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# recurrent point

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 closure of the set of recurrent points of $f$ is often denoted $R(f)$ and is called the *recurrent set* of $f$.

Every recurrent point is a nonwandering point, hence if $f$ is a homeomorphism and $X$ is compact, $R(f)$ is an invariant subset of $\Omega(f)$, which may be a proper subset.

Defines:

recurrent set

Related:

NonwanderingSet

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

37B20*no label found*

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