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## The existence of attractors of Weyl foliations modelled on pseudo-Riemannian manifolds

Journal of Physics: Conference Series. 2018. Vol. 990. No. 1. P. 1-15.

A foliation that admits a Weyl structure arising from a pseudo-Riemannian metric of any signature as its transverse structure is called a pseudo-Riemannian Weyl foliation or (for short) a Weyl foliation. We investigate codimension q ≥ 2 Weyl foliations on (not necessarily compact) manifolds. Different interpretations of their holonomy groups are given. We prove a criterion for a Weyl foliation to be pseudo-Riemannian. We find a condition on the holonomy groups which guarantees the existence of a transitive attractor of (M, F). Moreover, if the Weyl foliation is complete, this condition implies the existence of a global transitive attractor. We describe the structure of complete Weyl foliations modelled on Riemannian manifolds.

Zhukova N., Труды Московского физико-технического института 2017 Т. 9 № 4 С. 132-141

Complete transversely affine foliations are studied. The strong transversal equivalence of
complete affine foliations is investigated, which is a more refined notion than the transverse
equivalence of foliations in the sense of Molino. A global holonomy group of a complete
affine foliations is determined and it is proved that this group is the complete invariant
of the foliation relatively ...

Added: November 28, 2017

Bazaikin Y., Galaev A., Zhukova N., Chaos 2020 Vol. 30 P. 1-9

Chaotic foliations generalize Devaney's concept of chaos for
dynamical systems. The property of a foliation to
be chaotic is transversal. The existence problem of chaos for a Cartan foliation
is reduced to the corresponding problem for its holonomy pseudogroup of
local automorphisms of a transversal manifold. Chaotic foliations with transversal Cartan ...

Added: October 6, 2020

N. I. Zhukova, Journal of Mathematical Sciences 2016 Vol. 219 No. 1 P. 112-124

We consider a Cartan foliation (M,F) of an arbitrary codimension q admitting an
Ehresmann connection such that all leaves of (M,F) are embedded submanifolds of M.
We prove that for any foliation (M,F) there exists an open, not necessarily connected,
saturated, and everywhere dense subset M0 of M and a manifold L0 such that the induced
foliation (M0, FM0) ...

Added: October 21, 2016

Zhukova N., Differential Geometry and its Application 2021 Vol. 74 Article 101699

The existence problem for attractors of foliations with transverse linear connection
is investigated. In general foliations with transverse linear connection do not
admit attractors. A conditions that implies the existence of a global attractor which is
a minimal set, is specified. An application to transversely similar pseudo-Riemannian
foliations is obtained. The global structure of transversely similar Riemannian
foliations is described. ...

Added: October 20, 2020

Zhukova N., Chebochko N., Известия высших учебных заведений. Математика 2020 № 11 С. 87-92

The aim of this work is to describe the structure of complete Lorentzian foliations $(M, F)$ of codimension two
on $n$-dimensional closed manifolds. It is proved that $(M, F)$ is either Riemannian or has a constant
transversal curvature and its structure is described. For such foliations $(M, F)$, the criterion is obtained,
reducing the chaos problem in $(M, ...

Added: October 6, 2020

Zhukova N.I., K. I. Sheina, / Cornell University. Series math "arxiv.org". 2015. No. 1410.1144.

We get sufficient conditions for the full basic automorphism group of a complete
Cartan foliation to admit a unique (finite-dimensional) Lie group structure in the category
of Cartan foliations. In particular, we obtain sufficient conditions for this group
to be discrete. Emphasize that the transverse Cartan geometry may be noneffective.
Some estimates of the dimension of this group depending ...

Added: November 10, 2014

Zhukova N. I., Proceedings of the Steklov Institute of Mathematics 2012 Vol. 278 No. 1 P. 94-105

We prove that any compact manifold whose fundamental group contains an abelian normal subgroup of positive rank can be represented as a leaf of a structurally stable suspended foliation on a compact manifold. In this case, the role of a transversal manifold can be played by an arbitrary manifold. We construct examples of structurally stable ...

Added: October 19, 2014

Chernyshev V.L., Tolchennikov A. A., Ergodic Theory and Dynamical Systems 2018 Vol. 38 No. 5 P. 1697-1708

We study a topological space obtained from a graph via replacing vertices with smooth Riemannian manifolds, i.e. a decorated graph. We consider the following dynamical system on decorated graphs. Suppose that at the initial time we have a narrow wave packet on a one-dimensional edge. It can be thought of as a point moving along ...

Added: April 14, 2016

Nina I. Zhukova, Anna Yu. Dolgonosova .., Central European Journal of Mathematics 2013 Vol. 11 No. 12 P. 2076-2088

The category of foliations is considered. In this category
morphisms are differentiable mappings transforming leaves of one
foliation into leaves of the other foliation.
We proved that the automorphism group of the foliations
admitting a transverse linear connection is an infinite-dimensional
Lie group modeled on $LF$-spaces. This result extends the corresponding
result of Macias-Virgos and Sanmartin for Riemannian foliations.
In particular, our ...

Added: September 28, 2014

Н.И. Жукова, Mathematical Notes (Rusian Federation) 2013 Т. 93 № 6 С. 994-996

In this paper a unified method for studying foliations with transversal psrsbolic geometry of rank one is presented.
Ideas of Fraces' paper on parabolic geometry of rank one and of works of the author on conformal foliations
are developed. ...

Added: September 28, 2014

N. I. Zhukova, Sbornik Mathematics 2012 Vol. 203 No. 2 P. 380-405

We prove that every complete foliation (M, F) of codimension q > 1 is either Riemannian or a (Conf (S^q), S^q)-foliation. We further prove that if (M, 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 ...

Added: October 19, 2014

Zhukova N., Sheina K., Ufa Mathematical Journal 2022 Vol. 14 No. 1 P. 20-36

We study foliations of arbitrary codimension 𝑞 on 𝑛-dimensional smooth manifolds
admitting an integrable Ehresmann connection. The category of such foliations is
considered, where isomorphisms preserve both foliations and their Ehresman connections.
We show that this category can be considered as that of bifoliations covered by products.
We introduce the notion of a canonical bifoliation and we prove that ...

Added: March 23, 2022

Н.И. Жукова, Шеина К. И., Труды Математического центра им. Н.И. Лобачевского 2014 Т. 50 С. 74-76

We investigate Cartan foliations covered by fibrations. We obtain a sufficient condition for the full
basic automorphism group of a complete Cartan foliation covered by fibration to admit a
unique (finite-dimensional) Lie group structure in the category of
Cartan foliations. The explicit new formula for determining its basic automorphism
Lie group is given. Examples of computing the full basic ...

Added: November 12, 2014

Zhukova N., Журнал Средневолжского математического общества 2018 Т. 20 № 4 С. 395-407

It is shown that the structural theory of Molino for Riemannian foliations on compact
manifolds and complete Riemannian manifolds is generalized to Riemannian foliations with
Ehresmann connection. There are no restrictions on the codimension of the foliation
and the dimension of the foliated manifold.
For a Riemannian foliation $(M, F)$ with Ehresmann connection
it is proved that the closure of ...

Added: December 27, 2019

А.Ю. Долгоносова .., Н.И. Жукова, Труды Математического центра им. Н.И. Лобачевского 2013 Т. 47 С. 43-46

Different equivalent approaches to the notion of a foliation with transverse linear connection are
represented. ...

Added: October 18, 2014

N. I. Zhukova, Труды Математического института им. В.А. Стеклова РАН 2012 Т. 278 С. 102-113

We prove that any compact manifold whose fundamental group contains an abelian normal subgroup of positive rank can be represented as a leaf of a structurally stable suspended foliation on a compact manifold. In this case, the role of a transversal manifold can be played by an arbitrary manifold. We construct examples of structurally stable ...

Added: September 28, 2014

Sheina K., Известия высших учебных заведений. Поволжский регион. Физико-математические науки 2021 Т. 1 № 1 С. 49-65

The basic automorphism group of a Cartan foliation (M,F) is the quotient group of the automorphism group of (M, F ) by the normal subgroup, which preserves every leaf invariant. For Cartan foliations covered by fibrations, we find sufficient conditions for the existence of a structure of a finite-dimensional Lie group in their basic automorphism ...

Added: December 16, 2020

Zhukova N., Ufa Mathematical Journal 2019 Vol. 11 No. 3 P. 30-44

We study totally geodesic foliations (M, F) of arbitrary codimension q on n-dimensional pseudo-Riemannian manifolds, for which the induced metrics on leaves is nondegenerate.
We assume that the q-dimensional orthogonal distribution M to (M, F) is an
Ehresmann connection for this foliation. Since the usual graph G(F) is not Hausdorff
manifold in general, we study the graph GM(F) ...

Added: October 28, 2019

Zhukova N., Математический сборник 2012 Т. 203 № 3 С. 79-106

Доказано, что любое полное конформное слоение (M,F) коразмерности q> 2 является либо римановым, либо (Conf(S^q),S^q)-слоением. Если (M,F) не является римановым слоением, то оно имеет глобальный аттрактор, представляющий собой либо нетривиальное минимальное множество, либо один замкнутый слой или объединение двух замкнутых слоев. При этом компактность многообразия M не предполагается. В частности, каждое собственное полное конформное не риманово ...

Added: September 28, 2014

Zhukova N., Журнал Средневолжского математического общества 2017 Т. 19 № 4 С. 33-44

For any smooth orbifold $\mathcal N$ is constructed a foliated model, which is a foliation
with an Ehresmann, the leaf space of which is the same as $\mathcal N$. We investigate
the relationship relationship between some properties of orbifold and its foliated model.
The article discusses the application to Cartan orbifolds, that is orbifolds endowed with Cartan geometry. ...

Added: February 20, 2018

Zhukova N., Journal of Mathematical Sciences 2015 Vol. 208 No. 1 P. 115-130

We study the problem of classification of complete non-Riemannian conformal foliations
of codimension q > 2 with respect to transverse equivalence. It is proved that two
such foliations are transversally equivalent if and only if their global holonomy groups
are conjugate in the group of conformal transformations of the q-dimensional sphere
Conf (Sq). Moreover, any countable essential subgroup of ...

Added: December 11, 2017

Zhukova N. I., Mathematical notes 2013 Vol. 93 No. 5-6 P. 928-931

In this paper a unified method for studying foliations with transversal parabolic geometry of rank one is presented.
Ideas of Fraces' paper on parabolic geometry of rank one and of works of the author on conformal foliations
are developed. ...

Added: October 19, 2014

Nina. I. Zhukova, Galaev A., / Cornell University. Series math "arxiv.org". 2017.

The paper is focused on the existence problem of attractors for foliations. Since the existence of an attractor is a transversal property of the foliation, it is natural to consider foliations admitting transversal geometric structures. As transversal structures are chosen Cartan geometries due to their universality. The existence problem of an attractor on a complete ...

Added: March 23, 2017