An attempt toward a solution to the C^r-stability conjecture (r【greater than or equal】2)

尝试解决C^r稳定性猜想（r【大于或等于】2）

基本信息

批准号：
15540197
负责人：
SAKAI Kazuhiro
金额：
$ 1.34万
依托单位：
Utsunomiya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540197/
关键词：
shadowing property stability conjecture bifurcation theory chaos theory Pesin theory center manifold ergodic theory Lyapunov exponent Shadowing property shadowing property

项目摘要

The dynamical systems theory originated from two notions of hyperbolicity and structural stability, and researches toward a solution to the stability conjecture played an important role in the developments of the theory. The conjecture asserts every structurally stable system is hyperbolic, and in 1987 it was proved by Mane for r=1. In the proof, so-called Franks lemma was essential. Since the lemma does not work for the C^r-topology (r【greater than or equal】2), the conjecture is still open when r【greater than or equal】2. The purpose of this research project is to prove the hyperbolicity under the shadowing-C^r-open condition (r【greater than or equal】2) fusing with Pesin theory, and by applying the techniques obtained in this process, we try to solve the C^r-stability conjecture for r 【greater than or equal】 2.In 2003, we restrict ourselves to 2-dimensional dynamical systems and concentrated to prove the hyperbolicity of the system under the shadowing-C^r-open condition. In 2004, we continuously proceeded the above strategy, but there were noting special for publication. However, for some partial results obtained in this research, we have found some handles to generalize them for higher dimensions. In our opinion, the achieve percentage of this project might be evaluated 50%.Before to show the hyperbolicity of the dynamical systems, it is necessary to prove the hyperbolicity of the periodic points. Under the shadowing-C^r-open condition (r【greater than or equal】2), the head investigator proved the hyperbolicity of the periodic points etc., and making use of the facts, he also proved the hyperbolicity for 2-dimensional dynamical systems by assuming additional conditions (as was stated it turned out that this result can be generalized).A base of this research project has been completed. Hereafter we would like to do my best to complete the project.

动态系统理论源自双曲和结构稳定性的两个音符，并研究解决稳定概念的解决方案在理论的发展中起着重要作用。该概念断言每个结构稳定的系统都是双曲线的，在1987年，Mane证明了R = 1。在证据中，所谓的弗兰克斯引理至关重要。由于引理对C r-Topology不起作用（R [大于或相等] 2），因此当R大于或等于】2时，猜想仍然是打开的。 The purpose of this research project is to prove the hyperbolicity under the shadowing-C^r-open condition (r【greater than or equal】2) fusing with Pesin theory, and by applying the techniques obtained in this process, we try to solve the C^r-stability concept for r【greater than or equal】2.In 2003, we restrict ourselves to 2-dimensional dynamical systems and concentrated to prove the hyperbolicity of the system under the阴影c^r-open条件。在2004年，我们继续采取上述战略，但有特殊的出版物。但是，对于这项研究中获得的一些部分结果，我们发现了一些将它们推广到更高维度的处理方法。我们认为，该项目的成就百分比可能会被评估为50％。在显示动态系统的双曲线之前，有必要证明周期性点的透明度。在阴影-C^r-open条件（R [大于或相等] 2）下，首席调查员证明了周期点等的双透色性并利用事实，他还证明了通过假设其他条件的二维动态系统的夸张性（正如所指出的那样，结果证明这是该研究项目的基础）。此后，我们想尽力完成该项目。