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Existence of Exotic rotation domains and Herman rings for quadratic Hénon maps

Raphaël Krikorian

TL;DR

The paper proves the existence of exotic rotation domains for quadratic Hénon maps in both the a priori elliptic (conservative) and a priori hyperbolic (dissipative) regimes, and it establishes attracting Herman rings in the dissipative case. The authors develop a rigorous multi-scale framework combining resonant Birkhoff normal forms, vector-field reductions, invariant annulus theory, and a renormalization scheme based on commuting pairs, ultimately reducing the dynamics to a real-analytic vector field whose flow yields invariant annuli and rotating behavior. A KAM-Siegel type theorem for commuting pairs is used to linearize the reduced dynamics and produce invariant tori or annuli, while reversibility and symmetry constraints ensure the constructed objects persist under parameter variations. The work explains Ushiki’s numerical observations, provides a mechanism to locate Herman rings, and yields precise parameter regimes and localization results near $\tau\approx 1$ as well as more general locations. Overall, the results significantly advance the understanding of high-dimensional complex dynamics for Hénon maps by linking local normal-form structure to global invariant objects through a robust renormalization and KAM framework.

Abstract

A quadratic Hénon map is an automorphism of $\C^2$ of the form $h:(x,y)\mapsto (ł^{1/2} (x^2+c)-ły,x)$. It has a constant Jacobian equal to $ł$ and has two fixed points. If $λ$ is on the unit circle (one says $h$ is conservative) these fixed points can be both elliptic or both hyperbolic. In the elliptic case, under an additional Diophantine condition, a simple application of Siegel Theorem shows that $h$ admits quasi-periodic orbits with two frequencies in the neighborhood of its fixed points. Surprisingly, in some hyperbolic cases, Shigehiro Ushiki observed numerically what seems to be quasi-periodic orbits belonging to some ``Exotic rotation domains'' though no Siegel disk is associated to the fixed points. The aim of this paper is to explain and prove the existence of these ``Exotic rotation domains''. Our method also applies to the dissipative case ($|ł|<1$) and allows to prove the existence of attracting Herman rings. The theoretical framework we develop permits to produce numerically these Herman rings that were never observed before.

Existence of Exotic rotation domains and Herman rings for quadratic Hénon maps

TL;DR

The paper proves the existence of exotic rotation domains for quadratic Hénon maps in both the a priori elliptic (conservative) and a priori hyperbolic (dissipative) regimes, and it establishes attracting Herman rings in the dissipative case. The authors develop a rigorous multi-scale framework combining resonant Birkhoff normal forms, vector-field reductions, invariant annulus theory, and a renormalization scheme based on commuting pairs, ultimately reducing the dynamics to a real-analytic vector field whose flow yields invariant annuli and rotating behavior. A KAM-Siegel type theorem for commuting pairs is used to linearize the reduced dynamics and produce invariant tori or annuli, while reversibility and symmetry constraints ensure the constructed objects persist under parameter variations. The work explains Ushiki’s numerical observations, provides a mechanism to locate Herman rings, and yields precise parameter regimes and localization results near as well as more general locations. Overall, the results significantly advance the understanding of high-dimensional complex dynamics for Hénon maps by linking local normal-form structure to global invariant objects through a robust renormalization and KAM framework.

Abstract

A quadratic Hénon map is an automorphism of of the form . It has a constant Jacobian equal to and has two fixed points. If is on the unit circle (one says is conservative) these fixed points can be both elliptic or both hyperbolic. In the elliptic case, under an additional Diophantine condition, a simple application of Siegel Theorem shows that admits quasi-periodic orbits with two frequencies in the neighborhood of its fixed points. Surprisingly, in some hyperbolic cases, Shigehiro Ushiki observed numerically what seems to be quasi-periodic orbits belonging to some ``Exotic rotation domains'' though no Siegel disk is associated to the fixed points. The aim of this paper is to explain and prove the existence of these ``Exotic rotation domains''. Our method also applies to the dissipative case () and allows to prove the existence of attracting Herman rings. The theoretical framework we develop permits to produce numerically these Herman rings that were never observed before.

Paper Structure

This paper contains 143 sections, 115 theorems, 1257 equations, 23 figures.

Table of Contents

  1. Hénon maps, Exotic Rotation Domains and Herman Rings
  2. Hénon maps
  3. Dynamics of Hénon maps
  4. Real Hénon maps
  5. Complex Hénon maps
  6. Rotation domains
  7. Definition
  8. Classification
  9. Shigehiro Ushiki's numerical experiments
  10. Herman rings
  11. Results
  12. Existence of Exotic rotation domains
  13. On reversibility
  14. Elliptic vs. hyperbolic case
  15. Existence of Herman rings in the dissipative case
  16. ...and 128 more sections

Key Result

Theorem 1

A holomorphic germ $f:({\mathbb C}^2,(0,0))\righttoleftarrow$ of the form $f(z,w)=(e^{2\pi i \alpha_{1}}z,e^{2\pi i \alpha_{2}}w)+O^2(z,w)$ where $(\alpha_{1},\alpha_{2})\in {\mathbb R}^2$ satisfies a Diophantine condition ($C>0, \tau>0$) is linearizable in a neighborhood of $(0,0)$: there exists $g:({\mathbb C}^2,(0,0))\righttoleftarrow$ such that

Figures (23)

  • Figure 1: S. Ushiki's example. Iteration of the map $h_{\beta,c}$ with $\beta=0.327136$, $c=0.269343$. The curve represents (after the scaling $(z,w)\mapsto (20\times (z-0.5),20\times (w-0.58))$) $(\Re(z),\Re(w))$ after 5000 iterations. The initial condition is $(z_{*},w_{*})$ avec $z_*= 0.3512857-0.352772 \sqrt{-1}$, $w_*= 0.3856867+0.353207\sqrt{-1}$.
  • Figure 2: A Herman ring for the Hénon map $h:(x,y)\mapsto (e^{i\pi \beta}(x^2+c)-e^{2\pi i \beta} y,x)$, $\beta= 0.3289999+0.0043333\sqrt{-1}$, $c=0.2619897-0.0088858\sqrt{-1}$. Initial condition $(z_{*},w_{*})$, $z_{*}=0.44672099-0.16062292\sqrt{-1}$, $w_{*}= 0.3961953+0.149208\sqrt{-1}$. $N=5000$ iterations. The cyan curve is the projection $(\Im z,\Im w)$ and the red and blue curves (that coincide) the projections $(\Re z,\Im z)$, $(\Re w, \Im w)$.
  • Figure 3: A Herman ring for the Hénon map $h:(x,y)\mapsto (e^{i\pi \beta}(x^2+c)-e^{2\pi i \beta} y,x)$, $\beta= 0.33121126+0.00218737 \sqrt{-1}$$c= 0.2557783-0.00497994 \sqrt{-1}$ Initial condition $(z_{*},w_{*})$, $z_{*}= 0.471458035-0.113447719\sqrt{-1}$$w_{*}= 0.41305318+0.0975217\sqrt{-1}$ Number of iteration $N= 7000$. The cyan curve is the projection $(\Im z,\Im w)$ and the red and blue curves (that coincide and give the violet curve) the projections $(\Re z,\Im z)$, $(\Re w, \Im w)$. The picture is scaled by a factor 5
  • Figure 4: Domain of reversibility.
  • Figure 5: S. Ushiki's example after a change of coordinates (BNF and scaling). Parameters $\mathring{\beta}=(-1.8592)/3,\qquad \mathring{\alpha}=(-0.8846+2.67\sqrt{-1})/3$, $\delta=0.01$; initial condition $(z_{*},w_{*})$, $z_{*}=2.3+3.5\sqrt{-1}$, $w_{*}=-3.8+7.2\sqrt{-1}$. 5000 iterations. The red (resp. blue) curve is the projection of the orbit on the $z$-coordinate (resp. $w$-coordinate). Scaling factor of the picture $0.1$.
  • ...and 18 more figures

Theorems & Definitions (147)

  • Definition 1.1: Rotation domain
  • Theorem : Siegel
  • Definition 1.2
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • Theorem A: Existence of Exotic Rotation Domains, Elliptic case
  • Theorem A': Existence of Exotic Rotation Domains, Hyperbolic case
  • Theorem B: Existence of Herman Rings
  • Remark 3.1
  • ...and 137 more