1.  Read Chapters 1-4 in the book

2.  Define the ``shift'' function S(x) on the interval [0,1) by

that is, S(x) is the fractional part of 10x.  For example, S(0.3245) = 3.245 (mod 1) = 0.245.

1. Discuss the behavior of the orbit of 1/7 under S (use fractions, not decimal approximations).
2. Express each fraction in the orbit of 1/7 as a (repeating) decimal.  What is the pattern?
3. Why do you think S(x) is called the shift function?
4. Show that any point of the form k / 9, where k is an integer from 0 to 8, is a fixed point of S. What do the decimal expansions of these points look like?
5. Give an example of a point in [0,1) that lies on a 2-cycle for S.  (Hint:  the easiest way to do this is to use decimal expansions).
6. Give an example of a point in [0,1) that lies on a 3-cycle for S.
7. If a point is eventually fixed , what must its decimal expansion look like?
8. Explain why the orbit of every rational number is periodic or eventually periodic.