Qualitative properties of foliations on closed surfaces (Q1570903)

The aim of this paper is to present a survey of the results obtained by the author (and his collaborators) during the preceding 10 years, on the theory of qualitative properties of singular foliations on closed surfaces. The author presents several important basic results on the development of the theory, from its origin until recent times (he considers work of Poincaré, Kneser, Denjoy, Maier and Anosov). The results that are discussed in this paper concern the following problems: (1) The Poincaré-Bendixson theory and its generalizations. This is the study of the behaviour of individual leaves and semi-leaves, and the structure of their limit sets. (2) The ``Kneser problem'', that is, the problem of estimating the number of quasiminimal sets of a foliation. (3) The ``Anosov problem'', that is, the study of the connection between the asymptotic behaviour of leaves and the asymptotic behaviour of geodesic lines. In fact, in the 1960's, Anosov stated the problem of whether or not a closed surface can contain a semi-infinite curve without self-intersections whose lift to the universal cover has an asymptotic direction and which stays at a bounded distance from a geodesic, with respect to the Poincaré metric of the disk. (4) The topological classification of supertransitive foliations. (Here, a foliation \(F\) is said to be supertransitive if the following three conditions hold: (i) \(F\) is transitive, that is, there is a semi-leaf of \(F\) which is dense in the surface; (ii) \(F\) has a finite set of singularities, and each of these singularities is a saddle with a non-zero integral or semi-integral index; (iii) every leaf of \(F\) contains at most one singularity, and \(F\) has no cycles made up of singular leaves).