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Wikipedia / 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 (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.

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 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.

In other words the external rays define vertical foliation which is orthogonal to horizontal foliation defined by the level sets of potential.

Uniformization

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

denotes the extended complex plane denote the Boettcher map. is a uniformizing on the complement of the unit disk:

and

is called the Boettcher coordinate.

Formal definition of dynamic ray

The external ray is:

Properties

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."

Uniformization

be the mapping from the complement (exterior) of the closed unit disk to the complement of the Mandelbrot set.

, 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 :

  1. denotes the extended complex plane

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 is:

Definition of the Boettcher map

Douady and Hubbard define:

of dynamical plane

External angle

Angle is named external angle ( argument ).

Principal value of external angles are measured in turns modulo 1

  1. 1 turn = 360 degrees = 2 radians

Compare different types of angles :

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 ( for example exponential ) infinity is not a fixed point but an essential singularity and there is no Boettcher isomorphism.

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.

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

See also

References