FF-sensitivity and multi-sensitivity of hyperspatial dynamical systems

Let f   be a continuous self-map defined on a compact metric space XX and f¯ be a continuous self-map naturally induced by f   on the hyperspace K(X)K(X) of all nonempty compact subsets of XX endowed with a Hausdorff metric. Firstly, a few sufficient and necessary conditions to ensure a dynamical system be FF-sensitive or multi-sensitive are obtained. Then, the following results are proved:(1)If (X,f)(X,f) is a non-minimal M  -system, then (K(X),f¯) has FsFs-sensitive pairs almost everywhere.(2)If (K(X),f¯) or (K(Y),g¯) is FF-sensitive, then (K(X×Y),f×g¯) is FF-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)(X,f) or (Y,g)(Y,g) is multi-sensitive. Moreover, it is proved that f×gf×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].