Английская Википедия:External ray

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Шаблон:Cleanup rewrite 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 <math>K\,</math> 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 <math>K\,</math>.

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 <math>\Psi_c\,</math> be the conformal isomorphism from the complement (exterior) of the closed unit disk <math>\overline{\mathbb{D}}</math> to the complement of the filled Julia set <math>\ K_c </math>.

<math>\Psi_c: \hat{\Complex} \setminus \overline{\mathbb{D}} \to \hat{\Complex} \setminus K_c</math>

where <math>\hat{\Complex}</math> denotes the extended complex plane. Let <math>\Phi_c = \Psi_c^{-1}\,</math> denote the Boettcher map.[14] <math>\Phi_c\,</math> is a uniformizing map of the basin of attraction of infinity, because it conjugates <math>f_c</math> on the complement of the filled Julia set <math>K_c </math> to <math>f_0(z)=z^2</math> on the complement of the unit disk:

<math>\begin{align}

\Phi_c: \hat{\Complex} \setminus K_c &\to \hat{\Complex} \setminus \overline{\mathbb{D}}\\ 

z & \mapsto \lim_{n\to \infty} (f_c^n(z))^{2^{-n}}

\end{align}</math>

and

<math> \Phi_c \circ f_c \circ \Phi_c^{-1} = f_0 </math>

A value <math>w = \Phi_c(z)</math> is called the Boettcher coordinate for a point <math>z \in \hat{\Complex}\setminus K_c</math>.

Formal definition of dynamic ray
Файл:Erays.svg
Polar coordinate system and <math>\psi_c</math> for <math>c=-2</math>

The external ray of angle <math>\theta\,</math> noted as <math>\mathcal{R}^K _{\theta} </math>is:

  • the image under <math>\Psi_c\,</math> of straight lines <math>\mathcal{R}_{\theta} = \{\left(r\cdot e^{2\pi i \theta}\right) : \ r > 1 \}</math>
<math>\mathcal{R}^K _{\theta} = \Psi_c(\mathcal{R}_{\theta})</math>
  • set of points of exterior of filled-in Julia set with the same external angle <math>\theta</math>
<math>\mathcal{R}^K _{\theta} = \{ z\in \hat{\Complex} \setminus K_c : \arg(\Phi_c(z)) = \theta \}</math>
Properties

The external ray for a periodic angle <math>\theta\,</math> satisfies:

<math>f(\mathcal{R}^K _{\theta}) = \mathcal{R}^K _{2 \theta}</math>

and its landing point[15] <math>\gamma_f(\theta) </math> satisfies:

<math>f(\gamma_f(\theta)) = \gamma_f(2\theta) </math>

Parameter plane = c-plane

"Parameter rays are simply the curves that run perpendicular to the equipotential curves of the M-set."[16]

Uniformization
Файл:Jung200.png
Boundary of Mandelbrot set as an image of unit circle under <math>\Psi_M\,</math>
Файл:Jung50e.png
Uniformization of complement (exterior) of Mandelbrot set

Let <math>\Psi_M\,</math> be the mapping from the complement (exterior) of the closed unit disk <math>\overline{\mathbb{D}}</math> to the complement of the Mandelbrot set <math>\ M </math>.[17]

<math>\Psi_M:\mathbb{\hat{C}}\setminus \overline{\mathbb{D}}\to\mathbb{\hat{C}}\setminus M</math>

and Boettcher map (function) <math>\Phi_M\,</math>, which is uniformizing map[18] of complement of Mandelbrot set, because it conjugates complement of the Mandelbrot set <math>\ M </math> and the complement (exterior) of the closed unit disk

<math>\Phi_M: \mathbb{\hat{C}}\setminus M \to \mathbb{\hat{C}}\setminus \overline{\mathbb{D}}</math>

it can be normalized so that :

<math>\frac{\Phi_M(c)}{c} \to 1 \ as\ c \to \infty \,</math>[19]

where :

<math>\mathbb{\hat{C}}</math> denotes the extended complex plane

Jungreis function <math>\Psi_M\,</math> is the inverse of uniformizing map :

<math>\Psi_M = \Phi_{M}^{-1} \,</math>

In the case of complex quadratic polynomial one can compute this map using Laurent series about infinity[20][21]

<math>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} + ...\,</math>

where

<math>c \in \mathbb{\hat{C}}\setminus M</math>
<math>w \in \mathbb{\hat{C}}\setminus \overline{\mathbb{D}}</math>
Formal definition of parameter ray

The external ray of angle <math>\theta\,</math> is:

  • the image under <math>\Psi_c\,</math> of straight lines <math>\mathcal{R}_{\theta} = \{\left(r*e^{2\pi i \theta}\right) : \ r > 1 \}</math>
<math>\mathcal{R}^M _{\theta} = \Psi_M(\mathcal{R}_{\theta})</math>
  • set of points of exterior of Mandelbrot set with the same external angle <math>\theta</math>[22]
<math>\mathcal{R}^M _{\theta} = \{ c\in \mathbb{\hat{C}}\setminus M  : \arg(\Phi_M(c)) = \theta \}</math>
Definition of the Boettcher map

Douady and Hubbard define:

<math>\Phi_M(c) \ \overset{\underset{\mathrm{def}}{}}{=} \ \Phi_c(z=c)\,</math>

so external angle of point <math>c\,</math> of parameter plane is equal to external angle of point <math>z=c\,</math> of dynamical plane

External angle

Angle Шаблон:Mvar is named external angle ( argument ).[23]

Principal value of external angles are measured in turns modulo 1

1 turn = 360 degrees = 2 × Шаблон:Pi radians

Compare different types of angles :

external angle internal angle plain angle
parameter plane <math> \arg(\Phi_M(c)) \,</math> <math> \arg(\rho_n(c)) \,</math> <math> \arg(c) \, </math>
dynamic plane <math> \arg(\Phi_c(z)) \,</math> <math> \arg(z) \,</math>
Computation of external argument
  • argument of Böttcher coordinate as an external argument[24]
    • <math> \arg_M(c) = \arg(\Phi_M(c)) </math>
    • <math> \arg_c(z) = \arg(\Phi_c(z)) </math>
  • 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

Dynamic rays


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.

Parameter plane of the complex exponential family f(z)=exp(z)+c with 8 external ( parameter) rays

Programs that can draw external rays

See also

Шаблон:Commons

References

Шаблон:Reflist

External links

Шаблон:Wikibooks

Партнерские ресурсы
Криптовалюты
Магазины
Хостинг
Разное

  1. ↑ J. Kiwi : Rational rays and critical portraits of complex polynomials. Ph. D. Thesis SUNY at Stony Brook (1997); IMS Preprint #1997/15. Шаблон:Webarchive
  2. ↑ Шаблон:Cite journal
  3. ↑ Шаблон:Cite journal
  4. ↑ Шаблон:Cite arXiv
  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. ↑ Шаблон:Cite journal
  9. ↑ A. Douady, J. Hubbard: Etude dynamique des polynˆomes complexes. Publications math´ematiques d’Orsay 84-02 (1984) (premi`ere partie) and 85-04 (1985) (deuxi`eme partie).
  10. ↑ Шаблон:Cite arXiv
  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. ↑ Tessellation and Lyubich-Minsky laminations associated with quadratic maps I: Pinching semiconjugacies Tomoki Kawahira Шаблон:Webarchive
  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. ↑ Шаблон:Cite journal
  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
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