三维流形上的叶状结构理论与结构稳定动力系统

In this program, we will use foliation theory to study two classes of dynamical systems on three manifolds: Anosov flows and NSS diffeomorphisms. To Anosov flows, we will consider some questions related to the following four sides: (1) classifying Anosov flows on a kind of hyperbolic 3-manifolds; (2) isotopy, homotopy and homeomorphism classes of periodic orbits in Anosov flows on toroidal 3-manifolds; (3) constructing several Anosov flows on some given 3-manifolds; (4) admitting Anosov flows up to finite covering on fibered hyperbolic 3-manifolds. All of them are hot topics in this field. In particular, (3) and (4) are related to two open questions. To NSS diffeomorphisms, we will consider some questions related to the following two sides: (1) constructing structurally stable NSS diffeomorphisms by using some combinatorial ways; (2) building some determinations to decide the structural stability of NSS diffeomorphisms.

本项目拟利用叶状结构理论来研究三维流形上的两类动力系统：Anosov流与NSS微分自同胚。拟研究的有关Anosov流的问题涉及到如下四个方面：（1）一类双曲流形上三维Anosov流的分类；（2）toroidal三维流形Anosov流周期轨道的同痕、同伦、同胚类；（3）固定三维流形上多个Anosov流的构造；（4）可纤维化双曲三维流形在有限复叠下对Anosov流的承载。它们都是目前国际上这个课题的热点，特别是（3）和（4）涉及到两个公开问题。拟研究的有关NSS微分自同胚的问题主要涉及两个方面：（1）结构稳定NSS微分自同胚的组合构造；（2）结构稳定性的判别。

本项目主要按照原计划书，利用叶状结构研究三维流形中Anosov流的与分类相关的几个问题。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，首次对一类无穷个双曲三维流形上的Anosov 流完整分类，并证实承载套紧叶状结构但不承载Anosov流的（无穷个）双曲三维流形的存在性。该论文被国际著名数学杂志Duke.Math.J接收发表。并被该杂志审稿人们评价为：“solves a longstanding problem”、“very first progress in a subject that was considered very difficult.”.2，与他人合作，首次对含Franks-Williams流形的一类无穷个混合流形上非传递Anosov流分类。该工作发表在国际数学著名杂志PLMS上。.3，与他人合作，我们建立了通过粘合双曲块得到的Anosov流的符号动力系统。并将其应用于分类问题，以及构造承载多个非直线覆叠Anosov流的双曲三维流形。