La décomposition dynamique et la différentiabilité des feuilletages des surfaces. (The dynamic decomposition and the differentiability of foliations of surfaces) (Q1078855)

Let \({\mathcal F}\) be a singular foliation on a compact surface M. In order to analyse the dynamics of \({\mathcal F}\), one can canonically cut up M into subsurfaces bounded by curves transverse to \({\mathcal F}:\) the components of the recurrence of \({\mathcal F}\) (quasiminimal sets) are contained in the ''regions of recurrence'' and may be studied separately; on the other hand, the dynamics is trivial in the other regions (''regions of passage''). The paper also offers a definition of a singular foliation of class \(C^ r\) on M, and studies the topological and dynamical features of \(C^ 2\) (or \(C^{\infty})\) foliations.