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Mat. Sb., 2012, Volume 203, Number 3, Pages 79–106 (Mi msb7821)  

This article is cited in 7 scientific papers (total in 7 papers)

Global attractors of complete conformal foliations

N. I. Zhukova

N. I. Lobachevski State University of Nizhni Novgorod

Abstract: We prove that every complete conformal foliation $(M,\mathscr F)$ of codimension $q\ge 3$ is either Riemannian or a $(\operatorname{Conf}(S^q), S^q)$-foliation. We further prove that if $(M,\mathscr F)$ is not Riemannian, it has a global attractor which is either a nontrivial minimal set or a closed leaf or a union of two closed leaves. In this theorem we do not assume that the manifold $M$ is compact. In particular, every proper conformal non-Riemannian foliation $(M,\mathscr F)$ has a global attractor which is either a closed leaf or a union of two closed leaves, and the space of all nonclosed leaves is a connected $q$-dimensional orbifold. We show that every countable group of conformal transformations of the sphere $S^q$ can be realized as the global holonomy group of a complete conformal foliation. Examples of complete conformal foliations with exceptional and exotic minimal sets as global attractors are constructed.
Bibliography: 20 titles.

Keywords: conformal foliation, global holonomy group, minimal set, global attractor.

DOI: https://doi.org/10.4213/sm7821

Full text: PDF file (692 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2012, 203:3, 380–405

Bibliographic databases:

UDC: 514.77
MSC: Primary 37C85, 57R30; Secondary 22F05, 53C12
Received: 18.11.2010 and 12.05.2011

Citation: N. I. Zhukova, “Global attractors of complete conformal foliations”, Mat. Sb., 203:3 (2012), 79–106; Sb. Math., 203:3 (2012), 380–405

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Zhukova N.I., Chubarov G.V., “Obobschënnye nadstroechnye sloeniya”, Vestn. Nizhegorodskogo un-ta im. N. I. Lobachevskogo, 2012, no. 5-1, 157–164  elib
    2. N. I. Zhukova, “Attractors of Foliations with Transversal Parabolic Geometry of Rank One”, Math. Notes, 93:6 (2013), 928–931  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. Zhukova N.I., Dolgonosova A.Yu., “The automorphism groups of foliations with transverse linear connection”, Cent. Eur. J. Math., 11:12 (2013), 2076–2088  crossref  mathscinet  zmath  isi  elib  scopus
    4. N. I. Zhukova, “Transverse equivalence of complete conformal foliations”, Journal of Mathematical Sciences, 208:1 (2015), 115–130  crossref  zmath  scopus
    5. N. I. Zhukova, “Sloenye modeli dlya gladkikh orbifoldov i ikh primenenie”, Zhurnal SVMO, 19:4 (2017), 33–44  mathnet  crossref  elib
    6. Zhukova N.I., “The Existence of Attractors of Weyl Foliations Modelled on Pseudo-Riemannian Manifolds”, European Conference - Workshop Nonlinear Maps and Applications, Journal of Physics Conference Series, 990, IOP Publishing Ltd, 2018, UNSP 012014  crossref  mathscinet  isi  scopus
    7. N. I. Zhukova, “Struktura rimanovykh sloenii so svyaznostyu Eresmana”, Zhurnal SVMO, 20:4 (2018), 395–407  mathnet  crossref  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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