complex dynamics and crazy mathematics dynamics of three very different families of complex...
TRANSCRIPT
![Page 1: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/1.jpg)
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)
![Page 2: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/2.jpg)
We’ll investigate chaotic behavior inthe dynamical plane (the Julia sets)
z2 + c
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
exp(z) z2 + /z2
![Page 3: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/3.jpg)
As well as the structure of theparameter planes.
z2 + c exp(z) z3 + /z3
(the Mandelbrot set)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
![Page 4: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/4.jpg)
A couple of subthemes:
1. Some “crazy” mathematics
2. Great undergrad research topics
![Page 5: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/5.jpg)
The Fractal Geometryof
the Mandelbrot Set
![Page 6: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/6.jpg)
How to count
The Fractal Geometryof
the Mandelbrot Set
![Page 7: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/7.jpg)
The Fractal Geometryof
the Mandelbrot Set
How to add
How to count
![Page 8: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/8.jpg)
Many people know thepretty pictures...
![Page 9: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/9.jpg)
but few know the evenprettier mathematics.
![Page 10: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/10.jpg)
![Page 11: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/11.jpg)
![Page 12: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/12.jpg)
![Page 13: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/13.jpg)
![Page 14: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/14.jpg)
![Page 15: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/15.jpg)
![Page 16: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/16.jpg)
![Page 17: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/17.jpg)
![Page 18: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/18.jpg)
![Page 19: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/19.jpg)
![Page 20: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/20.jpg)
![Page 21: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/21.jpg)
![Page 22: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/22.jpg)
![Page 23: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/23.jpg)
Oh, that's nothing but the 3/4 bulb ....
![Page 24: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/24.jpg)
...hanging off the period 16 M-set.....
![Page 25: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/25.jpg)
...lying in the 1/7 antenna...
![Page 26: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/26.jpg)
...attached to the 1/3 bulb...
![Page 27: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/27.jpg)
...hanging off the 3/7 bulb...
![Page 28: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/28.jpg)
...on the northwest side of the main cardioid.
![Page 29: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/29.jpg)
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.
![Page 30: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/30.jpg)
Start with a function:
x + constant2
![Page 31: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/31.jpg)
Start with a function:
x + constant2
and a seed:
x0
![Page 32: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/32.jpg)
Then iterate:
x = x + constant1 02
![Page 33: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/33.jpg)
Then iterate:
x = x + constant1 02
x = x + constant2 12
![Page 34: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/34.jpg)
Then iterate:
x = x + constant1 02
x = x + constant2 12
x = x + constant3 2
2
![Page 35: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/35.jpg)
Then iterate:
x = x + constant1 02
x = x + constant2 12
x = x + constant3 2
2
x = x + constant4 3
2
![Page 36: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/36.jpg)
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.
![Page 37: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/37.jpg)
Example: x + 1 Seed 02
x = 00x = 1x = 2
x = 3
x = 4
x = 5
x = 6
![Page 38: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/38.jpg)
Example: x + 1 Seed 02
x = 00x = 11x =2
x = 3
x = 4
x = 5
x = 6
![Page 39: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/39.jpg)
Example: x + 1 Seed 02
x = 00x = 11x = 22
x = 3
x = 4
x = 5
x = 6
![Page 40: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/40.jpg)
Example: x + 1 Seed 02
x = 00x = 11x = 22
x = 53
x = 4
x = 5
x = 6
![Page 41: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/41.jpg)
Example: x + 1 Seed 02
x = 00x = 11x = 22
x = 53
x = 264
x = 5
x = 6
![Page 42: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/42.jpg)
Example: x + 1 Seed 02
x = 00x = 11x = 22
x = 53
x = 264
x = big5
x = 6
![Page 43: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/43.jpg)
Example: x + 1 Seed 02
x = 00x = 11x = 22
x = 53
x = 264
x = big5
x = BIGGER6
![Page 44: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/44.jpg)
Example: x + 1 Seed 02
x = 00x = 11x = 22
x = 53
x = 264
x = big5
x = BIGGER6
“Orbit tends to infinity”
![Page 45: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/45.jpg)
Example: x + 0 Seed 02
x = 00x = 1x = 2
x = 3
x = 4
x = 5
x = 6
![Page 46: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/46.jpg)
Example: x + 0 Seed 02
x = 00x = 01x = 2
x = 3
x = 4
x = 5
x = 6
![Page 47: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/47.jpg)
Example: x + 0 Seed 02
x = 00x = 01x = 02
x = 3
x = 4
x = 5
x = 6
![Page 48: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/48.jpg)
Example: x + 0 Seed 02
x = 00x = 01x = 02
x = 03
x = 4
x = 5
x = 6
![Page 49: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/49.jpg)
Example: x + 0 Seed 02
x = 00x = 01x = 02
x = 03
x = 04
x = 05
x = 06
“A fixed point”
![Page 50: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/50.jpg)
Example: x - 1 Seed 02
x = 00x = 1x = 2
x = 3
x = 4
x = 5
x = 6
![Page 51: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/51.jpg)
Example: x - 1 Seed 02
x = 00x = -11x = 2
x = 3
x = 4
x = 5
x = 6
![Page 52: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/52.jpg)
Example: x - 1 Seed 02
x = 00x = -11x = 02
x = 3
x = 4
x = 5
x = 6
![Page 53: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/53.jpg)
Example: x - 1 Seed 02
x = 00x = -11x = 02
x = -13
x = 4
x = 5
x = 6
![Page 54: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/54.jpg)
Example: x - 1 Seed 02
x = 00x = -11x = 02
x = -13
x = 04
x = 5
x = 6
![Page 55: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/55.jpg)
Example: x - 1 Seed 02
x = 00x = -11x = 02
x = -13
x = 04
x = -15
x = 06
“A two- cycle”
![Page 56: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/56.jpg)
Example: x - 1.1 Seed 02
x = 00x = 1x = 2
x = 3
x = 4
x = 5
x = 6
![Page 57: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/57.jpg)
Example: x - 1.1 Seed 02
x = 00x = -1.11x = 2
x = 3
x = 4
x = 5
x = 6
![Page 58: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/58.jpg)
Example: x - 1.1 Seed 02
x = 00x = -1.11x = 0.112
x = 3
x = 4
x = 5
x = 6
![Page 59: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/59.jpg)
Example: x - 1.1 Seed 02
x = 00x = -1.11x = 0.112
x = 3
x = 4
x = 5
x = 6
time for the computer!
![Page 60: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/60.jpg)
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.
![Page 61: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/61.jpg)
Complex Iteration
Iterate z + c2
complexnumbers
![Page 62: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/62.jpg)
Example: z + i Seed 02
z = 00z = 1z = 2
z = 3
z = 4
z = 5
z = 6
![Page 63: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/63.jpg)
Example: z + i Seed 02
z = 00z = i1z = 2
z = 3
z = 4
z = 5
z = 6
![Page 64: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/64.jpg)
Example: z + i Seed 02
z = 00z = i1z = -1 + i2
z = 3
z = 4
z = 5
z = 6
![Page 65: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/65.jpg)
Example: z + i Seed 02
z = 00z = i1z = -1 + i2
z = -i 3
z = 4
z = 5
z = 6
![Page 66: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/66.jpg)
Example: z + i Seed 02
z = 00z = i1z = -1 + i2
z = -i 3
z = -1 + i 4
z = 5
z = 6
![Page 67: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/67.jpg)
Example: z + i Seed 02
z = 00z = i1z = -1 + i2
z = -i 3
z = -1 + i 4
z = -i 5
z = 6
![Page 68: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/68.jpg)
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
![Page 69: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/69.jpg)
Example: z + i Seed 02
1-1
i
-i
![Page 70: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/70.jpg)
Example: z + i Seed 02
1-1
i
-i
![Page 71: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/71.jpg)
Example: z + i Seed 02
1-1
i
-i
![Page 72: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/72.jpg)
Example: z + i Seed 02
-i
-1 1
i
![Page 73: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/73.jpg)
Example: z + i Seed 02
1-1
i
-i
![Page 74: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/74.jpg)
Example: z + i Seed 02
-i
-1 1
i
![Page 75: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/75.jpg)
Example: z + i Seed 02
1-1
i
-i
![Page 76: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/76.jpg)
Example: z + i Seed 02
-i
-1 1
i
![Page 77: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/77.jpg)
Example: z + 2i Seed 02
z = 00z = 1z = 2
z = 3
z = 4
z = 5
z = 6
![Page 78: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/78.jpg)
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
![Page 79: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/79.jpg)
Same observation
Sometimes orbit of 0 goes to infinity, other times it does not.
![Page 80: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/80.jpg)
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?
![Page 81: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/81.jpg)
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.
![Page 82: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/82.jpg)
Algorithm for computing M
Start with a grid of complex numbers
![Page 83: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/83.jpg)
Algorithm for computing M
Each grid point is a complex c-value.
![Page 84: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/84.jpg)
Algorithm for computing M
Compute the orbitof 0 for each c. Ifthe orbit of 0 escapes,color that grid point.
red = fastest escape
![Page 85: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/85.jpg)
Algorithm for computing M
Compute the orbitof 0 for each c. Ifthe orbit of 0 escapes,color that grid point.
orange = slower
![Page 86: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/86.jpg)
Algorithm for computing M
Compute the orbitof 0 for each c. Ifthe orbit of 0 escapes,color that grid point.
yellowgreenblueviolet
![Page 87: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/87.jpg)
Algorithm for computing M
Compute the orbitof 0 for each c. Ifthe orbit of 0 does not escape, leave that grid pointblack.
![Page 88: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/88.jpg)
Algorithm for computing M
Compute the orbitof 0 for each c. Ifthe orbit of 0 does not escape, leave that grid pointblack.
![Page 89: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/89.jpg)
The eventual orbit of 0
![Page 90: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/90.jpg)
The eventual orbit of 0
![Page 91: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/91.jpg)
The eventual orbit of 0
3-cycle
![Page 92: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/92.jpg)
The eventual orbit of 0
3-cycle
![Page 93: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/93.jpg)
The eventual orbit of 0
3-cycle
![Page 94: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/94.jpg)
The eventual orbit of 0
3-cycle
![Page 95: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/95.jpg)
The eventual orbit of 0
3-cycle
![Page 96: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/96.jpg)
The eventual orbit of 0
3-cycle
![Page 97: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/97.jpg)
The eventual orbit of 0
3-cycle
![Page 98: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/98.jpg)
The eventual orbit of 0
3-cycle
![Page 99: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/99.jpg)
The eventual orbit of 0
3-cycle
![Page 100: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/100.jpg)
The eventual orbit of 0
![Page 101: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/101.jpg)
The eventual orbit of 0
![Page 102: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/102.jpg)
The eventual orbit of 0
4-cycle
![Page 103: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/103.jpg)
The eventual orbit of 0
4-cycle
![Page 104: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/104.jpg)
The eventual orbit of 0
4-cycle
![Page 105: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/105.jpg)
The eventual orbit of 0
4-cycle
![Page 106: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/106.jpg)
The eventual orbit of 0
4-cycle
![Page 107: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/107.jpg)
The eventual orbit of 0
4-cycle
![Page 108: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/108.jpg)
The eventual orbit of 0
4-cycle
![Page 109: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/109.jpg)
The eventual orbit of 0
4-cycle
![Page 110: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/110.jpg)
The eventual orbit of 0
![Page 111: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/111.jpg)
The eventual orbit of 0
![Page 112: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/112.jpg)
The eventual orbit of 0
5-cycle
![Page 113: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/113.jpg)
The eventual orbit of 0
5-cycle
![Page 114: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/114.jpg)
The eventual orbit of 0
5-cycle
![Page 115: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/115.jpg)
The eventual orbit of 0
5-cycle
![Page 116: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/116.jpg)
The eventual orbit of 0
5-cycle
![Page 117: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/117.jpg)
The eventual orbit of 0
5-cycle
![Page 118: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/118.jpg)
The eventual orbit of 0
5-cycle
![Page 119: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/119.jpg)
The eventual orbit of 0
5-cycle
![Page 120: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/120.jpg)
The eventual orbit of 0
5-cycle
![Page 121: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/121.jpg)
The eventual orbit of 0
5-cycle
![Page 122: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/122.jpg)
The eventual orbit of 0
5-cycle
![Page 123: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/123.jpg)
The eventual orbit of 0
2-cycle
![Page 124: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/124.jpg)
The eventual orbit of 0
2-cycle
![Page 125: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/125.jpg)
The eventual orbit of 0
2-cycle
![Page 126: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/126.jpg)
The eventual orbit of 0
2-cycle
![Page 127: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/127.jpg)
The eventual orbit of 0
2-cycle
![Page 128: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/128.jpg)
The eventual orbit of 0
fixed point
![Page 129: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/129.jpg)
The eventual orbit of 0
fixed point
![Page 130: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/130.jpg)
The eventual orbit of 0
fixed point
![Page 131: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/131.jpg)
The eventual orbit of 0
fixed point
![Page 132: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/132.jpg)
The eventual orbit of 0
fixed point
![Page 133: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/133.jpg)
The eventual orbit of 0
fixed point
![Page 134: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/134.jpg)
The eventual orbit of 0
fixed point
![Page 135: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/135.jpg)
The eventual orbit of 0
fixed point
![Page 136: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/136.jpg)
The eventual orbit of 0
goes to infinity
![Page 137: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/137.jpg)
The eventual orbit of 0
goes to infinity
![Page 138: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/138.jpg)
The eventual orbit of 0
goes to infinity
![Page 139: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/139.jpg)
The eventual orbit of 0
goes to infinity
![Page 140: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/140.jpg)
The eventual orbit of 0
goes to infinity
![Page 141: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/141.jpg)
The eventual orbit of 0
goes to infinity
![Page 142: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/142.jpg)
The eventual orbit of 0
goes to infinity
![Page 143: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/143.jpg)
The eventual orbit of 0
goes to infinity
![Page 144: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/144.jpg)
The eventual orbit of 0
goes to infinity
![Page 145: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/145.jpg)
The eventual orbit of 0
goes to infinity
![Page 146: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/146.jpg)
The eventual orbit of 0
goes to infinity
![Page 147: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/147.jpg)
The eventual orbit of 0
gone to infinity
![Page 148: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/148.jpg)
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.
![Page 149: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/149.jpg)
How understand the of the bulbs?periods
![Page 150: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/150.jpg)
How understand the of the bulbs?periods
![Page 151: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/151.jpg)
junction point
three spokes attached
![Page 152: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/152.jpg)
Period 3 bulb
junction point
three spokes attached
![Page 153: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/153.jpg)
![Page 154: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/154.jpg)
![Page 155: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/155.jpg)
Period 4 bulb
![Page 156: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/156.jpg)
![Page 157: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/157.jpg)
![Page 158: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/158.jpg)
Period 5 bulb
![Page 159: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/159.jpg)
![Page 160: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/160.jpg)
![Page 161: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/161.jpg)
Period 7 bulb
![Page 162: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/162.jpg)
![Page 163: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/163.jpg)
![Page 164: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/164.jpg)
![Page 165: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/165.jpg)
Period 13 bulb
![Page 166: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/166.jpg)
Filled Julia Set:
![Page 167: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/167.jpg)
Filled Julia Set:
Fix a c-value. The filled Julia set is all of the complex seeds whose orbits do NOT go to infinity.
![Page 168: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/168.jpg)
Example: z2
Seed:
0
In filled Julia set?
![Page 169: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/169.jpg)
Example: z2
Seed:
0 Yes
In filled Julia set?
![Page 170: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/170.jpg)
Example: z2
Seed:
0 Yes
1
In filled Julia set?
![Page 171: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/171.jpg)
Example: z2
Seed:
0 Yes
1 Yes
In filled Julia set?
![Page 172: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/172.jpg)
Example: z2
Seed:
0 Yes
1 Yes
-1
In filled Julia set?
![Page 173: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/173.jpg)
Example: z2
Seed:
0 Yes
1 Yes
-1 Yes
In filled Julia set?
![Page 174: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/174.jpg)
Example: z2
Seed:
0 Yes
1 Yes
-1 Yes
i
In filled Julia set?
![Page 175: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/175.jpg)
Example: z2
Seed:
0 Yes
1 Yes
-1 Yes
i Yes
In filled Julia set?
![Page 176: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/176.jpg)
Example: z2
Seed:
0 Yes
1 Yes
-1 Yes
i Yes
2i
In filled Julia set?
![Page 177: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/177.jpg)
Example: z2
Seed:
0 Yes
1 Yes
-1 Yes
i Yes
2i No
In filled Julia set?
![Page 178: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/178.jpg)
Example: z2
Seed:
0 Yes
1 Yes
-1 Yes
i Yes
2i No
5
In filled Julia set?
![Page 179: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/179.jpg)
Example: z2
Seed:
0 Yes
1 Yes
-1 Yes
i Yes
2i No
5 No way
In filled Julia set?
![Page 180: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/180.jpg)
Filled Julia Set for z 2
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
All seeds on and inside the unit circle.
i
1-1
![Page 181: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/181.jpg)
The Julia Set is the boundary of the filled Julia set
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
That’s where the map is “chaotic”
![Page 182: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/182.jpg)
The Julia Set is the boundary of the filled Julia set
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
That’s where the map is “chaotic”Nearby orbits behave very differently
![Page 183: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/183.jpg)
The Julia Set is the boundary of the filled Julia set
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
That’s where the map is “chaotic”Nearby orbits behave very differently
![Page 184: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/184.jpg)
The Julia Set is the boundary of the filled Julia set
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
That’s where the map is “chaotic”Nearby orbits behave very differently
![Page 185: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/185.jpg)
The Julia Set is the boundary of the filled Julia set
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
That’s where the map is “chaotic”Nearby orbits behave very differently
![Page 186: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/186.jpg)
The Julia Set is the boundary of the filled Julia set
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
That’s where the map is “chaotic”Nearby orbits behave very differently
![Page 187: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/187.jpg)
The Julia Set is the boundary of the filled Julia set
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
That’s where the map is “chaotic”Nearby orbits behave very differently
![Page 188: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/188.jpg)
The Julia Set is the boundary of the filled Julia set
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
That’s where the map is “chaotic”Nearby orbits behave very differently
![Page 189: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/189.jpg)
The Julia Set is the boundary of the filled Julia set
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
That’s where the map is “chaotic”Nearby orbits behave very differently
![Page 190: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/190.jpg)
The Julia Set is the boundary of the filled Julia set
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
That’s where the map is “chaotic”Nearby orbits behave very differently
![Page 191: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/191.jpg)
The Julia Set is the boundary of the filled Julia set
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
That’s where the map is “chaotic”Nearby orbits behave very differently
![Page 192: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/192.jpg)
Other filled Julia sets
![Page 193: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/193.jpg)
Other filled Julia sets
c = 0
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 194: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/194.jpg)
Other filled Julia sets
c = -1
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 195: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/195.jpg)
Other filled Julia sets
c = -1
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 196: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/196.jpg)
Other filled Julia sets
c = -1
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 197: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/197.jpg)
Other filled Julia sets
c = -1
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 198: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/198.jpg)
Other filled Julia sets
c = -1
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 199: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/199.jpg)
Other filled Julia sets
c = -1
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 200: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/200.jpg)
Other filled Julia sets
c = -1
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 201: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/201.jpg)
Other filled Julia sets
c = -1
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 202: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/202.jpg)
Other filled Julia sets
c = -.12+.75i
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 203: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/203.jpg)
Other filled Julia sets
c = -.12+.75i
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 204: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/204.jpg)
Other filled Julia sets
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
c = -.12+.75i
![Page 205: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/205.jpg)
Other filled Julia sets
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
c = -.12+.75i
![Page 206: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/206.jpg)
Other filled Julia sets
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
c = -.12+.75i
![Page 207: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/207.jpg)
Other filled Julia sets
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
c = -.12+.75i
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 208: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/208.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
If c is in the Mandelbrot set, then the filled Julia set is always a connected set.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 209: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/209.jpg)
Other filled Julia sets
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
But if c is not in the Mandelbrot set, then the filled Julia set is totally disconnected.
![Page 210: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/210.jpg)
Other filled Julia sets
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
c = .3
QuickTime™ and a decompressor
are needed to see this picture.
![Page 211: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/211.jpg)
Other filled Julia sets
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
c = .3
QuickTime™ and a decompressor
are needed to see this picture.
![Page 212: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/212.jpg)
Other filled Julia sets
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
c = .3
QuickTime™ and a decompressor
are needed to see this picture.
![Page 213: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/213.jpg)
Other filled Julia sets
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
c = .3
QuickTime™ and a decompressor
are needed to see this picture.
![Page 214: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/214.jpg)
Other filled Julia sets
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
c = .3
QuickTime™ and a decompressor
are needed to see this picture.
![Page 215: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/215.jpg)
Other filled Julia sets
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
c = -.8+.4i
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 216: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/216.jpg)
Another reason why we use the orbit ofthe critical point to plot the M-set:
Theorem: (Fatou & Julia) For z2 + c:
![Page 217: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/217.jpg)
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.
![Page 218: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/218.jpg)
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.
![Page 219: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/219.jpg)
Animations:
In and out of M
arrangementof the bulbs
Saddle node
Period doubling
Period 4 bifurcation
![Page 220: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/220.jpg)
How do we understand the arrangement of the bulbs?
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 221: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/221.jpg)
How do we understand the arrangement of the bulbs?
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Assign a fraction p/q to eachbulb hanging off the main cardioid.
q = period of the bulb
![Page 222: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/222.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Where is the smallest spoke in relationto the “principal spoke”?
p/3 bulb
principal spoke
![Page 223: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/223.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/3 bulb
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
principal spoke
The smallest spoke is located 1/3 ofa turn in the counterclockwise direction
from the principal spoke.
![Page 224: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/224.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/3 bulb
1/3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 225: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/225.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/3 bulb
1/3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 226: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/226.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/3 bulb
1/3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 227: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/227.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/3 bulb
1/3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 228: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/228.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/3 bulb
1/3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 229: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/229.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/3 bulb
1/3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 230: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/230.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/3 bulb
1/3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 231: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/231.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/3 bulb
1/3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 232: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/232.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/3 bulb
1/3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 233: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/233.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/3 bulb
1/3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 234: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/234.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
??? bulb
1/3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 235: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/235.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/4 bulb
1/3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 236: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/236.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/4 bulb
1/3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/4
![Page 237: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/237.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/4 bulb
1/3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/4
![Page 238: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/238.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/4 bulb
1/3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/4
![Page 239: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/239.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/4 bulb
1/3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/4
![Page 240: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/240.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/4 bulb
1/3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/4
![Page 241: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/241.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/4 bulb
1/3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/4
![Page 242: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/242.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/4 bulb
1/3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/4
![Page 243: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/243.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/4 bulb
1/3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/4
![Page 244: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/244.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/4 bulb
1/3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/4
![Page 245: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/245.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
??? bulb
1/3
1/4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 246: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/246.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/5 bulb
1/3
1/4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 247: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/247.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/5 bulb
1/3
1/4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/5
![Page 248: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/248.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/5 bulb
1/3
1/4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/5
![Page 249: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/249.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/5 bulb
1/3
1/4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/5
![Page 250: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/250.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/5 bulb
1/3
1/4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/5
![Page 251: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/251.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/5 bulb
1/3
1/4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/5
![Page 252: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/252.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
??? bulb
1/3
1/4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/5
![Page 253: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/253.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
3/7 bulb
1/3
1/4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/5
![Page 254: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/254.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
3/7 bulb
1/3
1/4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/5
3/7
![Page 255: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/255.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
3/7 bulb
1/3
1/4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/5
3/7
![Page 256: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/256.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
3/7 bulb
1/3
1/4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/5
3/7
![Page 257: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/257.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
3/7 bulb
1/3
1/4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/5
3/7
![Page 258: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/258.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
3/7 bulb
1/3
1/4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/5
3/7
![Page 259: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/259.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
3/7 bulb
1/3
1/4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/5
3/7
![Page 260: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/260.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
3/7 bulb
1/3
1/4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
3/7
2/5
![Page 261: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/261.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
??? bulb
1/3
1/43/7
2/5
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 262: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/262.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/2 bulb
1/3
1/43/7
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/2
2/5
![Page 263: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/263.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/2 bulb
1/3
1/43/7
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/2
2/5
![Page 264: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/264.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/2 bulb
1/3
1/43/7
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/2
2/5
![Page 265: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/265.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/2 bulb
1/3
1/43/7
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/2
2/5
![Page 266: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/266.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
??? bulb
1/3
1/43/7
1/2
2/5
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 267: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/267.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/3 bulb
1/3
1/43/7
1/2 QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/3
2/5
![Page 268: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/268.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/3 bulb
1/3
1/43/7
1/2 QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/3
2/5
![Page 269: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/269.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/3 bulb
1/3
1/43/7
1/2 QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/3
2/5
![Page 270: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/270.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/3 bulb
1/3
1/43/7
1/2 QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/3
2/5
![Page 271: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/271.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/3 bulb
1/3
1/43/7
1/2 QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/3
2/5
![Page 272: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/272.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/3 bulb
1/3
1/43/7
1/2 QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2/3
2/5
![Page 273: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/273.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
How to count
![Page 274: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/274.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/4
How to count
![Page 275: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/275.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/3
1/4
How to count
![Page 276: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/276.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/3
1/42/5
How to count
![Page 277: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/277.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/3
1/42/5
3/7
How to count
![Page 278: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/278.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/3
1/42/5
3/7
1/2
How to count
![Page 279: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/279.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/3
1/42/5
3/7
1/2
2/3
How to count
![Page 280: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/280.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
1/3
1/42/5
3/7
1/2
2/3
The bulbs are arranged in the exactorder of the rational numbers.
How to count
![Page 281: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/281.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
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
![Page 282: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/282.jpg)
Animations:
Mandelbulbs
Spiralling fingers
![Page 283: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/283.jpg)
How to add
![Page 284: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/284.jpg)
How to add
1/2
![Page 285: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/285.jpg)
How to add
1/2
1/3
![Page 286: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/286.jpg)
How to add
1/2
1/3
2/5
![Page 287: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/287.jpg)
How to add
1/2
1/3
2/5
3/7
![Page 288: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/288.jpg)
+ =
1/2 + 1/3 = 2/5
![Page 289: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/289.jpg)
+ =
1/2 + 2/5 = 3/7
![Page 290: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/290.jpg)
221/2
0/1
Here’s an interesting sequence:
![Page 291: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/291.jpg)
221/2
0/1
Watch the denominators
1/3
![Page 292: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/292.jpg)
221/2
0/1
Watch the denominators
1/3
2/5
![Page 293: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/293.jpg)
221/2
0/1
Watch the denominators
1/3
2/5
3/8
![Page 294: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/294.jpg)
221/2
0/1
Watch the denominators
1/3
2/5
3/85/13
![Page 295: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/295.jpg)
221/2
0/1
What’s next?
1/3
2/5
3/85/13
![Page 296: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/296.jpg)
221/2
0/1
What’s next?
1/3
2/5
3/85/13
8/21
![Page 297: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/297.jpg)
221/2
0/1
The Fibonacci sequence
1/3
2/5
3/85/13
8/2113/34
![Page 298: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/298.jpg)
The Farey Tree
€
0
1
€
1
1
![Page 299: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/299.jpg)
The Farey Tree
€
0
1
€
1
1
How get the fraction in betweenwith the smallest denominator?
![Page 300: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/300.jpg)
The Farey Tree
€
0
1
€
1
1
€
1
2
How get the fraction in betweenwith the smallest denominator?
Farey addition
![Page 301: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/301.jpg)
The Farey Tree
€
0
1
€
1
1
€
1
2
€
1
3
€
2
3
![Page 302: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/302.jpg)
The Farey Tree
€
0
1
€
1
1
€
1
2
€
1
3
€
2
3
€
2
5
€
1
4
€
3
5
€
3
4
![Page 303: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/303.jpg)
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
![Page 304: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/304.jpg)
Another sequence (denominatorsonly)
1
2
![Page 305: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/305.jpg)
Another sequence (denominatorsonly)
1
2
3
![Page 306: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/306.jpg)
Another sequence (denominatorsonly)
1
2
3
4
![Page 307: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/307.jpg)
Another sequence (denominatorsonly)
1
2
3
4
5
![Page 308: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/308.jpg)
Another sequence (denominatorsonly)
1
2
3
4
5
6
![Page 309: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/309.jpg)
Another sequence (denominatorsonly)
1
2
3
4
5
67
![Page 310: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/310.jpg)
sequence
1
2
3
4
5
67
Devaney
![Page 311: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/311.jpg)
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.
![Page 312: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/312.jpg)
Farey.qt
Farey tree
D-sequence
Continued fraction expansion
Far from rationals
Other topics
Website
![Page 313: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/313.jpg)
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:
![Page 314: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/314.jpg)
Continued fraction expansion
12
= 12
the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....
![Page 315: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/315.jpg)
Continued fraction expansion
13
= 12 + 1
1
the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....
![Page 316: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/316.jpg)
Continued fraction expansion
25
= 12 + 1
1 + 11
the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....
![Page 317: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/317.jpg)
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,.....
![Page 318: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/318.jpg)
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,.....
![Page 319: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/319.jpg)
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,.....
![Page 320: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/320.jpg)
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,.....
![Page 321: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/321.jpg)
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,.....
![Page 322: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/322.jpg)
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
![Page 323: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/323.jpg)
The real way to prove all this:
Need to measure: the size of bulbs the length of spokes the size of the “ears.”
![Page 324: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/324.jpg)
There is an external Riemann map : C - D C - Mtaking the exterior of the unit disk to the exterior of the Mandelbrot set.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
Φ
€
Φ
![Page 325: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/325.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
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
![Page 326: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/326.jpg)
€
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
![Page 327: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/327.jpg)
€
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.
![Page 328: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/328.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
Φ0
1/3
2/3
€
γ00 is fixed under angle doubling, so lands at the cusp of the main cardioid.
€
γ0
![Page 329: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/329.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
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
![Page 330: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/330.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
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
![Page 331: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/331.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
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
![Page 332: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/332.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
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
![Page 333: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/333.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
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
![Page 334: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/334.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
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....
![Page 335: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/335.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
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
![Page 336: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/336.jpg)
1/72/7
3/15 and 4/15 have period 4, andare between 1/7 and 2/7....
![Page 337: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/337.jpg)
1/72/7
3/15 and 4/15 have period 4, andare between 1/7 and 2/7....
3/154/15
![Page 338: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/338.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
So what do we know about M?
All rational external rays land at a single point in M.
![Page 339: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/339.jpg)
So what do we know about M?
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
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).
![Page 340: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/340.jpg)
So what do we know about M?
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
“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
![Page 341: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/341.jpg)
So what do we NOT know about M?
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
But we don't know if irrationals that are “close” to rationals land.
So we won't understandquadratic functions untilwe figure this out.
![Page 342: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/342.jpg)
MLC Conjecture:
The boundary of the M-setis “locally connected” ---if so, all rays land and we are in heaven!. But if not......
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 343: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/343.jpg)
The Dynamical Systems and Technology Project at Boston University
website: math.bu.edu/DYSYS
Have fun!
![Page 344: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/344.jpg)
A number is far from the rationals if:
€
θ
€
|θ − p /q |
€
>
![Page 345: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/345.jpg)
A number is far from the rationals if:
€
θ
€
|θ − p /q |
€
>
€
c /qk
![Page 346: Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))](https://reader038.vdocument.in/reader038/viewer/2022110206/56649cd85503460f949a052b/html5/thumbnails/346.jpg)
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