  We investigate the description of Fatou components for polynomial
skew-products in two complex variables. The non-existence of wandering domains
near a super-attracting invariant fiber was shown in [L], and the
geometrically-attracting case was studied in [PV] and [PS]. In [ABDPR] it was
proven that wandering domains can exist near a parabolic invariant fiber. In
this paper we study the remaining case, namely the dynamics near an elliptic
invariant fiber. We prove that the two-dimensional Fatou components near the
elliptic invariant fiber correspond exactly to the Fatou components of the
restriction to the fiber, under the assumption that the multiplier at the
elliptic invariant fiber satisfies the Brjuno condition and that the
restriction polynomial has no critical points on the Julia set. We also show
the description does not hold when the Brjuno condition is dropped. Our main
tool is the construction of expanding metrics on nearby fibers, and one of the
key steps in this construction is given by a local description of the dynamics
near a parabolic periodic cycle.
