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 <math>X</math> be a metric space, and let <math>f:X\rightarrow X</math> be a continuous function. The <math>\omega</math>-limit set of <math>x\in X</math> , denoted by <math>\omega(x,f)</math>, is the set of cluster points of the forward orbit <math>\{f^n(x)\}_{n\in \mathbb{N}}</math> of the iterated function <math>f</math>. Hence, <math>y\in \omega(x,f)</math> if and only if there is a strictly increasing sequence of natural numbers <math>\{n_k\}_{k\in \mathbb{N}}</math> such that <math>f^{n_k}(x)\rightarrow y</math> as <math>k\rightarrow\infty</math>. Another way to express this is

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

The points in the limit set are called recurrent points.

If <math>f</math> is a homeomorphism (that is, a bicontinuous bijection), then the <math>\alpha</math>-limit set is defined in a similar fashion, but for the backward orbit; i.e. <math>\alpha(x,f)=\omega(x,f^{-1})</math>.

Both sets are <math>f</math>-invariant, and if <math>X</math> is compact, they are compact and nonempty.

[edit] Definition for flows

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

Similarly, the <math>\alpha</math>-limit set <math>\alpha(x,\varphi)</math> is the <math>\omega</math>-limit set of the reversed flow (i.e. for <math>\psi(x,t) = \phi(x,-t)</math>). The <math>alpha</math> and <math>omega</math>-limit sets are invariant, and if <math>X</math> is compact, they are compact and nonempty. Furthermore,

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