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Fatou-Julia dichotomy and non-uniform hyperbolicity for holomorphic endomorphisms on P2(C)

Abstract : This thesis deals with two different aspects (polynomial skew products and postctitically finite endomorphisms) of holomorphic dynamics on projective plane P2. It contains the following three papers: I. Non-wandering Fatou components for strongly attracting polynomial skew products. (Published in The Journal of Geometric Analysis.) We prove a generalization of Sullivan's non-wandering domain theorem for polynomial skew products onC2. More precisely, we show that if f is a polynomial skew product with an invariant vertical line L, assume L is attracting and moreover the corresponding multiplier is sufficiently small, then there is no wandering Fatou component in the attracting basin of L. II. Non-uniform hyperbolicity in polynomial skew products. (Submitted for publication.) We show that if f is a polynomial skew product with an attracting invariant vertical line L, assume the restriction of f on L satisfies one of the following non-uniformly hyperbolic condition: 1. fjL is topological Collet-Eckmann and WeaklyRegular, 2. the Lyapunov exponent at every critical value point lying in the Julia set of fjL exist and is positive, and there is no parabolic cycle. Then the Fatou set in the attracting basin of L is union of basins of attracting cycles, and the Julia set in the attracting basin of L has Lebesgue measure zero. As a corollary, there are no wandering Fatou components in the attracting basin of L. III. Structure of Julia sets for post-critically finite endomorphisms on P2. (Preprint.) De Thélin proved that for post-critically finite endomorphism on P2, the Green current T is laminar in J1 n J2, where J1 denotes the Julia set, and J2 denotes the support of the measure of maximal entropy. We give a more explicit description of the dynamics on J1 n J2 for post-critically finite endomorphism on P2: either x is contained in an attracting basin of a critical component cycle, or there is a Fatou disk passing through x. We also prove that for post-critically finite endomorphism on P2 such that all branches of PC(f) are smooth and intersect transversally, J2 = P2 if and only if f is strictly post-critically finite. This gives a partial converse of a result of Jonsson. As an intermediate step of the proof, we show that for post-critically finite endomorphism on P2, J2 is the closure of the set of repelling cycles
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Submitted on : Tuesday, November 2, 2021 - 4:35:43 PM
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Zhuchao Ji. Fatou-Julia dichotomy and non-uniform hyperbolicity for holomorphic endomorphisms on P2(C). Complex Variables [math.CV]. Sorbonne Université, 2020. English. ⟨NNT : 2020SORUS334⟩. ⟨tel-03411953v2⟩

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