complex dynamics and crazy mathematics dynamics of three very different families of complex...
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Complex Dynamics and
Crazy Mathematics
Dynamics of three very different families of complex functions:
1. Polynomials (z2 + c)
2. Entire maps ( exp(z))
3. Rational maps (zn + /zn)
We’ll investigate chaotic behavior inthe dynamical plane (the Julia sets)
z2 + c
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exp(z) z2 + /z2
As well as the structure of theparameter planes.
z2 + c exp(z) z3 + /z3
(the Mandelbrot set)
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A couple of subthemes:
1. Some “crazy” mathematics
2. Great undergrad research topics
The Fractal Geometryof
the Mandelbrot Set
How to count
The Fractal Geometryof
the Mandelbrot Set
The Fractal Geometryof
the Mandelbrot Set
How to add
How to count
Many people know thepretty pictures...
but few know the evenprettier mathematics.
Oh, that's nothing but the 3/4 bulb ....
...hanging off the period 16 M-set.....
...lying in the 1/7 antenna...
...attached to the 1/3 bulb...
...hanging off the 3/7 bulb...
...on the northwest side of the main cardioid.
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.
Start with a function:
x + constant2
Start with a function:
x + constant2
and a seed:
x0
Then iterate:
x = x + constant1 02
Then iterate:
x = x + constant1 02
x = x + constant2 12
Then iterate:
x = x + constant1 02
x = x + constant2 12
x = x + constant3 2
2
Then iterate:
x = x + constant1 02
x = x + constant2 12
x = x + constant3 2
2
x = x + constant4 3
2
Then iterate:
x = x + constant1 02
x = x + constant2 12
x = x + constant3 2
2
x = x + constant4 3
2
Orbit of x0
etc.
Goal: understand the fate of orbits.
Example: x + 1 Seed 02
x = 00x = 1x = 2
x = 3
x = 4
x = 5
x = 6
Example: x + 1 Seed 02
x = 00x = 11x =2
x = 3
x = 4
x = 5
x = 6
Example: x + 1 Seed 02
x = 00x = 11x = 22
x = 3
x = 4
x = 5
x = 6
Example: x + 1 Seed 02
x = 00x = 11x = 22
x = 53
x = 4
x = 5
x = 6
Example: x + 1 Seed 02
x = 00x = 11x = 22
x = 53
x = 264
x = 5
x = 6
Example: x + 1 Seed 02
x = 00x = 11x = 22
x = 53
x = 264
x = big5
x = 6
Example: x + 1 Seed 02
x = 00x = 11x = 22
x = 53
x = 264
x = big5
x = BIGGER6
Example: x + 1 Seed 02
x = 00x = 11x = 22
x = 53
x = 264
x = big5
x = BIGGER6
“Orbit tends to infinity”
Example: x + 0 Seed 02
x = 00x = 1x = 2
x = 3
x = 4
x = 5
x = 6
Example: x + 0 Seed 02
x = 00x = 01x = 2
x = 3
x = 4
x = 5
x = 6
Example: x + 0 Seed 02
x = 00x = 01x = 02
x = 3
x = 4
x = 5
x = 6
Example: x + 0 Seed 02
x = 00x = 01x = 02
x = 03
x = 4
x = 5
x = 6
Example: x + 0 Seed 02
x = 00x = 01x = 02
x = 03
x = 04
x = 05
x = 06
“A fixed point”
Example: x - 1 Seed 02
x = 00x = 1x = 2
x = 3
x = 4
x = 5
x = 6
Example: x - 1 Seed 02
x = 00x = -11x = 2
x = 3
x = 4
x = 5
x = 6
Example: x - 1 Seed 02
x = 00x = -11x = 02
x = 3
x = 4
x = 5
x = 6
Example: x - 1 Seed 02
x = 00x = -11x = 02
x = -13
x = 4
x = 5
x = 6
Example: x - 1 Seed 02
x = 00x = -11x = 02
x = -13
x = 04
x = 5
x = 6
Example: x - 1 Seed 02
x = 00x = -11x = 02
x = -13
x = 04
x = -15
x = 06
“A two- cycle”
Example: x - 1.1 Seed 02
x = 00x = 1x = 2
x = 3
x = 4
x = 5
x = 6
Example: x - 1.1 Seed 02
x = 00x = -1.11x = 2
x = 3
x = 4
x = 5
x = 6
Example: x - 1.1 Seed 02
x = 00x = -1.11x = 0.112
x = 3
x = 4
x = 5
x = 6
Example: x - 1.1 Seed 02
x = 00x = -1.11x = 0.112
x = 3
x = 4
x = 5
x = 6
time for the computer!
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.
Complex Iteration
Iterate z + c2
complexnumbers
Example: z + i Seed 02
z = 00z = 1z = 2
z = 3
z = 4
z = 5
z = 6
Example: z + i Seed 02
z = 00z = i1z = 2
z = 3
z = 4
z = 5
z = 6
Example: z + i Seed 02
z = 00z = i1z = -1 + i2
z = 3
z = 4
z = 5
z = 6
Example: z + i Seed 02
z = 00z = i1z = -1 + i2
z = -i 3
z = 4
z = 5
z = 6
Example: z + i Seed 02
z = 00z = i1z = -1 + i2
z = -i 3
z = -1 + i 4
z = 5
z = 6
Example: z + i Seed 02
z = 00z = i1z = -1 + i2
z = -i 3
z = -1 + i 4
z = -i 5
z = 6
Example: z + i Seed 02
z = 00z = i1z = -1 + i2
z = -i 3
z = -1 + i 4
z = -i 5
z = -1 + i 6
2-cycle
Example: z + i Seed 02
1-1
i
-i
Example: z + i Seed 02
1-1
i
-i
Example: z + i Seed 02
1-1
i
-i
Example: z + i Seed 02
-i
-1 1
i
Example: z + i Seed 02
1-1
i
-i
Example: z + i Seed 02
-i
-1 1
i
Example: z + i Seed 02
1-1
i
-i
Example: z + i Seed 02
-i
-1 1
i
Example: z + 2i Seed 02
z = 00z = 1z = 2
z = 3
z = 4
z = 5
z = 6
Example: z + 2i Seed 02
z = 00z = 2i1z = -4 + 2i 2
z = 12 - 14i3
z = -52 + 336i 4
z = big 5
z = BIGGER 6
Off toinfinity
Same observation
Sometimes orbit of 0 goes to infinity, other times it does not.
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?
The Mandelbrot Set:
All c-values for which orbit of 0 does NOT go to infinity.
As we shall see, the orbit of the critical point determines just about everything for z2 + c.
0 is the critical point of z2 + c.
Algorithm for computing M
Start with a grid of complex numbers
Algorithm for computing M
Each grid point is a complex c-value.
Algorithm for computing M
Compute the orbitof 0 for each c. Ifthe orbit of 0 escapes,color that grid point.
red = fastest escape
Algorithm for computing M
Compute the orbitof 0 for each c. Ifthe orbit of 0 escapes,color that grid point.
orange = slower
Algorithm for computing M
Compute the orbitof 0 for each c. Ifthe orbit of 0 escapes,color that grid point.
yellowgreenblueviolet
Algorithm for computing M
Compute the orbitof 0 for each c. Ifthe orbit of 0 does not escape, leave that grid pointblack.
Algorithm for computing M
Compute the orbitof 0 for each c. Ifthe orbit of 0 does not escape, leave that grid pointblack.
The eventual orbit of 0
The eventual orbit of 0
The eventual orbit of 0
3-cycle
The eventual orbit of 0
3-cycle
The eventual orbit of 0
3-cycle
The eventual orbit of 0
3-cycle
The eventual orbit of 0
3-cycle
The eventual orbit of 0
3-cycle
The eventual orbit of 0
3-cycle
The eventual orbit of 0
3-cycle
The eventual orbit of 0
3-cycle
The eventual orbit of 0
The eventual orbit of 0
The eventual orbit of 0
4-cycle
The eventual orbit of 0
4-cycle
The eventual orbit of 0
4-cycle
The eventual orbit of 0
4-cycle
The eventual orbit of 0
4-cycle
The eventual orbit of 0
4-cycle
The eventual orbit of 0
4-cycle
The eventual orbit of 0
4-cycle
The eventual orbit of 0
The eventual orbit of 0
The eventual orbit of 0
5-cycle
The eventual orbit of 0
5-cycle
The eventual orbit of 0
5-cycle
The eventual orbit of 0
5-cycle
The eventual orbit of 0
5-cycle
The eventual orbit of 0
5-cycle
The eventual orbit of 0
5-cycle
The eventual orbit of 0
5-cycle
The eventual orbit of 0
5-cycle
The eventual orbit of 0
5-cycle
The eventual orbit of 0
5-cycle
The eventual orbit of 0
2-cycle
The eventual orbit of 0
2-cycle
The eventual orbit of 0
2-cycle
The eventual orbit of 0
2-cycle
The eventual orbit of 0
2-cycle
The eventual orbit of 0
fixed point
The eventual orbit of 0
fixed point
The eventual orbit of 0
fixed point
The eventual orbit of 0
fixed point
The eventual orbit of 0
fixed point
The eventual orbit of 0
fixed point
The eventual orbit of 0
fixed point
The eventual orbit of 0
fixed point
The eventual orbit of 0
goes to infinity
The eventual orbit of 0
goes to infinity
The eventual orbit of 0
goes to infinity
The eventual orbit of 0
goes to infinity
The eventual orbit of 0
goes to infinity
The eventual orbit of 0
goes to infinity
The eventual orbit of 0
goes to infinity
The eventual orbit of 0
goes to infinity
The eventual orbit of 0
goes to infinity
The eventual orbit of 0
goes to infinity
The eventual orbit of 0
goes to infinity
The eventual orbit of 0
gone to infinity
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.
How understand the of the bulbs?periods
How understand the of the bulbs?periods
junction point
three spokes attached
Period 3 bulb
junction point
three spokes attached
Period 4 bulb
Period 5 bulb
Period 7 bulb
Period 13 bulb
Filled Julia Set:
Filled Julia Set:
Fix a c-value. The filled Julia set is all of the complex seeds whose orbits do NOT go to infinity.
Example: z2
Seed:
0
In filled Julia set?
Example: z2
Seed:
0 Yes
In filled Julia set?
Example: z2
Seed:
0 Yes
1
In filled Julia set?
Example: z2
Seed:
0 Yes
1 Yes
In filled Julia set?
Example: z2
Seed:
0 Yes
1 Yes
-1
In filled Julia set?
Example: z2
Seed:
0 Yes
1 Yes
-1 Yes
In filled Julia set?
Example: z2
Seed:
0 Yes
1 Yes
-1 Yes
i
In filled Julia set?
Example: z2
Seed:
0 Yes
1 Yes
-1 Yes
i Yes
In filled Julia set?
Example: z2
Seed:
0 Yes
1 Yes
-1 Yes
i Yes
2i
In filled Julia set?
Example: z2
Seed:
0 Yes
1 Yes
-1 Yes
i Yes
2i No
In filled Julia set?
Example: z2
Seed:
0 Yes
1 Yes
-1 Yes
i Yes
2i No
5
In filled Julia set?
Example: z2
Seed:
0 Yes
1 Yes
-1 Yes
i Yes
2i No
5 No way
In filled Julia set?
Filled Julia Set for z 2
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All seeds on and inside the unit circle.
i
1-1
The Julia Set is the boundary of the filled Julia set
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That’s where the map is “chaotic”
The Julia Set is the boundary of the filled Julia set
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That’s where the map is “chaotic”Nearby orbits behave very differently
The Julia Set is the boundary of the filled Julia set
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That’s where the map is “chaotic”Nearby orbits behave very differently
The Julia Set is the boundary of the filled Julia set
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That’s where the map is “chaotic”Nearby orbits behave very differently
The Julia Set is the boundary of the filled Julia set
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That’s where the map is “chaotic”Nearby orbits behave very differently
The Julia Set is the boundary of the filled Julia set
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That’s where the map is “chaotic”Nearby orbits behave very differently
The Julia Set is the boundary of the filled Julia set
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That’s where the map is “chaotic”Nearby orbits behave very differently
The Julia Set is the boundary of the filled Julia set
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That’s where the map is “chaotic”Nearby orbits behave very differently
The Julia Set is the boundary of the filled Julia set
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That’s where the map is “chaotic”Nearby orbits behave very differently
The Julia Set is the boundary of the filled Julia set
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That’s where the map is “chaotic”Nearby orbits behave very differently
The Julia Set is the boundary of the filled Julia set
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That’s where the map is “chaotic”Nearby orbits behave very differently
Other filled Julia sets
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
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c = -.12+.75i
Other filled Julia sets
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c = -.12+.75i
Other filled Julia sets
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c = -.12+.75i
Other filled Julia sets
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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
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But if c is not in the Mandelbrot set, then the filled Julia set is totally disconnected.
Other filled Julia sets
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c = .3
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Other filled Julia sets
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c = .3
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Other filled Julia sets
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c = .3
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Other filled Julia sets
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c = .3
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Other filled Julia sets
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c = .3
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Other filled Julia sets
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c = -.8+.4i
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Another reason why we use the orbit ofthe critical point to plot the M-set:
Theorem: (Fatou & Julia) For z2 + c:
Another reason why we use the orbit ofthe 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.
Another reason why we use the orbit ofthe critical point to plot the M-set:
Theorem: (Fatou & Julia) For z2 + c:
But if the orbit of 0 does not go to infinity,the Julia set is connected (just one piece).
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.
Animations:
In and out of M
arrangementof the bulbs
Saddle node
Period doubling
Period 4 bifurcation
How do we understand the arrangement of the bulbs?
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How do we understand the arrangement of the bulbs?
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Assign a fraction p/q to eachbulb hanging off the main cardioid.
q = period of the bulb
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Where is the smallest spoke in relationto the “principal spoke”?
p/3 bulb
principal spoke
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1/3 bulb
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principal spoke
The smallest spoke is located 1/3 ofa 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
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1/4
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1/4 bulb
1/3
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1/4
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1/4 bulb
1/3
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1/4
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1/4 bulb
1/3
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1/4
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1/4 bulb
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1/4
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1/4
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1/4
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1/4
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??? bulb
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2/5 bulb
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2/5 bulb
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2/5
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2/5 bulb
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2/5
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2/5 bulb
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How to count
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How to count
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How to count
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How to count
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How to count
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How to count
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The bulbs are arranged in the exactorder of the rational numbers.
How to count
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1/3
1/42/5
3/7
1/2
2/3
The bulbs are arranged in the exactorder of the rational numbers.
1/101
32,123/96,787
How to count
Animations:
Mandelbulbs
Spiralling fingers
How to add
How to add
1/2
How to add
1/2
1/3
How to add
1/2
1/3
2/5
How to add
1/2
1/3
2/5
3/7
+ =
1/2 + 1/3 = 2/5
+ =
1/2 + 2/5 = 3/7
221/2
0/1
Here’s an interesting sequence:
221/2
0/1
Watch the denominators
1/3
221/2
0/1
Watch the denominators
1/3
2/5
221/2
0/1
Watch the denominators
1/3
2/5
3/8
221/2
0/1
Watch the denominators
1/3
2/5
3/85/13
221/2
0/1
What’s next?
1/3
2/5
3/85/13
221/2
0/1
What’s next?
1/3
2/5
3/85/13
8/21
221/2
0/1
The Fibonacci sequence
1/3
2/5
3/85/13
8/2113/34
The Farey Tree
€
0
1
€
1
1
The Farey Tree
€
0
1
€
1
1
How get the fraction in betweenwith the smallest denominator?
The Farey Tree
€
0
1
€
1
1
€
1
2
How get the fraction in betweenwith the smallest denominator?
Farey addition
The Farey Tree
€
0
1
€
1
1
€
1
2
€
1
3
€
2
3
The Farey Tree
€
0
1
€
1
1
€
1
2
€
1
3
€
2
3
€
2
5
€
1
4
€
3
5
€
3
4
The Farey Tree
€
0
1
€
1
1
€
1
2
€
1
3
€
2
3
€
2
5
€
1
4
€
3
5
€
3
4
€
3
8
€
5
13
....
essentially the golden number
Another sequence (denominatorsonly)
1
2
Another sequence (denominatorsonly)
1
2
3
Another sequence (denominatorsonly)
1
2
3
4
Another sequence (denominatorsonly)
1
2
3
4
5
Another sequence (denominatorsonly)
1
2
3
4
5
6
Another sequence (denominatorsonly)
1
2
3
4
5
67
sequence
1
2
3
4
5
67
Devaney
The Dynamical Systems and Technology Project at Boston University
website: math.bu.edu/DYSYS:
Have fun!
Mandelbrot set explorer;Applets for investigating M-set;Applets for other complex functions;Chaos games, orbit diagrams, etc.
Farey.qt
Farey tree
D-sequence
Continued fraction expansion
Far from rationals
Other topics
Website
Continued fraction expansion
Let’s rewrite the sequence:
1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34, ..... as a continued fraction:
Continued fraction expansion
12
= 12
the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....
Continued fraction expansion
13
= 12 + 1
1
the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....
Continued fraction expansion
25
= 12 + 1
1 + 11
the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....
Continued fraction expansion
38
= 12 + 1
1 + 11 1
1+
the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....
Continued fraction expansion
= 12 + 1
1 + 11 1
1+
11
+
513
the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....
Continued fraction expansion
= 12 + 1
1 + 11 1
1+
11
+
821
11
+
the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....
Continued fraction expansion
= 12 + 1
1 + 11 1
1+
11
+
1334
11
+
11
+
the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....
Continued fraction expansion
= 12 + 1
1 + 11 1
1+
11
+
1334
11
+
11
+
essentially the1/golden number
the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....
We understand what happens for
= 1a + 1
b + 1c 1
d+
1e
+
1f
+
1g
+
where all entries in the sequence a, b, c, d,.... are bounded above. But if that sequence grows too quickly, we’re in trouble!!!
etc.€
θ
The real way to prove all this:
Need to measure: the size of bulbs the length of spokes the size of the “ears.”
There is an external Riemann map : C - D C - Mtaking the exterior of the unit disk to the exterior of the Mandelbrot set.
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Φ
€
Φ
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€
Φ
€
Φ takes straight rays in C - D to the “external rays” in C - M
01/2
1/3
2/3 €
γ0
€
γ1/3
€
γ2/3€
γ1/2
external ray of angle 1/3
€
1
3→2
3→1
3→
1
7→2
7→4
7→1
7→
1
5→2
5→4
5→3
5→1
5→
Suppose p/q is periodic of period k under doubling mod 1:
period 2
period 3
period 4
€
1
3→2
3→1
3→
1
7→2
7→4
7→1
7→
1
5→2
5→4
5→3
5→1
5→
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/qlands at the “root point” of a period k bulb in the Mandelbrot set.
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Φ0
1/3
2/3
€
γ00 is fixed under angle doubling, so lands at the cusp of the main cardioid.
€
γ0
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Φ0
1/3
2/3
€
γ1/3
€
γ2/3
€
γ0
€
γ1/3
€
γ2/31/3 and 2/3 have period 2 under doubling, so and land at the root of the period 2 bulb.
2
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Φ0
1/3
2/3
€
γ1/3
€
γ2/3
€
γ0
€
γ1/3
€
γ2/3And if lies between 1/3 and 2/3,then lies between and .
2
€
θ
€
γθ
€
θ
€
γθ
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Φ0
1/3
2/3
€
γ1/3
€
γ2/3
€
γ0
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., 2/3-1/3=1/3.
2
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Φ0
1/3
2/3
€
γ1/3
€
γ2/3
€
γ0
1/15 and 2/15 have period 4, andare smaller than 1/7....
1/72/7
3/7
4/7
5/7
6/7
€
γ1/7
€
γ2/7
€
γ3/7
€
γ4 /7
€
γ5/7
€
γ6/7
2
3
3
1/15
2/15
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Φ0
1/3
2/3
€
γ1/3
€
γ2/3
€
γ0
1/15 and 2/15 have period 4, andare smaller than 1/7....
1/72/7
3/7
4/7
5/7
6/7
€
γ1/7
€
γ2/7
€
γ3/7
€
γ4 /7
€
γ5/7
€
γ6/7
2
3
3
1/15
2/15
€
γ1/15€
γ2/15
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Φ0
1/3
2/3
€
γ1/3
€
γ2/3
€
γ0
1/72/7
3/7
4/7
5/7
6/7
€
γ1/7
€
γ2/7
€
γ3/7
€
γ4 /7
€
γ5/7
€
γ6/7
2
3
3
1/15
2/15
€
γ1/15€
γ2/15
3/15 and 4/15 have period 4, andare between 1/7 and 2/7....
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Φ0
1/3
2/3
€
γ1/3
€
γ2/3
€
γ0
3/15 and 4/15 have period 4, andare between 1/7 and 2/7....
1/72/7
3/7
4/7
5/7
6/7
€
γ1/7
€
γ2/7
€
γ3/7
€
γ4 /7
€
γ5/7
€
γ6/7
2
3
3
1/15
2/15
€
γ1/15€
γ2/15
1/72/7
3/15 and 4/15 have period 4, andare between 1/7 and 2/7....
1/72/7
3/15 and 4/15 have period 4, andare between 1/7 and 2/7....
3/154/15
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So what do we know about M?
All rational external rays land at a single point in M.
So what do we know about M?
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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 raysland at Misiurewicz points(how we measure lengthof antennas).
So what do we know about M?
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“Highly irrational” rays also land at unique points, and we understand what goes on here.
“Highly irrational" = “far”from rationals, i.e.,
€
θ −pq>c
qk
So what do we NOT know about M?
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But we don't know if irrationals that are “close” to rationals land.
So we won't understandquadratic functions untilwe figure this out.
MLC Conjecture:
The boundary of the M-setis “locally connected” ---if so, all rays land and we are in heaven!. But if not......
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The Dynamical Systems and Technology Project at Boston University
website: math.bu.edu/DYSYS
Have fun!
A number is far from the rationals if:
€
θ
€
|θ − p /q |
€
>
A number is far from the rationals if:
€
θ
€
|θ − p /q |
€
>
€
c /qk
A number is far from the rationals if:
€
θ
€
|θ − p /q |
€
>
€
c /qk
This happens if the “continued fraction expansion” of has only bounded terms.
€
θ
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