弱双曲动力系统的遍历理论研究

The study of topological and ergodic properties on dynamics with weak hyperbolicity is the core research in modern differentiable dynamical systems, including the cases of non-uniform hyperbolicity, partial hyperbolicity and dominated splitting etc. By using tools such as Pesin theory, Liao theory and its development, approximation of Lyapunov exponents, we aim to study Oseledec ergodic average, various recurrence, SRB-like measure and develop the general theory of them for dynamics beyond uniform hyperbolicity. These will help people to understand more dynamical properties of general dynamics. More precisely, the concrete contents include: (1) Variational principle of level set in the sense of Oseledec ergodic average, the dynamical difference between different asymptotic recurrent behavior and their connection with Li-Yorke chaotic etc.; (2) Constructing examples on entropy dissipation of equivalent flows with dominated splitting; (3) Searching sufficient and necessary conditions for existence of maximal entropy measure and studying the dynamical structure of SRB-like measure in the case of dominated splitting or partial hyperbolicity.

探索具有弱双曲性的微分动力系统（包括非一致双曲、部分双曲、控制分解等情形）的拓扑和遍历性质是当前微分动力系统的核心研究内容。本项目旨在利用Pesin理论、廖理论及其最新发展、Lyapunov指数逼近等成果,探讨弱双曲动力系统的Oseledec乘法遍历平均、各种回复性、SRB-like测度等课题，拓展和丰富一致双曲之外动力系统的一般理论，帮助人们进一步理解大多数微分动力系统的各种动力学性态。具体地，本项目研究内容包括：1.探讨弱双曲系统Oseledec乘法遍历平均水平集的变分原理，通过熵等指标研究不同渐进回复行为之间的动力复杂性以及与Li-Yorke混沌等指标交叉起来的复杂性; 2. 构造控制分解条件下等价流熵消失的例子; 3. 对具有控制分解或部分双曲性的C^1 系统寻找最大熵测度存在的充要条件、解析SRB-like测度的结构及相关问题。

非一致双曲、部分双曲系统等微分系统是继一致双曲之后人们普遍关心的几类动力系统。本项目主要在遍历平均、各种回复性、SRB-like测度、熵等方面取得一些成果，丰富和拓展了动力系统特别是一致双曲之外动力系统的拓扑理论和遍历理论。例如，（1）对一些部分双曲系统找到了SRB测度或物理测度存在的判定准则，（2）对非SRB-like测度对应的轨道行为从熵角度进行了拓扑式描述等，（3）还从熵角度对各种遍历平均、回复性等研究课题进行了一些深入分析等。