  We study discrete dynamical systems through the topological concepts of limit
set, which consists of all points that can be reached arbitrarily late, and
asymptotic set, which consists of all adhering values of orbits. In particular,
we deal with the case when each of these are a singleton, or when the
restriction of the system is periodic on them, and show that this is equivalent
to some simple dynamics in the case of subshifts or cellular automata.
Moreover, we deal with the stability of these properties with respect to some
simulation notions.
