Recurrent point

Short description: Mathematical concept

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.

Definition

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 set. This means that for each neighborhood $U$ of $x$ there exists $n>0$ 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.<ref>{{citation

| last1 = Coven | first1 = Ethan M.
| doi = 10.1090/S0002-9939-1980-0565362-0 | doi-access=free
| jstor=2043258
| issue = 2
| journal = Proceedings of the American Mathematical Society
| mr = 565362
| pages = 316–318
| title = $\overline{P}=\overline{R}$ for maps of the interval
| volume = 79

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).

References

This article incorporates material from Recurrent point on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.