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.
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 no external ray braches. When the Julia set is not connected then some external rays brach
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." 
Maps.
Polynomials.
Dynamical plane = z-plane.
External rays are associated to a compact, full, connected subset formula_1 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 formula_1.
In other words the external rays define vertical foliation which is orthogonal to horizontal foliation defined by the level sets of potential.
Uniformization.
Let formula_5 be the conformal isomorphism from the complement (exterior) of the closed unit disk formula_6 to the complement of the filled Julia set formula_7.
where formula_9 denotes the extended complex plane.
Let formula_10 denote the Boettcher map.
formula_11 is a uniformizing map of the basin of attraction of infinity, because it conjugates formula_12 on the complement of the filled Julia set formula_13 to formula_14 on the complement of the unit disk:
and 
A value formula_17 is called the Boettcher coordinate for a point formula_18.
Formal definition of dynamic ray.
The external ray of angle formula_19 noted as formula_20is:
Properties.
The external ray for a periodic angle formula_19 satisfies:
and its landing point formula_28 satisfies:
Parameter plane = c-plane.
"Parameter rays are simply the curves that run perpendicular to the equipotential curves of the M-set."
Uniformization.
Let formula_30 be the mapping from the complement (exterior) of the closed unit disk formula_6 to the complement of the Mandelbrot set formula_32.
and Boettcher map (function) formula_34, which is uniformizing map of complement of Mandelbrot set, because it conjugates complement of the Mandelbrot set formula_32 and the complement (exterior) of the closed unit disk
it can be normalized so that :
formula_37
where :
Jungreis function formula_30 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 formula_19 is:
Definition of formula_50.
Douady and Hubbard define:
formula_51
so external angle of point formula_52 of parameter plane is equal to external angle of point formula_53 of dynamical plane
External angle.
Angle is named external angle ( argument ).
Principal value of external angles are measured in turns modulo 1
Compare different types of angles :
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.
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.