Kalpana Kalpana (Editor)

External ray

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit
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.

Contents

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

Notation

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.

Dynamical plane = z-plane

External rays are associated to a compact, full, connected subset K of the complex plane as :

  • the images of radial rays under the Riemann map of the complement of K
  • the gradient lines of the Green's function of K
  • 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 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 Ψ c be the mapping from the complement (exterior) of the closed unit disk D ¯ to the complement of the filled Julia set   K c .

    Ψ c : C ^ ∖ D ¯ → C ^ ∖ K c

    and Boettcher map(function) Φ c , which is uniformizing map of basin of attraction of infinity, because it conjugates complement of the filled Julia set   K c and the complement (exterior) of the closed unit disk

    Φ c : C ^ ∖ K c → C ^ ∖ D ¯

    where :

    C ^ denotes the extended complex plane

    Boettcher map Φ c is an isomorphism :

    Ψ c = Φ c − 1

    w = Φ c ( z ) = lim n → ∞ ( f c n ( z ) ) 2 − n

    where :

    z ∈ C ^ ∖ K c

    w ∈ C ^ ∖ D ¯

    w is a Boettcher coordinate

    Formal definition of dynamic ray

    The external ray of angle θ noted as R θ K is:

  • the image under Ψ c of straight lines R θ = { ( r ∗ e 2 π i θ ) :   r > 1 }
    • R θ K = Ψ c ( R θ )
  • set of points of exterior of filled-in Julia set with the same external angle θ
    • R θ K = { z ∈ C ^ ∖ K c : arg ⁡ ( Φ c ( z ) ) = θ }

    Properties

    External ray for periodic angle θ satisfies :

    f ( R θ K ) = R 2 θ K

    and its landing point γ f ( θ ) )  :

    f ( γ f ( θ ) ) = γ f ( 2 θ )

    Uniformization

    Let Ψ M be the mapping from the complement (exterior) of the closed unit disk D ¯ to the complement of the Mandelbrot set   M .

    Ψ M : C ^ ∖ D ¯ → C ^ ∖ M

    and Boettcher map (function) Φ 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

    Φ M : C ^ ∖ M → C ^ ∖ D ¯

    it can be normalized so that :

    Φ M ( c ) c → 1   a s   c → ∞

    where :

    C ^ denotes the extended complex plane

    Jungreis function Ψ M is the inverse of uniformizing map :

    Ψ M = Φ M − 1

    In the case of complex quadratic polynomial one can compute this map using Laurent series about infinity

    c = Ψ M ( w ) = w + ∑ m = 0 ∞ b m w − m = w − 1 2 + 1 8 w − 1 4 w 2 + 15 128 w 3 + . . .

    where

    c ∈ C ^ ∖ M w ∈ C ^ ∖ D ¯

    Formal definition of parameter ray

    The external ray of angle θ is:

  • the image under Ψ c of straight lines R θ = { ( r ∗ e 2 π i θ ) :   r > 1 }
    • R θ M = Ψ M ( R θ )
  • set of points of exterior of Mandelbrot set with the same external angle θ
    • R θ M = { c ∈ C ^ ∖ M : arg ⁡ ( Φ M ( c ) ) = θ }

    Definition of Φ M {displaystyle Phi _{M},}

    Douady and Hubbard define:

    Φ M ( c )   = d e f   Φ c ( z = c )

    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

    1 turn = 360 degrees = 2 * Pi radians

    Compare different types of angles :

  • external ( point of set's exterior )
  • internal ( point of component's interior )
  • plain ( argument of complex number )

    • Computation of external argument

  • argument of Böttcher coordinate as an external argument
  • a r g M ( c ) = a r g ( Φ M ( c ) )
  • a r g c ( z ) = a r g ( Φ c ( z ) )
  • kneading sequence as a binary expansion 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 :

  • connecting a point in an escaping set and infinity
  • lying in an escaping set

    • 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.

    References

    External ray Wikipedia
    (Text) CC BY-SA


    Similar Topics