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

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

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

Bibliographic databases:

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

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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• http://mi.mathnet.ru/eng/msb7821
• https://doi.org/10.4213/sm7821
• http://mi.mathnet.ru/eng/msb/v203/i3/p79

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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
2. N. I. Zhukova, “Attractors of Foliations with Transversal Parabolic Geometry of Rank One”, Math. Notes, 93:6 (2013), 928–931
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
4. N. I. Zhukova, “Transverse equivalence of complete conformal foliations”, Journal of Mathematical Sciences, 208:1 (2015), 115–130
5. N. I. Zhukova, “Sloenye modeli dlya gladkikh orbifoldov i ikh primenenie”, Zhurnal SVMO, 19:4 (2017), 33–44
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
7. N. I. Zhukova, “Struktura rimanovykh sloenii so svyaznostyu Eresmana”, Zhurnal SVMO, 20:4 (2018), 395–407
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