Dynamics on Hyperspaces

Given a compact metric space (X; \varrho) and a continuous function f:X\rightarrow X, we study the dynamics of the induced map \bar{f} on the hyperspace of the compact subsets of X. We show how the chain recurrent set of f and its components are related with the one of the induced map. The main result of the paper proves that, under mild conditions, the numbers of chain components of \bar{f} is greater than the ones of f. Showing the richness in the dynamics of \bar{f} which cannot be perceived by f.