Dynamical Systems     MATH 374

Homework Assignment #5

Due Friday, March 28, 5:00 pm

All problem numbers refer to the primary course text by Robert 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 your work. A nonempty subset will be graded. You are encouraged to work on these problems with other classmates, and it is ok to use internet sources for help if it's absolutely necessary; however, the solutions you turn in should be YOUR OWN WORK and written in YOUR OWN WORDS.

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

Chapter 7 (pp. 80 - 81)
Problems:   3, 5, 9, 10, 12, 13

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

Hint: For problem #20, there are two different approaches. One approach is to use the graphs of the higher iterates of D(x) that you computed for HW#1, 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 Qc and the shift map σ are topologically conjugate. Prove that the inverse of the itinerary map, S-1, is continuous. (See the class handout about the conjugacy for more details.)