We completely characterise the bounded sets that arise as components of the
Fatou and Julia sets of meromorphic functions. On the one hand, we prove that a
bounded domain is a Fatou component of some meromorphic function if and only if
it is regular. On the other hand, we prove that a planar continuum is a Julia
component of some meromorphic function if and only if it has empty interior. We
do so by constructing meromorphic functions with wandering continua using
approximation theory.Comment: 15 pages, 4 figures. V2: We have revised the introduction, and
introduced two new sections: Section 2 discusses and compare topological
properties of Fatou components, while Section 3 establishes that certain
bounded regular domains cannot arise as eventually periodic Fatou components
of meromorphic function
