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

TL;DR

This paper proves the existence of exotic rotation domains and Herman rings in quadratic Hénon maps, revealing new complex dynamics phenomena in both conservative and dissipative cases.

Contribution

It introduces a theoretical framework that explains and proves the existence of exotic rotation domains and Herman rings, including in dissipative cases, with numerical methods for their visualization.

Findings

Existence of exotic rotation domains in hyperbolic cases.

Existence of attracting Herman rings in dissipative cases.

Numerical production and visualization of Herman rings.

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.