  Let $f:\mathbb{C}^2\to \mathbb{C}^2$ be a polynomial skew product which
leaves invariant an attracting vertical line $ L $. Assume moreover $f$
restricted to $L$ is non-uniformly hyperbolic, in the sense that $f$ restricted
to $L$ satisfies one of the following conditions: 1. $f|_L$ satisfies
Topological Collet-Eckmann and Weak Regularity conditions. 2. The Lyapunov
exponent at every critical value point lying in the Julia set of $f|_L$ exist
and is positive, and there is no parabolic cycle. Under one of the above
conditions we show that the Fatou set in the basin of $L$ coincides with the
union of the basins of attracting cycles, and the Julia set in the basin of $L$
has Lebesgue measure zero. As an easy consequence there are no wandering Fatou
components in the basin of $L$.
