# External ray

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An external ray is a curve that runs from infinity toward a Julia or Mandelbrot set.[1] Although this curve is only rarely a half-line (ray) it is called a ray because it is an image of a 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

## Types

Criteria for classification :

• plane : parameter or dynamic
• map
• bifurcation of dynamic rays
• Stretching
• landing[2]

### plane

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.

### bifurcation

Dynamic ray can be:

• bifurcated = branched[3] = broken [4]
• smooth = unbranched = unbroken

When the filled Julia set is connected, there are no branching external rays. When the Julia set is not connected then some external rays branch.[5]

### stretching

Stretching rays were introduced by Branner and Hubbard:[6]

"The notion of stretching rays is a generalization of that of external rays for the Mandelbrot set to higher degree polynomials."[7]

### landing

Every rational parameter ray of the Mandelbrot set lands at a single parameter.[8][9]

## Maps

### Polynomials

#### Dynamical plane = z-plane

External rays are associated to a compact, full, connected subset ${\displaystyle K\,}$ 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 ${\displaystyle K\,}$.

In other words the external rays define vertical foliation which is orthogonal to horizontal foliation defined by the level sets of potential.[12]

#### Uniformization

Let ${\displaystyle \Psi _{c}\,}$ be the conformal isomorphism from the complement (exterior) of the closed unit disk ${\displaystyle {\overline {\mathbb {D} ))}$ to the complement of the filled Julia set ${\displaystyle \ K_{c))$.

${\displaystyle \Psi _{c}:{\hat {\mathbb {C} ))\setminus {\overline {\mathbb {D} ))\to {\hat {\mathbb {C} ))\setminus K_{c))$

where ${\displaystyle {\hat {\mathbb {C} ))}$ denotes the extended complex plane. Let ${\displaystyle \Phi _{c}=\Psi _{c}^{-1}\,}$ denote the Boettcher map.[13] ${\displaystyle \Phi _{c}\,}$ is a uniformizing map of the basin of attraction of infinity, because it conjugates ${\displaystyle f_{c))$ on the complement of the filled Julia set ${\displaystyle K_{c))$ to ${\displaystyle f_{0}(z)=z^{2))$ on the complement of the unit disk:

{\displaystyle {\begin{aligned}\Phi _{c}:{\hat {\mathbb {C} ))\setminus K_{c}&\to {\hat {\mathbb {C} ))\setminus {\overline {\mathbb {D} ))\\z&\mapsto \lim _{n\to \infty }(f_{c}^{n}(z))^{2^{-n))\end{aligned))}

and

${\displaystyle \Phi _{c}\circ f_{c}\circ \Phi _{c}^{-1}=f_{0))$

A value ${\displaystyle w=\Phi _{c}(z)}$ is called the Boettcher coordinate for a point ${\displaystyle z\in {\hat {\mathbb {C} ))\setminus K_{c))$.

#### Formal definition of dynamic ray

polar coordinate system and Ψc for c=−2

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

• the image under ${\displaystyle \Psi _{c}\,}$ of straight lines ${\displaystyle {\mathcal {R))_{\theta }=\{\left(r\cdot e^{2\pi i\theta }\right):\ r>1\))$
${\displaystyle {\mathcal {R))_{\theta }^{K}=\Psi _{c}({\mathcal {R))_{\theta })}$
• set of points of exterior of filled-in Julia set with the same external angle ${\displaystyle \theta }$
${\displaystyle {\mathcal {R))_{\theta }^{K}=\{z\in {\hat {\mathbb {C} ))\setminus K_{c}:\arg(\Phi _{c}(z))=\theta \))$
##### Properties

The external ray for a periodic angle ${\displaystyle \theta \,}$ satisfies:

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

and its landing point[14] ${\displaystyle \gamma _{f}(\theta )}$ satisfies:

${\displaystyle f(\gamma _{f}(\theta ))=\gamma _{f}(2\theta )}$

#### Parameter plane = c-plane

"Parameter rays are simply the curves that run perpendicular to the equipotential curves of the M-set."[15]

##### Uniformization
Boundary of Mandelbrot set as an image of unit circle under ${\displaystyle \Psi _{M}\,}$

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

${\displaystyle \Psi _{M}:\mathbb {\hat {C)) \setminus {\overline {\mathbb {D} ))\to \mathbb {\hat {C)) \setminus M}$

and Boettcher map (function) ${\displaystyle \Phi _{M}\,}$, which is uniformizing map[16] of complement of Mandelbrot set, because it conjugates complement of the Mandelbrot set ${\displaystyle \ M}$ and the complement (exterior) of the closed unit disk

${\displaystyle \Phi _{M}:\mathbb {\hat {C)) \setminus M\to \mathbb {\hat {C)) \setminus {\overline {\mathbb {D} ))}$

it can be normalized so that :

${\displaystyle {\frac {\Phi _{M}(c)}{c))\to 1\ as\ c\to \infty \,}$[17]

where :

${\displaystyle \mathbb {\hat {C)) }$ denotes the extended complex plane

Jungreis function ${\displaystyle \Psi _{M}\,}$ is the inverse of uniformizing map :

${\displaystyle \Psi _{M}=\Phi _{M}^{-1}\,}$

In the case of complex quadratic polynomial one can compute this map using Laurent series about infinity[18][19]

${\displaystyle c=\Psi _{M}(w)=w+\sum _{m=0}^{\infty }b_{m}w^{-m}=w-{\frac {1}{2))+{\frac {1}{8w))-{\frac {1}{4w^{2))}+{\frac {15}{128w^{3))}+...\,}$

where

${\displaystyle c\in \mathbb {\hat {C)) \setminus M}$
${\displaystyle w\in \mathbb {\hat {C)) \setminus {\overline {\mathbb {D} ))}$
##### Formal definition of parameter ray

The external ray of angle ${\displaystyle \theta \,}$ is:

• the image under ${\displaystyle \Psi _{c}\,}$ of straight lines ${\displaystyle {\mathcal {R))_{\theta }=\{\left(r*e^{2\pi i\theta }\right):\ r>1\))$
${\displaystyle {\mathcal {R))_{\theta }^{M}=\Psi _{M}({\mathcal {R))_{\theta })}$
• set of points of exterior of Mandelbrot set with the same external angle ${\displaystyle \theta }$[20]
${\displaystyle {\mathcal {R))_{\theta }^{M}=\{c\in \mathbb {\hat {C)) \setminus M:\arg(\Phi _{M}(c))=\theta \))$
##### Definition of ${\displaystyle \Phi _{M}\,}$

${\displaystyle \Phi _{M}(c)\ {\overset {\underset {\mathrm {def} }{)){=))\ \Phi _{c}(z=c)\,}$

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

#### External angle

Angle θ is named external angle ( argument ).[21]

Principal value of external angles are measured in turns modulo 1

1 turn = 360 degrees = 2 × π radians

Compare different types of angles :

external angle internal angle plain angle ${\displaystyle \arg(\Phi _{M}(c))\,}$ ${\displaystyle \arg(\rho _{n}(c))\,}$ ${\displaystyle \arg(c)\,}$ ${\displaystyle \arg(\Phi _{c}(z))\,}$ ${\displaystyle \arg(z)\,}$
##### Computation of external argument
• argument of Böttcher coordinate as an external argument[22]
• ${\displaystyle \arg _{M}(c)=\arg(\Phi _{M}(c))}$
• ${\displaystyle \arg _{c}(z)=\arg(\Phi _{c}(z))}$
• kneading sequence as a binary expansion of external argument[23][24][25]

### Transcendental maps

For transcendental maps ( for example exponential ) infinity is not a fixed point but an essential singularity and there is no Boettcher isomorphism.[26][27]

Here dynamic ray is defined as a curve :

## Images

### Dynamic rays

• Julia set for ${\displaystyle f_{c}(z)=z^{2}-1}$ with 2 external ray landing on repelling fixed point alpha
• Julia set and 3 external rays landing on fixed point ${\displaystyle \alpha _{c}\,}$
• Dynamic external rays landing on repelling period 3 cycle and 3 internal rays landing on fixed point ${\displaystyle \alpha _{c}\,}$
• Julia set with external rays landing on period 3 orbit

### Parameter rays

Mandelbrot set for complex quadratic polynomial with parameter rays of root points

• 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.
• External rays for angles of form  : n / ( 25 - 1) landing on the root points of period 5 components
• 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 parameter (external) rays of angles 1/7 and 2/7
• Internal ray for angle 1/3 of main cardioid made by conformal map from unit circle
• Mini Mandelbrot set with period 134 and 2 external rays
• Wakes near the period 3 island
• Wakes along the main antenna

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. Archived 2004-11-05 at the Wayback Machine
2. ^ Non-landing parameter rays of the multicorns by Hiroyuki Inou, Sabyasachi Mukherjee
3. ^ Atela, P. (1992). Bifurcations of dynamic rays in complex polynomials of degree two. Ergodic Theory and Dynamical Systems, 12(3), 401-423. doi:10.1017/S0143385700006854
4. ^ Periodic Points and Smooth Rays by Carsten L. Petersen, Saeed Zakeri
5. ^ Holomorphic Dynamics: On Accumulation of Stretching Rays by Pia B.N. Willumsen, see page 12
6. ^ The iteration of cubic polynomials Part I : The global topology of parameter by BODIL BRANNER and JOHN H. HUBBARD
7. ^ LANDING PROPERTY OF STRETCHING RAYS FOR REAL CUBIC POLYNOMIALS YOHEI KOMORI AND SHIZUO NAKANE. CONFORMAL GEOMETRY AND DYNAMICS An Electronic Journal of the American Mathematical Society Volume 8, Pages 87–114 (March 29, 2004) S 1088-4173(04)00102-X
8. ^ A. Douady, J. Hubbard: Etude dynamique des polynˆomes complexes. Publications math´ematiques d’Orsay 84-02 (1984) (premiere partie) and 85-04 (1985) (deuxieme partie).
9. ^ Rational parameter rays of the Mandelbrot set by Dierk Schleicher
10. ^ Video : The beauty and complexity of the Mandelbrot set by John Hubbard ( see part 3 )
11. ^ 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
12. ^ POLYNOMIAL BASINS OF INFINITY LAURA DEMARCO AND KEVIN M. PILGRIM
13. ^ How to draw external rays by Wolf Jung
14. ^
15. ^ Douady Hubbard Parameter Rays by Linas Vepstas
16. ^ Irwin Jungreis: The uniformization of the complement of the Mandelbrot set. Duke Math. J. Volume 52, Number 4 (1985), 935-938.
17. ^ Adrien Douady, John Hubbard, Etudes dynamique des polynomes complexes I & II, Publ. Math. Orsay. (1984-85) (The Orsay notes)
18. ^ 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,
19. ^ Weisstein, Eric W. "Mandelbrot Set." From MathWorld--A Wolfram Web Resource
20. ^ An algorithm to draw external rays of the Mandelbrot set by Tomoki Kawahira
21. ^ http://www.mrob.com/pub/muency/externalangle.html External angle at Mu-ENCY (the Encyclopedia of the Mandelbrot Set) by Robert Munafo
22. ^ Computation of the external argument by Wolf Jung
23. ^ A. DOUADY, Algorithms for computing angles in the Mandelbrot set (Chaotic Dynamics and Fractals, ed. Barnsley and Demko, Acad. Press, 1986, pp. 155-168).
24. ^ Adrien Douady, John H. Hubbard: Exploring the Mandelbrot set. The Orsay Notes. page 58
25. ^ Exploding the Dark Heart of Chaos by Chris King from Mathematics Department of University of Auckland
26. ^ Topological Dynamics of Entire Functions by Helena Mihaljevic-Brandt
27. ^ Dynamic rays of entire functions and their landing behaviour by Helena Mihaljevic-Brandt
• 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.)
• John Milnor, Dynamics in One Complex Variable, Third Edition, Princeton University Press, 2006, ISBN 0-691-12488-4
• Wolf Jung : Homeomorphisms on Edges of the Mandelbrot Set. Ph.D. thesis of 2002
External ray

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