The transitivity of induced maps

For a metric continuum X, we consider the hyperspaces 2 X and C ( X ) of the closed and nonempty subsets of X and of subcontinua of X, respectively, both with the Hausdorff metric. For a given map f : X → X we investigate the transitivity of the induced maps 2 f : 2 X → 2 X and C ( f ) : C ( X ) → C ( X ) . Among other results, we show that if X is a dendrite or a continuum of type λ and f : X → X is a map, then C ( f ) is not transitive. However, if X is the Hilbert cube, then there exists a transitive map f : X → X such that 2 f and C ( f ) are transitive.