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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][7]

"The notion of stretching rays is a generalization of that of external rays for the Mandelbrot set to higher degree polynomials."[8]

### landing

Every rational parameter ray of the Mandelbrot set lands at a single parameter.[9][10]

## 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.[13]

##### 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.[14] ${\displaystyle \Phi _{c}\,}$ 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

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[15] ${\displaystyle \gamma _{f}(\theta )}$ 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."[16]

##### Uniformization

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 ${\displaystyle \ M}$.[17]

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

and Boettcher map (function) ${\displaystyle \Phi _{M}\,}$, which is uniformizing map[18] 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 :

${\displaystyle {\frac {\Phi _{M}(c)}{c))\to 1\ as\ c\to \infty \,}$[19]

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[20][21]

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

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 ${\displaystyle \theta }$[22]
${\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 ).[23]

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[24]
• ${\displaystyle \arg _{M}(c)=\arg(\Phi _{M}(c))}$
• ${\displaystyle \arg _{c}(z)=\arg(\Phi _{c}(z))}$
• kneading sequence as a binary expansion of external argument[25][26][27]

### Transcendental maps

For transcendental maps ( for example exponential ) infinity is not a fixed point but an essential singularity and there is no Boettcher isomorphism.[28][29]

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. Archived 2004-11-05 at the Wayback Machine
2. ^ Inou, Hiroyuki; Mukherjee, Sabyasachi (2016). "Non-landing parameter rays of the multicorns". Inventiones Mathematicae. 204 (3): 869–893. arXiv:1406.3428. Bibcode:2016InMat.204..869I. doi:10.1007/s00222-015-0627-3. S2CID 253746781.
3. ^ Atela, Pau (1992). "Bifurcations of dynamic rays in complex polynomials of degree two". Ergodic Theory and Dynamical Systems. 12 (3): 401–423. doi:10.1017/S0143385700006854. S2CID 123478692.
4. ^ Petersen, Carsten L.; Zakeri, Saeed (2020). "Periodic Points and Smooth Rays". arXiv:2009.02788 [math.DS].
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. ^ Stretching rays for cubic polynomials by Pascale Roesch
8. ^ Komori, Yohei; Nakane, Shizuo (2004). "Landing property of stretching rays for real cubic polynomials" (PDF). Conformal Geometry and Dynamics. 8 (4): 87–114. Bibcode:2004CGDAM...8...87K. doi:10.1090/s1088-4173-04-00102-x.
9. ^ 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).
10. ^ Schleicher, Dierk (1997). "Rational parameter rays of the Mandelbrot set". arXiv:math/9711213.
11. ^ Video : The beauty and complexity of the Mandelbrot set by John Hubbard ( see part 3 )
12. ^ 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
13. ^ POLYNOMIAL BASINS OF INFINITY LAURA DEMARCO AND KEVIN M. PILGRIM
14. ^ How to draw external rays by Wolf Jung
15. ^
16. ^ Douady Hubbard Parameter Rays by Linas Vepstas
17. ^ John H. Ewing, Glenn Schober, The area of the Mandelbrot Set
18. ^ Irwin Jungreis: The uniformization of the complement of the Mandelbrot set. Duke Math. J. Volume 52, Number 4 (1985), 935-938.
19. ^ Adrien Douady, John Hubbard, Etudes dynamique des polynomes complexes I & II, Publ. Math. Orsay. (1984-85) (The Orsay notes)
20. ^ Bielefeld, B.; Fisher, Y.; Vonhaeseler, F. (1993). "Computing the Laurent Series of the Map Ψ: C − D → C − M". Advances in Applied Mathematics. 14: 25–38. doi:10.1006/aama.1993.1002.
21. ^ Weisstein, Eric W. "Mandelbrot Set." From MathWorld--A Wolfram Web Resource
22. ^ An algorithm to draw external rays of the Mandelbrot set by Tomoki Kawahira
23. ^ http://www.mrob.com/pub/muency/externalangle.html External angle at Mu-ENCY (the Encyclopedia of the Mandelbrot Set) by Robert Munafo
24. ^ Computation of the external argument by Wolf Jung
25. ^ A. DOUADY, Algorithms for computing angles in the Mandelbrot set (Chaotic Dynamics and Fractals, ed. Barnsley and Demko, Acad. Press, 1986, pp. 155-168).
26. ^ Adrien Douady, John H. Hubbard: Exploring the Mandelbrot set. The Orsay Notes. page 58
27. ^ Exploding the Dark Heart of Chaos by Chris King from Mathematics Department of University of Auckland
28. ^ Topological Dynamics of Entire Functions by Helena Mihaljevic-Brandt
29. ^ Dynamic rays of entire functions and their landing behaviour by Helena Mihaljevic-Brandt