Dynamical Systems     MATH 374

Homework Assignment #5

Due Thursday, March 25, START of Class

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.)