Set-valued discrete chaos

Given a continuous map f : X → X on a metric space (X, d), we characterize topological transitivity for the (set-valued) map f¯:K(X)→K(X) induced by f on the space K(X) of compact subsets of X, endowed with the Hausdorff distance. More precisely, f¯ is transitive if and only if f is weakly mixing. Some consequences are also derived for the dynamics on fractals and for (continuous and) linear maps on infinite-dimensional spaces.