Geometric Study of Complex Dynamical Systems

复杂动力系统的几何研究

• 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
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09440029/
• 关键词: Cpmplex dynamical systems; Julis set; Mandelbrot set; Fractal; Chaos; Teichmuller space; Renormalization; ジョリア集合; 力学系; エルゴード理論

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.

在这个项目中，我们研究了一维和高维复杂动力系统以及相关的真实动力系统的理论。一维动力学的结果：真实二次多项式族的新证明，通过将其简化为某些自的强收缩-通用 Teichmuller 空间的地图。通过埃卡勒复兴函数理论再现抛物线不动点的阿贝尔函数。复数 Ruelle-Perron-Frobebius 算子的特征值和特征函数的性质。装饰指数族的拓扑完整性证明。通过对 Teichmuller 空间的余切空间的作用证明实多项式族 x^<2n>+c 的单调性。导致更高维动力学：我们完全确定了 P^n 全纯映射的完全不变子类型。 P^n 的新临界有限全纯映射的构造。对于某些高维实展开图，存在绝对连续不变测度（勒贝格测度）。数值实验：我们通过计算机上的各种数值实验，研究了复杂动力系统的分岔、Julia 集、重正化和普适性现象。

项目成果

M.Tsujii: "A simple proof of monotonicity of entropy in the quadratic family" Ergodic theory and Dynamical Systems. (予定). (1999)

M.Tsujii：“二次族中熵单调性的简单证明”遍历理论和动力系统（计划）。

K.Matsuzaki and M.Taniguchi: "Hyperbolic manifolds and Kleunian groups" Oxford University Press, 253 (1998)

K.Matsuzaki 和 M.Taniguchi：“双曲流形和 Kleunian 群”，牛津大学出版社，253 (1998)

S.Nii: "An extension of the stability index for travelling wave solutions and its application for bifurcations" SIAM Journal on Mathematical Analysis. 28. 402-433 (1997)

S.Nii：“行波解稳定性指数的扩展及其在分岔中的应用”SIAM 数学分析杂志。

上田 哲生: "Complex Dynamics on P^n and Kobayashi metric" 数理解析研究所講究録. 998. 188-197 (1997)

Tetsuo Ueda：“P^n 和小林度量的复杂动力学”Kokyuroku，数学分析研究所 998. 188-197 (1997)。

H.Shiga & H.Tanigawa: "Projective structures on Riemann surfaces with discontinuous hotonomies" Trans.Amer.Math.Soc.(予定). (1999)

H.Shiga 和 H.Tanikawa：“具有不连续单调的黎曼曲面上的投影结构”Trans.Amer.Math.Soc（计划）。