Let ${\displaystyle X}$  be a Hausdorff space and ${\displaystyle f\colon X\to X}$  a function. A point ${\displaystyle x\in X}$  is said to be recurrent (for ${\displaystyle f}$ ) if ${\displaystyle x\in \omega (x)}$ , i.e. if ${\displaystyle x}$  belongs to its ${\displaystyle \omega }$ -limit set. This means that for each neighborhood ${\displaystyle U}$  of ${\displaystyle x}$  there exists ${\displaystyle n>0}$  such that ${\displaystyle f^{n}(x)\in U}$ .^[1]

The set of recurrent points of ${\displaystyle f}$  is often denoted ${\displaystyle R(f)}$  and is called the recurrent set of ${\displaystyle f}$ . Its closure is called the Birkhoff center of ${\displaystyle f}$ ,^[2] and appears in the work of George David Birkhoff on dynamical systems.^[3][4]

Every recurrent point is a nonwandering point,^[1] hence if ${\displaystyle f}$  is a homeomorphism and ${\displaystyle X}$  is compact, then ${\displaystyle R(f)}$  is an invariant subset of the non-wandering set of ${\displaystyle f}$  (and may be a proper subset).

This article incorporates material from Recurrent point on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.