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

众所周知，在封闭的c^∞歧管上具有阴影特性的差异性由动态系统空间中的一个大类组成。该研究项目的目的是表征具有差异几何形状的阴影特性的一组具有阴影特性的c^r间接（r≧2）。然后，通过我们的结果，我们试图为研究解决方案的研究做出贡献，用于r≧2的c^r稳定性猜想，并研究将分叉理论的泛化对更高维度的概括。Pesin理论是一种研究非热动力学系统的有力理论，该理论是根据测量理论和基本理论的研究，通常是研究重要的一项重要的角色。由于在我们的框架中，我们可以将Pesin理论应用于该研究对象。在该研究项目中，我们采用了以下策略来证明c^<r->内部差异内部中每个元素具有均匀的双曲线，具有……更多地掩盖了阴影特性：我们证明，通过将佩辛理论与阴影特性结合在一起。具体地，对于任何给定的Ergodic不变测量μ，由Oseledec定理μ。在与Lyapunov指数相对应的μ的支持下，切线束分裂。如果相对于测量，几乎所有地方的Lyapunov指数在几乎所有地方都非零，则通过佩辛理论，分裂是双曲线（但通常不是均匀双曲线）。这种测量称为双曲线度量。该项目的步骤是：我们找到一种双曲线度量，然后，我们通过阴影属性证明了统一的双曲线。该研究项目在2005年的主要目的是证明存在一种双曲线措施，在阴影c^<r->开放状态下具有足够的巨大支持，并与共同发射的人一起推动了这一问题，这是一个问题。不幸的是，在2006年3月31日，目前，值得出版时值得一提的结果。但是，在上述过程中，我们可以发现考虑向量字段的类似问题的必要和重要性。特别是，在2005年底，首席调查员取得了有关具有阴影特性的向量场的稳定性的结果。结果已经发表在《微分方程杂志》（Elsevier）上。当我们考虑该研究项目中有关向量领域的同一问题时，这个结果似乎是在调查中找到一种方法的关键。这是要特别考虑的要点。我们认为，该项目的成就百分比可以评估50％。较少的