External ray

 ( Complex Numbers ) 

An external ray is a curve that runs from infinity toward a Julia set or Mandelbrot set.J. Kiwi : Rational rays and critical portraits of complex polynomials. Ph. D. Thesis SUNY at Stony Brook (1997); IMS Preprint #1997/15. Although this curve is only rarely a half-line (ray) it is called a ray because it is an image of a ray.

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

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

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Types Criteria for classification:

• Plane: parameter or dynamic
• Map
• Bifurcation of dynamic rays
• Stretching
• Landing

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Plane External rays of (connected) 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.

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Bifurcation Dynamic rays can be:

• Bifurcated, branched, broken
• 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. Holomorphic Dynamics: On Accumulation of Stretching Rays by Pia B.N. Willumsen, see page 12

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Stretching Stretching rays were introduced by Branner and Hubbard: The iteration of cubic polynomials Part I : The global topology of parameter by BODIL BRANNER and JOHN H. HUBBARD 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."

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Landing Every rational parameter ray of the Mandelbrot set lands at a single parameter. 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).

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Maps

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Polynomials

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Dynamical plane = z-plane External rays are associated to a compact space, full, connected space 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 of the Green's function of $K\backslash ,$
• of Douady-Hubbard potential 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 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. POLYNOMIAL BASINS OF INFINITY LAURA DEMARCO AND KEVIN M. PILGRIM

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Uniformization Let $\backslash Psi\_c\backslash ,$ be the Conformal map isomorphism from the complement (exterior) of the 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 Riemann sphere. Let $\backslash Phi\_c\; =\; \backslash Psi\_c^\{-1\}\backslash ,$ denote the Boettcher map. 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 $z\; \backslash in\; \backslash hat\{\backslash Complex\}\backslash setminus\; K\_c$.

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Formal definition of dynamic ray 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 $\backslash theta$

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

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Properties 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 $\backslash gamma\_f(\backslash theta)$ satisfies:

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

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Parameter plane = c-plane "Parameter rays are simply the curves that run perpendicular to the equipotential curves of the M-set." Douady Hubbard Parameter Rays by Linas Vepstas

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Uniformization Let $\backslash Psi\_M\backslash ,$ be the mapping from the complement (exterior) of the unit disk $\backslash overline\{\backslash mathbb\{D\}\}$ to the complement of the Mandelbrot set $\backslash \; M$. 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 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 complement of the Mandelbrot set $\backslash \; M$ and the complement (exterior) of the 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 :

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

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 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

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

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

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Formal definition of parameter ray 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 $\backslash theta$ An algorithm to draw external rays of the Mandelbrot set by Tomoki Kawahira

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

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Definition of the Boettcher map 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

====External angle====

Angle 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 operation 1

1 turn = 360 degrees = 2 ×

Compare different types of angles :

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

! external angle ! internal angle ! plain angle

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Computation of external argument

• argument of Böttcher coordinate as an external argument 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). Adrien Douady, John H. Hubbard: Exploring the Mandelbrot set. The Orsay Notes. page 58 Exploding the Dark Heart of Chaos by Chris King from Mathematics Department of University of Auckland

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Transcendental maps For transcendental maps ( for example exponential ) infinity is not a fixed point but an essential singularity and there is no Boettcher isomorphism. Topological Dynamics of Entire Functions by Helena Mihaljevic-Brandt 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
• lying in an escaping set

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Images

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Dynamic rays 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 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

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Parameter rays 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

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

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Programs that can draw external rays

• Mandel - program by Wolf Jung written in C++ using Qt with source code available under the GNU General Public License
• Java applets by Evgeny Demidov ( code of mndlbrot::turn function by Wolf Jung has been ported to Java ) with free source code
• ezfract by Michael Sargent, uses the code by Wolf Jung
• OTIS by Tomoki KAWAHIRA - Java applet without source code
• Spider XView program by Yuval Fisher
• YABMP by Prof. Eugene Zaustinsky for MS-DOS without source code
• DH_Drawer by Arnaud Chéritat written for Windows 95 without source code
• Linas Vepstas C programs for Linux console with source code
• Program Julia by Curtis T. McMullen written in C and Linux commands for C shell console with source code
• 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)
• RatioField by Gert Buschmann, for windows with Pascal source code for Dev-Pascal 1.9.2 (with Free Pascal compiler )
• Mandelbrot program by Milan Va, written in Delphi with source code
• Power MANDELZOOM by Robert Munafo
• ruff by Claude Heiland-Allen

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See also

• 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

• 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 Stony Brook IMS Preprint in 1999, available as arXiV:math.DS/9905169.)
• John Milnor, Dynamics in One Complex Variable, Third Edition, Princeton University Press, 2006,
• Wolf Jung : Homeomorphisms on Edges of the Mandelbrot Set. Ph.D. thesis of 2002

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External links

• Hubbard Douady Potential, Field Lines by Inigo Quilez
• Intertwined Internal Rays in Julia Sets of Rational Maps by Robert L. Devaney
• Extending External Rays Throughout the Julia Sets of Rational Maps by Robert L. Devaney With Figen Cilingir and Elizabeth D. Russell
• John Hubbard's presentation, The Beauty and Complexity of the Mandelbrot Set, part 3.1
• videos by ImpoliteFruit

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