熵是动力系统和遍历理论的核心内容之一。正熵被广泛认为是混沌的标志，说明系统中存在复杂性。本项目旨在研究有正熵的系统的各种动力学性质，其中主要是与系统复杂性有关的性质。主要的研究内容包括拥有不同熵或维数的遍历测度或不变集的存在性，以及与轨道的发散速度有关的量的指数增长或衰减等等。其中研究的重点在于不变集和遍历测度的构造，以及各种关于熵和维数的计算和估计。主要的研究目标是证明满足一定条件的有正熵的微分动力系统中存在有任意测度熵的遍历测度。本项目研究涉及拓扑动力系统、微分动力系统、遍历论、分形和维数理论等多个领域，项目组有一定的研究基础并希望通过项目的实施使我们在相关问题上保持国际领先水平。

Entropy is the kernel concept in dynamical systems and ergodic theroy. Positive entropy is widely accepted as a sign of chaos，which represents some complexity. In this project we would like to study dynamical systems of positive entropy. We woule like to reveal more properties of such systems related to their complexity, especially existence of ergodic measures or invariant sets of different entropies or dimensions, and exponential decay of quantities about scattering speed of trajectories. The main tasks include construction of invariant sets and ergodic measures, as well as computation of entropy and dimension. The main goal of this project is to show that a differentiable dynamical system of positive topological entropy has ergodic measures of arbitrary intermediate entropies. This research involves many areas, such as topological dynamical systems, differentiable dyanmical systems, ergodic theory, fractal, dimension theory, and etc. Through this project, we expect to maintain a leading position in the research on related topics.