An external ray is a curve that runs from infinity toward a Julia or Mandelbrot set.[1] 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 :
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:
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."[8]
Every rational parameter ray of the Mandelbrot set lands at a single parameter.[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.[13]
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.[14] 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 .
The external ray of angle noted as is:
The external ray for a periodic angle satisfies:
and its landing point[15] satisfies:
"Parameter rays are simply the curves that run perpendicular to the equipotential curves of the M-set."[16]
Let be the mapping from the complement (exterior) of the closed unit disk to the complement of the Mandelbrot set .[17]
and Boettcher map (function) , 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 :
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[20][21]
where
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
collecting bits outwards
Binary decomposition of unrolled circle plane
binary decomposition of dynamic plane for f(z) = z^2
Angle θ is named external angle ( argument ).[23]
Principal value of external angles are measured in turns modulo 1
Compare different types of angles :
external angle | internal angle | plain angle | |
---|---|---|---|
parameter plane | |||
dynamic plane |
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 :
Julia set for with 2 external ray landing on repelling fixed point alpha
Julia set and 3 external rays landing on fixed point α c {\displaystyle \alpha _{c}\,}
Dynamic external rays landing on repelling period 3 cycle and 3 internal rays landing on fixed point
Julia set with external rays landing on period 3 orbit
Rays landing on parabolic fixed point for periods 2-40
Branched dynamic ray
Mandelbrot set for complex quadratic polynomial with parameter rays of root points
External rays for angles of the form : n / ( 21 - 1) (0/1; 1/1) landing on the point c= 1/4, which is cusp of main cardioid ( period 1 component)
External rays for angles of the form : n / ( 22 - 1) (1/3, 2/3) landing on the point c= - 3/4, which is root point of period 2 component
External rays for angles of the form : n / ( 23 - 1) (1/7,2/7) (3/7,4/7) landing on the point c= -1.75 = -7/4 (5/7,6/7) landing on the root points of period 3 components.
External rays for angles of form : n / ( 24 - 1) (1/15,2/15) (3/15, 4/15) (6/15, 9/15) landing on the root point c= -5/4 (7/15, 8/15) (11/15,12/15) (13/15, 14/15) landing on the root points of period 4 components.
External rays for angles of form : n / ( 25 - 1) landing on the root points of period 5 components
internal ray of main cardioid of angle 1/3: starts from center of main cardioid c=0, ends in the root point of period 3 component, which is the landing point of parameter (external) rays of angles 1/7 and 2/7
Internal ray for angle 1/3 of main cardioid made by conformal map from unit circle
Mini Mandelbrot set with period 134 and 2 external rays
Wakes near the period 3 island
Wakes along the main antenna
Parameter space of the complex exponential family f(z)=exp(z)+c. Eight parameter rays landing at this parameter are drawn in black.