2025/11/02 by Raphaël Krikorian, Krikorian, Raphaël
Mathematics · Physics and Astronomy · #Mathematical Dynamics and Fractals #Advanced Differential Equations and Dynamical Systems #Chaos control and synchronization
paper · pdf · doi:10.48550/arxiv.2511.01044
A quadratic Hénon map is an automorphism of \C2 of the form h:(x,y)↦ (ł1/2 (x2+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.