Chain recurrence

Summary

A point is chain recurrent when it can return to itself along a path that follows the dynamics up to arbitrarily small jumps. Such paths are called $\varepsilon$-chains or $\varepsilon$-pseudo-orbits. The chain recurrent set is the weakest of the usual notions of recurrence: it contains the non-wandering set. By Conley's fundamental theorem, outside the chain recurrent set the dynamics is gradient-like.