The limit set of iterations of entire functions on wandering domains

We first establish any continuum without interiors can be a limit set of iterations of an entire function on an oscillating wandering domain, and hence arise as a component of Julia sets. Recently, Luka Boc Thaler showed that every bounded connected regular open set, whose closure has a connected complement, is an oscillating or an escaping wandering domain of some entire function. A natural question is: What kind of domains can be realized as a periodic domain of some entire function? In this paper, we construct a sequence of entire functions whose invariant Fatou components can be approached to a regular domain.