Bifurcations of Dynamical Systems Satisfying the Pseudo-orbit Tracing Property

本研究的目的是分析含有一个满足伪轨跟踪性质（简称POTP）的自同态的单参数族的分支现象。注意，POTP也被称为跟踪性质。在2000年以前，我们已经刻画了满足POTP的非纯同态的C^1内部的非纯同态，并在2000年利用这些结果，得到了满足POTP的非纯同态的动力学，更准确地说，是满足POTP的非纯同态的稳定流形和不稳定流形的交的性质。（伪轨道）Lipschitz跟踪性（简称LSP）的特点。然后，在2001年，我们试图分析位于满足LSP的超同态集合的边界上的超同态的分支现象。不幸的是，我们的调查不可能取得如此辉煌的成就，但我坚信，长期劳工计划的结果将对解决问题发挥重要作用。此外，在2001年，我们注意到我们在这个项目中的方法也适用于C^1向量场。实际上，利用Hayashi的连接引理，我们刻画了具有拓扑稳定性的向量场集合的C^1内部（这也是本项目的一个显著结果）。一般来说，由于拓扑稳定性比POTP强，我们不能立刻刻画满足POTP的向量场的动力学。但是，通过修改证明中使用的技术，在不久的将来可能会描述动力学。

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.