On Dynamical Systems in the Étale Topology

We discuss $\mathcal{D}$-modules and dynamical systems in the étale topology. We introduce the differential scheme associated to a morphism $f:X\to S$ of schemes of the same dimension. We introduce differential inertia group $I_{diff}^i$ which act trivially on the special fibers of these schemes. As an application we discuss the problem of counting points on elliptic curves which we show is connected to vanishing cycles of singularities and to closed orbits of dynamical systems.