Limit Theorems in Dynamical Systems

动力系统中的极限定理

• 批准号: 1665046
• 负责人: Dmitry Dolgopyat
• 金额: $ 20.5万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1665046&HistoricalAwards=false
• 关键词: Limit; Theorems; Dynamical; Systems

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.

该数学研究项目研究确定性系统中的概率属性（或随机性）。由于对初始条件的敏感依赖性，许多确定性系统表现出随机行为。 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。 将特别注意描述短空间和长时间尺度上的动态。 该项目期间开发的方法将应用于多个领域，包括概率论、数论和统计物理学。研究人员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进行讲座和迷你课程、撰写调查论文以及指导研究生。研究人员将研究动力系统的极限定理。研究的主要方向将涉及（1）局部极限定理和相关渐近展开以及（2）托拉平移和其他零熵系统的极限定理。这些方法将借鉴动力系统和概率论。 特别是，研究将依赖于谱方法和重正化技术。 该项目两个部分之间的联系在于，该项目第一部分中开发的重正化动力学极限定理将在第二部分中发挥重要作用。

项目成果

Infinite measure renewal theorem and related results: INFINITE MEASURE RENEWAL THEOREM AND RELATED RESULTS

无限测度更新定理及相关结果： INFINITE MEASURE RENEWAL THEOREM AND RELATED RESULTS

• DOI: 10.1112/blms.12217
• 发表时间: 2018-11
• 期刊: Bulletin of the London Mathematical Society
• 影响因子: 0.9
• 作者: Dolgopyat, Dmitry;Nándori, Péter
• 通讯作者: Nándori, Péter

On mixing and the local central limit theorem for hyperbolic flows

关于双曲流的混合和局部中心极限定理

• DOI: 10.1017/etds.2018.29
• 发表时间: 2017-10-24
• 期刊: Ergodic Theory and Dynamical Systems
• 影响因子: 0.9
• 作者: D. Dolgopyat;P'eter N';ori;ori
• 通讯作者: ori

Central limit theorem for recurrent random walks on a strip with bounded potential

在有界势带上循环随机游走的中心极限定理

• DOI: 10.1088/1361-6544/aab89b
• 发表时间: 2018-07
• 期刊: Nonlinearity
• 影响因子: 1.7
• 作者: Dolgopyat, D;Goldsheid, I
• 通讯作者: Goldsheid, I

A rectangular billiard with moving slits

带有移动狭缝的长方形台球

• DOI: 10.1088/1361-6544/ab60d7
• 发表时间: 2020-04
• 期刊: Nonlinearity
• 影响因子: 1.7
• 作者: Zhou; Jing
• 通讯作者: Jing