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## Summary

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 Polar coordinate system and ψ c {\displaystyle \psi _{c}}   for c = − 2 {\displaystyle c=-2}

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

• the image under $\Psi _{c}\,$  of straight lines ${\mathcal {R}}_{\theta }=\{\left(r\cdot e^{2\pi i\theta }\right):\ r>1\}$
${\mathcal {R}}_{\theta }^{K}=\Psi _{c}({\mathcal {R}}_{\theta })$
• 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 $\Phi _{M}\,$ $\Phi _{M}(c)\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \Phi _{c}(z=c)\,$

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

#### External angle

Angle 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 angle internal angle plain angle $\arg(\Phi _{M}(c))\,$ $\arg(\rho _{n}(c))\,$ $\arg(c)\,$ $\arg(\Phi _{c}(z))\,$ $\arg(z)\,$ ##### 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 :

## Images

### 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.