## Exam #2, 2001

1. True/False.

• a. Any dense subest of the interval 0\leq x \leq 1 must contain rational numbers.

• b. The Sierpinski Tetrahedron has topological dimension 2.

• c. The Sierpinski triangle has fractal dimension \log (3/2).

• d. If a continuous function on the real line has a periodic point of period 56, then it must also have a periodic point of period 128.

• e.If a function has negative Schwarzian derivative, then it has an attracting cycle.

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

• a. The Schwarzian min-max principle states:

• b. Give an example of a function which has a period three cycle and no other fixed or periodic points.

• c. If the chaos game is played with four vertices at the corners of a square and contraction factor 2, the resulting image is:

• d. Sketch the complex square root of the figure below Picture of a face in the right half plane.

• e. The Schwarzian derivative of F(x) = cos x is:

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.

• Fixed Point Property. Suppose G is continuous and the intervals I and J satisfy I is a subset of J....

• Subinterval Property. Suppose I and J are intervals and G(I) contains J .....

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

• a. Fractal.

• b. Conjugacy

• c. Fractal Dimension

• d. The complex derivative

• e. Topological Dimension k (both k=0 and k>0).

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