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Concept
External ray
Summary
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.
External rays were introduced in Douady and Hubbard's study of the Mandelbrot set
Criteria for classification :
plane : parameter or dynamic
map
bifurcation of dynamic rays
Stretching
landing
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.
Dynamic ray 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 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."
Every rational parameter ray of the Mandelbrot set lands at a single parameter.
External rays are associated to a compact, full, connected subset of the complex plane as :
the images of radial rays under the Riemann map of the complement of
the gradient lines of the Green's function of
field lines of Douady-Hubbard potential
an integral curve of the gradient vector field of the Green's function on neighborhood of infinity
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.
Let be the conformal isomorphism from the complement (exterior) of the closed unit disk to the complement of the filled Julia set .
where denotes the extended complex plane.
Let denote the Boettcher map.
Official source
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