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

动力系统理论起源于双曲性和结构稳定性两个概念，对稳定性猜想解的研究对该理论的发展起了重要作用。这个猜想断言每一个结构稳定的系统都是双曲的，1987年由Mane在r=1时证明了这个猜想。在证明中，所谓的弗兰克引理是必不可少的。由于这个引理不适用于C^r拓扑（r【大于等于】2），所以当r【大于等于】2时，这个猜想仍然是开放的。本课题的目的是利用Pesin理论证明阴影-C^r-开条件（r【大于等于】2）下的双曲性，并利用在此过程中获得的技术，尝试求解r【大于等于】2时的C^r-稳定性猜想。2003年，我们将研究对象限制在二维动力系统上，集中证明了系统在阴影- c ^r开条件下的双曲性。2004年，我们继续执行上述策略，但没有特别发表。然而，对于本研究中得到的部分结果，我们找到了一些将其推广到更高维度的句柄。在我们看来，这个项目的完成率可以评估为50%。在证明动力系统的双曲性之前，必须先证明周期点的双曲性。在阴影- c ^r-开条件（r【大于等于】2）下，证明了周期点等的双曲性，并利用这些事实，在假定附加条件的情况下，证明了二维动力系统的双曲性（如上所示，结果可以推广）。本课题的研究基础已经完成。今后我们将尽我最大的努力完成这个项目。

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.