中心一维部分双曲系统

Partial hyperbolicity is one of the most important subjects of differentiable dynamics. Its research originated from the study of structural stability conjecture, Palis density conjecture and stable ergodicity conjecture. In this project, we will use Liao theory, perturbation techniques, ergodic theory, invariant manifold in differentiable dynamics and foliation theory to study the geometric theory of partially hyperbolic diffeomorphisms with one dimensional center bundle and partially hyperbolic flows with 2-dimensional center bundle. Precisely, we will focus on the following five problems: 1. Is the center foliation of any partially hyperbolic diffeomorphism with one dimensional center bundle plaque expansive when its center bundle is integrable? 2. Is any skew-product partially hyperbolic diffeomorphism on 3-torus tame? 3. Is any partially hyperbolic diffeomorphism homotopic to Anosov (chain) transitive? 4. Minimality of the strong stable manifolds of partially hyperbolic Anosov diffeomorphisms on 3-torus with 2 dimensional stable bundles. 5. Palis weak density conjecture for 4-dimensional flows: every 4-dimensional flow can be C^1 approximated by Morse-Smale systems or by systems with Smale horseshoes. Here we have to study if every nontrivial singular hyperbolic chain recurrent class with singularity contains a periodic orbit.

部分双曲是当前微分动力系统最重要的方向之一，其研究来源于稳定性猜测，Palis稠密性猜测以及稳定遍历猜测。本项目旨在运用微分动力系统中的廖理论、扰动理论、不变流形理论、遍历论及叶状结构理论等工具研究中心一维部分双曲微分同胚以及中心二维部分双曲流的几何理论。具体而言，我们将研究如下五个问题：1、中心一维部分双曲微分同胚在中心可积时，其中心叶状结构是否总是片可扩的；2、三维环面上部分双曲的斜积系统是否为驯顺系统；3、三维环面上同伦于Anosov微分同胚的部分双曲系统的（链）传递性；4、部分双曲的三维Anosov微分同胚（二维压缩）强稳定流形的极小性；5、四维流的Palis弱稠密性猜测，即任意四维流都可以被Morse-Smale系统或具有Smale马蹄的系统C^1逼近，其中一个问题是要研究（通有地）含奇点的奇异双曲非平凡鞍型链回复类是否一定包含周期轨。

部分双曲系统，尤其是中心一维的部分双曲系统，是当前微分动力系统的重要研究领域。本项目主要研究中心一维部分双曲微分同胚与中心二维部分双曲流的拓扑与遍历性质。主要成果如下：证明了三维环面上同伦于 Anosov 系统的部分双曲系统的遍历性；刻画了三维环面上同伦于 Anosov 系统的部分双曲系统的中心化子；系统研究了Kan（型）自同态/微分同胚的混合型与传递型；证明了中心一维部分双曲系统的Cr(r>1)封闭引理；在高维流的弱Palis稠密性猜测研究中取得进展。