MATH 845, Arithmetic of Dynamical Systems -- Course materials

Resources: The following are available. Let me know if you find something appropriate to add.
  • I have published the Sage worksheet for classical Fatou and Julia sets on the public sage notebook server. You can either make an account there and play with the worksheet or download it and put it in another sage server.
Nr Exercises Due date
1 1.3, 1.12, 1.20 Jan. 20
  • What happens to Theorem 21 if d=1?
  • 2.8, 2.16, 2.21, 2.23
  • Correlate 2.11 with 1.3
Feb. 3
  • 3.35, 3.42
  • This question concerns Theorem 3.43. First, let S be a finite set of primes. We say that a rational number is an S-integer (element of ZS) if its denominator consists entirely of prime factors from S. Can we modify the prove the statement of Theorem 3.43 for S-integers as well?
  • This question concerns the proof of Theorem 3.35. Consider the equation X^4-3Y^4=B. If we adjoin sqrt(3), we can write X^4-3Y^4=(X^2-sqrt(3)Y^2)(X^2+sqrt(3)Y^2), so perhaps we can avoid using Roth's theorem and use the same argument that applies if the form factors over Z already. What goes wrong?
Mar. 1
4 4.1, 4.6 March 31
5 6.1, 6.8, 6.10 April 14