三维流形上的Anosov流与双曲块

In this program, we will study an important kind of flows (Anosov flows) and some related topics (for instance, hyperbolic plugs) on three manifolds. More precisely, we will focus on the following sub-topics. (1) We will try to build a bridge between expanding attractors and Pseudo-Anosov flows on three manifolds by DPA (Derived from Anosov) surgeries. (2) We will provide a combinatorial proof for spectral-like decomposition of Anosov flows on three manifolds. (3) We will study the topologically equivalent classes of hyperbolic plugs on three manifolds. In particular, first we will focus on the cases when the three manifold is homeomorphic to the complement space of figure eight knot or Seifert manifold. (4)We will try to classify (up to topological equivalence) non-transitive Anosov flows on Franks-Williams manifold. (5)We will try to think about an open question: whether there exists a three manifold admitting infinitely many Anosov flows.

本项目拟对三维流形中一类重要的流（Anosov流）及其相关课题（双曲块）展开研究。 具体而言： (1)拟通过DPA手术，对两个在动力系统和叶状结构理论都很重要的课题扩张吸引子和伪Anosov流建立桥梁； (2)拟对Anosov流的类谱分解通过train track等技术进行组合的证明； (3)拟对一些紧带边三维流形上的双曲块的拓扑等价分类做一些探讨，特别是八字结的补空间和Seifert流形; (4)拟对Franks-Williams流形上不可迁Anosov流拓扑等价分类进行探讨； (4)拟对一个公开问题（是否存在流形上存在无穷多Anosov流）作一些尝试。

本项目主要按照原计划书，利用双曲块研究三维流形中的Anosov流；另外，作为项目的延伸，我们也对三维流形的叶状结构分类进行研究，并用其研究奇异吸引子。现将本项目完成的主要学术成果及其科学意义简述如下。.1，与他人合作，给出了三维Anosov流的一个构造性定理，并构造了一些含新动力学现象的Anosov流, 其中回答了Katok与Barbot-Fenley的两个公开问题。得到国际同行的正面评价，如“new, powerful and vast constructions”、” answer some of the open questions”等。并被一些重要综述（如ICM2018邀请报告）及重要工作（发表在Invent. Math）引用。.2，与他人合作，发现了三维Anosov流存在类谱分解。也被ICM2018邀请报告综述引用。.3，解决了三维流形中Hirsch叶状结构分类问题。建立螺线圈吸引子与叶状结构间联系。