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NON-UNIFORM HYPERBOLICITY IN POLYNOMIAL SKEW PRODUCTS

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-02285663
Contributor : Zhuchao Ji <>
Submitted on : Thursday, October 10, 2019 - 10:47:05 PM
Last modification on : Friday, April 10, 2020 - 5:13:33 PM

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  • HAL Id : hal-02285663, version 2
  • ARXIV : 1909.06084

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Zhuchao Ji. NON-UNIFORM HYPERBOLICITY IN POLYNOMIAL SKEW PRODUCTS. 2019. ⟨hal-02285663v2⟩

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