1.[N] Apply the Mandelbrot map

x_{n+1} = x_{n}^{2} - y_{n}^{2} + a

y_{n+1} = 2*x_{n}*y_{n} + b

three times to each of these points, always starting with
x_{0} = 0, y_{0} = 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
x_{0} = 0, y_{0} = 0.

(a) Compute the distance between successive
x_{n}: x_{1} - x_{0}, x_{2} - x_{1}, ...,
and x_{10} - x_{9}. What do you observe?
Answer

(b) Now compute the distance between successive odd x_{n}, and between
successive even xn: x_{2} - x_{0},
x_{4} - x_{2}, ..., and x_{3} - x_{1},
x_{5} - x_{3}, ... . 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 x_{0} = 0 and
y_{0} = 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 x_{0} = 0 and
y_{0} = 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, x_{0},
y_{0}, x_{1}, y_{1}, ..., x_{n}, y_{n}, and
x'_{0}, y'_{0}, x'_{1}, y'_{1}, ...,
x'_{n}, y'_{n}, with x_{0} = y_{0} =
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 x_{1} = x'_{1} and y_{1} = -y'_{1}. Now
show x_{2} = x'_{2} and y_{2} = -y'_{2}.
Assuming x_{n} = x'_{n} and y_{n} = -y'_{n}, show
x_{n+1} = x'_{n+1} and y_{n+1} = -y'_{n+1}.
How does this tell you the Mandelbrot Set is symmetric above and below the a-axis?
Answer

Return to Chapter 7 Exercises

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