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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2511.01044 |
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| _version_ | 1866915592388214784 |
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| author | Krikorian, Raphaël |
| author_facet | Krikorian, Raphaël |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_01044 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence of Exotic rotation domains and Herman rings for quadratic Hénon maps Krikorian, Raphaël Dynamical Systems 37 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. |
| title | Existence of Exotic rotation domains and Herman rings for quadratic Hénon maps |
| topic | Dynamical Systems 37 |
| url | https://arxiv.org/abs/2511.01044 |