Complex and Arithmetic Dynamics
Tutorial Instructor: Jake & Zijian
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)
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.
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
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
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)