Recurrence of Transitive Points in Dynamical Systems with the Specification Property
Recurrence of Transitive Points in Dynamical Systems with the Specification Property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.