Some transcendental meromorphic functions with buried singleton components

Abstract

In this paper, we give a new sufficient condition which guarantees that transcendental meromorphic functions have buried singleton components, which entire functions in the Eremenko–Lyubich class cannot have, and construct some transcendental meromorphic functions with buried singleton components with this condition. The result is an extension of the results of Domínguez, Fagella, Ng, Zheng, and Choi which assure the existence of buried singleton components. The main tool of the proof is the theory of the disk-annulus surgery developed by Pilgrim and Tan for rational maps. We modify this technique into our transcendental case. We also see that this surgery enables us to understand dynamical properties of some meromorphic functions from those of some entire functions in the Eremenko–Lyubich class. We give sufficient conditions for some transcendental meromorphic functions, which assure that their Julia sets contain Cantor bouquets, Cremer points, or Jordan curves which are boundaries of some Siegel disks, or that their escaping sets are of Lebesgue measure zero. We also construct various kinds of transcendental meromorphic functions which satisfy those conditions and which can be chosen to have a wide variety of properties: Julia sets of positive or zero Lebesgue measure, buried unbounded continua, buried Cantor bouquets, buried escaping sets, buried sets of all repelling periodic points, no buried points, or Siegel disks whose boundaries are Jordan curves or unbounded.