View source for External ray

{{Cleanup rewrite|date=December 2021}}
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.] {{Webarchive|url=https://web.archive.org/web/20041105030845/http://www.math.sunysb.edu/cgi-bin/preprint.pl?ims97-15 |date=2004-11-05 }}</ref>
Although this curve is only rarely a [[Line (mathematics)#Ray|half-line (ray)]] it is called a [[Line (mathematics)#Ray|ray]] because it is an image of a ray.

External rays are used in [[complex analysis]], particularly in [[complex dynamics]] and [[geometric function theory]].

==History==
External rays were introduced in [[Adrien Douady|Douady]] and [[John H. Hubbard|Hubbard]]'s study of the [[Mandelbrot set]]

==Types==
Criteria for classification :
* [[#plane|plane]] : parameter or dynamic
* [[#Maps|map]]
* bifurcation of dynamic rays
* Stretching
* landing<ref>{{cite journal | arxiv=1406.3428 | doi=10.1007/s00222-015-0627-3 | title=Non-landing parameter rays of the multicorns | year=2016 | last1=Inou | first1=Hiroyuki | last2=Mukherjee | first2=Sabyasachi | journal=Inventiones Mathematicae | volume=204 | issue=3 | pages=869–893 | bibcode=2016InMat.204..869I | s2cid=253746781 }}</ref> 

===plane ===
External rays of (connected) [[Julia set]]s on [[Complex quadratic polynomial#Dynamical plane|dynamical plane]] are often called '''dynamic rays'''.

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'''.

===bifurcation===
Dynamic ray can be:
* bifurcated = branched<ref>{{Cite journal |doi=10.1017/S0143385700006854 |title=Bifurcations of dynamic rays in complex polynomials of degree two |year=1992 |last1=Atela |first1=Pau |journal=Ergodic Theory and Dynamical Systems |volume=12 |issue=3 |pages=401–423 |s2cid=123478692 }}</ref>  = broken <ref>{{cite journal | arxiv=2009.02788 | last1=Petersen | first1=Carsten L. | last2=Zakeri | first2=Saeed | title=Periodic points and smooth rays | journal=Conformal Geometry and Dynamics of the American Mathematical Society | year=2020 | volume=25 | issue=8 | pages=170–178 | doi=10.1090/ecgd/364 }}</ref>  
* smooth = unbranched =  unbroken


When the [[filled Julia set]] is connected, there are no branching external rays. When the Julia set is not connected then some external rays branch.<ref>[https://orbit.dtu.dk/en/publications/holomorphic-dynamics-on-accumulation-of-stretching-rays Holomorphic Dynamics: On Accumulation of Stretching Rays by Pia B.N. Willumsen, see page 12]</ref>

===stretching===
Stretching rays were introduced by Branner and Hubbard:<ref>[http://pi.math.cornell.edu/~hubbard/IterationCubics1.pdf The iteration of cubic polynomials Part I : The global topology of parameter by BODIL BRANNER and JOHN H. HUBBARD]</ref><ref>[https://www.youtube.com/watch?v=DyJDt4EyiBA&list=PL53AB2CAE70F31F2A&index=29 Stretching rays for cubic polynomials by Pascale Roesch]</ref>

"The notion of stretching rays is a generalization of that of external rays for the Mandelbrot set to higher degree polynomials."<ref>{{cite journal|url=https://www.ams.org/journals/ecgd/2004-08-04/S1088-4173-04-00102-X/S1088-4173-04-00102-X.pdf|doi=10.1090/s1088-4173-04-00102-x |title=Landing property of stretching rays for real cubic polynomials |year=2004 |last1=Komori |first1=Yohei |last2=Nakane |first2=Shizuo |journal= Conformal Geometry and Dynamics |volume=8 |issue=4 |pages=87–114 |bibcode=2004CGDAM...8...87K }}</ref>

===landing===

Every rational parameter ray of the Mandelbrot set lands at a single parameter.<ref>[https://web.archive.org/web/20160526222635/http://www.math.cornell.edu/~hubbard/OrsayFrench.pdf A. Douady, J. Hubbard: Etude dynamique des polynˆomes complexes. Publications math´ematiques d’Orsay 84-02 (1984) (premi`ere partie) and 85-04 (1985) (deuxi`eme partie).]</ref><ref>{{cite arXiv | eprint=math/9711213 | last1=Schleicher | first1=Dierk | title=Rational parameter rays of the Mandelbrot set | year=1997 }}</ref>

==Maps==
===Polynomials===

====Dynamical plane = z-plane ====
'''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 :
* the images of radial rays under the [[Riemann mapping theorem|Riemann map]] of the complement of <math>K\,</math>
* the [[gradient|gradient lines]] of the [[Green's function]] of <math>K\,</math>
* [[field line]]s of Douady-Hubbard potential<ref>[https://www.youtube.com/watch?v=N3ah6iTupIg&t=2652s Video : The beauty and complexity of the Mandelbrot set by John Hubbard ( see part 3 )]</ref>
* 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>

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>.

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.northwestern.edu/~demarco/basins.pdf POLYNOMIAL BASINS OF INFINITY LAURA DEMARCO AND KEVIN M. PILGRIM]</ref>

=====Uniformization=====
Let <math>\Psi_c\,</math> be the [[Conformal map|conformal]] [[isomorphism]] 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>\ K_c </math>.

:<math>\Psi_c: \hat{\Complex} \setminus \overline{\mathbb{D}} \to \hat{\Complex} \setminus K_c</math>

where <math>\hat{\Complex}</math> denotes the [[Riemann sphere|extended complex plane]].
Let <math>\Phi_c = \Psi_c^{-1}\,</math> denote the '''[[Böttcher's equation|Boettcher map]]'''.<ref>[http://www.mndynamics.com/indexp.html How to draw external rays by Wolf Jung]</ref>
<math>\Phi_c\,</math> is a [[Uniformization theorem|uniformizing]] map of the basin of attraction of infinity, because it conjugates <math>f_c</math> on the complement of the filled Julia set <math>K_c </math> to <math>f_0(z)=z^2</math> on the complement of the unit disk:

<!-- Φ_c(z) = lim n→∞ (f_c^n(z))^{1/{2^n}} -->
:<math>\begin{align}
\Phi_c: \hat{\Complex} \setminus K_c &\to \hat{\Complex} \setminus \overline{\mathbb{D}}\\
 z & \mapsto \lim_{n\to \infty} (f_c^n(z))^{2^{-n}}
\end{align}</math>

and 

:<math> \Phi_c \circ f_c \circ \Phi_c^{-1} = f_0 </math>

A value <math>w = \Phi_c(z)</math> is called the '''[[Böttcher's equation|Boettcher coordinate]]''' for a point <math>z \in \hat{\Complex}\setminus K_c</math>.

=====Formal definition of dynamic ray=====
[[Image:Erays.svg|right|thumb|Polar coordinate system and <math>\psi_c</math> for <math>c=-2</math>]]
<!-- :<math>\mathbb{\hat{C}}\setminus \overline{\mathbb{D}} = { w\setin</math>\mathbb{\hat{C}}:abs(w)>1}</math>
  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 /> -->

The '''external ray''' of angle <math>\theta\,</math>  noted as <math>\mathcal{R}^K  _{\theta} </math> is:
* the image under <math>\Psi_c\,</math> of straight lines <math>\mathcal{R}_{\theta} = \{\left(r\cdot e^{2\pi i \theta}\right) :  \ r > 1 \}</math>

:<math>\mathcal{R}^K  _{\theta} = \Psi_c(\mathcal{R}_{\theta})</math>

*set of points of exterior of filled-in Julia set with the same external angle  <math>\theta</math>

:<math>\mathcal{R}^K _{\theta} = \{ z\in \hat{\Complex} \setminus K_c : \arg(\Phi_c(z)) = \theta \}</math>

======Properties======

The external ray for a periodic angle <math>\theta\,</math> satisfies:

:<math>f(\mathcal{R}^K  _{\theta}) =  \mathcal{R}^K  _{2 \theta}</math>

and its landing point<ref>[http://eprintweb.org/S/article/math/0609280 Tessellation and Lyubich-Minsky laminations associated with quadratic maps I: Pinching semiconjugacies Tomoki Kawahira] {{webarchive|url=https://web.archive.org/web/20160303233036/http://eprintweb.org/S/article/math/0609280 |date=2016-03-03 }}</ref> <math>\gamma_f(\theta) </math> satisfies:

:<math>f(\gamma_f(\theta)) = \gamma_f(2\theta) </math>

====Parameter plane = c-plane ====
"Parameter rays are simply the curves that run perpendicular to the equipotential curves of the M-set."<ref>[http://linas.org/art-gallery/escape/phase/phase.html Douady Hubbard Parameter Rays by Linas Vepstas]</ref>
=====Uniformization=====
[[File:Jung200.png|right|thumb|Boundary of [[Mandelbrot set]] as [[Image (mathematics)|an image]] of [[unit circle]] under <math>\Psi_M\,</math>]]
[[File:Jung50e.png|right|thumb| [[Uniformization theorem|Uniformization]] of [[Complement (set theory)|complement (exterior)]] of [[Mandelbrot set]]]]
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>.<ref>[http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN001185500 John H. Ewing, Glenn Schober, The area of the Mandelbrot Set]</ref>

:<math>\Psi_M:\mathbb{\hat{C}}\setminus \overline{\mathbb{D}}\to\mathbb{\hat{C}}\setminus M</math>

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]]

:<math>\Phi_M: \mathbb{\hat{C}}\setminus M \to   \mathbb{\hat{C}}\setminus \overline{\mathbb{D}}</math>

it can be normalized so that :

<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>

where :
:<math>\mathbb{\hat{C}}</math> denotes the [[Riemann sphere|extended complex plane]]

Jungreis function <math>\Psi_M\,</math> is the inverse of [[Uniformization theorem|uniformizing]] map :

:<math>\Psi_M = \Phi_{M}^{-1} \,</math>

In the case of [[complex quadratic polynomial]] one can compute this map using [[Laurent series]] about [[infinity]]<ref>{{cite journal | doi=10.1006/aama.1993.1002 | title=Computing the Laurent Series of the Map Ψ: C − D → C − M | year=1993 | last1=Bielefeld | first1=B. | last2=Fisher | first2=Y. | last3=Vonhaeseler | first3=F. | journal=Advances in Applied Mathematics | volume=14 | pages=25–38 | doi-access=free }}</ref><ref>[http://mathworld.wolfram.com/MandelbrotSet.html Weisstein, Eric W. "Mandelbrot Set." From MathWorld--A Wolfram Web Resource]</ref>

:<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>

where

:<math>c \in \mathbb{\hat{C}}\setminus M</math>

:<math>w \in \mathbb{\hat{C}}\setminus \overline{\mathbb{D}}</math>

=====Formal definition of parameter ray=====

<!-- :<math>\mathbb{\hat{C}}\setminus \overline{\mathbb{D}} = { w\setin</math>\mathbb{\hat{C}}:abs(w)>1}</math>
  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 /> -->

The '''external ray''' of angle <math>\theta\,</math> is:
*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>

:<math>\mathcal{R}^M  _{\theta} = \Psi_M(\mathcal{R}_{\theta})</math>

*set of points of exterior of Mandelbrot set with the same external angle  <math>\theta</math><ref>[http://www.math.titech.ac.jp/~kawahira/programs/mandel-exray.pdf An algorithm to draw external rays of the Mandelbrot set by Tomoki Kawahira ]</ref>

:<math>\mathcal{R}^M  _{\theta} = \{ c\in \mathbb{\hat{C}}\setminus M  : \arg(\Phi_M(c)) =  \theta \}</math>

=====Definition of the Boettcher map =====

Douady and Hubbard define:

<math>\Phi_M(c) \ \overset{\underset{\mathrm{def}}{}}{=} \ \Phi_c(z=c)\,</math>

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

====External angle====
<gallery>
2015-03-04 exray binary.gif|collecting bits outwards
Binary decomposition.png|Binary decomposition of unrolled circle plane
Binary decomposition of dynamic plane for f0(z) = z^2.png|binary decomposition of dynamic plane for f(z) = z^2

</gallery>
Angle {{mvar|&theta;}} is named '''external angle''' ( [[Complex number#Complex plane|argument]] ).<ref>http://www.mrob.com/pub/muency/externalangle.html External angle at Mu-ENCY (the Encyclopedia of the Mandelbrot Set) by Robert Munafo</ref>

[[Principal value]] of external angles are [[Angle#Units|measured]] in [[Turn (geometry)|turns]] [[Modulo operation|modulo]] 1

:1  [[Turn (geometry)|turn]] = 360 [[Degree (angle)|degrees]] = 2 &times; {{Pi}} [[radian]]s

Compare different types of angles :
* external ( point of set's exterior )
* internal ( point of component's interior  )
* plain ( [[Arg (mathematics)|argument of complex number]] )

{| class="wikitable"
|-
|
! external angle
! internal angle
! plain angle
|-
! parameter plane
| <math> \arg(\Phi_M(c))  \,</math>
| <math> \arg(\rho_n(c)) \,</math>
| <math> \arg(c) \, </math>
|-
! dynamic plane
| <math> \arg(\Phi_c(z)) \,</math>
| 
| <math> \arg(z) \,</math>
|-
|}

=====Computation of external argument=====
*  argument of Böttcher coordinate as an external argument<ref>[http://www.mndynamics.com/indexp.html Computation of the external argument by Wolf Jung]</ref>
** <math> \arg_M(c) = \arg(\Phi_M(c)) </math>
** <math> \arg_c(z) = \arg(\Phi_c(z)) </math>
* 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>

===Transcendental maps===

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 [[Böttcher's equation|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>

Here dynamic ray is defined as a curve :
* connecting a point in an [[escaping set]] and [[Point at infinity|infinity]] {{Clarify|date=March 2009}}
* lying in an [[escaping set]]

==Images==

===Dynamic rays===
<gallery caption="unbranched" perrow="6">
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
JuliaRay3.png|Julia set and 3 [[w:external ray|external rays]] landing on  fixed point <math>\alpha_c\,</math>
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>
Julia-p9.png|Julia set with external rays landing on period 3 orbit
Parabolic rays landing on fixed point.ogv|Rays landing on parabolic fixed point for periods 2-40
</gallery>


<gallery caption="branched" perrow="6">
Dynamical plane with branched periodic external ray 0 for map f(z) = z*z + 0.35.png| Branched dynamic ray
</gallery>

===Parameter rays===

[[Mandelbrot set]] for [[complex quadratic polynomial]] with parameter rays of root points

<gallery perrow="5">
File:Mandelbrot set for complex quadratic polynomial with parameter rays of root points.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)
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
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.
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.
Image:Man5period.jpg| External rays for angles of form  :   n / ( 2<SUP>5</SUP> - 1)  landing on the root points of period 5 components
</gallery>
<gallery>
Image:Mandel ie 1 3.jpg|internal ray of main cardioid of angle 1/3: starts from center of main cardioid c=0, ends in the root point of period 3 component, which is the landing point of parameter (external) rays of angles  1/7 and 2/7
Image:Iray.png|Internal ray for angle 1/3 of main cardioid made by conformal map from unit circle
</gallery>

<gallery>
File:Smiley mini Mandelbrot set with external rays.png| Mini Mandelbrot set with period 134 and 2 external rays
File:Part of parameter plane with external 5 rays landing on the Mandelbrot set.png
File:One arm spiral - part of Mandelbrot set.png
File:Mini Mandelbrot set period=68 with external rays.png
File:Wakes near the period 3 island in the Mandelbrot set.png|Wakes near the period 3 island
File:Wakes along the main antenna in the Mandelbrot set.png|Wakes along the main antenna
</gallery>

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.

[[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]]

==Programs that can draw external rays==
*[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]]
**[https://web.archive.org/web/20080509103942/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]]
** [http://www.uvm.edu/~msargent/programs/index.html ezfract by Michael Sargent], uses the code by Wolf Jung
*[https://web.archive.org/web/20141030192550/http://www.math.nagoya-u.ac.jp/~kawahira/programs/otis.html OTIS  by Tomoki KAWAHIRA ] - [[Java applet]]  without [[source code]]
*[https://web.archive.org/web/20060602113914/http://inls.ucsd.edu/%7Efisher/Complex/ Spider XView program by Yuval Fisher ]
*[http://archives.math.utk.edu/software/msdos/fractals/yabmp097/.html YABMP by Prof. Eugene Zaustinsky] {{Webarchive|url=https://web.archive.org/web/20060615042830/http://archives.math.utk.edu/software/msdos/fractals/yabmp097/.html |date=2006-06-15 }} for [[DOS]] without [[source code]]
*[http://www.picard.ups-tlse.fr/~cheritat/e_index.html DH_Drawer] {{Webarchive|url=https://web.archive.org/web/20081021094744/http://picard.ups-tlse.fr/~cheritat/e_index.html |date=2008-10-21 }} by [[Arnaud Chéritat]] written for Windows 95 without [[source code]]
*[http://linas.org/art-gallery/ Linas Vepstas C programs ]  for [[Linux]] [[Command-line interface|console]] with [[source code]]
*[https://web.archive.org/web/20080705115700/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]]
*[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)
*[http://www.juliasets.dk/RatioField.htm 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 )
*Mandelbrot program by Milan Va, written in Delphi with source code
* [http://www.mrob.com/pub/muency/externalangle.html Power MANDELZOOM by Robert Munafo]
* [https://web.archive.org/web/20120120133504/http://claudiusmaximus.goto10.org/cm/2011-10-09_ruff-0.2_and_gruff-0.2_released.html ruff by Claude Heiland-Allen]

==See also==
{{commons|Category:External rays}}
*external rays of [[Misiurewicz point]]
*[[Orbit portrait]]
*[[Periodic points of complex quadratic mappings]]
*[[Prouhet-Thue-Morse constant]]
*[[Carathéodory's theorem (conformal mapping)|Carathéodory's theorem]]
*[[Julia set#Field lines|Field lines of Julia sets]]

==References==
{{Reflist}}
*[[Lennart Carleson]] and Theodore W. Gamelin, ''Complex Dynamics'', Springer 1993
*Adrien Douady and John H. Hubbard, ''Etude dynamique des polynômes complexes'', Prépublications mathémathiques d'Orsay 2/4 (1984 / 1985)
*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 [https://web.archive.org/web/20060424085751/http://www.math.sunysb.edu/preprints.html Stony Brook IMS Preprint] in 1999, available as [https://arxiv.org/abs/math.DS/9905169 arXiV:math.DS/9905169].)
* [[John Milnor]], ''Dynamics in One Complex Variable'', Third Edition, Princeton University Press, 2006, {{ISBN|0-691-12488-4}}
*[https://web.archive.org/web/20040828174339/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]

==External links==
{{Wikibooks|Fractals }}
*[http://www.iquilezles.org/trastero/fieldlines/ Hubbard Douady Potential, Field Lines by Inigo Quilez ]{{dead link|date=January 2018 |bot=InternetArchiveBot |fix-attempted=yes }}
*[http://math.bu.edu/people/bob/papers.html Intertwined Internal Rays in Julia Sets of Rational Maps by Robert L. Devaney]
*[http://math.bu.edu/people/bob/papers.html Extending External Rays Throughout the Julia Sets of Rational Maps by Robert L. Devaney With Figen Cilingir and Elizabeth D. Russell]
*[http://www.revver.com/video/91465/mandelbrot-p31/ John Hubbard's presentation, The Beauty and Complexity of the Mandelbrot Set, part 3.1 ] {{Webarchive|url=https://web.archive.org/web/20080226171703/http://revver.com/video/91465/mandelbrot-p31/ |date=2008-02-26 }}
*[https://www.youtube.com/user/ImpoliteFruit videos by ImpoliteFruit]
*{{cite web|url=http://sweb.cz/milan_va/Mandelbrot/|archive-url=https://archive.today/20130210042915/http://sweb.cz/milan_va/Mandelbrot/|url-status=dead|archive-date=February 10, 2013|author=Milan Va|title=Mandelbrot set drawing|accessdate=2009-06-15}}

{{DEFAULTSORT:External Ray}}
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