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Mandelbrot and Sierpinski arcs and spirals E. Chang Department of Mathematics and Statistics Boston University TCD2015 E. Chang (Boston University) Mandelbrot and Sierpinski arcs and spirals TCD2015 1 / 59 Outline Introduction and


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SLIDE 1

Mandelbrot and Sierpinski arcs and spirals

  • E. Chang

Department of Mathematics and Statistics Boston University

TCD2015

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 1 / 59

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SLIDE 2

Outline

1

Introduction and Exploration

2

Classification of the parameter plane

3

Exploration

4

The setup

5

The payoff

6

n = 4, d = 3

7

Payoff part 2

8

The future

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 2 / 59

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SLIDE 3

Outline

1

Introduction and Exploration

2

Classification of the parameter plane

3

Exploration

4

The setup

5

The payoff

6

n = 4, d = 3

7

Payoff part 2

8

The future

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 3 / 59

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SLIDE 4

The Rational Map

Consider the function Fλ(z) = zn + λ zd , z, λ ∈ C (1)

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 4 / 59

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SLIDE 5

The Rational Map

Consider the function Fλ(z) = zn + λ zd , z, λ ∈ C (1) Fλ most likely behaves differently for different λ. How can we visualize that information?

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 4 / 59

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SLIDE 6

The Rational Map

Consider the function Fλ(z) = zn + λ zd , z, λ ∈ C (1) Fλ most likely behaves differently for different λ. How can we visualize that information? Let’s take a look, for n = 2, d = 3.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 4 / 59

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The parameter plane for n = 2, d = 3

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 5 / 59

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The parameter plane for n = 2, d = 3

Some program drew this parameter plane arranged into differently colored regions. What do they correspond to?

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 5 / 59

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SLIDE 9

The parameter plane for n = 2, d = 3

Some program drew this parameter plane arranged into differently colored regions. What do they correspond to? The parameter plane has Re(λ) and Im(λ) for axes, and different values of λ probably result in different behavior for Fλ.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 5 / 59

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SLIDE 10

The parameter plane for n = 2, d = 3

Some program drew this parameter plane arranged into differently colored regions. What do they correspond to? The parameter plane has Re(λ) and Im(λ) for axes, and different values of λ probably result in different behavior for Fλ. Let’s see what the dynamical plane looks like as we fix λ at different values:

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 5 / 59

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λ outside the spaceship thing

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 6 / 59

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λ outside the spaceship thing

That’s kind of neat

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 6 / 59

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λ in an orange region

  • E. Chang (Boston University)

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λ in an orange region

Pretty cool.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 7 / 59

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λ in the same orange region

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 8 / 59

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λ in the same orange region

That looks more or less the same.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 8 / 59

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λ in the same orange region

That looks more or less the same. We could say something about topological conjugacy

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 8 / 59

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λ in the same orange region

That looks more or less the same. We could say something about topological conjugacy but we won’t.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 8 / 59

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λ in a black spot

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 9 / 59

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λ in a black spot

What is happening here?

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 9 / 59

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SLIDE 21

Where to begin?

If only we had a way to classify the regions in the parameter plane.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 10 / 59

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SLIDE 22

Where to begin?

If only we had a way to classify the regions in the parameter plane. Such a way exists, based on the Julia set and orbits of the critical values of Fλ.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 10 / 59

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SLIDE 23

Where to begin?

If only we had a way to classify the regions in the parameter plane. Such a way exists, based on the Julia set and orbits of the critical values of Fλ. Apologies if the following is boring to you:

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 10 / 59

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SLIDE 24

Terminology

For Fλ(z) = z2 + λ z3 , z, λ ∈ C

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 11 / 59

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Terminology

For Fλ(z) = z2 + λ z3 , z, λ ∈ C There are 5 critical points given by cλ = 3λ 2 1/5 .

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 11 / 59

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Terminology

For Fλ(z) = z2 + λ z3 , z, λ ∈ C There are 5 critical points given by cλ = 3λ 2 1/5 . Each has a corresponding critical value vλ = 5λ2/5 33/522/5 .

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 11 / 59

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Terminology

For Fλ(z) = z2 + λ z3 , z, λ ∈ C There are 5 critical points given by cλ = 3λ 2 1/5 . Each has a corresponding critical value vλ = 5λ2/5 33/522/5 . There are also 5 prepoles given by pλ = (−λ)1/5.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 11 / 59

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SLIDE 28

Terminology

For Fλ(z) = z2 + λ z3 , z, λ ∈ C There are 5 critical points given by cλ = 3λ 2 1/5 . Each has a corresponding critical value vλ = 5λ2/5 33/522/5 . There are also 5 prepoles given by pλ = (−λ)1/5. When |z| is large, |Fλ(z)| > |z| and so the point at ∞ is an attracting fixed point in the Riemann sphere. We denote the immediate basin of attraction of ∞ by Bλ.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 11 / 59

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SLIDE 29

Terminology

For Fλ(z) = z2 + λ z3 , z, λ ∈ C There are 5 critical points given by cλ = 3λ 2 1/5 . Each has a corresponding critical value vλ = 5λ2/5 33/522/5 . There are also 5 prepoles given by pλ = (−λ)1/5. When |z| is large, |Fλ(z)| > |z| and so the point at ∞ is an attracting fixed point in the Riemann sphere. We denote the immediate basin of attraction of ∞ by Bλ. There is a pole at the origin, so there is a neighborhood of the origin that is mapped into Bλ. If the preimage of Bλ surrounding the origin is disjoint from Bλ, we call this region the trap door and denote it by Tλ.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 11 / 59

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More Terminology

The Julia set of Fλ, denoted J (Fλ), has several equivalent

  • definitions. J (Fλ) is the set of all points at which the family of

iterates of Fλ fails to be a normal family in the sense of Montel. Equivalently, J (Fλ) is the closure of the set of repelling periodic points of Fλ, and it is also the boundary of the set of points whose

  • rbits tend to ∞ under iteration of Fλ.
  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 12 / 59

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SLIDE 31

More Terminology

The Julia set of Fλ, denoted J (Fλ), has several equivalent

  • definitions. J (Fλ) is the set of all points at which the family of

iterates of Fλ fails to be a normal family in the sense of Montel. Equivalently, J (Fλ) is the closure of the set of repelling periodic points of Fλ, and it is also the boundary of the set of points whose

  • rbits tend to ∞ under iteration of Fλ.

J (Fλ) is where the dynamical behavior is interesting.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 12 / 59

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SLIDE 32

More Terminology

The Julia set of Fλ, denoted J (Fλ), has several equivalent

  • definitions. J (Fλ) is the set of all points at which the family of

iterates of Fλ fails to be a normal family in the sense of Montel. Equivalently, J (Fλ) is the closure of the set of repelling periodic points of Fλ, and it is also the boundary of the set of points whose

  • rbits tend to ∞ under iteration of Fλ.

J (Fλ) is where the dynamical behavior is interesting. The Fatou Set, or F(Fλ), is the complement of J (Fλ) in the Riemann sphere.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 12 / 59

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More Terminology

The Julia set of Fλ, denoted J (Fλ), has several equivalent

  • definitions. J (Fλ) is the set of all points at which the family of

iterates of Fλ fails to be a normal family in the sense of Montel. Equivalently, J (Fλ) is the closure of the set of repelling periodic points of Fλ, and it is also the boundary of the set of points whose

  • rbits tend to ∞ under iteration of Fλ.

J (Fλ) is where the dynamical behavior is interesting. The Fatou Set, or F(Fλ), is the complement of J (Fλ) in the Riemann sphere. For us, it’s not that interesting.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 12 / 59

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SLIDE 34

More Terminology

The Julia set of Fλ, denoted J (Fλ), has several equivalent

  • definitions. J (Fλ) is the set of all points at which the family of

iterates of Fλ fails to be a normal family in the sense of Montel. Equivalently, J (Fλ) is the closure of the set of repelling periodic points of Fλ, and it is also the boundary of the set of points whose

  • rbits tend to ∞ under iteration of Fλ.

J (Fλ) is where the dynamical behavior is interesting. The Fatou Set, or F(Fλ), is the complement of J (Fλ) in the Riemann sphere. For us, it’s not that interesting. So we want to look at the behavior of the critical values of Fλ for different λ. The dynamical plane is symmetric under rotation, so it is enough to look at any one critical value to see the behavior of all of them.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 12 / 59

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Outline

1

Introduction and Exploration

2

Classification of the parameter plane

3

Exploration

4

The setup

5

The payoff

6

n = 4, d = 3

7

Payoff part 2

8

The future

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 13 / 59

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Cantor set locus

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 14 / 59

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Cantor set locus

vλ lies in Bλ. In this case it is known that J (Fλ) is a Cantor set.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 14 / 59

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Cantor set locus

vλ lies in Bλ. In this case it is known that J (Fλ) is a Cantor set. The corresponding set of λ-values in the parameter plane is called the Cantor set locus.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 14 / 59

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SLIDE 39

Sierpinski holes

  • E. Chang (Boston University)

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Sierpinski holes

vλ enters Tλ at iteration 2 or higher. In this case it is known that J (Fλ) is a Sierpinski curve, i.e. a set that is homeomorphic to the Sierpinski carpet fractal.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 15 / 59

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Sierpinski holes

vλ enters Tλ at iteration 2 or higher. In this case it is known that J (Fλ) is a Sierpinski curve, i.e. a set that is homeomorphic to the Sierpinski carpet fractal. The corresponding set of λ-values in the parameter plane are regions that we call Sierpinski holes.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 15 / 59

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The connectedness locus

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 16 / 59

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The connectedness locus

vλ does not escape to ∞.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 16 / 59

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The connectedness locus

vλ does not escape to ∞. The corresponding set of λ-values in the parameter plane includes the Mandelbrot sets. Together with the Sierpinski holes, this region is called the connectedness locus, as J (Fλ) is a connected set for all λ in the locus.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 16 / 59

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Outline

1

Introduction and Exploration

2

Classification of the parameter plane

3

Exploration

4

The setup

5

The payoff

6

n = 4, d = 3

7

Payoff part 2

8

The future

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 17 / 59

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SLIDE 46

Sierpinski hole of higher escape time

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 18 / 59

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Sierpinski hole of higher escape time

For a λ in the next Sierpinski hole to the left:

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 18 / 59

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SLIDE 48

Sierpinski hole of higher escape time

For a λ in the next Sierpinski hole to the left: vλ enters Tλ at iteration 3.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 18 / 59

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Sierpinski hole of higher escape time

For a λ in the next Sierpinski hole to the left: vλ enters Tλ at iteration 3. The next Sierpinski hole along the negative real axis probably has escape time 4.

  • E. Chang (Boston University)

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Sierpinski hole of higher escape time

For a λ in the next Sierpinski hole to the left: vλ enters Tλ at iteration 3. The next Sierpinski hole along the negative real axis probably has escape time 4. This idea of increasingly higher escape time Sierpinski holes might be interesting... let’s look around some more.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 18 / 59

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SLIDE 51

More Mandelbrot sets

  • E. Chang (Boston University)

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SLIDE 52

More Mandelbrot sets

There is the clearly visible principal Mandelbrot set.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 19 / 59

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SLIDE 53

More Mandelbrot sets

There is the clearly visible principal Mandelbrot set. Also two baby Mandelbrot sets.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 19 / 59

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SLIDE 54

More Mandelbrot sets

There is the clearly visible principal Mandelbrot set. Also two baby Mandelbrot sets. Six more baby Mandelbrot sets. Are there others?

  • E. Chang (Boston University)

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SLIDE 55

Zooming In

  • E. Chang (Boston University)

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SLIDE 56

Zooming In

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 20 / 59

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SLIDE 57

Zooming In

There is one between the two Sierpinski holes.

  • E. Chang (Boston University)

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SLIDE 58

Further along the negative real axis

  • E. Chang (Boston University)

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SLIDE 59

Further along the negative real axis

  • E. Chang (Boston University)

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SLIDE 60

Further along the negative real axis

Looks like another one between the next pair of Sierpinski holes. Is there a pattern?

  • E. Chang (Boston University)

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SLIDE 61

Conjecture

  • E. Chang (Boston University)

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SLIDE 62

Conjecture

There are infinitely many Sierpinski holes along the negative real axis

  • f the parameter plane.
  • E. Chang (Boston University)

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SLIDE 63

Conjecture

  • E. Chang (Boston University)

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SLIDE 64

Conjecture

Between each pair of Sierpinski holes is a Mandelbrot set, though it might be hard to see.

  • E. Chang (Boston University)

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SLIDE 65

If it works the first 3 times, it works all the time

  • E. Chang (Boston University)

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SLIDE 66

If it works the first 3 times, it works all the time

We can’t keep zooming in for each of the (infinite number of) Mandelbrot sets.

  • E. Chang (Boston University)

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SLIDE 67

If it works the first 3 times, it works all the time

Is there a way to prove the existence of this alternating arc of infinite Sierpinski holes and Mandelbrot sets using the dynamical plane?

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 25 / 59

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Outline

1

Introduction and Exploration

2

Classification of the parameter plane

3

Exploration

4

The setup

5

The payoff

6

n = 4, d = 3

7

Payoff part 2

8

The future

  • E. Chang (Boston University)

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SLIDE 69

The dynamical plane for n = 2, d = 3

  • E. Chang (Boston University)

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SLIDE 70

The dynamical plane for n = 2, d = 3

This is the dynamical plane for n = 2, d = 3.

  • E. Chang (Boston University)

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SLIDE 71

The dynamical plane for n = 2, d = 3

This is the dynamical plane for n = 2, d = 3. To construct the objects in the Sierpinski Mandelbrot arc we will need to consider some closed sets in the dynamical plane.

  • E. Chang (Boston University)

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SLIDE 72

The dynamical plane for n = 2, d = 3

This is the dynamical plane for n = 2, d = 3. To construct the objects in the Sierpinski Mandelbrot arc we will need to consider some closed sets in the dynamical plane. We will also restrict attention to an annular region in the parameter

  • plane. The details aren’t that important for this talk.
  • E. Chang (Boston University)

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SLIDE 73

The left wedge

  • E. Chang (Boston University)

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SLIDE 74

The left wedge

Let Lλ be the closed portion of the wedge with inner boundary in the trapdoor, outer boundary in the basin, and straight line boundaries that are part of the two adjacent prepole rays as shown.

  • E. Chang (Boston University)

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SLIDE 75

The left wedge

Let Lλ be the closed portion of the wedge with inner boundary in the trapdoor, outer boundary in the basin, and straight line boundaries that are part of the two adjacent prepole rays as shown. There is one critical point is in the interior of Lλ.

  • E. Chang (Boston University)

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SLIDE 76

The right wedge

  • E. Chang (Boston University)

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SLIDE 77

The right wedge

Let Rλ be the symmetric right wedge. The straight line boundaries are part of two adjacent critical point rays. There is one prepole in the interior of Rλ.

  • E. Chang (Boston University)

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SLIDE 78

The right wedge

Let Rλ be the symmetric right wedge. The straight line boundaries are part of two adjacent critical point rays. There is one prepole in the interior of Rλ. The critical value corresponding to the critical point in the interior of Lλ is in Rλ.

  • E. Chang (Boston University)

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SLIDE 79

The subset of the trapdoor

  • E. Chang (Boston University)

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The subset of the trapdoor

Let T λ be a closed subset of the trapdoor containing 0 such that Lλ ∪ T λ ∪ Rλ are connected, and they only intersect along boundaries. This union will be referred to informally as the bowtie.

  • E. Chang (Boston University)

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SLIDE 81

Proposition

There are more parts to the proposition for the paper in the works, but the part we care about for now is:

  • E. Chang (Boston University)

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SLIDE 82

Proposition

There are more parts to the proposition for the paper in the works, but the part we care about for now is:

Proposition

For each λ in some roughly annular region the details of which I skipped

  • ver:
  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 31 / 59

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SLIDE 83

Proposition

There are more parts to the proposition for the paper in the works, but the part we care about for now is:

Proposition

For each λ in some roughly annular region the details of which I skipped

  • ver:
  • 1. Fλ maps Rλ in 1-1 fashion onto a region that contains the interior of

Lλ ∪ T λ ∪ Rλ.

  • E. Chang (Boston University)

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SLIDE 84

Proposition

There are more parts to the proposition for the paper in the works, but the part we care about for now is:

Proposition

For each λ in some roughly annular region the details of which I skipped

  • ver:
  • 1. Fλ maps Rλ in 1-1 fashion onto a region that contains the interior of

Lλ ∪ T λ ∪ Rλ.

  • 2. Wait for the paper.
  • E. Chang (Boston University)

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SLIDE 85

Proposition

There are more parts to the proposition for the paper in the works, but the part we care about for now is:

Proposition

For each λ in some roughly annular region the details of which I skipped

  • ver:
  • 1. Fλ maps Rλ in 1-1 fashion onto a region that contains the interior of

Lλ ∪ T λ ∪ Rλ.

  • 2. Wait for the paper.
  • 3. See part 2.
  • E. Chang (Boston University)

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SLIDE 86

“Proof” of part 1

  • E. Chang (Boston University)

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SLIDE 87

“Proof” of part 1

The critical point rays map to the prepole rays.

  • E. Chang (Boston University)

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SLIDE 88

“Proof” of part 1

The critical point rays map to the prepole rays. The boundary of Rλ in Bλ maps to the outer arc on the right.

  • E. Chang (Boston University)

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SLIDE 89

“Proof” of part 1

The critical point rays map to the prepole rays. The boundary of Rλ in Bλ maps to the outer arc on the right. The boundary of Rλ in T λ maps to the outer arc on the left.

  • E. Chang (Boston University)

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SLIDE 90

“Proof” of part 1

The critical point rays map to the prepole rays. The boundary of Rλ in Bλ maps to the outer arc on the right. The boundary of Rλ in T λ maps to the outer arc on the left. Then the image of Rλ properly contains the interiors of both Rλ and Lλ.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 32 / 59

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SLIDE 91

“Proof” of part 1

The critical point rays map to the prepole rays. The boundary of Rλ in Bλ maps to the outer arc on the right. The boundary of Rλ in T λ maps to the outer arc on the left. Then the image of Rλ properly contains the interiors of both Rλ and Lλ. In other words, inside Rλ is a bowtie which consists of a preimage of Lλ, a preimage of T λ, and a preimage of Rλ.

  • E. Chang (Boston University)

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SLIDE 92

Outline

1

Introduction and Exploration

2

Classification of the parameter plane

3

Exploration

4

The setup

5

The payoff

6

n = 4, d = 3

7

Payoff part 2

8

The future

  • E. Chang (Boston University)

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SLIDE 93

Drawing a picture

  • E. Chang (Boston University)

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SLIDE 94

Drawing a picture

Let’s dress that dynamical plane up with a bowtie.

  • E. Chang (Boston University)

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SLIDE 95

Bowties in bowties

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SLIDE 96

Bowties in bowties

That bowtie contains a preimage of Rλ, which should have another bowtie in it. Which should have another bowtie in it. Which should have another bowtie in it.

  • E. Chang (Boston University)

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SLIDE 97

Bowties in bowties

That bowtie contains a preimage of Rλ, which should have another bowtie in it. Which should have another bowtie in it. Which should have another bowtie in it.

  • E. Chang (Boston University)

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SLIDE 98

Bowties in bowties

That bowtie contains a preimage of Rλ, which should have another bowtie in it. Which should have another bowtie in it. Which should have another bowtie in it.

  • E. Chang (Boston University)

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SLIDE 99

Bowties in bowties

That bowtie contains a preimage of Rλ, which should have another bowtie in it. Which should have another bowtie in it. Which should have another bowtie in it.

  • E. Chang (Boston University)

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SLIDE 100

Back to the parameter plane

It turns out each preimage of Lλ and each preimage of T λ corresponds to a Mandelbrot set and a Sierpinski hole, respectively, in the parameter plane.

  • E. Chang (Boston University)

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SLIDE 101

Back to the parameter plane

It turns out each preimage of Lλ and each preimage of T λ corresponds to a Mandelbrot set and a Sierpinski hole, respectively, in the parameter plane. Justification for this claim is at the end of the talk.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 36 / 59

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SLIDE 102

Back to the parameter plane

It turns out each preimage of Lλ and each preimage of T λ corresponds to a Mandelbrot set and a Sierpinski hole, respectively, in the parameter plane. Justification for this claim is at the end of the talk. So if you believe me, we have proven the existence of a set of infinitely many alternating Sierpinski holes and Mandelbrot sets in the parameter plane by finding a set of infinitely many alternating preimages of Lλ and T λ!

  • E. Chang (Boston University)

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SLIDE 103

Back to the parameter plane

It turns out each preimage of Lλ and each preimage of T λ corresponds to a Mandelbrot set and a Sierpinski hole, respectively, in the parameter plane. Justification for this claim is at the end of the talk. So if you believe me, we have proven the existence of a set of infinitely many alternating Sierpinski holes and Mandelbrot sets in the parameter plane by finding a set of infinitely many alternating preimages of Lλ and T λ! There’s definitely more going on for this function, but I want to talk about the case where n = 4, d = 3.

  • E. Chang (Boston University)

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SLIDE 104

Outline

1

Introduction and Exploration

2

Classification of the parameter plane

3

Exploration

4

The setup

5

The payoff

6

n = 4, d = 3

7

Payoff part 2

8

The future

  • E. Chang (Boston University)

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SLIDE 105

The parameter plane for n = 4, d = 3

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SLIDE 106

The parameter plane for n = 4, d = 3

Looks like we still have a set of infinitely many alternating Sierpinski holes and Mandelbrot sets along the negative real axis.

  • E. Chang (Boston University)

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SLIDE 107

The parameter plane for n = 4, d = 3

Looks like we still have a set of infinitely many alternating Sierpinski holes and Mandelbrot sets along the negative real axis. We should be able to prove that by a similar argument to n = 2, d = 3.

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SLIDE 108

By analogy

There are now 7 critical points, critical values, and prepoles.

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SLIDE 109

By analogy

There are now 7 critical points, critical values, and prepoles. Since there seems to be no reason we can’t, let’s construct Lλ, T λ, and Rλ as before.

  • E. Chang (Boston University)

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SLIDE 110

By analogy

There are now 7 critical points, critical values, and prepoles. Since there seems to be no reason we can’t, let’s construct Lλ, T λ, and Rλ as before.

  • E. Chang (Boston University)

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SLIDE 111

By analogy

There are now 7 critical points, critical values, and prepoles. Since there seems to be no reason we can’t, let’s construct Lλ, T λ, and Rλ as before. In fact, we can make another right wedge above the existing one. Let’s refer to the wedge symmetric to Lλ as Rλ

0 , and to the new one

as Rλ

1 . Looks like our bowtie is now lopsided.

  • E. Chang (Boston University)

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SLIDE 112

By analogy

There are now 7 critical points, critical values, and prepoles. Since there seems to be no reason we can’t, let’s construct Lλ, T λ, and Rλ as before. In fact, we can make another right wedge above the existing one. Let’s refer to the wedge symmetric to Lλ as Rλ

0 , and to the new one

as Rλ

1 . Looks like our bowtie is now lopsided.

The existence of Mandelbrot sets and Sierpinski holes based on sets in the dynamical plane depends on being able to vary Arg(λ) by a certain amount, and adding more right wedges in the n = 2, d = 3 case would have violated that.

  • E. Chang (Boston University)

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SLIDE 113

(Lopsided) bowties in (lopsided) bowties

It’s reasonable to assume that Rλ

0 contains a (now lopsided) bowtie as

before.

  • E. Chang (Boston University)

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SLIDE 114

(Lopsided) bowties in (lopsided) bowties

It’s reasonable to assume that Rλ

0 contains a (now lopsided) bowtie as

before.

  • E. Chang (Boston University)

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SLIDE 115

(Lopsided) bowties in (lopsided) bowties

It’s reasonable to assume that Rλ

0 contains a (now lopsided) bowtie as

before. The mapping seems to bear that out.

  • E. Chang (Boston University)

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SLIDE 116

(Lopsided) bowties in (lopsided) bowties

It’s reasonable to assume that Rλ

0 contains a (now lopsided) bowtie as

before. The mapping seems to bear that out. Let’s go back to calling them bowties - lopsided bowtie takes too long to type.

  • E. Chang (Boston University)

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SLIDE 117

(Lopsided) bowties in (lopsided) bowties

It’s reasonable to assume that Rλ

0 contains a (now lopsided) bowtie as

before. The mapping seems to bear that out. Let’s go back to calling them bowties - lopsided bowtie takes too long to type. What about Rλ

1 ?

  • E. Chang (Boston University)

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SLIDE 118

Bowties in bowties

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SLIDE 119

Bowties in bowties

So inside Rλ

0 is a bowtie which consists of a preimage of Lλ, a

preimage of T λ, a preimage of Rλ

0 , and a preimage of Rλ 1 .

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SLIDE 120

Bowties in bowties

So inside Rλ

0 is a bowtie which consists of a preimage of Lλ, a

preimage of T λ, a preimage of Rλ

0 , and a preimage of Rλ 1 .

Inside Rλ

1 is a bowtie which consists of a preimage of Lλ, a preimage

  • f T λ, a preimage of Rλ

0 , and a preimage of Rλ 1 , but “rotated” π

radians before being placed in Rλ

1 . Note the orientation is preserved.

  • E. Chang (Boston University)

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SLIDE 121

Bowties in bowties

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SLIDE 122

Bowties in bowties

  • E. Chang (Boston University)

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SLIDE 123

Bowties in bowties

That bowtie contains a preimage of Rλ, which should have another bowtie in it. Which should have another bowtie in it. Which should have another bowtie in it. We’ll need to zoom in a bunch:

  • E. Chang (Boston University)

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SLIDE 124

Bowties in bowties

That bowtie contains a preimage of Rλ, which should have another bowtie in it. Which should have another bowtie in it. Which should have another bowtie in it. We’ll need to zoom in a bunch:

  • E. Chang (Boston University)

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SLIDE 125

Bowties in bowties

That bowtie contains a preimage of Rλ, which should have another bowtie in it. Which should have another bowtie in it. Which should have another bowtie in it. We’ll need to zoom in a bunch:

  • E. Chang (Boston University)

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SLIDE 126

Bowties in bowties

That bowtie contains a preimage of Rλ, which should have another bowtie in it. Which should have another bowtie in it. Which should have another bowtie in it. We’ll need to zoom in a bunch:

  • E. Chang (Boston University)

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SLIDE 127

Symolic dynamics

At this point it would be useful to be able to name the preimages of Lλ and T λ.

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SLIDE 128

Symolic dynamics

At this point it would be useful to be able to name the preimages of Lλ and T λ. We will use sequences of 0’s and 1’s, ending with L or T to represent

  • preimages. 0 represents a choice of the Rλ

0 preimage, and 1 the Rλ 1

  • preimage. If the region is a preimage of Lλ or T λ, the sequence will

end with L or T respectively.

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SLIDE 129

Symolic dynamics

At this point it would be useful to be able to name the preimages of Lλ and T λ. We will use sequences of 0’s and 1’s, ending with L or T to represent

  • preimages. 0 represents a choice of the Rλ

0 preimage, and 1 the Rλ 1

  • preimage. If the region is a preimage of Lλ or T λ, the sequence will

end with L or T respectively. Under this naming scheme, our diagram would be labeled:

  • E. Chang (Boston University)

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SLIDE 130

Symolic dynamics

At this point it would be useful to be able to name the preimages of Lλ and T λ. We will use sequences of 0’s and 1’s, ending with L or T to represent

  • preimages. 0 represents a choice of the Rλ

0 preimage, and 1 the Rλ 1

  • preimage. If the region is a preimage of Lλ or T λ, the sequence will

end with L or T respectively. Under this naming scheme, our diagram would be labeled:

  • E. Chang (Boston University)

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SLIDE 131

What a mess

We can see that trying to draw this diagram to scale quickly becomes unmanageable.

  • E. Chang (Boston University)

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SLIDE 132

What a mess

We can see that trying to draw this diagram to scale quickly becomes unmanageable. The L and T for the same sequence go together. We can simply label the L preimage and drop the L from the sequence without losing information.

  • E. Chang (Boston University)

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SLIDE 133

What a mess

We can see that trying to draw this diagram to scale quickly becomes unmanageable. The L and T for the same sequence go together. We can simply label the L preimage and drop the L from the sequence without losing information. Another interpretation of the sequence is that the first number denotes the R wedge in which it is located. The remaining numbers are its image, and the final letter its eventual destination. In this sense, the sequence can be thought of as an itinerary.

  • E. Chang (Boston University)

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SLIDE 134

What a mess

We can see that trying to draw this diagram to scale quickly becomes unmanageable. The L and T for the same sequence go together. We can simply label the L preimage and drop the L from the sequence without losing information. Another interpretation of the sequence is that the first number denotes the R wedge in which it is located. The remaining numbers are its image, and the final letter its eventual destination. In this sense, the sequence can be thought of as an itinerary. One can verify that for λ inside a Sierpinski hole corresponding to a sequence ending in T, the critical value has that itinerary before escaping through the trapdoor.

  • E. Chang (Boston University)

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SLIDE 135

What a mess

We can see that trying to draw this diagram to scale quickly becomes unmanageable. The L and T for the same sequence go together. We can simply label the L preimage and drop the L from the sequence without losing information. Another interpretation of the sequence is that the first number denotes the R wedge in which it is located. The remaining numbers are its image, and the final letter its eventual destination. In this sense, the sequence can be thought of as an itinerary. One can verify that for λ inside a Sierpinski hole corresponding to a sequence ending in T, the critical value has that itinerary before escaping through the trapdoor. A more stylized depiction of the two wedges would then look like:

  • E. Chang (Boston University)

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SLIDE 136

Stylized depiction of the right wedges

  • E. Chang (Boston University)

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SLIDE 137

Stylized depiction of Rλ

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SLIDE 138

Stylized depiction of Rλ

0 with labeled L preimages

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SLIDE 139

Outline

1

Introduction and Exploration

2

Classification of the parameter plane

3

Exploration

4

The setup

5

The payoff

6

n = 4, d = 3

7

Payoff part 2

8

The future

  • E. Chang (Boston University)

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SLIDE 140

Arc madness

We can guess that every sequence ending in a 0 corresponds to an arc

  • f infinitely many alternating Sierpinski holes and Mandelbrot sets in

the parameter plane.

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SLIDE 141

Arc madness

We can guess that every sequence ending in a 0 corresponds to an arc

  • f infinitely many alternating Sierpinski holes and Mandelbrot sets in

the parameter plane. We still have the 0,00,000,0000,... arc analogous to the original arc from the n = 2, d = 3 case.

  • E. Chang (Boston University)

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SLIDE 142

Arc madness

We can guess that every sequence ending in a 0 corresponds to an arc

  • f infinitely many alternating Sierpinski holes and Mandelbrot sets in

the parameter plane. We still have the 0,00,000,0000,... arc analogous to the original arc from the n = 2, d = 3 case. We also its preimage in Rλ

1 : the 10,100,1000,10000,... arc.

  • E. Chang (Boston University)

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SLIDE 143

Arc madness

We can guess that every sequence ending in a 0 corresponds to an arc

  • f infinitely many alternating Sierpinski holes and Mandelbrot sets in

the parameter plane. We still have the 0,00,000,0000,... arc analogous to the original arc from the n = 2, d = 3 case. We also its preimage in Rλ

1 : the 10,100,1000,10000,... arc.

We also have the 10,100,1000,10000,... arc’s preimage in Rλ

0 : the

010,0100,01000,010000,... arc.

  • E. Chang (Boston University)

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SLIDE 144

Arc madness

We can guess that every sequence ending in a 0 corresponds to an arc

  • f infinitely many alternating Sierpinski holes and Mandelbrot sets in

the parameter plane. We still have the 0,00,000,0000,... arc analogous to the original arc from the n = 2, d = 3 case. We also its preimage in Rλ

1 : the 10,100,1000,10000,... arc.

We also have the 10,100,1000,10000,... arc’s preimage in Rλ

0 : the

010,0100,01000,010000,... arc. We also have the 10,100,1000,10000,... arc’s preimage in Rλ

1 : the

110,1100,11000,110000,... arc.

  • E. Chang (Boston University)

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SLIDE 145

Arc madness

We can guess that every sequence ending in a 0 corresponds to an arc

  • f infinitely many alternating Sierpinski holes and Mandelbrot sets in

the parameter plane. We still have the 0,00,000,0000,... arc analogous to the original arc from the n = 2, d = 3 case. We also its preimage in Rλ

1 : the 10,100,1000,10000,... arc.

We also have the 10,100,1000,10000,... arc’s preimage in Rλ

0 : the

010,0100,01000,010000,... arc. We also have the 10,100,1000,10000,... arc’s preimage in Rλ

1 : the

110,1100,11000,110000,... arc. You get the idea.

  • E. Chang (Boston University)

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SLIDE 146

Sequences ending in 1

I’m still not sure how to describe this in words - here is a crude attempt at displaying the information visually.

  • E. Chang (Boston University)

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SLIDE 147

Sequences ending in 1

I’m still not sure how to describe this in words - here is a crude attempt at displaying the information visually.

  • E. Chang (Boston University)

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SLIDE 148

More than one spiral

Actually, there is more than the 1,11,111,1111,... spiral in Rλ

1 . How

should that look?

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SLIDE 149

More than one spiral

Actually, there is more than the 1,11,111,1111,... spiral in Rλ

1 . How

should that look?

  • E. Chang (Boston University)

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SLIDE 150

How many?

This suggests that each element in the set {1, 10, 100, 1000, ...} has a unique spiral in the dynamical plane that corresponds to a spiral in the parameter plane consisting of infinitely many arcs of infinitely many alternating S-holes and M-sets that accumulate in infinitely many S-holes along the spiral.

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SLIDE 151

How many?

This suggests that each element in the set {1, 10, 100, 1000, ...} has a unique spiral in the dynamical plane that corresponds to a spiral in the parameter plane consisting of infinitely many arcs of infinitely many alternating S-holes and M-sets that accumulate in infinitely many S-holes along the spiral. Since each element in that set has a preimage in Rλ

0 , there are that

many corresponding spirals in Rλ

0 as well!

  • E. Chang (Boston University)

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SLIDE 152

Back to the parameter plane

The 0 arc is still along the negative real axis in the parameter plane. Where are the other ones?

  • E. Chang (Boston University)

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SLIDE 153

Back to the parameter plane

The 0 arc is still along the negative real axis in the parameter plane. Where are the other ones? For that matter, where are the spirals?

  • E. Chang (Boston University)

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SLIDE 154

Back to the parameter plane

The 0 arc is still along the negative real axis in the parameter plane. Where are the other ones? For that matter, where are the spirals? That’s a good question...

  • E. Chang (Boston University)

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SLIDE 155

Speculation

  • E. Chang (Boston University)

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SLIDE 156

Outline

1

Introduction and Exploration

2

Classification of the parameter plane

3

Exploration

4

The setup

5

The payoff

6

n = 4, d = 3

7

Payoff part 2

8

The future

  • E. Chang (Boston University)

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SLIDE 157

What’s next?

Finding the current structures in the parameter plane.

  • E. Chang (Boston University)

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SLIDE 158

What’s next?

Finding the current structures in the parameter plane. Are there other structures?

  • E. Chang (Boston University)

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SLIDE 159

What’s next?

Finding the current structures in the parameter plane. Are there other structures? What happens when we increase to n = 6? d = 5? Can we generalize?

  • E. Chang (Boston University)

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SLIDE 160

Thanks!

I had a great time here! Happy turkey day!

  • E. Chang (Boston University)

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SLIDE 161

Thanks!

I had a great time here! Happy turkey day! Long Live Catalonia!

  • E. Chang (Boston University)

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SLIDE 162

Justification for a Sierpinski hole

Given a specific point zk

λ that varies analytically with λ and for which

F k

λ (zk λ) = 0,

  • E. Chang (Boston University)

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SLIDE 163

Justification for a Sierpinski hole

Given a specific point zk

λ that varies analytically with λ and for which

F k

λ (zk λ) = 0, we need to find a disk D in the parameter plane for which a

critical value winds once around zk

λ as λ winds once around the boundary

  • f D.
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SLIDE 164

Justification for a Sierpinski hole

Given a specific point zk

λ that varies analytically with λ and for which

F k

λ (zk λ) = 0, we need to find a disk D in the parameter plane for which a

critical value winds once around zk

λ as λ winds once around the boundary

  • f D. Then there is a unique λ for which vλ = zk

λ, i.e., F k+1 λ

(cλ) = 0.

  • E. Chang (Boston University)

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SLIDE 165

Justification for a Sierpinski hole

Given a specific point zk

λ that varies analytically with λ and for which

F k

λ (zk λ) = 0, we need to find a disk D in the parameter plane for which a

critical value winds once around zk

λ as λ winds once around the boundary

  • f D. Then there is a unique λ for which vλ = zk

λ, i.e., F k+1 λ

(cλ) = 0. This unique λ is the “center” of a Sierpinski hole.

  • E. Chang (Boston University)

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SLIDE 166

Justification for a Mandelbrot set

Find an open disk of parameters D satisfying, for each λ in D, there are

  • pen disks Uλ ⊂ Vλ which move analytically with λ
  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 59 / 59

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SLIDE 167

Justification for a Mandelbrot set

Find an open disk of parameters D satisfying, for each λ in D, there are

  • pen disks Uλ ⊂ Vλ which move analytically with λ and F k

λ : Uλ → Vλ is

two-to-one,

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 59 / 59

slide-168
SLIDE 168

Justification for a Mandelbrot set

Find an open disk of parameters D satisfying, for each λ in D, there are

  • pen disks Uλ ⊂ Vλ which move analytically with λ and F k

λ : Uλ → Vλ is

two-to-one, and, as λ winds around the boundary of D, vλ circles once around Uλ in Uλ \ Vλ.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 59 / 59

slide-169
SLIDE 169

Justification for a Mandelbrot set

Find an open disk of parameters D satisfying, for each λ in D, there are

  • pen disks Uλ ⊂ Vλ which move analytically with λ and F k

λ : Uλ → Vλ is

two-to-one, and, as λ winds around the boundary of D, vλ circles once around Uλ in Uλ \ Vλ. Such a family of maps is called a polynomial-like family.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 59 / 59

slide-170
SLIDE 170

Justification for a Mandelbrot set

Find an open disk of parameters D satisfying, for each λ in D, there are

  • pen disks Uλ ⊂ Vλ which move analytically with λ and F k

λ : Uλ → Vλ is

two-to-one, and, as λ winds around the boundary of D, vλ circles once around Uλ in Uλ \ Vλ. Such a family of maps is called a polynomial-like

  • family. So F (k+1)

λ

is polynomial-like of degree 2 on Uk

λ, and this produces

an M-set.

  • E. Chang (Boston University)

Mandelbrot and Sierpinski arcs and spirals TCD2015 59 / 59