External ray

An external ray is a curve that runs from infinity toward a Julia or Mandelbrot set.J. Kiwi : Rational rays and critical portraits of complex polynomials. Ph. D. Thesis SUNY at Stony Brook (1997); IMS Preprint #1997/15. 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

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 rays can be: Bifurcated, branched, broken 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.Holomorphic Dynamics: On Accumulation of Stretching Rays by Pia B.N. Willumsen, see page 12

Stretching Stretching rays were introduced by Branner and Hubbard:The iteration of cubic polynomials Part I : The global topology of parameter by BODIL BRANNER and JOHN H. HUBBARDStretching rays for cubic polynomials by Pascale Roesch "The notion of stretching rays is a generalization of that of external rays for the Mandelbrot set to higher degree polynomials."

Landing Every rational parameter ray of the Mandelbrot set lands at a single parameter.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).

Maps Polynomials

Dynamical plane = z-plane External rays are associated to a compact, full, connected subset of the complex plane as : the images of radial rays under the Riemann map of the complement of the gradient lines of the Green's function of field lines of Douady-Hubbard potentialVideo : The beauty and complexity of the Mandelbrot set by John Hubbard ( see part 3 ) an integral curve of the gradient vector field of the Green's function on neighborhood of infinityYunping 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

External rays together with equipotential lines of Douady-Hubbard potential ( level sets) form a new polar coordinate system for exterior ( complement ) of .

In other words the external rays define vertical foliation which is orthogonal to horizontal foliation defined by the level sets of potential.POLYNOMIAL BASINS OF INFINITY LAURA DEMARCO AND KEVIN M. PILGRIM

Uniformization Let be the conformal isomorphism from the complement (exterior) of the closed unit disk to the complement of the filled Julia set .

where denotes the extended complex plane. Let denote the Boettcher map.How to draw external rays by Wolf Jung is a uniformizing map of the basin of attraction of infinity, because it conjugates on the complement of the filled Julia set to on the complement of the unit disk:

and

A value is called the Boettcher coordinate for a point .

Formal definition of dynamic ray

The external ray of angle noted as is: the image under of straight lines

set of points of exterior of filled-in Julia set with the same external angle

Properties

The external ray for a periodic angle satisfies:

and its landing point satisfies:

Parameter plane = c-plane "Parameter rays are simply the curves that run perpendicular to the equipotential curves of the M-set."Douady Hubbard Parameter Rays by Linas Vepstas Uniformization

Let be the mapping from the complement (exterior) of the closed unit disk to the complement of the Mandelbrot set .John H. Ewing, Glenn Schober, The area of the Mandelbrot Set

and Boettcher map (function) , which is uniformizing map Irwin Jungreis: The uniformization of the complement of the Mandelbrot set. Duke Math. J. Volume 52, Number 4 (1985), 935-938. of complement of Mandelbrot set, because it conjugates complement of the Mandelbrot set and the complement (exterior) of the closed unit disk

it can be normalized so that :

Adrien Douady, John Hubbard, Etudes dynamique des polynomes complexes I & II, Publ. Math. Orsay. (1984-85) (The Orsay notes)

where : denotes the extended complex plane

Jungreis function is the inverse of uniformizing map :

In the case of complex quadratic polynomial one can compute this map using Laurent series about infinityWeisstein, Eric W. "Mandelbrot Set." From MathWorld--A Wolfram Web Resource

where

Formal definition of parameter ray

The external ray of angle is: the image under of straight lines

set of points of exterior of Mandelbrot set with the same external angle An algorithm to draw external rays of the Mandelbrot set by Tomoki Kawahira

Definition of the Boettcher map

Douady and Hubbard define:

so external angle of point of parameter plane is equal to external angle of point of dynamical plane

External angle

Angle is named external angle ( argument ).http://www.mrob.com/pub/muency/externalangle.html External angle at Mu-ENCY (the Encyclopedia of the Mandelbrot Set) by Robert Munafo

Principal value of external angles are measured in turns modulo 1

1 turn = 360 degrees = 2 × radians

Compare different types of angles : external ( point of set's exterior ) internal ( point of component's interior ) plain ( argument of complex number )

external angle internal angle plain angle parameter plane dynamic plane

Computation of external argument argument of Böttcher coordinate as an external argumentComputation of the external argument by Wolf Jung

kneading sequence as a binary expansion of external argumentA. DOUADY, Algorithms for computing angles in the Mandelbrot set (Chaotic Dynamics and Fractals, ed. Barnsley and Demko, Acad. Press, 1986, pp. 155-168).Adrien Douady, John H. Hubbard: Exploring the Mandelbrot set. The Orsay Notes. page 58 Exploding the Dark Heart of Chaos by Chris King from Mathematics Department of University of Auckland

Transcendental maps

For transcendental maps ( for example exponential ) infinity is not a fixed point but an essential singularity and there is no Boettcher isomorphism.Topological Dynamics of Entire Functions by Helena Mihaljevic-BrandtDynamic rays of entire functions and their landing behaviour by Helena Mihaljevic-Brandt

Here dynamic ray is defined as a curve : connecting a point in an escaping set and infinity lying in an escaping set

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.

Programs that can draw external rays Mandel - program by Wolf Jung written in C++ using Qt with source code available under the GNU General Public License Java applets by Evgeny Demidov ( code of mndlbrot::turn function by Wolf Jung has been ported to Java ) with free source code ezfract by Michael Sargent, uses the code by Wolf Jung OTIS by Tomoki KAWAHIRA - Java applet without source code Spider XView program by Yuval Fisher YABMP by Prof. Eugene Zaustinsky for MS-DOS without source code DH_Drawer by Arnaud Chéritat written for Windows 95 without source code Linas Vepstas C programs for Linux console with source code Program Julia by Curtis T. McMullen written in C and Linux commands for C shell console with source code mjwinq program by Matjaz Erat written in delphi/windows without source code ( For the external rays it uses the methods from quad.c in julia.tar by Curtis T McMullen) RatioField by Gert Buschmann, for windows with Pascal source code for Dev-Pascal 1.9.2 (with Free Pascal compiler ) Mandelbrot program by Milan Va, written in Delphi with source code Power MANDELZOOM by Robert Munafo ruff by Claude Heiland-Allen

External ray

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