Advances in Dynamics Seminar
December 11, 2025 9pm - 12am Beijing Time
Zoom ID: 827 8743 3327
Passcode: 692502
9pm - 10pm: Raphaël Krikorian (Ecole Polytechnique)
Title: Exotic rotation domains and Herman rings for quadratic Hénon maps
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 mathematically 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.
10pm - 11pm: Bertrand Deroin (CNRS - CYU)
Title: Towards a Structural Stability Theory for Holomorphic Foliations on Algebraic Complex Surfaces
Abstract: I'll review work done in collaboration with Aurélien Alvarez, aiming at developing a theory of structural stability for holomorphic foliations on compact complex surfaces. Important new examples are the Jouanolou foliations of the complex projective plane, and the fundamental properties they satisfy allow us to define a more general family of foliations on arbitrary algebraic surfaces, which we call Jouanolou-type foliations. I will present these conditions, as well as some of their properties, and, time permitting, I will state a number of conjectures.
11pm - 12am: Giovanni Forni (CYU - University of Maryland)
Title: On the dynamics of billiards in polygons
Abstract: In this talk we will survey several results on the dynamics of billiards in polygons. These include results on their ergodic theory (weak mixing), KAM-type results on stability of invariant surfaces of rational billiards under perturbations and existence of periodic orbits. Part of the work is in collaboration with F. Arana Herrera and J. Chaika.
