# 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 (http://planetmath.org/OmegaLimitSet3) 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.

Title recurrent point RecurrentPoint 2013-03-22 14:29:53 2013-03-22 14:29:53 Koro (127) Koro (127) 10 Koro (127) Definition msc 37B20 NonwanderingSet recurrent set