Set-valued characterizations of periodic density for discrete systems

Let (X, d) be a metric space and f : X rarr X a continuous function. If we consider the space (K(X),H) of all nonempty compact subsets of X endowed with the Hausdorff metric H induced by d, and f macr : K(X) rarr K(X) the natural extension of f to K(X) defined by f(A) = {f(alpha)/alpha isin A}, then the aim of this work is to connect the periodic density of f on X with the periodic density of f restricted to some distinguished subspaces of K(X).