Dynamic properties for the induced maps in the symmetric products

For a metric continuum X, let Fn(X) be the hyperspace of nonempty subsets of X with at most n points. Given a map f:X → X we consider the induced map fn:Fn(X) → Fn(X) given by fn(A) = f(A) (the image of A under f  ). Given a dynamic property PP, we study the implication: if A ∈ Fn(X  ) has property PP in the system (Fn(X), fn), then each element of A   has property PP in the system (X, f), and we also study the converse implication. We consider properties as periodicity, recurrence, regular recurrence, etc.