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Mixing and proximal cells along sequences. (English) Zbl 1055.37014

Summary: A dynamical system \((X,T)\) is \({\mathcal F}\)-transitive if for each pair of open and nonempty subsets \(U\) and \(V\) of \(X\), \(N(U,V)= \{n\in\mathbb{Z}_+: U\cap T^{-n}\neq \emptyset\}\in{\mathcal F}\), where \({\mathcal F}\) is a collection of subsets of \(\mathbb{Z}_+\) that is hereditary upward. \((X,T)\) is \({\mathcal F}\)-mixing if \((X\times X,\,T\times T)\) is \({\mathcal F}\)-transitive. For a subset \(S\) of \(\mathbb{Z}_+\), \((x,y)\in X\times X\) is \(S\)-proximal if \(\liminf_{S\ni n\to+\infty} d(T^n(x), T^n(y))= 0\) and the \(S\)-proximal cell \(P_S(x)\) is the set of points that are \(S\)-proximal to \(x\in X\). We show that if \((X,T)\) is \({\mathcal F}\)-mixing, then for each \(S\in k{\mathcal F}\) (the dual family of \({\mathcal F}\)) and \(x\in X\), \(P_S(x)\) is a dense \(G_\delta\) subset of \(X\), and when \((X,T)\) is minimal and \({\mathcal F}\) is a filter the reciprocal is true. Moreover, other conditions under which the reciprocal is true are obtained. Finally, the structure of proximal cells for \({\mathcal F}\)-mixing systems is discussed, and a new and simpler proof of the Xiong-Yang theorem is presented.

MSC:

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
54H20 Topological dynamics (MSC2010)
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