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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.

我们研究了与动力系统相关的Perron-Frobenius算子的谱。该方法首先定义与动力系统相关的生成函数，然后在生成函数上构造更新方程。然后我们可以确定一个矩阵我们称之为弗雷德霍姆矩阵。该矩阵行列式的零点决定了动力系统的遍历性，如遍历性、混合性和相关关系的衰减率。用这种方法可以研究动力系统的旋转数或大的偏差。我们甚至可以计算由动力系统产生的分形的豪斯多夫维数。特别地，我们研究了由动力系统产生的随机数的差异。对于一维情况，我们可以构造一个判定随机数的差异的定理。利用这一定理，我们可以确定一维动力系统产生的低差序列。近年来，随着数学金融等领域的发展，我们需要高维低差序列。然而，对于高维情况，我们只构造了抽象定理来确定Perron-Frobenius算子的谱。因此，我们尝试了低差异序列的例子，并得到了二维和三维的情况。