarly article
title: On the configuration of Herman rings of meromorphic functions
zbMATH Open document ID: 1272.37023
author: Núria Fagella
author: Jörn Peter
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-08-01T00:00:00Z
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 37F50
Mathematics Subject Classification ID: 37F30
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 37E45
zbMATH DE Number: 6062834
zbMATH Keywords: complex dynamical systems
zbMATH Keywords: Herman rings
zbMATH Keywords: Julia set
zbMATH Keywords: Fatou set
zbMATH Keywords: invariant curves
zbMATH Keywords: configuration
zbMATH Keywords: quasiconformal surgery
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.05.005
OpenAlex ID: W2052374999
cites work: The components of a Julia set
cites work: Q4264327
cites work: Rational iteration: complex analytic dynamical systems
cites work: Iteration of meromorphic functions
cites work: Q4210915
cites work: Iterates of meromorphic functions: I
cites work: Iterates of Meromorphic Functions II: Examples of Wandering Domains
cites work: Iterates of meromorphic functions III: Preperiodic domains
cites work: Iterates of meromorphic functions. IV: Critically finite functions
cites work: Dynamical properties of some classes of entire functions
cites work: A finiteness theorem for a dynamical class of entire functions
cites work: Exemples de fractions rationnelles ayant une orbite dense sur la sphère de Riemann
cites work: Small denominators. I. Mappings of the circumference onto itself
cites work: On the quasiconformal surgery of rational functions
cites work: Existence of herman rings for meromorphic functions
cites work: Remarks on Herman rings of transcendental meromorphic functions
cites work: Trees associated with the configuration of Herman rings
cites work: Q5512337
cites work: Q4884413
cites work: Quasiconformal Surgery in Holomorphic Dynamics
cites work: Conformally natural extension of homeomorphisms of the circle
DOI: 10.1016/J.JMAA.2012.05.005
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review text: Let \(f\) be a transcendental meromorphic map. There is a well known classification of the connected components of the Fatou set \(\mathcal{F}\) of \(f\). One of the possible types are Herman rings. A Herman ring is a doubly connected \(p\)-periodic Fatou component \(A\) such that there exists \(\theta\in[0,1]\setminus\mathbb{Q}\) (the rotation number of \(A\)), \(0<r<1\) and a conformal change of variables \(L:\{z\in\mathbb{C}:r<|z|<1\}\to A\) which conjugates \(f^p\) (the \(p\)-th iterate of \(f\)) to the rotation \(z\mapsto e^{2\pi i\theta}z\).NEWLINENEWLINE\textit{M. Shishikura} showed that for any \(p>0\) and any \(\theta\) of Brjuno type there exists a rational function with a \(p\)-cycle of Herman rings of rotation number \(\theta\) [Ann. Sci. Éc. Norm. Supér. (4) 20, No. 1, 1--29 (1987; Zbl 0621.58030)]. The main contribution of the paper in review is to show that every configuration of Herman rings which is possible for rational maps is also possible for transcendental meromorphic maps. Thereby, a configuration is an equivalence class which provides the relative position, orientation and dynamics of a cylce of Herman rings.NEWLINENEWLINEFurthermore, the authors prove that every bounded component of \(\mathbb{C}\setminus\bigcup_{i=1}^{n}A_i\) contains a pole, where \(A_1,\dots,A_n\) are the invariant Herman rings of a transcendental meromorphic map. It is also proven that one pole is sufficient to obtain cycles of Herman rings of arbitrary period.NEWLINENEWLINEThe main technique used in the paper is quasiconformal surgery.
description: scientific article; zbMATH DE number 6062834