External ray


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 Edit

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

Types Edit

Criteria for classification :

  • plane : parameter or dynamic
  • map
  • bifurcation of dynamic rays
  • Stretching
  • landing[2]

plane Edit

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 Edit

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 Edit

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]

landing Edit

Every rational parameter ray of the Mandelbrot set lands at a single parameter.[9][10]

Maps Edit

Polynomials Edit

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.[13]

Uniformization Edit

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 Edit
Polar coordinate system and   for  

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 Edit

The external ray for a periodic angle   satisfies:


and its landing point[15]   satisfies:


Parameter plane = c-plane Edit

"Parameter rays are simply the curves that run perpendicular to the equipotential curves of the M-set."[16]

Uniformization Edit
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 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  [22]
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 ).[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 Edit
  • argument of Böttcher coordinate as an external argument[24]
  • kneading sequence as a binary expansion of external argument[25][26][27]

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.[28][29]

Here dynamic ray is defined as a curve :

Images Edit

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.


Programs that can draw external rays Edit

See also Edit

References Edit

  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. ^ http://www.mrob.com/pub/muency/externalangle.html 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
  • 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

  • Hubbard Douady Potential, Field Lines by Inigo Quilez [permanent dead link]
  • Intertwined Internal Rays in Julia Sets of Rational Maps by Robert L. Devaney
  • Extending External Rays Throughout the Julia Sets of Rational Maps by Robert L. Devaney With Figen Cilingir and Elizabeth D. Russell
  • John Hubbard's presentation, The Beauty and Complexity of the Mandelbrot Set, part 3.1 Archived 2008-02-26 at the Wayback Machine
  • videos by ImpoliteFruit
  • Milan Va. "Mandelbrot set drawing". Retrieved 2009-06-15.