9 COMPLEX DYNAMICS, THE JULIA SET. 43 Properties of the Julia Set for Q • The Julia set contains all repelling periodic points. • Repelling periodic points are dense in the Julia set. • Q is supersentive at any point in the Julia set. We will prove the first property above, with help from the following: Cauchy Estimate. If P (z) is a polynomial such that |P (z)| ≤ M on the disk |z − z0| ≤ r, then ¡ M |P (z0)| < . r Theorem. If z0 is a repelling fixed point for Q , then z0 is in the Julia set for Q . ¡ ¢ Proof. Let z0 be a repelling fixed point with period n. Then |(Q ) (z0)| = λ > 1. Then by the chain rule, we have for each k, £ £ ¡ ¢ |(Q ) (z0)| = λ → ∞, as k tends to infinity. Since the orbit of z0 is periodic, it does not tend to infinity and so z0 must be in the filled Julia set for Q . Assume, however, that z0 is not in the Julia set (the boundary of the filled Julia set). Then there is a disk |z − z0| ≤ r that lies entirely in the filled Julia set. Then for each z in the disk, and any k, we have £ ¢ |Q (z)| ≤ M = max{|c|, 2}, £ ¢ by the escape criterion. Since Q is a polynomial, the Cauchy Estimate from Complex Analysis gives us £ ¡ ¢M |(Q ) (z0)| < . r This bound holds for all k → ∞, so we have a contraction. Thus z0 is in the Julia set for Q . 9 COMPLEX DYNAMICS, THE JULIA SET. 44 Since Q is supersensitive on the Julia set, the orbit of any point arbitrarily close to the Julia set will eventually go everywhere in the complex plane. Hence, if we pick any point in the plane and iterate backwards, we will eventually ”find” points in the Julia set. Algorithm for the Julia Set for Q . Choose any point z ∈ C. Compute and plot the preimages of z (points on the ”backwards orbit” of z), but don’t plot the first N points. Write your own computer program to compute Julia sets. There is a multi- functional applet that uses both of the algorithms we have described to compute Filled Julia sets and Julia sets available at http://www.math.nagoya-u.ac.jp/ kawahira/programs/otis.html 10 THE MANDELBROT SET 45 10 The Mandelbrot Set After looking at many different filled Julia sets, you will notice that the filled Julia set is either connected, or totally disconnected. We want to see if there is a relationship between the parameter c and the connectedness of the filled Julia set. The Mandelbrot Set M consists of all c-values for which the filled Julia set for Q is connected. That is, ¢ M = {c| |Q (0)| 6→ ∞} We have the following two theorems: Theorem If |c| > 2, then the Julia set is a Cantor set, meaning it consists of an uncountable number of totally disconnected points. Therefore, the Mandelbrot set is contained inside |c| ≤ 2. 2 Theorem Let Q0(z) = z + c. 1. If the orbit of 0 remains bounded, the filled Julia set is connected. 2. If the orbit of 0 escapes to infinity, the filled Julia set is totally disconnected. We will use the Theorem to develop an algorithm for plotting the Mandelbrot set. Algorithm for the Mandelbrot Set. Choose a maximum number of iterations N. For each c in the grid, compute the first N points on the orbit of 0, under 2 £ Q = z + c. If |Q (0)| > 2 for some k ≤ N, then stop iterating and color c £ white. If |Q (0)| ≤ 2 for all k ≤ N, then color c black. Exercise. Write a computer program to plot the Mandelbrot set..