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

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

If $\phi :ℝ\times X\to X$ 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,

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