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.
