The buried points on the Julia sets of rational and entire functions
Let \(f\) be a rational function of degree larger than one or entire transcendental. By \(\mathcal F\) resp. \(\mathcal J\) we denote the Fatou set resp. the Julia set of \(f\). A buried component is defined to be a component of \(\mathcal J\), which is not on the boundary of any component of \(\mathcal F\). The first example for buried components has been given by \textit{C. McMullen} [Holomorphic functions and moduli I, Proc. Workshops, Berkeley/Calif. 1986, Publ., Math. Sci. Res. Inst. 10, 31-60 (1988; Zbl 0692.30035)]. McMullen showed (Prop. 7.2) that for sufficiently small \(\lambda\) the functions \(f_\lambda(z)= z^2+ \lambda/z^3\) has uncountably many buried components and one unique invariant buried component \(K\), i.e. \(f(K)= K\). \textit{A. F. Beardon} [Ann. Acad. Sci. Fenn., Ser. A I 16, 173-177 (1991; Zbl 0757.30034)] has shown (Corollary) that if for a rational function \(\mathcal J\) is disconnected and all components of \(\mathcal F\) are of finite connectivity, then \(\mathcal J\) has buried components. Using this result, the author of the paper under review gives the following more general result (Theorem 1) for rational functions with nonempty Fatou set: \(\mathcal J\) contains buried components if and only if \(\mathcal J\) is disconnected and \(\mathcal F\) has no completely invariant component. For the well-known subclass of transcendental entire functions of finite type (i.e. the inverse has only finitely many singularities) -- this class has a lot of properties in common with rational functions, e.g., there are no Baker domains or wandering domains -- with nonempty Fatou set, the author gives the following result (Theorem 2): \(\mathcal J\) contains buried points if and only if \(\mathcal F\) is disconnected.
scientific article; zbMATH DE number 850853