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

我们研究了与动力系统相关的Perron-Frobenius算子的谱。方法是定义与动力系统相关的母函数，然后构造关于母函数的更新方程，从而确定一个矩阵，我们称之为Fredholm矩阵。这个矩阵的行列式的零点决定了动力系统的遍历性，如动力系统关联性的遍历性、混合性和衰减率，利用这个方法可以研究动力系统的旋转数或大偏差。我们甚至可以计算动力系统生成的分形数的Hausdorff维，特别是我们研究了动力系统生成的随机数的差异性。对于一维情形，我们可以构造一个如何确定随机数差的定理。利用这个定理，我们可以确定一维动力系统生成的低偏差序列。近年来，随着数学金融等的进步，我们需要高维的低偏差序列。然而，对于高维情形，我们只构造了抽象定理来确定Perron-Frobenius算子的谱。因此，我们尝试了低偏差序列的例子，并得到了二维和三维情况。

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.