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The filled-in Julia set
K
(
f
)
{\displaystyle K(f)}
of a polynomial
f
{\displaystyle f}
is a
Julia set and its
interior ,
non-escaping set .
Formal definition
The filled-in
Julia set
K
(
f
)
{\displaystyle K(f)}
of a polynomial
f
{\displaystyle f}
is defined as the set of all points
z
{\displaystyle z}
of the dynamical plane that have
bounded
orbit with respect to
f
{\displaystyle f}
K
(
f
)
=
d
e
f
{
z
∈
C
:
f
(
k
)
(
z
)
↛
∞
as
k
→
∞
}
{\displaystyle K(f){\overset {\mathrm {def} }{{}={}}}\left\{z\in \mathbb {C} :f^{(k)}(z)\not \to \infty ~{\text{as}}~k\to \infty \right\}}
where:
Relation to the Fatou set
The filled-in Julia set is the
(absolute) complement of the
attractive basin of
infinity .
K
(
f
)
=
C
∖
A
f
(
∞
)
{\displaystyle K(f)=\mathbb {C} \setminus A_{f}(\infty )}
The
attractive basin of
infinity is one of the
components of the Fatou set .
A
f
(
∞
)
=
F
∞
{\displaystyle A_{f}(\infty )=F_{\infty }}
In other words, the filled-in Julia set is the
complement of the unbounded
Fatou component :
K
(
f
)
=
F
∞
C
.
{\displaystyle K(f)=F_{\infty }^{C}.}
Relation between Julia, filled-in Julia set and attractive basin of infinity
Wikibooks has a book on the topic of:
Fractals
The
Julia set is the common
boundary of the filled-in Julia set and the
attractive basin of
infinity
J
(
f
)
=
∂
K
(
f
)
=
∂
A
f
(
∞
)
{\displaystyle J(f)=\partial K(f)=\partial A_{f}(\infty )}
where:
A
f
(
∞
)
{\displaystyle A_{f}(\infty )}
denotes the
attractive basin of
infinity = exterior of filled-in Julia set = set of escaping points for
f
{\displaystyle f}
A
f
(
∞
)
=
d
e
f
{
z
∈
C
:
f
(
k
)
(
z
)
→
∞
a
s
k
→
∞
}
.
{\displaystyle A_{f}(\infty )\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \{z\in \mathbb {C} :f^{(k)}(z)\to \infty \ as\ k\to \infty \}.}
If the filled-in Julia set has no
interior then the
Julia set coincides with the filled-in Julia set. This happens when all the critical points of
f
{\displaystyle f}
are pre-periodic. Such critical points are often called
Misiurewicz points .
Spine
The most studied polynomials are probably
those of the form
f
(
z
)
=
z
2
+
c
{\displaystyle f(z)=z^{2}+c}
, which are often denoted by
f
c
{\displaystyle f_{c}}
, where
c
{\displaystyle c}
is any complex number. In this case, the spine
S
c
{\displaystyle S_{c}}
of the filled Julia set
K
{\displaystyle K}
is defined as
arc between
β
{\displaystyle \beta }
-fixed point and
−
β
{\displaystyle -\beta }
,
S
c
=
−
β
,
β
{\displaystyle S_{c}=\left[-\beta ,\beta \right]}
with such properties:
spine lies inside
K
{\displaystyle K}
.
[1] This makes sense when
K
{\displaystyle K}
is connected and full
[2]
spine is invariant under 180 degree rotation,
spine is a finite topological tree,
Critical point
z
c
r
=
0
{\displaystyle z_{cr}=0}
always belongs to the spine.
[3]
β
{\displaystyle \beta }
-fixed point is a landing point of
external ray of angle zero
R
0
K
{\displaystyle {\mathcal {R}}_{0}^{K}}
,
−
β
{\displaystyle -\beta }
is landing point of
external ray
R
1
/
2
K
{\displaystyle {\mathcal {R}}_{1/2}^{K}}
.
Algorithms for constructing the spine:
detailed version is described by A. Douady
[4]
Simplified version of algorithm:
connect
−
β
{\displaystyle -\beta }
and
β
{\displaystyle \beta }
within
K
{\displaystyle K}
by an arc,
when
K
{\displaystyle K}
has empty interior then arc is unique,
otherwise take the shortest way that contains
0
{\displaystyle 0}
.
[5]
Curve
R
{\displaystyle R}
:
R
=
d
e
f
R
1
/
2
∪
S
c
∪
R
0
{\displaystyle R{\overset {\mathrm {def} }{{}={}}}R_{1/2}\cup S_{c}\cup R_{0}}
divides dynamical plane into two components.
Images
Filled Julia set for f
c , c=1−φ=−0.618033988749…, where φ is the
Golden ratio
Filled Julia with no interior = Julia set. It is for c=i.
Filled Julia set for c=−1+0.1*i. Here Julia set is the boundary of filled-in Julia set.
Filled Julia set for c = −0.8 + 0.156i.
Filled Julia set for c = 0.285 + 0.01i.
Filled Julia set for c = −1.476.
Names
Notes
^
Douglas C. Ravenel : External angles in the Mandelbrot set: the work of Douady and Hubbard. University of Rochester
Archived 2012-02-08 at the
Wayback Machine
^
John Milnor : Pasting Together Julia Sets: A Worked Out Example of Mating. Experimental Mathematics Volume 13 (2004)
^
Saaed Zakeri: Biaccessiblility in quadratic Julia sets I: The locally-connected case
^ A. Douady, “Algorithms for computing angles in the Mandelbrot set,” in Chaotic Dynamics and Fractals, M. Barnsley and S. G. Demko, Eds., vol. 2 of Notes and Reports in Mathematics in Science and Engineering, pp. 155–168, Academic Press, Atlanta, Georgia, USA, 1986.
^
K M. Brucks , H Bruin :
Topics from One-Dimensional Dynamics Series: London Mathematical Society Student Texts (No. 62) page 257
^
The Mandelbrot Set And Its Associated Julia Sets by Hermann Karcher
References
Peitgen Heinz-Otto, Richter, P.H. : The beauty of fractals: Images of Complex Dynamical Systems. Springer-Verlag 1986.
ISBN
978-0-387-15851-8 .
Bodil Branner : Holomorphic dynamical systems in the complex plane. Department of Mathematics Technical University of Denmark,
MAT-Report no. 1996-42 .