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

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 ${\displaystyle K\,}$ 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.[4]

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[5]. ${\displaystyle \Phi _{c}\,}$ 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[6] ${\displaystyle \gamma _{f}(\theta )}$ satisfies:

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

Parameter plane = c-plane

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[7] 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 :

${\displaystyle {\frac {\Phi _{M}(c)}{c}}\to 1\ as\ c\to \infty \,}$[8]

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[9][10]

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

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 ${\displaystyle \theta }$[11]
${\displaystyle {\mathcal {R}}_{\theta }^{M}=\{c\in \mathbb {\hat {C}} \setminus M:\arg(\Phi _{M}(c))=\theta \}}$

Definition of ${\displaystyle \Phi _{M}\,}$

Douady and Hubbard define:

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

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 ).[12]

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[13]
• ${\displaystyle \arg _{M}(c)=\arg(\Phi _{M}(c))}$
• ${\displaystyle \arg _{c}(z)=\arg(\Phi _{c}(z))}$
• kneading sequence as a binary expansion of external argument[14][15][16]

Transcendental maps

For transcendental maps ( for example exponential ) infinity is not a fixed point but an essential singularity and there is no Boettcher isomorphism.[17][18]

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. ^ Video : The beauty and complexity of the Mandelbrot set by John Hubbard ( see part 3 )
3. ^ 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
4. ^ POLYNOMIAL BASINS OF INFINITY LAURA DEMARCO AND KEVIN M. PILGRIM
5. ^ How to draw external rays by Wolf Jung
6. ^
7. ^ Irwin Jungreis: The uniformization of the complement of the Mandelbrot set. Duke Math. J. Volume 52, Number 4 (1985), 935-938.
8. ^ Adrien Douady, John Hubbard, Etudes dynamique des polynomes complexes I & II, Publ. Math. Orsay. (1984-85) (The Orsay notes)
9. ^ 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,
10. ^ Weisstein, Eric W. "Mandelbrot Set." From MathWorld--A Wolfram Web Resource
11. ^ An algorithm to draw external rays of the Mandelbrot set by Tomoki Kawahira
12. ^ http://www.mrob.com/pub/muency/externalangle.html External angle at Mu-ENCY (the Encyclopedia of the Mandelbrot Set) by Robert Munafo
13. ^ Computation of the external argument by Wolf Jung
14. ^ A. DOUADY, Algorithms for computing angles in the Mandelbrot set (Chaotic Dynamics and Fractals, ed. Barnsley and Demko, Acad. Press, 1986, pp. 155-168).
15. ^ Adrien Douady, John H. Hubbard: Exploring the Mandelbrot set. The Orsay Notes. page 58
16. ^ Exploding the Dark Heart of Chaos by Chris King from Mathematics Department of University of Auckland
17. ^ Topological Dynamics of Entire Functions by Helena Mihaljevic-Brandt
18. ^ 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