Multi-transitivity Of Semigroup Actions

In 2010,notions of multi-transitivity,weakly mixing and Δ-transitivity for cascades are introduced by Moothathu,and then generalized to topological dynamic systems ofsemigroup actions by Zeng Tiaoying in 2017.This thesis studies multi-transitivity and its related properties for dynamical systems of semigroup actions,and mainly contains the following three sections:(1)Study the relations between thickly transitivity,syndetically transitivity and multiransitivity of semigroup actions.Our main result for this section as the following:Let(S,X)be a weakly mixing and minimal dynamical system,where S’ is Abelian and every element of S is a surjective map on X.If Sn is s-syndetic subsemigroup of S for every n∈N,then(S,X)is multi-transitive.(2)In 2010,Moothathu proved that multi-transitivity,weakly mixing and Δ-transitivity are equivalent for minimal homeomorphisms.Here in this thesis,we extend this result to Abelian group actions.(3)Consider multi-transitivity and chaoticity for dynamical systems of semigroup actions.The main result for this part is that a multi-transitive system of Abelian group actions is Ki-York ε-chaotic for some ε>0...