Furstenberg families and sensitivity
Summary: We introduce and study some concepts of sensitivity via Furstenberg families. A dynamical system \((X,f)\) is \({\mathcal F}\)-sensitive if there exists a positive \(\varepsilon\) such that for every \(x\in X\) and every open neighborhood \(U\) of \(x\) there exists \(y\in U\) such that the pair \((x,y)\) is not \({\mathcal F}\)-\(\varepsilon\)-asymptotic; that is, the time set \(\{n:d(f^n(x),f^n(y))>\varepsilon\}\) belongs to \({\mathcal F}\), where \({\mathcal F}\) is a Furstenberg family. A dynamical system \((X,f)\) is \(({\mathcal F}_1,{\mathcal F}_2)\)-sensitive if there is a positive \(\varepsilon\) such that every \(x\in X\) is a limit of points \(y\in X\) such that the pair \((x,y)\) is \({\mathcal F}_1\)-proximal but not \({\mathcal F}_2\)-\(\varepsilon\)-asymptotic; that is, the time set \(\{n:d(f^n(x),f^n(y))<\delta\}\) belongs to \({\mathcal F}_1\) for any positive \(\delta\) but the time set \(\{n:d(f^n(x),f^n(y))>\varepsilon\}\) belongs to \({\mathcal F}_2\), where \({\mathcal F}_1\) and \({\mathcal F}_2\) are Furstenberg families.
scientific article; zbMATH DE number 5701490