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| In [[mathematics]], especially in the study of [[dynamical system]]s, a '''limit set''' is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they can be used to understand the long term behavior of a dynamical system.
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| ==Types ==
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| * [[Fixed point (mathematics)|fixed point]]s
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| * [[periodic orbit]]s
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| * [[limit cycle]]s
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| * [[attractor]]s.
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| In general limits sets can be very complicated as in the case of [[strange attractor]]s, but for 2-dimensional dynamical systems the [[Poincaré–Bendixson theorem]] provides a simple characterization of all possible limit sets as a union of fixed points and periodic orbits.
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| ==Definition for iterated functions==
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| 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
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| :<math>\omega(x,f) = \bigcap_{n\in \mathbb{N}} \overline{\{f^k(x): k>n\}},</math>
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| where <math>\overline{S}</math> denotes the ''closure'' of set <math>S</math>. The closure is here needed, since we have not assumed that the underlying metric space of interest to be a [[complete metric space]]. The points in the limit set are non-wandering (but may not be '''[[recurrent point]]s'''). This may also be formulated as the outer limit ([[limsup]]) of a sequence of sets, such that
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| :<math>\omega(x,f) = \bigcap_{n=1}^\infty \overline{\bigcup_{k=n}^\infty \{f^k(x)\}}.</math> | |
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| 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>.
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| Both sets are <math>f</math>-invariant, and if <math>X</math> is [[compact space|compact]], they are compact and nonempty.
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| ==Definition for flows== | |
| Given a [[real dynamical system]] (''T'', ''X'', φ) with [[flow (mathematics)|flow]] <math>\varphi:\mathbb{R}\times X\to X</math>, a point ''x'' and an [[orbit (dynamics)|orbit]] γ through ''x'', we call a point ''y'' an ω-'''limit point''' of γ if there exists a sequence <math>(t_n)_{n \in \mathbb{N}}</math> in '''R''' so that
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| :<math>\lim_{n \to \infty} t_n = \infty</math>
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| :<math>\lim_{n \to \infty} \varphi(t_n, x) = y </math>.
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| Analogously we call ''y'' an α-'''limit point''' if there exists a sequence <math>(t_n)_{n \in \mathbb{N}}</math> in '''R''' so that
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| :<math>\lim_{n \to \infty} t_n = -\infty</math>
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| :<math>\lim_{n \to \infty} \varphi(t_n, x) = y </math>.
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| The set of all ω-limit points (α-limit points) for a given orbit γ is called ω-'''limit set''' (α-'''limit set''') for γ and denoted lim<sub>ω</sub> γ (lim<sub>α</sub> γ).
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| If the ω-limit set (α-limit set) is disjoint from the orbit γ, that is lim<sub>ω</sub> γ ∩ γ = ∅ (lim<sub>α</sub> γ ∩ γ = ∅), we call lim<sub>ω</sub> γ (lim<sub>α</sub> γ) a '''[[ω-limit cycle]]''' ('''[[α-limit cycle]]''').
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| Alternatively the limit sets can be defined as
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| :<math>\lim_\omega \gamma := \bigcap_{n\in \mathbb{R}}\overline{\{\varphi(x,t):t>n\}} </math>
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| and
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| :<math>\lim_\alpha \gamma := \bigcap_{n\in \mathbb{R}}\overline{\{\varphi(x,t):t<n\}}.</math>
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| === Examples ===
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| * For any [[periodic orbit]] γ of a dynamical system, lim<sub>ω</sub> γ = lim<sub>α</sub> γ = γ
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| * For any [[Fixed point (mathematics)|fixed point]] <math>x_0</math> of a dynamical system, lim<sub>ω</sub> <math>x_0</math> = lim<sub>α</sub> <math>x_0</math> = <math>x_0</math>
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| === Properties ===
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| * lim<sub>ω</sub> γ and lim<sub>α</sub> γ are [[closed set|closed]]
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| * if ''X'' is compact then lim<sub>ω</sub> γ and lim<sub>α</sub> γ are [[nonempty]], [[compact set|compact]] and [[Connected space|connected]]
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| * lim<sub>ω</sub> γ and lim<sub>α</sub> γ are φ-invariant, that is φ('''R''' × lim<sub>ω</sub> γ) = lim<sub>ω</sub> γ and φ('''R''' × lim<sub>α</sub> γ) = lim<sub>α</sub> γ
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| ==See also==
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| * [[Julia set]]
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| * [[Stable manifold|Stable set]]
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| * [[Limit cycle]]
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| * [[Periodic point]]
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| * [[Non-wandering set]]
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| * [[Kleinian group]]
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| ==References==
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| * {{cite book| last = Teschl| given = Gerald|authorlink=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[[American Mathematical Society]]| place = [[Providence, Rhode Island|Providence]]| year = 2012| isbn= 978-0-8218-8328-0| url = http://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}
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| {{PlanetMath attribution|id=4316|title=Omega-limit set}}
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| [[Category:Limit sets| ]]
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