recurrent point

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 $x\in \omega (x)$, 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 $\mathrm{\Omega }(f)$, which may be a proper subset.