External ray

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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.


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


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.


Dynamical plane = z-plane[edit]

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

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.[4]


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[5]. 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:


A value is called the Boettcher coordinate for a point .

Formal definition of dynamic ray[edit]

polar coordinate system and Ψc for c=−2

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


The external ray for a periodic angle satisfies:

and its landing point[6] satisfies:

Parameter plane = c-plane[edit]


Boundary of Mandelbrot set as an image of unit circle under

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[7] 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 :


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[9][10]


Formal definition of parameter ray[edit]

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 [11]

Definition of [edit]

Douady and Hubbard define:

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

External angle[edit]

Angle θ is named external angle ( argument ).[12]

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
parameter plane
dynamic plane

Computation of external argument[edit]

  • argument of Böttcher coordinate as an external argument[13]
  • kneading sequence as a binary expansion of external argument[14][15][16]

Transcendental maps[edit]

For transcendental maps ( for example exponential ) infinity is not a fixed point but an essential singularity and there is no Boettcher isomorphism.[17][18]

Here dynamic ray is defined as a curve :


Dynamic rays[edit]

Parameter rays[edit]

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.

Parameter plane of the complex exponential family f(z)=exp(z)+c with 8 external ( parameter) rays

Programs that can draw external rays[edit]

See also[edit]


  1. ^ J. Kiwi : Rational rays and critical portraits of complex polynomials. Ph. D. Thesis SUNY at Stony Brook (1997); IMS Preprint #1997/15.
  2. ^ Video : The beauty and complexity of the Mandelbrot set by John Hubbard ( see part 3 )
  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
  5. ^ How to draw external rays by Wolf Jung
  6. ^ Tessellation and Lyubich-Minsky laminations associated with quadratic maps I: Pinching semiconjugacies Tomoki Kawahira Archived 2016-03-03 at the Wayback Machine
  7. ^ Irwin Jungreis: The uniformization of the complement of the Mandelbrot set. Duke Math. J. Volume 52, Number 4 (1985), 935-938.
  8. ^ Adrien Douady, John Hubbard, Etudes dynamique des polynomes complexes I & II, Publ. Math. Orsay. (1984-85) (The Orsay notes)
  9. ^ 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,
  10. ^ Weisstein, Eric W. "Mandelbrot Set." From MathWorld--A Wolfram Web Resource
  11. ^ An algorithm to draw external rays of the Mandelbrot set by Tomoki Kawahira
  12. ^ http://www.mrob.com/pub/muency/externalangle.html External angle at Mu-ENCY (the Encyclopedia of the Mandelbrot Set) by Robert Munafo
  13. ^ Computation of the external argument by Wolf Jung
  14. ^ A. DOUADY, Algorithms for computing angles in the Mandelbrot set (Chaotic Dynamics and Fractals, ed. Barnsley and Demko, Acad. Press, 1986, pp. 155-168).
  15. ^ Adrien Douady, John H. Hubbard: Exploring the Mandelbrot set. The Orsay Notes. page 58
  16. ^ Exploding the Dark Heart of Chaos by Chris King from Mathematics Department of University of Auckland
  17. ^ Topological Dynamics of Entire Functions by Helena Mihaljevic-Brandt
  18. ^ 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[edit]