In mathematics, a recurrent point for a function f is a point that is in its own limit set by f. Any neighborhood containing the recurrent point will also contain (a countable number of) iterates of it as well.
Definition Let X {\displaystyle X} be a Hausdorff space and f : X → X {\displaystyle f\colon X\to X} a function. A point x ∈ X {\displaystyle x\in X} is said to be recurrent (for f {\displaystyle f} ) if x ∈ ω ( x ) {\displaystyle x\in \omega (x)} , i.e. if x {\displaystyle x} belongs to its ω {\displaystyle \omega } -limit set. This means that for each neighborhood U {\displaystyle U} of x {\displaystyle x} and ∀ N {\displaystyle {\forall N}} there exists n > N {\displaystyle n>N} such that f n ( x ) ∈ U {\displaystyle f^{n}(x)\in U} . The set of recurrent points of f {\displaystyle f} is often denoted R ( f ) {\displaystyle R(f)} and is called the recurrent set of f {\displaystyle f} . Its closure is called the Birkhoff center of f {\displaystyle f} , and appears in the work of George David Birkhoff on dynamical systems. Every recurrent point is a nonwandering point, hence if f {\displaystyle f} is a homeomorphism and X {\displaystyle X} is compact, then R ( f ) {\displaystyle R(f)} is an invariant subset of the non-wandering set of f {\displaystyle f} (and may be a proper subset) and it's already closed.
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This article incorporates material from Recurrent point on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.