MA 471-671

Exam #2, 2001

1. True/False.

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

3. State Sarkovskii's Theorem (including the Sarkovskii ordering):

In the proof of this theorem, we used both of the properties below. Complete the statement of each of them.

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

5. Prove that all orbits of the complex function F(z) = lambda z are cycles when lambda = exp(2 pi i (p/q)).

6. In a brief essay, discuss the meaning of the ``black lines'' that you see in the orbit diagram. Include a brief discussion of why this occurs.

7. Here is a target and a starting point from the chaos game with vertices R, G, and B. What sequences of R, G, and B's would you use to hit the interior of this target in exactly 4 moves.

Picture of chaos game at Novice level