Homework should be turned in at the BEGINNING OF CLASS. All problem numbers refer to the primary textbook by Devaney. Unless otherwise indicated, all parts of a problem (a), (b), etc. should be completed. You should write up solutions neatly to all problems, making sure to SHOW ALL YOUR WORK. A nonempty subset will be graded. You are strongly encouraged to work on these problems with other classmates, although the solutions you turn in should be YOUR OWN WORK.

** Note: ** Please list the names of any students or faculty who you worked with
on the assignment.

**Strange Attractors**

Read *Strange Attractors*, the fifth chapter of
Gleick's book *Chaos*.

What is a strange attractor? In particular, what is the difference between an attractor and
a *strange* attractor? Who were the two scientists to come up with the term?
Who found the first strange attractor in their research before the term even existed?

**Chapter 7 (pp. 80 - 81) **

Problems: 3, 5, 8, 9, 10, 12, 13, 14

**Hint:** For #14, try and find the ternary expansion for points in Gamma.
A rigorous argument involves finding a formula for the ternary expansion of T(x)
for two different cases: x in [0,1/2] and x in (1/2,1]. It also helps to review
your class notes on the middle-thirds Cantor set.

**Chapter 10 (pp. 130 - 132) **

Problems: 3, 4, 6, 8, 16, 20

**Hint:** For problem #20, there are two different approaches. One is to use the
graphs of the higher iterates of D(x) which you computed in HW#2, Ch. 3, Exercise 13
and prove that D(x) satisfies all three properties of chaos directly. The second approach
is to use binary notation for x and D(x) and see what map D(x) reminds you of.

**Additional Problem:**

Let **S** be the itinerary map used to show that Q_c and the shift map
are topologically conjugate. Prove that the inverse of the itinerary
map, **S^{-1}**, is continuous. (See class notes from 3/16 for details.)