On Transitive Compactness And Sensitivity In Dynamical Systems Of Semigroup Actions

In this paper,we introduced and studied the concepts of the Lyapanov Num-bers and transitive compactness in dynamical systems of semigroup actions.We analyzed the relationship among the Lyapanov Numbers in the topological tran-sitive systems or weakly mixing systems and studied the relationship among tran-sitive compactness,weakly mixing,elasticity and sensitivity.The details are as follows:In Chapter 1,we briefly described the development and branch of the dy-namical systems and introduced the research status of the dynamical properties in dynamical systems of semigroup actions.In Chapter 2,we introduced the basic concepts of the Lyapanov Numbers,transitive compactness and elasticity,and the knowledge in dynamical systems of semigroup actions.In Chapter 3,we studied the relationship among the Lyapanov Numbers.We proved that for a sensitive topologically transitive dynamical system(S,X),where S is an Abelian semigroup,(X,d)is a compact metric space,then the second Lyapanov Number and the fourth Lyapanov Number are the same(see theorem 3.6).In Chapter 4,we mainly studied the knowledge of transitive compactness and other transitivities in dynamical systems of semigroup actions.In section 4.1,we proved the following conclusion:Let(S,X)be a dynamical system,where S is Abelian,every s? is a surjective of X onto itself,(S,X)is transitive compact if and only if(S,X)is topologically transitive and for any point x? X,there exists a point z? X,such that N(x,Gz)? N(W,s-1W)?0 for any neighborhood Gz of z,opene W in X and s ? S(see proposition 4.1.3).In section 4.2,we proved the following conclusion:Let(S,X)be a dynamical system,where S is an Abelian semigroup.If(S,X)is weakly mixing,then(S,X)is transitive compact(see proposition 4.2.4).In section 4.3,we proved the following conclusion:Let(S,X)be a dynamical system,where every element of S is a surjective of X onto itself.If(S,X)is elastic,then(S,X)is totally transitive(see proposition 4.3.5).Finally,we discussed the relationship between transitive compactness and Li-Yorke sensitivity.We proved the following conclusion:Let(S,X)be a dynamical system where(X,d)is a compact metric space.If(S,X)is transitive compact,then(S,X)is Li-Yorke sensitive(see proposition 5.5).