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2013
Impact Factor

    Nina Zhukova

    B. Pecherskaja, 25/12, Nizhny Novgorod, 603155, Ru
    National Research University Higher School of Econ

    Publications:

    Zhukova N. I.
    Abstract
    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.
    Keywords: Reeb foliation, Reeb component, attractor of a foliation, category of foliations
    Citation: Zhukova N. I.,  Foliations of codimension one on a three-dimensional sphere with a countable family of compact attractor leaves, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 4, pp.  579–584
    DOI:10.20537/nd1704010
    Zhukova N. I.
    Weil foliations
    2010, Vol. 6, No. 1, pp.  219-231
    Abstract
    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.
    Keywords: Weil foliation, minimal set, attractor, holonomy group
    Citation: Zhukova N. I.,  Weil foliations, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 1, pp.  219-231
    DOI:10.20537/nd1001017

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