Qualitative theory of foliations on closed surfaces

Abstract

S. Kh. Aranson, I. U. Bronstein, E. V. Zhuzhoma, and I.V. Nikolaev UDC 517.987.5; 517.933 Preface Foliations on surfaces are natural generalizations of flows (continuous-time dynamical systems) on sur- faces. According to the classical theory, in the neighborhood of a nonsingular point (i.e., a point different from the equilibrium state), the trajectories of a flow are a family of parallel straight lines. This is what served as a starting point for defining a foliation. Let M be a closed surface (a closed two-dimensional manifold). The foliation F on M with a set of singularities Sing (F) is the partitioning M - Sing (F) into nonintersecting curves (called fibers) which are locally homeomorphic to the family of straight lines. As to the set of singularities Sing (F), this set must be described separately for every class of fibers. In the sequel, unless otherwise specified, we shall consider the set Sing (F) to be finite. Foliations occupy an intermediate place between flows and arbitrary families of curves on surfaces. Foli- ations which can be embedded into a flow are said to be orientable; otherwise they are nonorientable. The origination of a qualitative theory of foliations goes back to the works by H. Kneser, G. Reeb, A. Haefliger, and S. P. Novikov. The theory of foliations attracted special attention early in the sixties in connection with the study of Y-flows and Y-diffeomorphisms introduced by D. V. Anosov. The technique of foliations allowed G. Franks to classify Y-diffeomorphisms of codimension one the nonwandering set of whose points coincides with the whole manifold. The application of a "surgical operation" to Y-diffeomorphisms of codimension one leads to nontrivial base sets of codimension one (attractors and repellers) the profound results in the study of whose geometry and topology belong to P. V. Plykin and V. Z. Grines. A more general approach was suggested by Ya. G. Sinai, who introduced a class of dynamical systems with two invariant transversal fibers. When we use this approach, we neglect such properties of the systems as the estimates of the contraction and extension of the fibers of invariant foliations, the everywhere density of the periodic trajectories in a nonwandering set, and others, and leave only those properties which are closely connected with the topology' of the manifolds. A new impetus to the study of foliations and homeomorphisms with invariant foliations was given by the works of W. Thurston, in which he complemented the homotopic classification of homeomorphisms of surfaces obtained by d. Nielson in 1920-30 and gave a new interpretation to it. The introduction by Thurston of the concept of pseudo-Anosov homeomorphism, which generalized the concept of the Anosov diffeomorphism, stimulated further investigations in this direction based on the study of the action of homeomorphisms in a fundamental group. Beginning in the 1980s, the geometry and topology of foliations with saddle-point singularities were studied by H. Rosenberg (the construction of labyrinths), G. Levitt (equivalence in the sense of Whitehead), G. Papandopoulos, K. Dantoni, and others. At the same time, many questions concerning the qualitative theory of foliations (the topological and smooth classifications, the structural stability, typicalness and so on) remained open. This review is devoted to the contemporary state of the qualitative theory of foliations with singularities on closed surfaces, tile main emphasis being placed on the development of the theory of Poincard-Bendixon Translated from ltogi Nauki i Tekhniki, Seriya Sovreme,l,mya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 32, Dynanlical Systems-5, 1996. 1072-3374/98/9003-2111520.00 9 Plenum Publishing Corporation 2111