一致性遍历定理和不连续几乎周期势的动力系统量

Ergodic theorems play a very important role in studying quantites of dynamical systems (rotation numbers and Lyapunov exponents). This project will introduce a kind of discontinuous skew-product flows. Its feature is that the measure of set of discontinuous points is zero. Then with help of knowledge of dynamical systems, measure theory, functional analysis and point-set topology, we will study the properties of the set of invariant measures for this system, and establish the uniformly ergodic theorems. It will be a complete development of results for classical flows. As an application of the above theory, combining qualitative theory of differential equations, we will establish quantites of dynamical systems for the linear Schrodinger equation with discontinuous almost periodic potentials, and give their ergodic formulae. In particular, we will establish the rotation number and study the relationship between rotation number and potential. Because quantites of dynamical systems describe the long-time limit behavior for solutions of differential equations, we will then study the spectral problem of the above Schrodinger equation, for example, the spectral theory of Schrodinger operators.

研究动力系统量（旋转数和Lyapunov指数），遍历定理是极为重要的基础和工具。本项目将首先引入一类不连续的斜积流，其特点是，不连续点集为零测度。然后综合运用动力系统、测度论、泛函分析、点集拓扑等多方面的理论和方法，研究此系统的不变测度集合的性质，建立相应的一致性遍历定理。这将是经典流结果的完整继承和发展。作为上述理论的应用，我们将结合方程定性理论，对具有不连续几乎周期势函数的线性薛定谔方程建立动力系统量，给出相应的遍历表达式。特别是，旋转数概念的建立以及旋转数关于势函数的依赖关系。由于动力系统量描述了方程解的长时间极限行为，我们将以此为工具，研究上述方程的谱问题，如薛定谔算子的谱理论。

本项目研究了动力系统和微分方程中的三个问题。.1，考虑当势能为不连续的几乎周期函数时，对薛定谔方程建立旋转数的概念。并给出了旋转数关于位势的最优估计。.2，利用链回复性研究连续映射迭代形成的半群上利普希茨各态历经和广义各态历经，分别给出了利普希茨各态历经和广义各态历经的充分必要条件，并举例说明各态历经，广义各态历经和利普希茨各态历经性的关系。.3，利用Poincare-Birkhoff环域定理和三阶近似方法，研究了一类二阶非线性方程的周期解的存在性和稳定性，它模拟了卫星在围绕其质心的椭圆轨道上的平面振荡。