Nina Zhukova

In this paper we present an explicit construction of a continuum family of smooth pairwise nonisomorphic foliations of codimension one on a standard three-dimensional sphere, each of which has a countable set of compact attractors which are leaves diffeomorphic to a torus. As it was proved by S.P.Novikov, every smooth foliation of codimension one on a standard three-dimensional sphere contains a Reeb component. Changing this foliation only in the Reeb component by the method presented, we get a continuum family of smooth pairwise nonisomorphic foliations containing a countable set of compact attractor leaves diffeomorphic to a torus which coincides with the original foliation outside this Reeb component.

A foliation that admits a Weil geometry as its transverse structure is called by us a Weil foliation. We proved that there exists an attractor for any Weil foliation that is not Riemannian foliation. If such foliation is proper, there exists an attractor coincided with a closed leaf. The above assertions are proved without assumptions of compactness of foliated manifolds and completeness of the foliations.

We proved also that an arbitrary complete Weil foliation either is a Riemannian foliation, with the closure of each leaf forms a minimal set, or it is a trasversally similar foliation and there exists a global attractor. Any proper complete Weil foliation either is a Riemannian foliation, with all their leaves are closed and the leaf space is a smooth orbifold, or it is a trasversally similar foliation, and it has a unique closed leaf which is a global attractor of this foliation.