Complex Dynamics

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Complex Dynamics and Crazy Mathematics
Dynamics of three very different families of
complex functions

• Polynomials (z2 c)
• 2. Entire maps (? exp(z))
• 3. Rational maps (zn ?/zn)

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Well investigate chaotic behavior in the
dynamical plane (the Julia sets)
exp(z)
z2 c
z2 ?/z2
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As well as the structure of the parameter planes.
? exp(z)
z3 ?/z3
z2 c
(the Mandelbrot set)
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A couple of subthemes

1. Some crazy mathematics
2. Great undergrad research topics

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The Fractal Geometry of the Mandelbrot Set
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The Fractal Geometry of the Mandelbrot Set
How to count
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The Fractal Geometry of the Mandelbrot Set
How to count
How to add
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Many people know the pretty pictures...
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but few know the even prettier mathematics.
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Oh, that's nothing but the 3/4 bulb ....
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...hanging off the period 16 M-set.....
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...lying in the 1/7 antenna...
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...attached to the 1/3 bulb...
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...hanging off the 3/7 bulb...
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...on the northwest side of the main cardioid.
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Oh, that's nothing but the 3/4 bulb, hanging
off the period 16 M-set, lying in the 1/7
antenna of the 1/3 bulb attached to the 3/7
bulb on the northwest side of the main cardioid.
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Start with a function
2
x constant
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Start with a function
2
x constant
and a seed
x
0
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Then iterate
2
x x constant
1
0
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Then iterate
2
x x constant
1
0
2
x x constant
2
1
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Then iterate
2
x x constant
1
0
2
x x constant
2
1
2
x x constant
3
2
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Then iterate
2
x x constant
1
0
2
x x constant
2
1
2
x x constant
3
2
2
x x constant
4
3
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Then iterate
2
x x constant
1
0
2
x x constant
2
1
Orbit of x
2
0
x x constant
3
2
2
x x constant
4
3
etc.
Goal understand the fate of orbits.
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2
Example x 1 Seed 0
x 0
0
x
1
x
2
x
3
x
4
x
5
x
6
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2
Example x 1 Seed 0
x 0
0
x 1
1
x
2
x
3
x
4
x
5
x
6
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2
Example x 1 Seed 0
x 0
0
x 1
1
x 2
2
x
3
x
4
x
5
x
6
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2
Example x 1 Seed 0
x 0
0
x 1
1
x 2
2
x 5
3
x
4
x
5
x
6
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2
Example x 1 Seed 0
x 0
0
x 1
1
x 2
2
x 5
3
x 26
4
x
5
x
6
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2
Example x 1 Seed 0
x 0
0
x 1
1
x 2
2
x 5
3
x 26
4
x big
5
x
6
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2
Example x 1 Seed 0
x 0
0
x 1
1
x 2
2
x 5
3
x 26
4
x big
5
x BIGGER
6
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2
Example x 1 Seed 0
x 0
0
x 1
1
x 2
2
Orbit tends to infinity
x 5
3
x 26
4
x big
5
x BIGGER
6
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2
Example x 0 Seed 0
x 0
0
x
1
x
2
x
3
x
4
x
5
x
6
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2
Example x 0 Seed 0
x 0
0
x 0
1
x
2
x
3
x
4
x
5
x
6
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2
Example x 0 Seed 0
x 0
0
x 0
1
x 0
2
x
3
x
4
x
5
x
6
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2
Example x 0 Seed 0
x 0
0
x 0
1
x 0
2
x 0
3
x
4
x
5
x
6
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2
Example x 0 Seed 0
x 0
0
x 0
1
x 0
2
A fixed point
x 0
3
x 0
4
x 0
5
x 0
6
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2
Example x - 1 Seed 0
x 0
0
x
1
x
2
x
3
x
4
x
5
x
6
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2
Example x - 1 Seed 0
x 0
0
x -1
1
x
2
x
3
x
4
x
5
x
6
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2
Example x - 1 Seed 0
x 0
0
x -1
1
x 0
2
x
3
x
4
x
5
x
6
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2
Example x - 1 Seed 0
x 0
0
x -1
1
x 0
2
x -1
3
x
4
x
5
x
6
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2
Example x - 1 Seed 0
x 0
0
x -1
1
x 0
2
x -1
3
x 0
4
x
5
x
6
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2
Example x - 1 Seed 0
x 0
0
x -1
1
x 0
2
x -1
A two- cycle
3
x 0
4
x -1
5
x 0
6
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2
Example x - 1.1 Seed 0
x 0
0
x
1
x
2
x
3
x
4
x
5
x
6
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2
Example x - 1.1 Seed 0
x 0
0
x -1.1
1
x
2
x
3
x
4
x
5
x
6
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2
Example x - 1.1 Seed 0
x 0
0
x -1.1
1
x 0.11
2
x
3
x
4
x
5
x
6
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2
Example x - 1.1 Seed 0
x 0
0
x -1.1
1
x 0.11
2
x
3
time for the computer!
x
4
x
5
x
6
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Observation
For some real values of c, the orbit of 0 goes
to infinity, but for other values, the orbit of
0 does not escape.
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Complex Iteration
2
Iterate z c
complex numbers
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2
Example z i Seed 0
z 0
0
z
1
z
2
z
3
z
4
z
5
z
6
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2
Example z i Seed 0
z 0
0
z i
1
z
2
z
3
z
4
z
5
z
6
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2
Example z i Seed 0
z 0
0
z i
1
z -1 i
2
z
3
z
4
z
5
z
6
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2
Example z i Seed 0
z 0
0
z i
1
z -1 i
2
z -i
3
z
4
z
5
z
6
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2
Example z i Seed 0
z 0
0
z i
1
z -1 i
2
z -i
3
z -1 i
4
z
5
z
6
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2
Example z i Seed 0
z 0
0
z i
1
z -1 i
2
z -i
3
z -1 i
4
z -i
5
z
6
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2
Example z i Seed 0
z 0
0
z i
1
z -1 i
2
z -i
3
z -1 i
4
z -i
5
2-cycle
z -1 i
6
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2
Example z i Seed 0
i
1
-1
-i
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2
Example z i Seed 0
i
1
-1
-i
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2
Example z i Seed 0
i
1
-1
-i
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2
Example z i Seed 0
i
1
-1
-i
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2
Example z i Seed 0
i
1
-1
-i
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2
Example z i Seed 0
i
1
-1
-i
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2
Example z i Seed 0
i
1
-1
-i
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2
Example z i Seed 0
i
1
-1
-i
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2
Example z 2i Seed 0
z 0
0
z
1
z
2
z
3
z
4
z
5
z
6
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2
Example z 2i Seed 0
z 0
0
z 2i
1
z -4 2i
2
Off to infinity
z 12 - 14i
3
z -52 336i
4
z big
5
z BIGGER
6
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Same observation
Sometimes orbit of 0 goes to infinity, other
times it does not.
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The Mandelbrot Set
All c-values for which orbit of 0 does NOT go to
infinity.
Why do we care about the orbit of 0?
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The Mandelbrot Set
All c-values for which orbit of 0 does NOT go to
infinity.
0 is the critical point of z2 c.
As we shall see, the orbit of the critical point
determines just about everything for z2 c.
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Algorithm for computing M
Start with a grid of complex numbers
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Algorithm for computing M
Each grid point is a complex c-value.
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Algorithm for computing M
Compute the orbit of 0 for each c. If the orbit
of 0 escapes, color that grid point.
red fastest escape
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Algorithm for computing M
Compute the orbit of 0 for each c. If the orbit
of 0 escapes, color that grid point.
orange slower
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Algorithm for computing M
Compute the orbit of 0 for each c. If the orbit
of 0 escapes, color that grid point.

yellow green blue violet
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Algorithm for computing M
Compute the orbit of 0 for each c. If the orbit
of 0 does not escape, leave that grid point
black.
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Algorithm for computing M
Compute the orbit of 0 for each c. If the orbit
of 0 does not escape, leave that grid point
black.
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The eventual orbit of 0
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The eventual orbit of 0
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The eventual orbit of 0
3-cycle
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The eventual orbit of 0
3-cycle
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The eventual orbit of 0
3-cycle
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The eventual orbit of 0
3-cycle
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The eventual orbit of 0
3-cycle
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The eventual orbit of 0
3-cycle
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The eventual orbit of 0
3-cycle
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The eventual orbit of 0
3-cycle
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The eventual orbit of 0
3-cycle
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The eventual orbit of 0
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The eventual orbit of 0
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The eventual orbit of 0
4-cycle
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The eventual orbit of 0
4-cycle
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The eventual orbit of 0
4-cycle
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The eventual orbit of 0
4-cycle
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The eventual orbit of 0
4-cycle
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The eventual orbit of 0
4-cycle
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The eventual orbit of 0
4-cycle
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The eventual orbit of 0
4-cycle
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The eventual orbit of 0
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The eventual orbit of 0
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The eventual orbit of 0
5-cycle
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The eventual orbit of 0
5-cycle
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The eventual orbit of 0
5-cycle
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The eventual orbit of 0
5-cycle
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The eventual orbit of 0
5-cycle
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The eventual orbit of 0
5-cycle
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The eventual orbit of 0
5-cycle
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The eventual orbit of 0
5-cycle
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The eventual orbit of 0
5-cycle
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The eventual orbit of 0
5-cycle
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The eventual orbit of 0
5-cycle
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The eventual orbit of 0
2-cycle
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The eventual orbit of 0
2-cycle
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The eventual orbit of 0
2-cycle
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The eventual orbit of 0
2-cycle
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The eventual orbit of 0
2-cycle
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The eventual orbit of 0
fixed point
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The eventual orbit of 0
fixed point
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The eventual orbit of 0
fixed point
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The eventual orbit of 0
fixed point
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The eventual orbit of 0
fixed point
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The eventual orbit of 0
fixed point
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The eventual orbit of 0
fixed point
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The eventual orbit of 0
fixed point
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The eventual orbit of 0
goes to infinity
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The eventual orbit of 0
goes to infinity
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The eventual orbit of 0
goes to infinity
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The eventual orbit of 0
goes to infinity
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The eventual orbit of 0
goes to infinity
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The eventual orbit of 0
goes to infinity
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The eventual orbit of 0
goes to infinity
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The eventual orbit of 0
goes to infinity
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The eventual orbit of 0
goes to infinity
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The eventual orbit of 0
goes to infinity
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The eventual orbit of 0
goes to infinity
147
The eventual orbit of 0
gone to infinity
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One reason for the importance of the critical
orbit
If there is an attracting cycle for z2 c, then
the orbit of 0 must tend to it.
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periods
How understand the of the bulbs?
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periods
How understand the of the bulbs?
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junction point
three spokes attached
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junction point
three spokes attached
Period 3 bulb
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Period 4 bulb
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Period 5 bulb
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Period 7 bulb
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Period 13 bulb
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Filled Julia Set
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Filled Julia Set
Fix a c-value. The filled Julia set is all of
the complex seeds whose orbits do NOT go to
infinity.
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2
Example z
Seed
In filled Julia set?
0
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2
Example z
Seed
In filled Julia set?
0
Yes
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2
Example z
Seed
In filled Julia set?
0
Yes
1
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2
Example z
Seed
In filled Julia set?
0
Yes
1
Yes
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2
Example z
Seed
In filled Julia set?
0
Yes
1
Yes
-1
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2
Example z
Seed
In filled Julia set?
0
Yes
1
Yes
-1
Yes
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2
Example z
Seed
In filled Julia set?
0
Yes
1
Yes
-1
Yes
i
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2
Example z
Seed
In filled Julia set?
0
Yes
1
Yes
-1
Yes
i
Yes
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2
Example z
Seed
In filled Julia set?
0
Yes
1
Yes
-1
Yes
i
Yes
2i
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2
Example z
Seed
In filled Julia set?
0
Yes
1
Yes
-1
Yes
i
Yes
No
2i
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2
Example z
Seed
In filled Julia set?
0
Yes
1
Yes
-1
Yes
i
Yes
No
2i
5
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2
Example z
Seed
In filled Julia set?
0
Yes
1
Yes
-1
Yes
i
Yes
No
2i
5
No way
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Filled Julia Set for z
2
i
1
-1
All seeds on and inside the unit circle.
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The Julia Set is the boundary of the filled
Julia set
Thats where the map is chaotic
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The Julia Set is the boundary of the filled
Julia set
Thats where the map is chaotic Nearby orbits
behave very differently
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The Julia Set is the boundary of the filled
Julia set
Thats where the map is chaotic Nearby orbits
behave very differently
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The Julia Set is the boundary of the filled
Julia set
Thats where the map is chaotic Nearby orbits
behave very differently
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The Julia Set is the boundary of the filled
Julia set
Thats where the map is chaotic Nearby orbits
behave very differently
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The Julia Set is the boundary of the filled
Julia set
Thats where the map is chaotic Nearby orbits
behave very differently
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The Julia Set is the boundary of the filled
Julia set
Thats where the map is chaotic Nearby orbits
behave very differently
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The Julia Set is the boundary of the filled
Julia set
Thats where the map is chaotic Nearby orbits
behave very differently
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The Julia Set is the boundary of the filled
Julia set
Thats where the map is chaotic Nearby orbits
behave very differently
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The Julia Set is the boundary of the filled
Julia set
Thats where the map is chaotic Nearby orbits
behave very differently
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The Julia Set is the boundary of the filled
Julia set
Thats where the map is chaotic Nearby orbits
behave very differently
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Other filled Julia sets
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Other filled Julia sets
c 0
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Other filled Julia sets
c -1
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Other filled Julia sets
c -1
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Other filled Julia sets
c -1
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Other filled Julia sets
c -1
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Other filled Julia sets
c -1
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Other filled Julia sets
c -1
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Other filled Julia sets
c -1
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Other filled Julia sets
c -1
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Other filled Julia sets
c -.12.75i
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Other filled Julia sets
c -.12.75i
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Other filled Julia sets
c -.12.75i
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Other filled Julia sets
c -.12.75i
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Other filled Julia sets
c -.12.75i
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Other filled Julia sets
c -.12.75i
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If c is in the Mandelbrot set, then the filled
Julia set is always a connected set.
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Other filled Julia sets
But if c is not in the Mandelbrot set, then the
filled Julia set is totally disconnected.
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Other filled Julia sets
c .3
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Other filled Julia sets
c .3
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Other filled Julia sets
c .3
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Other filled Julia sets
c .3
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Other filled Julia sets
c .3
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Other filled Julia sets
c -.8.4i
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Another reason why we use the orbit of the
critical point to plot the M-set
Theorem (Fatou Julia) For z2 c
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Another reason why we use the orbit of the
critical point to plot the M-set
Theorem (Fatou Julia) For z2 c
If the orbit of 0 goes to infinity, the Julia
set is a Cantor set (totally disconnected,
fractal dust, a scatter of uncountably many
points.
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Another reason why we use the orbit of the
critical point to plot the M-set
Theorem (Fatou Julia) For z2 c
If the orbit of 0 goes to infinity, the Julia
set is a Cantor set (totally disconnected,
fractal dust, a scatter of uncountably many
points.
But if the orbit of 0 does not go to
infinity, the Julia set is connected (just one
piece).
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Animations
In and out of M
Saddle node
Period doubling
Period 4 bifurcation
arrangement of the bulbs
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How do we understand the arrangement of the
bulbs?
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How do we understand the arrangement of the
bulbs?
Assign a fraction p/q to each bulb hanging off
the main cardioid. q period of the bulb
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p/3 bulb
principal spoke
Where is the smallest spoke in relation to the
principal spoke?
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1/3 bulb
principal spoke
The smallest spoke is located 1/3 of a turn in
the counterclockwise direction from the principal
spoke.
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1/3 bulb
1/3
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1/3 bulb
1/3
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1/3 bulb
1/3
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1/3 bulb
1/3
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1/3 bulb
1/3
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1/3 bulb
1/3
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1/3 bulb
1/3
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1/3 bulb
1/3
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1/3 bulb
1/3
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1/3 bulb
1/3
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??? bulb
1/3
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1/4 bulb
1/3
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1/4 bulb
1/3
1/4
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1/4 bulb
1/3
1/4
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1/4 bulb
1/3
1/4
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1/4 bulb
1/3
1/4
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1/4 bulb
1/3
1/4
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1/4 bulb
1/3
1/4
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1/4 bulb
1/3
1/4
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1/4 bulb
1/3
1/4
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1/4 bulb
1/3
1/4
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??? bulb
1/3
1/4
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2/5 bulb
1/3
1/4
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2/5 bulb
1/3
2/5
1/4
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2/5 bulb
1/3
2/5
1/4
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2/5 bulb
1/3
2/5
1/4
250
2/5 bulb
1/3
2/5
1/4
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2/5 bulb
1/3
2/5
1/4
252
??? bulb
1/3
2/5
1/4
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3/7 bulb
1/3
2/5
1/4
254
3/7 bulb
1/3
2/5
1/4
3/7
255
3/7 bulb
1/3
2/5
1/4
3/7
256
3/7 bulb
1/3
2/5
1/4
3/7
257
3/7 bulb
1/3
2/5
1/4
3/7
258
3/7 bulb
1/3
2/5
1/4
3/7
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3/7 bulb
1/3
2/5
1/4
3/7
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3/7 bulb
1/3
2/5
1/4
3/7
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??? bulb
1/3
2/5
1/4
3/7
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1/2 bulb
1/3
2/5
1/4
3/7
1/2
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1/2 bulb
1/3
2/5
1/4
3/7
1/2
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1/2 bulb
1/3
2/5
1/4
3/7
1/2
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1/2 bulb
1/3
2/5
1/4
3/7
1/2
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??? bulb
1/3
2/5
1/4
3/7
1/2
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2/3 bulb
1/3
2/5
1/4
3/7
1/2
2/3
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2/3 bulb
1/3
2/5
1/4
3/7
1/2
2/3
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2/3 bulb
1/3
2/5
1/4
3/7
1/2
2/3
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2/3 bulb
1/3
2/5
1/4
3/7
1/2
2/3
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2/3 bulb
1/3
2/5
1/4
3/7
1/2
2/3
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2/3 bulb
1/3
2/5
1/4
3/7
1/2
2/3
273
How to count
274
How to count
1/4
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How to count
1/3
1/4
276
How to count
1/3
2/5
1/4
277
How to count
1/3
2/5
1/4
3/7
278
How to count
1/3
2/5
1/4
3/7
1/2
279
How to count
1/3
2/5
1/4
3/7
1/2
2/3
280
How to count
1/3
2/5
1/4
3/7
1/2
2/3
The bulbs are arranged in the exact order of the
rational numbers.
281
How to count
1/3
32,123/96,787
2/5
1/4
3/7
1/101
1/2
2/3
The bulbs are arranged in the exact order of the
rational numbers.
282
Animations
Mandelbulbs
Spiralling fingers
283
How to add
284
How to add
1/2
285
How to add
1/3
1/2
286
How to add
1/3
2/5
1/2
287
How to add
1/3
2/5
3/7
1/2
288
1/2 1/3 2/5


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1/2 2/5 3/7


290
Heres an interesting sequence
2
2
1/2
0/1
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Watch the denominators
1/3
2
2
1/2
0/1
292
Watch the denominators
1/3
2/5
2
2
1/2
0/1
293
Watch the denominators
1/3
3/8
2/5
2
2
1/2
0/1
294
Watch the denominators
1/3
3/8
5/13
2/5
2
2
1/2
0/1
295
Whats next?
1/3
3/8
5/13
2/5
2
2
1/2
0/1
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Whats next?
8/21
1/3
3/8
5/13
2/5
2
2
1/2
0/1
297
The Fibonacci sequence
13/34
8/21
1/3
3/8
5/13
2/5
2
2
1/2
0/1
298
The Farey Tree
299
The Farey Tree
How get the fraction in between with the smallest
denominator?
300
The Farey Tree
How get the fraction in between with the smallest
denominator?
Farey addition
301
The Farey Tree
302
The Farey Tree
303
The Farey Tree
....
essentially the golden number
304
Another sequence
(denominators only)
2
1
305
Another sequence
(denominators only)
3
2
1
306
Another sequence
(denominators only)
3
4
2
1
307
Another sequence
(denominators only)
3
4
5
2
1
308
Another sequence
(denominators only)
3
4
5
2
6
1
309
Another sequence
(denominators only)
3
4
5
2
6
7
1
310
sequence
Devaney
3
4
5
2
6
7
1
311
The Dynamical Systems and Technology
Project at Boston University
website math.bu.edu/DYSYS
Mandelbrot set explorer Applets for
investigating M-set Applets for other complex
functions Chaos games, orbit diagrams, etc.
Have fun!
312
Other topics
Farey.qt
Farey tree
D-sequence
Far from rationals
Continued fraction expansion
Website
313
Continued fraction expansion
Lets rewrite the sequence 1/2, 1/3,
2/5, 3/8, 5/13, 8/21, 13/34, .....
as a continued fraction
314
Continued fraction expansion
1 2
1 2

the sequence 1/2, 1/3, 2/5, 3/8, 5/13, 8/21,
13/34,.....
315
Continued fraction expansion
1 3
1 2


1 1
the sequence 1/2, 1/3, 2/5, 3/8, 5/13, 8/21,
13/34,.....
316
Continued fraction expansion
2 5
1 2


1 1

1 1
the sequence 1/2, 1/3, 2/5, 3/8, 5/13, 8/21,
13/34,.....
317
Continued fraction expansion
3 8
1 2


1 1

1 1

1 1
the sequence 1/2, 1/3, 2/5, 3/8, 5/13, 8/21,
13/34,.....
318
Continued fraction expansion
5

1 2


1 1
13

1 1

1 1

1 1
the sequence 1/2, 1/3, 2/5, 3/8, 5/13, 8/21,
13/34,.....
319
Continued fraction expansion
8

1 2


1 1
21

1 1

1 1

1 1

1 1
the sequence 1/2, 1/3, 2/5, 3/8, 5/13, 8/21,
13/34,.....
320
Continued fraction expansion

13
1 2


1 1
34

1 1

1 1

1 1

1 1

1 1
the sequence 1/2, 1/3, 2/5, 3/8, 5/13, 8/21,
13/34,.....
321
Continued fraction expansion

13
1 2


1 1
34

1 1

1 1

1 1

1 1

1 1
essentially the 1/golden number
the sequence 1/2, 1/3, 2/5, 3/8, 5/13, 8/21,
13/34,.....
322
We understand what happens for
1 a


1 b

1 c

1 d

1 e

1 f

1 g
etc.
where all entries in the sequence a, b, c,
d,.... are bounded above. But if that
sequence grows too quickly, were in trouble!!!
323
The real way to prove all this
Need to measure the size of bulbs the
length of spokes the size of the ears.
324
There is an external Riemann map
C - D C - M taking the exterior of the
unit disk to the exterior of the Mandelbrot set.
325
takes straight rays in C - D to the
external rays in C - M
external ray of angle 1/3
1/3
0
1/2
2/3
326
Suppose p/q is periodic of period k
under doubling mod 1
period 2
period 3
period 4
327
Suppose p/q is periodic of period k
under doubling mod 1
period 2
period 3
period 4
Then the external ray of angle p/q lands at the
root point of a period k bulb in the
Mandelbrot set.
328
0 is fixed under angle doubling, so lands at the
cusp of the main cardioid.
1/3
0
2/3
329
1/3 and 2/3 have period 2 under doubling, so
and land at the root of the period 2
bulb.
1/3
2
0
2/3
330
And if lies between 1/3 and 2/3, then
lies between and .
1/3
2
0
2/3
331
So the size of the period 2 bulb is, by
definition, the length of the set of rays
between the root point rays, i.e., 2/3-1/31/3.
1/3
2
0
2/3
332
1/15 and 2/15 have period 4, and are smaller than
1/7....
1/3
2/7
1/7
3
3/7
2/15
1/15
2
0
3
4/7
6/7
2/3
5/7
333
1/15 and 2/15 have period 4, and are smaller than
1/7....
1/3
2/7
1/7
3
3/7
2/15
1/15
2
0
3
4/7
6/7
2/3
5/7
334
3/15 and 4/15 have period 4, and are between 1/7
and 2/7....
1/3
2/7
1/7
3
3/7
2/15
1/15
2
0
3
4/7
6/7
2/3
5/7
335
3/15 and 4/15 have period 4, and are between 1/7
and 2/7....
1/3
2/7
1/7
3
3/7
2/15
1/15
2
0
3
4/7
6/7
2/3
5/7
336
3/15 and 4/15 have period 4, and are between 1/7
and 2/7....
1/7
2/7
337
3/15 and 4/15 have period 4, and are between 1/7
and 2/7....
3/15
4/15
1/7
2/7
338
So what do we know about M?
All rational external rays land at a single
point in M.
339
So what do we know about M?
All rational external rays land at a single
point in M.
Rays that are periodic under doubling land at
root points of a bulb.
Non-periodic rational rays land at Misiurewicz
points (how we measure length of antennas).
340
So what do we know about M?
Highly irrational rays also land at unique
points, and we understand what goes on
here. Highly irrational" far from
rationals, i.e.,
341
So what do we NOT know about M?
But we don't know if irrationals that are
close to rationals land. So we won't
understand quadratic functions until we figure
this out.
342
MLC Conjecture
The boundary of the M-set is locally connected
--- if so, all rays land and we are in heaven!.
But if not......
343
The Dynamical Systems and Technology
Project at Boston University
website math.bu.edu/DYSYS
Have fun!
344
A number is far from the rationals if
345
A number is far from the rationals if
346
A number is far from the rationals if
This happens if the continued fraction
expansion of has only bounded terms.