权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Various problems in Hamiltonian dynamical systems and related topics in geometry and analysis

哈密顿动力系统中的各种问题以及几何和分析中的相关主题

基本信息

批准号：
09440050
负责人：
ITO Hidekazu
金额：
$ 2.69万
依托单位：
TOKYO INSTITUTE OF TECHNOLOGY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

项目摘要

The following is the abstract for the main results obtained under this research project.1. In the research of integrable systems, Miyaoka proved that all isoparametric hypersurfaces in the sphere with six principal curvatures are homogeneous. For the proof, she used the isospectrality of the family of the shape operators on the focal set of isoparametric hypersurfaces. This is a remarkable result to solve the conjecture by Yau, and shows close connection between the theory of hypersurfaces and that of integrable systems.2. In the research of ergodic theory, using transfer operators method, Morita obtained Fredholm determinant representation for the Selberg zeta function Z(s) of closed geodesics on hyperbolic Riemann surface with finite area. He investigated the spectral properties of the transfer operators, and then obtained some analytic information of Z(s).3. By using variational methods, Tanaka studied unbounded solutions of singular Hamiltonian systems of the two-body type. Namely he proved the existence of a hyperbolic-like solutions for a class of singular potentials with the strong force alpha > 2. Also, he studied the prescribed energy problem for Hamiltonian systems of the same form with alpha = 2, and showed variationally the existence of periodic solutions on the zero energy surface. This led to an existence theorem of closed geodesics on noncompact Riemannian manifold.4. In the research of symplectic geometry, Ono solved the Arnold's conjecture for general closed symplectic manifold by constructing Floer homology for periodic Hamiltonian and Gromov-Witten invariant. He generalized this approach further and studied constructions of Floer homology for Lagrangian intersection.5. In the research of complex dynamical systems, Shiga showed some similar properties between limit sets of Kleinian group and Julia sets in theory of complex dynamical systems.

以下是本研究项目取得的主要成果摘要。在可积系统的研究中，Miyaoka证明了球面上所有具有六个主曲率的等参超曲面都是齐次的。为了证明，她使用了等参超曲面的焦点集上的形状算子族的等谱性。这是解决丘氏猜想的一个显著结果，表明了超曲面理论与可积系统理论之间的密切联系.在遍历理论的研究中，Morita利用转移算子方法，得到了有限面积双曲Riemann曲面上闭测地线的Selberg zeta函数Z（s）的Fredholm行列式表示.他研究了转移算子的谱性质，从而得到了Z（s）的一些解析信息.田中利用变分方法研究了二体型奇异汉密尔顿系统的无界解。也就是说，他证明了存在的双曲型解决方案一类奇异的潜力与强大的力量α> 2。此外，他研究了规定的能源问题的哈密顿系统的相同形式与α = 2，并显示变分存在的周期性解决方案的零能量表面。这导致了非紧黎曼流形上的闭测地线的存在定理。在辛几何的研究中，Ono通过构造周期Hamilton和Gromov-Witten不变量的Floer同调，解决了一般闭辛流形的Arnold猜想。他进一步推广了这种方法，并研究了拉格朗日相交的Floer同调结构。5.在复杂动力系统的研究中，滋贺指出了Kleinian群的极限集与复杂动力系统理论中Julia集的一些相似性质。