Non-wandering set

Summary

The non-wandering set of a continuous map consists of the points $x$ whose every neighbourhood meets one of its own later images. It is the complement of the set of wandering points from the notion of wandering set. It is closed and invariant, and it contains every periodic orbit, every recurrent point and every $\omega$-limit set. In smooth dynamics, hyperbolicity conditions such as Smale's Axiom A are imposed on the non-wandering set.