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Section 4.4 Exercises


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 \(e^{\pi i/4}\text{.}\)

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.

Let \(f(z)=2z^5-z^4+3z^3-8z^2+z-1\text{.}\) What is the escape radius guaranteed by theorem TheoremĀ 4.1.1?