Dynamical Systems     MATH 374

Homework Assignment #6

Due Friday, Nov. 11, 5:00 pm

All problem numbers refer to the course text 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.

Chapter 10 (pp. 130 - 132)
Problems:   3, 4, 6, 8, 11, 16, 20, 24

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 and see what map D(x) reminds you of.

Chapter 11 (pp. 151 - 153)
Problems:   1, 2, 3, 4

Hint: For problem #3, note that there is no assumption that F is continuous.

Additional Problem:
Prove the Transitivity Proposition, that is, prove that if f and g are topologically conjugate and f is toplogically transitive, then g is topologically transistive. (See class notes from 11/7 for hints and details.)