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Section 4.4 Exercises

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?

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