Furstenberg family and chaos (Q2465131)

Let \(\mathcal{P}\) be the collection of all subsets of \(\mathbb{Z}^+.\) A family \(\mathcal{F}\subset \mathcal{P}\) is called a Furstenberg family if it is hereditary upwards, that is, \(F_1\subset F_2\) and \(F_1\in\mathcal{F}\) imply \(F_2\in\mathcal{F}.\) In the first part of the paper the authors present basic facts concerning Furstenberg families and related notions to be used later. In the second part, given a discrete dynamical system \((X,f),\) they define a Furstenberg family \(\mathcal{F}\) to be compatible with it if for every open set \(V\) of \(X\) the set of \(\mathcal{F}\)-attaching points of \(V\) is a \(G_{\delta}\)-set (in turn, an \(\mathcal{F}\)-attaching point \(x\in X\) of \(V\) verifies that if \(f^{n_i}(x)\in V,\) then \(n_i\in \mathcal{F}\)). In Theorem 3.2 the authors find sufficient conditions for a Furstenberg family to be compatible with every system. Next, for a discrete dynamical system and a Furstenberg family \(\mathcal{F}\) they define the notion of \(\mathcal{F}\)- scrambled and \(\mathcal{F}\)-\(\varepsilon\)-scrambled pair of points in \(X\) (too involved to be described here). In Theorem 5.2 they prove that a scrambled pair in the Li-Yorke sense [see \textit{T. Y. Li} and \textit{J. A. Yorke}, Am. Math. Monthly 82, 985--992 (1975; Zbl 0351.92021)] or in the distribution sense [see \textit{B. Schweizer} and \textit{J. Smítal}, Trans. Am. Math. Soc. 344, 737--754 (1994; Zbl 0812.58062)] is an \(\mathcal{F}\)-scrambled pair for a suitable Furstenberg family. Finally, they introduce the notion of a system \((X,f)\) being generically-(strongly) \(\mathcal{F}\)-chaotic [\(\equiv\) the set of all \(\mathcal{F}\)-(\(\mathcal{F}\)-\(\varepsilon\))-scrambled pairs of points of \(X\times X\) is a \(G_\delta\)-set dense in \(X\times X\)], and obtain a criterion for a system to be generically-strongly \(\mathcal{F}\)-chaotic for a particular class of compatible Furstenberg families.