Regularity and Scale Invariant Properties of Critical Dynamic Systems: Harmonic Analysis and Numerical Implementations

这一建议的重点是发展分析函数和测度精细结构的数值方法，并将它们应用于动力系统理论中的几个关键对象的研究。该项目结合了调和分析(Littlewood-Paley理论、多分辨率分析、小波)和理论物理(重整化、热力学形式)的思想，分析了关键对象的规律性和标度特性。临界函数的重要性是因为它们可以作为临界现象的模型，如向湍流和相变的转变，在物理系统的规则行为和混沌行为之间提供“障碍”，等等。研究人员和他的合作者计划研究的关键对象是(临界)圆映射之间的共轭，保面积映射的临界不变圆，以及Siegel圆盘的边界。该项目的目标如下：(A)由PI及其合作者开发的计算函数的全局正则性的数学技术将应用于各种科学问题。(B)小波理论的最新结果以及现有的方法将被用来数值研究临界函数的局部正则性，以及相关不变度量的标度性质。(C)将计算准确的傅立叶谱和小波谱，并利用调和分析和重整化方法分析它们的结构。拟议的研究将提供准确的经验数据，使物理学家更好地理解临界现象，并将向纯数学家提出具有挑战性的问题。从长远来看，开发、实施和测试新的数值方法将为研究人员提供强大的工具，用于规律性和标度特性的数值研究，这在许多科学和工程领域(特别是在大气科学、地球物理、信号处理、数据网络流量)中是重要的。它将推动对临界函数和自相似度量理论的新研究--这是现代动力系统理论的中心问题。本次活动将为理工科不同专业的学生提供研究机会。由于该项目是高度跨学科的，它将不仅培养学生的学术能力，还将培养学生的其他科学应用能力，并将促进学生与不同领域的科学家之间的联系。

This proposal is focused on developing numerical methods for analysis of the fine structure of functions and measures, and applying them to study several critical objects in theory of dynamical systems. The project brings together ideas from harmonic analysis (Littlewood-Paley theory, multi-resolution analysis, wavelets) and theoretical physics (renormalization, thermodynamic formalism) to analyze the regularity and scaling properties of critical objects. The importance of critical functions is due to the fact that they serve as models of critical phenomena like transition to turbulence and phase transitions, provide "barriers" between regular and chaotic behavior of physical systems, etc. The critical objects that the investigator and his collaborators plan to study are conjugacies between (critical) circle maps, critical invariant circles of area-preserving maps, and boundaries of Siegel disks. The objectives of the project are the following: (A) Mathematical techniques for computation of global regularity of functions developed by the PI and collaborators will be applied to a wide variety of problems of scientific interest. (B) Recent results in theory of wavelets, as well as existing methods, will be implemented to investigate numerically the local regularity of critical functions, and the scaling properties of the associated invariant measures. (C) Accurate Fourier and wavelet spectra will be computed, and their structure will be analyzed by utilizing techniques from harmonic analysis and renormalization methods.The proposed research will supply accurate empirical data that will provide physicists with better understanding of critical phenomena and will pose challenging problems for pure mathematicians. In the long term, developing, implementing, and testing new numerical methods will provide researchers with robust tools for numerical studies of regularity and scaling properties, which are important in many areas of science and engineering (in particular, in atmospheric science, geophysics, signal processing, data network traffic). It will motivate new research in the theory of critical functions and self-similar measures -- a central problem of modern theory of dynamical systems. This activity will provide research opportunities for students majoring in different branches of science and engineering. Since the project is highly interdisciplinary, it will train students not only for academic, but also for other scientific applications, and will stimulate contacts between students and scientists in different areas.