Disintegrated differential equations and mixing Anosov flows
By Rémi Jaoui
The classification of minimal types in differentially closed fields is beautiful evidence for the effectiveness of geometric stability theory in the study of some noetherian geometric contexts that were previously inaccessible. However for a given nonintegrable differential equation, it is often hard to determine the properies of the correspoinding definable set in DCF0. On the other hand, for many interesting examples of noninntegrable equations, Anosov has isolated some very concrete properties reflecting the “chaotic” nature of the corresponding flow acting on the set of initial conditions. Starting with a differential equation over the field of real numbers, I will explain how to witness disintegration of its generic type (in DCF0) from the dynamical properties of the associated flow. Then I will apply this technique to study certain differential equations describing the geodesic motion on an algebraically presented Riemannian manifold with negative curvature. These results were obtained during my Ph. D. and shortly after.