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

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 $f$ ,^[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 $f$ 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).