Dynamical Systems     MATH 374

Prof. Gareth Roberts

Homework Assignment #6 (Last One!)

Due Friday, November 10, 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: For this final assignment, you are allowed to work in groups of up to 4 people. Please turn in one assignment per group.

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

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

Chapter 12 (pp. 161 - 163)
Problems:   1b, 1c, 2, 7, 8

Additional Problem:
Prove the Transitivity Proposition, that is, prove that if f and g are topologically conjugate and f is topologically transitive, then g is also topologically transitive.
Hint: If h is the conjugacy, use h-1. You may assume, since h is continuous, that h-1(U) is an open set whenever U is an open set. First prove that for any two sets A and B, h(A ∩ B) ⊂ h(A) ∩ h(B).