External ray

From Wikipedia, the free encyclopedia

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.

1 Formal definition of dynamic ray
2 Images
3 Root point, external and internal ray
4 References
5 Programs that can draw external rays:
6 Programs that can find roots of components
7 External links

[edit] Formal definition of dynamic ray

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

$\Psi_c:\mathbb{C}\setminus \overline{\mathbb{D}}\to\mathbb{C}\setminus Kc$
$\Psi_c: d = te^{2\pi i \theta}\mapsto z = \Psi_c\left(te^{2\pi i \theta}\right) for \ t > 1$
where :
$\mathbb{C}$ denotes the Complex plane
d and z are complex numbers

If $Ψ c$ is unique conformal isomorphism whose leading Laurent coefficient at infinity is real and positive
then the external ray of angle $\theta\,$ is:

the image under $Ψ c$ of straight lines $\mathcal{R}_{\theta} = \{\left(t*e^{2\pi i \theta}\right) : \ t > 1 \}$
set of points of exterior of filled-in Julia set with the same angle $θ$

$\mathcal{R}^K _{\theta} = \{ \Psi_c\left(t*e^{2\pi i \theta}\right) : \ t > 1 \}$

Angle $θ$ is named external angle ( argument ).
Rays are labeled by the coresponding angles.

[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:
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] Root point, external and internal ray

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] References

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.)