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

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

Uniformization

Let \Psi_c, be the conformal isomorphism from the complement (exterior) of the closed unit disk to the complement of the filled Julia set \ K_c. where denotes the extended complex plane. Let denote the Boettcher map. \Phi_c, is a uniformizing map of the basin of attraction of infinity, because it conjugates f_c on the complement of the filled Julia set K_c to f_0(z)=z^2 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 \theta, noted as is:

Properties

The external ray for a periodic angle \theta, 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

Let \Psi_M, be the mapping from the complement (exterior) of the closed unit disk to the complement of the Mandelbrot set \ M. and Boettcher map (function) \Phi_M,, which is uniformizing map of complement of Mandelbrot set, because it conjugates complement of the Mandelbrot set \ M and the complement (exterior) of the closed unit disk it can be normalized so that : where : Jungreis function \Psi_M, 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 \theta, is:

Definition of the Boettcher map

Douady and Hubbard define: so external angle of point c, of parameter plane is equal to external angle of point z=c, of dynamical plane

External angle

Angle θ is named external angle ( argument ). Principal value of external angles are measured in turns modulo 1 Compare different types of angles :

Computation 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