Bad reduction of rational maps

We show that the reduction of a projective endomorphism modulo a discrete valuation naturally takes the form of a set-theoretic correspondence. This raises the possibility of classifying "reduction types" of such dynamical systems, reminiscent of the additive/multiplicative dichotomy for elliptic curves. These correspondences facilitate the exact evaluation of certain integrals of dynamical Green's functions, which arise as local factors in the context of counting rational points ordered by the Call-Silverman canonical height. No prior knowledge of arithmetic dynamics will be assumed.