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，而对稳定性猜想的求解在动力系统理论的发展中起着重要的作用。这个猜想断言每个结构稳定的系统都是双曲的，并且在1987年由马内证明了r=1。在证明中，所谓的弗兰克斯引理是必要的。由于引理不适用于C^r-拓扑（r[大于或等于]2），所以当r[大于或等于]2时，猜想仍然是开放的。本研究项目的目的是在阴影-C ^r-开条件下证明双曲性（r[大于或等于]2）与Pesin理论相融合，并应用在这一过程中得到的技巧，我们试图解决r [大于或等于] 2的C^r-稳定性猜想。我们把自己限制在2维动力系统，并集中于证明系统的双曲性下的shadowing-C^r-开条件。2004年，我们继续推进上述战略，但没有特别的出版物。然而，对于在本研究中得到的一些部分结果，我们已经找到了一些处理，以推广到更高的维度。在证明动力系统的双曲性之前，首先要证明周期点的双曲性。在阴影-C ^r-开条件（r[大于或等于]2）下，首席研究员证明了周期点等的双曲性，并利用这一事实，通过附加条件的假设，证明了二维动力系统的双曲性（如前所述，证明了这一结果是可以推广的），为这一研究项目奠定了基础。今后我们将尽最大努力完成这项工程。

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.