# External ray

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

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

## Notation

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

### Dynamical plane = z-plane

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

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

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 $\mathcal{R}_{\theta} = \{\left(r*e^{2\pi i \theta}\right) : \ r > 1 \}$
$\mathcal{R}^K _{\theta} = \Psi_c(\mathcal{R}_{\theta})$
• set of points of exterior of filled-in Julia set with the same external angle $\theta$
$\mathcal{R}^K _{\theta} = \{ z\in \mathbb{\hat{C}}\setminus Kc : \arg(\Phi_c(z)) = \theta \}$

#### Properities

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

#### Uniformization

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

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 $\Phi_M \,$

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

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

### External angle

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

• 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

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

### Parameter rays

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.

## References

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