拟周期微分动力系统在动力系统和数学物理中具有重要作用，描述了物理中许多有趣和基本的现象，如准晶体结构、量子霍尔效应和多频振子的振动等。其中系统的定性理论和相应算子的谱性质是动力系统中研究的基本课题。. 自20世纪50年代发展至今，KAM理论在近可积保守动力系统的研究中发挥了重要作用：KAM理论说明了完全可积系统中携带拟周期运动的不变环面，在扰动下大部分仍然被保持。现在，KAM理论已经成为对许多动力系统进行系统分析的强有力的工具。然而，由于小分母问题的出现，经典的KAM理论只能处理丢番图频率的系统，在刘维尔频率的拟周期系统中的研究成果相对较少，有待进一步发展。本项目拟利用KAM理论对刘维尔频率的拟周期系统的约化问题和相应算子的谱性质方面的几个重要问题进行深入研究。

Quasi-periodic differential dynamical systems play an important role in dynamical systems and mathematical physics as a source of examples of interesting and fundamental phenomena in physics, such as the structure of quasi-crystal, quantum Hall effect and oscillation of oscillators forced with two or more incommensurate frequencies. The qualitative theory of the system and the spectral property of the corresponding operator are fundamental topics in dynamical systems.. KAM theory was developed since the 1950's for conservative dynamical systems that are nearly integrable. KAM theory establishes persistence of most invariant tori carrying quasi-periodic motion in completely integrable systems under small perturbations. Nowadays, KAM theory provides powerful tools for the systematic analysis of many dynamical systems. However, due to the occurrence of small divisor problem, the classical KAM theory can only deal with the systems with Diophantine frequencies. There are just a few results on the application of KAM theory in quasi-periodic systems with Liouvillean frequencies, and therefore need further development. This project aims at deep study of some important problems on the reducibility of quasi-periodic systems with Liouvillean frequencies and the spectral properties of the corresponding operators.