External ray
An external ray is a curve that runs from infinity toward a Julia or Mandelbrot set.
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 Julia sets on dynamical plane are often called dynamic rays.External rays of the Mandelbrot set on parameter plane are called parameter rays.
Bifurcation
Dynamic rays can be:- Bifurcated, branched, broken
- Smooth, unbranched, unbroken
Stretching
Stretching rays were introduced by Branner and Hubbard: "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.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 potential
- an integral curve of the gradient vector field of the Green's function on neighborhood of infinity
In other words the external rays define vertical foliation which is orthogonal to horizontal foliation defined by the level sets of potential.
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.
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
[Image:Erays.svg|right|thumb|Polar coordinate system and for ]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
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."Uniformization
Let be the mapping from the complement (exterior) of the closed unit disk to the complement of the Mandelbrot set .and Boettcher map, which is uniformizing map 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 :
where :
Jungreis function is the inverse of uniformizing map :
In the case of complex quadratic polynomial one can compute this map using Laurent series about infinity
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
Definition of the Boettcher map
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.Principal value of external angles are measured in turns modulo 1
Compare different types of angles :
- external
- internal
- plain
| external angle | internal angle | plain angle | |
| parameter plane | |||
| dynamic plane |
Computation of external argument
- argument of Böttcher coordinate as an external argument
- *
- *
- kneading sequence as a binary expansion of external argument
Transcendental maps
For transcendental maps infinity is not a fixed point but an essential singularity and there is no Boettcher isomorphism.Here dynamic ray is defined as a curve :
- connecting a point in an escaping set and infinity
- lying in an escaping set
Images
Parameter rays
Mandelbrot set for complex quadratic polynomial with parameter rays of root pointsParameter 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
- - program by Wolf Jung written in C++ using Qt with source code available under the GNU General Public License
- * by Evgeny Demidov with free source code
- *, uses the code by Wolf Jung
- - Java applet without source code
- for MS-DOS without source code
- by Arnaud Chéritat written for Windows 95 without source code
- for Linux console with source code
- by Curtis T. McMullen written in C and Linux commands for C shell console with source code
- written in delphi/windows without source code
- , for windows with Pascal source code for
- Mandelbrot program by Milan Va, written in Delphi with source code