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

该建议的重点是发展用于分析函数和测度的精细结构的数值方法，并将其应用于研究动力系统理论中的几个关键对象。 该项目汇集了调和分析（Littlewood-Paley理论，多分辨率分析，小波）和理论物理（重整化，热力学形式主义）的想法，以分析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，如向湍流和相变的过渡，在物理系统的规则和混沌行为之间提供“屏障”等。 研究者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，面积保持映射的临界不变圆，以及Siegel圆盘的边界。该项目的目标如下：（A）由PI和合作者开发的计算函数全局正则性的数学技术将应用于各种各样的科学问题。 (B)最近的结果在小波理论，以及现有的方法，将实施调查数值的局部正则性的临界函数，和相关的不变措施的标度特性。 (C)精确的傅立叶谱和小波谱将被计算出来，并且它们的结构将被利用谐波分析和重整化方法的技术来分析。拟议的研究将提供精确的经验数据，这将为物理学家提供更好地理解临界现象，并将对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（特别是大气科学，地球物理学，信号处理，数据网络流量）中非常重要。 它将激发新的研究理论的临界函数和自相似措施-一个中心问题的现代理论的动力系统。 这项活动将为科学和工程不同分支专业的学生提供研究机会。 由于该项目是高度跨学科的，它将不仅为学术培训学生，而且还为其他科学应用培训学生，并将促进不同领域的学生和科学家之间的接触。

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.