External ray

An external ray is a curve that runs from infinity toward a Julia or Mandelbrot set.

External rays were introduced in Douady and Hubbard's study of the Mandelbrot set Criteria for classification : 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.

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.

[5] Stretching rays were introduced by Branner and Hubbard:[6][7] "The notion of stretching rays is a generalization of that of external rays for the Mandelbrot set to higher degree polynomials.

[9][10] 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 ( 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.

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.

is a uniformizing map of the basin of attraction of infinity, because it conjugates

is called the Boettcher coordinate for a point

is: The external ray for a periodic angle

satisfies: "Parameter rays are simply the curves that run perpendicular to the equipotential curves of the M-set.

be the mapping from the complement (exterior) of the closed unit disk

, which is uniformizing map[18] 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 :

is the inverse of uniformizing map : In the case of complex quadratic polynomial one can compute this map using Laurent series about infinity[20][21] where The external ray of angle

of parameter plane is equal to external angle of point

[23] Principal value of external angles are measured in turns modulo 1 Compare different types of angles : For transcendental maps ( for example exponential ) infinity is not a fixed point but an essential singularity and there is no Boettcher isomorphism.

[28][29] Here dynamic ray is defined as a curve :

Mandelbrot set for complex quadratic polynomial with parameter rays of root points Parameter space of the complex exponential family f(z)=exp(z)+c.