【作者】 代万基；

【导师】 王军；

【作者基本信息】 大连理工大学 ， 计算数学， 2008， 博士

【摘要】 本文考虑的动力系统是一个紧致的拓扑空间到其自身的连续映射所生成的迭代系统。因为这个空间是紧致的,所以存在一个有限开覆盖。如果用一个有限字母表中的字母去标记这些开覆盖,则产生一个符号动力系统,这个系统的状态空间或相空间就是符号序列空间,动力学则由移位操作实现.由原空间中初值的序关系可相应地定义其对应的符号序列的一个序关系。由单位区间到自身的单峰映射及立方映射等生成的系统的符号序列的序关系不同于普通的字典序,而是一种“奇偶字典序”(parity-lexicographic ordering),这个序关系使得相应的符号序列具有丰富的组合性质。例如,Louck于1997年通过一些实例发现单峰映射的周期符号序列(MSS序列)具有某种“奇偶交错性质”并作为公开问题提出,后来Chen,Louck,Wang用α-序列的语言给予证实。本文用字的语言研究单峰映射和反对称立方映射的周期符号序列的组合性质,论文共三章。第一章首先回顾符号动力系统和字上组合的发展历史,然后引进一些必要的概念和记号。第二章讨论单峰映射的周期揉序列(即MSS序列)的组合性质。我们用字的语言给出相邻的两个MSS序列的显式的构造,由此重新得到其奇偶交错性质。第三章讨论反对称立方映射的周期揉序列的组合性质。我们确定给定长度的最小周期揉序列,建立其相邻关系并证明其奇偶交错性质。更多还原

【Abstract】 A dynamical system concerned here is an iterative system generated by a continued map from a compact topological space into itself. Since the space is compact, there exists a finite open covering of the space. We label each covering by a letter from a finite alphabet. Then a discrete, or symbolic dynamical system is formed: the space is the set of all possible symbolic sequences, or words, made of these letters, and the dynamics is performed by a shift operator on the sequences. According to the order relation on the initial values in the original space we may define an related order on the corresponding symbolic sequences. Different from the usual lexicographic ordering, the orderings defined on the symbolic sequences of unimodal maps or antisymmetric cubic maps defined on a interval are a kind "parity-lexicographic ordering" , which make the corresponding symbolic sequences having more combinatorial properties. For example, Louck discovered that the periodic symbolic sequences of unimodal maps (MSS sequences) have a certain parity-alternative property, and he posed it as an open problem in 1997. This problem was solved later by Chen, Louck and Wang in the language ofα-sequences. This thesis will use the language of words to investigate the combinatorial properties of the periodic symbolic sequences of unimodal maps and antisymmetric cubic maps. It consists of three chapters.In Chapter 1, we first look back the history of symbolic dynamics and combinatorics on words, and then introduce some necessary concepts and notations.Chapter 2 contributes the combinatotial properties of periodic kneading sequences of unimodal maps (MSS sequences). We shall give an explicit structure of two adjacent MSS sequences of the same length in the language of word, and then reobtain the parity-alternative property.Chapter 3 contributes the combinatotial properties of periodic kneading sequences of antisymmetric cubic maps. We shall determine the least words of given lengths, give an explicit structure of two adjacent periodic kneading sequences, as well as the parity-alternative property.更多还原