傅立葉係數, 李阿波諾夫指數和不變測度三者之間的關係

Abstract

一個複雜和不可預測的頻率譜長久以來在物理和工程上都用了作為一個訊號是否是混沌的一個重要指標。仍而都沒有嚴格的數學證明，一直到最近，在陳、許、黃和Roque-Sol的工作[1]，他們證明了nf(合成n次)的Fourier係數若滿足某些充分條件，可得到f的拓樸熵(topological entropy)是正的。這是一篇非常好的理論論文，然而實際卻不容易用，因為要計算nf的Fourier係數在絕大多數的情況下幾乎是不可能的。同時，f的拓樸熵是正的也無法得證f在一個測度為正的不變集上的動態形為是混沌的。例如()(1)fxx x μ=.，在μ是3.839的附近f有「window」的現象出現，[0,1]區間中有measure是1的點集皆為一period three的軌道吸入，此f在一Cantor set上(measure為0)的形為是混沌的。 此三年的計畫，第一年希望進一步探討Fourier coefficients of f (而不是nf)和混沌其他特徵量，例如：Lyapunove exponent, Invariant measure and Rotation number。第二年我們希望探討這些基本量和另一些well-known的orthornormal expansions如Harr basis and wavelets的關係。此部分的動機是因為Fourier coefficients of f是不穩定當f在局部作改變。前二年的計畫侷限於一堆的實函數。第三年我們將討論高維度函數的Fourier係數和Wavelets係數和這些混沌特徵量的關係。 這個計畫需要用的數學含Ergodic Theory, Invariant Measures, 和Wavelets。 References： 1. G. Chen, S.-B. Hsu, Y. Huang, M. A. Roque-Sol, Mathematical Analysis of the Fourier Spectrum of Chaotic Time Series, preprint. 2. D. Walters, An Introduction to Ergodic Theory Springer, New York, 2000. 3. A. Boyarshy & P. Gora, Laws of Chaos, Invariant Measures and Dynamical Systems in One Dimension, Birkhauser, 1997. 4. J. J. Benedtto & M. W. Frazier, Wavelets：Mathematics and Applications, CRC Press, 1994.

A complex and unpredictable frequency spectrum of a signal has long been seen in physics and engineering as an indication of a chaotic signal. However, no mathematical study of such connection until the worth of Chen, Hsu, Huang and Roque-Sol [1]. Specifically, they found various sufficient conditions on the Fourier coefficients of the n-th iterate nf of an interval map for which the topological entropy of f is positive. This is a nice piece of theoretical work. However, in practical, it is difficult to compute the Fourier coefficients of the n-th iterate nf. Moreover, the positivity of the topological entropy of f does not guarantee the chaotical behavior of f on an invariant set whose measure is positive. Consider the quadratic map ()(1)fxx x μ=. on [0,1]. Clearly, the topological entropy of f is positive given the μ in a small neighborhood of 3.839. However, f has an attracting period three orbits whose attracting set has a measure of 1. Thus, no computer can pick up the chaotic behavior of such f. Inspired by their work [1], the purpose of this three years project is to explore the relationship between other characteristics of a chaotic map with its Fourier coefficients. On the first year, we are to study the connection between Lyapunove exponent, (absolutely continuous) invariant measure, the rotation number of a map and its Fourier coefficients. On the second year, we are to study the connection between these mentioned characteristics：Lyapunove exponent, invariant measure, the rotation number of a map and its other orthornormal expansion, such as Harr basis and wavelets. The motivation for such study is due to the fact that a local perturbation of f may significantly affect all Fourier coefficients of f. On the third year, we are to study those connections mentioned in the first two years for maps defined in a metric space. In particular, we will concentrate on the high dimensional map. The mathematical tools needed to complete this project are Ergodic Theory [2], Invariant Measures [3] and Wavelets [4]. References： 1. G. Chen, S.-B. Hsu, Y. Huang, M. A. Roque-Sol, Mathematical Analysis of the Fourier Spectrum of Chaotic Time Series, preprint. 2. D. Walters, An Introduction to Ergodic Theory Springer, New York, 2000. 3. A. Boyarshy & P. Gora, Laws of Chaos, Invariant Measures and Dynamical Systems in One Dimension, Birkhauser, 1997. 4. J. J. Benedtto & M. W. Frazier, Wavelets：Mathematics and Applications, CRC Press, 1994.