【作者】 李春雷；

【导师】 史恩慧；

【作者基本信息】 苏州大学 ， 基础数学， 2012， 硕士

【摘要】 共轭分类是动力系统的基本问题之一.这方面最经典的结果当属Poincare利用旋转数对圆周保向同胚给出的分类定理.近期,该结果已经被法国数学家Ghys推广到一般群作用.本文,我们讨论交换群Zn在星型树上拓扑传递作用的共轭分类问题.第一章介绍了拓扑动力系统理论和群论中的一些基本概念和定义.第二章考虑区间上拓扑传递Zn-作用的共轭分类.首先,对于任意一个无理数α∈(0,1)以及每个n≥2,我们构造了一个紧传递Zn-作用φα,n：R→R.然后,对于任意两个如此构造的同胚作用φα,n与φβ,n,我们给出了它们共轭的充要条件.最后,我们证明了：对于直线上有可数多个非传递点的紧传递作用均共轭于我们所构造的某个作用φα,n.第三章考虑三个分支的星型树Y上拓扑传递Z2-作用共轭分类.类似于第二章的构造方法,对任一无理数α∈(0,1),我们构造了Y上的传递Z2-作用Hα.最后我们证明了：对于星型树Y上任意一个传递Z2-作用均与我们构造的某个作用Hα共轭.更多还原

【Abstract】 The conjugate classifcation is one of the basic problems of topological dynamicalsystems. Without doubt, the most classical result is that Poincar′e established theclassifcation theorem for orientation preserving homeomorphisms on a circle by usingrotation numbers. Recently, French mathematician Ghys has extended that result intogeneral group actions.In this paper, We mainly discuss the conjugate classifcation problem of topologi-cally transitive actions of Abelian group Znon star-trees.In Chapter one, we introduce some basic notions and defnitions in topologicaldynamical systems and group theory.In Chapter two, we consider the conjugate classifcation of topologically transitiveZn-actions on the interval. Firstly, for any irrational number α∈(0,1) and each n≥2,we construct a tightly transitive Zn-action φα,n: R→R on the line. Then, we givea necessary and sufcient condition for conjugacy of the two actions φα,nand φβ,nwhich we constructed. Finally, we prove that every tightly transitive Zn-actions onthe line with countably many non-transitive points is conjugate to some φα,nthat weconstructed.In Chapter three, we consider the conjugate classifcation of some topologicallytransitive Z2-actions on the star-tree Y with three branches. At frst, using the methodsimilar to that we use in Chapter two, for any irrational number α∈(0,1) we constructa topologically transitive Z2-action Hαon Y. Finally, it is shown that any topologicallytransitive Z2-action on the star-tree Y is conjugate to some action Hα.更多还原