External ray

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.