On completely invariant Fatou components
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 their boundary is the Julia set. The main result in the paper states that a meromorphic function having one pole which is an omitted value, if it has a completely invariant Fatou component, then in any other component of \(F(f)\), \(f\) is either invariant or a two-fold map. Also it is shown that if \(f\) is a transcendental analytic self-map of \(\mathbb C^*\), then it can have at most one completely invariant component; which also provides a positive answer to Baker's and Bergweiler's problems relating meromorphic functions having invariant components.
scientific article; zbMATH DE number 2057394
scientific article