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

Matt Ziemke Fractals and the Mandelbrot Set What is a Fractal?

”My personal feeling is that the definition of a ’fractal’ should be regarded 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 geometrical language. 3.) In most cases, defined in a very simple way. 4.) Often exibits some form of self-similarity.

Matt Ziemke Fractals and the Mandelbrot Set The Koch Curve- 10 Iterations

Matt Ziemke Fractals and the Mandelbrot Set 5-Iterations

Matt Ziemke Fractals and the Mandelbrot Set The Minkowski Fractal- 5 Iterations

Matt Ziemke Fractals and the Mandelbrot Set 5 Iterations

Matt Ziemke Fractals and the Mandelbrot Set 8 Iterations

Matt Ziemke Fractals and the Mandelbrot Set Heighway’s Dragon

Matt Ziemke Fractals and the Mandelbrot Set Julia Fractal 1.1

Matt Ziemke Fractals and the Mandelbrot Set Julia Fractal 1.2

Matt Ziemke Fractals and the Mandelbrot Set Julia Fractal 1.3

Matt Ziemke Fractals and the Mandelbrot Set Julia Fractal 1.4

Matt Ziemke Fractals and the Mandelbrot Set Matt Ziemke Fractals and the Mandelbrot Set Matt Ziemke Fractals and the Mandelbrot Set Julia Fractals

2 Step 1: Let fc : C → C where f (z) = z + c. ∞ Step 2: For each w ∈ C, recursively deﬁne the sequence {wn}n=0 ∞ where w0 = w and wn = f (wn−1). The sequence wnn=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 : sup |wn| ≤ M, for some M > 0} n∈N

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

Matt Ziemke Fractals and the Mandelbrot 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 + 0.335i w2 = f (w1) = f (0.365 + 0.335i) = 0.396 + 0.5796i w20 ≈ 0.014 + 0.026i ∞ In fact, {wn}n=0 does not converge but it is bounded by 2. So 0.1i ∈ Kc . Consider x = 1. Then, x1 ≈ 1.375 + 0.335i x2 ≈ 2.153 + 1.256i x3 ≈ 3.434 + 5.745i x4 ≈ −20.843 + 39.794i x5 ≈ −1148.782 − 1658.450i So looks as though 1 ∈/ Kc .

Matt Ziemke Fractals and the Mandelbrot Set Julia Fractal - Example, Image 1

Matt Ziemke Fractals and the Mandelbrot Set Julia Fractal - Example, Image 2

Matt Ziemke Fractals and the Mandelbrot Set Julia Fractal - Example, Image 3

Why the colors?

Matt Ziemke Fractals and the Mandelbrot Set c=-1.145+0.25i

Matt Ziemke Fractals and the Mandelbrot Set c=-0.110339+0.887262i

Matt Ziemke Fractals and the Mandelbrot Set c=0.06+0.72i

Matt Ziemke Fractals and the Mandelbrot Set c=-0.022803-0.672621i

Matt Ziemke Fractals and the Mandelbrot Set 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., if and only if 0 ∈ Kc .

Matt Ziemke Fractals and the Mandelbrot Set The Mandelbrot Set cont.

A natural question to ask is... What does

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

look like?

Matt Ziemke Fractals and the Mandelbrot Set The Mandelbrot Set cont.

Matt Ziemke Fractals and the Mandelbrot Set M is a ”catalog” for the connected Julia sets.

Matt Ziemke Fractals and the Mandelbrot Set Interesting Facts about M

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

Matt Ziemke Fractals and the Mandelbrot Set Open questions about M

1.) What’s the area of M? (n) ∞ 2.) Are there any points c ∈ M so that {fc (0)}n=1 is not attracted to a cycle? 3.) Is µ(δ(M)) > 0? Where µ is the Lebesgue measure.

Matt Ziemke Fractals and the Mandelbrot Set