1

Consider iteration of the function \(f(z)=z^3\text{.}\)

1. Show that zero is a super-attractive fixed point of \(f\text{.}\)
2. Show that the orbit of \(z_0\) tends to zero whenever \(|z_0| \lt 1\) but diverges to \(\infty\)
3. Explain precisely why \(f\) displays sensitive dependence on initial conditions.
4. Compute the orbits of \(e^{\pi i/3}\) and \(e^{\pi i/4}\text{.}\)

2

Let \(f(z)=z^3-z-1\text{.}\)

1. Determine the fixed points of \(f\) and classify as attractive, repelling, or neutral.
2. Plot the filled Julia set of \(f\) and indicate the locations of the the fixed points.

3

Let \(f(z)=2z^5-z^4+3z^3-8z^2+z-1\text{.}\) What is the escape radius guaranteed by theorem Theorem 3.5.1?

4

Show that \(z_0=1\) is a neutral fixed point of \(f(z) = e^{z-1}\text{.}\)

5

In section Section 8, we examined the function \(f(z)=z+z^5\) motivated, in part, by dissatisfaction with figure Figure 3.8.1. Specifically, we asked - "Does the stable region contain the origin?" Using the family \(f_r(z)=rz+z^5\text{,}\) show that it's possible for a function to have a Julia set that qualitatively looks like figure Figure 3.8.1, yet is stable at zero.

6

Consider iteration of the function \(f(z)=z-z^3\text{.}\) Note that zero is a neutral fixed point. Show that real points near zero move towards zero under iteration of \(f\) but that imaginary points near zero move away from zero under iteration of \(f\text{.}\)

7

Let \(f(z) = z^2 + 1\) and let \(N(z)\) be the corresponding Newton method iteration function. Show by direct computation that \(i\) is a super-attractive fixed point for \(N\text{.}\)