动力系统的 Banach 上链上同调方程解的存在性和正则性研究

The existence and the regularity of solutions of the cohomological equation is an important topic in the study of dynamical systems. It is closely related to the rigidity problem in dynamical systems and to the problem when a topological conjugacy in dynamical systems will be a smooth conjugacy. Recently, progress on this issue has been made concerning different systems and cocycles, but there are still many improtant open problems. .In this subject, we study the existence of solutions of cohomological equations for Banach cocycles, and the regularity of the solutions. We first investigate the existence of the measurable solution of the cohomological equations over non-uniformly hyperbolic systems, that is, the Livsic theorem for Banach cocycles over non-uniformly hyperbolic systems. And then, if the dynamical system is uniformly hyperbolic or partially hyperbolic, we study the problem whether a measurable solution can be extended to a Hölder continuous one. .The study of this project is conducive to reveal the relationship beween cohomological equations, Lyapunov expoents and periodic points.

上同调方程解的存在性与正则性问题是微分动力系统的重要课题之一，它与微分动力系统的拓扑共轭在什么条件下是光滑共轭的以及动力系统的刚性等问题具有非常密切的联系。近年来，针对不同的系统及不同的上链，上同调方程问题有着很大的进展，但仍有很多问题亟待解决。.本项目主要研究在不同系统下关于Banach上链的上同调方程的解的存在性和正则性问题：对于非一致双曲系统，研究上同调方程可测解的存在性问题；对于一致双曲系统和部分双曲系统，研究上同调方程的解的正则性问题，也即研究上同调方程的可测解能否延拓为Hölder连续解的问题。.本项目的研究有助于进一步揭示关于Banach上链的上同调方程与Lyapunov指数、周期点等之间的关系。

上同调⽅程是当前微分动⼒系统的⼀个重要研究课题。本项⽬主要研究关于Banach上链的上同调⽅程解的存在性和正则性问题。主要成果如下：对于⾮⼀致双曲系统，给出了Banach上链上同调⽅程可测解存在的⼀个充分条件；对于⼀致双曲系统，证明了上同调⽅程的可测解可以延拓为Hölder的；对于具有控制分解性质的C1⾮⼀致双曲系统，研究了其指数的马蹄逼近性质；证明了矩阵上链次可加拓扑压的连续性。