Non-Uniform Hyperbolicity in Polynomial Skew Products

Let f : C-2 -> C-2 be a polynomial skew product that 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) exists 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.