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| An external ray is a [[curve]] that runs from [[Point at infinity|infinity]] toward a [[Julia set|Julia]] or [[Mandelbrot set]].<ref>J. Kiwi : Rational rays and critical portraits of complex polynomials. Ph. D. Thesis SUNY at Stony Brook (1997); [http://www.math.sunysb.edu/cgi-bin/preprint.pl?ims97-15 IMS Preprint #1997/15.]</ref>
| | Let me initial begin by introducing myself. My name is Boyd Butts although it is not the title on my beginning certificate. Supervising is my profession. California is exactly where her house is but she needs to move simply because of her family. To do aerobics is a factor that I'm completely addicted to.<br><br>my web page ... [http://203.250.78.160/zbxe/?document_srl=810243 home std test] |
| This curve is only sometimes a [[Line_(mathematics)#Ray|half-line ( ray )]] but is called [[Line_(mathematics)#Ray|ray]] because it is image of ray.
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| External rays are used in [[complex analysis]], particularly in [[complex dynamics]] and [[geometric function theory]],
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| ==History==
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| External rays were introduced in [[Adrien Douady|Douady]] and [[John H. Hubbard|Hubbard]]'s study of the [[Mandelbrot set]]
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| ==Notation==
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| External rays of (connected) [[Julia set]]s on [[Complex_quadratic_polynomial#Dynamical_plane|dynamical plane]] are often called '''dynamic rays'''.
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| External rays of the Mandelbrot set (and similar one-dimensional [[connectedness locus|connectedness loci]]) on [[Complex_quadratic_polynomial#Parameter_plane|parameter plane]] are called '''parameter rays'''.
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| ==Polynomials==
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| ===Dynamical plane = z-plane ===
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| '''External rays''' are associated to a [[compact space|compact]], [[full set (topology)|full]], [[connected space|connected]] subset <math>K\,</math> of the [[complex plane]] as :
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| * the images of radial rays under the [[Riemann mapping theorem|Riemann map]] of the complement of <math>K\,</math>
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| * the [[gradient lines]] of the [[Green's function]] of <math>K\,</math>
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| * [[field line]]s of Douady-Hubbard potential
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| * an integral curve of the gradient vector field of the [[Green's function]] on neighborhood of [[Point at infinity|infinity]]<ref>[http://qcpages.qc.cuny.edu/~yjiang/HomePageYJ/Download/2004MandLocConn.pdf Yunping Jing : Local connectivity of the Mandelbrot set at certain infinitely renormalizable points ] Complex Dynamics and Related Topics, New Studies in Advanced Mathematics, 2004, The International Press, 236-264</ref>
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| External rays together with equipotential lines of Douady-Hubbard potential ( level sets) form a new [[polar coordinate system]] for [[Exterior (topology)|exterior]] ( [[Complement (set theory)|complement]] ) of <math>K\,</math>.
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| In other words the external rays define vertical [[foliation]] which is orthogonal to horizontal foliation defined by the level sets of potential.<ref>[http://www.math.uic.edu/~demarco/basins.pdf POLYNOMIAL BASINS OF INFINITY LAURA DEMARCO AND KEVIN M. PILGRIM]</ref>
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| ==== Uniformization ====
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| Let <math>\Psi_c\,</math> be the mapping from the [[Complement (set theory)|complement (exterior)]] of the [[unit disk|closed unit disk]] <math>\overline{\mathbb{D}}</math> to the complement of the [[filled Julia set]] <math>\ Kc </math>.
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| :<math>\Psi_c:\mathbb{\hat{C}}\setminus \overline{\mathbb{D}}\to\mathbb{\hat{C}}\setminus Kc</math>
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| and '''[[Böttcher's equation|Boettcher map]]'''<ref>[http://www.mndynamics.com/indexp.html How to draw external rays by Wolf Jung]</ref>(function) <math>\Phi_c\,</math>, which is [[Uniformization theorem|uniformizing]] map of basin of attraction of infinity, because it conjugates complement of the [[filled Julia set]] <math>\ Kc </math> and the [[Complement (set theory)|complement (exterior)]] of the [[unit disk|closed unit disk]]
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| :<math>\Phi_c: \mathbb{\hat{C}}\setminus Kc \to \mathbb{\hat{C}}\setminus \overline{\mathbb{D}}</math>
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| where : | |
| :<math>\mathbb{\hat{C}}</math> denotes the [[Riemann sphere|extended complex plane]]
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| Boettcher map <math>\Phi_c\,</math> is an [[isomorphism]] :
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| :<math>\Psi_c = \Phi_{c}^{-1} \,</math>
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| <!-- Φc(z) = limn→∞ (fcn(z))1/2n -->
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| <math>w = \Phi_c(z) = \lim_{n\rightarrow \infty} (f_c^n(z))^{2^{-n}}</math>
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| where :
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| <math>z \in \mathbb{\hat{C}}\setminus K_c</math>
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| <math>w \in \mathbb{\hat{C}}\setminus \overline{\mathbb{D}}</math>
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| <math>w\,</math> is a '''[[Böttcher's equation|Boettcher coordinate]]'''
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| ====Formal definition of dynamic ray====
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| [[Image:Erays.png|right|thumb|polar coordinate system and Psi_c for c=-2]]
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| <!-- :<math>\mathbb{\hat{C}}\setminus \overline{\mathbb{D}} = { w\setin</math>\mathbb{\hat{C}}:abs(w)>1}</math>
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| If <math>\Psi_c\,</math> is [[Riemann mapping theorem|unique conformal isomorphism]] whose leading [[Coefficients of Laurent series|Laurent coefficient]] at infinity is real and positive <br /> -->
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| The '''external ray''' of angle <math>\theta\,</math> noted as <math>\mathcal{R}^K _{\theta} </math>is:
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| *the image under <math>\Psi_c\,</math> of straight lines <math>\mathcal{R}_{\theta} = \{\left(r*e^{2\pi i \theta}\right) : \ r > 1 \}</math>
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| :<math>\mathcal{R}^K _{\theta} = \Psi_c(\mathcal{R}_{\theta})</math>
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| *set of points of exterior of [[Filled Julia set|filled-in Julia set]] with the same external angle <math>\theta</math>
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| :<math>\mathcal{R}^K _{\theta} = \{ z\in \mathbb{\hat{C}}\setminus Kc : \arg(\Phi_c(z)) = \theta \}</math>
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| ====Properities====
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| External ray for periodic angle <math>\theta\,</math> satisfies :
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| :<math>f(\mathcal{R}^K _{\theta}) = \mathcal{R}^K _{2 \theta}</math>
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| and its landing point <math>\gamma_f(\theta)) </math> :<ref>[http://eprintweb.org/S/article/math/0609280 Tessellation and Lyubich-Minsky laminations associated with quadratic maps I: Pinching semiconjugacies Tomoki Kawahira]</ref>
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| :<math>f(\gamma_f(\theta)) = \gamma_f(2\theta) </math>
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| ===Parameter plane = c-plane ===
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| ==== Uniformization ====
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| [[File:Jung200.png|right|thumb|Boundary of [[Mandelbrot set]] as [[Image (mathematics)|an image]] of [[unit circle]] under <math>\Psi_M\,</math>]]
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| [[File:Jung50e.png|right|thumb| [[Uniformization theorem|Uniformization]] of [[Complement (set theory)|complement (exterior)]] of [[Mandelbrot set]]]]
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| Let <math>\Psi_M\,</math> be the mapping from the [[Complement (set theory)|complement (exterior)]] of the [[unit disk|closed unit disk]] <math>\overline{\mathbb{D}}</math> to the complement of the [[Mandelbrot set]] <math>\ M </math>.
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| :<math>\Psi_M:\mathbb{\hat{C}}\setminus \overline{\mathbb{D}}\to\mathbb{\hat{C}}\setminus M</math>
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| and Boettcher map (function) <math>\Phi_M\,</math>, which is [[Uniformization theorem|uniformizing]] map<ref>[http://projecteuclid.org/euclid.dmj/1077304731 Irwin Jungreis: The uniformization of the complement of the Mandelbrot set. Duke Math. J. Volume 52, Number 4 (1985), 935-938.]</ref> of complement of Mandelbrot set, because it [[w:Topological conjugation|conjugates]] complement of the [[Mandelbrot set]] <math>\ M </math> and the [[Complement (set theory)|complement (exterior)]] of the [[unit disk|closed unit disk]]
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| :<math>\Phi_M: \mathbb{\hat{C}}\setminus M \to \mathbb{\hat{C}}\setminus \overline{\mathbb{D}}</math>
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| it can be normalized so that :
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| <math>\frac{\Phi_M(c)}{c} \to 1 \ as\ c \to \infty \,</math><ref>[http://www.math.cornell.edu/~hubbard/OrsayEnglish.pdf Adrien Douady, John Hubbard, Etudes dynamique des polynomes complexes I & II, Publ. Math. Orsay. (1984-85) (The Orsay notes)]</ref>
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| where :
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| :<math>\mathbb{\hat{C}}</math> denotes the [[Riemann sphere|extended complex plane]]
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| Jungreis function <math>\Psi_M\,</math> is the inverse of [[Uniformization theorem|uniformizing]] map :
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| :<math>\Psi_M = \Phi_{M}^{-1} \,</math>
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| In the case of [[complex quadratic polynomial]] one can compute this map using [[Laurent series]] about [[infinity]]<ref>Computing the Laurent series of the map Psi: C-D to C-M. Bielefeld, B.; Fisher, Y.; Haeseler, F. V. Adv. in Appl. Math. 14 (1993), no. 1, 25--38,</ref><ref>[http://mathworld.wolfram.com/MandelbrotSet.html Weisstein, Eric W. "Mandelbrot Set." From MathWorld--A Wolfram Web Resource]</ref>
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| :<math>c = \Psi_M (w) = w + \sum_{m=0}^{\infty} b_m w^{-m} = w -\frac{1}{2} + \frac{1}{8w} - \frac{1}{4w^2} + \frac{15}{128w^3} + ...\,</math>
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| where
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| :<math>c \in \mathbb{\hat{C}}\setminus M</math>
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| :<math>w \in \mathbb{\hat{C}}\setminus \overline{\mathbb{D}}</math>
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| ====Formal definition of parameter ray====
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| <!-- :<math>\mathbb{\hat{C}}\setminus \overline{\mathbb{D}} = { w\setin</math>\mathbb{\hat{C}}:abs(w)>1}</math>
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| If <math>\Psi_c\,</math> is [[Riemann mapping theorem|unique conformal isomorphism]] whose leading [[Coefficients of Laurent series|Laurent coefficient]] at infinity is real and positive <br /> -->
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| The '''external ray''' of angle <math>\theta\,</math> is:
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| *the image under <math>\Psi_c\,</math> of straight lines <math>\mathcal{R}_{\theta} = \{\left(r*e^{2\pi i \theta}\right) : \ r > 1 \}</math>
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| :<math>\mathcal{R}^M _{\theta} = \Psi_M(\mathcal{R}_{\theta})</math>
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| *set of points of exterior of Mandelbrot set with the same external angle <math>\theta</math><ref>[http://www.math.nagoya-u.ac.jp/~kawahira/programs/mandel-exray.pdf An algorithm to draw external rays of the Mandelbrot set by Tomoki Kawahira ]</ref>
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| :<math>\mathcal{R}^M _{\theta} = \{ c\in \mathbb{\hat{C}}\setminus M : \arg(\Phi_M(c)) = \theta \}</math>
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| ====Definition of <math>\Phi_M \,</math>====
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| Douady and Hubbard define:
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| <math>\Phi_M(c) \ \overset{\underset{\mathrm{def}}{}}{=} \ \Phi_c(z=c)\,</math>
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| so external angle of point <math>c\,</math> of parameter plane is equal to external angle of point <math>z=c\,</math> of dynamical plane
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| ===External angle===
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| Angle <math>\theta\,</math> is named '''external angle''' ( [[Complex number#Complex plane|argument]] ).<ref>http://www.mrob.com/pub/muency/externalangle.html External angle at Mu-ency by Robert Munafo</ref>
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| [[Principal value]] of external angles are [[Angle#Units|measured]] in [[Turn (geometry)|turns]] [[Modulo operation|modulo]] 1
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| 1 [[Turn (geometry)|turn]] = 360 [[Degree (angle)|degrees]] = 2 * Pi [[radian]]s
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| Compare different types of angles :
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| * external ( point of set's exterior )
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| * internal ( point of component's interior )
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| * plain ( [[Arg (mathematics)|argument of complex number]] )
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| {| class="wikitable"
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| |-
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| !
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| ! external angle
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| ! internal angle
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| ! plain angle
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| |-
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| |parameter plane
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| | <math> arg(\Phi_M(c)) \,</math>
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| | <math> arg(\rho_n(c)) \,</math>
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| | <math> arg(c) \, </math>
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| |-
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| |dynamic plane
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| | <math> arg(\Phi_c(z)) \,</math>
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| | <math> arg(z) \,</math>
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| |-
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| |}
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| ===Computation of external argument===
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| * argument of Böttcher coordinate as an external argument<ref>[http://www.mndynamics.com/indexp.html Computation of the external argument by Wolf Jung]</ref>
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| ** <math> arg_M(c) = arg(\Phi_M(c)) \,</math>
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| **<math> arg_c(z) = arg(\Phi_c(z)) \,</math>
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| * kneading sequence as a binary expansion of external argument<ref>A. DOUADY, Algorithms for computing angles in the Mandelbrot set (Chaotic Dynamics and Fractals, ed. Barnsley and Demko, Acad. Press, 1986, pp. 155-168).</ref><ref>[http://www.math.cornell.edu/~hubbard/OrsayEnglish.pdf Adrien Douady, John H. Hubbard: Exploring the Mandelbrot set. The Orsay Notes. page 58 ]</ref><ref>[http://www.dhushara.com/DarkHeart/DarkHeart.htm Exploding the Dark Heart of Chaos by Chris King from Mathematics Department of University of Auckland]</ref>
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| ==Transcendental maps==
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| For [[Transcendental function|transcendental]] maps ( for example [[Exponential map (discrete dynamical systems)|exponential]] ) [[Point at infinity|infinity]] is not a fixed point but an [[essential singularity]] and there is no Boettcher isomorphism.<ref>[http://pcwww.liv.ac.uk/~helenam/Poster.pdf Topological Dynamics of Entire Functions by Helena Mihaljevic-Brandt]</ref><ref>[http://pcwww.liv.ac.uk/~helenam/slides_manchester.pdf Dynamic rays of entire functions and their landing behaviour by Helena Mihaljevic-Brandt]</ref>
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| Here dynamic ray is defined as a curve :
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| * connecting a point in an [[escaping set]] and [[Point at infinity|infinity]] {{Clarify|date=March 2009}}
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| * lying in an [[escaping set]]
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| == Images ==
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| Dynamic rays
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| <gallery perrow="6">
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| File:JuliaRay 1 3.png|Julia set for <math>f_c(z) = z^2 -1</math> with 2 external ray landing on repelling fixed point alpha
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| File:JuliaRay3.png|Julia set and 3 [[w:external ray|external rays]] landing on fixed point <math>\alpha_c\,</math>
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| File:Dynamic internal and external rays .svg|Dynamic external rays landing on repelling period 3 cycle and 3 internal rays landing on fixed point <math>\alpha_c\,</math>
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| File:Julia-p9.png|Julia set with external rays landing on period 3 orbit
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| File:Parabolic rays landing on fixed point.ogv|Rays landing on parabolic fixed point for periods 2-40
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| </gallery>
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| [[Mandelbrot set]] for [[complex quadratic polynomial]] with parameter rays of root points
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| <gallery perrow="6">
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| Image:Man1.jpg|External rays for angles of the form : n / ( 2<SUP>1</SUP> - 1) (0/1; 1/1) landing on the point c= 1/4, which is cusp of main cardioid ( period 1 component)
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| Image:Man2period.jpg|External rays for angles of the form : n / ( 2<SUP>2</SUP> - 1) (1/3, 2/3) landing on the point c= - 3/4, which is root point of period 2 component
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| Image:Man3period.jpg|External rays for angles of the form : n / ( 2<SUP>3</SUP> - 1) (1/7,2/7) (3/7,4/7) landing on the point c= -1.75 = -7/4 (5/7,6/7) landing on the root points of period 3 components.
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| Image:Man4period.jpg|External rays for angles of form : n / ( 2<SUP>4</SUP> - 1) (1/15,2/15) (3/15, 4/15) (6/15, 9/15) landing on the root point c= -5/4 (7/15, 8/15) (11/15,12/15) (13/15, 14/15) landing on the root points of period 4 components.
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| Image:Man5period.jpg| External rays for angles of form : n / ( 2<SUP>5</SUP> - 1) landing on the root points of period 5 components
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| </gallery>
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| Parameter space of [[Exponential map (discrete dynamical systems)|the complex exponential family f(z)=exp(z)+c]]. Eight parameter rays landing at this parameter are drawn in black.
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| [[Image:Exponential Parameter Space Detail PSP Rays.png|Parameter plane of the complex exponential family f(z)=exp(z)+c with 8 external ( parameter) rays]]
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| ==Center, root, external and internal ray==
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| [[Image:Mandel ie 1 3.jpg|right|Internal ray of cardioid]]
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| [[Image:Iray.png|right|400px|Internal ray for angle 1/3 of main cardioid made by conformal map from unit circle]]
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| internal ray of main cardioid of angle 1/3:<br>
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| starts from center of main cardioid c=0<br>
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| ends in the root point of period 3 component<br>
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| which is the landing point of parameter (external) rays of angles 1/7 and 2/7
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| ==Programs that can draw external rays==
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| *[http://www.mndynamics.com/indexp.html Mandel ] - program by Wolf Jung written in [[C++]] using [[Qt (framework)|Qt]] with [[source code]] available under the [[GNU General Public License]]
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| *[http://www.ibiblio.org/e-notes/MSet/external.htm Java applets] by Evgeny Demidov ( code of mndlbrot::turn function by Wolf Jung has been ported to Java ) with free [[source code]]
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| *[http://www.math.nagoya-u.ac.jp/~kawahira/programs/aboutotis.htm OTIS by Tomoki KAWAHIRA ] - [[Java applet]] without [[source code]]
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| *[http://inls.ucsd.edu/%7Efisher/Complex/ Spider XView program by Yuval Fisher ]
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| *[http://archives.math.utk.edu/software/msdos/fractals/yabmp097/.html YABMP by Prof. Eugene Zaustinsky] for [[DOS]] without [[source code]]
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| *[http://www.picard.ups-tlse.fr/~cheritat/e_index.html DH_Drawer by Arnaud Chéritat] written for Windows 95 without [[source code]]
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| *[http://linas.org/art-gallery/ Linas Vepstas C programs ] for [[Linux]] [[Command-line interface|console]] with [[source code]]
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| *[http://abel.math.harvard.edu/~ctm/programs.html Program Julia by Curtis T McMullen ] written in C and [[Linux commands]] for [[C shell]] [[Command-line interface|console]] with [[source code]]
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| *[http://www.mathematik.hu-berlin.de/~erat/mj/ mjwinq program by Matjaz Erat ] written in delphi/windows without [[source code]] ( For the external rays it uses the methods from quad.c in julia.tar by Curtis T McMullen)
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| *[http://www.juliasets.dk/RatioField RatioField by Gert Buschmann], for windows with [[Pascal (programming language)|Pascal]] source code for [http://www.bloodshed.net/devpascal.html Dev-Pascal 1.9.2] (with [[Free Pascal]] compiler )
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| *Mandelbrot program by Milan Va, written in Delphi with source code
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| * [http://www.mrob.com/pub/muency/externalangle.html Power MANDELZOOM by Robert Munafo]
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| * [http://claudiusmaximus.goto10.org/cm/2011-10-09_ruff-0.2_and_gruff-0.2_released.html ruff by Claude Heiland-Allen]
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| ==See also==
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| {{commons}}
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| *external rays of [[Misiurewicz point]]
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| *[[Orbit portrait]]
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| *[[Periodic points of complex quadratic mappings]]
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| *[[Prouhet-Thue-Morse constant]]
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| *[[Carathéodory's theorem (conformal mapping)|Carathéodory's theorem]]
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| *[[Julia_set#Field_lines|Field lines of Julia sets]]
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| ==References==
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| {{Reflist}}
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| *[[Lennart Carleson]] and Theodore W. Gamelin, ''Complex Dynamics'', Springer 1993
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| *Adrien Douady and John H. Hubbard, ''Etude dynamique des polynômes complexes'', Prépublications mathémathiques d'Orsay 2/4 (1984 / 1985)
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| *John W. Milnor, ''Periodic Orbits, External Rays and the Mandelbrot Set: An Expository Account''; Géométrie complexe et systèmes dynamiques (Orsay, 1995), Astérisque No. 261 (2000), 277–333. (First appeared as a [http://www.math.sunysb.edu/preprints.html Stony Brook IMS Preprint] in 1999, available as [http://arxiv.org/abs/math.DS/9905169 arXiV:math.DS/9905169].)
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| * John Milnor, ''Dynamics in One Complex Variable'', Third Edition, Princeton University Press, 2006, ISBN 0-691-12488-4
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| *[http://www.math.sunysb.edu/cgi-bin/thesis.pl?thesis02-3 Wolf Jung : Homeomorphisms on Edges of the Mandelbrot Set. Ph.D. thesis of 2002]
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| ==External links==
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| {{Wikibooks|Fractals }}
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| *[http://www.iquilezles.org/trastero/fieldlines/ Hubbard Douady Potential, Field Lines by Inigo Quilez ]
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| *[http://republika.pl/fraktal/mset_jungreis.html Drawing Mc by Jungreis Algorithm]
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| *[http://republika.pl/fraktal/internalAngleMset.html Internal rays of components of Mandelbrot set]
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| *[http://www.revver.com/video/91465/mandelbrot-p31/ John Hubbard's presentation, The Beauty and Complexity of the Mandelbrot Set, part 3.1 ]
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| *[http://pl.youtube.com/user/ImpoliteFruit videos by ImpoliteFruit]
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| *{{cite web|url=http://sweb.cz/milan_va/Mandelbrot/|author=Milan Va|title=Mandelbrot set drawing|accessdate=2009-06-15}}
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| {{DEFAULTSORT:External Ray}}
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| [[Category:Complex numbers]]
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| [[Category:Fractals]]
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| [[Category:Polynomials]]
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