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## Chaotic Topological Foliations

Russian Mathematics. 2022. Vol. 66. No. 8. P. 66-70.

We call a foliation (M, F) on a manifold M chaotic if it is topologically transitive and the

union of closed leaves is dense in M. The chaotic topological foliations of arbitrary codimension on

n-dimensional manifolds can be considered as a multidimensional generalization of chaotic dynamical

systems in the Devaney sense. For topological foliations (M, F) covered by bundles, we prove that

a foliation (M, F) is chaotic if and only if its global holonomy group is chaotic. Applying the method

of suspension, a new countable family of pairwise nonisomorphic chaotic topological foliations of

codimension two on 3-dimensional closed and nonclosed manifolds is constructed.

Zhukova N., Levin G. S., Tonysheva N. S., Russian Mathematics 2022 Vol. 66 No. 8 P. 66-70

We call a foliation (M; F) on a manifold M chaotic if it is topologically transitive and the
union of closed leaves is dense in M. The chaotic topological foliations of arbitrary codimension on
n-dimensional manifold can be considered as multidimensional generalization of chaotic dynamical
systems in the sense of Devaney. For topological foliations covered by brations we ...

Added: January 31, 2023

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., Chebochko N., Итоги науки и техники. Современная математика и ее приложения. Тематические обзоры 2021 Т. 203 С. 17-38

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: November 17, 2021

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

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

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

Nina I. Zhukova, Journal of Geometry and Physics 2024 Vol. 199 Article 105166

We consider smooth codimension q foliations on n-dimensional manifolds where 0<q<n. We use Ehresmann connections as a technical tool to introduce the notion of sensitivity to initial conditions for foliations. We extend Devaney's definition of chaos for cascades to foliations with Ehresmann connection. Our main result states that sensitivity to initial conditions of a foliation ...

Added: March 10, 2024

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

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

N. I. Zhukova, N. S. Tonysheva, Journal of Mathematical Sciences 2023 Vol. 276 No. 1 P. 74-97

We construct infinite countable families of pairwise topologically nonconjugate free
chaotic groups of homeomorphisms on closed topological manifolds of different dimension.
We find new invariants of suspended foliations in some complete subcategory of
the category of foliations and use these results to construct infinite countable families of
pairwise nonisomorphic chaotic topological foliations of an arbitrary even codimension
on closed manifolds. ...

Added: October 9, 2023

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

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

Glutsyuk A., Ergodic Theory and Dynamical Systems 2024 Vol. 44 No. 5 P. 1418-1467

Reflection in a strictly convex bounded planar billiard acts on the space of oriented lines and preserves a standard area form. A caustic is a curve C whose tangent lines are reflected by the billiard to lines tangent to C. The famous Birkhoff conjecture states that the only strictly convex billiards with a foliation by ...

Added: December 29, 2023

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

Khoroshkin A., Transformation Groups 2015 P. 1-40

We prove the conjecture by Feigin, Fuchs and Gelfand describing the Lie algebra cohomology of formal vector fields on an n-dimensional space with coefficients in symmetric powers of the coadjoint representation. We also compute the cohomology of the Lie algebra of formal vector fields that preserve a given flag at the origin. The latter encodes ...

Added: April 9, 2015

Zhukova N., В кн. : Дифференциальные уравнения и их приложения в математическом моделировании: материалы XIII Международной научной конференции (Саранск, 12–16 июля 2017 г.). : Саранск : Средневолжское математическое общество (СВМО), 2017. С. 403-407.

Complete affine foliations (i.e., foliations admitting the affine geometry as the transversal structure) are investigated. The strong transversal equivalence of complete affine foliations is considered, which is a more refined notion than the transverse equivalence of foliations in the sense of Molino. The classification of complete affine foliations with respect to the strong transversal equivalence ...

Added: November 13, 2017

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

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 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 ...

Added: April 1, 2018

Zhukova N. I., Chubarov G. V., Journal of Mathematical Sciences 2022 Vol. 261 No. 3 P. 410-425

We study the graph G(F, Q) of a foliation F with Ehresmann connection Q. We prove that for any suspended foliation (M,F) the induced foliation on the graph G(F, Q) is also suspended. We describe the structure of the graph G(F, Q), as well as leaves of the induced foliation. We show that the induced ...

Added: March 14, 2022

N.I. Zhukova, Journal of Mathematical Physics, Analysis, Geometry 2013 Vol. 9 No. 3 P. 400-420

The equivalence between local stability and completeness and Quasi-analyticity is proved for an arbitrary compact foliation. We prove that a compact foliation Locally stable if and only if it admits an Ehresmann connection and has a quasi-analytical holonomy pseudo-group. As an application we prove the local stability of complete compact foliations with a rigid transverse ...

Added: October 2, 2014

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

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