Dynamical Systems     MATH 374

Prof. Gareth Roberts

Homework Assignment #5

Due Friday, November 3, 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:   1, 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 Problems:

  1. Recall that a set A is countable if there exists a bijection (1-1, onto function) between A and the natural numbers N.

    (a) Prove that the union of two countable sets is countable. Hint: Find a way to count the union, making sure that all elements are counted.

    (b) Prove that the set of rational numbers is countable. Start by listing the positive rational numbers in a clever way.

    (c) Watch the YouTube video on Cantor's diagonalization proof by Cory Chang. Suppose that you were to use Cantor's diagonalization argument on the set of rational numbers between 0 and 1. Where does the proof break down? (It has to break down because the rationals are countable.)

    (d) Using parts (a) and (b), show that the set of irrational numbers are uncountable. You may assume that the set of real numbers is uncountable.

  2. 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.)