Polynomial skew-products in dimension 2: Bulging and Wandering Fatou components

Raissy, Jasmin

doi:10.1007/s40574-016-0101-1

Cited by 5 publications

(4 citation statements)

References 26 publications

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“…The first example was given by Astorg, Buff, Dujardin, Peters, and Raissy [2], who found a polynomial skew-product possessing a wandering Fatou component as the quasi-conformal methods break down. The reader can refer to the survey [63] for more details about other relevant work on polynomial skew-product [42,60,59,61]. For complex Hénon maps 2 , a recent paper by Leandro Arosio, Anna Miriam Benini, John Erik Fornaess, and Han Peters [1] found a transcendental Hénon map exhibiting a wandering domain.…”

Section: Introductionmentioning

confidence: 99%

Nonexistence of Wandering Domains for Infinitely Renormalizable Hénon Maps

2017

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This article extends the theorem of the absence of wandering domains from unimodal maps to infinitely period-doubling renormalizable Hénon-like maps in the strongly dissipative (area contracting) regime. The theorem solves an open problem proposed by several authors [64,44], and covers a class of maps in the nonhyperbolic higher dimensional setting. The classical proof for unimodal maps breaks down in the Hénon settings, and two techniques, "the area argument" and "the good region and the bad region", are introduced to resolve the main difficulty.The theorem also helps to understand the topological structure of the heteroclinic web for such kind of maps: the union of the stable manifolds for all periodic points is dense.Acknowledgements The author thanks Marco Martens for many discussions on Hénon maps, careful reading the manuscript, and useful comments on the results. The author also thanks Department of Mathematics, Stony Brook University for financial support.

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Section: Introductionmentioning

confidence: 99%

Nonexistence of Wandering Domains for Infinitely Renormalizable Hénon Maps

2017

Preprint

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“…Polynomial skew products, for example, [7,8], have been useful test cases for complex dynamics in two dimensions. This allows one to use one variable results in one of the variables: Suppose F is a polynomial skew product, F (z, w) = (P (z), Q(z, w)), where P (z), Q(z, w) are two polynomials of degree m 1 , m 2 ≥ 2 on C and P (0) = 0, Q(0, 0) = 0.…”

Section: Introductionmentioning

confidence: 99%

Dynamics inside Fatou sets in higher dimensions

Mi¹

2023

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In this paper, we investigate the behavior of orbits inside attracting basins in higher dimensions. Suppose F (z, w) = (P (z), Q(w)), whereLet Ω be the immediate attracting basin of F (z, w). Then there is a constant C such that for every point (z 0 , w 0 ) ∈ Ω, there exists a point (z, w) ∈ ∪ k F −k (0, 0), k ≥ 0 so that d Ω (z 0 , w 0 ), (z, w) ≤ C, d Ω is the Kobayashi distance on Ω. However, for many other cases, this result is invalid.

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“…In contrast to the complex one-dimensional case, Astorg et al [2] found a polynomial skew-product possessing a wandering Fatou component. The reader can refer to the survey [62] for more details about other relevant works. For complex Hénon maps, 2 a recent paper by Arosio et al [1] found a transcendental Hénon map exhibiting a wandering domain.…”

Section: Introductionmentioning

confidence: 99%