NON-UNIFORM HYPERBOLICITY IN POLYNOMIAL SKEW PRODUCTS

Zhuchao Ji (1)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

Let f: C^2- > 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. 2019. ⟨hal-02285663v1⟩