Inducing sensitivity on hyperspaces

Let ( X , d ) be a compact metric space and ( K ( X ) , d H ) be the space of all non-empty compact subsets of X equipped with the Hausdorff metric d H . The dynamical system ( X , f ) induces another dynamical system ( K ( X ) , f ¯ ) . We study the relations between the various forms of sensitivity of the systems ( X , f ) and ( K ( X ) , f ¯ ) . We prove that all forms of sensitivity of ( K ( X ) , f ¯ ) partly imply the same for ( X , f ) , and the converse holds in some cases.