Recurrent point

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

The set of recurrent points of $$f$$ is often denoted $$R(f)$$ and is called the recurrent set of $$f$$. Its closure is called the Birkhoff center of $$f$$, and appears in the work of George David Birkhoff on dynamical systems.

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