环上离散动力系统的周期问题研究

A discrete dynamical system consists of a nonempty set X with some additional algebraic structure and a function f from X into itself. For a given element a contained in X, if there exists a positive integer n such that a is fixed after n times iterations of f, then a is called a periodic point of the dynamical system, and such smallest positive integer n is called the period of a. Discrete dynamical systems have been found in information science and computer science. The core of studying dynamical systems is to understand the structure of its period and periodic point. In particular, for a given dynamical system, we want to find out whether there exists periodic points, how many are the periodic points, what is distribution of length of period, what is the maximal and minimal length of period. In applications we want to give effective algorithms for determining whether a point is a periodic point of a dynamical system. We also want to find out special period point when X has interesting structure. In this case we also concern about the algebraic or arithmetic property of periodic point. For example, what is the algebraic degree and average of periodic points.

一个离散动力系统(X,f)通常由一个具有某些代数结构的非空集合X和X到本身的一个映射f构成。如果对X中的一个元素a通过f的n次迭代映射回自身，则称a是该动力系统的一个周期点，并且此时把满足条件的最小的正整数n称为a的周期。离散动力系统在信息科学，计算机工程中有着广泛的应用背景。离散动力系统的核心研究内容是刻画周期和周期点的性质。例如我们希望得到周期点的个数，周期长度的分布，特别地最大最小周期分别是多少。在实际应用中则希望得到判断某个点是否该动力系统的周期点的判断准则或有效算法。对于一些具有代数结构上的离散动力系统，我们还关心周期点代数性质和算术性质，如周期点的代数次数，均值等等。

本项目主要研究了一些有限群上的幂方动力系统,得到了这类幂方动力系统中顶点入度的分布，圈长的结构，连接到每一点的根树的结构，并给出了此类动力系统同构的充分必要条件。此外还利用代数的方法研究了有限交换环上线性动力系统的高度的分布，得到了一类线性动力系统的高度分布随着环的大小是一个拟线性函数，我们猜测所有这类线性动力系统的高度都是如此。