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Devaney’s chaos for maps on \(G\)-spaces. (English) Zbl 1401.37019

Summary: We study the notion of sensitivity on \(G\)-spaces and through examples observe that \(G\)-sensitivity neither implies nor is implied by sensitivity. Further, we obtain necessary and sufficient conditions for a map to be \(G\)-sensitive. Next, we define the notion of Devaney’s chaos on \(G\)-space and show that \(G\)-sensitivity is a redundant condition in the definition.

MSC:

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37D05 Dynamical systems with hyperbolic orbits and sets
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
54H20 Topological dynamics (MSC2010)
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