External ray

In complex analysis, particularly in complex dynamics and geometric function theory, external rays are associated to a compact, full, connected subset $K,$ of the complex plane as the images of radial rays under the Riemann map of the complement of $K,$. Equivalently, they are the gradient lines of the Green's function of $K,$ or field lines of Douady-Hubbard potential .

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

External rays are particularly useful in the dynamical study of complex polynomials, where they were introduced in Douady and Hubbard's study of the Mandelbrot set. External rays of (connected) Julia sets on dynamical plane are often called dynamic rays, while external rays of the Mandelbrot set on parameter plane (and similar one-dimensional connectedness loci) are called parameter rays.

=Dynamical plane = z-plane =

Uniformization

Let $Psi\_c,$ be the mapping from the complement (exterior) of the closed unit disk $overline\{mathbb\{D$ to the complement of the filled Julia set $Kc$.

:$Psi\_c:mathbb\{hat\{Csetminus\; overline\{mathbb\{D\; omathbb\{hat\{Csetminus\; Kc$

and Boettcher map [ [http://www.mndynamics.com/indexp.html How to draw external rays by Wolf Jung] ] (function) $Phi\_c,$, which is uniformizing map of basin of attraction of infinity , because it conjugates complement of the filled Julia set $Kc$ and the complement (exterior) of the closed unit disk

:$Phi\_c:\; mathbb\{hat\{Csetminus\; Kc\; o\; mathbb\{hat\{Csetminus\; overline\{mathbb\{D$

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

Map $Psi\_c,$ is the inverse of uniformizing

:$Psi\_c\; =\; Phi\_\{c\}^\{-1\}\; ,$

$w\; =\; Phi\_c(z)\; =\; lim\_\{n\; ightarrow\; infty\}\; (f\_c^n(z))^\{2^\{-n$

where :

$z\; in\; mathbb\{hat\{Csetminus\; K\_c$

$w\; in\; mathbb\{hat\{Csetminus\; overline\{mathbb\{D$

Formal definition of dynamic ray

The external ray of angle $heta,$ is:

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

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

*set of points of exterior of filled-in Julia set with the same external angle $heta$

:$mathcal\{R\}^K\; \_\{\; heta\}\; =\; \{\; zin\; mathbb\{hat\{Csetminus\; Kc\; :\; arg(Phi\_c(z))\; =\; heta\; \}$

=Parameter plane = c-plane =

Uniformization

Let $Psi\_M,$ be the mapping from the complement (exterior) of the closed unit disk $overline\{mathbb\{D$ to the complement of the Mandelbrot set $M$.

:$Psi\_M:mathbb\{hat\{Csetminus\; overline\{mathbb\{D\; omathbb\{hat\{Csetminus\; M$

and Boettcher map (function) $Phi\_M,$, 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 $M$ and the complement (exterior) of the closed unit disk

:$Phi\_M:\; mathbb\{hat\{Csetminus\; M\; o\; mathbb\{hat\{Csetminus\; overline\{mathbb\{D$

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

Map $Psi\_M,$ is the inverse of uniformizing

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

On can compute this map using Laurent series

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

where

:$c\; in\; mathbb\{hat\{Csetminus\; M$

:$w\; in\; mathbb\{hat\{Csetminus\; overline\{mathbb\{D$

Formal definition of parameter ray

The external ray of angle $heta,$ is:

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

:$mathcal\{R\}^M\; \_\{\; heta\}\; =\; Psi\_c(mathcal\{R\}\_\{\; heta\})$

*set of points of exterior of Mandelbrot set with the same external angle $heta$

:$mathcal\{R\}^M\; \_\{\; heta\}\; =\; \{\; cin\; mathbb\{hat\{Csetminus\; M\; :\; arg(Phi\_M(c))\; =\; heta\; \}$

External angle

Angle $heta,$ is named external angle ( argument ).

External angles are measured in turns modulo 1

1 turn = 360 degrees = 2 * Pi radians

Images

Mandelbrot set for map: Z(n+1)=Z(n)*Z(n) +C

Center, root, external and internal ray

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 external rays of angles 1/7 and 2/7

Programs that can draw external 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
* [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
* [http://www.math.nagoya-u.ac.jp/~kawahira/programs/aboutotis.htm OTIS - Java applet by Tomoki KAWAHIRA ]
* [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 ]
* [http://www.picard.ups-tlse.fr/~cheritat/e_index.html DH_Drawer by Arnaud Chéritat ]
* [http://linas.org/art-gallery/ Linas Vepstas C programs]

ee also

*external rays of Misiurewicz point
*Orbit portrait
*Periodic points of complex quadratic mappings
*Prouhet-Thue-Morse constant

External links

* [http://rgba.scenesp.org/iq/trastero/fieldlines/ Hubbard Douady Potential, Field Lines by Inigo Quilez ]
* [http://www.mrob.com/pub/muency/externalangle.html External angle at Mu-ency by Robert Munafo]
* [http://republika.pl/fraktal/mset_jungreis.html Drawing Mc by Jungreis Algorithm]
* [http://republika.pl/fraktal/internalAngleMset.html Internal rays of components of Mandelbrot set]
* [http://www.revver.com/video/91465/mandelbrot-p31/ John Hubbard's presentation, The Beauty and Complexity of the Mandelbrot Set, part 3.1 ]
* [http://pl.youtube.com/user/ImpoliteFruit videos by ImpoliteFruit]

References

*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 [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] .)
* John Milnor, "Dynamics in One Complex Variable", Third Edition, Princeton University Press, 2006, ISBN 0-691-12488-4
* [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]