傅利葉係數，黎阿普諾夫指數，不變測度及渾沌

Abstract

長久以來，在物理及工程上，常利用對一個複雜且不可預測的信號作光譜分析來判斷此信號是否渾沌。首先將此現象做數學分析的是陳鞏老師等人。他們是希望尋求一種關於渾沌動態系統以及傅利葉係數之間的關係。陳鞏老師等人找到了許多關於一個系統做n次疊代之後的傅利葉係數，可以使得這個系統的拓樸熵大於零的充分條件。在這篇論文當中，我們創新出一個針對定義在一個區間的函數，傅利葉係數，黎阿普諾夫指數和不變測度的關係。尤其我們是針對一個定義在馬可夫分割上的片段線性函數以及二次函數來討論這三種特徵量。

A complex and unpredictable frequency spectrum of a signal has long been seen in physics and engineering as an indication of a chaotic signal. The first step to understand such phenomenon mathematically was taken up by Chen, Hsu, Huang and Roque-Sol. In particular, they look for possible connections between chaotic dynamical systems and the behavior of its Fourier coefficients. Among other things, they found variety of sufficient conditions on the Fourier coefficients of the -th iterate of an interval map , for which the topological entropy of is positive. In this thesis, we explore the relationship between the Fourier coefficients of an interval map and its Lyapunov exponent and invariant measure. Specifically, the relationships between those three quantities of two family of interval maps, piecewise linear maps admitting a Markov partition and quadratic family, are considered.