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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:
(a)
“minimal” if $$\overline{Tx}=X$$ for every $$x\in X$$;
(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 $$T$$;
(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 $$t\in T$$;
(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 $$d(tx,ty)>c$$;
(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 $$\lim_{n\to\infty} d(t_nx, t_ny)=0$$.
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:
(1)
each $$t\in T$$ acts on $$O$$ either as an $$n$$-cyclic permutation, or as the identity map;
(2)
not all elements $$t\in T$$ act on $$O$$ as the identity map.
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:
 37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) 54H20 Topological dynamics (MSC2010)
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