Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z)) 3. Rational maps (z n + /z n )
We’ll investigate chaotic behavior in the dynamical plane (the Julia sets) z 2 + c exp(z) z 2 + /z 2
As well as the structure of the parameter planes. z 2 + c exp(z)z 3 + /z 3 (the Mandelbrot set)
A couple of subthemes: 1.Some “crazy” mathematics 2.Great undergrad research topics
The Fractal Geometry of the Mandelbrot Set
How to count The Fractal Geometry of the Mandelbrot Set
The Fractal Geometry of the Mandelbrot Set How to add How to count
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 + constant 2
Start with a function: x + constant 2 and a seed: x 0
Then iterate: x = x + constant 10 2
Then iterate: x = x + constant
Then iterate: x = x + constant
Then iterate: x = x + constant
Then iterate: x = x + constant Orbit of x 0 etc. Goal: understand the fate of orbits.
Example: x + 1 Seed 0 2 x = 0 0 x =
Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x =
Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x =
Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 4 5 6
Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 26 4 x = 5 6
Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 26 4 x = big 5 x = 6
Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 26 4 x = big 5 x = BIGGER 6
Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 26 4 x = big 5 x = BIGGER 6 “Orbit tends to infinity”
Example: x + 0 Seed 0 2 x = 0 0 x =
Example: x + 0 Seed 0 2 x = x =
Example: x + 0 Seed 0 2 x = x =
Example: x + 0 Seed 0 2 x = x = 4 5 6
Example: x + 0 Seed 0 2 x = “A fixed point”
Example: x - 1 Seed 0 2 x = 0 0 x =
Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x =
Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x = 0 2 x =
Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x = 0 2 x = -1 3 x = 4 5 6
Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x = 0 2 x = -1 3 x = 0 4 x = 5 6
Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x = 0 2 x = -1 3 x = 0 4 x = -1 5 x = 0 6 “A two- cycle”
Example: x Seed 0 2 x = 0 0 x =
Example: x Seed 0 2 x = 0 0 x = x =
Example: x Seed 0 2 x = 0 0 x = x = x =
Example: x Seed 0 2 x = 0 0 x = x = x = 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 + c 2 complex numbers
Example: z + i Seed 0 2 z = 0 0 z =
Example: z + i Seed 0 2 z = 0 0 z = i 1 z =
Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z =
Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z = -i 3 z = 4 5 6
Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z = -i 3 z = -1 + i 4 z = 5 6
Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z = -i 3 z = -1 + i 4 z = -i 5 z = 6
Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z = -i 3 z = -1 + i 4 z = -i 5 z = -1 + i 6 2-cycle
Example: z + i Seed i -i
Example: z + i Seed i -i
Example: z + i Seed i -i
Example: z + i Seed 0 2 -i 1 i
Example: z + i Seed i -i
Example: z + i Seed 0 2 -i 1 i
Example: z + i Seed i -i
Example: z + i Seed 0 2 -i 1 i
Example: z + 2i Seed 0 2 z = 0 0 z =
Example: z + 2i Seed 0 2 z = 0 0 z = 2i 1 z = i 2 z = i 3 z = i 4 z = big 5 z = BIGGER 6 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. As we shall see, the orbit of the critical point determines just about everything for z 2 + c. 0 is the critical point of z 2 + 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
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
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
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 z 2 + 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:
Fix a c-value. The filled Julia set is all of the complex seeds whose orbits do NOT go to infinity.
Example: z 2 Seed: 0 In filled Julia set?
Example: z 2 Seed: 0Yes In filled Julia set?
Example: z 2 Seed: 0Yes 1 In filled Julia set?
Example: z 2 Seed: 0Yes 1 In filled Julia set?
Example: z 2 Seed: 0Yes 1 In filled Julia set?
Example: z 2 Seed: 0Yes 1 Yes In filled Julia set?
Example: z 2 Seed: 0Yes 1 Yes i In filled Julia set?
Example: z 2 Seed: 0Yes 1 Yes i In filled Julia set?
Example: z 2 Seed: 0Yes 1 Yes i 2i In filled Julia set?
Example: z 2 Seed: 0Yes 1 Yes i 2i No In filled Julia set?
Example: z 2 Seed: 0Yes 1 Yes i 2i No 5 In filled Julia set?
Example: z 2 Seed: 0Yes 1 Yes i 2i No 5No way In filled Julia set?
Filled Julia Set for z 2 All seeds on and inside the unit circle. i 1
The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic”
The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently
The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently
The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently
The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently
The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently
The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently
The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently
The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently
The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently
The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently
Other filled Julia sets
c = 0
Other filled Julia sets c = -1
Other filled Julia sets c = -1
Other filled Julia sets c = -1
Other filled Julia sets c = -1
Other filled Julia sets c = -1
Other filled Julia sets c = -1
Other filled Julia sets c = -1
Other filled Julia sets c = -1
Other filled Julia sets c = i
Other filled Julia sets c = i
Other filled Julia sets c = i
Other filled Julia sets c = i
Other filled Julia sets c = i
Other filled Julia sets c = i
If c is in the Mandelbrot set, then the filled Julia set is always a connected set.
Other filled Julia sets But if c is not in the Mandelbrot set, then the filled Julia set is totally disconnected.
Other filled Julia sets c =.3
Other filled Julia sets c =.3
Other filled Julia sets c =.3
Other filled Julia sets c =.3
Other filled Julia sets c =.3
Other filled Julia sets c = i
Another reason why we use the orbit of the critical point to plot the M-set: Theorem: (Fatou & Julia) For z 2 + c:
Another reason why we use the orbit of the critical point to plot the M-set: Theorem: (Fatou & Julia) For z 2 + 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 z 2 + 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 arrangement of the bulbs Saddle node Period doubling Period 4 bifurcation
How do we understand the arrangement of the bulbs?
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
Where is the smallest spoke in relation to the “principal spoke”? p/3 bulb principal spoke
1/3 bulb principal spoke The smallest spoke is located 1/3 of a turn in the counterclockwise direction from the principal spoke.
1/3 bulb 1/3
1/3 bulb 1/3
1/3 bulb 1/3
1/3 bulb 1/3
1/3 bulb 1/3
1/3 bulb 1/3
1/3 bulb 1/3
1/3 bulb 1/3
1/3 bulb 1/3
1/3 bulb 1/3
??? bulb 1/3
1/4 bulb 1/3
1/4 bulb 1/3 1/4
1/4 bulb 1/3 1/4
1/4 bulb 1/3 1/4
1/4 bulb 1/3 1/4
1/4 bulb 1/3 1/4
1/4 bulb 1/3 1/4
1/4 bulb 1/3 1/4
1/4 bulb 1/3 1/4
1/4 bulb 1/3 1/4
??? bulb 1/3 1/4
2/5 bulb 1/3 1/4
2/5 bulb 1/3 1/4 2/5
2/5 bulb 1/3 1/4 2/5
2/5 bulb 1/3 1/4 2/5
2/5 bulb 1/3 1/4 2/5
2/5 bulb 1/3 1/4 2/5
??? bulb 1/3 1/4 2/5
3/7 bulb 1/3 1/4 2/5
3/7 bulb 1/3 1/4 2/5 3/7
3/7 bulb 1/3 1/4 2/5 3/7
3/7 bulb 1/3 1/4 2/5 3/7
3/7 bulb 1/3 1/4 2/5 3/7
3/7 bulb 1/3 1/4 2/5 3/7
3/7 bulb 1/3 1/4 2/5 3/7
3/7 bulb 1/3 1/4 3/7 2/5
??? bulb 1/3 1/4 3/7 2/5
1/2 bulb 1/3 1/4 3/7 1/2 2/5
1/2 bulb 1/3 1/4 3/7 1/2 2/5
1/2 bulb 1/3 1/4 3/7 1/2 2/5
1/2 bulb 1/3 1/4 3/7 1/2 2/5
??? bulb 1/3 1/4 3/7 1/2 2/5
2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5
2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5
2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5
2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5
2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5
2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5
How to count
1/4 How to count
1/3 1/4 How to count
1/3 1/4 2/5 How to count
1/3 1/4 2/5 3/7 How to count
1/3 1/4 2/5 3/7 1/2 How to count
1/3 1/4 2/5 3/7 1/2 2/3 How to count
1/3 1/4 2/5 3/7 1/2 2/3 The bulbs are arranged in the exact order of the rational numbers. How to count
1/3 1/4 2/5 3/7 1/2 2/3 The bulbs are arranged in the exact order of the rational numbers. 1/101 32,123/96,787 How to count
Animations: Mandelbulbs Spiralling fingers
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
22 1/2 0/1 Here’s an interesting sequence:
22 1/2 0/1 Watch the denominators 1/3
22 1/2 0/1 Watch the denominators 1/3 2/5
22 1/2 0/1 Watch the denominators 1/3 2/5 3/8
22 1/2 0/1 Watch the denominators 1/3 2/5 3/8 5/13
22 1/2 0/1 What’s next? 1/3 2/5 3/8 5/13
22 1/2 0/1 What’s next? 1/3 2/5 3/8 5/13 8/21
22 1/2 0/1 The Fibonacci sequence 1/3 2/5 3/8 5/13 8/21 13/34
The Farey Tree
How get the fraction in between with the smallest denominator?
The Farey Tree How get the fraction in between with the smallest denominator? Farey addition
The Farey Tree
.... essentially the golden number
Another sequence (denominators only) 1 2
Another sequence (denominators only) 1 2 3
Another sequence (denominators only)
Another sequence (denominators only)
Another sequence (denominators only)
Another sequence (denominators only)
sequence 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 1212 = 1212 the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....
Continued fraction expansion 1313 = the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....
Continued fraction expansion 2525 = the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....
Continued fraction expansion 3838 = the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....
Continued fraction expansion = the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....
Continued fraction expansion = the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....
Continued fraction expansion = the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....
Continued fraction expansion = 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 = 1a1a + 1b1b + 1c1c 1d1d + 1e1e + 1f1f + 1g1g + 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.
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.
takes straight rays in C - D to the “external rays” in C - M 01/2 1/3 2/3 external ray of angle 1/3
Suppose p/q is periodic of period k under doubling mod 1: period 2 period 3 period 4
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.
0 1/3 2/3 0 is fixed under angle doubling, so lands at the cusp of the main cardioid.
0 1/3 2/3 1/3 and 2/3 have period 2 under doubling, so and land at the root of the period 2 bulb. 2
0 1/3 2/3 And if lies between 1/3 and 2/3, then lies between and. 2
0 1/3 2/3 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
0 1/3 2/3 1/15 and 2/15 have period 4, and are smaller than 1/ /7 2/7 3/7 4/7 5/7 6/ /15 2/15
0 1/3 2/3 1/15 and 2/15 have period 4, and are smaller than 1/ /7 2/7 3/7 4/7 5/7 6/ /15 2/15
0 1/3 2/3 1/7 2/7 3/7 4/7 5/7 6/ /15 2/15 3/15 and 4/15 have period 4, and are between 1/7 and 2/7....
0 1/3 2/3 3/15 and 4/15 have period 4, and are between 1/7 and 2/ /7 2/7 3/7 4/7 5/7 6/ /15 2/15
1/72/7 3/15 and 4/15 have period 4, and are between 1/7 and 2/7....
1/72/7 3/15 and 4/15 have period 4, and are between 1/7 and 2/ /154/15
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? 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).
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.,
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.
MLC Conjecture: The boundary of the M-set is “locally connected” --- if so, all rays land and we are in heaven!. But if not......
The Dynamical Systems and Technology Project at Boston University website: math.bu.edu/DYSYS Have fun!
A number is far from the rationals if:
This happens if the “continued fraction expansion” of has only bounded terms.