MA 471-671
Final Exam --- 1999
Do all problems and show all work.
1. True/False
- F(x) = x^{2}-1 has an eventual cycle at x
= 2^{1/2}.
- The set of endpoints of the Cantor
Middle-Thirds set is dense in the Cantor set itself.
- The Julia set of F(z)=z^{2}+1 is conncected.
- The Mandelbrot set consists of all
seeds whose orbits do not go to infinity under z^{2} + c.
- The fractal dimension of the Menger
sponge is log 20/log 3.
2. Quickies. Answers only -- no partial credit.
- Let (111....) and (001001001....)
be two sequences in the sequence space Sigma. Then the distance between
these two points is:
- The set of points whose orbits go to infinity under
iteration of x^{2} -1 is:
- Draw a picture of the complex square root of the square
whose vertices lie at
(2,2),(-2,2),(-2,-2),(2, -2)
- List all points that lie on a cycle of prime period 4
for the shift map.
- The complex function F(z) = z^{2} + c has an attracting
fixed point for which complex c-values.
- The family of functions F_{k}(x) = k x (1-x) has
what kind of bifurcation at k = 3?
3. Definitions. Give the precise definitions of each of
the following.
- Fractal
- The function F is continuous at x_{0}.
- The sequence space Sigma
- Q is a dense subset of the set T
4. Does the function F(z) = |z| have a
complex derivative at z_{0}=1= 1+0i? If so, prove it.
If not, prove it.
5. First define what is meant be a function being
chaotic. Then prove that the shift map is chaotic on the sequence
space.
6. The Julia set. Write an essay describing the
Julia sets of z^{2} + c. In your essay you should discuss the
the geometry of these sets together with whatever you can say about
orbits on the Julia set. Also discuss the Julia sets for the special
cases c=0 and c=-2.