The Mandelbrot Set

Andrew Brown

April 14, 2008

The Mandelbrot Set and other Fractals are Cool

But What are They?

To understand Fractals, we must first understand some things

about iterated polynomials on C.

Let f : C→ C be a polynomial. Then nth iterate of f is

fn(z) = f(f(f(· · · f(z) · · · )))︸ ︷︷ ︸n times

Note: The n here is not an exponent.

The orbit launched from z0 ∈ C is the sequence

Of(z0) = fn(z0).

The Julia Set of a Polynomial f(z)

The Basin of Attraction for ∞ is the set

Af(∞) = z ∈ C | Of(z)→∞.

The Julia Set of f(z) is the the boundary of Af(∞),

J (f) = δAf(∞).

Example: Take f(z) = zm. Then J (f) is the unit circle.

The Mandelbrot Set

For the Mandelbrot set, we consider polynomials of the form

pc(z) = z2 + c,

with parameter c ∈ C. We’ll write Opc(z) = Oc(z) for the orbits

of pc.

The Mandelbrot Set is

M = c ∈ C | J (pc) is connected.

This definition appears simple enough, but Julia Sets are not

simple objects.

J (pc) for c = 3+6i10

J (pc) for c = −1 + 3i10

J (pc) for c = i

J (pc) Connectedness Characterization

Luckily, there is a nice characterization of J (pc) being connected.

Theorem. The Julia Set, J (pc), is connected if and only if there

is an R ∈ R such that

|pnc (0)| ≤ R for all n.

If the Julia set is disconnected, it is totally disconnected.

This characterization shows that c 6∈ M ⇔ Oc(0)→∞.

A useful Fact

Lemma. Let |pnc (0)| > 2, and |pnc (0)| ≥ |c| for some n ≥ 1.

Then Oc(0)→∞.

Proof Take n be the smallest such integer. We have that

|pn+1c (0)| = |pnc (0)2 + c| ≥ |pnc (0)|2 − |c| ≥ (|pnc (0)| − 1)|pnc (0)|.

Now, if |pn+kc (0)| ≥ (|pnc (0)| − 1)k|pnc (0)|, then

|pn+k+1c (0)| ≥ (|pn+k

c (0)| − 1)|pn+kc (0)| ≥ (|pnc (0)| − 1)k+1|pnc (0)|.

Hence, by induction, we have that |pnc (0)| → ∞.

Characterization of M

Theorem. c ∈M ⇔ |pnc (0)| ≤ 2 for all n ≥ 1.

Proof ⇐: By the J (pc) Connectedness Characterization, |pnc (0)| ≤2 for all n ≥ 1⇒ c ∈M.

⇒: Say |pkc(0)| > 2 for some k. If |c| > 2, then |pnc (0)| ≥ |c| > 2.If |c| ≤ 2, |pnc (0)| ≥ |c|.

In either case, we can apply the above Lemma for some n to getthat |pkc(0)| → ∞. Hence, c 6∈ M.

Corollary. M⊂ D2, the disc of radius 2.

Proof |pc(0)| = |c|.

This bound is sharp. J (p−2) = [−2,2] is connected, so −2 ∈M.

Compactness

Theorem. M is compact.

Proof Let Mn be the set of parameter values with |pnc (0)| ≤ 2.

Let ck ⊂Mn be a sequence in Mn, and ck → c.Then |pnck(0)| → |pnc (0)|. Since |pnck(0)| ≤ 2, |pnc (0)| ≤ 2, so foreach n, Mn is a closed set.

The Characterization of M above gives that

M =∞⋂n=1

Mn,

so M is also closed.

M is closed and bounded, so M is compact.

The Mandelbrot Set is Connected?

We outline the Proof given by Douady and Hubbard that M is

a connected subset of C.

Our first stop is a result of Bottcher’s that underlies the proof.

Theorem. Let f(z) be a polynomial of degree n ≥ 2. Then

there is an conformal change of coordinates w = ψ(z) such that

ψ f ψ−1 : w 7→ wn

on some neighbourhood of ∞. ψ is unique up to multiplication

by an n−1 root of unity.

We say that f is conformally conjugate to zn, with conjugacy ψ.

The Bottcher Coordinate for pc

The above Theorem gives that pc is conformally conjugate to the

map z 7→ z2 in some neighbourhood, Uc, of ∞ and the conjugacy

is unique.

Douady and Hubbard go further and calculate the explicit form

of the conjugacy.

Theorem. Let Bc : Uc → C \ DR be the conformal conjugacy

associated with the polynomial pc. Then

Bc(z) = lim[pnc (z)

]1/2n,

where the root on the RHS is choosen so that[pnc (z)

]1/2n∼ z.

Moreover, Bc(z) ∼ z near ∞.

Analytic Continuation of the Bottcher Coordinate for pc

It is straightforward to show that the Bottcher coordinate obeysthe following formula.

Bc[pc(z)

]=

[Bc(z)

]2.

We can see that Bc obeys the conjugacy relationship with pc.

Now, if the critical point c is not in Uc, then pc has an analyticinverse that sends Uc to the pre-image p−1

c (Uc), by the InverseFunction Theorem.

The above equation then gives us a way to conformally extendBc to the bigger domain p−1

c (Uc).

If c is not in p−1c (Uc) either, we can extend to p−2

c (Uc), and soon, until c ∈ p−kc (Uc).

Analytic Continuation of the Bottcher Coordinate for pc

These extensions give us the the conformal map

Bc : Ωc → C \ DRc,

where the new domain Ωc depends on J(pc).

If J (pc) is connected, c 6∈ A(∞), so the extending process out-lined for Bc can be repeated infinitely, so Ωc = A(∞).

If J (pc) is disconnected, c ∈ A(∞), so the extended domain willeventually contain c. The Inverse Function Theorem then failsto provide an inverse.

Though it is not obvious from this discussion, Rc = 1 when J (pc)is connected, and Rc > 1 otherwise.

The Mandelbrot IS Connected.

We now consider the function

Φ : C∗ \M → C∗ \ D1, where c 7→ Bc(c).

Φ is well defined as c ∈ Ωc ⇔ J (pc) disconnected ⇔ c 6∈ M,

and B∞(∞) =∞.

Φ is also conformal. (not proved here)

Hence, C∗ \M and C∗ \ D1 are conformally equivalent via Φ, so

C∗ \M is simply connected.

Thus, M is connected.