On generic and dense chaos for maps induced on hyperspaces

A continuous map f on a compact metric space X induces in a natural way the map f on the hyperspace K(X) of all closed non-empty subsets of X. We study the question of transmission of chaos between f and f. We deal with generic, generic e-, dense and dense e-chaos for interval maps. We prove that all four types of chaos transmit from f to f, while the converse transmission from f to f is true for generic, generic e-and dense e-chaos. Moreover, the transmission of dense eand generic e-chaos from f to f is true for maps on general compact metric spaces.