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

以下是本课题主要研究成果的摘要。在可积系统的研究中，Miyaoka证明了六主曲率球面上的所有等参超曲面都是齐次的。为了证明这一点，她使用了等参超曲面焦点集上形状算子族的等谱性。这是解决丘氏猜想的一个显著结果，显示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利用传递算子方法，得到了有限面积双曲Riemann曲面上封闭测大地Selberg zeta函数Z(s)的Fredholm行列式表示。他研究了传递算子的谱性质，得到了Z(s)的一些解析信息。Tanaka利用变分方法研究了两体型奇异哈密顿系统的无界解。也就是说，他证明了一类具有强作用力的奇异势的双曲解的存在性。他还研究了α = 2的同形式哈密顿系统的规定能量问题，并变分地证明了零能量表面上周期解的存在性。由此得到了非紧黎曼流形上闭测地线的存在性定理。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。他进一步推广了这一方法，并研究了拉格朗日交点的花同调构造。在复杂动力系统的研究中，Shiga证明了Kleinian群的极限集与复杂动力系统理论中的Julia集之间的一些相似性质。

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.