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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，在闭C^∞流形上具有跟踪性质的超同态是动力系统空间中的一大类。本研究项目的目的是从微分几何的角度刻画具有跟踪性质的微分同胚集合的C^r-内部（r ≥ 2），并证明其双曲性。Pesin理论是研究非双曲动力系统的测度理论和遍历理论的一个强有力的理论，并且，通常，该理论在Henon映射的研究中起着重要的作用.在我们的框架中，我们可以将Pesin理论应用于这个研究对象。在这个研究项目中，我们采用了以下策略来证明双同态波塞的C^内部的每个元素的一致双曲性<r-> ...更多信息跟踪性质的研究：将Pesin理论与跟踪性质相结合，证明了集合中每个元素的一致双曲性。具体地说，通过Oseledec定理，对任何给定的遍历不变测度μ。对应于李雅普诺夫指数，在μ的支撑上切丛存在分裂。如果李雅普诺夫指数在关于测度的几乎所有地方都是非零的，则根据Pesin理论，分裂是双曲的（但一般不是一致双曲的）。这种度量称为双曲度量。这个项目的步骤是：找到一个双曲测度，然后利用跟踪性质证明一致双曲性，2005年的这个研究项目的主要目的是证明在shadowing-C^开条件下存在一个具有足够大支集的双曲测度<r->，我们和合作研究者一直在这个问题上大力推进。不幸的是，在2006年3月31日，现在，没有什么特别的结果值得公布。然而，在上述过程中，我们可以发现考虑向量场的类似问题的必要性和重要性。特别是在2005年底，主要研究者得到了一个关于具有跟踪性质的向量场的稳定性的结果。该结果已发表在Journal of Differential Equations（Elsevier）上。这一结果似乎是我们在本研究项目中考虑关于向量场的相同问题时找到研究方法的关键。这是要特别考虑的几点。在我们看来，这个项目的完成率可以评估为50%。少