Chain transitivity in hyperspaces

Given a non-empty compact metric space X and a continuous function f: X → X, we study the dynamics of the induced maps on the hyperspace of non-empty compact subsets of X and on various other invariant subspaces thereof, in particular symmetric products. We show how some important dynamical properties transfer across induced systems. These amongst others include, chain transitivity, chain (weakly) mixing, chain recurrence, exactness by chains. From our main theorem we derive an ε-chain version of Furstenberg's celebrated 2 implies n Theorem. We also show the implications our results have for dynamics on continua.