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.

In this work, by a dynamical system we mean a pair $(S, \,X)$, where $S$ is either a pseudogroup of local diffeomorphisms, or a transformation group, or a smooth foliation of the manifold $X$. The groups of transformations can be both discrete and nondiscrete. We define the concepts of attractor and global attractor of the dynamical system $(S, \,X)$ and investigate the properties of attractors and the problem of the existence of attractors of dynamical systems $(S, \,X)$. Compactness of attractors and ambient manifolds is not assumed. A property of the dynamical system is called transverse if it can be expressed in terms of the orbit space or the leaf space (in the case of foliations). It is shown that the existence of an attractor of a dynamical system is a transverse property. This property is applied by us in proving two subsequent criteria for the existence of an attractor (and global attractor): for foliations of codimension $q$ on an $n$-dimensional manifold, $0 < q < n$, and for foliations covered by fibrations. A criterion for the existence of an attractor that is a minimal set for an arbitrary dynamical system is also proven. Various examples of both regular attractors and attractors of transformation groups that are fractals are constructed.