Research on Complex Dynamics

在这个研究项目中，我们研究了复杂动力学及其相关领域的各种问题，如实动力学、函数理论、Kleininan群和Teichmuller空间。在前人的研究中，证明了Mandelbrot集的边界以及从该边界出发的一般参数的二次Julia集的边界具有Hausdorff维数2。下一个自然的问题是这些集合的勒贝格测度是否为零。当所有周期点相互排斥且映射不是无限可重整时，我们可以肯定地回答这个问题。主要方法有：Yoccoz谜题分割法、tau函数组合分析法和环空模面积不等式法。利用类似的思想，我们还研究了二次多项式的刚性问题。我们证明了实映射的刚性问题可以简化为通用Teichmuller空间的自映射问题，并且证明了自映射在基点位移上的一致收缩和先验界足以保证无限可重整的实数二次多项式的刚性。从复点的角度研究了实类二次映射的单调性问题。讨论了全纯二次微分和法图坐标作用下的Ruelle算子无限振荡的可能性。对于保护空间上的全纯映射，给出了2维和3维全不变型的分类。并构造了一个将latte的例子推广到高维的例子。这给出了一类新的临界有限映射的例子。与S. Matsumoto （Nihon university）合作，研究了无限大旋转上的一类歪斜积映射，并对具有最小集的映射进行了刻画。为了对高维复杂动力学的研究有一个新的视角，我们邀请了Serge Cantat教授（法国雷恩大学）就K3表面上的动力学进行了一系列的演讲。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，这有助于对Julia集合、Mandelbrot集合和Yoccoz谜题等的动力学和可视化的理解。少

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