Existence of Exotic rotation domains and Herman rings for quadratic H\'enon maps

Abstract

A quadratic H\'enon map is an automorphism of $\C^2$ of the form $h:(x,y)\mapsto (\l^{1/2} (x^2+c)-\l y,x)$. It has a constant Jacobian equal to $\l$ and has two fixed points. If $\lambda$ 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 ($|\l|<1$) 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.