PlanetMath: $\omega$-limit set

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$\omega$ -limit set

(Definition)

Let 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

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

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 -invariant, and if 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 of 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 is compact they are compact and nonempty. Furthermore,

$\displaystyle \omega(x,f) = \bigcap_{n\in \mathbb{N}}\overline{\{\varphi(x,t):t>n\}}.$

" $\omega$ -limit set" is owned by Koro.

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