| Budget Amount *help |
¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 2000: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 1999: ¥700,000 (Direct Cost: ¥700,000)
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| Research Abstract |
The purpose of this research project is to analyze the bifurcation phenomena of 1-parameter family containing a diffeomorphism which satisfies the pseudo-orbit tracing property (abbr.POTP). Remark that the POTP is also well known as the shadowing property. Before 2000, we had characterized diffeomorphisms in the C^1 interior of diffeomorphisms satisfying the POTP, and in 2000, by making use of those results the dynamics, more precisely, the property of the intersection of the stable and unstable manifolds of diffeomorphisms satisfying the (pseudo-orbit) Lipschitz shadowing property (abbr.LSP) was characterized.
Then, in 2001 we tried to analyze the bifurcation phenomena of diffeomorphisms lying in the boundary of the set of diffeomorphisms satisfying the LSP.Unfortunately, we could not produce so splendid achievements in the investigation, but I am strongly convinced that the results on the LSP will play an important role to solve the problem. Furthermore, in 2001 we had noticed that our method in this project also work for C^1 vector fields. Actually, by Hayashi's connecting lemma we have characterized the C^1 interior of the set of vector fields having the topological stability (this is also a remarkable result of this project). In general, since the topological stability is stronger than the POTP, we cannot characterize the dynamics of vector fields satisfying the POTP at once. But, by modifying the techniques used in the proof it might be possible to characterize the dynamics in the near future.
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