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.

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  $$\ K_c $$.


 * $$\Psi_c: \hat{\Complex} \setminus \overline{\mathbb{D}} \to \hat{\Complex} \setminus K_c$$

where $$\hat{\Complex}$$ 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{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


 * $$ \Phi_c \circ f_c \circ \Phi_c^{-1} = f_0 $$

A value $$w = \Phi_c(z)$$ is called the Boettcher coordinate for a point $$z \in \hat{\Complex}\setminus K_c$$.

Formal definition of dynamic ray


The external ray of angle $$\theta\,$$ noted as $$\mathcal{R}^K  _{\theta} $$is:
 * the image under $$\Psi_c\,$$ of straight lines $$\mathcal{R}_{\theta} = \{\left(r\cdot e^{2\pi i \theta}\right) : \ r > 1 \}$$


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


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


 * $$\mathcal{R}^K _{\theta} = \{ z\in \hat{\Complex} \setminus K_c : \arg(\Phi_c(z)) = \theta \}$$

Properties
The external ray for a periodic angle $$\theta\,$$ satisfies:


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

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 $$\mathcal{R}_{\theta} = \{\left(r*e^{2\pi i \theta}\right) : \ r > 1 \}$$


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


 * set of points of exterior of Mandelbrot set with the same external angle $$\theta$$


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

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
Angle $&theta;$ is named external angle ( argument ).

Principal value of external angles are measured in turns modulo 1


 * 1 turn = 360 degrees = 2 &times; $\pi$ 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

 * 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

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

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 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