Dynamics of functions meromorphic outside a small set (Q2730704)

In this paper the theory of Fatou and Julia is extended to include the dynamics of functions \(f\) which are meromorphic in \(\widehat{\mathbb{C}}\) outside a totally disconnected compact set \(E(f)\) at whose points the cluster set of \(f\) is \(\widehat{\mathbb{C}}\). NEWLINENEWLINENEWLINEIn \S 2 the authors discuss the class \({\mathbf M}\) of functions: \({\mathbf M}=\{f : \) there is a compact totally disconnected set \(E=E(f)\) such that \(f\) is meromorphic in \(E^c=\widehat{\mathbb{C}} \setminus E\) and the cluster set \(C(f, E^c, z_0)=\widehat{\mathbb{C}}\) for all \(z_0\in E. \) If \(E=\emptyset\), \(f\) is neither constant nor univalent in \(\widehat{\mathbb{C}}\}\), where \(C(f, E^c, z_0)=\{w: w=\lim_n f(z_n)\) for some \(z_n\in E^c\) with \(z_n\to z_0\}\), and define the Fatou set \(F(f)\) and the Julia set \(J(f)\) of such a function. In \S 3 the standard properties of these sets are given and the classification of the different types of components of the open set \(F(f)\) according to the behaviour of the sequence of iterates of \(f\) in them is described in \S 5. In \S 4 the authors extend a result obtained first by \textit{A. E. Eremenko} in 1989 [Banach Center Publ. 23, 339-345 (1989; Zbl 0692.30021)] for transcendental entire functions and later by \textit{P. Domínguez} in 1998 [Ann. Acad. Sci. Fenn., Math. 23, No. 1, 225-250 (1998; Zbl 0892.30025)] for meromorphic functions in \(\mathbb{C}\). NEWLINENEWLINENEWLINEIn \S 6 the authors introduce the subclass \textbf{MSR} of function \(f\) whose totally disconnected set of singularities \(E(f)\) satisfies a technical condition and extend Sullivan's theorem to show that for functions in \textbf{MSR} the Fatou set has no wandering components. In addition, in \S 7 they show for the functions in \textbf{MSR} there are no Fatou components in which a limit function of iterates belongs to \(E(f)\) (no `Baker domains'). NEWLINENEWLINENEWLINEFinally, the relation between singular orbits and Fatou domains is discussed in \S 8 and a class of functions for which the Julia set is totally disconnected is given in \S 9.