Non-uniform hyperbolicity in polynomial skew products

Abstract: 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 $L$.