__Claremont Graduate University__ | __Claremont McKenna__ | __Harvey Mudd__ | __Pitzer__ | __Pomona__ | __Scripps__

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When

Start: 01/18/2016 - 3:45pm

End : 01/18/2016 - 4:45pm

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