On the Dynamical Meaning of the Picard–Vessiot Theory

In this paper we present a dynamical interpretation of the Differential Galois Theory of Linear Differential Equations (also called the Picard$ndash;Vessiot Theory). The key point is that when a linear differential equation is not solvable in closed form then by a theorem of Tits the monodromy group for fuchsian equations (or a generalization of it for irregular singularities: the Ramis monodromy group) contains a free non-abelian group. Roughly this free group gives us a very complicated dynamics on some suitable spaces.