Study of higher dimensional complex dynamical systems

高维复杂动力系统的研究

基本信息

批准号：
09640232
负责人：
NISHIMURA Yasuichiro
金额：
$ 0.7万
依托单位：
Osaka Medical College
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
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-09640232/
关键词：
complex dynamics iteration Julia set Fatou set Henon map iteration Fatcu集合 Heron写像

项目摘要

Nishimura and Yasuda studied dynamical systems of birational maps of two-dimensional complex projective space P^2. In the study of dynamics of rational maps, it appear some phen, omena which do not appeal in the study of dynamics of holomorphic maps, and these phenomena cause some problems, that is, tlie problem of distribution of inderterminate points and the problem of degree lowering of homogeneous polynomial representation. In order to understand these phenomena, we investigated in detail the family of birational polynomial quadratic maps psi and their inverse maps psi. In the case of maps psi, we expressed the mechanism of the way how the degree lowering occurs and we described concretely the distribution of inderminate points. We have successfully picked out the algebraic curves C_n of common divisers which appears when the degree lowering occurs at the n-times iteration psi^n. We also proved that the set of all indeterminate points of the iterated maps psi^n coincides with the set of intersection points of all pairs of the algebraic curves C_m and C_n.Though our research is of purely mathrmatical, Nishimura implemented a computer program of two dimensional complex dynamical systems which help the pure mathematical research. The computation of iteration by using the homogeneous coordinate system of projective space was tried.Ueda studied the complex dynamics on the n-dimensional projective space P^n. Specifically, the case of critically finite maps, that is the case when the orbits of the branch points constitute an algebraic sets, were studied deeply. In this case, he proved that the Fatou maps are constant maps, the Julia set coincides the whole space P^n and that the set of repelling periodic points is dense in the whole space P^n Furthermore, he classified the quadratic maps of P^2 and constructed some examples of critically finite maps of P^2.

Nishimura和Yasuda研究了二维复合物射击空间P^2的生育图的动力学系统。在理性图动力学的研究中，它似乎是某种现象，在研究全体形态图动力学的研究中没有吸引人，而这些现象引起了一些问题，即，降低了非终端点的分布问题以及降低同质多项式表示的程度问题。为了理解这些现象，我们详细研究了Birational多项式二次地图PSI及其倒数映射PSI的家族。在地图psi的情况下，我们表达了降低程度降低方式的机制，并且我们巧妙地描述了倒线点的分布。我们已经成功挑选了公共分裂器的代数曲线C_N，当降低度降低时出现在n-times Iteration psi^n时。我们还证明了迭代地图的所有不确定点的集合psi^n与代数曲线和C_N对所有对的一组相交点相吻合。尝试了使用投影空间的均匀坐标系来计算迭代的计算。Ueda研究了N维射击空间p^n上的复杂动力学。具体而言，深入研究了分支点的轨道构成代数集时，严重有限地图的情况是这种情况。在这种情况下，他证明了fatou地图是恒定的地图，朱莉娅集合整个空间p^n，并且在整个空间p^n中，一组排斥周期点的浓度是密集的，他对P^2的二次图进行了分类，并构建了一些杰出的示例P^2的有限限制映射。