External ray

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In complex analysis, particularly in complex dynamics and geometric function theory, external rays are associated to a compact, full, connected subset K\, of the complex plane as the images of radial rays under the Riemann map of the complement of K\,. Equivalently, they are the gradient lines of the Green's function of K\, or field lines of Douady-Hubbard potential .

External rays together with equipotential lines of Douady-Hubbard potential form a new polar coordinate system for exterior ( complement ) of K\,.

External rays are particularly useful in the dynamical study of complex polynomials, where they were introduced in Douady and Hubbard's study of the Mandelbrot set. External rays of (connected) Julia sets of polynomials are often called dynamic rays, while external rays of the Mandelbrot set (and similar one-dimensional connectedness loci) are called parameter rays.

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[edit] Dynamical plane = z-plane

[edit] Uniformization

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 [1](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


Map \Psi_c\, is the inverse of uniformizing map :

\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}}

[edit] Formal definition of dynamic ray

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}^K _{\theta} = \Psi_c(\mathcal{R}_{\theta})
  • set of points of exterior of filled-in Julia set with the same external angle θ
\mathcal{R}^K _{\theta} = \{ z\in \mathbb{\hat{C}}\setminus Kc  : \arg(\Phi_c(z)) = \theta \}

[edit] Parameter plane = c-plane

[edit] Uniformization

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[2] 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}}

where :

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


Map \Psi_M\, is the inverse of uniformizing map :

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

On can compute this map using Laurent series

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} + ...\,[3]

where

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

[edit] Formal definition of parameter ray

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_c(\mathcal{R}_{\theta})
  • set of points of exterior of Mandelbrot set with the same external angle θ
\mathcal{R}^M _{\theta} = \{ c\in \mathbb{\hat{C}}\setminus M  : \arg(\Phi_M(c)) = \theta \}

[edit] External angle

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

External angles are measured in turns modulo 1

1 turn = 360 degrees = 2 * Pi radians

[edit] Images

Mandelbrot set for map: Z(n+1)=Z(n)*Z(n) +C

External rays for angles of the form  : n / ( 21 - 1) (0/1; 1/1) landing on the point c= 1/4, which is cusp of main cardioid ( period 1 component)

External rays for angles of the form  : n / ( 22 - 1) (1/3, 2/3) landing on the point c= - 3/4, which is root point of period 2 component

External rays for angles of the form : n / ( 23 - 1) (1/7,2/7) (3/7,4/7) landing on the point c= -1.75 = -7/4 (5/7,6/7) landing on the root points of period 3 components.

External rays for angles of form  : n / ( 24 - 1) (1/15,2/15) (3/15, 4/15) (6/15, 9/15) landing on the root point c= -5/4 (7/15, 8/15) (11/15,12/15) (13/15, 14/15) landing on the root points of period 4 components.Notice that 5/15 = 1*5/3*5 = 1/3; 10/15 = 2*5/3*5 = 2/3 so these are angles of rays landing on root point of period 2 component

External rays for angles of form  : n / ( 25 - 1) (1/31,2/31) (3/31,4/31) (5/31, 6/31) (7/31,8/31) (9/31,10/31)(13/31,18/31) (14/31,17/31) (15/31,16/31) (19/31,20/31) (21/31,22/31) (23/31,24/31) (25/31,26/31) (27/31,28/31)(29/31,30/31) landing on the root points of period 5 components

[edit] Center, root, external and internal ray

Image:Mandel_ie_1_3.jpg
internal ray of main cardioid of angle 1/3:
starts from center of main cardioid c=0
ends in the root point of period 3 component
which is the landing point of external rays of angles 1/7 and 2/7


[edit] Programs that can draw external rays

[edit] See also

[edit] External links

[edit] References

  1. ^ How to draw external rays by Wolf Jung
  2. ^ Irwin Jungreis: The uniformization of the complement of the Mandelbrot set. Duke Math. J. Volume 52, Number 4 (1985), 935-938.
  3. ^ 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,