Recurrent point

In mathematics, a recurrent point for function "f" is a point that is in the limit set of the iterated function "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 $fcolon\; X\; o\; X$ a function. A point $xin\; X$ is said to be recurrent (for $f$) if $xin\; omega(x)$, "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, then $R(f)$ is an invariant subset of the non-wandering set of $f$ (and may be a proper subset).