Nonexistence of wandering domains for strongly dissipative infinitely renormalizable Hénon maps at the boundary of chaos

Ou, Dyi-Shing

doi:10.1007/s00222-019-00902-4

Cited by 6 publications

(3 citation statements)

References 60 publications

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“…(1) for every x ∈ C, the limit set of the orbit of x contains a horseshoe K, 4. The renormalizations we will deal with have unbounded combinatorics, in the case of bounded combinatorics, Ou [Ou19] showed recently the nonexistence of wandering stable components.…”

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%

Emergence of wandering stable components

Berger¹,

Biebler²

2020

Preprint

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“…Later, Friedland and Milnor [18] proved that every polynomial diffeomorphism is conjugate to a composition of Hénon maps or an elementary transformation. The fact shows the importance of Hénon maps for the dynamics of polynomial diffeomorphisms that are of special interests [1–3, 8, 31, 32, 36, 38]. Since the Jacobian determinant of is equal to b , the map is area preserving if .…”

Section: Two Important Two-dimensional Mappings In Natural Sciencementioning

confidence: 99%

Iterative roots of two-dimensional mappings

Yu¹,

Li²,

Matkowski³

2023

Proceedings of the Edinburgh Mathematical Society

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As a weak version of embedding flow, the problem of iterative roots is studied extensively in one dimension, especially in monotone case. There are few results in high dimensions because the constructive method dealing with monotone mappings is unavailable. In this paper, by introducing a kind of partial order, we define the monotonicity for two-dimensional mappings and then present some results on the existence of iterative roots for linear mappings, triangle-type mappings, and co-triangle-type mappings, respectively. Our theorems show that even the property of monotonicity for iterative roots of monotone mappings, which is a trivial result in one dimension, does not hold anymore in high dimensions. At the end of this paper, the problem of iterative roots for two well-known planar mappings, that is, Hénon mappings and coupled logistic mappings, are also discussed.

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“…Parametric families of maps in the plane, and especially the Hénon and Lozi maps, have been widely studied in many different contexts and a lot of important results have been proved; see for example [5], [6], [10], [17], [18], [19], [23], [26], [29], [32].…”

Section: Introductionmentioning

confidence: 99%

Densely branching trees as models for Hénon-like and Lozi-like attractors

Boroński¹,

Štimac²

2021

Preprint

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Inspired by a recent work of Crovisier and Pujals on mildly dissipative diffeomorphisms of the plane, we show that Hénon-like and Lozi-like maps on their strange attractors are conjugate to natural extensions (a.k.a. shift homeomorphisms on inverse limits) of maps on metric trees with dense set of branch points. In consequence, these trees very well approximate the topology of the attractors, and the maps on them give good models of the dynamics. To the best of our knowledge, these are the first examples of canonical two-parameter families of attractors in the plane for which one is guaranteed such a 1-dimensional locally connected model tying together topology and dynamics of these attractors. For Hénon maps this applies to Benedicks-Carleson positive Lebesgue measure parameter set, and sheds more light onto the result of Barge from 1987, who showed that there exist parameter values for which Hénon maps on their attractors are not natural extensions of any maps on branched 1-manifolds. For Lozi maps the result applies to an open set of parameters given by Misiurewicz in 1980. In general, under mild dissipation, conjugacy holds if the attractor does not have an arc in common with a stable manifold, and the density of branch points holds when the attractor is a homoclinic class. Our result can be seen as a generalization to the non-uniformly hyperbolic world of a classical result of Williams from 1967.

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Nonexistence of wandering domains for strongly dissipative infinitely renormalizable Hénon maps at the boundary of chaos

Abstract: This article generalizes the nonexistence of wandering domains from unimodal maps to infinitely period-doubling renormalizable Hénon-like maps in the strongly dissipative (area contracting) regime. This solves an open problem proposed by van Strien (

Cited by 6 publications

References 60 publications

Emergence of wandering stable components

Emergence of wandering stable components

Iterative roots of two-dimensional mappings

Densely branching trees as models for Hénon-like and Lozi-like attractors

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