If is a dynamical system, then the hyperspace dynamical system is defined by on the collection of all subsets of . The relation of different concepts of chaotic behaviour on and 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 such that for any and any there is a with and an with . In this paper the authors introduce the notion of collective sensitivity. This means that there is a such that for finitely many and any there are with for all and there is an and a such that for all or for all .
It is proved that is sensitive if and only if is collectively sensitive. Here is endowed with the hit-or-miss topology. Moreover, also the conditions is sensitive and is sensitive are equivalent to is sensitive, where is the collection of all nonempty compact subsets of and is the collection of all nonempty finite subsets of , 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.