Distinguished Lecture——Exotic rotation domains and Herman rings for quadratic Hénon maps.
报告人：Raphaël Krikorian (École Polytechnique)
时间：2024-03-26 10:30-11:30
地点：智华楼四元厅224/225
Abstract: Quadratic Hénon maps are polynomial automorphism of $\mathbb{C}^2$ of the form $h:(x, y) \mapsto (\lambda^{1/2} (x^2+c)-\lambda y, x)$. They have constant Jacobian equal to $\lambda$ and they admit two fixed points. If $\lambda$ 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 many quasi-periodic orbits with two frequencies in the neighborhood of its fixed points. Surprisingly, in some hyperbolic cases, S. Ushiki observed some years ago what seems to be quasi-periodic orbits (though no Siegel disks exist). I will explain why this is the case. This theoretical framework also predicts (and proves), in the dissipative case ($\lambda$ of module less than 1), the existence of (attractive) Herman rings. These Herman rings, which were not observed before, can be produced in numerical experiments.
BIO: Raphaël Krikorian, is professor of the École Polytechnique in France. He was a Member of the University Institute of France. He is a leading expert on Hamiltonian dynamical systems and spectral theory of quasiperiodic Schrodinger operator. For example, collaborating with A. Avila, he established the global dichotomy of quasi-periodic cocycles, solved the zero-measure spectrum problem for the critical almost Mathieu operator; he proved the generic divergence of the Birkhoff normal form of a real analytic symplectic diffeomorphism on a two-dimensional disk. His papers have been published in renowned journals such as Annals of Mathematics, Inventiones Mathematicae, Publ. Math. IHES, and he was an invited speaker at the 2018 International Congress of Mathematicians.
