星型哈密顿能量面上的周期轨道

给定能量面上的周期轨道(称之为闭特征)是哈密顿动力系统中重要的研究对象。紧星型超曲面上闭特征的多重性和稳定性还有许多未解决的问题,它们也是动力系统、辛几何、非线性分析等数学领域关心的重要问题。本项目主要研究紧星型超曲面上闭特征的多重性和稳定性：当n大于2时，2n维欧式空间中紧星型超曲面上是否至少存在两个几何不同的闭特征；什么情形下存在椭圆的闭特征；利用Maslov型指标与其迭代理论，能否得到更好的闭特征数目下界估计；当紧星型超曲面是对称时，以上问题会得出何种结果。对于闭特征，建立类似于闭测地线研究中的Hingston定理，并结合近期建立的紧星型超曲面上闭特征的共振恒等式进一步考虑闭特征问题。此外，本项目还将利用Floer同调及切触同调的方法，并从微分动力系统、拓扑动力系统和遍历理论的角度去考虑闭特征问题。

Periodic orbits(called closed characteristics) on given energy surfaces are important objects in Hamiltonian system. There are many open problems about the multiplicity and stability of closed characteristics on compact star-shaped hypersurfaces, which are also important problems concerned in Dynamical System、Symplectic Geometry、Nonlinear Analysis, etc. In this project we study the multiplicity and stability of closed characteristics on compact star-shaped hypersurfaces: When n is great than 2, do there exist at least two geometrically distinct closed characteristics on every compact star-shaped hypersurface in the 2n-dimensional Euclidean space；Under what cases there is an elliptic closed characteristic；Using Maslov index and its iteration theory, can we obtain a more better lower bound of the number of closed characteristics；How about the results when the compact star-shaped hypersurface is symmetric. For closed characteristics, we shall establish a theorem similar to Hingston' which was established for closed geodesics, and combine it with the recently established resonance identities of closed characteristics on compact star-shaped hypersurfaces to consider the problem further. In addition, in this project we shall also use the methods of Floer Homology, Contact Homology and consider problems in view of Differential Dynamical System、Topological Dynamical System and Ergodic Theory.