Research on Complex Dynamics

复杂动力学研究

• 批准号: 11440053
• 负责人: SHISHIKURA Mitsuhiro
• 金额: $ 8.51万
• 依托单位: KYOTO UNIVERSITY (2001)Hiroshima University (1999-2000)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11440053/
• 关键词: Complex dynamics; Fractal; Chaos; Mandelbrot set; Julia set; renormalization; rigidity; bifurcation; 力学系; 単峰写像

In this research project, we studied various problems in complex dynamics and related fields, such as real dynamics, function theory, Kleininan groups and Teichmuller spaces.In previous research, it was proved that the boundary of the Mandelbrot set as well as quadratic Julia sets for generic parameters from the boundary have Hausdorff dimension two. The next natural question is whether these sets have Lebesgue measure zero. We were able to answer this question affirmatively when all periodic points are repelling and the map is not infinitely renormalizable. The main methods are Yoccoz puzzle partion, combinatorial analysis through tau-functions and the modulus-area inequality for annuli.Using a similar idea, we also studied the rigidity problem of quadratic polynomials. We showed that the rigidity problem for real maps can be reduced to the problem of a self map of the universal Teichmuller space, and that the uniform contraction and apriori bound on the displacement of the base point … More for the self map is enough to ensure the rigidity for infinitely renormalizable real quadratic polynmials.The monotonicity problem of real quadratic-like maps was also studied via comple point of view. The possibility of infinite oscillation was discussed in connection with Ruelle operators acting of holomorphic quadratic differentials and Fatou coordinates.For holomorphic maps on protective spaces, the classification of totally invariant varieties is given for the case of dimension 2 and 3. A generalization of Lattes example to higher dimension was also constructed. This gives a new class of examples of critically finite maps.With S. Matsumoto (Nihon Univ.), we studied a class of skew product map on an infinite annulus over an irrational rotation and characterized those maps which have minimal sets.In order to have perspaectives on the research on higher dimensional complex dynamics, we invied Prof. Serge Cantat (Universte de Rennes, France) to give a series of talks on the dynamics on K3 surfaces.Numerical experiments on varius complex dynamical systems including quadratic polynomials have been done, and this contributed the understanding of dynamics and visualization of Julia sets, Mandelbrot set and Yoccoz puzzles etc. Less

本课题研究了复动力学及其相关领域中的各种问题，如真实的动力学、函数论、Kleinininan群和Teichmuller空间，在以前的研究中，我们证明了Mandelbrot集的边界以及从边界出发的一般参数的二次Julia集的Hausdorff维数为2。下一个自然的问题是这些集合是否具有勒贝格测度为零。当所有的周期点都是相斥的，并且映射不是无限可重正化的时，我们能够肯定地回答这个问题。主要方法有Yoccoz难题分解、τ-函数组合分析和环的模面积不等式，并利用类似的思想研究了二次多项式的刚性问题。证明了真实的映射的刚性问题可归结为泛Teichmuller空间的自映射问题，并证明了基点位移的一致压缩和先验界 ...更多信息 对于自映射，证明了自映射的单调性足以保证无穷可重正化真实的二次多项式的刚性，并从复的观点研究了真实的类二次映射的单调性问题.讨论了全纯二次微分和Fatou坐标作用下的Ruelle算子无穷振动的可能性，给出了保护空间上全纯映射的2维和3维全不变簇的分类.还构造了Lattes例子到高维的推广。这给出了一类新的临界有限映射的例子。松本（日本大学），本文研究了无限环上一类无理旋转上的斜积映射，并刻画了这类映射的极小集.为了对高维复动力学的研究有所启发，我们邀请了SergeCantat教授（雷恩大学，法国）的对K_3曲面上的动力学进行了一系列的讨论，并对包括二次多项式在内的各种复杂动力系统进行了数值实验，完成，这有助于动力学的理解和可视化的朱莉娅集，Mandelbrot集和Yoccoz难题等。减

项目成果

Mitsuhiro Shishikura: "On a theorew of Mary Ress for the matings of pdyuomials"Tan Lei,Ed.TheMandelbrot set-The theme and variation. (in press). (2000)

Mitsuhiro Shishikura：“论玛丽·雷斯（Mary Ress）关于 pdyuomials 交配的理论”Tan Lei，Ed.The Mandelbrot set-主题和变奏。

S.Morosawa: "Julea sets of subhypertoolic rational functions"Complex varables. (掲載予定). (2000)

S.Morosawa：“Julea 集亚超工具有理函数”复变量（即将出版）。

S.Morosawa: "Holomorphic Dynamics"Cambridge University Press. 338 (2000)

S.Morosawa：《全纯动力学》剑桥大学出版社。

S.Morosawa: "Caratheodory convergence of Fatou compouents of Polynomids.--"Proceedings of the second congress ISACC. (掲載予定). (2000)

S.Morosawa：“多项式的 Fatou 组件的 Caratheodory 收敛。-”第二届 ISACC 大会论文集（待出版）。

Y.Akuyama: "Non-linearizability of polynomids at irrationally indifecat fixed points"Kodai Math.J.. 22. 56-65 (1999)

Y.Akuyama：“多项式在无理无理不动点处的非线性”Kodai Math.J.. 22. 56-65 (1999)