Juan Morales-Ruiz

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.

We give a review about the integrability of complex analytical dynamical systems started with the works of Kovalevskaya, Liapounov and Painleve as well as by Picard and Vessiot at the end of the XIX century. In particular, we state a new result which generalize a theorem of Ramis and the author. This last theorem is itself a generalization of Ziglin's non-integrability theorem about the monodromy group of the first order variational equation. Also we try to point out some ideas about the connection of the above results with the Painleve property.