Dynamical Systems from a Number Theory Perspective

When
Start: 01/18/2016 - 3:45pm
End  : 01/18/2016 - 4:45pm

Category
Colloquium

Speaker
Joe Silverman

Abstract

Dynamics is the study of iteration.  Classically one takes a rational map f(z)=F(z)/G(z) and attempts to describe the behavior of points under interation f^n(z) = f(f(f(...f(f(z))...))). The *f-orbit* of a point b is the set of images of b under the repeated iteration of f, Orbit of b = { b, f(b), f^2(b), f^3(b), ... }. The points with finite orbit are called *preperiodic points*. They play a particularly important role in the dynamics of f. For a number theorist, it is natural to take F(z) and G(z) to be polynomials with integer coefficients and to study the orbits of rational numbers b. In this talk I will survey some of the known results and some of the outstanding conjectures related to this number-theoretic view of dynamics.

Typical problems include: 

(1) How many preperiodic points can be rational numbers?

(2) For which rational maps f can the orbit of a rational number b contain infinitely many integers?

 

Where
Emmy Noether Room, Millikan 1021, Pomona College