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 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 Julia sets on dynamical plane are often called dynamic rays.
External rays of the Mandelbrot set on parameter plane are called parameter rays.

Polynomials

Dynamical plane = z-plane

External rays are associated to a compact, full, connected subset of the complex plane as :
External rays together with equipotential lines of Douady-Hubbard potential form a new polar coordinate system for exterior of.
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 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

The external ray of angle noted as is:
The external ray for a periodic angle satisfies:
and its landing point satisfies:

Parameter plane = c-plane

Uniformization

Let be the mapping from the complement 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 complement of the Mandelbrot set and the complement 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:
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.
Principal value of external angles are measured in turns modulo 1
Compare different types of angles :
external angleinternal angleplain angle
parameter plane
dynamic plane

Computation of external argument

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 :

Dynamic rays

Parameter rays

for complex quadratic polynomial with parameter rays of root points
Parameter space of the complex exponential family f=exp+c. Eight parameter rays landing at this parameter are drawn in black.

Programs that can draw external rays

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