Wednesday October 16th, 2024 | |
| Time: | 2:05 PM - 3:05 PM |
| Title: | Exotic rotation domains and Herman rings for quadratic Hénon maps. |
| Speaker: | Raphael Krikorian, Ecole Polytechnique |
| Abstract: | |
| Quadratic H'enon maps are polynomial automorphism of $\mathbb{C}^2$ of the form $h\colon (x,y) → (λ^{1/2}(x^2+c)-λ y, x)$. They have constant Jacobian equal to $lambda$ and they admit two fixed points. If $λ$ is on the unit circle (one says the map $h$ is conservative) these fixed points can be elliptic or hyperbolic. In the elliptic case, a simple application of Siegel Theorem shows (under a Diophantine assumption) that $h$ admits an open set of quasi-periodic orbits with two frequencies in the neighborhood of its fixed points. Fundamental results by Bedford-Smillie and Barrett-Bedford-Dadok suggest that there could exist more general ``Rotation domains'', i.e. open sets filled with quasi-periodic motions, but without fixed points (the rotation domains are then said to be ``exotic''). In some hyperbolic cases, Shigehiro Ushiki observed numerically some years ago, what seems to be such ``Exotic rotation domains'' (quasi-periodic orbits though no Siegel disks exist). I will give a proof of the existence of such ERDs and provide a mathematical explanation for S. Ushiki's discovery. In the dissipative case ($λ$ of module less than 1), the same theoretical framework, predicts, and provides a proof of, the existence of (attracting) Herman rings. These Herman rings, which were not observed before, can be systematically produced in numerical experiments. | |