Research on Spectra of Perron-Frobenius operator generated by dynamical systm and random numbers

动力系统与随机数生成Perron-Frobenius算子谱的研究

基本信息

批准号：
16540121
负责人：
MORI Makoto
金额：
$ 2.27万
依托单位：
Nihon University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16540121/
关键词：
Perron-Frobenius operatur ergodic theory low discrepancy spectrum 疑似乱数擬似乱数 rotation number 大偏差原理 van der Corput列 discrepancy

项目摘要

We have studied the spectra of the Perron-Frobenius operator associated with dynamical systems. The method is to define the generating function associated with the dynamical systems, then construct renewal equations on generating functions.Then we can determine a matrix which we call Fredholm matrix. The zeros of the determinant of this matrix determine the ergodic property of the dynamical system, such as ergodicity, mixingity and decay rate of correlations of dynamical systems.Using this method, we could study rotaion numbers of dynamical system or large deviations. We can even calculate the Hausdorff dimension of fractals generated by dynamical systems.Especially, we have studied the discrepancy of random numbers generated by dynamical systems. For one dimensional cases, we can construct a theorem how to determine the discrepancy of random numbers. Using this theorem, we can determine the low discrepancy sequences generated by one dimensional dynamical systems.Recently, the progress of mathematical finance and so on, we need the higher dimensional low discrepancy sequences. However, for higher dimensional cases, we have only constructed abstract theorem to determine the spectra of the Perron-Frobenius operateor. Therefore, we have tried examples of low discrepancy sequences, and get two and three dimensional cases.

我们研究了与动态系统相关的Perron-Frobenius操作员的光谱。该方法是定义与动态系统关联的生成函数，然后在生成函数上构造更新方程。然后，我们可以确定一个称为Fredholm矩阵的矩阵。该矩阵决定因素的零确定动态系统的厄乳元特性，例如千古，混合性和动力系统矛盾的衰减速率。使用此方法，我们可以研究动力学系统的旋转数量或大偏差。我们甚至可以计算动态系统生成的分形的Hausdorff尺寸。尤其是我们研究了动态系统生成的随机数的差异。对于一个维情况，我们可以构建一个定理如何确定随机数的差异。使用该定理，我们可以确定由一个维动力系统生成的低差异序列。此外，数学金融的进展等等，我们需要更高维度的低差异序列。但是，对于较高维度的情况，我们仅构建了抽象定理来确定perron-frobenius操作器的光谱。因此，我们尝试了低差异序列的示例，并获得了两个和三维案例。