Devaney chaos of a set-valued map and its inverse limit

We study relationships between a set-valued map and its inverse limit about the notion of periodic point set, transitivity, sensitivity and Devaney chaos. We show that periodic point set of a set-valued map is dense if and only if periodic point set of the inverse limit with the set-valued map is dense. Sensitivity of a set-valued map and its inverse limit are independent of each other. If the inverse limit with the set-valued map is chaotic in the sense of Devaney (respectively, transitive), then the set-valued map is chaotic in the sense of Devaney (respectively, transitive).