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

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证明了具有六个主曲率的球面上的所有等参超曲面都是齐次的。作为证明，她使用了等参超曲面焦点集上形状算子族的等谱性质。这是解决Yau猜想的一个显著结果，显示了超曲面理论与可积系统理论的密切联系。在遍历理论的研究中，利用转移算符方法，Morita得到了有限面积双曲黎曼曲面上闭测地线的Selberg Zeta函数Z(S)的Fredholm型行列式表示。他研究了转移算子的谱性质，得到了Z(S)的一些解析信息。利用变分方法，田中研究了二体型奇异哈密顿系统的无界解。即证明了一类具有强力α&gt；2的奇异位势的类双曲解的存在性。他还研究了具有α=2的相同形式的哈密顿系统的规定能量问题，并变分地证明了零能面上周期解的存在性。这就引出了非紧黎曼流形上的闭测地线存在定理。在辛几何的研究中，Ono通过构造周期哈密顿量和Gromov-Witten不变量的Floer同调，解决了一般闭辛流形的Arnold猜想。他进一步推广了这一方法，并研究了拉格朗日交的Floer同调的构造。在复杂动力系统的研究中，Shiga在复杂动力系统理论中表现出了Kleian群的极限集与Julia集之间的一些相似的性质。

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.