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We study discrete dynamical systems through the topological concepts of limitset, which consists of all points that can be reached arbitrarily late, andasymptotic 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 therestriction of the system is periodic on them, and show that this is equivalentto some simple dynamics in the case of subshifts or cellular automata.Moreover, we deal with the stability of these properties with respect to somesimulation notions.

Asymptotic behavior of dynamical systems and cellular automata