On completely invariant Fatou components
type: Item
instance of: scholarly article
title: On completely invariant Fatou components
zbMATH Open document ID: 1034.37024
DOI: 10.1007/BF02390814
author: Chunlei Cao
author: Yuefei Wang
published in: Arkiv för Matematik
publication date: 2004-03-15T00:00:00Z
review text: The paper focuses on the components of the Fatou set for some special classes of functions, the components which are completely invariant. It had been proved by I. N. Baker, J. Kotus and Y. Lü [loc. cit. below] that for any meromorphic function \(f\), any completely invariant Fatou component (i.e. a component \(U\) of the Fatou set of \(f, F(f)\) which has the property that \(f(U)\subset U\) and \(f^{-1}(U)\subset U)\) has connectivity 1 or \(\infty\). Moreover, if the Fatou set \(F(f)\) has more than two completely invariant components, then all are simply connected. This result is proved in the article, using a similar argument found in the book of \textit{A. F. Beardon} [Iteration of rational functions. Complex analytic dynamical systems, Graduate Texts in Mathematics, 132, New York etc.: Springer-Verlag (1991; Zbl 0742.30002)] which extends the theorem to transcendental meromorphic functions. The main problem arises in relation to the structure of the Fatou set. It is well known that \(F(f)\) is itself a completely invariant set, the question then is how many components of different classes does it have? In this article, one treats those components which are completely invariant and proves that for a meromorphic function which has finitely many critical and asympotic values, the Fatou set can have at most two completely invariant components. However, it appears that both the result and the proof can be found in the paper by \textit{I. N. Baker}, \textit{J. Kotus}, and \textit{Y. Lü} [Ergodic Theory Dyn. Syst. 11, 603--618 (1991; Zbl 0774.30023)]. Moreover, there is a recent result for this class of meromorphic functions, obtained by W. Bergweiler and A. E. Eremenko who found that a meromorphic function with finitely many critical and asymptotic values and which has two completely invariant Fatou components, the Julia set \(J(f)\) is a Jordan curve. Their approach is similar: the starting point is the simple connectivity of these components and the fact that thei…
reviewed by: Chungchun Yang
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 30D30
Mathematics Subject Classification ID: 37F50
zbMATH DE Number: 2057394
zbMATH Keywords: Fatou set
zbMATH Keywords: completely invariant components
zbMATH Keywords: meromorphic functions
MaRDI profile type: Publication
cites work: Repulsive fixpoints of entire functions
cites work: Wandering domains for maps of the punctured plane
cites work: Iterates of Meromorphic Functions II: Examples of Wandering Domains
cites work: Iterates of meromorphic functions: I
cites work: Iterates of meromorphic functions III: Preperiodic domains
cites work: Iterates of meromorphic functions. IV: Critically finite functions
cites work: Q4002072
cites work: Iteration of meromorphic functions
cites work: Dynamical properties of some classes of entire functions
cites work: Are there critical points on the boundaries of singular domains?
cites work: Frontiers in complex dynamics
cites work: On the Iteration of analytic functions .
cites work: A class of meromorphic functions with no wandering domains
cites work: Q4294403
description: scientific article; zbMATH DE number 2057394