## 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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