Complex and Arithmetic Dynamics

Tutorial Instructor: Jake & Zijian

Schedule

Jun. 20      5pm-7pm      Introduction to dynamics (Jake) []; Maps defined by rational functions (Zijian) [Silverman Chapter 1.1 & 1.2]
Jun. 22      6pm-8pm      Riemann surfaces, uniformization and normal families (Jake) [Milnor Chapter 1, 2, 3]; Periodic and exceptional points (Zijian) [Silverman Chapter 1.3 - 1.6]

Jun. 27      5pm-7pm      Dynamics over p-adic numbers (Zijian) [Silverman 2.1-2.3]
Jun. 29      6pm-8pm      Good reduction; beginning of height (Zijian) [Silverman 2.5-2.6; 3.1-3.2]

Jul. 6        6pm-8pm      Height; dynamics over rational numbers (Zijian) [Silverman 3.1 - 3.5]
Jul. 8        5pm-7pm      TBA (Zijian) []

Jul. 11      5pm-7pm      Conformal metric and hyperbolic surfaces (Jake)
Jul. 13      6pm-8pm      TBA (Jake)

Jul. 18      5pm-7pm      TBA (Jake)
Jul. 20      6pm-8pm      TBA(Jake)

Jul. 25      5pm-7pm      Student Presentations I (Jake)
Jul. 27      6pm-8pm      Student Presentations II (Zijian)

Additional discussions

TBA

Suggested Homework

Week 2:
Basic: Learn about p-adic numbers and projective lines; [Silverman] 1.1; 1.14.
Not so basic: [Silverman] 1.8; 1.10; 1.27; 1.30.
(Slightly more) Challenging: As stated in class, find a point in the square whose orbit (module the integral lattice) is dense under the multiplication by 2 map.

Main References

1. [Milnor] John Milnor,    "Dynamics in one complex variable"    Princeton University Press, 2006.
2. [McMullen] Curt McMullen,    "Riemann surfaces, dynamics, and geometry"    accessible here
3. [Silverman] Silverman,    "The Arithmetic of Dynamical Systems"    Springer, 2007

Additional References

1).   on p-adic numbers (this is required reading)
     Introduction to p-adic numbers
     Another introduction to p-adic numbers

2).   on the projective line
     TBA

3).   mapping class group (if you are interested)
Benson Farb and Dan Margalit,    "A Primer on Mapping Class Groups",    Princeton University Press, 2011.

4).   Renormalization (if you are interested)
"Complex Dynamics and Renormalization"    Princeton University Press, 1994.

5).   Billiards (if you are interested)
     Billiards