The Mandelbrot Set. Andrew Brown. April 14, 2008

Transcription

1 The Mandelbrot Set Andrew Brown April 14, 2008

2 The Mandelbrot Set and other Fractals are Cool

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6 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 )}.

7 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.

8 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.

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

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

11 J (p c ) for c = i

12 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).

13 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).

14 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.

15 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.

16 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.

17 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.

18 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 ).

19 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.

20 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.