MAT 335 Practice Homework Problems for Julia and Mandelbrot Sets
* * * Not to be handed in * * *
- Explain the shapes of the main bodies of the "Mandelbrot" sets
of the functions z^3 +c and z^4 + c (cf. page 8 of the handout).
Also explain the symmetries of the sets.
- Suppose q_c has a stable period 3 orbit. Make a sketch of what
the Julia set J_c looks like in general. Explain the shape of J_c (or P_c)
in terms of the stable set ( = basin of attraction) of the perioodic orbit.
(See April 9 notes and Figure 14.18).
- What to the Julia sets look like for those values of c where
the periodic orbit is parabolic (indifferent, neutral)? Where are these
c-values located on the Mandelbrot set?
- Derive the formula for the threshold radius r(c) of q_c.
- Derive the formulae describing the main cardioid region of the
Mandelbrot set and the period 2 bulbs (including an explanation of
why we are considering stable periodic orbits of q_c).
- In what sense does the Mandelbrot set contain copies of Julia sets?
- Explain why the Julia set is a fixed point of an "IFS". Begin by
defining the "IFS".
- Explain how one could use the chaos game to draw Julia sets.
- Sketch the encirclements Q_o^(-n) for the Julia set q_o(z) = z^2.
- Calculate the Jacobian matrix of the transformation W(w) = sqrt(w-c)
(i.e., one of the inverse transformations of q_c(z) = z^2 + c = w) and
determine the region in the complex plane where W contracts area.
Note: It is easier to calculate the determinant of
the Jacobian matrix of q_c(z) and determine the regions where q_c
expands area. Then the regions where W(w) contracts areas would
be those (since W(w) is an inverse of q_c). To do this calculation
write q_c(z) as a two-dimensional function F(x,y) where z = x + iy.