The Fractal Geometry of the Mandelbrot Set.

The Fractal Geometry of the Mandelbrot Set

The Fractal Geometry of the Mandelbrot Set How to count

The Fractal Geometry of the Mandelbrot Set How to count How to add

Many people know the pretty pictures...

but few know the even prettier mathematics.

Oh, that's nothing but the 3/4 bulb ....

...hanging off the period 16 M-set.....

...lying in the 1/7 antenna...

...attached to the 1/3 bulb...

...hanging off the 3/7 bulb...

...on the northwest side of the main cardioid.

Oh, that's nothing but the 3/4 bulb, hanging off the period 16 M-set, lying in the 1/7 antenna of the 1/3 bulb attached to the 3/7 bulb on the northwest side of the main cardioid.

Start with a function: 2 x constant

Start with a function: 2 x constant and a seed: x

Then iterate: 2 x = x constant 1

Then iterate: 2 x = x constant 1 2 x = x constant 2 1

Then iterate: x = x + constant x = x + constant x = x + constant 2 1 2
2 x = x constant 2 1 2 x = x constant 3 2

Then iterate: x = x + constant x = x + constant x = x + constant
2 x = x constant 1 2 x = x constant 2 1 2 x = x constant 3 2 2 x = x constant 4 3 etc.

Then iterate: x = x + constant x = x + constant x = x + constant
2 x = x constant 1 2 x = x constant 2 1 Orbit of x 2 x = x constant 3 2 2 x = x constant 4 3 etc. Goal: understand the fate of orbits.

2 Example: x Seed 0 x = 0 x = 1 x = 2 x = 3 x = 4 x = 5 x = 6

2 Example: x Seed 0 x = 0 x = 1 1 x = 2 x = 3 x = 4 x = 5 x = 6

2 Example: x Seed 0 x = 0 x = 1 1 x = 2 2 x = 3 x = 4 x = 5 x = 6

Example: x + 1 Seed 0 x = 0 x = 1 x = 2 x = 5 x = x = x = 2 1 2 3 4 5
x = 1 1 x = 2 2 x = 5 3 x = 4 x = 5 x = 6

Example: x + 1 Seed 0 x = 0 x = 1 x = 2 x = 5 x = 26 x = x = 2 1 2 3 4
x = 1 1 x = 2 2 x = 5 3 x = 26 4 x = 5 x = 6

Example: x + 1 Seed 0 x = 0 x = 1 x = 2 x = 5 x = 26 x = big x = 2 1 2
x = 1 1 x = 2 2 x = 5 3 x = 26 4 x = big 5 x = 6

Example: x + 1 Seed 0 x = 0 x = 1 x = 2 x = 5 x = 26 x = big
x = 1 1 x = 2 2 x = 5 “Orbit tends to infinity” 3 x = 26 4 x = big 5 x = BIGGER 6

2 Example: x Seed 0 x = 0 x = 0 1 x = 2 x = 3 x = 4 x = 5 x = 6

2 Example: x Seed 0 x = 0 x = 0 1 x = 0 2 x = 3 x = 4 x = 5 x = 6

Example: x + 0 Seed 0 x = 0 x = 0 x = 0 x = 0 x = x = x = 2 1 2 3 4 5
x = 0 1 x = 0 2 x = 0 3 x = 4 x = 5 x = 6

Example: x + 0 Seed 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0
2 Example: x Seed 0 x = 0 x = 0 1 x = 0 2 “A fixed point” x = 0 3 x = 0 4 x = 0 5 x = 0 6

2 Example: x Seed 0 x = 0 x = -1 1 x = 2 x = 3 x = 4 x = 5 x = 6

2 Example: x Seed 0 x = 0 x = -1 1 x = 0 2 x = 3 x = 4 x = 5 x = 6

Example: x - 1 Seed 0 x = 0 x = -1 x = 0 x = -1 x = x = x = 2 1 2 3 4
x = -1 1 x = 0 2 x = -1 3 x = 4 x = 5 x = 6

Example: x - 1 Seed 0 x = 0 x = -1 x = 0 x = -1 x = 0 x = -1 x = 0
2 Example: x Seed 0 x = 0 x = -1 1 x = 0 2 x = -1 “A two- cycle” 3 x = 0 4 x = -1 5 x = 0 6

Example: x - 1.1 Seed 0 x = 0 x = -1.1 x = x = x = x = x = 2 1 2 3 4 5
x = 1 x = 2 x = 3 x = 4 x = 5 x = 6

Example: x - 1.1 Seed 0 x = 0 x = -1.1 x = 0.11 x = x = x = x = 2 1 2
x = 1 x = 2 x = 3 x = 4 x = 5 x = 6

Example: x - 1.1 Seed 0 x = 0 x = -1.1 x = 0.11 x = x = x = x =
2 Example: x Seed 0 x = 0 x = 1 x = 2 x = 3 time for the computer! x = 4 x = 5 x = 6

Observation: For some real values of c, the orbit
of 0 goes to infinity, but for other values, the orbit of 0 does not escape.

Complex Iteration 2 Iterate z + c complex numbers

2 Example: z + i Seed 0 z = 0 z = 1 z = 2 z = 3 z = 4 z = 5 z = 6

2 Example: z + i Seed 0 z = 0 z = i 1 z = 2 z = 3 z = 4 z = 5 z = 6

Example: z + i Seed 0 z = 0 z = i z = -1 + i z = z = z = z = 2 1 2 3 4
z = i 1 z = i 2 z = 3 z = 4 z = 5 z = 6

Example: z + i Seed 0 z = 0 z = i z = -1 + i z = -i z = z = z = 2 1 2
z = i 1 z = i 2 z = -i 3 z = 4 z = 5 z = 6

Example: z + i Seed 0 z = 0 z = i z = -1 + i z = -i z = -1 + i z = z =
2 Example: z + i Seed 0 z = 0 z = i 1 z = i 2 z = -i 3 z = i 4 z = 5 z = 6

Example: z + i Seed 0 z = 0 z = i z = -1 + i z = -i z = -1 + i z = -i
2 Example: z + i Seed 0 z = 0 z = i 1 z = i 2 z = -i 3 z = i 4 z = -i 5 z = 6

Example: z + i Seed 0 z = 0 z = i z = -1 + i z = -i z = -1 + i z = -i
2 Example: z + i Seed 0 z = 0 z = i 1 z = i 2 z = -i 3 z = i 4 z = -i 5 2-cycle z = i 6

2 Example: z + i Seed 0 i -1 1 -i

2 Example: z i Seed 0 z = 0 z = 1 z = 2 z = 3 z = 4 z = 5 z = 6

Example: z + 2i Seed 0 z = 0 z = 2i z = -4 + 2i z = 12 - 14i
z = 2i 1 z = i 2 Off to infinity z = i 3 z = i 4 z = big 5 z = BIGGER 6

Same observation Sometimes orbit of 0 goes to
infinity, other times it does not.

The Mandelbrot Set: All c-values for which orbit
of 0 does NOT go to infinity.

Algorithm for computing M
Start with a grid of complex numbers

Each grid point is a complex c-value.

Compute the orbit of 0 for each c. If the orbit of 0 escapes, color that grid point. red = fastest escape

Compute the orbit of 0 for each c. If the orbit of 0 escapes, color that grid point. orange = slower

Compute the orbit of 0 for each c. If the orbit of 0 escapes, color that grid point. yellow green blue violet

Compute the orbit of 0 for each c. If the orbit of 0 does not escape, leave that grid point black.

The eventual orbit of 0 Eventual orbit

The eventual orbit of 0 3-cycle

The eventual orbit of 0

The eventual orbit of 0 fixed point

The eventual orbit of 0 goes to infinity

The eventual orbit of 0 gone to infinity

How understand the of the bulbs? periods

junction point three spokes attached

junction point three spokes attached Period 3 bulb

Period 4 bulb

Period 5 bulb

Period 7 bulb

Period 13 bulb

Period ?? bulb

Period 3,145,183 bulb well, I may be off a bit...

Filled Julia Set of F: Fix a c-value. The filled Julia set K(F)
is the set of all of the complex seeds whose orbits do NOT go to infinity.

Filled Julia Set of F: Julia Set of F:
Fix a c-value. The filled Julia set K(F) is the set of all of the complex seeds whose orbits do NOT go to infinity. Julia Set of F: Several equivalent definitions of J(F):

Fix a c-value. The filled Julia set K(F) is the set of all of the complex seeds whose orbits do NOT go to infinity. Julia Set of F: 1. The boundary of the filled Julia set

Fix a c-value. The filled Julia set K(F) is the set of all of the complex seeds whose orbits do NOT go to infinity. Julia Set of F: 1. The boundary of the filled Julia set 2. The closure of the set of repelling periodic points

Fix a c-value. The filled Julia set K(F) is the set of all of the complex seeds whose orbits do NOT go to infinity. Julia Set of F: 1. The boundary of the filled Julia set 2. The closure of the set of repelling periodic points The set of points at which the family of iterates is not a normal family in the sense of Montel

Fix a c-value. The filled Julia set K(F) is the set of all of the complex seeds whose orbits do NOT go to infinity. Julia Set of F: 1. The boundary of the filled Julia set 2. The closure of the set of repelling periodic points The set of points at which the family of iterates is not a normal family in the sense of Montel 4. The chaotic set

2 Example: z Seed: In filled Julia set?

2 Example: z Seed: In filled Julia set? Yes

2 Example: z Seed: In filled Julia set? Yes 1

2 Example: z Seed: In filled Julia set? Yes 1 Yes

2 Example: z Seed: In filled Julia set? Yes 1 Yes -1

2 Example: z Seed: In filled Julia set? Yes 1 Yes -1 Yes

2 Example: z Seed: In filled Julia set? Yes 1 Yes -1 Yes i

2 Example: z Seed: In filled Julia set? Yes 1 Yes -1 Yes i Yes

2 Example: z Seed: In filled Julia set? Yes 1 Yes -1 Yes i Yes 2i

2 Example: z Seed: In filled Julia set? Yes 1 Yes -1 Yes i Yes 2i No

2 Example: z Seed: In filled Julia set? Yes 1 Yes -1 Yes i Yes 2i No 5

Example: z Seed: In filled Julia set? Yes 1 Yes -1 Yes i Yes 2i No 5
Yes 1 Yes -1 Yes i Yes 2i No 5 No way

All seeds on or inside the unit circle.
Filled Julia Set for z2 i -1 1 All seeds on or inside the unit circle.

2 Example: z Seed: In the Julia set? 1 -1 i 2i 5

Example: z Seed: In the Julia set? No 1 Yes -1 Yes i Yes 2i No 5
No 1 Yes -1 Yes i Yes 2i No 5 No way

All seeds on the unit circle.
The Julia Set for z2 i -1 1 All seeds on the unit circle.

Given any point p in the Julia set,
The Julia Set for z2 i p -1 1 Given any point p in the Julia set,

Given any point p in the Julia set, and any neighborhood N of p,
The Julia Set for z2 i N p -1 1 Given any point p in the Julia set, and any neighborhood N of p,

Given any point p in the Julia set, and any
The Julia Set for z2 i N p -1 1 Given any point p in the Julia set, and any neighborhood N of p, the union of the forward images of N fills the entire plane (except 0), so z2 + c is chaotic on the Julia set.

Other filled Julia sets

c = 0

c = -1

c = i

The Julia set of z2 + c is always a fractal, except
when c = 0 (J = S1) or c = -2 (J = [-2,2]).

If c is in the Mandelbrot set, then the
filled Julia set is always a connected set.

But if c is not in the Mandelbrot set, then the filled Julia set is totally disconnected.

c = .3

c = i

Amazingly, the orbit of 0 knows it all:
Theorem: For z2 + c: If the orbit of 0 goes to infinity, the Julia set is a Cantor set (totally disconnected, “fractal dust”), and c is not in the Mandelbrot set. But if the orbit of 0 does not go to infinity, the Julia set is connected (just one piece), and c is in the Mandelbrot set.

Furthermore, if z2 + c has an attracting cycle,
then the orbit of 0 tends to this cycle. That is why we can attach a period to each of the bulbs in the Mandelbrot set.

z2 + c undergoes a “saddle node” bifurcation when c = 1/4:
y = x c = 1/4 1/2 c = 1/4: a single neutral fixed point at 1/2, and the orbit of 0 tends to this fixed point

y = x c = 1/4 c = 0.1 q p -3/4 < c < 1/4: a pair of fixed points: an attracting fixed point q and a repelling fixed point p, and the orbit of 0 tends to q

y = x c = 1/4 c = 0.1 q p c > 1/4: no fixed points on the real axis and the orbit of 0 tends to ∞

So the Julia set of z2 + c suddenly changes
from a connected set to a Cantor set when c passes through 1/4 y = x q p

Animations: In and out of M Saddle-node arrangement of bulbs

How do we understand the arrangement of the bulbs?

How do we understand the arrangement of the bulbs?
Assign a fraction p/q to each bulb hanging off the main cardioid. q = period of the bulb

??? bulb shortest spoke principal spoke

1/3 bulb

1/3 bulb 1/3

??? bulb 1/3

1/4 bulb 1/3

1/4 bulb 1/3 1/4

??? bulb 1/3 1/4

2/5 bulb 1/3 1/4

2/5 bulb 1/3 2/5 1/4

??? bulb 1/3 2/5 1/4

3/7 bulb 1/3 2/5 1/4

3/7 bulb 1/3 2/5 1/4 3/7

??? bulb 1/3 2/5 1/4 3/7

1/2 bulb 1/3 2/5 1/4 3/7 1/2

??? bulb 1/3 2/5 1/4 3/7 1/2

2/3 bulb 1/3 2/5 1/4 3/7 1/2 2/3

How to count The bulbs are arranged in the exact
1/3 2/5 1/4 3/7 1/2 2/3 The bulbs are arranged in the exact order of the rational numbers.

How to count The bulbs are arranged in the exact
1/3 32,123/96,787 2/5 1/4 3/7 1/101 1/2 2/3 The bulbs are arranged in the exact order of the rational numbers.

Mandelbulbs Spiralling fingers

The real way.... Along the boundary of the main
cardioid, z2 + c has a neutral fixed point.

cardioid, z2 + c has a neutral fixed point. So: z2 + c = z 2z = exp(2πiθ)

cardioid, z2 + c has a neutral fixed point. So: z2 + c = z 2z = exp(2πiθ) Or: c = z - z2 z = (exp(2πiθ))/2

cardioid, z2 + c has a neutral fixed point. So: z2 + c = z 2z = exp(2πiθ) Or: c = z - z2 z = (exp(2πiθ))/2 c = e2πiθ/2 - e4πiθ/4 This gives the equation of the boundary of the main cardioid

The real way.... θ=1/3 Along the boundary of the main
cardioid, z2 + c has a neutral fixed point. So: z2 + c = z 2z = exp(2πiθ) Or: c = z - z2 z = (exp(2πiθ))/2 c = e2πiθ/2 - e4πiθ/4 This gives the equation of the boundary of the main cardioid θ=2/5 θ=0 θ=1/2 Same arrangement as before.

Sub-bulbs

Sub-bulbs Period three bulb

Sub-bulbs Per 2·3 Per 3·3 Per 4·3 Period three bulb

Sub-bulbs Per 4·3 Period three bulb

Sub-bulbs 3 spokes Per 4·3 4 spokes Period 4·3 bulb Period three bulb

Sub-bulbs 3 spokes Per 5·3 5 spokes Period 5·3 bulb Period three bulb

How to add

How to add 1/2

How to add 1/3 1/2

How to add 1/3 2/5 1/2

How to add 1/3 2/5 3/7 1/2

1/2 + 1/3 = 2/5 + =

1/2 + 2/5 = 3/7 + = Farey addition

The Fibonacci Sequence
2 2

2 2 1/2 0/1

1/3 2 2 1/2 0/1

1/3 2/5 2 2 1/2 0/1

1/3 3/8 2/5 2 2 1/2 0/1

1/3 5/13 3/8 2/5 2 2 1/2 0/1

8/21 1/3 5/13 3/8 2/5 2 2 1/2 0/1

The Farey Tree

The Farey Tree The Farey parents

The Farey Tree The Farey child

The Farey Tree Farey grandchildren

The Farey Tree ... produces at each stage the fraction with
the smallest denominator between the parents

The Farey Tree .... essentially the golden number

Another sequence (denominators only) 2 1

Another sequence (denominators only) 3 2 1

Another sequence (denominators only) 3 4 2 1

Another sequence (denominators only) 3 4 5 2 1

Another sequence (denominators only) 3 4 5 2 6 1

Another sequence (denominators only) 3 4 5 2 6 7 1

Devaney sequence 3 4 5 2 6 7 1

The Dynamical Systems and Technology Project at Boston University
website: math.bu.edu/DYSYS: Mandelbrot set explorer; Applets for investigating M-set; Applets for other complex functions; Chaos games, orbit diagrams, etc. Have fun!

Other topics Farey.qt Farey tree D-sequence Far from rationals Continued Fraction Expansion How to measure antennas Website

The real way to prove all this:
Need to measure: the size of bulbs the length of spokes the size of the “ears.”

There is an external Riemann map
: C - D C - M taking the exterior of the unit disk to the exterior of the Mandelbrot set.

takes straight rays in C - D to the “external rays” in C - M
external rays of angle 0 and 1/2 1/2

takes straight rays in C - D to the “external rays” in C - M
external rays of angle 1/3 and 2/3 1/3 1/2 2/3

Suppose p/q is periodic of period
k under doubling mod 1: period 2 period 3 period 4

Suppose p/q is periodic of period
k under doubling mod 1: period 2 period 3 period 4 Then the external ray of angle p/q lands at the “root point” of a period k bulb in the Mandelbrot set.

0 is fixed under angle doubling, so
lands at the cusp of the main cardioid. 1/3 2/3

1/3 and 2/3 have period 2 under doubling, so
and land at the root of the period 2 bulb. 1/3 2 2/3

And if lies between 1/3 and 2/3, then lies between and .
2/3

So the size of the period 2 bulb is, by
definition, the length of the set of rays between the root point rays, i.e., 2/3-1/3=1/3. 1/3 2 2/3

1/7 and 2/7 have period 3, and are between 0 and 1/3.... 2/7 1/3 1/7 2
2/3

on this period three bulb, whose
So the 1/7 and 2/7 rays land on this period three bulb, whose size is 2/7 - 1/7 = 1/7. 2/7 1/3 1/7 3 2

Same with the 3/7, 4/7, 5/7 and 6/7 rays. 2/7 1/3 1/7 3/7 3 2 3 4/7
4/7 3 6/7 2/3 5/7

1/15 and 2/15 have period 4, and are between 0 and 1/7.... 2/7 1/3 1/7
3

So the 1/15 and 2/15 rays land on this period 4 bulb. 2/7 1/3 1/7 2/15
3

3/15 and 4/15 have period 4, and are between 1/7 and 2/7.... 4/15 2/7
1/3 1/7 3 2 3

3/15 and 4/15 have period 4, and are between 1/7 and 2/7.... 2/7 1/7

3/15 and 4/15 have period 4, and are between 1/7 and 2/7.... 4/15 3/15

So what do we know about M?
All rational external rays land at a single point in M.

So what do we know about M?
All rational external rays land at a single point in M. Rays that are periodic under doubling land at root points of a bulb. Non-periodic rational rays land at Misiurewicz points (how we measure length of antennas).

Misiurewicz Points A Misiurewicz point is a c-value where the orbit of
0 is eventually periodic. c = -2 For example, c = -2 is a Misiurewicz point. We have which is fixed and the Julia set is the closed interval [-2, 2].

c = i is also a Misiurewicz
Misiurewicz Points c = i is also a Misiurewicz point. We have 0 i -1+i -i -1+I -i which has period 2. c = i

Misiurewicz Points c = i is also a Misiurewicz point. We have
0 i -1+i -i -1+I -i which has period 2. Now the Julia set is a “dendrite” which always happens in the Misiurewicz case (except when c = -2).

Highly irrational rays
also land at unique points, and we understand what goes on here. “Highly irrational" = “far” from rationals, i.e.,

Specifically a “highly irrational” ray corresponds to a
Brjuno number θ whose continued fraction expansion has a sequence of convergents pn/qn that satisfy For such a map, the neutral fixed point is surrounded by an open disk on which the map is conjugate to the irrational rotation by angle θ. This disk is called a Siegel disk.

A Siegel disk for z2 + c All orbits rotate around the fixed point
as an irrational rotation. Here c is the golden mean (the most highly irrational θ-value).

So what do we NOT know about M?
But we don't know if irrationals that are “close” to rationals land. So we won't understand quadratic functions until we figure this out.

MLC Conjecture: The boundary of the M-set is “locally connected” ---
if so, all rays land and we are in heaven!. But if not......

The Dynamical Systems and Technology Project at Boston University
website: math.bu.edu/DYSYS Have fun!

A number is far from the rationals if:

A number is far from the rationals if:
This happens if the “continued fraction expansion” of has only bounded terms.

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:

1 2 1 2 = the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

1 3 1 2 = + 1 the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

2 5 1 2 = + 1 + 1 the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

3 8 1 2 = + 1 + 1 + 1 the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

5 1 2 = 13 + 1 + 1 + 1 + 1 the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

8 1 2 = 21 + 1 + 1 + 1 + 1 + 1 the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

13 1 2 = 34 + 1 + 1 + 1 + 1 + 1 + 1 the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

13 1 2 = 34 + 1 + 1 + 1 + 1 + 1 + 1 essentially the 1/golden number the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

We understand what happens for
1 a = + 1 b + 1 c + 1 d + 1 e + 1 f + 1 g etc. where all entries in the sequence a, b, c, d,.... are bounded above. But if that sequence grows too quickly, we’re in trouble!!!

The Fractal Geometry of the Mandelbrot Set.

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The Fractal Geometry of the Mandelbrot Set.

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