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'The Paths of Lovers Cross in the Line of Duty.'
THE THEORY BEHIND 'THE CELL AND THE WOMB'
Julia Sets
Copyright (C) 1989 by Homer Wilson Smith
All Rights Reserved
A Julia Set is a closed (connected) boundary in the Z plane that
separates the Z's that go off to infinity under forward iteration and
the Z's that go inward towards a single or multiple cycle fixed
points. In rare cases Z can forward orbit into a chaotic cycle called
a Siegel disk. Points directly on the Julia Set (the boundary)
forward iterate to other points directly on the Julia Set.
Each C taken from the Mandelbrot Set has a characteristic Julia
Set. For C's taken from outside the M Set, the Julia sets are open
(dusty) and are called Cantor Sets. In this case the inside and outside
of the Cantor sets are contiguous and all starting Z's go to infinity
except those directly on the Cantor Set which still go to other points
on the Cantor Set.
In case anyone wonders how people get the names for these things,
Mandelbrot, Julia and Cantor were all people. Just as were Hertz,
Volt, Ampere, Ohm, Coulomb, Faraday and Gauss.
Thus for each Julia Set there are 3 regions of interest each
having its own fixed point. A fixed point is a starting Z that
forward iterates directly to itself. For Z = Z*Z+ C, Since every Z
has TWO points that forward iterate TO it, (every Z has two backward
images), a fixed point also has two backward images one of which is
itself, and the other is another point somewhere on the Z plane.
The first region of the Julia Set is OUTSIDE where Z's go off to
infinity. Infinity is an attractive fixed point because it attracts
under forward iteration all Z's in its general area. It is a fixed
point because INFINITY**2 + C is infinity. It's backward images are
plus and minus infinity both of which go to infinity under forward
iteration.
The second region of the Julia Set is ON the Julia Set where Z's
forward iterate to other points ON the Julia Set. For this region you
find the fixed points by solving the quadratic equation:
Z = Z*Z + C or 0 = Z*Z - Z + C.
Using the quadratic formula this becomes
Z = (1 + SQRT(1 - 4*C))/2 or Z = (1 - SQRT(1 - 4*C))/2
For the 1 cycle case (C chosen from the main cardioid of the M set)
one of these fixed points is ON the Julia Set, and the other is INSIDE
the Julia Set.
The fixed point ON the Julia is a REPULSIVE fixed point as it
repels all Z's away from it except those ON the Julia set, (which
forward interate to other points on the Julia set.)
Take for example the number 1 on the real axis. 1*1 is 1. So 1
is a fixed point. But 1.1*1.1 is 1.21 which is further away. And
.9*.9 is .81 which is also further away in the opposite direction.
When considering the complex plane the number 1 becomes the
entire Julia Set. Points ON the Julia Set iterate to points ON the
Julia Set just as 1 iterates to itself. Points INSIDE the Julia Set
iterate to points further inside, and points OUTSIDE the Julia Set
iterate to points further outside. This is repulsive behavior. The
Julia Set is basically a repulsive item, attractive only to itself.
This is very much like a mountain range where a marble balanced
on the top stays put, but off to any side starts to roll down the
mountain way from the top. Repulsiveness is the mark of unstability.
Attractiveness is the mark of stability. The second fixed point
inside the Julia Set is attractive and sits inside something that is
very much like a valley or a basin, in fact the area immediately
surrounding this attractive fixed point is called a basin of attraction.
The marble when placed in the center stays put, and when put up the side
of the slope a bit, rolls right back to the center.
Slight perturbations to the marble on the top of the mountain
will cause it to loose its position entirely to which it will never
return. This is unstability. Slight perturbations to the marble in
the basin of the valley will cause the marble to settle back down to
where it was. This is stability.
In summary therefore let's consider the case of C = 0 for only
the real number line using Z = Z*Z + C.
Z = 0 is an attractive fixed point because 0*0 + 0 = 0 and all
points near by are attracted to it. Infinity is an attractive fixed
point because INFINITY*INFINITY + 0 = INFINITY and also attracts all
points near by. 1 is a repulsive fixed point because 1*1 + 0 = 1 and
repels all points near by to either 0 or infinity.
If C is chosen from one of the other balls on the Mandelbrot Set
that are NOT in the main cardioid, then the forward orbits of Z have a
period cycle greater than 1. For example if C = -1 and Z starts at 0
then 0*0 - 1 = -1, and -1*-1 - 1 = 0. Thus every TWO iterates Z comes
back to where it stated. Observing this on the Julia plane, we see
that there are two fixed points inside the Julia Set, namely 0 and -1,
and any Z starting off inside the Julia set will forward orbit towards
BOTH of them alternately.
Therefore in general every C taken from the Mandelbrot Set has
associated with it a compete Julia Set, and each Julia Set (if closed)
will have an attractive set of one or more 'fixed points' inside it that
iterates will go to if they start off inside the Julia Set. If iterates
start of outside the Julia Sets they will go to infinity and of course
this is true for all Julia sets whether open or closed.
In order to find these multi period fixed points inside a closed
Julia Set you first must know what period you are looking for. This is
determined by knowing which ball of the Mandelbrot Set you have taken
your value of C from.
Once you know the expected period of the Julia Set you can easily
find the exact values of the period points themselves.
Take for example C from the 2 ball of the M set where the cycle is
2, then you find the equation that is equivalent to 2 iterations of the
primary equation.
Since,
F(Z) = Z**2 + C
it must follow that two iterates of this is
F( F(Z) ) = (Z**2 + C)**2 + C
= Z**4 + 2*C*Z*Z + C*C + C
Remembering that a fixed point is a value of Z that iterates right
back to itself after (in this case) two iterates we can write the
required equation as follows.
Z = Z**4 + 2*C*Z*Z + C*C + C
which is the same as
0 = Z**4 + 2*C*Z*Z - Z + C*C + C
Being a 4th degree equation this has 4 answers. Two of these
answers are the previously discussed ONE cycle fixed points found in the
earlier discussion. This is because a one cycle fixed point is also a
two cycle fixed point. It comes back to itself after one cycle so it
certainly comes back to itself after two cycles!
The other two answers are the 2 cycle fixed points that lie INSIDE
the Julia Set. They return to themselves after two cycles. Thus we
call them 2 cycle fixed points.
Each ball on the Mandelbrot Set has its own cycle count, and
distinctive Julia pattern. The Julia Set is the boundary of the basins
of attraction that contain the fixed point cycles in the middle of them.
Thus a 5 cycle Julia Set will have a 5 fold basin of attraction and this
will determine the basic shape of the Julia Set.
Thus each ball of the Mandelbrot Set (meaning the C's taken from
the balls) has its own Julia pattern. Because each ball has other balls
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connected to them and further balls connected to THEM the Julia patterns
can get very complex. However each ball retains its own distinctive
pattern.
This is so much so that Julia Sets taken off of balls off of balls
will have both patterns in a clearly recognizable mix. For example the
2 ball off the 3 ball has the 2 ball pattern inside the 3 ball pattern
and total cycle of 6. Likewise the 3 ball off of the 2 ball has the 3
ball pattern inside the 2 ball pattern and also a total cycle of 6.
This of course has to be seen to be believed, and the reader is
directed to the sheet entitled 'Mandelbrot Sets and Julia Sets' for a
visual confirmation of these facts.
Being able to draw Julia Sets, their 1 cycle fixed points, and
their period cycle 'fixed points' is much more important than being able
to draw the Mandelbrot Set. It is the Julia Set where all the life is
and it is the Julia Set that determines the nature of the Mandelbrot
Set.
Julia Sets taken from properly chosen points on the Mandelbrot Set
can be stunningly beautiful. Julia Sets however tend to be very self-
similar and scale independant. Thus zooms tend to be boring after a
point. Once you have seen part of it you have seen all of it.
Mandelbrot Sets however are not strictly self-similar and so zooms
take you into ever new territory.