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Geometric Study of Complex Dynamical Systems

复杂动力系统的几何研究

基本信息

批准号：
09440029
负责人：
SHISHIKURA Mitsuhiro
金额：
$ 5.18万
依托单位：
The University of Tokyo
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

In this project, we studied theory of one-dimensional and higher dimensional complex dynamical systems and related real dynamical systems.Results in one-dimensional dynamics : A new proof for the real quadratic polynomial family, by reducing it to the strong contraction of certain self-map of the universal Teichmuller space. A reproduction of the Abel functions for parabolic fixed points via Ecalle's theory of resurgent functions. The properties of eigenvalues and eigenfunctions of complexified Ruelle-Perron-Frobebius operator. Proof of the topological completeness of decorated exponential famillies. Proof of the monotonicity of the real polynomial family x^<2n>+c via an action onto the cotangent space of Teichmuller space.Results in higher dimensional dynamics : We completely determined completely invariant subvarieties of holomorphic mappings of P^n. A construction of a new critically finite holomorphic mapping of P^n. The existence of absolutely continuous invariant measure (wrt Lebesgue measure) for certain real expanding map in higher dimension.Numerical experiments : We studied the bifurcation of complex dynamical systems, Julia sets, renormalization and the universality phenomena, via various numerical experiments on computers.

本项目研究了一维和高维复杂动力系统及相关的真实的动力系统的理论，在一维动力学中得到了：通过将真实的二次多项式族归结为泛Teichmuller空间的某个自映射的强压缩，得到了该多项式族的一个新的证明。通过Ecalle的复活函数理论再现抛物不动点的Abel函数。复化Ruelle-Perron-Frobebius算子的特征值和特征函数的性质修饰指数族拓扑完备性的证明。证明了真实的多项式族x^<2n>+c在Teichmuller空间的余切空间上的作用的单调性。结果在高维动力学中：我们完全确定了P^n的全纯映射的完全不变子簇。P^n中一个新的临界有限全纯映射的构造在高维空间中某些真实的扩张映射的绝对连续不变测度（wrt Lebesgue测度）的存在性。数值实验：通过计算机上的各种数值实验，我们研究了复杂动力系统的分支、Julia集、重整化和普适性现象。