External ray

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An external ray is a curve that runs from infinity toward a Julia or Mandelbrot set.[1] This curve is only sometimes a half-line ( ray ) but is called 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 K\, 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 K\,.

In other words the external rays define vertical foliation which is orthogonal to horizontal foliation defined by the level sets of potential.[3]


Let \Psi_c\, be the mapping from the complement (exterior) of the closed unit disk \overline{\mathbb{D}} to the complement of the filled Julia set \ Kc .

\Psi_c:\mathbb{\hat{C}}\setminus \overline{\mathbb{D}}\to\mathbb{\hat{C}}\setminus Kc

and Boettcher map[4](function) \Phi_c\,, which is uniformizing map of basin of attraction of infinity, because it conjugates complement of the filled Julia set \ Kc and the complement (exterior) of the closed unit disk

\Phi_c: \mathbb{\hat{C}}\setminus Kc \to   \mathbb{\hat{C}}\setminus \overline{\mathbb{D}}

where :

\mathbb{\hat{C}} denotes the extended complex plane

Boettcher map \Phi_c\, is an isomorphism :

\Psi_c = \Phi_{c}^{-1} \,

w = \Phi_c(z) = \lim_{n\rightarrow \infty} (f_c^n(z))^{2^{-n}}

where :

z \in \mathbb{\hat{C}}\setminus K_c

w \in \mathbb{\hat{C}}\setminus \overline{\mathbb{D}}

w\, is a Boettcher coordinate

Formal definition of dynamic ray[edit]

polar coordinate system and Psi_c for c=-2

The external ray of angle \theta\, noted as \mathcal{R}^K  _{\theta} is:

  • the image under \Psi_c\, of straight lines \mathcal{R}_{\theta} = \{\left(r*e^{2\pi i \theta}\right) :  \ r > 1 \}
\mathcal{R}^K  _{\theta} = \Psi_c(\mathcal{R}_{\theta})
\mathcal{R}^K  _{\theta} = \{ z\in \mathbb{\hat{C}}\setminus Kc  : \arg(\Phi_c(z)) =  \theta \}


External ray for periodic angle \theta\, satisfies :

f(\mathcal{R}^K  _{\theta}) =  \mathcal{R}^K  _{2 \theta}

and its landing point \gamma_f(\theta))  :[5]

f(\gamma_f(\theta)) = \gamma_f(2\theta)

Parameter plane = c-plane[edit]


Boundary of Mandelbrot set as an image of unit circle under \Psi_M\,

Let \Psi_M\, be the mapping from the complement (exterior) of the closed unit disk \overline{\mathbb{D}} to the complement of the Mandelbrot set \ M .

\Psi_M:\mathbb{\hat{C}}\setminus \overline{\mathbb{D}}\to\mathbb{\hat{C}}\setminus M

and Boettcher map (function) \Phi_M\,, which is uniformizing map[6] of complement of Mandelbrot set, because it conjugates complement of the Mandelbrot set \ M and the complement (exterior) of the closed unit disk

\Phi_M: \mathbb{\hat{C}}\setminus M \to   \mathbb{\hat{C}}\setminus \overline{\mathbb{D}}

it can be normalized so that :

\frac{\Phi_M(c)}{c} \to 1 \ as\ c \to \infty \,[7]

where :

\mathbb{\hat{C}} denotes the extended complex plane

Jungreis function \Psi_M\, is the inverse of uniformizing map :

\Psi_M = \Phi_{M}^{-1} \,

In the case of complex quadratic polynomial one can compute this map using Laurent series about infinity[8][9]

c = \Psi_M (w)  =  w + \sum_{m=0}^{\infty} b_m w^{-m} = w -\frac{1}{2} + \frac{1}{8w} - \frac{1}{4w^2} + \frac{15}{128w^3} + ...\,


c \in \mathbb{\hat{C}}\setminus M
w \in \mathbb{\hat{C}}\setminus \overline{\mathbb{D}}

Formal definition of parameter ray[edit]

The external ray of angle \theta\, is:

  • the image under \Psi_c\, of straight lines \mathcal{R}_{\theta} = \{\left(r*e^{2\pi i \theta}\right) :  \ r > 1 \}
\mathcal{R}^M  _{\theta} = \Psi_M(\mathcal{R}_{\theta})
  • set of points of exterior of Mandelbrot set with the same external angle \theta[10]
\mathcal{R}^M  _{\theta} = \{ c\in \mathbb{\hat{C}}\setminus M  : \arg(\Phi_M(c)) =  \theta \}

Definition of \Phi_M \,[edit]

Douady and Hubbard define:

\Phi_M(c) \ \overset{\underset{\mathrm{def}}{}}{=} \ \Phi_c(z=c)\,

so external angle of point c\, of parameter plane is equal to external angle of point z=c\, of dynamical plane

External angle[edit]

Angle \theta\, is named external angle ( argument ).[11]

Principal value of external angles are measured in turns modulo 1

1 turn = 360 degrees = 2 * Pi radians

Compare different types of angles :

external angle internal angle plain angle
parameter plane  arg(\Phi_M(c))  \,  arg(\rho_n(c)) \,  arg(c) \,
dynamic plane  arg(\Phi_c(z)) \,  arg(z) \,

Computation of external argument[edit]

  • argument of Böttcher coordinate as an external argument[12]
    •  arg_M(c) = arg(\Phi_M(c)) \,
    •  arg_c(z) = arg(\Phi_c(z))  \,
  • kneading sequence as a binary expansion of external argument[13][14][15]

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.[16][17]

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. ^ 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
  4. ^ How to draw external rays by Wolf Jung
  5. ^ Tessellation and Lyubich-Minsky laminations associated with quadratic maps I: Pinching semiconjugacies Tomoki Kawahira
  6. ^ Irwin Jungreis: The uniformization of the complement of the Mandelbrot set. Duke Math. J. Volume 52, Number 4 (1985), 935-938.
  7. ^ Adrien Douady, John Hubbard, Etudes dynamique des polynomes complexes I & II, Publ. Math. Orsay. (1984-85) (The Orsay notes)
  8. ^ 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,
  9. ^ Weisstein, Eric W. "Mandelbrot Set." From MathWorld--A Wolfram Web Resource
  10. ^ An algorithm to draw external rays of the Mandelbrot set by Tomoki Kawahira
  11. ^ http://www.mrob.com/pub/muency/externalangle.html External angle at Mu-ency by Robert Munafo
  12. ^ Computation of the external argument by Wolf Jung
  13. ^ A. DOUADY, Algorithms for computing angles in the Mandelbrot set (Chaotic Dynamics and Fractals, ed. Barnsley and Demko, Acad. Press, 1986, pp. 155-168).
  14. ^ Adrien Douady, John H. Hubbard: Exploring the Mandelbrot set. The Orsay Notes. page 58
  15. ^ Exploding the Dark Heart of Chaos by Chris King from Mathematics Department of University of Auckland
  16. ^ Topological Dynamics of Entire Functions by Helena Mihaljevic-Brandt
  17. ^ Dynamic rays of entire functions and their landing behaviour by Helena Mihaljevic-Brandt

External links[edit]