MA 471-671

Final Exam --- 1999

Do all problems and show all work.

1. True/False

• F(x) = x²-1 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)=z²+1 is conncected.

• The Mandelbrot set consists of all seeds whose orbits do not go to infinity under z² + 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 x² -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) = z² + 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 x₀.

• The sequence space Sigma

• Q is a dense subset of the set T

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 z² + 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.