Galois theory of differential equations with an action of an endomorphisms.

I'll explain how one can construct a Galois theory for differential equations that takes into account the action of a difference operator,i.e., an endomorphisms, on the solutions.The theory attaches a group scheme to a differential equation, which encodes the algebraic difference relations among the solutions of the differential equation.This is typically the case of $p$-adic differential equation with a Frobenius structure. This is a joint work with C. Hardouin and M. Wibmer.