Weak mixing of semigroup actions on hyperspaces. (Chinese. English summary) Zbl 1265.37002
Summary: This paper deals with weak mixing of semigroup actions on hyperspaces. We show that the following statements are equivalent:
(1) \( (S,X)\) is weakly mixing;
(2) \( (S, 2^X)\) is weakly mixing, where \( (S, 2^X)\) is a hyperspace dynamical system induced from \( (S, X)\);
(3) for every closed subset \(K\) of \(X\) with nonempty interior there exists a sequence \(\{s_n\}\subset S\) such that \(\lim\limits_{n\to \infty}s_n (K)=X\) in \( (S, 2^X)\).
(1) \( (S,X)\) is weakly mixing;
(2) \( (S, 2^X)\) is weakly mixing, where \( (S, 2^X)\) is a hyperspace dynamical system induced from \( (S, X)\);
(3) for every closed subset \(K\) of \(X\) with nonempty interior there exists a sequence \(\{s_n\}\subset S\) such that \(\lim\limits_{n\to \infty}s_n (K)=X\) in \( (S, 2^X)\).
MSC:
| 37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |
| 54H20 | Topological dynamics (MSC2010) |
