Dynamical cancellation

Let X be a projective variety and let f be a dominant endomorphism of f, both of which are defined over a number field K. We consider the question of when there is some integer n, depending only on X and K, such that whenever x and y are K-points of X with the property that some iterate of f maps x and y to the same point, we necessarily have that the n-th iterate of f also achieves this.  We consider this an instance of “dynamical cancellation’’ and we show that such a cancellation result holds for etale morphisms of projective varieties as well as self-maps of smooth projective curves.  As a result we are able to prove a general cancellation result for semigroups of polynomials: if f_1, … , f_r are polynomials of degree at least two then there is a proper closed subset of P^1 x P^1 with the property that for any a, b in K satisfying phi(a)=phi(b) for some phi in the semigroup generated by f_1, … f_r under composition, we necessarily have (a,b) lies in this closed subset.  Moreover, we show that this Z can be taken to be the union of the diagonal with a finite set of points for “non-exceptional” semigroups.  This is joint work with Matt Satriano and Yohsuke Matsusawa.

Go to  http://people.math.harvard.edu/~demarco/AlgebraicDynamics/ for more information