Recurrent point ($T^{n_j}x\to x,n_j\to \infty$)

A point of a dynamical system is recurrent if some sequence of positive times tending to infinity returns its orbit arbitrarily close to the point. In a compact metric space, this is equivalent to the displayed condition. A periodic point is recurrent, but a recurrent point need not be periodic.

Every continuous self-map of a nonempty compact metric space has a recurrent point. Choose a minimal subsystem. The forward orbit of every point in it is dense, and the same is true of every tail of that orbit; consequently each neighborhood is revisited at arbitrarily large times. A nonempty subsystem is all that is guaranteed: the identity map on a singleton has no larger subsystem.