Analysis of dynamical systems and related topics in geometry

以下是本研究项目取得的主要成果摘要。在哈密顿系统的研究中，伊藤把哈密顿系统的完全可积性推广到一般向量场的完全可积性。他证明，可积性的分析向量场是相当于存在一个收敛的正规化变换附近的一个equilibrium点是非共振和椭圆。它给出了Poincare中心问题的一个解答.在遍历理论的研究中，Morita研究了二维散射台球问题中的zeta函数。他成功地将它亚纯地推广到一个半平面，其真实的部分大于一些负常数。在动力系统分支理论的研究中，Kokubu研究了Conley指标理论在奇摄动向量场慢-快系统中的推广。他定义了当慢变量为一维时的转移矩阵，并得到了证明周期或异宿轨道存在性的一般方法.田中利用奇异汉密尔顿系统的变分方法证明了以下轨道的存在性：（1）散射型;（2）-1/γ^2型扰动类下的周期轨道;（3）势有两个奇点的系统的无界和混沌运动。5.在辛/接触几何的研究中，Ono成功地构造了整系数的Floer同调。Nakai从叶理理论和网几何的角度研究了一阶偏微分方程。特别是，他定义了仿射连接的那些偏微分方程的有限型，并利用它们来研究奇点与叶理定义的解决方案。

The following is the abstract for the main results obtained under this research project.1. In the research of Hamiltonian systems, Ito generalized the notion of complete integrability for Hamiltonian systems to that for general vector fields. He proved that the integrability of an analytic vector field is equivalent to the existence of a convergent normalizing transformation near an equlibrium point that are non-resonant and elliptic. It gives an answer to the Poincare center problem.2. In the research of ergodic theory, Morita studied the zeta function associated with two dimensional scattering billiards problem. He succeeded in extending it meromorphically to a half plane with its real part greater than some negative constant.3. In the research of bifurcation theory of dynamical systems, Kokubu studied the generalization of Conley index theory to slow-fast systems which are singularly perturbed vector fields. He defined transition matrices when the slow variables are of dimension one, and obtained a general method for proving the existence of periodic or heteroclinic orbits.4. By using variational method for singular Hamiltonian systems, Tanaka proved the existence of orbits such as (1) scattering type ; (2) periodic orbits under the class of perturbation of type -1/γ^2 ; (3) unbounded and chaotic motions for systems whose potential have two singular points.5. In the research of symplectic/contact geometry, Ono succeeded in constructing the Floer homology with integer coefficients. Nakai studied 1st order PDE's from the viewpoint of foliation theory and Web geometry. In particular, he defined affine connections for those PDE's with finite type, and used them to study the singularities associated with the foliation defined by their solutions.