Sensitivity, Chaos And Transitivity Of Semigroup Action

In this thesis, we study mainly the sensitivity, chaos and transitivity of semigroupaction.In Chapter 1, we describe the origination and development of the theory of dy-namical systems , and give a brief survey of the backgroud and recent progress of thedynamics on group action.In Chapter 2, the basic notions and properties on topological dynamical systemand the topological dynamical system on semigroup action are recalled.In Chapter 3, we deal with sensitivity and chaos of semigroup actions. Thenotions of syndetic transitivity is introduced and investigated, It turns out that asyndetic-transitive system (X,d) is either minimal and equicontinuous or sensitive,where (X,d) is a Polish space and S is a C-semigroup. Besides we prove the follow-ing result: Let (X,d) be a Polish space and S be an abelian monoid. Suppose that thefollowing conditions hold: (1) (S,X) has a transitive point x and an n-periodic orbitO; (2) Hx is perfect where H = {s∈S : s|O is an identity map }. Then (S,X) ischaotic.In Chapter 4, we study the transitivity of semigroup action. The notions of thicktransitivity and ( syndetic)? transitivity are introduced and investigated, it turns outthat a system is thick transitive if and only if it is weakly mixing, where the semi-group that acts on the space is an abelian monoid in which every map is surjective .Additionally, we show that a ( syndetic)? transitive system with dense almost pe-riod points is weakly mixing, where the semigroup that acts on the space is an abeliansemigroup in which every map is surjective.