Limit Theorems in Dynamical Systems

这个数学研究项目处理确定性系统中的概率特性（或随机性）。许多确定性系统由于对初始条件的敏感依赖而表现出随机行为。本项目的目标是开发使用统计方法预测此类系统行为的方法。将特别注意在短空间和长时间尺度上描述动力学。在这个项目中开发的方法将应用于几个领域，包括概率论，数论和统计物理。研究者还将继续他的协同活动，包括组织会议，就动力学的最新进展举办讲座和迷你课程，撰写调查论文，指导研究生。研究者将研究动力系统的极限定理。主要研究方向为：(1)局部极限定理及相关的渐近展开式；(2)总平移及其他零熵系统的极限定理。方法将从动力系统和概率论中得出。特别是调查将依赖于光谱方法和重整化技术。项目两部分之间的联系是，本项目第一部分中发展的重整化动力学的极限定理将在第二部分中发挥重要作用。

This mathematical research project deals with probabilistic properties (or stochasticity) in deterministic systems. Many deterministic systems exhibit random behavior due to sensitive dependence on initial conditions. The goal of the present project is to develop methods of predicting the behavior of such systems using a statistical approach. Particular attention will be paid to describing the dynamics on short spatial and long temporal scales. The methods developed during this project will have applications to several fields, including probability theory, number theory, and statistical physics. The investigator also will continue his synergistic activities including organizing conferences, giving lectures and minicourses on recent advances in dynamics, writing survey papers, and mentoring graduate students.The investigator will work on limit theorems for dynamical systems. The main directions of research will concern (1) local limit theorems and related asymptotic expansions and (2) limit theorems for toral translations and other zero entropy systems. The methods will be drawn from dynamical systems and probability theory. In particular the investigations will rely on a spectral approach and renormalization techniques. The connection between the two parts of the project is that the limit theorems for renormalization dynamics developed in the first part of this project will play important role in the second part.