  Given a self-map of a compact metric space $X$, we study periodic points of
the map induced on the hyperspace of closed subsets of $X$. We give some
necessary conditions on admissible sets of periods for these maps. Seemingly
unrelated to this, we construct an almost totally minimal homeomorphism of the
Cantor set. We also apply our theory to give a full description of admissible
period sets for induced maps of the interval maps. The description of
admissible periods is also given for maps induced on symmetric products.
