A study on the characterization of C^r-diffeomorphisms possessing the shadowing property

具有遮蔽性质的C^r-微分同胚的表征研究

基本信息

批准号：
17540187
负责人：
SAKAI Kazuhiro
金额：
$ 1.09万
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：
2005
资助国家：
日本
起止时间：
2005 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17540187/
关键词：
dynamical systems shadowing property hyperbolic structural stability Pesin theory hyperbolic measures

项目摘要

It is known that diffeomorphisms possessing the shadowing property on closed C^∞ manifolds consists of a large class in the space of dynamical systems. The purpose of this research project is to characterize the C^r-interior (r≧2) of the set of diffeomorphisms possessing the shadowing property in view of differential geometry and to prove the hyperbolicity. Then, by means of our results we try to contribute to the study of a solution for the C^r-stability conjecture for r≧2 and the study of a generalization of bifurcation theory to higher dimensions.Pesin theory is a powerful theory to study non-hyperbolic dynamical systems in terms of measure theory and ergodic theory, and, usually, the theory plays an important role in the investigation of Henon maps. We can apply Pesin theory to this research object since r≧2 in our framework. In this research project, we adopted the following strategy to prove the uniform hyperbolicity for each element in the C^<r->interior of diffeomorphisms posse … More ssing the shadowing property : we prove the uniform hyperbolicity for each element of the set by combining Pesin theory with the shadowing property. Concretely, by Oseledec's theorem, for any given ergodic invariant measure μ. there is a splitting of the tangent bundle on the support of μ corresponding to the Lyapunov exponents. If the Lyapunov exponents are non-zero in almost everywhere with respect to the measure, then the splitting is hyperbolic (but not uniformly hyperbolic in general) by Pesin theory. Such measure it is called to be a hyperbolic measure. The steps of this project are : we find a hyperbolic measure, then, we prove the uniform hyperbolicity by means of the shadowing property.The main purpose of this research project in 2005 is to prove the existence of a hyperbolic measure which has a sufficiently huge support under the shadowing-C^<r->open condition, and we, together with co-investigator, have been push strongly forward with this problem. Unfortunately, in March 31, 2006, the present, there are noting special results worth while to publish. However, in the above process, we could find a necessity and an importance to consider the similar problem for vector fields. Especially, in the end of 2005, the head investigator got a result concerning the stability of vector fields possessing the shadowing property. The result has already published in the Journal of Differential Equations (Elsevier). This result seems to be the key to find a method in the investigation when we consider the same problem in this research project with respect to vector fields. This is the points to be specially considered. In our opinion, the achieve percentage of this project might be evaluated 50 %. Less

已知闭 C^∞ 流形上具有遮蔽性质的微分同胚由动力系统空间中的一大类组成。本研究项目的目的是表征集合的 C^r 内部 (r≧2)。然后，通过我们的结果，我们尝试为 C^r 稳定性的解决方案的研究做出贡献。 r≧2的猜想以及分岔理论向更高维度的推广的研究。Pesin理论是测度论和遍历理论方面研究非双曲动力系统的有力理论，通常，该理论发挥着重要作用在Henon图的研究中，我们可以将Pesin理论应用到这个研究对象中，因为在我们的框架中r≧2，我们采用了以下策略来证明每个元素的一致双曲性。 C^<r->微分同胚的内部具有阴影性质：我们通过将 Pesin 理论与阴影性质相结合，具体证明对于任何给定的遍历不变测度 μ ，集合中每个元素的一致双曲性。对应于 Lyapunov 指数，在 μ 的支持上存在切束分裂。对于该测度几乎处处非零，则根据 Pesin 理论，分裂是双曲的（但通常不是一致双曲的），该测度被称为双曲测度。该项目的步骤是：我们找到一个。 2005年这个研究项目的主要目的是证明在双曲测度下有足够大的支持的双曲测度的存在性。不幸的是，在 2006 年 3 月 31 日，目前，有一些值得发表的特殊结果。然而，在上述过程中，我们可以发现考虑矢量场的类似问题的必要性和重要性，特别是在2005年底，首席研究员得到了关于具有阴影特性的矢量场稳定性的结果。有当我们在这个研究项目中考虑向量场的相同问题时，这个结果似乎是寻找方法的关键。这是需要特别考虑的点。我们认为，该项目的完成率可能评估为50%以下。