MA 471671
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
x^{2}+y^{2}+z^{2} = 1) in
three dimensional space has
topological dimension $3$.
 The function F(x) = x^{2}  1 is
chaotic on the interval [1,0].
 The quadratic functions x^{2}1 and
x^{2}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 MiddleFifths Cantor set is derived by removing the open
middle fifth of an interval and then iterating this procedure.
This set has topological dimension
 The MiddleFifths Cantor set has fractal dimension
 The length of the Koch curve is
 The Schwarzian MinMax principle states that:
3. Prove that the doubling function on the circle is
semiconjugate to x^{2}2 on 2<= x<= 2 via the
semiconjugacy
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.