Limit Theorems in Dynamical Systems

众所周知，许多确定性系统表现出随机行为。一个这样的系统的最简单的例子是一个球在两个无限重的周期性移动的墙壁之间移动。这个例子是费米和乌拉姆在上个世纪中期研究的。 更复杂的例子包括等离子体中带电粒子的运动和小行星带中小行星的运动。本项目的目的是利用混沌动力学的最新进展，理解这类系统的随机性（运动的随机性）的起源，并对它们的行为进行精确预测。PI将结合概率论和动力系统的方法，继续进行动力系统极限定理的研究。主要研究方向如下：（1）绝热不变量的演化;（2）局部极限定理。该建议将集中在获得精确的渐近性的方法上，该方法允许在长时间尺度和短空间尺度上控制极限分布。拟议活动的更广泛影响将是将动力系统工具应用于其他领域，包括概率论，数论和统计物理。PI将继续他的协同活动，包括组织会议，就动态的最新进展进行讲座和小型课程，撰写调查论文和指导研究生。

It is well known that many deterministic systems exhibit random behavior. The simplest example of one such system is provided by a ball moving between two infinitely heavy periodically moving walls. This example was studied by Fermi and Ulam in the middle of the last century. More complicated examples include the motion of charged particles in plasma and the motion of the asteroids in the asteroid belt. The aim of this proposal is to understand the origin of stochasticity (randomness of motion) of such systems using recent advances in chaotic dynamics, with the goals of making precise predictions about their behavior.The PI will continue his work on limit theorems for dynamical systems combing the methods of probability theory and dynamical systems. The main directions of research will be the following: (1) Evolution of adiabatic invariants, and (2) Local Limit Theorems. This proposal will concentrate on the methods of obtaining precise asymptotics which allows control of the limit distributions on long temporal scales and short spatial scales. The broader impact of the proposed activity will be in applying dynamical systems tools to other fields including probability theory, number theory, and statistical physics. The PI will continue his synergistic activities including organizing conferences, giving lectures and minicourses on recent advances in dynamics, writing survey papers and mentoring graduate students.