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| {{Unreferenced|date=October 2009}}
| | Hi there. Allow me start by introducing the author, her name is Myrtle Cleary. California is our beginning location. To do aerobics is a thing that I'm completely addicted to. I am a meter reader but I strategy on changing it.<br><br>my blog post :: [http://jewelrycase.co.kr/xe/Ring/11593 jewelrycase.co.kr] |
| In [[mathematics]], a '''recurrent point''' for a function ''f'' is a point that is in the [[limit set]] of the [[iterated function]] ''f''. Any [[neighborhood]] containing the recurrent point will also contain (a [[countable]] number of) iterates of it as well.
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| ==Definition==
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| Let <math>X</math> be a [[Hausdorff space]] and <math>f\colon X\to X</math> a function. A point <math>x\in X</math> is said to be recurrent (for <math>f</math>) if <math>x\in \omega(x)</math>, ''i.e.'' if <math>x</math> belongs to its <math>\omega</math>-[[limit set]]. This means that for each [[neighborhood]] <math>U</math> of <math>x</math> there exists <math>n>0</math> such that <math>f^n(x)\in U</math>.
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| The [[closure (topology)|closure]] of the set of recurrent points of <math>f</math> is often denoted <math>R(f)</math> and is called the '''recurrent set''' of <math>f</math>.
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| Every recurrent point is a [[nonwandering point]], hence if <math>f</math> is a [[homeomorphism]] and <math>X</math> is [[Compactness|compact]], then <math>R(f)</math> is an [[Invariant set|invariant subset]] of the non-wandering set of <math>f</math> (and may be a [[proper subset]]).
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| {{PlanetMath attribution|id=6034|title=Recurrent point}}
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| [[Category:Limit sets]]
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Latest revision as of 09:35, 24 May 2014
Hi there. Allow me start by introducing the author, her name is Myrtle Cleary. California is our beginning location. To do aerobics is a thing that I'm completely addicted to. I am a meter reader but I strategy on changing it.
my blog post :: jewelrycase.co.kr