Poincaré-Bendixson theory for leaves of codimension one

Abstract:

The level of a local minimal set of a ${C^2}$ codimension-one foliation of a compact manifold is a nonnegative integer defined inductively, level zero corresponding to the minimal sets in the usual sense. Each leaf of a local minimal set at level $k$ is at level $k$. The authors develop a theory of local minimal sets, level, and how leaves at level $k$ asymptotically approach leaves at lower level. This last generalizes the classical Poincaré-Bendixson theorem and provides information relating growth, topological type, and level, e.g. if $L$ is a totally proper leaf at level $k$ then $L$ has exactly polynomial growth of degree $k$ and topological type $k - 1$.