Abstract: Let (X,d)be a compact metric space, f: X→X a continuous map, and (K(X),H) a compact metric space consisting of all nonempty compact subsets of X. It has been proved that the topological ergodicity of setvalued map is equivalent to the topological double ergodicityof f by studying the relation between the motion of points and the motion of sets; moreover, a compactsystem has been constructed which has zero topological entropy and no chaotic property, but the inducedset valued map of which has infinite to pological entropy and distributional chaos, this implies that the topological complexity of could be far greater than that of f.

Key words: setvalued map, topological ergodicity, topological entropy, distributional chaos