Some Dynamical Properties Of The Discrete Group Action

This thesis is concerned with the dynamics of discrete group actions, especially we are interested in Zd actions.In chapter 1, we introduce some background of the dynamical system of discrete group actions and the main results in this thesis.In chapter 2, we consider the existence of expansive Zd actions on continua. We prove that there exists no expansive Z2 actions on graphs, but the free product Z * Z can act on the closed interval expansively.In chapter 3, The properties of chaotic group actions are considered. It is proved that the space which contains a free arc does not admit any chaotic group action. The structure of chaotic G x F actions are also studied, where F is a finite group. At last, we construct a topological space which admits chaotic group actions but does not admit any chaotic homeomorphism.In chapter 4, we consider the ergodic properties of group actions. We find a simple relation between the maximal ergodic subgroup and distal properties of algebraic actions and characterize the distal properties of Zd algebraic actions. We also prove that if the group action is equicontinuous then ergodicity and topological transitivity are equivalent.In chapter 5, we are interested in the topological contractions for automorphisms of Lie groups. We prove that the strong and weak topological contractions for automorphisms of connected Lie groups are equivalent. We also get that the connected Lie group which admits a topological contractive automorphism must be a nilpotent Lie group.