One Dimensional Maps And Two Dimensional Entropy

• Main Authors: Shih-Feng Shieh, 謝世峰
• Other Authors: Jonq Juang
• Format: Others
• Language: en_US
• Published: 2001
• Online Access: http://ndltd.ncl.edu.tw/handle/21263474949022416188

博士 === 國立交通大學 === 應用數學系 === 89 === 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.