Geometric Study of Complex Dynamical Systems

在这个项目中，我们研究了一维和高维复动力系统及相关的实动力系统的理论。结果在一维动力学中：通过将实二次多项式族归结为泛Teichmuller空间的某一自映射的强压缩，给出了实二次多项式族的一个新的证明。利用Ecalle的复活函数理论，给出了抛物不动点的Abel函数的一个复现。复化的Ruelle-Perron-Frobebius算子的特征值和特征函数的性质。装饰指数族拓扑完备性的证明。通过在TeichMuller空间的余切空间上的作用证明了实多项式族x^&lt；2n&gt；+c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。构造了一个新的临界有限的P^n的全纯映射。对于某些高维的实扩张映射，证明了绝对连续不变测度(WRT勒贝格测度)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。

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.