Carathèodory convergence for Leau and Baker domains

ABSTRACT Consider a meromorphic function f with a countable compact set of essential singularities and whose Fatou set consists only of finitely many cycles of Leau domains and finitely many families of invariant Baker domains and their preimages. We provide sufficient conditions on a sequence of meromorphic functions with only Leau domains and uniformly convergent on compact sets to f, so that the Julia sets of converge in the Hausdorff metric to the Julia set of f. In particular, the Leau domains of approximate in the sense of Carathèodory, either the Leau or the Baker domains of f.