The Weakly Mixing, Recurrence And Chaos Of Semigroup Actions

In this thesis, we mainly study the weakly mixing, chaos and recurrence of M-system of semigroup actions.In chapter1, we briefly describe the developed history and the branch of dy-namical systems, and emphatically introduce the development of the dynamics onsemigroup actions.In chapter2, we recall some basic notions and properties on topological dynam-ical system of semigroup action.In chapter3, We mainly study the equivalent conditions of weakly mixing ofsemigroup action. We show that (1)(S, X) is weakly mixing iff there exists a se-quence {s_n} S, such that limnâ†'∞sˉ_n(K)=X, for every closed subset K of Xwith nonempty interior, where (Sˉ,2~X) is the hyperspace dynamical system inducedfrom (S, X).(2)(S, X) is weakly mixing iff it has a uniform positive sequence en-tropy and it is transitive, where X is a compact Hausdorff space and S is an abelianmonoid.In chapter4, at first, we study the recurrence of transitive points of M-systemin the dynamical of semigroup actions. We proved that if S is countable,(S, X) is aM-system iff N(x, U) is a piecewise syndetic set, for every transitive point x andevery neighborhood U of x. Finally, we obtained chaotic theorem in the dynamicalsystem of semigroup actions.