频繁超循环半群的(弱)混合性

对于单个算子而言,所有频繁超循环算子都是弱混合的,满足频繁超循环准则的算子都是拓扑混合的.在单个频繁超循环算子的研究成果的基础上,再结合单个算子弱混合和混合的研究方法,进一步对单个频繁超循环算子和频繁超循环半群的相关性质进行了对比分析,主要讨论了频繁超循环C_0-半群的相关性质.首先,把Erd?s-Sárk?zy定理推广到了在实数集上,给出了判定正实数集合是syndetic集的一个充分条件,即已知一个正实数集合有正的下密度,则这个集合的差集是syndetic的.其次,证明了任意频繁超循环C_0-半群是弱混合的.最后,给出了判定C_0-半群是混合的一个充分条件.利用泛函分析的方法,证明了满足频繁超循环准则的C_0-半群是混合的.

The (Weakly) Mixing Property of Frequently Hypercyclic Semigroups

For a single operator, frequently hypercyclic operators are weakly mixing, and operators satisfying the frequent hypercyclcity criterion are topologically mixing. Based on the results in the researches of single frequently hypercyclic operators, we use the research methods of weak mixing and mixing to make a comparative analysis of the related properties of frequently hypercyclic operators with frequently hypercyclic semigroups, with the focus of discussion placed on the properties of frequently hypercyclic C₀-semigroups. First, we generalize the Erdös-Sárközy theorem to real number sets, and give a sufficient condition for judging positive real number sets to be syndetic sets, that is, if a known positive real number set has positive lower density, then its difference set is syndetic. Next, we prove that any frequent hypercyclic C₀-semigroup is weak mixing. Finally, we give a sufficient condition for C₀-semigroupsto be mixing. With the method of functional analysis, we prove that C₀-semigroups satisfying the frequent hypercyclcity criterion are mixing.