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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 : parameter or dynamic
map
bifurcation of dynamic rays
Stretching
landing
= 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:
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
=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."
= landing
=Every rational parameter ray of the Mandelbrot set lands at a single parameter.
Maps
= Polynomials
=Dynamical plane = z-plane
External rays are associated to a compact, full, connected subset
K
{\displaystyle K\,}
of the complex plane as :
the images of radial rays under the Riemann map of the complement of
K
{\displaystyle K\,}
the gradient lines of the Green's function of
K
{\displaystyle 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
{\displaystyle 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
{\displaystyle \Psi _{c}\,}
be the conformal isomorphism from the complement (exterior) of the closed unit disk
D
¯
{\displaystyle {\overline {\mathbb {D} }}}
to the complement of the filled Julia set
K
c
{\displaystyle \ K_{c}}
.
Ψ
c
:
C
^
∖
D
¯
→
C
^
∖
K
c
{\displaystyle \Psi _{c}:{\hat {\mathbb {C} }}\setminus {\overline {\mathbb {D} }}\to {\hat {\mathbb {C} }}\setminus K_{c}}
where
C
^
{\displaystyle {\hat {\mathbb {C} }}}
denotes the extended complex plane.
Let
Φ
c
=
Ψ
c
−
1
{\displaystyle \Phi _{c}=\Psi _{c}^{-1}\,}
denote the Boettcher map.
Φ
c
{\displaystyle \Phi _{c}\,}
is a uniformizing map of the basin of attraction of infinity, because it conjugates
f
c
{\displaystyle f_{c}}
on the complement of the filled Julia set
K
c
{\displaystyle K_{c}}
to
f
0
(
z
)
=
z
2
{\displaystyle f_{0}(z)=z^{2}}
on the complement of the unit disk:
Φ
c
:
C
^
∖
K
c
→
C
^
∖
D
¯
z
↦
lim
n
→
∞
(
f
c
n
(
z
)
)
2
−
n
{\displaystyle {\begin{aligned}\Phi _{c}:{\hat {\mathbb {C} }}\setminus K_{c}&\to {\hat {\mathbb {C} }}\setminus {\overline {\mathbb {D} }}\\z&\mapsto \lim _{n\to \infty }(f_{c}^{n}(z))^{2^{-n}}\end{aligned}}}
and
Φ
c
∘
f
c
∘
Φ
c
−
1
=
f
0
{\displaystyle \Phi _{c}\circ f_{c}\circ \Phi _{c}^{-1}=f_{0}}
A value
w
=
Φ
c
(
z
)
{\displaystyle w=\Phi _{c}(z)}
is called the Boettcher coordinate for a point
z
∈
C
^
∖
K
c
{\displaystyle z\in {\hat {\mathbb {C} }}\setminus K_{c}}
.
= Formal definition of dynamic ray =
The external ray of angle
θ
{\displaystyle \theta \,}
noted as
R
θ
K
{\displaystyle {\mathcal {R}}_{\theta }^{K}}
is:
the image under
Ψ
c
{\displaystyle \Psi _{c}\,}
of straight lines
R
θ
=
{
(
r
⋅
e
2
π
i
θ
)
:
r
>
1
}
{\displaystyle {\mathcal {R}}_{\theta }=\{\left(r\cdot e^{2\pi i\theta }\right):\ r>1\}}
R
θ
K
=
Ψ
c
(
R
θ
)
{\displaystyle {\mathcal {R}}_{\theta }^{K}=\Psi _{c}({\mathcal {R}}_{\theta })}
set of points of exterior of filled-in Julia set with the same external angle
θ
{\displaystyle \theta }
R
θ
K
=
{
z
∈
C
^
∖
K
c
:
arg
(
Φ
c
(
z
)
)
=
θ
}
{\displaystyle {\mathcal {R}}_{\theta }^{K}=\{z\in {\hat {\mathbb {C} }}\setminus K_{c}:\arg(\Phi _{c}(z))=\theta \}}
Properties
The external ray for a periodic angle
θ
{\displaystyle \theta \,}
satisfies:
f
(
R
θ
K
)
=
R
2
θ
K
{\displaystyle f({\mathcal {R}}_{\theta }^{K})={\mathcal {R}}_{2\theta }^{K}}
and its landing point
γ
f
(
θ
)
{\displaystyle \gamma _{f}(\theta )}
satisfies:
f
(
γ
f
(
θ
)
)
=
γ
f
(
2
θ
)
{\displaystyle f(\gamma _{f}(\theta ))=\gamma _{f}(2\theta )}
Parameter plane = c-plane
"Parameter rays are simply the curves that run perpendicular to the equipotential curves of the M-set."
= Uniformization =
Let
Ψ
M
{\displaystyle \Psi _{M}\,}
be the mapping from the complement (exterior) of the closed unit disk
D
¯
{\displaystyle {\overline {\mathbb {D} }}}
to the complement of the Mandelbrot set
M
{\displaystyle \ M}
.
Ψ
M
:
C
^
∖
D
¯
→
C
^
∖
M
{\displaystyle \Psi _{M}:\mathbb {\hat {C}} \setminus {\overline {\mathbb {D} }}\to \mathbb {\hat {C}} \setminus M}
and Boettcher map (function)
Φ
M
{\displaystyle \Phi _{M}\,}
, which is uniformizing map of complement of Mandelbrot set, because it conjugates complement of the Mandelbrot set
M
{\displaystyle \ M}
and the complement (exterior) of the closed unit disk
Φ
M
:
C
^
∖
M
→
C
^
∖
D
¯
{\displaystyle \Phi _{M}:\mathbb {\hat {C}} \setminus M\to \mathbb {\hat {C}} \setminus {\overline {\mathbb {D} }}}
it can be normalized so that :
Φ
M
(
c
)
c
→
1
a
s
c
→
∞
{\displaystyle {\frac {\Phi _{M}(c)}{c}}\to 1\ as\ c\to \infty \,}
where :
C
^
{\displaystyle \mathbb {\hat {C}} }
denotes the extended complex plane
Jungreis function
Ψ
M
{\displaystyle \Psi _{M}\,}
is the inverse of uniformizing map :
Ψ
M
=
Φ
M
−
1
{\displaystyle \Psi _{M}=\Phi _{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
+
.
.
.
{\displaystyle c=\Psi _{M}(w)=w+\sum _{m=0}^{\infty }b_{m}w^{-m}=w-{\frac {1}{2}}+{\frac {1}{8w}}-{\frac {1}{4w^{2}}}+{\frac {15}{128w^{3}}}+...\,}
where
c
∈
C
^
∖
M
{\displaystyle c\in \mathbb {\hat {C}} \setminus M}
w
∈
C
^
∖
D
¯
{\displaystyle w\in \mathbb {\hat {C}} \setminus {\overline {\mathbb {D} }}}
= Formal definition of parameter ray =
The external ray of angle
θ
{\displaystyle \theta \,}
is:
the image under
Ψ
c
{\displaystyle \Psi _{c}\,}
of straight lines
R
θ
=
{
(
r
∗
e
2
π
i
θ
)
:
r
>
1
}
{\displaystyle {\mathcal {R}}_{\theta }=\{\left(r*e^{2\pi i\theta }\right):\ r>1\}}
R
θ
M
=
Ψ
M
(
R
θ
)
{\displaystyle {\mathcal {R}}_{\theta }^{M}=\Psi _{M}({\mathcal {R}}_{\theta })}
set of points of exterior of Mandelbrot set with the same external angle
θ
{\displaystyle \theta }
R
θ
M
=
{
c
∈
C
^
∖
M
:
arg
(
Φ
M
(
c
)
)
=
θ
}
{\displaystyle {\mathcal {R}}_{\theta }^{M}=\{c\in \mathbb {\hat {C}} \setminus M:\arg(\Phi _{M}(c))=\theta \}}
= Definition of the Boettcher map =
Douady and Hubbard define:
Φ
M
(
c
)
=
d
e
f
Φ
c
(
z
=
c
)
{\displaystyle \Phi _{M}(c)\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \Phi _{c}(z=c)\,}
so external angle of point
c
{\displaystyle c\,}
of parameter plane is equal to external angle of point
z
=
c
{\displaystyle 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 × π 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
arg
M
(
c
)
=
arg
(
Φ
M
(
c
)
)
{\displaystyle \arg _{M}(c)=\arg(\Phi _{M}(c))}
arg
c
(
z
)
=
arg
(
Φ
c
(
z
)
)
{\displaystyle \arg _{c}(z)=\arg(\Phi _{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
Images
= Dynamic rays
=unbranched
branched
= 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.
Programs that can draw external rays
Mandel - program by Wolf Jung written in C++ using Qt with source code available under the GNU General Public License
Java applets by Evgeny Demidov ( code of mndlbrot::turn function by Wolf Jung has been ported to Java ) with free source code
ezfract by Michael Sargent, uses the code by Wolf Jung
OTIS by Tomoki KAWAHIRA - Java applet without source code
Spider XView program by Yuval Fisher
YABMP by Prof. Eugene Zaustinsky Archived 2006-06-15 at the Wayback Machine for DOS without source code
DH_Drawer Archived 2008-10-21 at the Wayback Machine by Arnaud Chéritat written for Windows 95 without source code
Linas Vepstas C programs for Linux console with source code
Program Julia by Curtis T. McMullen written in C and Linux commands for C shell console with source code
mjwinq program by Matjaz Erat written in delphi/windows without source code ( For the external rays it uses the methods from quad.c in julia.tar by Curtis T McMullen)
RatioField by Gert Buschmann, for windows with Pascal source code for Dev-Pascal 1.9.2 (with Free Pascal compiler )
Mandelbrot program by Milan Va, written in Delphi with source code
Power MANDELZOOM by Robert Munafo
ruff by Claude Heiland-Allen
See also
external rays of Misiurewicz point
Orbit portrait
Periodic points of complex quadratic mappings
Prouhet-Thue-Morse constant
Carathéodory's theorem
Field lines of Julia sets
References
Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer 1993
Adrien Douady and John H. Hubbard, Etude dynamique des polynômes complexes, Prépublications mathémathiques d'Orsay 2/4 (1984 / 1985)
John W. Milnor, Periodic Orbits, External Rays and the Mandelbrot Set: An Expository Account; Géométrie complexe et systèmes dynamiques (Orsay, 1995), Astérisque No. 261 (2000), 277–333. (First appeared as a Stony Brook IMS Preprint in 1999, available as arXiV:math.DS/9905169.)
John Milnor, Dynamics in One Complex Variable, Third Edition, Princeton University Press, 2006, ISBN 0-691-12488-4
Wolf Jung : Homeomorphisms on Edges of the Mandelbrot Set. Ph.D. thesis of 2002
External links
Hubbard Douady Potential, Field Lines by Inigo Quilez
Intertwined Internal Rays in Julia Sets of Rational Maps by Robert L. Devaney
Extending External Rays Throughout the Julia Sets of Rational Maps by Robert L. Devaney With Figen Cilingir and Elizabeth D. Russell
John Hubbard's presentation, The Beauty and Complexity of the Mandelbrot Set, part 3.1 Archived 2008-02-26 at the Wayback Machine
videos by ImpoliteFruit
Milan Va. "Mandelbrot set drawing". Retrieved 2009-06-15.