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

If $\left(E,f\right)$ is a dynamical system, then the hyperspace dynamical system $\left(\stackrel{^}{E},\stackrel{^}{f}\right)$ is defined by $\stackrel{^}{f}\left(A\right):=f\left(A\right)$ on the collection $\stackrel{^}{E}$ of all subsets of $E$. The relation of different concepts of chaotic behaviour on $\left(E,f\right)$ and $\left(\stackrel{^}{E},\stackrel{^}{f}\right)$ 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 $\epsilon >0$ there is a $y\in E$ with $d\left(y,x\right)<\epsilon$ and an $n\in ℕ$ with $d\left({f}^{n}\left(y\right),{f}^{n}\left(x\right)\right)\ge \delta$. In this paper the authors introduce the notion of collective sensitivity. This means that there is a $\delta >0$ such that for finitely many ${x}_{1},{x}_{2},\cdots ,{x}_{k}\in E$ and any $\epsilon >0$ there are ${y}_{1},{y}_{2},\cdots ,{y}_{k}\in E$ with $d\left({y}_{j},{x}_{j}\right)<\epsilon$ for all $j\in \left\{1,2,\cdots ,k\right\}$ and there is an $n\in ℕ$ and a $u\in \left\{1,2,\cdots ,k\right\}$ such that $d\left({f}^{n}\left({y}_{j}\right),{f}^{n}\left({x}_{u}\right)\right)\ge \delta$ for all $j\in \left\{1,2,\cdots ,k\right\}$ or $d\left({f}^{n}\left({x}_{j}\right),{f}^{n}\left({y}_{u}\right)\right)\ge \delta$ for all $j\in \left\{1,2,\cdots ,k\right\}$.

It is proved that $\left(\stackrel{^}{E},\stackrel{^}{f}\right)$ is sensitive if and only if $\left(E,f\right)$ is collectively sensitive. Here $\stackrel{^}{E}$ is endowed with the hit-or-miss topology. Moreover, also the conditions $\left(𝒞,\stackrel{^}{f}\right)$ is sensitive and $\left(ℱ,\stackrel{^}{f}\right)$ is sensitive are equivalent to $\left(\stackrel{^}{E},\stackrel{^}{f}\right)$ is sensitive, where $𝒞$ is the collection of all nonempty compact subsets of $E$ and $ℱ$ 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.

##### MSC:
 37B05 Transformations and group actions with special properties 54B20 Hyperspaces (general topology) 54H20 Topological dynamics