Recurrence of Transitive Points in Dynamical Systems with the Specification Property
Recurrence of Transitive Points in Dynamical Systems with the Specification Property
复制标题
DOI：
10.1007/s10114-018-7534-7
复制
发表时间：
2018-06
期刊：
影响因子：
--
通讯作者：
Xiao Yi Wang;Yu Huang
中科院分区：
文献类型：
--
作者：
Xiao Yi Wang;Yu Huang
设T：X→ X是紧度量空间X上的连续映射.一个点x ∈ X称为Banach回归点，如果对所有邻域Vofx，{n∈ N：Tn（x）∈V}有正的上Banach密度.用Tr（T），W（T），QW（T），BR（T）表示（X，T）的传递点集，弱概周期点集，拟弱概周期点集和Banach回归点集.若（X，T）具有规范性质，则证明了每个传递点都是Banach常返的，并且证明了其中W *（T）是与Zhou和Feng提出的一个公开问题有关的常返点集.具体地说，集合Tr（T）| W *（T）| W（T）是X中的残差。构造了一个点x ∈BR\QWin符号动力系统，证明了该动力系统的集合W（T），QW（T）和BR（T）都是Borel集.
LetT:X→Xbe a continuous map of a compact metric spaceX. A pointx∈Xis called Banach recurrent point if for all neighborhoodVofx, {n∈ ℕ:Tn(x) ∈V} has positive upper Banach density. Denote byTr(T),W(T),QW(T) andBR(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 ∅ ≠W(T) ∩Tr(T) ⫋W* (T) ∩Tr(T) ⫋QW(T) ∩Tr(T) ⫋BR(T) ∩Tr(T), in whichW* (T) is a recurrent points set related to an open question posed by Zhou and Feng. Specifically the set Tr(T) ∩ W * (T) \W(T) is residual inX. Moreover, we construct a pointx∈BR\QWin symbol dynamical system, and demonstrate that the setsW(T),QW(T) andBR(T) of a dynamical system are all Borel sets.