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 in
the dynamical plane (the Julia sets)
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z2 + c
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exp(z)
z2 + /z2
As well as the structure of the
parameter planes.
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z2 + c
(the Mandelbrot set)
 exp(z)
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z3 + /z3
A couple of subthemes:
1. Some “crazy” mathematics
2. Great undergrad research topics
The Fractal Geometry
of
the Mandelbrot Set
The Fractal Geometry
of
the Mandelbrot Set
How to count
The Fractal Geometry
of
the Mandelbrot Set
How to count
How to add
Many people know the
pretty pictures...
but few know the even
prettier 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
2
+ constant
Start with a function:
x
2
+ constant
and a seed:
x0
Then iterate:
x1 = x
2
0
+ constant
Then iterate:
x1 = x
2
0
+ constant
x2 = x
2
1
+ constant
Then iterate:
x1 = x
2
0
+ constant
x2 = x
2
1
+ constant
x3 = x
2
2
+ constant
Then iterate:
x1 = x
2
0
+ constant
x2 = x
2
1
+ constant
x3 = x
2
2
+ constant
x4 = x
2
3
+ constant
Then iterate:
x1 = x
2
0
+ constant
x2 = x
2
1
+ constant
x3 = x
2
2
+ constant
x4 = x
2
3
+ constant
Orbit of x 0
etc.
Goal: understand the fate of orbits.
2
Example: x + 1
x0 = 0
x1 =
x2 =
x3 =
x4 =
x5 =
x6 =
Seed 0
2
Example: x + 1
x0 = 0
x1 = 1
x2 =
x3 =
x4 =
x5 =
x6 =
Seed 0
2
Example: x + 1
x0 = 0
x1 = 1
x2 = 2
x3 =
x4 =
x5 =
x6 =
Seed 0
2
Example: x + 1
x0 = 0
x1 = 1
x2 = 2
x3 = 5
x4 =
x5 =
x6 =
Seed 0
2
Example: x + 1
x0 = 0
x1 = 1
x2 = 2
x3 = 5
x 4 = 26
x5 =
x6 =
Seed 0
2
Example: x + 1
x0 = 0
x1 = 1
x2 = 2
x3 = 5
x 4 = 26
x 5 = big
x6 =
Seed 0
2
Example: x + 1
Seed 0
x0 = 0
x1 = 1
x2 = 2
x3 = 5
x 4 = 26
x 5 = big
x 6 = BIGGER
2
Example: x + 1
Seed 0
x0 = 0
x1 = 1
x2 = 2
x3 = 5
x 4 = 26
x 5 = big
x 6 = BIGGER
“Orbit tends
to infinity”
2
Example: x + 0
x0 = 0
x1 =
x2 =
x3 =
x4 =
x5 =
x6 =
Seed 0
2
Example: x + 0
x0 = 0
x1 = 0
x2 =
x3 =
x4 =
x5 =
x6 =
Seed 0
2
Example: x + 0
x0 = 0
x1 = 0
x2 = 0
x3 =
x4 =
x5 =
x6 =
Seed 0
2
Example: x + 0
x0 = 0
x1 = 0
x2 = 0
x3 = 0
x4 =
x5 =
x6 =
Seed 0
2
Example: x + 0
x0 = 0
x1 = 0
x2 = 0
x3 = 0
x4 = 0
x5 = 0
x6 = 0
Seed 0
“A fixed
point”
2
Example: x - 1
x0 = 0
x1 =
x2 =
x3 =
x4 =
x5 =
x6 =
Seed 0
2
Example: x - 1
x0 = 0
x 1 = -1
x2 =
x3 =
x4 =
x5 =
x6 =
Seed 0
2
Example: x - 1
x0 = 0
x 1 = -1
x2 = 0
x3 =
x4 =
x5 =
x6 =
Seed 0
2
Example: x - 1
x0 = 0
x 1 = -1
x2 = 0
x 3 = -1
x4 =
x5 =
x6 =
Seed 0
2
Example: x - 1
x0 = 0
x 1 = -1
x2 = 0
x 3 = -1
x4 = 0
x5 =
x6 =
Seed 0
2
Example: x - 1
Seed 0
x0 = 0
x 1 = -1
x2 = 0
x 3 = -1
x4 = 0
x 5 = -1
x6 = 0
“A twocycle”
2
Example: x - 1.1
x0 = 0
x1 =
x2 =
x3 =
x4 =
x5 =
x6 =
Seed 0
2
Example: x - 1.1
x0 = 0
x 1 = -1.1
x2 =
x3 =
x4 =
x5 =
x6 =
Seed 0
2
Example: x - 1.1
x0 = 0
x 1 = -1.1
x 2 = 0.11
x3 =
x4 =
x5 =
x6 =
Seed 0
2
Example: x - 1.1
Seed 0
x0 = 0
x 1 = -1.1
x 2 = 0.11
x3 =
x4 =
x5 =
x6 =
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
2
Iterate z + c
complex
numbers
2
Example: z + i
z0 = 0
z1 =
z2 =
z3 =
z4 =
z5 =
z6 =
Seed 0
2
Example: z + i
z0 = 0
z1 = i
z2 =
z3 =
z4 =
z5 =
z6 =
Seed 0
2
Example: z + i
Seed 0
z0 = 0
z1 = i
z 2 = -1 + i
z3 =
z4 =
z5 =
z6 =
2
Example: z + i
Seed 0
z0 = 0
z1 = i
z 2 = -1 + i
z 3 = -i
z4 =
z5 =
z6 =
2
Example: z + i
Seed 0
z0 = 0
z1 = i
z 2 = -1 + i
z 3 = -i
z 4 = -1 + i
z5 =
z6 =
2
Example: z + i
Seed 0
z0 = 0
z1 = i
z 2 = -1 + i
z 3 = -i
z 4 = -1 + i
z 5 = -i
z6 =
2
Example: z + i
Seed 0
z0 = 0
z1 = i
z 2 = -1 + i
z 3 = -i
z 4 = -1 + i
z 5 = -i
z 6 = -1 + i
2-cycle
2
Example: z + i
Seed 0
i
1
-1
-i
2
Example: z + i
Seed 0
i
1
-1
-i
2
Example: z + i
Seed 0
i
1
-1
-i
2
Example: z + i
Seed 0
i
1
-1
-i
2
Example: z + i
Seed 0
i
1
-1
-i
2
Example: z + i
Seed 0
i
1
-1
-i
2
Example: z + i
Seed 0
i
1
-1
-i
2
Example: z + i
Seed 0
i
1
-1
-i
2
Example: z + 2i
z0 = 0
z1 =
z2 =
z3 =
z4 =
z5 =
z6 =
Seed 0
2
Example: z + 2i
Seed 0
z0 = 0
z 1 = 2i
z 2 = -4 + 2i
z 3 = 12 - 14i
z 4 = -52 + 336i
z 5 = big
z 6 = BIGGER
Off to
infinity
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.
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.
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 orbit
of 0 for each c. If
the orbit of 0 escapes,
color that grid point.
red = fastest escape
Algorithm for computing M
Compute the orbit
of 0 for each c. If
the orbit of 0 escapes,
color that grid point.
orange = slower
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
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.
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.
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
periods of the bulbs?
How understand the
periods of the bulbs?
junction point
three spokes
attached
junction point
three spokes
attached
Period 3 bulb
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: z
Seed:
0
2
In filled Julia set?
Example: z
2
Seed:
In filled Julia set?
0
Yes
Example: z
2
Seed:
In filled Julia set?
0
Yes
1
Example: z
2
Seed:
In filled Julia set?
0
Yes
1
Yes
Example: z
2
Seed:
In filled Julia set?
0
Yes
1
Yes
-1
Example: z
2
Seed:
In filled Julia set?
0
Yes
1
Yes
-1
Yes
Example: z
2
Seed:
In filled Julia set?
0
Yes
1
Yes
-1
Yes
i
Example: z
2
Seed:
In filled Julia set?
0
Yes
1
Yes
-1
Yes
i
Yes
Example: z
2
Seed:
In filled Julia set?
0
Yes
1
Yes
-1
Yes
i
Yes
2i
Example: z
2
Seed:
In filled Julia set?
0
Yes
1
Yes
-1
Yes
i
Yes
2i
No
Example: z
2
Seed:
In filled Julia set?
0
Yes
1
Yes
-1
Yes
i
Yes
2i
No
5
Example: z
2
Seed:
In filled Julia set?
0
Yes
1
Yes
-1
Yes
i
Yes
2i
No
5
No way
Filled Julia Set for z
2
i
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-1
1
All seeds on and inside the unit circle.
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
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c=0
Other filled Julia sets
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c = -1
Other filled Julia sets
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c = -1
Other filled Julia sets
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c = -1
Other filled Julia sets
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c = -1
Other filled Julia sets
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c = -1
Other filled Julia sets
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c = -1
Other filled Julia sets
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c = -1
Other filled Julia sets
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c = -1
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
Other filled Julia sets
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c = -.12+.75i
Other filled Julia sets
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c = -.12+.75i
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
Other filled Julia sets
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c = .3
Other filled Julia sets
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c = .3
Other filled Julia sets
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c = .3
Other filled Julia sets
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c = .3
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c = -.8+.4i
Another reason why we use the orbit of
the critical point to plot the M-set:
Theorem: (Fatou & Julia)
For z2 + c:
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.
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).
Animations:
In and out of M
Saddle node
Period doubling
Period 4 bifurcation
arrangement
of the bulbs
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 each
bulb hanging off the main cardioid.
q = period of the bulb
p/3 bulb
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Where is the smallest spoke in relation
to the “principal spoke”?
1/3 bulb
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principal
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The smallest spoke is located 1/3 of
a turn in the counterclockwise direction
from the principal spoke.
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
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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 exact
order of the rational numbers.
How to count
1/3
32,123/96,787
2/5
1/4
3/7
1/101
1/2
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The bulbs are arranged in the exact
order of the rational numbers.
Animations:
Mandelbulbs
Spiralling fingers
How to add
How to add
1/2
How to add
1/3
1/2
How to add
1/3
2/5
1/2
How to add
1/3
2/5
3/7
1/2
1/2 + 1/3 = 2/5
+
=
1/2 + 2/5 = 3/7
+
=
Here’s an interesting sequence:
1/2
22
0/1
Watch the denominators
1/3
1/2
22
0/1
Watch the denominators
1/3
2/5
1/2
22
0/1
Watch the denominators
3/8
1/3
2/5
1/2
22
0/1
Watch the denominators
5/13
3/8
1/3
2/5
1/2
22
0/1
What’s next?
5/13
3/8
1/3
2/5
1/2
22
0/1
What’s next?
5/13
8/21
3/8
1/3
2/5
1/2
22
0/1
The Fibonacci sequence
13/34
5/13
8/21
3/8
1/3
2/5
1/2
22
0/1
The Farey Tree
0
1
1
1

The Farey Tree
0
1
1
1
How get the fraction in between

with the smallest denominator?
The Farey Tree
0
1
1
2
How get the fraction in between

with the smallest denominator?

Farey addition
1
1
The Farey Tree
0
1
1
2
1
3
2
3


1
1

The Farey Tree
0
1
1
1
1
2
1
3
1
4

3
5
2
5

2
3




3
4
The Farey Tree
0
1
1
1
1
2
1
3
1
4

3
8



3
5
2
5
5
13 
2
3
3
4


essentially the golden number
Another sequence
(denominators
only)
2
1
Another sequence
(denominators
only)
3
2
1
(denominators
only)
Another sequence
3
4
2
1
(denominators
only)
Another sequence
3
4
2
5
1
(denominators
only)
Another sequence
3
4
2
5
6
1
(denominators
only)
Another sequence
3
4
2
5
6
7
1
Devaney sequence
3
4
2
5
6
7
1
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!
Other topics
Farey.qt
Farey tree
D-sequence
Far from rationals
Continued fraction expansion
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
1
2
= 1
2
the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....
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,.....
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,.....
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,.....
Continued fraction expansion
5
13
= 1
2 + 1
1 + 1
1 + 1
1 + 1
1
the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....
Continued fraction expansion
8
21
= 1
2 + 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,.....
Continued fraction expansion
13
34
= 1
2 + 1
1 + 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,.....
Continued fraction expansion
13
34
= 1
2 + 1
1 + 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,.....

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, we’re in trouble!!!
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 - M
taking the exterior of the unit disk to
the exterior of the Mandelbrot set.



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
takes straight rays in C - D to
the “external rays” in C - M
 1/ 3

external ray of angle 1/3
1/3


1/2
0
2/3

 1/2
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

2/3
0
Suppose p/q is periodic of period
k under doubling mod 1:
1
2
1
  
3
3
3
1
2
4
1
   
7
7
7
7
1
2
4
3 1
    
5
5
5
5
5

period 2
period 3
period 4
Suppose p/q is periodic of period
k under doubling mod 1:
1
2
1
  
3
3
3
1
2
4
1
   
7
7
7
7
1
2
4
3 1
    
5
5
5
5
5
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.
0
0 is fixed under angle doubling, so
lands at the cusp of the main cardioid.

1/3

0
2/3

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
0
 1/ 3
1/3 and 2/3 have period 2 under doubling, so
and  2 / 3land at the root of the period 2 bulb.
 1/ 3


1/3


0
2/3
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

2/3
0
And if  lies between 1/3 and 2/3,
and
1/ 3
then   lies between
. 2 / 3




 1/ 3

1/3


0
2/3

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

2/3
0
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.
 1/ 3
1/3


0
2/3
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

2/3
0
1/15 and 2/15 have period 4, and
are smaller than 1/7....
 1/ 3
1/3
2/7
 1/ 7
2/7
1/7
3/7
2/15

 3/7
1/15


4/7

2/3



0
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0
 4 /7
3

6/7
5/7
3
2/3
5/7
6/7
1/15 and 2/15 have period 4, and
are smaller than 1/7....
 1/ 3
1/3
2/7
 1/ 7
 2 /15
2/7
1/7
3/7
2/15

1/15
 3/7


4/7

2/3




0
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0

 4 /7
3

6/7
5/7
 1/15
3
2/3
5/7
6/7
3/15 and 4/15 have period 4, and
are between 1/7 and 2/7....
 1/ 3
1/3
2/7
 1/ 7
 2 /15
2/7
1/7
3/7
2/15

1/15
 3/7


4/7

2/3




0
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0

 4 /7
3

6/7
5/7
 1/15
3
2/3
5/7
6/7
3/15 and 4/15 have period 4, and
are between 1/7 and 2/7....
 1/ 3
1/3
2/7
 1/ 7
 2 /15
2/7
1/7
3/7
2/15

1/15
 3/7


4/7

2/3




0
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0

 4 /7
3

6/7
5/7
 1/15
3
2/3
5/7
6/7
3/15 and 4/15 have period 4, and
are between 1/7 and 2/7....
2/7
1/7
3/15 and 4/15 have period 4, and
are between 1/7 and 2/7....
4/15
2/7
3/15
1/7
So what do we know about M?
All rational external rays
land at a single point in M.
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So what do we know about M?
All rational external rays
land at a single point in M.
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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).
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.,
p
c
  k
q q
So what do we NOT know about M?
But we don't know if
irrationals that are “close”
to rationals land.
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So we won't understand
quadratic functions until
we figure this out.
MLC Conjecture:
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The boundary of the M-set
is “locally connected” --if so, all rays land and we are
in heaven!. But if not......
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
The Dynamical Systems and
Technology Project
at Boston University
website: math.bu.edu/DYSYS
Have fun!
A number
is far from the rationals if:
|  p /q | 



A number
is far from the rationals if:
k

|  p /q |
c/q


 
A number
is far from the rationals if:
k

|  p /q |
c/q

This happens if the “continued fraction
expansion” of has only bounded terms.
 

