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Chaos via Furstenberg family couple. (English) Zbl 1161.37019

A family \({\mathcal F}\) of subsets of positive integers is called a Furstenberg family if it is hereditary upwards: if \(F_1 \in {\mathcal F}\) and \(F_1 \subset F_2\) then \(F_2 \in {\mathcal F}\). Starting with a couple \(({\mathcal F}_1, {\mathcal F}_2)\) of Furstenberg sets the author define a notion of the \(({\mathcal F}_1, {\mathcal F}_2)\)-chaos for a topological dynamical system \((X,f)\) where \(X\) is a complete metric space dense in itself and \(f:X \to X\) is a countinuous map. They also show that some other definitions of chaos due to Li–Yorke and Schweizer–Smital (distributional chaos) are particular cases of their general definition. For instance, a system is Li–Yorke chaotic iff it is \(({\mathcal B},{\mathcal B})\)-chaotic where \({\mathcal B}\) is the Furstenberg family formed by all infinite subsets of positive integers. They derived some criteria for such a chaoticity and use them to show that the second type of the distributional chaos does not imply the first type of distributional chaos.

MSC:

37B40 Topological entropy
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B10 Symbolic dynamics
54H20 Topological dynamics (MSC2010)
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References:

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