# $\omega$-limit set

Let $X$ be a metric space, and let $f:X\rightarrow X$ 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 $f^{n_{k}}(x)\rightarrow y$ as $k\rightarrow\infty$.

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. $\alpha(x,f)=\omega(x,f^{-1})$.

Both sets are $f$-invariant, and if $X$ is compact, they are compact and nonempty.

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

 $\omega(x,f)=\bigcap_{n\in\mathbb{N}}\overline{\{\varphi(x,t):t>n\}}.$
Title $\omega$-limit set omegalimitSet 2013-03-22 13:39:37 2013-03-22 13:39:37 Koro (127) Koro (127) 6 Koro (127) Definition msc 37B99 omega-limit set NonwanderingSet $\alpha$-limit alpha-limit $\omega$-limit omega-limit