Arithmetic dynamics of algebraic groups

The Artin-Mazur zeta function is a formal power series that counts the periodic points of a discrete dynamical system. Hinkannen established its rationality for any rational map on the projective line over an algebraically closed field of characteristic

zero. The situation in positive characteristic is more delicate, and much less is known. We consider particular classes of discrete dynamical systems arising from endomorphisms of algebraic groups in characteristic p. By studying complex analytic properties of the zeta function, we show that, in some cases, rationality is rather the exceptional case.