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Zbl 1172.37006
Wang, Yangeng; Wei, Guo; Campbell, William H.
Sensitive dependence on initial conditions between dynamical systems and their induced hyperspace dynamical systems. (English)
[J] Topology Appl. 156, No. 4, 803-811 (2009). ISSN 0166-8641

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 $n\in{\Bbb N}$ with $d(f\sp{n}(y),f\sp{n}(x))\ge \delta$. In this paper the authors introduce the notion of collective sensitivity. This means that there is a $\delta>0$ such that for finitely many $x\sb{1},x\sb{2},\dots ,x\sb{k}\in E$ and any $\varepsilon>0$ there are $y\sb{1},y\sb{2},\dots , y\sb{k}\in E$ with $d(y\sb{j},x\sb{j})<\varepsilon$ for all $j\in\{1,2,\dots ,k\}$ and there is an $n\in{\Bbb N}$ and a $u\in\{1,2,\dots ,k\}$ such that $d(f\sp{n}(y\sb{j}),f\sp{n}(x\sb{u}))\ge\delta$ for all $j\in\{1,2,\dots ,k\}$ or $d(f\sp{n}(x\sb{j}),f\sp{n}(y\sb{u}))\ge\delta$ for all $j\in\{1,2,\dots ,k\}$. \par 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 $({\Cal C},\widehat{f})$ is sensitive and $({\Cal F},\widehat{f})$ is sensitive are equivalent to $(\widehat{E},\widehat{f})$ is sensitive, where ${\Cal C}$ is the collection of all nonempty compact subsets of $E$ and ${\Cal F}$ 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.
[Peter Raith (Wien)]

MSC 2000:: *37B05 Transformations and group actions with special properties
54B20 Hyperspaces
54H20 Topological dynamics

Keywords: hyperspace dynamical system; sensitive dependance on initial conditions; collective sensitivity; hit-or-miss topology

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Zentralblatt MATH Berlin [Germany]