## Final Exam --- 2001

Do all problems and show all work.

1. True/False

• a. F(x) = x2+2 is chaotic on the interval [-2,2].

• b.F(z) = z2 - 1 has a connected filled Julia set.

• c. F(x) = kx(1-x) has a saddle node bifurcation at k=1.

• d. The subset of the sequence space that consists of all sequences that contain infinitely many 0's is dense.

• e. If a continuous function on the real line has a periodic point of period 60, then it also ha a periodic point of period 40.

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

• a. Give an example of a sequence in Sigma whose distance from the sequence (010101...) is exactly 1/8.

• b. Sketch the 2/7-bulb in the Mandelbrot set.

• c. List all periodic points of period 3 for the doubling function on the unit circle (i.e., theta --> 2 theta).

• d. Give an example of a point in Sigma whose orbit under the shift is dense.

• e. The complex function F(z) = z2 + c has an attracting fixed point for which complex c-values?

• f. Consider the Cantor set obtained by removing middle fifths subintervals from the unit interval at each stage. What is the fractal dimension of this set?

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

• a. The filled Julia set of z2 + c.

• b. The function F is continuous at x0.

• c. H is a conjugacy between F and G.

• d. Q is a dense subset of the set T

• e. The function F is chaotic.

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.