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Recurrent point

Mathematical concept

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Source updated August 23, 2026

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.

01Definition

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 $x$ and ${\forall N}$ there exists $n>N$ 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) and it's already closed.

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This article is adapted from the Wikipedia article “Recurrent point”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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