JULIA SETS
AND MANDELBROT SET
For Quadratic Maps
i
2
JULIA SETS
Quadratic Maps
zn+i = f(z„) =	+ c, c = 3tc+ iQc
Xn+l = xl-yl + ^tc,
Уп-\Л. %xnyn ^ЛС,
Fixed points
Fixed point z of the map zn+i = f(zn) is a root of equation
/(«) = z
By analogy with equilibrium: stable, unstable, neutral
Fixed point is
•	attractive if < 1
•	repulsive if |/'(i)| > 1
•	neutral if — 1
= /(г) + (г -
since f(z) = z, then near z the map zn+1 — f(zn) takes a form
zn+1- z = (zn- z)f'(z).
\zn — z\ is a distance to fixed point
з
Cycles
Cycle of period 2 consists of two points G and G
f(Ci) = C2 /(C2) = Ci-
Ci and C2 are fixed points of the map
Zn+1 = /(/(z„))	/(2)(z„).
Proof: let we apply f to /(Ci) = C2 and use /(C2) = Ci, then
/(/(Ci)) = /(C2) = Ci-
Similarily one can show that /(/(G)) = G-
Stable or unstable cycles
/(2)'(С1) = /'(/(C1))/'(C1) = /'(C2)/'(C1) = /(2) '(G)-
Cycle of period n consists of n points G, G? ••• G-
They all are fixed points of the function
/<">(z) = /(...(/(/(*)))...)
4
The case c = 0: f(z) = г2; zn+1 = г2
There are 2 fixed points: z\ = 0 and z^ — 1. The first one is
attractive since f'(zi) = 0, and the second one is repulsive since
/Ш = 2.
We just square the number at each iteration
zo -» z0 -» z0 -» z0 -» ...
There are three possibilities for the sequence, depending on zq
1.	The numbers become smaller and smaller, their sequence ap-
proaches zero. We say that zero is an attractor for the process
z z2. All points less than a distance of 1 from this attractor
are drawn into it
2.	The numbers become larger and larger, tending towards infin-
ity. We say that infinity is also an attractor for this process.
All points farther than a distance of 1 from zero are drawn
into it.
3.	The points are at a distance of 1 from zero and stay there.
Their sequence lies on the boundary between the two domains
of attraction, in this case the unit circle around zero.
|zq I <1 — prisoner set, P
| го | >1 — escape set, E
| го | =1 — Julia set, J
5
Graphical Iteration for
z —»z2
Initial points zg with |zg| < 1
rapidly converge to the origin, while
points with |zb| > I escape to infin-
ity.
6
Julia Set for c = 0
7
The basin of attractive fixed point
c= -0.12375 + 0.56508 i
8
Prisoner Set for
c = —0.5 4- 0.5г
The prisoner set for z —► z2 4- c,
c = —0.5 4- 0.5t is shown in black.
Points outside escape to infinity.
The framed region is enlarged in fig-
ure 13.16.
Figure 13.15

Blowups of Prisoner Set for
c = —0.5 -I- 0.5г
The prisoner set for z —► z2 4- c,
c = -0.54-0.5t from figure 13.15 is
successively enlarged near a bound-
ary point. Each picture (from left to
right) is a computation of the small
framed region in the previous one.
Figure 13.16
Basin of Attractive Fixed
Points
Julia sets that bound the basin of at-
traction of an attractive fixed point
marked by the dot. Note that the
other fixed point is in the Julia set
and is repelling. The parameters
are c = —0.55 — 0.3t (left) and
c = 0.28 4- 0.2t (right).
%п+1 — %n “b
z1 — z + c = 0,
21-2 - 2 ±
9
с = 0.0 + 0i
с = -0.25 + Oi
с = -0.5 +0i
с = -0.7+01
с = -0.75 + Oi
с = -0.8 +0i
с = -0.9 +0i
Figure 14.21 : Starting from the Julia set for с = 0 (the circle) we decrease the parameter to c = — 1. The Julia
set develops a pinching point for c = -0.75 and is the boundary of a period-2 attractor for the remaining plots.
A fractal rabbit, c = —0.12 + 0.74г
The basin of attractive cycle of period 3
io
11
Fig. 3.7: The Julia sets of a. z2 — 1, b. z2 + .25, c. z2 —0.9 + 0.12», andd. z2 — 0.3. The
region shown is — 2 < Re z < 2, — 2 < Im z < 2
The Fatou Dust, c = 0.11031 - 0.67037 г
12
13
Figure 1: с = 0.34522 + 0.08838 г.
Figure 2: с = 0.373239 + 0.1471817 i.
14
Inverse Iteration Methods
Julia set is a repeller with regard to the transformation z —> г2+c.
Therefore it is an attractor with regard to the inverse transforma-
tion.

Nonlinear Chaos Game
ЛИ = +V^~c,
ЛИ =	- c
Pi = P2 = 0.5
Figure 3: Inverse iteration method, c = —0.12375 + 0.56508 г.
15
ч
FRHETHL WHJIJ fRRnn.
MRCM for Julia Set
The MRCM with the two nonlinear
lenses ±y/to — c with c = — 1 is ap-
plied to an initial image consisting
of the sequence of letters ‘FRAC-
TAL’. In each step two deformed
copies of the input image are com-
posed which rapidly converge to the
corresponding Julia set.

Figure 13.35
16

Figure 4: Inverse iteration method (fractal rabbit), c = —0.12 + 0.74г.

Figure 5: Inverse iteration method, c = 0.4.
17
Progress of the Chaos Game
Computation of the Julia Set for
c = 0.12+0.74i (termed the rabbit
by Douady and Hubbard). Although
the performance of the Chaos Game
is in this case not too satisfactory, a
first overview of the Julia set ap-
pears rather rapidly. The top left
image shows 1,000 points, the top
right one shows 10,000. In the bot-
tom left one even 100,000 points of
the Chaos Game are plotted. For the
bottom right image the Modified In-
verse Iteration Method was used. It
requires only 4,750 points.
18
The distribution do not uniformly cover Jc
Fig. 4.5: Julia set and histogram of density distribution forUM
19
Self-Similarity of a Julia Set
Take any small section of the Julia set. Then we apply the iteration
z —> z2 + c to every point in this section. We obtain a new,
typically larger, subset of the Julia set. Iterating this procedure a
finite number of times will result in the complete Julia set! This
says that the immensely complicated global structure of the Julia
set is already contained in any arbitrarily small section of it.
Self-Similarity of a Julia Set
The self-similarity of the Julia sets.
These two pictures show how a very
small section of the Julia set, de-
noted by R-i, is transformed sev-
eral times. In each transformation
the covered portion of the Julia set
indicated by the bold black parts la-
beled R-6 to A_|, increases. After
six iterations the result R-t is al-
ready one half of the Julia set; one
more application of z —» z2 + c
yields the whole set Ro.
20
The Mandelbrot Set: Ordering the Julia Sets
Depending on c there are two possibilities
•	Julia set is connected (one piece)
•	Julia set is totally disconnected (dust)
M = {с e C I Jc is connected}.
The Mandelbrot Set —
Dichotomy of Julia Sets
Any point in the c-plane, interpreted
as a parameter c for the iteration
of z —» z2 + c, corresponds to
a Julia set. The point is colored
black, if the corresponding Julia set
is connected, and white if the set is
disconnected. This is the essence
of Mandelbrot's experiment from
1979.
The Mandelbrot Set — Old
and New Rendering
The insert shows an original print-
out from Mandelbrot’s experiment.
We have produced the large Man-
delbrot set using a modem laser
printer and a more accurate math-
ematical algorithm.
21
The Mandelbrot Set is Connected!
1.2
1.0
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
-2.4	-2.0	-1.6	-1.2	-0.6	-0.2 0 0.2 0.4 0.6 0.6
Two simple definitions of the Mandelbrot set
A point in [c] complex plane belong to the Mandelbrot set if
and only if the corresponding Julia set is connected.
A point in [c] complex plane belong to the Mandelbrot set
if and only if the process of iterations started at the origin
(z$ — 0): 0 —> c —> с2 + c —* (с2 + с)2 + c —» ... does not go
to oo (|zn| < 2).
The Heart of the Mandelbrot Set
cos ф
x = ———
sin<^
22
Figure 14.4 : A connected and a disconnected Julia set.
Journey Around The
Mandelbrot Set
Journey around the Mandelbrot set
with locations of the individual im-
ages being marked on the initial
one.
23
Zoom into the Mandelbrot set
d
24
The Mandelbrot Set and its
Atoms
The buds of the Mandelbrot set cor-
respond to Julia sets that bound
basins of attraction of periodic or-
bits. The numbers in the figure in-
dicate the periods of these orbits.
Two Julia Sets from the Big
Bud of M
Julia sets that bound the basin
of attraction of an attractive cycle
(marked by large dots) of period 2.
Left c = —1 (the super attractive
case), right c = —0.83 + 0.16i.
Two Julia Sets from the Next
Buds of M
Julia sets that bound the basin of at-
traction of an attractive cycle of pe-
riod 3 and 4. Left c = —0.13 +
0.76г, right c = 0.28 + 0.53i.
25
Parabolic Fixed Points
Parabolic Fixed Points
Julia sets corresponding to parabolic
fixed points, c = 0.75 (left) and
c = -0.125 + 0.64925» (right), the
point where the ‘period-three bud* is
attached to the heart-shaped center
of M.
-0.481762- 0.531657 i
c = -1.25
c — 0.27334 + 0.00742 i
26
Siegel disk
Dendrit c = i
27
Misiurewicz Point i
The Mandelbrot set near the Misi-
urewicz point c = i.
Dendrites
Julia sets for c = —2 (left), a line
segment, and c = i (right), a more
typical dendrite.
28
Small Copies of the Mandelbrot Set
»
Miniature М-Set I
Enlargement of a secondary Man-
delbrot set in the upper region of
the Mandelbrot set. The Julia set is
for c = 0.159789 + 1.03332i.
Miniature М-Set II
Enlargement of a secondary Man-
delbrot set in the left region of the
Mandelbrot set (the tip). The Julia
set is for c = —1.77578.
29
Dendrite with beads
Julia set for c-value from secondary Mandelbrot set
Julia set for a c-value from seahorse valley
30
Fig. 4.23: “Image compression” in lhe Mandelbrot set demonstrated for the c-value —0.745429+
0.1130081. The top left shows a close up view of the Mandelbrot set around this c-value. The
vertical size of the window is 0.000060. The Julia set for the c-value is shown in the bottom right
image. The close up view of this Julia set in the bottom left is centered around the same above
c-value. It reveals the same double spiral as in the picture above . The vertical window size is
0.000433, and the Julia set has been rotated by 55° counter clockwise around the center of the
window.
31
Enlargement of Mandelbrot
Set
An enlargement centered at c =
-0.7454285 + 0.1130089г. The
width of the figure is 0.000006.
Figure 14.32
Enlargement of Julia Set
An enlargement for the Julia set for
c = —0.7454285+0.1130089г. The
figure is centered at c and has width
0.000045.
Figure 14.33
32
Asymptotic self-similarity at a point
Figure 14.29
Zoom into the Mandelbrot Set
In these 9 images a zoom into the
boundary of the Mandelbrot set is
shown. The final magnification is
300,000,000 fold.
Zoom into a Julia Set
A parameter c is chosen from the
center of the last image in the previ-
ous figure 14.29. We compute suc-
cessive enlargements centered about
this parameter c as for the Mandel-
brot set in figure 14.29. Note how
similar these Julia set sections are
to the Mandelbrot set closeups. In
the final images the objects are prac-
tically indistinguishable except for
the scale and a rotation.
Figure 14.30
33
Pollution With Small M-Sets
This sequence of enlargements
zooms in on a small copy of the
Mandelbrot set. Its diameter is
about 10~*. At the center of the
spiral in the upper left, there is a
Misiurewicz point.
Figure 14.44
Two Spirals
An enlargement of the Julia set
Jc with c = -0.77568377 +
0.13646737*, a Misiurewicz point,
is shown on the left. The image is
centered at c and has width 0.00036.
Pictured on the right is an enlarge-
ment of the Mandelbrot set at the
same Misiurewicz point (the width
is 0.00048). The double spirals are
almost identical, although the right
one must contain infinitely many
small copies of the Mandelbrot set,
while the left one must not have any
such copies in it (see figure 14.44.
Figure 14.45
34
Escape Time Algorithm
and Equipotentials of the M Set.
Encirclement of the
Mandelbrot Set
The Mandelbrot set M and its ap-
proximation by encirclements Mo
through M-ю.
Figure 14.5
Equipotentials and Field
Lines of Mandelbrot Set
The system of equipotentials and
field lines provides a polar coordi-
nate system for the complement of
the Mandelbrot set.
Figure 14.6
35
Complex Newton Basins
Cayley’s Problem — 1879
Tangent Method for solving the equation f(x) = 0
!/=- ®i) +
/(г„)
36
Solving the Equation z3 = 1 (f(z) = z3 — 1)
z3 — 1
3г2
Figure 6: ’’Newton pie”, zt = 1, z2 = —1/2 + iy/3/2, z3 = —1/2 г'л/3/2.
37
Hubbard Solution — 1977
38
39