Non-uniform hyperbolicity in polynomial skew products
The dynamics of Topological Collet-Eckmann rational maps on Riemann sphere are well understood, due to the work of Przytycki, Rivera-Letelier and Smirnov. In this talk we study the dynamics of polynomial skew products of C^2. Let f be a polynomial skew products with an attracting invariant line L such that f restricted on L satisfies Topological Collet-Eckmann condition and a Weak Regularity condition. We show that the the Fatou set of f in the basin of L equals to the union of the basins of attracting cycles, and the Julia set of f in the basin of L has Lebesgue measure zero. As a consequence there are no wandering Fatou components in the basin of L (We remark that for some polynomial skew products with a parabolic invariant line L, there can exist a wandering Fatou component in the basin of L).
Steiner symmetrization and its applications in convex geometry
Steiner symmetization was introduced by Steiner in the 18th century. Many (affine) isoperimetric inequalities in convex geometry that characterize ellipsoids can be established by using this approach. In this talk, we will present some new developments and applications of Steiner’s approach, including the affine inequalities for sets of finite perimeter, and general affine invariances related to Mahler volume.
A criterion for the existence of physical measures for partially hyperbolic attractors
In the partially hyperbolic setting, Pesin-Sinai showed that Gibbs u-states always exist. However, the existence of physical measures or SRB measures is delicate. In this talk, I will present a criterion for the existence of physical measures for partially hyperbolic attractors with one dimensional center. The talk is based on a joint work with S. Crovisier and D. Yang.
Variational construction for homoclinic and heteroclinic orbits in the N-center problem
It is well-known that the N-center problem is chaotic when N ≥ 3. By regularizing collisions, one can associate the dynamics with a symbolic dynamical system which yields infinitely many periodic and chaotic orbits, possibly with collisions. it is a challenging task to construct chaotic orbits without any collision. In this talk we consider the planar N-center problem with collinear centers and N ≥ 3, and show that, for any fixed nonnegative energy and certain types of periodic free-time minimizers, there are infinitely many collision-free heteroclinic orbits connecting them. Our approach is based on minimization of a normalized action functional over paths within certain topological classes, and the exclusion of collision is based on some recent advances on local deformation methods. This is a joint work with Kuo-Chang Chen.
Dispersion for the discrete operators with absolutely continuous spectrum
In contrast to localization, dispersion for the discrete linear operator with absolutely continuous spectrum is related to some transport properties and the dispersive estimates are important in studying the corresponding nonlinear equations. We recall classical results and present some recent works for discrete Schrödinger operators.
Steady concentrated vorticities of the 2-D Euler equation and their stability
In this talk, we will consider the existence and uniqueness of steady concentrated vorticities of the 2-D incompressible Euler equation on smooth bounded domains and study their stability. Given steady non degenerate point vortices configurations, we construct such steady piece wisely constant vortex flows and study their linear stability. Steady concentrated Lipschitz continuous vorticities are also been considered. Both of them are highly concentrated near the given steady vortex points. This talk is mainly based on a joint work with Prof. Yiming Long and Prof. Chongchun Zeng.