真拟弱几乎周期点和拟正则点

Proper quasi-weakly almost periodic points and quasi-regular points

T为紧致度量空间X上的连续映射,M(X)为X上所有Borel概率测度.设x∈X,记Mx(T)为概率测度序列{1n∑n 1i=0δTi(x)}在M(X)中的极限点的集合,其中δx表示支撑集是{x}的点测度.记W(T)和QW(T)分别为T的弱几乎周期点和拟弱几乎周期点集.本文证明,如果(X,T)非平凡且满足specifcation性质,则存在x,y∈QW(T)\W(T)(称为真拟弱几乎周期点),分别满足μ∈Mx(T),x∈Supp(μ)和ν∈My(T),y∈/Supp(ν),回答了周作领等提出的公开问题.Mx(T)在弱拓扑中是紧致连通集,所以,要么是单点集,要么是不可数集.如果x∈QW(T)\W(T),则Mx(T)是不可数集.一个自然的问题是,怎么刻画M x(T)是单点集的点x(这时x称为拟正则点).本文给出M x(T)是单点集的充要条件.

Let X be a compact metric space and T ： X → X be a continuous map. Denote by M（X） the set of all Borel probability measures on X. For x ∈ X, let Mx（T） be the set of all limit points of the sequence 1 n ∑n 1 i=0 δ Ti（x） in M（X）, where δx is the atomic probability measure with support {x}. Denote by W（T） and QW（T） the sets of all weakly almost periodic points and quasi-weakly almost periodic points of T, respectively. We proved that if T has specification property, then there exist x, y ∈ QW（T） ／ W（T）, respectively, with the properties that μ ∈ Mx（T）, x ∈ Supp（μ） and ν ∈ My（T）, y ∈/ Supp（ν）. This answers the open problem proposed by Zhou and Feng. On the other hand, it is well known that Mx（T） is a nonempty closed connected subset of M（X） in the weak topology. If x ∈ QW（T） ／ W（T）, Mx（T） will be uncountable. A natural problem is how to characterize the point x with Mx（T） being a singleton. We give a characterization of such x′s.