Non-Uniform Hyperbolicity in Polynomial Skew Products

Abstract Let $f:\mathbb {C}^2\to \mathbb {C}^2$ 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 satisfies one of following conditions: (1) $f|_L$ topological Collet–Eckmann and weak regularity conditions. (2) The Lyapunov exponent at every critical value point lying Julia set $f|_{L}$ exists positive, there no parabolic cycle. Under above conditions we show Fatou basin coincides with union basins cycles, has Lebesgue measure zero. As easy consequence are wandering components $L$.