# External ray

## From Wikipedia, the free encyclopedia

An **external ray** is a curve that runs from infinity toward a Julia or Mandelbrot set.^{[1]}
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

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
^{[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

Stretching rays were introduced by Branner and Hubbard:^{[6]}

"The notion of stretching rays is a generalization of that of external rays for the Mandelbrot set to higher degree polynomials."^{[7]}

### landing

Every rational parameter ray of the Mandelbrot set lands at a single parameter.^{[8]}^{[9]}

## Maps

### Polynomials

#### Dynamical plane = z-plane

**External rays** are associated to a compact, full, connected subset of the complex plane as :

- the images of radial rays under the Riemann map of the complement of
- the gradient lines of the Green's function of
- field lines of Douady-Hubbard potential
^{[10]} - an integral curve of the gradient vector field of the Green's function on neighborhood of infinity
^{[11]}

External rays together with equipotential lines of Douady-Hubbard potential ( level sets) form a new polar coordinate system for exterior ( complement ) of .

In other words the external rays define vertical foliation which is orthogonal to horizontal foliation defined by the level sets of potential.^{[12]}

#### Uniformization

Let be the conformal isomorphism from the complement (exterior) of the closed unit disk to the complement of the filled Julia set .

where denotes the extended complex plane.
Let denote the **Boettcher map**.^{[13]}
is a uniformizing map of the basin of attraction of infinity, because it conjugates on the complement of the filled Julia set to on the complement of the unit disk:

and

A value is called the **Boettcher coordinate** for a point .

#### Formal definition of dynamic ray

The **external ray** of angle noted as is:

- the image under of straight lines

- set of points of exterior of filled-in Julia set with the same external angle

##### Properties

The external ray for a periodic angle satisfies:

and its landing point^{[14]} satisfies:

#### Parameter plane = c-plane

"Parameter rays are simply the curves that run perpendicular to the equipotential curves of the M-set."^{[15]}

##### Uniformization

Let be the mapping from the complement (exterior) of the closed unit disk to the complement of the Mandelbrot set .

and Boettcher map (function) , which is uniformizing map^{[16]} of complement of Mandelbrot set, because it conjugates complement of the Mandelbrot set and the complement (exterior) of the closed unit disk

it can be normalized so that :

^{[17]}

where :

- denotes the extended complex plane

Jungreis function is the inverse of uniformizing map :

In the case of complex quadratic polynomial one can compute this map using Laurent series about infinity^{[18]}^{[19]}

where

##### Formal definition of parameter ray

The **external ray** of angle is:

- the image under of straight lines

- set of points of exterior of Mandelbrot set with the same external angle
^{[20]}

##### Definition of

Douady and Hubbard define:

so external angle of point of parameter plane is equal to external angle of point of dynamical plane

#### External angle

Angle θ is named **external angle** ( argument ).^{[21]}

Principal value of external angles are measured in turns modulo 1

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 | |||

dynamic plane |

##### Computation of external argument

- argument of Böttcher coordinate as an external argument
^{[22]} - kneading sequence as a binary expansion of external argument
^{[23]}^{[24]}^{[25]}

### Transcendental maps

For transcendental maps ( for example exponential ) infinity is not a fixed point but an essential singularity and there is no Boettcher isomorphism.^{[26]}^{[27]}

Here dynamic ray is defined as a curve :

- connecting a point in an escaping set and infinity
^{[clarification needed]} - lying in an escaping set

## Images

### 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.

## 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

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