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，对稳定性猜想的求解研究对该理论的发展起到了重要作用。这个猜想断言每个结构稳定的系统都是双曲的，1987年Mane证明了r=1。在证明中，所谓的Franks引理是必不可少的。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结合Pesin理论证明跟踪-C^r-开条件(r[大于或等于]2)下的双曲性，并利用在此过程中获得的技巧来解决r[大于或等于]2下的C^r-稳定性猜想。2003年，我们将自己局限于二维动力系统，集中证明了在跟踪-C^r-开条件下系统的双曲性。2004年，我们一直在进行上述战略，但没有特别的发表。然而，对于本研究中得到的一些部分结果，我们发现了一些处理方法来将它们推广到更高的维度。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，有必要证明周期点的双曲性。在影子-C^r-开条件(r[大于或等于]2)下，主要研究者证明了周期点的双曲性等，并利用这些事实，通过假设附加条件证明了二维动力系统的双曲性(如上所述，这一结果是可以推广的)，为本研究奠定了基础。今后我们会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.