Research on the spectrum of Perron-Frobenius operator and pseudo random number associated with higher dimensional dynamical system

Perron-Frobenius算子的本质谱半径在1维情况下取最小值，计算比较容易，但在高维情况下，考虑非本质谱半径取最小值的情况是很重要的，要构造本质谱半径取最小值的动力系统是不容易的.利用代数方法构造了这样一个除1外没有特征值的动力系统。从这个动力系统，我们构造低差异序列在任何维度。这为计算机科学如随机积分中的模拟和数值积分的研究带来了新的进展。另一方面，本研究也揭示了构造具有最小本质谱半径的动力系统的困难，以及在高维情形下构造“好”随机数的困难。

In 1-dimensional cases, the essential spectrum radius of thePerron-Frobenius operator takes its minimum value, and it is easy to calculate it.However, in higher dimensional cases, it is important to consider the cases when its unessential spectrum radius takes its minimum value, and it is not easy to construct dynamical systems for which the essential spectrum radius attains its minimum value. We construct such a dynamical system which has no eigenvalue except 1 by using algebraic method. From this dynamical system, we construct low discrepancy sequences in any dimension. This makes a new progression to computer science such as simulation and numerical integral in stochastic integrals. On the other hand, this research suggests the difficulty of constructing dynamical system which has minimal essential spectrum radius, and the difficulty of constructing “good” random numbers in higher dimensional cases.