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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


Criteria for classification :


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:

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]



Dynamical plane = z-plane

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.[13]


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.[14] 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
Polar coordinate system and for

The external ray of angle noted as is:


The external ray for a periodic angle satisfies:

and its landing point[15] satisfies:

Parameter plane = c-plane

"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
Uniformization of complement (exterior) of Mandelbrot set

Let be the mapping from the complement (exterior) of the closed unit disk to the complement of the Mandelbrot set .[17]

and Boettcher map (function) , which is uniformizing map[18] 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[20][21]


Formal definition of parameter ray

The external ray of angle is:

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 ).[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 angle internal angle plain angle
parameter plane
dynamic plane
Computation 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.[28][29]

Here dynamic ray is defined as a curve :


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.

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

Programs that can draw external rays

See also


  1. ^ 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
  2. ^ Inou, Hiroyuki; Mukherjee, Sabyasachi (2016). "Non-landing parameter rays of the multicorns". Inventiones Mathematicae. 204 (3): 869–893. arXiv:1406.3428. Bibcode:2016InMat.204..869I. doi:10.1007/s00222-015-0627-3. S2CID 253746781.
  3. ^ Atela, Pau (1992). "Bifurcations of dynamic rays in complex polynomials of degree two". Ergodic Theory and Dynamical Systems. 12 (3): 401–423. doi:10.1017/S0143385700006854. S2CID 123478692.
  4. ^ Petersen, Carsten L.; Zakeri, Saeed (2020). "Periodic Points and Smooth Rays". arXiv:2009.02788 [math.DS].
  5. ^ Holomorphic Dynamics: On Accumulation of Stretching Rays by Pia B.N. Willumsen, see page 12
  6. ^ The iteration of cubic polynomials Part I : The global topology of parameter by BODIL BRANNER and JOHN H. HUBBARD
  7. ^ Stretching rays for cubic polynomials by Pascale Roesch
  8. ^ Komori, Yohei; Nakane, Shizuo (2004). "Landing property of stretching rays for real cubic polynomials" (PDF). Conformal Geometry and Dynamics. 8 (4): 87–114. Bibcode:2004CGDAM...8...87K. doi:10.1090/s1088-4173-04-00102-x.
  9. ^ A. Douady, J. Hubbard: Etude dynamique des polynˆomes complexes. Publications math´ematiques d’Orsay 84-02 (1984) (premi`ere partie) and 85-04 (1985) (deuxi`eme partie).
  10. ^ Schleicher, Dierk (1997). "Rational parameter rays of the Mandelbrot set". arXiv:math/9711213.
  11. ^ Video : The beauty and complexity of the Mandelbrot set by John Hubbard ( see part 3 )
  12. ^ 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
  14. ^ How to draw external rays by Wolf Jung
  15. ^ Tessellation and Lyubich-Minsky laminations associated with quadratic maps I: Pinching semiconjugacies Tomoki Kawahira Archived 2016-03-03 at the Wayback Machine
  16. ^ Douady Hubbard Parameter Rays by Linas Vepstas
  17. ^ John H. Ewing, Glenn Schober, The area of the Mandelbrot Set
  18. ^ Irwin Jungreis: The uniformization of the complement of the Mandelbrot set. Duke Math. J. Volume 52, Number 4 (1985), 935-938.
  19. ^ Adrien Douady, John Hubbard, Etudes dynamique des polynomes complexes I & II, Publ. Math. Orsay. (1984-85) (The Orsay notes)
  20. ^ Bielefeld, B.; Fisher, Y.; Vonhaeseler, F. (1993). "Computing the Laurent Series of the Map Ψ: C − D → C − M". Advances in Applied Mathematics. 14: 25–38. doi:10.1006/aama.1993.1002.
  21. ^ Weisstein, Eric W. "Mandelbrot Set." From MathWorld--A Wolfram Web Resource
  22. ^ An algorithm to draw external rays of the Mandelbrot set by Tomoki Kawahira
  23. ^ External angle at Mu-ENCY (the Encyclopedia of the Mandelbrot Set) by Robert Munafo
  24. ^ Computation of the external argument by Wolf Jung
  25. ^ A. DOUADY, Algorithms for computing angles in the Mandelbrot set (Chaotic Dynamics and Fractals, ed. Barnsley and Demko, Acad. Press, 1986, pp. 155-168).
  26. ^ Adrien Douady, John H. Hubbard: Exploring the Mandelbrot set. The Orsay Notes. page 58
  27. ^ Exploding the Dark Heart of Chaos by Chris King from Mathematics Department of University of Auckland
  28. ^ Topological Dynamics of Entire Functions by Helena Mihaljevic-Brandt
  29. ^ Dynamic rays of entire functions and their landing behaviour by Helena Mihaljevic-Brandt