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Sensitive dependence on initial conditions between dynamical systems and their induced hyperspace dynamical systems. (English) Zbl 1172.37006

If \((E,f)\) is a dynamical system, then the hyperspace dynamical system \((\widehat{E},\widehat{f})\) is defined by \(\widehat{f}(A):=f(A)\) on the collection \(\widehat{E}\) of all subsets of \(E\). The relation of different concepts of chaotic behaviour on \((E,f)\) and \((\widehat{E},\widehat{f})\) has been investigated in several papers. A map is said to depend sensitively on initial conditions (this property is briefly called sensitivity), if there is a \(\delta>0\) such that for any \(x\in E\) and any \(\varepsilon>0\) there is a \(y\in E\) with \(d(y,x)<\varepsilon\) and an \(n\in{\mathbb N}\) with \(d(f^{n}(y),f^{n}(x))\geq \delta\). In this paper the authors introduce the notion of collective sensitivity. This means that there is a \(\delta>0\) such that for finitely many \(x_{1},x_{2},\dots ,x_{k}\in E\) and any \(\varepsilon>0\) there are \(y_{1},y_{2},\dots , y_{k}\in E\) with \(d(y_{j},x_{j})<\varepsilon\) for all \(j\in\{1,2,\dots ,k\}\) and there is an \(n\in{\mathbb N}\) and a \(u\in\{1,2,\dots ,k\}\) such that \(d(f^{n}(y_{j}),f^{n}(x_{u}))\geq\delta\) for all \(j\in\{1,2,\dots ,k\}\) or \(d(f^{n}(x_{j}),f^{n}(y_{u}))\geq\delta\) for all \(j\in\{1,2,\dots ,k\}\).
It is proved that \((\widehat{E},\widehat{f})\) is sensitive if and only if \((E,f)\) is collectively sensitive. Here \(\widehat{E}\) is endowed with the hit-or-miss topology. Moreover, also the conditions \(({\mathcal C},\widehat{f})\) is sensitive and \(({\mathcal F},\widehat{f})\) is sensitive are equivalent to \((\widehat{E},\widehat{f})\) is sensitive, where \({\mathcal C}\) is the collection of all nonempty compact subsets of \(E\) and \({\mathcal F}\) is the collection of all nonempty finite subsets of \(E\), both endowed with the Hausdorff metric (which is equivalent to the Vietoris topology in this case). The authors also prove that weak mixing implies collective sensitivity.
Reviewer: Peter Raith (Wien)

MSC:

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
54B20 Hyperspaces in general topology
54H20 Topological dynamics (MSC2010)
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