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External ray

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Source updated September 14, 2026

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.

01History

External rays were introduced in Douady and Hubbard's study of the Mandelbrot set.

02Types

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 rays 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.

Boundary of Mandelbrot set as an image of unit circle under

03Maps

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

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

04Images

Dynamic rays

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.

Uniformization of complement (exterior) of Mandelbrot set

05Programs 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 MS-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

Watch videos about External rayExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “External ray”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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