Sturm theory, Ghys theorem on zeroes of the Schwarzian derivative and flattening of Legendrian curves

Etienne Ghys has recently discovered a beautiful theorem: given a diffeomorphism of the projective line, there exist at least four distinct points in which the diffeomorphism is unusually well approximated by projective transformations [8]. The points in question are the ones in which the 3-jet of the diffeomorphism is that of a projective transformation; in a generic point the order of approximation is only 2. In other words, the Schwarzian derivative of every diffeomorphism of RP has at least four distinct zeroes. The theorem of Ghys is analogous to the classical four vertex theorem: a closed convex plane curve has at least four curvature extrema [11]. The proof presented by E. Ghys was purely geometrical: it was inspired by the Kneser theorem on osculating circles of a plane curve [10]. We will prove an amazing strengthening of the classical Sturm comparison theorem and deduce Ghys’ theorem from it. Sturm theory is related to the geometry of curves (see [2],[3],[4],[5],[9],[14], [15]). This relation is based on the following general idea. A geometrical problem is reformulated as the problem on the least number of zeroes of