Jason P. Bell

Automorphisms of varieties and critical density

Let sigma be an automorphism of a quasi-projective variety X. We look at the images of sigma^i(p) as i ranges over all integers, where p is a point in X. Suppose that we are working in characteristic 0 and sigma^i(p) lies in a fixed proper subvariety of X for infinitely many i. The conclusion we wish to draw is that there is some arithmetic progression such that for all i in this arithmetic progression sigma^i(p) is in our subvariety. This is not true in positive characteristic, but is conjectured to be true in characteristic 0 (by Dan Rogalski). We look at using p-adic methods as a means of proving results in characteristic 0. Using these p-adic methods, we show the Rogalski conjecture is true for affine n-space, Fano varieties, and other classes of varieties.