External ray

An external ray is a curve that runs from infinity toward a Julia or Mandelbrot set.^[1] 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.

History

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

Notation

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.

Polynomials

Dynamical plane = z-plane

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 $K\,$
the gradient lines of the Green's function of $K\,$
field lines of Douady-Hubbard potential
an integral curve of the gradient vector field of the Green's function on neighborhood of infinity^[2]

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

In other words the external rays define vertical foliation which is orthogonal to horizontal foliation defined by the level sets of potential.^[3]

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 {C}}}}\setminus \overline {{\mathbb {D}}}\to {\mathbb {{\hat {C}}}}\setminus Kc

and Boettcher map^[4](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 {C}}}}\setminus Kc\to {\mathbb {{\hat {C}}}}\setminus \overline {{\mathbb {D}}}

where :

{\mathbb {{\hat {C}}}}

denotes the extended complex plane

Boettcher map $\Phi _{c}\,$ is an isomorphism :

\Psi _{c}=\Phi _{{c}}^{{-1}}\,

$w=\Phi _{c}(z)=\lim _{{n\rightarrow \infty }}(f_{c}^{n}(z))^{{2^{{-n}}}}$

where :

$z\in {\mathbb {{\hat {C}}}}\setminus K_{c}$

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

$w\,$ is a Boettcher coordinate

Formal definition of dynamic ray

polar coordinate system and Psi_c for c=-2

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

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

{\mathcal {R}}_{{\theta }}^{K}=\Psi _{c}({\mathcal {R}}_{{\theta }})

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

{\mathcal {R}}_{{\theta }}^{K}=\{z\in {\mathbb {{\hat {C}}}}\setminus Kc:\arg(\Phi _{c}(z))=\theta \}

Properties

External ray for periodic angle $\theta \,$ satisfies :

f({\mathcal {R}}_{{\theta }}^{K})={\mathcal {R}}_{{2\theta }}^{K}

and its landing point $\gamma _{f}(\theta ))$ :^[5]

f(\gamma _{f}(\theta ))=\gamma _{f}(2\theta )

Parameter plane = c-plane

Uniformization

Boundary of Mandelbrot set as an image of unit circle under

\Psi _{M}\,

Uniformization of complement (exterior) of Mandelbrot set

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 {C}}}}\setminus \overline {{\mathbb {D}}}\to {\mathbb {{\hat {C}}}}\setminus M

and Boettcher map (function) $\Phi _{M}\,$ , which is uniformizing map^[6] 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 {C}}}}\setminus M\to {\mathbb {{\hat {C}}}}\setminus \overline {{\mathbb {D}}}

it can be normalized so that :

${\frac {\Phi _{M}(c)}{c}}\to 1\ as\ c\to \infty \,$ ^[7]

where :

{\mathbb {{\hat {C}}}}

denotes the extended complex plane

Jungreis function $\Psi _{M}\,$ is the inverse of uniformizing map :

\Psi _{M}=\Phi _{{M}}^{{-1}}\,

In the case of complex quadratic polynomial one can compute this map using Laurent series about infinity^[8]^[9]

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}}}+...\,

where

c\in {\mathbb {{\hat {C}}}}\setminus M

w\in {\mathbb {{\hat {C}}}}\setminus \overline {{\mathbb {D}}}

Formal definition of parameter ray

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

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

{\mathcal {R}}_{{\theta }}^{M}=\Psi _{M}({\mathcal {R}}_{{\theta }})

set of points of exterior of Mandelbrot set with the same external angle $\theta$ ^[10]

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

Definition of $\Phi _{M}\,$

Douady and Hubbard define:

$\Phi _{M}(c)\ {\overset {{\underset {{\mathrm {def}}}{}}}{=}}\ \Phi _{c}(z=c)\,$

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

External angle

Angle $\theta \,$ is named external angle ( argument ).^[11]

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 )

	external angle	internal angle	plain angle
parameter plane	$arg(\Phi _{M}(c))\,$	$arg(\rho _{n}(c))\,$	$arg(c)\,$
dynamic plane	$arg(\Phi _{c}(z))\,$		$arg(z)\,$

Computation of external argument

argument of Böttcher coordinate as an external argument^[12]
- $arg_{M}(c)=arg(\Phi _{M}(c))\,$
- $arg_{c}(z)=arg(\Phi _{c}(z))\,$
kneading sequence as a binary expansion of external argument^[13]^[14]^[15]

Transcendental maps

For transcendental maps ( for example exponential ) infinity is not a fixed point but an essential singularity and there is no Boettcher isomorphism.^[16]^[17]

Here dynamic ray is defined as a curve :

connecting a point in an escaping set and infinity
lying in an escaping set

Images

Dynamic rays

Julia set for $f_{c}(z)=z^{2}-1$ with 2 external ray landing on repelling fixed point alpha
Julia set and 3 external rays landing on fixed point $\alpha _{c}\,$
Dynamic external rays landing on repelling period 3 cycle and 3 internal rays landing on fixed point $\alpha _{c}\,$
Julia set with external rays landing on period 3 orbit
Rays landing on parabolic fixed point for periods 2-40

Parameter rays

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

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)
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
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.
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.
External rays for angles of form : n / ( 2⁵ - 1) landing on the root points of period 5 components

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
Internal ray for angle 1/3 of main cardioid made by conformal map from unit circle

Mini Mandelbrot set with period 134 and 2 external rays
Wakes near the period 3 island
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.

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
OTIS by Tomoki KAWAHIRA - Java applet without source code
Spider XView program by Yuval Fisher
YABMP by Prof. Eugene Zaustinsky for 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

References

↑ J. Kiwi : Rational rays and critical portraits of complex polynomials. Ph. D. Thesis SUNY at Stony Brook (1997); IMS Preprint #1997/15.
↑ 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
↑ POLYNOMIAL BASINS OF INFINITY LAURA DEMARCO AND KEVIN M. PILGRIM
↑ How to draw external rays by Wolf Jung
↑ Tessellation and Lyubich-Minsky laminations associated with quadratic maps I: Pinching semiconjugacies Tomoki Kawahira
↑ Irwin Jungreis: The uniformization of the complement of the Mandelbrot set. Duke Math. J. Volume 52, Number 4 (1985), 935-938.
↑ Adrien Douady, John Hubbard, Etudes dynamique des polynomes complexes I & II, Publ. Math. Orsay. (1984-85) (The Orsay notes)
↑ 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,
↑ Weisstein, Eric W. "Mandelbrot Set." From MathWorld--A Wolfram Web Resource
↑ An algorithm to draw external rays of the Mandelbrot set by Tomoki Kawahira
↑ http://www.mrob.com/pub/muency/externalangle.html External angle at Mu-ency by Robert Munafo
↑ Computation of the external argument by Wolf Jung
↑ A. 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
↑ Topological Dynamics of Entire Functions by Helena Mihaljevic-Brandt
↑ Dynamic rays of entire functions and their landing behaviour by Helena Mihaljevic-Brandt

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, ISBN 0-691-12488-4
Wolf Jung : Homeomorphisms on Edges of the Mandelbrot Set. Ph.D. thesis of 2002

External links

Wikibooks has a book on the topic of: Fractals

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External ray

History

Notation

Polynomials

Dynamical plane = z-plane

Uniformization

Formal definition of dynamic ray

Properties

Parameter plane = c-plane

Uniformization

Formal definition of parameter ray

Definition of

External angle

Computation of external argument

Transcendental maps

Images

Dynamic rays

Parameter rays

Programs that can draw external rays

See also

References

External links

Definition of $\Phi _{M}\,$