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Sensitivity and chaos of semigroup actions. (English) Zbl 1267.37010
A semiflow $(T,X,\pi)$, shortly $(T,X)$, is a triple consisting of a topological monoid $T$, a metric space $(X,d)$ and a continuous action $\pi:T\times X\to X$ of $T$ on $X$. The semiflow $(T,X)$ is said to be: {\parindent=6mm \item{(a)} “minimal” if $\overline{Tx}=X$ for every $x\in X$; \item{(b)} “syndetically transitive” if for any nonempty open subsets $U,V$ of $X$, the set $N(U,V)=\{t\in T\,|\,U\cap t^{-1}V\neq \emptyset\}$ is a syndetic subset of $T$; \item{(c)} “pointwise equicontinuous” if every point $x\in X$ is an equicontinuity point, meaning that for every $\varepsilon>0$ there is $\delta>0$ such that $d(x,y)<\delta$ implies $d(tx,ty)<\varepsilon$ for every $t\in T$; \item{(d)} sensitive if there exists $c>0$ (a sensitivity constant) such that for every $x\in X$ and every open neighborhood $U$ of $x$ there exists $y\in U$ and $t\in T$ with $d(tx,ty)>c$; \item{(e)} Li-Yorke chaotic if there is an uncountable scrambled subset $Y$ of $X$, meaning that for any two distinct points $x,y\in Y$ there are two infinite sequences $(s_n)$, $(t_n)$ in $T$ such that $\lim_{n\to\infty} d(s_nx, s_ny)>0$ and $\lim_{n\to\infty} d(t_nx, t_ny)=0$. \par} A point $x\in X$ is a “transitivity point” of $(T,X)$ if $Tx$ is dense in $X$. A subset $O=\{x_1, \dots, x_n\}$ of $X$ is an $n$-periodic orbit of $(T,X)$ if the following conditions hold: {\parindent=6mm \item{(1)} each $t\in T$ acts on $O$ either as an $n$-cyclic permutation, or as the identity map; \item{(2)} not all elements $t\in T$ act on $O$ as the identity map. \par} A topological semigroup $T$ is called a C-semigroup if for every $t\in T$ the subset $\overline{T\setminus Tt}$ of $T$ is compact. In the paper under review the two main results are the following theorems. Theorem 1. Let $(T,X)$ be a semiflow in which $X$ is a Polish space and $T$ is a C-semigroup. If $(T,X)$ is syndetically transitive, then it is either (i) minimal and pointwise equicontinuous, or, (ii) sensitive. Theorem 2. Let $(T,X)$ be a semiflow in which $X$ is a Polish space and $T$ is abelian. Suppose that $(T,X)$ has a transitive point $x$ and an $n$-periodic orbit $O$. Let $S$ be the set of all $t\in T$ which act on $O$ as the identity map and suppose that the set $\overline{Sx}$ has no isolated points. Then $(T,X)$ is Li-Yorke chaotic. The paper is well-written and interesting.

MSC:
37B05Transformations and group actions with special properties
54H20Topological dynamics
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Full Text: DOI
References:
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