Individual chaos implies collective chaos for weakly mixing discrete dynamical systems

Let X be a metric space, (X, f) a discrete dynamical system, where f : X → X   is a continuous function. Let f¯ denote the natural extension of f to the space of all non-empty compact subsets of X endowed with Hausdorff metric induced by d. In this paper we investigate some dynamical properties of f   and f¯. It is proved that f   is weakly mixing (mixing) if and only if f¯ is weakly mixing (mixing, respectively). From this, we deduce that weak-mixing of f   implies transitivity of f¯, further, if f is mixing or weakly mixing, then chaoticity of f   (individual chaos) implies chaoticity of f¯ (collective chaos) and if X   is a closed interval then f¯ is chaotic (in the sense of Devaney) if and only if f is weakly mixing.