Extremal Configurations in Arithmetic Dynamics

Joseph Silverman
Brown University

Let f : P^N --> P^N be a morphism of degree at least 2, let x be a point, and let O_f(x) = { f^n(x) : n = 0,1,2,... } be the forward orbit of x under iteration of f. The dynamical Mordell-Lang conjecture asserts that the intersection of O_f(x) with a subvariety Y in P^N is finite unless f^k(Y) is contained in Y for some k > 0. We consider the complementary question of whether O_f(x) can contain infinitely many co-linear triples, or more generally, infinitely many m-tuples lying in "smaller than expected" linear subspaces of P^N. (Joint work with Bianca Viray.)

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