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.


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)
Full Text: DOI


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