Chaos via Furstenberg family couple
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.
scientific article; zbMATH DE number 5522070