拟周期Cocycles是动力系统非常活跃的一个领域，近年来在Cocycles的可约性，Lyapunov指数和双曲性等方面取得了重大进展。Cocycles的基本概念与Schrodinger算子谱的基本概念存在着对应关系。人们将拟周期cocycle动力学理论应用于拟周期Schrodinger算子的谱理论，获得了巨大的成功，在Anderson localization, 纯点谱或纯绝对连续谱的存在性，相变现象，谱集的Cantor结构等方面取得了一系列前所未有的成果。不过仍有大量问题不清楚。本项目我们将进一步研究拟周期cocycle的Lyapynov指数的连续性，正则性，正性，以及拟周期Schrodinger算子的Anderson localization和Cantor谱的通有性等。

Quasiperodic cocycles theory is a very active field in dynamical system，many developments on the dynamical properties of it, including reducibility,continuity of Lyapunov exponent,hyperbolicity, etc. have been achieved in recent years. There exists some relationship between the concepts of cocycles and those of Schroidnger opertor. Applying the results in quasiperiodic cocycles in the spectrum of quasiperodic Schrodinger operators, people obtained a lot of results on the properties of Schrodinger operator's spectrum, such as Anderson Localiation, pure point spectrum, pure absolutely continuous spectrum, Metal-insulator transition, the geometric sturcture of spectrum set, and so on. However, there are still more many problems unclear. In this program, we will focus on the following problems: 1. The continuity,regularity, positivity of Lyapunov exponents of quasiperiodic cocycles. 2 Anderson localization and the genericity of Cantor spectrum for quasiperiodic Schrodinger operators.