The Mandelbrot Set. Andrew Brown. April 14, 2008

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 f n (z) = f(f(f( f(z) ))) }{{} n times Note: The n here is not an exponent. The orbit launched from z 0 C is the sequence O f (z 0 ) = {f n (z 0 )}.

The Julia Set of a Polynomial f(z) The Basin of Attraction for is the set A f ( ) = {z C O f (z) }. The Julia Set of f(z) is the the boundary of A f ( ), J (f) = δa f ( ). Example: Take f(z) = z m. Then J (f) is the unit circle.

The Mandelbrot Set For the Mandelbrot set, we consider polynomials of the form p c (z) = z 2 + c, with parameter c C. We ll write O pc (z) = O c (z) for the orbits of p c. The Mandelbrot Set is M = {c C J (p c ) is connected}. This definition appears simple enough, but Julia Sets are not simple objects.

J (p c ) for c = 3+6i 10

J (p c ) for c = 1 + 3i 10

J (p c ) for c = i

J (p c ) Connectedness Characterization Luckily, there is a nice characterization of J (p c ) being connected. Theorem. The Julia Set, J (p c ), is connected if and only if there is an R R such that p n c (0) R for all n. If the Julia set is disconnected, it is totally disconnected. This characterization shows that c M O c (0).

A useful Fact Lemma. Let p n c (0) > 2, and pn c (0) c for some n 1. Then O c (0). Proof Take n be the smallest such integer. We have that p n+1 c (0) = p n c (0) 2 + c p n c (0) 2 c ( p n c (0) 1) p n c (0). Now, if p n+k c (0) ( p n c (0) 1) k p n c (0), then p n+k+1 c (0) ( p n+k c (0) 1) p n+k c (0) ( p n c (0) 1) k+1 p n c (0). Hence, by induction, we have that p n c (0).

Characterization of M Theorem. c M p n c (0) 2 for all n 1. Proof : By the J (p c ) Connectedness Characterization, p n c (0) 2 for all n 1 c M. : Say p k c (0) > 2 for some k. If c > 2, then p n c (0) c > 2. If c 2, p n c (0) c. In either case, we can apply the above Lemma for some n to get that p k c (0). Hence, c M. Corollary. M D 2, the disc of radius 2. Proof p c (0) = c. This bound is sharp. J (p 2 ) = [ 2, 2] is connected, so 2 M.

Compactness Theorem. M is compact. Proof Let M n be the set of parameter values with p n c (0) 2. Let {c k } M n be a sequence in M n, and c k c. Then p n c k (0) p n c (0). Since p n c k (0) 2, p n c (0) 2, so for each n, M n is a closed set. The Characterization of M above gives that M = so M is also closed. n=1 M n, 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 Böttcher 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 w n 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 z n, with conjugacy ψ.

The Böttcher Coordinate for p c The above Theorem gives that p c is conformally conjugate to the map z z 2 in some neighbourhood, U c, of and the conjugacy is unique. Douady and Hubbard go further and calculate the explicit form of the conjugacy. Theorem. Let B c : U c C \ D R be the conformal conjugacy associated with the polynomial p c. Then B c (z) = lim [ p n c (z) ] 1/2 n, where the root on the RHS is choosen so that [ p n c (z) ] 1/2 n z. Moreover, B c (z) z near.

Analytic Continuation of the Böttcher Coordinate for p c It is straightforward to show that the Böttcher coordinate obeys the following formula. B c [ pc (z) ] = [ B c (z) ] 2. We can see that B c obeys the conjugacy relationship with p c. Now, if the critical point c is not in U c, then p c has an analytic inverse that sends U c to the pre-image p 1 c (U c ), by the Inverse Function Theorem. The above equation then gives us a way to conformally extend B c to the bigger domain p 1 c (U c ). If c is not in p 1 c (U c ) either, we can extend to p 2 c (U c ), and so on, until c p k c (U c ).

Analytic Continuation of the Böttcher Coordinate for p c These extensions give us the the conformal map B c : Ω c C \ D Rc, where the new domain Ω c depends on J(p c ). If J (p c ) is connected, c A( ), so the extending process outlined for B c can be repeated infinitely, so Ω c = A( ). If J (p c ) is disconnected, c A( ), so the extended domain will eventually contain c. The Inverse Function Theorem then fails to provide an inverse. Though it is not obvious from this discussion, R c = 1 when J (p c ) is connected, and R c > 1 otherwise.

The Mandelbrot IS Connected. We now consider the function Φ : C \ M C \ D 1, where c B c (c). Φ is well defined as c Ω c J (p c ) disconnected c M, and B ( ) =. Φ is also conformal. (not proved here) Hence, C \ M and C \ D 1 are conformally equivalent via Φ, so C \ M is simply connected. Thus, M is connected.