Algebraic Dynamics: Dynamical cancellation
ALGEBRAIC DYNAMICS, SEMINARS
Speaker:
Jason Bell - University of Waterloo
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.
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