  In the paper we study relations of rigidity, equicontinuity and pointwise
recurrence between a t.d.s. $(X,T)$ and the t.d.s. $(K(X),T_K)$ induced on the
hyperspace $K(X)$ of all compact subsets of $X$, and provide some
characterizations.
  Among other examples, we construct a minimal, non-equicontinuous, distal and
uniformly rigid t.d.s. and a t.d.s. which has dense small periodic sets but
does not have dense distal points, solving that way open questions existing in
the literature.
