1. True/False.

- a.
**F(x) = x**has a fixed point at^{2}**x= -1**. - b. Let
**(000....)**and**(010101.....)**be two sequences in the sequence space**Sigma**. Then the distance between these two points is 1. - c. The Cantor Middle-Thirds Set is
dense in the interval
**[0,1]**. - d. The function
**F(x) = -1**has no fixed points. - e. The function
**G**defined on the sequence space**Sigma**by**G(s_0 s_1 s_2 .....) = (000....)**if**s_0 = 0**, or**G(s_0 s_1 s_2 .....) = (111....)**if**s_0 = 1$**is continuous at all points of**Sigma**.

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

- a. The function
**F(x) = 3x(1-x)**has an indifferent fixed point at _______. - b. The family of functions
**F(x) = x**has a ___________ bifurcation at^{2}+ x + C**C = 0**. - c. For which values of
**c**does the function**F(x) = x**have no cycles or fixed points whatsoever?^{2}+ c - d. The shift map has how many cycles of
**prime**period 4? - e. The family of functions
**S(x) = A sin x**has an attracting fixed point at the origin for which**A**values?

3. Use graphical analysis to give a complete orbit
analysis of the function **F(x) = -x ^{2}**. List all fixed
points and
cycles and tell if they are attracting or repelling. List all points
whose orbits tend to cycles or fixed points and all points whose orbits tend
to + or - infinity.

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

- a. Chaos.
- b.
**F**is continuous at**x_0**. - c. The shift map
**sigma**on the sequence space. - d.
**Q**is a dense subset of the set**T**.

5.In an essay, describe the bifurcation that occurs at
**c = -3/4** for the family of functions **F _{c}(x) = x^{2}
+ c**. Your essay
should include enough graphs and bifurcation
diagrams to explain this bifurcation fully.

6. In an essay, describe the sequence space **Sigma**.
Define the metric on **Sigma**. Discuss what it means for two points to
be close together in **Sigma**. Give examples of nearby points and
points that are far apart.
You will be graded on
the accuracy of your statements as well as the use of English in your
essay (spelling, grammar, etc.)

7. Consider the subset of **Sigma** that consists of all
sequences that have no strings of the form **0110** in them. Is
this set dense? If so, prove it. If not, prove it. And, most
certainly, have fun in either case.