Dynamics of transcendental Hénon maps

Arosio, Leandro; Benini, Anna Miriam; Fornæss, John Erik; Peters, Hans

doi:10.48550/arxiv.1705.09183

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“…The parabola is the graph of a unimodal map. The points p(0), p (1) , and p (2) are defined as in Definition 3.4.…”

Section: Unimodal Mapsmentioning

confidence: 99%

“…Definition 3.4 Assume that f ∈ U has a unique fixed point p(0) ∈ I with a negative multiplier. Let p (1) = p(0) and p (2) be the point such that f (p (2) ) = p (1) and p (2) > c (0) . Define A = (−1, p (1) ) ∪ (p (2) , 1), B = (p (1) , p(0)), and C = (p(0), p (2) ).…”

Section: The Renormalization Of a Unimodal Mapmentioning

confidence: 99%

“…Let p (1) = p(0) and p (2) be the point such that f (p (2) ) = p (1) and p (2) > c (0) . Define A = (−1, p (1) ) ∪ (p (2) , 1), B = (p (1) , p(0)), and C = (p(0), p (2) ). The sets A = A( f ), B = B( f ), and C = C( f ) form a partition of the domain D ≡ I.…”

Section: The Renormalization Of a Unimodal Mapmentioning

confidence: 99%

“…g is concave on [−c (1) , c (1) ]. 4. g(c (1) ) = − 1 λ c (1) and g (c (1) ) = −λ . 5. g has negative Schwarzian derivative.…”

Section: The Fixed Point Of the Renormalization Operatormentioning

confidence: 99%

“…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. Nevertheless, the problem is still unsolved [5] for complex polynomial Hénon maps [33,34].…”

Section: Introductionmentioning

confidence: 99%

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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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“…The parabola is the graph of a unimodal map. The points p(0), p (1) , and p (2) are defined as in Definition 3.4.…”

Section: Unimodal Mapsmentioning

confidence: 99%

Section: The Renormalization Of a Unimodal Mapmentioning

confidence: 99%

Section: The Renormalization Of a Unimodal Mapmentioning

confidence: 99%

“…g is concave on [−c (1) , c (1) ]. 4. g(c (1) ) = − 1 λ c (1) and g (c (1) ) = −λ . 5. g has negative Schwarzian derivative.…”

Section: The Fixed Point Of the Renormalization Operatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nonexistence of Wandering Domains for Infinitely Renormalizable Hénon Maps

2017

Preprint

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Dynamics of transcendental Hénon maps

Cited by 1 publication

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Nonexistence of Wandering Domains for Infinitely Renormalizable Hénon Maps

Nonexistence of Wandering Domains for Infinitely Renormalizable Hénon Maps

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