## Section4.4Exercises

###### 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 4.1.1?