论文标题
三个manifolds上的星形向量场是多单明的双曲线
Star vector fields on three-manifolds are multi-singular hyperbolic
论文作者
论文摘要
链条及传递集合中的奇异性和常规轨道的共存一直是理解鲁棒动力学双曲/部分双曲线性质的主要障碍。值得注意的是，在没有奇异性的情况下，具有所有周期性轨道的矢量场是双曲线的。莫拉莱斯（Morales），Pacifico和Pujals提出了一种称为“奇异双曲线”的部分双曲线，该性表征了三维星流的开放式且密集的子集。在较高的维度中，Bonatti和Da Luz通过多线性双曲线表征了一组开放而密集的星向量场。在本文中，我们证明了一个相同索引表现出所有周期性轨道的矢量场是多单明的双曲线，从而推广了先前的结果。作为推论，我们获得了所有三维恒星流都是多线双曲线的。此外，如果同一类中的所有奇异性都表现出相同的索引，则恒星流量是奇异的双曲线。此外，在任何维度上都具有稳健链复发类别的恒星流是多尖的双曲线。
The coexistence of singularities and regular orbits in chain transitive sets has been a major obstacle in understanding the hyperbolic/partial hyperbolic nature of robust dynamics. Notably, the vector fields with all periodic orbits robustly hyperbolic (star flows), are hyperbolic in absence of singularities. Morales, Pacifico and Pujals proposed a partial hyperbolicity called "singular hyperbolicity" that characterizes an open and dense subset of three dimensional star flows. In higher dimensions, Bonatti and da Luz characterize an open and dense set of star vector fields by multi-singular hyperbolicity. In this article, we prove that a vector field exhibiting all periodic orbits robustly of the same index is multi-singular hyperbolic, generalizing the previous results. As a corollary, we obtained that all three-dimensional star flows are multi-singular hyperbolic. Moreover, if all singularities in the same class exhibit the same index, the star flow is singular hyperbolic. Additionally, star flows with robust chain recurrence classes in any dimension are multisingular hyperbolic.
