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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.

A recurrent point of a discrete dynamical system returns arbitrarily close to its starting state at arbitrarily late positive times. In a metric space, this means that there is a strictly increasing sequence $n_j\to \infty$ withEquivalently, every neighborhood of $x$ contains $T^{n}x$ for arbitrarily large $n$. A finite early return near the point does not suffice.

Every periodic point is recurrent: if $T^{p}x=x$, take $n_j=jp$. An irrational rotation of the circle gives recurrent points that are not periodic. For $T(\theta )=\theta +\alpha (\mathrm{mod}1)$ with irrational $\alpha$, the pigeonhole principle applied to $0,\alpha ,\dots ,N\alpha$ modulo one supplies an integer $1\le q\le N$ with distance from $q\alpha$ to an integer at most $1/N$. These approximating $q$ must be unbounded, since no fixed positive multiple of an irrational number is an integer. An increasing subsequence therefore gives the required returns for every starting angle.

Recurrence concerns returning close, whereas convergence to an attracting equilibrium point need not give recurrence of any other starting state. The map $T(x)=x/2$ has zero as a recurrent point, but a nonzero $x$ converges to zero rather than back to itself.