The Cohomological Equation for Area - Preservingflows on Compact

We study the equation Xu = f where X belongs to a class of area-preserving vector elds, having saddle-type singularities, on a compact orientable surface M of genus g 2. For a \full measure" set of such vector elds we prove the existence, for any suuciently smooth complex valued function f in a nite codimensional subspace, of a nitely diierentiable solution u. The loss of derivativesis nite, but the codimensionincreases as the diierentia-bility required for the solution increases, so that there are a countable number of necessary and suucient conditions which must be imposed on f, in addition to innnite diierentiability, to obtain innnitely diierentiable solutions. This is related to the fact that the "Keane conjecture" (proved by several authors such which implies for "almost all" X the unique ergodicity of the ow generated by X on the complement of its singularity set, does not extend to distributions. Indeed, our approach proves that, for \almost all" X, the vector space of invariant distributions not supported at the singularities has innnite (countable) dimension , while according to the Keane conjecture the cone of invariant measures is generated by the invariant area form !. x1. Introduction In this announcement we describe results on the cohomological equation Xu = f , where X is a smooth area-preserving vector eld on a compact orientable surface M of genus g 2. Topological reasons force X to have singularities, which will be assumed to be of a canonical polynomialsaddle type (not necessarily non-degenerate). The question we answer can be stated as follows: given a smooth complex valued function f on M, is it possible to nd a (smooth) solution u on M to the equation Xu = f? The study of cohomological equations is mainly motivated by the problem of describing time-changes for ows. In fact, the time-change induced in the ow generated by a vector eld X by a positive function f is trivial if and only if the equation Xu = f ? 1 has (smooth) solutions. In this case the ow produced by the time-change is (smoothly) conjugated to the original one K-H, x2.2]. The rst observation on cohomological equations is that each invariant measure for the ow X t generated by a vector eld X on a compact manifold M gives a necessary