  We discuss $\mathcal{D}$-modules and dynamical systems in the \'etale
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.
