A transcendental Hénon map with an oscillating wandering Short $\mathbb{C}^2$

Arosio, Leandro; Thaler, Luka Boc; Peters, Hans

doi:10.48550/arxiv.1901.07394

Cited by 1 publication

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“…Secondly, the Jacobian of an automorphism is constant and so the existence of a parabolic fixed point implies that the Jacobian is 1, which is incompatible with their construction (a real disk is shrunk along its orbit). In the same topic, let us mention the example of a transcendental biholomorphic map in C 2 with a wandering Fatou component oscillating to infinity by Fornaess and Sibony in [FS98] and other examples of wandering domains for transcendental mappings in higher dimension in [ABTP19a]. About the non-existence of wandering Fatou components in higher dimension, a few cases study have been done proved [Lil04,Ji18] in some particular cases (skew products with a super-attracting invariant fiber).…”

Section: Introduction: State Of the Art And Main Resultsmentioning

confidence: 99%

Emergence of wandering stable components

Berger¹,

Biebler²

2020

Preprint

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We prove the existence of a locally dense set of real polynomial automorphisms of C 2 displaying a wandering Fatou component; in particular this solves the problem of their existence, reported by Bedford and Smillie in 1991. These Fatou components have non-empty real trace and their statistical behavior is historical with high emergence. The proof is based on a geometric model for parameter families of surface real mappings. At a dense set of parameters, we show that the dynamics of the model displays a historical, high emergent, stable domain. We show that this model can be embedded into families of Hénon maps of explicit degree and also in an open and dense set of 5-parameter C r -families of surface diffeomorphisms in the Newhouse domain, for every 2 ≤ r ≤ ∞ and r = ω. This implies a complement of the work of Kiriki and Soma ( 2017), a proof of the last Taken's problem in the C ∞ and C ω -case. The main difficulty is that here perturbations are done only along finite-dimensional parameter families. The proof is based on the multi-renormalization introduced in [Ber18]. Contents Introduction: State of the art and main results 0.1. Wandering Fatou components 0.2. Statistical complexity of the dynamics: Emergence 0.3. Emergence of wandering stable components in the Newhouse domain 0.4. Outline of the proof and organization of the manuscript 1. The geometric model 1.1. System of type (A, C) 1.2. Unfolding of wild type 2. Examples of families displaying the geometric model 2.1. A simple example of unfolding of wild type (A, C) 2.2. Natural examples satisfying the geometric model 2.3. Proof of Theorem 2.10 3. Sufficient conditions for a wandering stable domain 3.1. Implicit representations and initial bounds 3.2. Normal form and definitions of the sets B j and Bj 3.3. Proof of Theorems 1.32 and 1.34 4. Parameter selection

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Section: Introduction: State Of the Art And Main Resultsmentioning

confidence: 99%