## Final Exam --- 2000

Do FIVE of the following problems. In each case, show all work. Each essay should be approximately 3 pages in length. It should be written in proper English and accompanied by relevant graphs or pictures.

1. Bifurcations. In an essay, discuss the concept of bifurcation. Describe in detail at least two different bifurcations with specific examples. Include graphs and other relevant diagrams.

2. Conjugacy. In an essay, discuss the notion of conjugacy. Why is conjugacy important in dynamical systems? In the essay, you should show explicitly that the functions \$F(z) = z^2\$ on the unit circle and F(x) = x2 -2 on the interval -2 <= x <= 2 are (semi)-conjugate.

3. Define dense subset. Give several examples of dense and non-dense subsets. Then prove or disprove: The set of all sequences in the sequence space that end with all zeroes is dense in the sequence space.

4. The Mandelbrot Set. In an essay, discuss the structure and meaning of the Mandelbrot set. Be sure to include a discussion of how you can "read off" the periods of the primary bulbs from these bulbs' geometry. Finally, find a formula for the boundary of the main cardioid.

5. Cantor set. In an essay discuss the Cantor Middle Thirds set. Describe its construction and other features of this set. Prove that this set is uncountable. Discuss the fractal dimension of the set.

6. In an essay, discuss the concept of a filled Julia set of the quadratic function \$F_c(z) = z^2 + c\$. Discuss the escape criterion for these functions. Include in your essay a clear discussion of the Fundamental Dichotomy. Give a brief description of the proof that, if the orbit of \$0\$ escapes, then the filled Julia set consists of infinitely many pieces.

7. Chaos. Define chaos. Prove that the shift map on the sequence space is chaotic.