Limit sets and global dynamic for 2-D divergence-free vector fields

The global structure of divergence-free vector fields on closed surfaces is investigated. We prove that if M is a closed surface and V is a divergence-free C-vector field with finitely many singularities on M then every orbit L of V is one of the following types: (i) a singular point, (ii) a periodic orbit, (iii) a closed (non periodic) orbit in M∗ = M − Sing(V), (iv) a locally dense orbit, where Sing(V) denotes the set of singular points of V . On the other hand, we show that the complementary in M of periodic components and minimal components is a compact invariant subset consisting of singularities and closed (non compact) orbits in M∗. These results extend those of T. Ma and S. Wang in [Discrete Contin. Dynam. Systems, 7 (2001), 431–445] established when the divergence-free vector field V is regular that is all its singular points are non-degenerate. 2010 Mathematics Subject Classification. Primary: 37A, 37B20, 37C10, 37E35