Limit set

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In mathematics, a limit set is the set of cluster points of an iterated function. The ω-limit set is the set of cluster points in the forward-iterated function; the α-limit set is similar, but for the reverse iteration.

[edit] Definition for iterated functions

Let $X$ be a metric space, and let $f:X\rightarrow X$ be a continuous function. The $ω$ -limit set of $x\in X$ , denoted by $ω(x, f)$ , is the set of cluster points of the forward orbit $\{f^n(x)\}_{n\in \mathbb{N}}$ of the iterated function $f$ . 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 points in the limit set are called recurrent points.

If $f$ is a homeomorphism (that is, a bicontinuous bijection), then the $α$ -limit set is defined in a similar fashion, but for the backward orbit; i.e. $α(x, f) = ω(x, f - 1)$ .

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

[edit] Definition for flows

If $\varphi:\mathbb{R}\times X\to X$ is a continuous flow, the definition of the $ω$ -limit set is similar: $\omega(x,\varphi)$ consists of those elements $y$ of $X$ for which there exists a strictly increasing sequence ${t n}$ of real numbers such that $t_n\rightarrow \infty$ and $\varphi(x,t_n) \rightarrow y\,$ as $n\rightarrow\infty$ . In this case, the term limit cycle is often used as a synonym.

Similarly, the $α$ -limit set $\alpha(x,\varphi)$ is the $ω$ -limit set of the reversed flow (i.e. for $ψ(x, t) = φ(x, - t)$ ). The $a l p h a$ and $o m e g a$ -limit sets are invariant, and if $X$ is compact, they are compact and nonempty. Furthermore,