微分遍历论指微分动力系统（主要是非一致双曲系统和部分双曲系统）的遍历理论。在这个理论领域我们已经有20余篇研究论文. 本项目研究这个领域的四个重要课题： 1. 非一致双曲确定系统和非一致双曲随机系统的Lyapunov 指数的连续变化性质；2. 特定的部分双曲系统的跟踪性质和specification 性质；3. 带奇点的非一致双曲系统的熵的上半连续性质；4. 带奇点的特定部分双曲系统的周期轨道指数增长率.

Differentiable ergodic theory is on ergodic properties for differentiable dynamicail systems that consisits mailly of non-uniformly dynamical systems and partially hyperbolic systems. We have more than 20 research papers on this theory. In the present proposal we investigate 4 important topics in this theory: 1. The Lyapunov expnents and its continuity for non-uniformly hyperbolic deterministic systems and stochastic systems; 2. Shaodowing property and specification property for certian partially hyperbolic systems; 3. Entropy and upper semi-continity of entropy for non-uniformly hyprbolic systems with singularity; 4. Growth rate of periodic orbits for certian partially hyperbolic systems with singulatity.