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 image of 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

Notation[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.

Polynomials[edit]

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]

Uniformization[edit]

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 \}

Properities[edit]

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]

Uniformization[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} + ...\,

where

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 :

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.

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]

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.
  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
  3. ^ POLYNOMIAL BASINS OF INFINITY LAURA DEMARCO AND KEVIN M. PILGRIM
  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]