Properties Of Frequently Hypercyclic C₀-semigroup

In this paper,we first introduce the concepts,examples and basic conclusions of the hypercyclicity,weak mixing,topological mixing,frequently hypercyclicity of discrete and continuous semigroups.Meanwhile,it also gives some examples and basic conclusion of these concepts.According to the basis of single frequently hypercyclic operators and the study methods of weak mixing and mixing,furthermore comparing the properties of frequently hypercyclic operators with frequently hypercyclic semigroups,we discuss the properties of frequently hypercyclic semigroups.This text mainly draws the following conclusions:Firstly,we give a sufficient condition for the positive real number set to be syndetic set and a sufficient condition for the C0-semigroup(Tt)t?0 to be weakly mixing.Theorem 3.2.1 in this paper reveals that if a set of positive real numbers is known to have a positive lower density,the difference set is syndetic.By using this theorem,we prove that the frequently hypercyclic semigroups are weakly mixing.Secondly,a sufficient condition is given to determine that a C0-semigroup is mixing.Utilizing the proving method of functional analysis,we prove that C0-semigroups satisfying the frequent hypercyclicity criterion are mixing,and give counter example to show that the inverse proposition of this theorem is not valid.Finally,it summarizes the whole paper and shows the prospect of issues still need to continue research.