External ray

From Wikipedia, the free encyclopedia

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

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

[edit] History

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

[edit] 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.

[edit] Polynomials

[edit] 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 form a new polar coordinate system for exterior ( complement ) of $K\,$ .

[edit] 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 ^[3](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

[edit] Formal definition of dynamic ray

polar coordinate system and Psi_c for c=-2

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

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

$\mathcal{R}^K _{\theta} = \Psi_c(\mathcal{R}_{\theta})$

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

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

[edit] Parameter plane = c-plane

[edit] 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^[4] 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 :

$\frac{\Phi_M(c)}{c} \to 1 \ as\ c \to \infty \,$ ^[5]

where :

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

Map $\Psi_M\,$ 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 ^[6]^[7]

$c = \Psi_M (w) = w + \sum_{m=0}^{\infty} b_m w^{-m} = w -\frac{1}{2} + \frac{1}{8w} - \frac{1}{4w^2} + \frac{15}{128w^3} + ...\,$

where

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

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

[edit] Formal definition of parameter ray

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

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

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

set of points of exterior of Mandelbrot set with the same external angle $θ$ ^[8]

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

[edit] 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

[edit] External angle

Angle $\theta\,$ is named external angle ( argument ).^[9]

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	$arg(\Phi_c(z)) \,$		$arg(z) \,$

[edit] Computation of external argument

argument of Böttcher coordinate as an external argument ^[10]
- $arg_M(c) = arg(\Phi_M(c)) \,$
- $arg_c(z) = arg(\Phi_c(z)) \,$
kneading sequence as a binary expansion of external argument ^[11]^[12]^[13]

[edit] Transcendental maps

For transcendental maps ( for example exponential ) infinity is not a fixed point but an essential singularity and there is no Boettcher isomorphism ^[14]^[15].

Here dynamic ray is defined as a curve :

connecting a point in an escaping set and infinity^{[clarification needed]}
lying in an escaping set

[edit] Images

Julia set and some of its external rays that land on fixed point $\alpha_c\,$

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

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

[edit] Center, root, external and internal ray

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

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

[edit] 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 without source code
Mandelbrot program by Milan Va, written in Delphi with source code

[edit] See also

Wikimedia Commons has media related to: External ray

Wikibooks has a book on the topic of

Fractals

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

[edit] 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
^ How to draw external rays by Wolf Jung
^ Irwin Jungreis: The uniformization of the complement of the Mandelbrot set. Duke Math. J. Volume 52, Number 4 (1985), 935-938.
^ [http://www.math.cornell.edu/~hubbard/OrsayEnglish.pdf Adrien Douady, John Hubbard, Etudes dynamique des polynomes comples 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

[edit] External links

[0] J. Kiwi : Rational rays and critical portraits of complex polynomials. Ph. D. Thesis SUNY at Stony Brook (1997); IMS Preprint #1997/15.

[1] 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

[2] How to draw external rays by Wolf Jung

[3] Irwin Jungreis: The uniformization of the complement of the Mandelbrot set. Duke Math. J. Volume 52, Number 4 (1985), 935-938.

[4] [http://www.math.cornell.edu/~hubbard/OrsayEnglish.pdf Adrien Douady, John Hubbard, Etudes dynamique des polynomes comples I & II, Publ. Math. Orsay. (1984-85) (The Orsay notes)]

[5] 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,

[6] Weisstein, Eric W. "Mandelbrot Set." From MathWorld--A Wolfram Web Resource

[7] An algorithm to draw external rays of the Mandelbrot set by Tomoki Kawahira

[8] ttp://www.mrob.com/pub/muency/externalangle.html External angle at Mu-ency by Robert Munafo

[9] Computation of the external argument by Wolf Jung

[10] A. DOUADY, Algorithms for computing angles in the Mandelbrot set (Chaotic Dynamics and Fractals, ed. Barnsley and Demko, Acad. Press, 1986, pp. 155-168).

[11] Adrien Douady, John H. Hubbard: Exploring the Mandelbrot set. The Orsay Notes. page 58

[12] Exploding the Dark Heart of Chaos by Chris King from Mathematics Department of University of Auckland

[13] Topological Dynamics of Entire Functions by Helena Mihaljevic-Brandt

[14] Dynamic rays of entire functions and their landing behaviour by Helena Mihaljevic-Brandt

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External ray

From Wikipedia, the free encyclopedia

Contents

[edit] History

[edit] Notation

[edit] Polynomials

[edit] Dynamical plane = z-plane

[edit] Uniformization

[edit] Formal definition of dynamic ray

[edit] Parameter plane = c-plane

[edit] Uniformization

[edit] Formal definition of parameter ray

[edit] Definition of $\Phi_M \,$

[edit] External angle

[edit] Computation of external argument

[edit] Transcendental maps

[edit] Images

[edit] Center, root, external and internal ray

[edit] Programs that can draw external rays

[edit] See also

[edit] References

[edit] External links

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