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数学进展 - 2018, Vol. 47(2): 207-214
Dense Small Periodic Sets and Total Transitivity of Semigroup Actions

WANG Huoyun1,*, ZHU Guifang1,**, WU Hongying2,***}

1. 广州大学数学与信息科学学院, 广州, 广东, 510006;
2. 怀化学院数学与计算科学学院, 怀化, 湖南, 418008
1. School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong, 510006, P. R. China;
2. School of Mathematics and Computing Science, Huaihua University, Huaihua, Hunan, 418008, P. R. China

收稿日期: 2016-09-25
出版日期: 2018-05-16
2018, Vol. 47(2): 207-214
DOI: 10.11845/sxjz.2016114b

[216 KB]

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摘要 称一个动力系统~$(S,X)$ 具有稠密g-小周期集, 如果对任意非空开集~$U\subset X$, 存在非空闭子集~$Y\subset U$ 和~$S$ 的一个g-syndetic 子半群~$T$, 使得~$TY\subset Y$; 称一个传递的动力系统~$(S,X)$ 是g-完全传递的, 如果对~$S$ 的每一个g-syndetic 子半群~$T$,~$(T,X)$ 都是传递的. 本文指出, 每一个具有稠密 g-小周期集的g-完全传递系统~$(S,X)$ 不交于任何极小系统, 其中~$S$ 是一个可数交换半群,~$S$ 最多只有可数个g-syndetic 子半群且~$S$ 中的每一个元~$S$都为$X$到自身的满射.
关键词 稠密g-小周期集g-完全传递g-syndetic子半群半群作用    
Abstract:A dynamical system $(S,X)$ is defined as a system with dense g-small periodic sets, if for every nonempty open subset $U$ of $X$ there exist a nonempty closed subset $Y$ of $U$ and a g-syndetic subsemigroup $T$ of $S$ such that $TY\subset Y$. A transitive dynamical system $(S,X)$ is called g-totally transitive, if $(T,X)$ is transitive for every g-syndetic subsemigroup $T$ of $S$. In this article, we point out that every g-totally transitive dynamical system $(S,X)$ with dense g-small periodic sets is disjoint from all minimal systems, where $S$ is a countable abelian semigroup, $S$ has at most countably many g-syndetic subsemigroups and every $s$ of $S$ is a surjective map from $X$ onto itself.
Key wordsdense g-small periodic sets    g-total transitivity    g-syndetic subsemigroup    semigroup actions
PACS:  O189.11  
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