Fractals and the Mandelbrot Set - Drexel University · Outline 1.Fractals 2.Julia Fractals 3.The Mandelbrot Set 4.Properties of the Mandelbrot Set 5.Open Questions Matt Ziemke Fractals

Fractals and the Mandelbrot Set

Matt Ziemke

October, 2012

Matt Ziemke Fractals and the Mandelbrot Set

Outline

1. Fractals

2. Julia Fractals

3. The Mandelbrot Set

4. Properties of the Mandelbrot Set

5. Open Questions


What is a Fractal?

”My personal feeling is that the definition of a ’fractal’ should beregarded in the same way as the biologist regards the definition of ’life’.”- Kenneth Falconer

Common Properties

1.) Detail on an arbitrarily small scale.2.) Too irregular to be described using traditional geometricallanguage.3.) In most cases, defined in a very simple way.4.) Often exibits some form of self-similarity.


The Koch Curve- 10 Iterations


5-Iterations


The Minkowski Fractal- 5 Iterations


5 Iterations


5 Iterations


8 Iterations


Heighway’s Dragon


Julia Fractal 1.1


Julia Fractal 1.2


Julia Fractal 1.3


Julia Fractal 1.4




Julia Fractals

Step 1: Let fc : C→ C where f (z) = z2 + c .Step 2: For each w ∈ C, recursively define the sequence {wn}∞n=0

where w0 = w and wn = f (wn−1). The sequence wn∞n=0 is referred

to as the orbit of w.Step 3: ”Collect” all the w ∈ C whose orbit is bounded, i.e., let

Kc = {w ∈ C : supn∈N|wn| ≤ M, for some M > 0}

and let Jc = δ(Kc) where δ(K ) is the boundary of K . Jc is called aJulia set.


Julia Fractals - Example

Let c = 0.375 + i(0.335).Consider w = 0.1i . Then,w1 = f (w0) = f (0.1i) = (0.1i) = 0.365 + 0.335iw2 = f (w1) = f (0.365 + 0.335i) = 0.396 + 0.5796iw20 ≈ 0.014 + 0.026iIn fact, {wn}∞n=0 does not converge but it is bounded by 2. So0.1i ∈ Kc .Consider x = 1. Then,x1 ≈ 1.375 + 0.335ix2 ≈ 2.153 + 1.256ix3 ≈ 3.434 + 5.745ix4 ≈ −20.843 + 39.794ix5 ≈ −1148.782− 1658.450iSo looks as though 1 /∈ Kc .


Julia Fractal - Example, Image 1





Why the colors?


c=-1.145+0.25i


c=-0.110339+0.887262i


c=0.06+0.72i


c=-0.022803-0.672621i


The Mandelbrot Set

Theorem of Julia and Fatou (1920)

Every Julia set is either connected or totally disconnected.

Brolin’s Theorem

Jc is connected if and only if the orbit of zero is bounded, i.e., ifand only if 0 ∈ Kc .


The Mandelbrot Set cont.

A natural question to ask is...What does

M = {c ∈ C : Jc is connected } = {c ∈ C : {f (n)c (0)}∞n=0 is bounded}

look like?














M is a ”catalog” for the connected Julia sets.


Interesting Facts about M

1.)If Jc is totally disconnected then Jc is homeomorphic to theCantor set.2.) fc : Jc → Jc is chaotic.3.) Julia fractals given by c-values in a given ”bulb” of M arehomeomorphic.4.) M is compact.5.) The Hausdorff dimension of δ(M) is two.


Open questions about M

1.) What’s the area of M?

2.) Are there any points c ∈ M so that {f (n)c (0)}∞n=1 is not

attracted to a cycle?3.) Is µ(δ(M)) > 0? Where µ is the Lebesgue measure.


Fractals and the Mandelbrot Set - Drexel University · Outline 1.Fractals 2.Julia Fractals 3.The Mandelbrot Set 4.Properties of the Mandelbrot Set 5.Open Questions Matt Ziemke Fractals

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