MA 471-671

Final Exam --- 2001

Do all problems and show all work.

1. True/False

2. Quickies. Answers only -- no partial credit.

3. Definitions. Give the precise definitions of each of the following.

4. Give exact formulas for the fixed points and the period two points of the function F(z) = z2 + c.

5. Cantor set. In an essay discuss the Cantor Middle Thirds set. Prove that this set is uncountable. Discuss the fractal dimension of the set.

6. In an essay, describe why the filled Julia set of F(z) = z2 + c consists of infinitely many pieces when the orbit of 0 tends to infinity.

7. In an essay, discuss the role of the critical orbit (the orbit of 0) in the dynamics of F(x) = x2 + c. Include both the real and complex case.