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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:
“minimal” if \(\overline{Tx}=X\) for every \(x\in X\);
“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 \(T\);
“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 \(t\in T\);
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 \(d(tx,ty)>c\);
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:
each \(t\in T\) acts on \(O\) either as an \(n\)-cyclic permutation, or as the identity map;
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.

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
54H20 Topological dynamics (MSC2010)
Full Text: DOI
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