| Budget Amount *help |
¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2006: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2005: ¥1,000,000 (Direct Cost: ¥1,000,000)
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| Research Abstract |
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
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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
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