Chaos for finitely generated semigroup actions. (English) Zbl 1412.54043
Summary: In this paper, we define and study Li-Yorke chaos and distributional chaos along a sequence for finitely generated semigroup actions. Let \(X\) be a compact space with metric \(d\) and \(G\) be a semigroup generated by \(f_1, f_2,\dots, f_m\) which are finitely many continuous mappings from \(X\) to itself. Then we show if \((X,G)\) is transitive and there exists a common fixed point for all the above mappings, then \((X,G)\) is chaotic in the sense of Li-Yorke. And we give a sufficient condition for \((X,G)\) to be uniformly distributionally chaotic along a sequence and chaotic in the strong sense of Li-Yorke. At the end of this paper, an example on the one-sided symbolic dynamical system for \((X,G)\) to be chaotic in the strong sense of Li-Yorke and uniformly distributionally chaotic along a sequence is given.
MSC:
| 54H20 | Topological dynamics (MSC2010) |
| 22B99 | Locally compact abelian groups (LCA groups) |
| 37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |
Keywords:
Li-Yorke chaos; distributional chaos along a sequence; finitely generated semigroup actionsReferences:
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