Homework #17

Reading: pg. 769-840

1. Pick a c in the period 1 bulb of M and iterate the origin. Repeat for the period 2 and both period 3 bulbs off the main cardioid; submit a plot of the orbit in the complex plane for one of the period 3 cases. Comment.

2. Find the equation for the main cardioid of M. (Hint: This is precisely those c for which Q(z)=z^2+c has a hyperbolic attracting fixed point. Find the fixed point and see where the derivative has modulus at most one.) Sketch it.

3. Discuss the dynamics of the quadratic iterator for c=1/4. Note that there is at least a fixed point and a period two cycle.

4. Discuss the change in the dynamics of the quadratic iterator as c passes through c=-.75 on the real axis. (See pg.868 of the text.)

5. Describe the Julia set of the quadratic iterator for c=-2. What happens as c passes through this point (along any path in the complex plane)?

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