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.)