External ray

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.

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 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]

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

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]

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 ${\displaystyle z\in {\hat {\mathbb {C} }}\setminus K_{c}}$.

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

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 ${\displaystyle \theta }$

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

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

Boundary of Mandelbrot set as an image of unit circle under ${\displaystyle \Psi _{M}\,}$

Uniformization of complement (exterior) of Mandelbrot set

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

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

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

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]

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

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

• 
  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 )

	external angle	internal angle	plain angle
parameter plane	${\displaystyle \arg(\Phi _{M}(c))\,}$	${\displaystyle \arg(\rho _{n}(c))\,}$	${\displaystyle \arg(c)\,}$
dynamic plane	${\displaystyle \arg(\Phi _{c}(z))\,}$		${\displaystyle \arg(z)\,}$

• 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]

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
• lying in an escaping set

• 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 ${\displaystyle \alpha _{c}\,}$
• 
  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

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.