F-sensitivity and multi-sensitivity of hyperspatial dynamical systems

Let f be a continuous self-map defined on a compact metric space X and f be a continuous self-map naturally induced by f on the hyperspace K(X) of all nonempty compact subsets of X endowed with a Hausdorff metric. Firstly, a few sufficient and necessary conditions to ensure a dynamical system be F-sensitive or multi-sensitive are obtained. Then, the following results are proved:. (1)If (X,f) is a non-minimal M-system, then (K(X),f) has F_s-sensitive pairs almost everywhere.(2)If (K(X),f) or (K(Y),g) is F-sensitive, then (K(X×Y),f×g) is F-sensitive.(3)(K(X×Y),f×g) is multi-sensitive if and only if (K(X),f) or (K(Y),g) is multi-sensitive, if and only if (X,f) or (Y,g) is multi-sensitive. Moreover, it is proved that f× g is multi-sensitive if and only if f or g is multi-sensitive. This is a positive answer to a question posed in R. Li and X. Zhou (2013) [18].