1.[N] Apply the Mandelbrot map
xn+1 = xn2 - yn2 + a
yn+1 = 2*xn*yn + b
three times to each of these points, always starting with x0 = 0, y0 = 0.
(a) a = 1, b = 0 Answer
(b) a = -1, b = 0 Answer
(c) a = 1, b = 1 Answer
(d) a = -1, b = 1 Answer
(e) a = -1, b = 0.25 Answer
(f) a = -1, b = -0.25. Answer
2.[N] For each application of the Mandelbrot map in exercise 1, compute the distance between the point and the origin. Answer
3.[N] Using a cut-off time of 3, which points of exercise 1 lie in the Mandelbrot Set. (Hint: the distance calculations were done in exercise 2.) Answer
4.[N] Take a = -1.1 and b = 0. Using a calculator, apply the Mandelbrot map 10 times, as usual starting with x0 = 0, y0 = 0.
(a) Compute the distance between successive xn: x1 - x0, x2 - x1, ..., and x10 - x9. What do you observe? Answer
(b) Now compute the distance between successive odd xn, and between successive even xn: x2 - x0, x4 - x2, ..., and x3 - x1, x5 - x3, ... . What do you observe? Answer
5.[N] In this exercise, we investigate the claim that the Mandelbrot Set is symmetric above and below the a-axis.
(a) Take a = -1 and b = 1. Starting from x0 = 0 and y0 = 0, apply the Mandelbrot map three times.
(b) Repeat (a), this time taking a = -1 and b = -1.
(c) Compare the results of (a) and (b). Specifically, how are the dynamics related?
(d) Take a = 0.25 and b = 0.1. Starting from x0 = 0 and y0 = 0, use a calculator to apply the Mandelbrot map three times.
(e) Repeat (d), this time taking a = 0.25 and b = -0.1.
(f) Compare the results of (d) and (e). Specifically, how are the dynamics related?
(g)[A] In general, suppose we have two sequences, x0, y0, x1, y1, ..., xn, yn, and x'0, y'0, x'1, y'1, ..., x'n, y'n, with x0 = y0 = x'0 = y'0 = 0. Suppose the first sequence is produced with controls a and b, while the second sequence is produced with controls a and -b. Show x1 = x'1 and y1 = -y'1. Now show x2 = x'2 and y2 = -y'2. Assuming xn = x'n and yn = -y'n, show xn+1 = x'n+1 and yn+1 = -y'n+1. How does this tell you the Mandelbrot Set is symmetric above and below the a-axis? Answer
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