# $\omega $-limit set

Let $X$ be a metric space, and let $f:X\to X$ be a homeomorphism^{}.
The *$\omega $-limit set* of $x\in X$, denoted by $\omega (x,f)$, is the set of cluster points^{} of the forward orbit ${\{{f}^{n}(x)\}}_{n\in \mathbb{N}}$.
Hence, $y\in \omega (x,f)$ if and only if there is a strictly increasing sequence of natural numbers^{} ${\{{n}_{k}\}}_{k\in \mathbb{N}}$ such that ${f}^{{n}_{k}}(x)\to y$ as $k\to \mathrm{\infty}$.

Another way to express this is

$$\omega (x,f)=\bigcap _{n\in \mathbb{N}}\overline{\{{f}^{k}(x):k>n\}}.$$ |

The *$\alpha $-limit set* is defined in a similar^{} fashion, but for the backward orbit; i.e. $\alpha (x,f)=\omega (x,{f}^{-1})$.

If $\phi :\mathbb{R}\times X\to X$ is a continuous flow, the definition is similar: $\omega (x,\phi )$ consists of those elements $y$ of $X$ for which there exists a strictly increasing sequnece $\{{t}_{n}\}$ of real numbers such that ${t}_{n}\to \mathrm{\infty}$ and $\phi (x,{t}_{n})\to y$ as $n\to \mathrm{\infty}$. Similarly, $\alpha (x,\phi )$ is the $\omega $-limit set of the reversed flow (i.e. $\psi (x,t)=\varphi (x,-t)$). Again, these sets are invariant and if $X$ is compact they are compact and nonempty. Furthermore,

$$\omega (x,f)=\bigcap _{n\in \mathbb{N}}\overline{\{\phi (x,t):t>n\}}.$$ |

Title | $\omega $-limit set |
---|---|

Canonical name | omegalimitSet |

Date of creation | 2013-03-22 13:39:37 |

Last modified on | 2013-03-22 13:39:37 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 6 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 37B99 |

Synonym | omega-limit set |

Related topic | NonwanderingSet |

Defines | $\alpha $-limit |

Defines | alpha-limit |

Defines | $\omega $-limit |

Defines | omega-limit |