一維函數與二維的熵

Abstract

本論文主要可以分為兩個部分，第一個部分標題為 Interval Maps, Total Variation and Chaos。在Huang and Chen [1]的文章中提供了關於total variation的概念稱之為H_1條件，H_1條件可用已判斷一個一維函數f的混沌行為，他們並且證明H_1條件與敏感性等價。在本論文中得出H_1條件則函數具有非2 power週期並且其拓樸熵為正，配合Chen的結果得出一個函數的敏感性則拓樸熵為正。 本論文的第二個部分標題為On The Two Dimensional Entropy Of The Golden Mean Matrices，首先我們證明倘若某一transition matrix為rank one，則entropy可以正確計算出來。接著我們證明假設給定一irreducible的transition matrix A，並由A造出一片斷線性函數$f_{\bf A,x}$，則我們可以得出$\sup_{{\bf x}:partition}\lambda({\bf x})=h_{top}(f_{\bf A,x})$,其中$\lambda({\bf x})$為$f_{\bf A,x}$的Liapunov Exponent，$h_{top}(f_{\bf A,x})$為$f_{\bf A,x}$的拓樸熵，最後我們由上述結果得出二維golden mean的一nontrivial下界。

My dissertation contains two parts. The subtitle of Part I is ``Interval Maps, Total Variation and Chaos". In a paper by Huang and Chen [1], a concept related to total variation termed ${\mathcal H}_1$ condition was proposed to characterize the chaotic behavior of an interval map $f$. They proved that for a piecewise-monotone continuous map $f$, ${\mathcal H}_1$ condition is equivalent to the sensitivity of $f$ on initial data. They also showed that such map $f$ has periodic points of period $2^n$ for all $n\in {\mathbb N}$. In this paper, we show that for a piecewise-monotone continuous map, ${\mathcal H}_1$ condition also gives the positivity of the topological entropy of $f$. Consequently, $f$ has a periodic point whose period is not a power of 2. The Part II is entitled ``On The Two Dimensional Entropy Of The Golden Mean Matrices". Our main results here in Part I are the following. First, we show that if either of the transition matrices is rank-one, then the associated exact entropy can be explicitly obtained. Second, let ${\bf A}$ be an irreducible transition matrix, and $f_{\bf A,x}$ be a piecewise linear map induced by ${\bf A}$ and a partition ${\bf x}$ of $[0,1]$. We then prove that $\sup_{{\bf x}:partition}\lambda({\bf x})=h_{top}(f_{\bf A,x})$, where $\lambda(\bf x)$ is the Liapunov exponent of $f_{\bf A,x}$, and $h_{top}(f_{\bf A,x})$ is the topological entropy of $f_{\bf A,x}$. Third, we combine the above results estimates a nontrivial lower bound of the spatial entropy of two-dimensional gold mean.