Classifcation Of Topologically Transitive Z~n-actions On Star-trees

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: Râ†'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.