Fractals and Julia sets; iteration gone wild
Randall Pyke, Dept of Mathematics, SFU
gone
wild
gone wild
gone wild
gone wild
gone wild
gone wild
gone wild
gone wild
gone wild
What‟s in a formula?
“The set
“The set of complex numbers
“The set of complex numbers such that
“The set of complex numbers such that the supremum („maximum‟) over all natural
numbers n
“The set of complex numbers such that the supremum („maximum‟) over all natural
numbers n of the absolute value of the nth iterate of
is finite”
“The set of complex numbers such that the supremum („maximum‟) over all natural
numbers n of the absolute value of the nth iterate of
is finite”
“The set of complex numbers such that the supremum („maximum‟) over all natural
numbers n of the absolute value of the nth iterate of
is finite”
“The set of complex numbers such that the supremum („maximum‟) over all natural
numbers n of the absolute value of the nth iterate of
is finite”
What is iteration?
is the nth iteration (composition) of the function f;
So,
is the nth iteration of f(z), starting at z=0;
For the quadratic function
and so,
For example, if c=1 then,
,
this becomes;
Thus, the iterates
for each n.
define a sequence of numbers depending on c, one number
Thus, the iterates
for each n.
define a sequence of numbers depending on c, one number
The numbers c are complex numbers; they are two dimensional numbers;
The real numbers (this line)
We can represent (visualize) the complex numbers by sketching them on the
complex plane;
The number 0
The complex plane;
The complex plane;
The Mandelbrot set is also a set of complex numbers in the complex plane;
The complex plane;
The Mandelbrot set is also a set of complex numbers in the complex plane;
How do you
decide which
complex numbers
are in the
Mandelbrot set?
The complex plane;
The Mandelbrot set is also a set of complex numbers in the complex plane;
How do you
decide which
complex numbers
are in the
Mandelbrot set?
Through iteration
Procedure to decide if a complex number c is in the Mandelbrot set or not:
So, the Mandelbrot set are those complex numbers c such that the iterations
of the quadratic function
starting at z=0 never become too large;
The Mandelbrot set is bizarre;
Iterating the MRCM
(Multiple Reduction Copy Machine)
3 copies of original
at ½ the size
1st iteration:
Start with
this shape:
Iterating the MRCM
This is the image when you
start with the 1st image
1st iteration:
Start with
this shape:
2nd iteration
Iterating the MRCM
1st iteration:
Start with
this shape:
2nd iteration
Note the hole
Iterating the MRCM
1st iteration:
Start with
this shape:
2nd iteration
3rd iteration
Iterating the MRCM
1st iteration:
Start with
this shape:
4th iteration
2nd iteration
3rd iteration
Iterating the MRCM
1st iteration:
2nd iteration
Start with
this shape:
4th iteration
5th iteration
3rd iteration
Iterating the MRCM
1st iteration:
2nd iteration
3rd iteration
Start with
this shape:
4th iteration
5th iteration
Many iterations
Can start the iteration with any shape!
And still obtain the Sierpinski triangle
Another MRCM and its iterations (4 lenses);
Start with
2nd
1st iteration
3rd
An MRCM with 2 lenses:
Initial image
1st iteration
2nd iteration
3rd iteration
4th iteration
5th iteration
Produces the Cantor set
Two important
properties of these
MRCMs;
1) Beginning with any
initial image, the
same final image
is produced
through iteration,
2) The final image is
the only one that
doesn’t change
through iteration
The final image is thus called the fixed point of the MRCM
Iterated Function Systems (IFSs) as MRCMs
An IFS is a function W that acts on images, producing another image;
We can visualize the IFS W through its blueprint;
the image of a square by W
We can visualize the IFS W through its blueprint;
the image of a square by W
The Sierpinski IFS
blueprint;
3 images of the
square
We can visualize the IFS W through its blueprint;
the image of a square by W
The Sierpinski IFS
blueprint;
3 images of the
square
Now iterate…..
1st iteration (blueprint)
1st iteration (blueprint)
2nd iteration
1st iteration (blueprint)
3rd iteration
2nd iteration
1st iteration (blueprint)
3rd iteration
2nd iteration
8th iteration
Is the fixed point
of W
Is the fixed point
of W
Fractal geometry. . . .
Iteration of functions f: R  R
Graphical iteration
y=x
f(x)=mx+b
p
p
Fixed point p; f(p)=p. Where the graph of f(x)
crosses the diagonal line y=x.
Graphical iteration
x1 = f(x0)
x2 = f(x1)
x3 = f(x2)
p
x0
Starting at any x0, iterates will converge to p
Graphical iteration
x1 = f(x0)
x2 = f(x1)
x3 = f(x2)
p
x0
Starting at any x0, iterates will converge to p
Graphical iteration
x1 = f(x0)
x2 = f(x1)
x3 = f(x2)
p
x0
Starting at any x0, iterates will converge to p
Graphical iteration
x1 = f(x0)
x2 = f(x1)
x3 = f(x2)
p
x1
x0
Starting at any x0, iterates will converge to p
Graphical iteration
x1 = f(x0)
x2 = f(x1)
x3 = f(x2)
p
x1
x0
Starting at any x0, iterates will converge to p
Graphical iteration
x1 = f(x0)
x2 = f(x1)
x3 = f(x2)
p
x2
x1
x0
Starting at any x0, iterates will converge to p
Graphical iteration
x1 = f(x0)
x2 = f(x1)
x3 = f(x2)
p
x2
x1
x0
Starting at any x0, iterates will converge to p
Graphical iteration
x1 = f(x0)
x2 = f(x1)
x3 = f(x2)
p
x3
x2
x1
x0
Starting at any x0, iterates will converge to p
Graphical iteration
x1 = f(x0)
x2 = f(x1)
x3 = f(x2)
p
x3
x2
x1
x0
Starting at any x0, iterates will converge to p
Graphical iteration
x1 = f(x0)
x2 = f(x1)
x3 = f(x2)
x2
x0
x1
p
x3
x2
x1
x0
Starting at any x0, iterates will converge to p
Not all iteration finds fixed points….
Graphical iteration
f(x) = mx+b
x1 = f(x0)
x2 = f(x1)
x3 = f(x2)
y=x
p
p
f(p)=p
Graphical iteration
f(x) = mx+b
x1 = f(x0)
x2 = f(x1)
x3 = f(x2)
y=x
p
x0
Starting at any x0, iterates will diverge away from p
Graphical iteration
f(x)=mx+b
x1 = f(x0)
x2 = f(x1)
x3 = f(x2)
p
x0
Starting at any x0, iterates will diverge away from p
Graphical iteration
f(x)=mx+b
x1 = f(x0)
x2 = f(x1)
x3 = f(x2)
p
x0
Starting at any x0, iterates will diverge away from p
Graphical iteration
f(x)=mx+b
x1 = f(x0)
x2 = f(x1)
x3 = f(x2)
p
x0
x1
Starting at any x0, iterates will diverge away from p
Graphical iteration
f(x)=mx+b
x1 = f(x0)
x2 = f(x1)
x3 = f(x2)
p
x0
x1
Starting at any x0, iterates will diverge away from p
Graphical iteration
f(x)=mx+b
x1 = f(x0)
x2 = f(x1)
x3 = f(x2)
p
x0
x1
x2
Starting at any x0, iterates will diverge away from p
Graphical iteration
f(x)=mx+b
x1 = f(x0)
x2 = f(x1)
x3 = f(x2)
x2
x1
x0
p
x0
x1
x2
Starting at any x0, iterates will diverge away from p
Prisoner and escape sets
Define:
• The prisoner set P of f(x) is those points that do not go off to infinity
under iteration by f
• The escape set E of f(x) is those points that do go off to infinity under
iteration.
(All points are either in P or E)
We‟ve seen:
• For linear f(x) with |m|<1, P is all points, E is empty
• For linear f(x) with |m|>1, P is just the fixed point p, and E are all
the other points
For nonlinear f(x) the sets P and E can both contain many points, and
it can be difficult to determine exactly what P and E are . . .
y=x
P = [-1,1]
E = (-∞,-1) U (1, ∞)
c=1/4
y=x
c=1/4
y=x
Period 1 orbit; 0  0  0  0  …
Period 2 orbit; 0  -1  0  -1  ….
Period 4 orbit
Period 8 orbit
Period 3 orbit
Prisoner set = {-p, p, . . . }
••• ••••••• •• • ••••••• ••
••• ••••••• •• • ••••••• ••
Conclusion;
Prisoner set P is an interval if -2 < c < ¼.
Otherwise P is empty or just points.
••• ••••••• •• • ••••••• ••
Conclusion;
Prisoner set P is an interval if -2 < c < ¼.
Otherwise P is empty or just points.
The Mandelbrot sits between -2 and ¼;
••• ••••••• •• • ••••••• ••
Conclusion;
Prisoner set P is an interval if -2 < c < ¼.
Otherwise P is empty or just points.
The Mandelbrot sits between -2 and ¼;
The Mandelbrot set is related to the
shape of the prisoner set…..
c = -1/2 + 1/2i
Self-similarity of Prisoner sets
The Julia Set
is the boundary (edge) of the prisoner set.
Some Julia sets in 2 dimensions (complex numbers):
Some Julia sets in 2 dimensions (complex numbers)
Zoom into a Julia set
The other (and original) definition of the Mandelbrot set;
The Mandelbrot set are those c such that the Julia set is one piece
One piece
Dust
The Mandelbrot set are those c such that the Julia set is one piece
One piece
Dust
Julia sets in colour!
Julia sets in colour!
How many iterations before the point „escapes‟?
10
20
30
40
50
100
> 200; the prisoner set
Theorem of Julia and Fatou (c.a. 1900);
“The Prisoner set is one piece if and only if the iterates of
are bounded”
And so we can define the Mandelbrot set as;
“The set of complex numbers such that the supremum („maximum‟) over all natural
numbers n of the absolute value of the nth iteration of fc(0) is finite”
The Mandelbrot set
The Mandelbrot set in colour
10
20
30
40
50
100
200
> 200; the Mandelbrot set
The Mandelbrot set in colour
The Mandelbrot set in other colours
The Mandelbrot set in other colours
The Mandelbrot set in other colours
The Mandelbrot set in other colours
The Mandelbrot set in other colours
Zoom into the Mandelbrot set….
http://wmi.math.u-szeged.hu/xaos/doku.php
XaoS
For more information:
• This presentation:
www.sfu.ca/~rpyke/julia2013.pdf
• More info:
www.sfu.ca/~rpyke/  “Fractals”
• Email: rpyke@sfu.ca