External ray#Uniformization

External rays were introduced in Douady and Hubbard's study of the Mandelbrot set

Criteria for classification :

• plane : parameter or dynamic
• map
• bifurcation of dynamic rays
• Stretching
• landing{{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 }}

External rays of (connected) Julia sets on dynamical plane are often called dynamic rays.

External rays of the Mandelbrot set (and similar one-dimensional connectedness loci) on parameter plane are called parameter rays.

Dynamic ray can be:

• bifurcated = branched{{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 }} = broken {{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 }}
• 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.[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]

Stretching rays were introduced by Branner and Hubbard:[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][https://www.youtube.com/watch?v=DyJDt4EyiBA&list=PL53AB2CAE70F31F2A&index=29 Stretching rays for cubic polynomials by Pascale Roesch]

"The notion of stretching rays is a generalization of that of external rays for the Mandelbrot set to higher degree polynomials."{{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 }}

Every rational parameter ray of the Mandelbrot set lands at a single parameter.[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).]{{cite arXiv | eprint=math/9711213 | last1=Schleicher | first1=Dierk | title=Rational parameter rays of the Mandelbrot set | year=1997 }}

External rays are associated to a compact, full, connected subset $K\backslash ,$ of the complex plane as :

• the images of radial rays under the Riemann map of the complement of $K\backslash ,$
• the gradient lines of the Green's function of $K\backslash ,$
• field lines of Douady-Hubbard potential[https://www.youtube.com/watch?v=N3ah6iTupIg&t=2652s Video : The beauty and complexity of the Mandelbrot set by John Hubbard ( see part 3 )]
• an integral curve of the gradient vector field of the Green's function on neighborhood of infinity[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

External rays together with equipotential lines of Douady-Hubbard potential ( level sets) form a new polar coordinate system for exterior ( complement ) of $K\backslash ,$.

In other words the external rays define vertical foliation which is orthogonal to horizontal foliation defined by the level sets of potential.[http://www.math.northwestern.edu/~demarco/basins.pdf POLYNOMIAL BASINS OF INFINITY LAURA DEMARCO AND KEVIN M. PILGRIM]

Let $\backslash Psi\_c\backslash ,$ be the conformal isomorphism from the complement (exterior) of the closed unit disk $\backslash overline\{\backslash mathbb\{D\}\}$ to the complement of the filled Julia set $\backslash \; K\_c$.

:$\backslash Psi\_c:\; \backslash hat\{\backslash Complex\}\; \backslash setminus\; \backslash overline\{\backslash mathbb\{D\}\}\; \backslash to\; \backslash hat\{\backslash Complex\}\; \backslash setminus\; K\_c$

where $\backslash hat\{\backslash Complex\}$ denotes the extended complex plane.

Let $\backslash Phi\_c\; =\; \backslash Psi\_c^\{-1\}\backslash ,$ denote the Boettcher map.[http://www.mndynamics.com/indexp.html How to draw external rays by Wolf Jung]

$\backslash Phi\_c\backslash ,$ is a uniformizing map of the basin of attraction of infinity, because it conjugates $f\_c$ on the complement of the filled Julia set $K\_c$ to $f\_0(z)=z^2$ on the complement of the unit disk:

:$\backslash begin\{align\}$

\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}

and

:$\backslash Phi\_c\; \backslash circ\; f\_c\; \backslash circ\; \backslash Phi\_c^\{-1\}\; =\; f\_0$

A value $w\; =\; \backslash Phi\_c(z)$ is called the Boettcher coordinate for a point $z\; \backslash in\; \backslash hat\{\backslash Complex\}\backslash setminus\; K\_c$.

Image:Erays.svg

The external ray of angle $\backslash theta\backslash ,$ noted as $\backslash mathcal\{R\}^K\; \_\{\backslash theta\}$ is:

• the image under $\backslash Psi\_c\backslash ,$ of straight lines $\backslash mathcal\{R\}\_\{\backslash theta\}\; =\; \backslash \{\backslash left(r\backslash cdot\; e^\{2\backslash pi\; i\; \backslash theta\}\backslash right)\; :\; \backslash \; r\; >\; 1\; \backslash \}$

:$\backslash mathcal\{R\}^K\; \_\{\backslash theta\}\; =\; \backslash Psi\_c(\backslash mathcal\{R\}\_\{\backslash theta\})$

• set of points of exterior of filled-in Julia set with the same external angle $\backslash theta$

:$\backslash mathcal\{R\}^K\; \_\{\backslash theta\}\; =\; \backslash \{\; z\backslash in\; \backslash hat\{\backslash Complex\}\; \backslash setminus\; K\_c\; :\; \backslash arg(\backslash Phi\_c(z))\; =\; \backslash theta\; \backslash \}$

The external ray for a periodic angle $\backslash theta\backslash ,$ satisfies:

:$f(\backslash mathcal\{R\}^K\; \_\{\backslash theta\})\; =\; \backslash mathcal\{R\}^K\; \_\{2\; \backslash theta\}$

and its landing point{{usurped|1=[https://web.archive.org/web/20160303233036/http://eprintweb.org/S/article/math/0609280 Tessellation and Lyubich-Minsky laminations associated with quadratic maps I: Pinching semiconjugacies Tomoki Kawahira]}} $\backslash gamma\_f(\backslash theta)$ satisfies:

:$f(\backslash gamma\_f(\backslash theta))\; =\; \backslash gamma\_f(2\backslash theta)$

"Parameter rays are simply the curves that run perpendicular to the equipotential curves of the M-set."[http://linas.org/art-gallery/escape/phase/phase.html Douady Hubbard Parameter Rays by Linas Vepstas]

File:Jung200.png as an image of unit circle under $\backslash Psi\_M\backslash ,$]]

File:Jung50e.png of complement (exterior) of Mandelbrot set]]

Let $\backslash Psi\_M\backslash ,$ be the mapping from the complement (exterior) of the closed unit disk $\backslash overline\{\backslash mathbb\{D\}\}$ to the complement of the Mandelbrot set $\backslash \; M$.[http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN001185500 John H. Ewing, Glenn Schober, The area of the Mandelbrot Set]

:$\backslash Psi\_M:\backslash mathbb\{\backslash hat\{C\}\}\backslash setminus\; \backslash overline\{\backslash mathbb\{D\}\}\backslash to\backslash mathbb\{\backslash hat\{C\}\}\backslash setminus\; M$

and Boettcher map (function) $\backslash Phi\_M\backslash ,$, which is uniformizing map[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.] of complement of Mandelbrot set, because it conjugates complement of the Mandelbrot set $\backslash \; M$ and the complement (exterior) of the closed unit disk

:$\backslash Phi\_M:\; \backslash mathbb\{\backslash hat\{C\}\}\backslash setminus\; M\; \backslash to\; \backslash mathbb\{\backslash hat\{C\}\}\backslash setminus\; \backslash overline\{\backslash mathbb\{D\}\}$

it can be normalized so that :

$\backslash frac\{\backslash Phi\_M(c)\}\{c\}\; \backslash to\; 1\; \backslash \; as\backslash \; c\; \backslash to\; \backslash infty\; \backslash ,$[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)]

where :

:$\backslash mathbb\{\backslash hat\{C\}\}$ denotes the extended complex plane

Jungreis function $\backslash Psi\_M\backslash ,$ is the inverse of uniformizing map :

:$\backslash Psi\_M\; =\; \backslash Phi\_\{M\}^\{-1\}\; \backslash ,$

In the case of complex quadratic polynomial one can compute this map using Laurent series about infinity{{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 }}[http://mathworld.wolfram.com/MandelbrotSet.html Weisstein, Eric W. "Mandelbrot Set." From MathWorld--A Wolfram Web Resource]

:$c\; =\; \backslash Psi\_M\; (w)\; =\; w\; +\; \backslash sum\_\{m=0\}^\{\backslash infty\}\; b\_m\; w^\{-m\}\; =\; w\; -\backslash frac\{1\}\{2\}\; +\; \backslash frac\{1\}\{8w\}\; -\; \backslash frac\{1\}\{4w^2\}\; +\; \backslash frac\{15\}\{128w^3\}\; +\; ...\backslash ,$

where

:$c\; \backslash in\; \backslash mathbb\{\backslash hat\{C\}\}\backslash setminus\; M$

:$w\; \backslash in\; \backslash mathbb\{\backslash hat\{C\}\}\backslash setminus\; \backslash overline\{\backslash mathbb\{D\}\}$

The external ray of angle $\backslash theta\backslash ,$ is:

• the image under $\backslash Psi\_c\backslash ,$ of straight lines $\backslash mathcal\{R\}\_\{\backslash theta\}\; =\; \backslash \{\backslash left(r*e^\{2\backslash pi\; i\; \backslash theta\}\backslash right)\; :\; \backslash \; r\; >\; 1\; \backslash \}$

:$\backslash mathcal\{R\}^M\; \_\{\backslash theta\}\; =\; \backslash Psi\_M(\backslash mathcal\{R\}\_\{\backslash theta\})$

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

:$\backslash mathcal\{R\}^M\; \_\{\backslash theta\}\; =\; \backslash \{\; c\backslash in\; \backslash mathbb\{\backslash hat\{C\}\}\backslash setminus\; M\; :\; \backslash arg(\backslash Phi\_M(c))\; =\; \backslash theta\; \backslash \}$

Douady and Hubbard define:

$\backslash Phi\_M(c)\; \backslash \; \backslash overset\{\backslash underset\{\backslash mathrm\{def\}\}\{\}\}\{=\}\; \backslash \; \backslash Phi\_c(z=c)\backslash ,$

so external angle of point $c\backslash ,$ of parameter plane is equal to external angle of point $z=c\backslash ,$ of dynamical plane

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

Angle {{mvar|θ}} is named external angle ( argument ).http://www.mrob.com/pub/muency/externalangle.html External angle at Mu-ENCY (the Encyclopedia of the Mandelbrot Set) by Robert Munafo

Principal value of external angles are measured in turns modulo 1

:1 turn = 360 degrees = 2 × {{Pi}} radians

Compare different types of angles :

• external ( point of set's exterior )
• internal ( point of component's interior )
• plain ( argument of complex number )

• argument of Böttcher coordinate as an external argument[http://www.mndynamics.com/indexp.html Computation of the external argument by Wolf Jung]
• $\backslash arg\_M(c)\; =\; \backslash arg(\backslash Phi\_M(c))$
• $\backslash arg\_c(z)\; =\; \backslash arg(\backslash Phi\_c(z))$
• kneading sequence as a binary expansion of external argumentA. DOUADY, Algorithms for computing angles in the Mandelbrot set (Chaotic Dynamics and Fractals, ed. Barnsley and Demko, Acad. Press, 1986, pp. 155-168).[http://www.math.cornell.edu/~hubbard/OrsayEnglish.pdf Adrien Douady, John H. Hubbard: Exploring the Mandelbrot set. The Orsay Notes. page 58 ][http://www.dhushara.com/DarkHeart/DarkHeart.htm Exploding the Dark Heart of Chaos by Chris King from Mathematics Department of University of Auckland]

For transcendental maps ( for example exponential ) infinity is not a fixed point but an essential singularity and there is no Boettcher isomorphism.[http://pcwww.liv.ac.uk/~helenam/Poster.pdf Topological Dynamics of Entire Functions by Helena Mihaljevic-Brandt][http://pcwww.liv.ac.uk/~helenam/slides_manchester.pdf Dynamic rays of entire functions and their landing behaviour by Helena Mihaljevic-Brandt]

Here dynamic ray is defined as a curve :

• connecting a point in an escaping set and infinity {{Clarify|date=March 2009}}
• lying in an escaping set

JuliaRay 1 3.png|Julia set for $f\_c(z)\; =\; z^2\; -1$ with 2 external ray landing on repelling fixed point alpha

JuliaRay3.png|Julia set and 3 external rays landing on fixed point $\backslash alpha\_c\backslash ,$

Dynamic internal and external rays .svg|Dynamic external rays landing on repelling period 3 cycle and 3 internal rays landing on fixed point $\backslash alpha\_c\backslash ,$

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

Dynamical plane with branched periodic external ray 0 for map f(z) = z*z + 0.35.png| Branched dynamic ray

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

File:Mandelbrot set for complex quadratic polynomial with parameter rays of root points.jpg|External rays for angles of the form : n / ( 2¹ - 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 / ( 2² - 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 / ( 2³ - 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 / ( 2⁴ - 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 / ( 2⁵ - 1) landing on the root points of period 5 components

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

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

Parameter space of the complex exponential family f(z)=exp(z)+c. Eight parameter rays landing at this parameter are drawn in black.

Parameter plane of the complex exponential family f(z)=exp(z)+c with 8 external ( parameter) rays

• [http://www.mndynamics.com/indexp.html Mandel ] - program by Wolf Jung written in C++ using 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 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 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 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]

{{commons|Category:External rays}}

• external rays of Misiurewicz point
• Orbit portrait
• Periodic points of complex quadratic mappings
• Prouhet-Thue-Morse constant
• Carathéodory's theorem
• Field lines of Julia sets

{{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]

{{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}}

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