## You are here

Homerecurrent point

## Primary tabs

# 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*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Apr 15

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier