Transitivity, mixing and chaos for a class of set-valued mappings

AbstractConsider the continuous map →: X → X and the continuous map $$\bar f$$ of K(X) into itself induced by →, where X is a metric space and K(X) the space of all non-empty compact subsets of X endowed with the Hausdor. metric. According to the questions whether the chaoticity of → implies the chaoticity of $$\bar f$$ posed by Román-Flores and when the chaoticity of → implies the chaoticity of $$\bar f$$ posed by Fedeli, we investigate the relations between → and $$\bar f$$ in the related dynamical properties such as transitivity, weakly mixing and mixing, etc. And by using the obtained results, we give the satisfied answers to Román-Flores's question and Fedeli's question.