An explicit scattering, non-weakly mixing example and weak disjointness
An explicit scattering, non-weakly mixing example and weak disjointness
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DOI：
10.1088/0951-7715/15/3/320
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发表时间：
2002-05
期刊：
影响因子：
1.7
通讯作者：
Wen Huang;X. Ye
中科院分区：
文献类型：
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作者：
Wen Huang;X. Ye
我们所说的动力系统是指一对$(X,T)$，其中$X$是一个紧致度量空间，$T:X\to X$是满射且连续的。我们研究拓扑动力系统中的弱不交性。$(X,T)$是散射的，当且仅当它与所有极小系统弱不交；$(X,T)$是强散射的，当且仅当它与所有$E$-系统弱不交，即具有全支撑不变测度的传递系统。显然，弱混合系统是强散射的，而强散射系统是散射的。Akin和Glasner（2001《数学分析杂志》84卷，243 - 286页）给出了散射的存在性证明以及一个非弱混合的例子。在本文中，我们将给出一个强散射但非弱混合的明确例子。我们还定义了极端散射、弱散射，并研究了各种定义之间的关系。对于一个比传递性更强的动力性质$P$，令$\overline{P}$为这样一个性质：一个系统具有$\overline{P}$当且仅当它与任何具有$P$的系统弱不交。我们证明了$\overline{P}=P$。此外，我们证明了（厚 syndetic - 传递）= 逐段 syndetic - 传递以及（逐段 syndetic - 传递）= 厚 syndetic - 传递。
By a dynamical system we mean a pair (X,T), where X is a compact metric space and T:X→X is surjective and continuous. We study weak disjointness in topological dynamics. (X,T) is scattering iff it is weakly disjoint from all minimal systems and (X,T) is strongly scattering iff it is weakly disjoint from all E-systems, i.e. transitive systems having invariant measures with full support. It is clear that a weakly mixing system is strongly scattering and the latter is scattering. An existential proof of scattering and a non-weakly mixing example is obtained by Akin and Glasner (2001 J. Anal. Math. 84 243-86). In this paper, we will give an explicit example which is strongly scattering and not weakly mixing. We also define extreme scattering, weak scattering and study the relationships of the various definitions. For a dynamical property P stronger than transitivity, let P be the property such that a system has P iff it is weakly disjoint from any system having P. We show that P = P. Moreover, we prove that (thickly syndetic-transitive) = piecewise-syndetic-transitive and (piecewise-syndetic-transitive) = thickly syndetic-transitive.