It was a fun class for me to teach, hope it was at least half as fun for you to take it. Have a great summer and hope to see you next semester!
Also, please let me know as soon as you can if you'd like to present in the morning (between 10:15 and 11:15) or in the afternoon (between 3 and 4) on Monday, May 12, and if you need an LCD projector. We'll be meeting in MH 331B both at 10:15 and at 3.
And also: HW 7 deadline has now shifted to Monday, April 28.
Finally, I just posted a handout entitled "Problem 6.1.10" (and an application of the Baire category theorem).
Remember that we do have a problem session tomorrow, April 14 at 10:15 (in MH 331B). I will talk about the bouncing ball system from the handout.
Regarding the survey paper, please give me a first draft by May 5 so I can give you some feedback before you present it. The presentations will be on May 12, in the morning from 10:15 to 11:10 (instead of the usual problem session) and afternoon 3-4.
Two other potential topics for the survey paper are billiards and the Lorentz attractor.
Enjoy your spring break and see you on April 2!
Feel free to suggest other topics.
Also, here's the midterm. Don't hesitate to email me with questions.
Here are some hints. Exercise 3.3.2: use Prop. 3.3.3 and a property of norms in a Euclidean space. Exercise 3.3.3: first show the limit on the RHS exists and is independent of norm, then show that LHS is less than or equal to the RHS (use the property that if k is an eigenvalue of A than k^n is an eigenvalue of A^n), then finally show that the RHS is less than or equal to the LHS using again Prop. 3.3.3. Exercise 3.3.4: use a well-known formula for the trace of A in terms of the eigenvalues of A. Hope this helps!
More importantly, since it seems that it would work with all the registered students, I will try to move the class to start at 2:50. Please let me know ASAP if you have any objections. I'll let you know if and when I get us a new classroom.
A hint for part of the problem 2.2.13, to show that for every x the sequence of iterates (x_n) of f converges to the unique fixed point x_0, consider the set A of all accumulation (i.e., limit) points of A. This is the set of limits of all convergent subsequences of (x_n). Show that f(A) is necessarily contained in A. Show that the sequence |x_n - x_0| converges, say to some number d and that A is contained in the set with two elements x_0 - d, x_0 + d. Then use the weak contractive property of f.
A hint for exercise 2.4.6: suppose p is a periodic point of period 3, so that p, f(p), f(f(p)) are all distinct. Consider all possible cases: p < f(p) < f(f(p)), f < f(f(p)) < f(p), etc. and in each one derive a contradiciton.
If you are still not sure if you want to take the class, feel free to email me for more information.
The course will be fairly self-contained (i.e., we won't be assuming too much from Math 134).
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Boris Hasselblatt and Anatole Katok, A First Course in Dynamics: with a Panorama of Recent Developments, Cambridge University Press, 2003 |
We will also use other books, but they are not required for the class:
The exam schedule:
| # | Due date | Assignment | Solutions |
|---|---|---|---|
| 1 | 2/4 | Exercise 2.2.6, problems 2.2.13, 2.2.14 (HK, p. 45) (pdf). | HW 1 |
| 2 | 2/11 | Exercise 2.3.4, 2.4.6, 2.5.3, 2.5.4 (pdf). | HW 2 |
| 3 | 2/25 | Exercise 3.3.2, 3.3.3, 3.3.4. | HW 3 |
| 4 | 3/3 | Exercise 4.1.3, 4.1.8, 4.1.9. | HW 4 |
| 5 | 3/12 | Homework 5 (pdf) | HW 5 |
| 6 | 4/9 | Exercise 6.1.1, 6.1.2, 6.1.4-6.1.7 | HW 6 |
| 7 | 4/23 | Exercise 7.1.2, 7.2.1, 7.2.3 | HW 7 |
| 8 | 5/12 | Exercise 7.2.5, 7.3.1, 7.2.11 | HW 8 |
Last modified: Tue May 20 14:06:36 PDT 2008