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^[2]

plane Edit

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.

bifurcation Edit

Dynamic ray can be:

• bifurcated = branched^[3] = broken ^[4]
• 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.^[5]

stretching Edit

Stretching rays were introduced by Branner and Hubbard:^[6][7]

"The notion of stretching rays is a generalization of that of external rays for the Mandelbrot set to higher degree polynomials."^[8]

landing Edit

Every rational parameter ray of the Mandelbrot set lands at a single parameter.^[9][10]

Polynomials Edit

Dynamical plane = z-plane Edit

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

• the images of radial rays under the Riemann map of the complement of ${\displaystyle K\,}$ 
• the gradient lines of the Green's function of ${\displaystyle K\,}$ 
• field lines of Douady-Hubbard potential^[11]
• an integral curve of the gradient vector field of the Green's function on neighborhood of infinity^[12]

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

In other words the external rays define vertical foliation which is orthogonal to horizontal foliation defined by the level sets of potential.^[13]

Uniformization Edit

Let ${\displaystyle \Psi _{c}\,}$  be the conformal isomorphism from the complement (exterior) of the closed unit disk ${\displaystyle {\overline {\mathbb {D} }}}$  to the complement of the filled Julia set ${\displaystyle \ K_{c}}$ .

${\displaystyle \Psi _{c}:{\hat {\mathbb {C} }}\setminus {\overline {\mathbb {D} }}\to {\hat {\mathbb {C} }}\setminus K_{c}}$ 

where ${\displaystyle {\hat {\mathbb {C} }}}$  denotes the extended complex plane. Let ${\displaystyle \Phi _{c}=\Psi _{c}^{-1}\,}$  denote the Boettcher map.^[14]${\displaystyle \Phi _{c}\,}$  is a uniformizing map of the basin of attraction of infinity, because it conjugates ${\displaystyle f_{c}}$  on the complement of the filled Julia set ${\displaystyle K_{c}}$  to ${\displaystyle f_{0}(z)=z^{2}}$  on the complement of the unit disk:

${\displaystyle {\begin{aligned}\Phi _{c}:{\hat {\mathbb {C} }}\setminus K_{c}&\to {\hat {\mathbb {C} }}\setminus {\overline {\mathbb {D} }}\\z&\mapsto \lim _{n\to \infty }(f_{c}^{n}(z))^{2^{-n}}\end{aligned}}}$ 

and

${\displaystyle \Phi _{c}\circ f_{c}\circ \Phi _{c}^{-1}=f_{0}}$ 

A value ${\displaystyle w=\Phi _{c}(z)}$  is called the Boettcher coordinate for a point ${\displaystyle z\in {\hat {\mathbb {C} }}\setminus K_{c}}$ .

Formal definition of dynamic ray Edit

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

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

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

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

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

Properties Edit

The external ray for a periodic angle ${\displaystyle \theta \,}$  satisfies:

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

and its landing point^[15] ${\displaystyle \gamma _{f}(\theta )}$  satisfies:

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

Parameter plane = c-plane Edit

"Parameter rays are simply the curves that run perpendicular to the equipotential curves of the M-set."^[16]

Uniformization Edit

Let ${\displaystyle \Psi _{M}\,}$  be the mapping from the complement (exterior) of the closed unit disk ${\displaystyle {\overline {\mathbb {D} }}}$  to the complement of the Mandelbrot set ${\displaystyle \ M}$ .^[17]

${\displaystyle \Psi _{M}:\mathbb {\hat {C}} \setminus {\overline {\mathbb {D} }}\to \mathbb {\hat {C}} \setminus M}$ 

and Boettcher map (function) ${\displaystyle \Phi _{M}\,}$ , which is uniformizing map^[18] of complement of Mandelbrot set, because it conjugates complement of the Mandelbrot set ${\displaystyle \ M}$  and the complement (exterior) of the closed unit disk

${\displaystyle \Phi _{M}:\mathbb {\hat {C}} \setminus M\to \mathbb {\hat {C}} \setminus {\overline {\mathbb {D} }}}$ 

it can be normalized so that :

${\displaystyle {\frac {\Phi _{M}(c)}{c}}\to 1\ as\ c\to \infty \,}$ ^[19]

where :

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

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

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

In the case of complex quadratic polynomial one can compute this map using Laurent series about infinity^[20][21]

${\displaystyle c=\Psi _{M}(w)=w+\sum _{m=0}^{\infty }b_{m}w^{-m}=w-{\frac {1}{2}}+{\frac {1}{8w}}-{\frac {1}{4w^{2}}}+{\frac {15}{128w^{3}}}+...\,}$ 

where

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

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

Formal definition of parameter ray Edit

The external ray of angle ${\displaystyle \theta \,}$  is:

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

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

• set of points of exterior of Mandelbrot set with the same external angle ${\displaystyle \theta }$ ^[22]

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

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

Douady and Hubbard define:

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

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

External angle Edit

•  

  collecting bits outwards

•  

  Binary decomposition of unrolled circle plane

•  

  binary decomposition of dynamic plane for f(z) = z^2

Angle θ is named external angle ( argument ).^[23]

Principal value of external angles are measured in turns modulo 1

1 turn = 360 degrees = 2 × π radians

Compare different types of angles :

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

Computation of external argument Edit

• argument of Böttcher coordinate as an external argument^[24]
  • ${\displaystyle \arg _{M}(c)=\arg(\Phi _{M}(c))}$ 
  • ${\displaystyle \arg _{c}(z)=\arg(\Phi _{c}(z))}$ 
• kneading sequence as a binary expansion of external argument^[25][26][27]

Transcendental maps Edit

For transcendental maps ( for example exponential ) infinity is not a fixed point but an essential singularity and there is no Boettcher isomorphism.^[28][29]

Here dynamic ray is defined as a curve :

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

Dynamic rays Edit

• unbranched
•  

  Julia set for ${\displaystyle f_{c}(z)=z^{2}-1}$  with 2 external ray landing on repelling fixed point alpha

•  

  Julia set and 3 external rays landing on fixed point α c {\displaystyle \alpha _{c}\,}    

  Dynamic external rays landing on repelling period 3 cycle and 3 internal rays landing on fixed point ${\displaystyle \alpha _{c}\,}$ 

•  

  Julia set with external rays landing on period 3 orbit

• Rays landing on parabolic fixed point for periods 2-40

• branched
•  

  Branched dynamic ray

Parameter rays Edit

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.

• 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 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

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

• Hubbard Douady Potential, Field Lines by Inigo Quilez ^{[permanent dead link]}
• 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 Archived 2008-02-26 at the Wayback Machine
• videos by ImpoliteFruit
• Milan Va. "Mandelbrot set drawing". Retrieved 2009-06-15.