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 , this becomes;
and so,
For example, if c=1 then,
Thus, the iterates define a sequence of numbers depending on c, one number for each n. Thus, the iterates define a sequence of numbers depending on c, one number for each n.
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: 2nd iteration
Start with this shape: Iterating the MRCM
1st iteration: 2nd iteration
Start with Note the hole this shape: Iterating the MRCM
1st iteration: 2nd iteration 3rd iteration
Start with this shape: Iterating the MRCM
1st iteration: 2nd iteration 3rd iteration
Start with this shape:
4th iteration Iterating the MRCM
1st iteration: 2nd iteration 3rd iteration
Start with this shape:
4th iteration 5th 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 1st iteration
2nd 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) 2nd iteration
3rd iteration 1st iteration (blueprint) 2nd iteration
3rd 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) y = x x3 = f(x2)
p
p f(p)=p Graphical iteration f(x) = mx+b x1 = f(x0) y = x 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
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: [email protected]