Classification Of Transitive Group Actions

Topological transitivity is a significant property of dynamical systems,and the classification of transitive systems has been one of the central topics in the study of topological dynamical systems.Up to now,lots of progresses has been made in the study of transitivity.In this thesis,we mainly study the classification of transitive group actions.We investigate systematically several topological transitivity and mixing concepts for group actions via weak disjointness,return time sets,topological complexity functions and point transitivity.The thesis is organized as follows:In Chapter 1,we introduce the background of the development of dynamical systems,the research progress of topological transitivity,and the main research work of this thesis.In Chapter 2,we mainly recall some basic notions and properties in topological dynamical systems,ergodic theory and Furstenberg family of groups which will be used in this thesis.In Chapter 3,we will investigate some notions of topological transitivity and mixing of group actions via return time sets and weak disjointness.We show that a standard dynamical system is transitive if and only if it is weakly disjoint from any strongly mixing system;a group action is totally transitive if and only if it is weakly disjoint from any transitive periodic system;scattering if and only if it is weakly disjoint from any M-system if and only if the intersection of the return time sets of any two non-empty open sets and the difference sets of any piecewise syndetic sets is non-empty;strongly scattering if and only if the return time set of any two non-empty open sets is measurable recurrence(equivalently,the intersection of it and the difference sets of any positive upper Banach density sets is non-empty);elastic if and only if the return time set of any two non-empty open sets is thick;mild mixing implies the intersection of the return time sets of any two non-empty open sets and the difference sets of infinite IP-sets is nonempty.In Chapter 4,we shall continue to develop the notion of topological complexity function for general group actions,and use it to characterize mild mixing,strong scattering and scattering.We show that a group action is scattering(resp.,strongly scattering)if and only if each non-trivial finite open cover has unbounded complexity along every syndetic set(resp.,positive upper Banach density sets);mild mixing implies each non-trivial finite open cover has unbounded complexity along every infinite IP-sets.In Chapter 5,we study transitivity,strong mixing,weak mixing,dense small periodic sets,multi-transitivity and Δ-transitivity of group actions via point transitivity.