Recurrence of Transitive Points in Dynamical Systems with the Specification Property

作     者：Xiao Yi WANG Yu HUANG 

作者机构：School of Mathematics and Statistics Zhaoqing University Zhaoqing 526061 P. R. China Department of Mathematics Sun Yat-sen University Guangzhou 510275 P. R. China 

出 版 物：《Acta Mathematica Sinica,English Series》 (数学学报（英文版）)

年 卷 期：2018年第34卷第12期

页      面：1879-1891页

核心收录：

学科分类：0711[理学-系统科学] 07[理学] 070104[理学-应用数学] 0701[理学-数学] 071101[理学-系统理论] 

基　　金：Supported by National Natural Science Foundation of China,Tian Yuan Special Foundation(Grant No.11426198) the Natural Science Foundation of Guangdong Province,China(Grant No.2015A030310166) 

主　　题：Specification property invariant measures recurrent points measure center 

摘      要：Let T ： X →X be a continuous map of a compact metric space X. A point x E X is called Banach recurrent point if for all neighborhood V of x, （n ∈ N ： T^n（x） ∈ V} has positive upper Banach density. Denote by Tr（T）, W（T）, QW（T） and BR（T） the sets of transitive points, weakly almost periodic points, quasi-weakly almost periodic points and Banach recurrent points of （X, T）. If （X, T） has the specification property, then we show that every transitive point is Banach recurrent and O≠ W（T） n Tr（T） ≠ W＊（T） ∩ Tr（T） ≠ QW（T） ∩ Tr（T） ≠ BR（T） ∩ Tr（T）, in which W＊（T） is a recurrent points set related to an open question posed by Zhou and Feng. Specifically the set Tr（T） M W＊（T） ／ W（T） is residual in X. Moreover, we construct a point x E BR ／ QW in symbol dynamical system, and demonstrate that the sets W（T）, QW（T） and BR（T） of a dynamical system are all Borel sets.