# External ray

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

## Types

Criteria for classification :
* plane : parameter or dynamic
* map
* bifurcation of dynamic rays
* Stretching
* landing

### plane

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

Dynamic ray 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.

### stretching

Stretching rays were introduced by Branner and Hubbard:
"The notion of stretching rays is a generalization of that of external rays for the Mandelbrot set to higher degree polynomials."

### landing

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

## Maps
### 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
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.

##### Uniformization

Let $\Psi _{c}\,$ be the conformal isomorphism from the complement (exterior) of the closed unit disk ${\overline {\mathbb {D} }}$ to the complement of the filled Julia set $\ K_{c}$.
    $\Psi _{c}:{\hat {\mathbb {C} }}\setminus {\overline {\mathbb {D} }}\to {\hat {\mathbb {C} }}\setminus K_{c}$
where ${\hat {\mathbb {C} }}$ denotes the extended complex plane. Let $\Phi _{c}=\Psi _{c}^{-1}\,$ denote the Boettcher map. $\Phi _{c}\,$ 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:
    ${\begin{aligned}\Phi _{c}:{\hat {\mathbb {C} }}\setminus K_{c}&\to {\hat {\mathbb {C} }}\setminus {\overline {\mathbb {D} }}\\z&\mapsto \lim _{n\to \infty }(f_{c}^{n}(z))^{2^{-n}}\end{aligned}}$
and
    $\Phi _{c}\circ f_{c}\circ \Phi _{c}^{-1}=f_{0}$
A value $w=\Phi _{c}(z)$ is called the Boettcher coordinate for a point $z\in {\hat {\mathbb {C} }}\setminus K_{c}$.

##### Formal definition of dynamic ray

The external ray of angle $\theta \,$ noted as ${\mathcal {R}}_{\theta }^{K}$ is:
* the image under $\Psi _{c}\,$ of straight lines ${\mathcal {R}}_{\theta }=\{\left(r\cdot e^{2\pi i\theta }\right):\ r>1\}$
    ${\mathcal {R}}_{\theta }^{K}=\Psi _{c}({\mathcal {R}}_{\theta })$
* set of points of exterior of filled-in Julia set with the same external angle $\theta$
    ${\mathcal {R}}_{\theta }^{K}=\{z\in {\hat {\mathbb {C} }}\setminus K_{c}:\arg(\Phi _{c}(z))=\theta \}$

###### Properties

The external ray for a periodic angle $\theta \,$ satisfies:
    $f({\mathcal {R}}_{\theta }^{K})={\mathcal {R}}_{2\theta }^{K}$
and its landing point $\gamma _{f}(\theta )$ satisfies:
    $f(\gamma _{f}(\theta ))=\gamma _{f}(2\theta )$

#### Parameter plane = c-plane

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

##### 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 $\ M$.
    $\Psi _{M}:\mathbb {\hat {C}} \setminus {\overline {\mathbb {D} }}\to \mathbb {\hat {C}} \setminus M$
and Boettcher map (function) $\Phi _{M}\,$, which is uniformizing map 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 :
${\frac {\Phi _{M}(c)}{c}}\to 1\ as\ c\to \infty \,$
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
    $c=\Psi _{M}(w)=w+\sum _{m=0}^{\infty }b_{m}w^{-m}=w-{\frac {1}{2}}+{\frac {1}{8w}}-{\frac {1}{4w^{2}}}+{\frac {15}{128w^{3}}}+...\,$
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 $\theta$
    ${\mathcal {R}}_{\theta }^{M}=\{c\in \mathbb {\hat {C}} \setminus M:\arg(\Phi _{M}(c))=\theta \}$

##### Definition of the Boettcher map

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

* 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 ).
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 )
| | 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
** $\arg _{M}(c)=\arg(\Phi _{M}(c))$
** $\arg _{c}(z)=\arg(\Phi _{c}(z))$
* kneading sequence as a binary expansion of external argument

### Transcendental maps

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

* unbranched
* 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
* branched
* Branched dynamic ray

### Parameter rays

Mandelbrot set for complex quadratic polynomial with parameter rays of root points
* External rays for angles of the form : n / ( 2$^1$ - 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 / ( 2$^2$ - 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 / ( 2$^3$ - 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 / ( 2$^4$ - 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 / ( 2$^5$ - 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
** 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 Archived 2006-06-15 at the Wayback Machine for DOS without source code
* DH_Drawer Archived 2008-10-21 at the Wayback Machine 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