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 : parameter or dynamic
- map
- bifurcation of dynamic rays
- Stretching

### 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
^{[2]}= broken^{[3]} - unbranched = smooth

When the filled Julia set is connected no external ray braches. When the Julia set is not connected then some external rays brach^{[4]}

### stretching

Streching rays were introduced by Branner and Hubbard^{[5]}

" the notion of stretching rays is a generalization of that of external rays for the Mandelbrot set to higher degree polynomials." ^{[6]}

## 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
^{[7]} - an integral curve of the gradient vector field of the Green's function on neighborhood of infinity
^{[8]}

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.^{[9]}

#### 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**.^{[10]} 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^{[11]} satisfies:

#### Parameter plane = c-plane

"Parameter rays are simply the curves that run perpendicular to the equipotential curves of the M-set."^{[12]}

##### 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^{[13]} 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 :

^{[14]}

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^{[15]}^{[16]}

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
^{[17]}

##### 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 ).^{[18]}

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
^{[19]} - kneading sequence as a binary expansion of external argument
^{[20]}^{[21]}^{[22]}

### Transcendental maps

For transcendental maps ( for example exponential ) infinity is not a fixed point but an essential singularity and there is no Boettcher isomorphism.^{[23]}^{[24]}

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

Julia set and 3 external rays landing on fixed point

Rays landing on parabolic fixed point for periods 2-40

### 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

## See also

Wikimedia Commons has media related to .Category:External rays |

- external rays of Misiurewicz point
- Orbit portrait
- Periodic points of complex quadratic mappings
- Prouhet-Thue-Morse constant
- Carathéodory's theorem
- Field lines of Julia sets

## References

**^**J. Kiwi : Rational rays and critical portraits of complex polynomials. Ph. D. Thesis SUNY at Stony Brook (1997); IMS Preprint #1997/15. Archived 2004-11-05 at the Wayback Machine**^**Atela, P. (1992). Bifurcations of dynamic rays in complex polynomials of degree two. Ergodic Theory and Dynamical Systems, 12(3), 401-423. doi:10.1017/S0143385700006854**^**Periodic Points and Smooth Rays by Carsten L. Petersen, Saeed Zakeri**^**Holomorphic Dynamics: On Accumulation of Stretching Rays by Pia B.N. Willumsen, see page 12**^**The iteration of cubic polynomials Part I : The global topology of parameter by BODIL BRANNER and JOHN H. HUBBARD**^**LANDING PROPERTY OF STRETCHING RAYS FOR REAL CUBIC POLYNOMIALS YOHEI KOMORI AND SHIZUO NAKANE. CONFORMAL GEOMETRY AND DYNAMICS An Electronic Journal of the American Mathematical Society Volume 8, Pages 87–114 (March 29, 2004) S 1088-4173(04)00102-X**^**Video : The beauty and complexity of the Mandelbrot set by John Hubbard ( see part 3 )**^**Yunping Jing : Local connectivity of the Mandelbrot set at certain infinitely renormalizable points Complex Dynamics and Related Topics, New Studies in Advanced Mathematics, 2004, The International Press, 236-264**^**POLYNOMIAL BASINS OF INFINITY LAURA DEMARCO AND KEVIN M. PILGRIM**^**How to draw external rays by Wolf Jung**^**Tessellation and Lyubich-Minsky laminations associated with quadratic maps I: Pinching semiconjugacies Tomoki Kawahira Archived 2016-03-03 at the Wayback Machine**^**Douady Hubbard Parameter Rays by Linas Vepstas**^**Irwin Jungreis: The uniformization of the complement of the Mandelbrot set. Duke Math. J. Volume 52, Number 4 (1985), 935-938.**^**Adrien Douady, John Hubbard, Etudes dynamique des polynomes complexes I & II, Publ. Math. Orsay. (1984-85) (The Orsay notes)**^**Computing the Laurent series of the map Psi: C-D to C-M. Bielefeld, B.; Fisher, Y.; Haeseler, F. V. Adv. in Appl. Math. 14 (1993), no. 1, 25--38,**^**Weisstein, Eric W. "Mandelbrot Set." From MathWorld--A Wolfram Web Resource**^**An algorithm to draw external rays of the Mandelbrot set by Tomoki Kawahira**^**http://www.mrob.com/pub/muency/externalangle.html External angle at Mu-ENCY (the Encyclopedia of the Mandelbrot Set) by Robert Munafo**^**Computation of the external argument by Wolf Jung**^**A. DOUADY, Algorithms for computing angles in the Mandelbrot set (Chaotic Dynamics and Fractals, ed. Barnsley and Demko, Acad. Press, 1986, pp. 155-168).**^**Adrien Douady, John H. Hubbard: Exploring the Mandelbrot set. The Orsay Notes. page 58**^**Exploding the Dark Heart of Chaos by Chris King from Mathematics Department of University of Auckland**^**Topological Dynamics of Entire Functions by Helena Mihaljevic-Brandt**^**Dynamic rays of entire functions and their landing behaviour by Helena Mihaljevic-Brandt

- Lennart Carleson and Theodore W. Gamelin,
*Complex Dynamics*, Springer 1993 - Adrien Douady and John H. Hubbard,
*Etude dynamique des polynômes complexes*, Prépublications mathémathiques d'Orsay 2/4 (1984 / 1985) - John W. Milnor,
*Periodic Orbits, External Rays and the Mandelbrot Set: An Expository Account*; Géométrie complexe et systèmes dynamiques (Orsay, 1995), Astérisque No. 261 (2000), 277–333. (First appeared as a Stony Brook IMS Preprint in 1999, available as arXiV:math.DS/9905169.) - John Milnor,
*Dynamics in One Complex Variable*, Third Edition, Princeton University Press, 2006, ISBN 0-691-12488-4 - Wolf Jung : Homeomorphisms on Edges of the Mandelbrot Set. Ph.D. thesis of 2002

## External links

Wikibooks has a book on the topic of: Fractals |

- Hubbard Douady Potential, Field Lines by Inigo Quilez
^{[permanent dead link]} - Drawing Mc by Jungreis Algorithm
- Internal rays of components of Mandelbrot set
- John Hubbard's presentation, The Beauty and Complexity of the Mandelbrot Set, part 3.1
- videos by ImpoliteFruit
- Milan Va. "Mandelbrot set drawing". Retrieved 2009-06-15.