Arithmetic Dynamics – Mathematics 2720 (CRN:24630)
Brown University – Spring, 2014
Professor Joseph Silverman
| Course Description: |
Dynamical systems is the study of iteration of a map
f:S→S.
Writing fn for the n'th iterate,
one attempts to classify the points of x∈S in
terms of the behavior of their orbit
Of(x) = {fn(x) :
n ≥ 0 }. Arithmetic dynamics is a relatively new field
in which the primary interest is the number theoretic properties
of orbits. We take f to be a polynomial or
rational function with rational, algebraic, or p-adic
coefficients, and then the orbits of analogously chosen points
lie in arithmetically interesting fields. In order to illustrate
the sorts of problems that one studies, here is one conjecture
and two theorems that we will discuss in this course.
Theorem 1: Let f(x)∈Q(x) be a rational function of degree d≥2 with rational coefficients. Then there are only finitely many initial rational points β∈Q whose f-orbit Of(β) is finite. (Such points are called pre-periodic.) Conjecture 2: With notation as in Theorem 1, there is a constant C(d), depending only on the degree of f, such that f has at most C(d) preperiodic points β∈Q. (This is currently not known even for quadratic polynomials.) Theorem 3: Let f(x)∈Q(x) be a rational function of degree d≥2 with rational coefficients, and assume that the second iterate f2(x) is not a polynomial. Then for every starting point β∈Q, the f-orbit of β contains only finitely many integers, i.e., Of(β) ∩ Z is finite. |
|---|---|
| Text |
The Arithmetic of Dynamical Systems
Joseph H. Silverman Springer-Verlag, ISBN: 978-0-387-69903-5, © 2011 |
| Office | Mathematics Department, Kassar House, Room 202 |
| Phone | 863-1124 |
| jhs@math.brown.edu | |
| Web Site | www.math.brown.edu/~jhs/MA0272/MA0272HomePage.html |
| Office Hours |
TBA (Or send me an email to make an appointment. I tend to be in on MWF and not on TTh.) |
| Course Time | MWF 11:00–11:50am (D hour) |
| Course Location | Kassar 105 |
| Homework |
Homework assignments will be posted online. (I'll also try
to announce them in class.)
Click here to go to the Math 272 Homework Page. |
| Note on Using Computers in Math 272 | I tend to use a computer program called PARI-GP to do number theory calculations. The good news about PARI is that it is free and very fast and powerful at doing number theoretic computations. The bad news is that it's not tremendouly user friendly. You can download PARI by clicking here. Another way to use PARI to do short calculations is to use the SAGE web site. You'll need to create a (free) account. Then you'll be able to type one or more PARI commands and type Shift-Return to perform the computation. Or you can use SAGE itself, which has a large collection of number theoretic algorithms, including an ongoing project to create an arithmetic dynamics package. |
Tentative Syllabus: