NON-UNIFORM HYPERBOLICITY IN POLYNOMIAL SKEW PRODUCTS

Zhuchao Ji (1, 2)

1 School of Mathematical Sciences [Fudan]
2 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

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 $L$.

Mots clés: Non-uniform hyperbolicity; Topological Collet-Eckmann condition; polynomial skew product; Fatou component; non-wandering domain theorem

Domaines: Systèmes dynamiques [math.DS]; Variables complexes [math.CV]

Zhuchao Ji. NON-UNIFORM HYPERBOLICITY IN POLYNOMIAL SKEW PRODUCTS. International Mathematics Research Notices, 2022. ⟨hal-02285663v4⟩