Non-Uniform Hyperbolicity in Polynomial Skew Products

Abstract: Let

f : m a t h b b C^{2} o m a t h b b 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

.