## Exam #2, 1997

1. True/False.

• If a continuous function on the real line has a periodic point of period 60, then it must also have a periodic point of period 72.

• If the Schwarzian derivative of a function is negative, then the graph of that function is concave down.

• The surface of the unit sphere (i.e., the locus of points satisfying x2+y2+z2 = 1) in three dimensional space has topological dimension \$3\$.

• The function F(x) = x2 - 1 is chaotic on the interval [-1,0].

• The quadratic functions x2-1 and x2-2 are conjugate.

• The doubling function has a dense orbit in the interval 0<= x < 1.

• There is no function that has only periodic points of period three and no other periods.

• The Sierpinski tetrahedron has topological dimension 1 and fractal dimension 2.

• The real line is a fractal.

• A dense subset of the interval 0 <= x <= 1 can have total length less than 1/2.

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

• The Middle-Fifths Cantor set is derived by removing the open middle fifth of an interval and then iterating this procedure. This set has topological dimension

• The Middle-Fifths Cantor set has fractal dimension

• The length of the Koch curve is

• The Schwarzian Min-Max principle states that:

3. Prove that the doubling function on the circle is semi-conjugate to x2-2 on -2<= x<= 2 via the semi-conjugacy H(x) = 2 cos x.

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

• Fractal.

• Conjugacy.

• Schwarzian Derivative of F.

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

5. State Sarkovskii's Theorem, including the Sarkovskii ordering.

6. Describe the relationship between the Sierpinski triangle and Pascal's triangle.

7. In an essay, discuss the chaos games that were played to create the movie that you see on the screen. Be sure to give the number of vertices used in each frame, the compression ratios (magnification factors), rotations, and how these items were changed to produce the animation.