  In this paper, we show that there exist transcendental meromorphic functions
with a cycle of 2-periodic Fatou components, where one is simply connected
while the other is doubly connected. In particular, the doubly connected Fatou
component can be an attracting, parabolic, or Baker domain. Thus, this settles
the problem of whether a doubly connected periodic Fatou component must be a
Herman ring.
  We also prove that there exists a transcendental meromorphic function with a
wandering domain that has no eventual connectivity. In addition, we show that
the connectivity sequence of this wandering domain is periodic with period two.
This solves a problem about the nonexistence of the eventual connectivity of
wandering domains and gives a different example constructed by Ferreira [J.
London Math. Soc. (2022), DOI:10.1112/jlms.12613].
