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## Attractors of foliations with transversal parabolic geometry of rank one

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.

Zhukova N., 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

Zhukova N., Sheina K., Basic automorphism groups of complete Cartan foliations covered by fibrations / 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., 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., Galaev A. S., Attractors of Cartan foliation / 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

Sheina K., Basic automorphism of Cartan foliations covered by fibrations / Cornell University. Series arXiv "math". 2020. No. 04348v1.

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 groups. Estimates ...

Added: December 9, 2020

Popov V., The Jordan property for Lie groups and automorphism groups of complex spaces / Cornell University. Series math "arxiv.org". 2018. No. 1804.00323v1.

We prove that the family of all connected n-dimensional real Lie groups is uniformly Jordan for every n. This implies that all algebraic groups (not necessarily affine) over fields of cha\-racte\-ristic zero and some transformation groups of complex spaces and Riemannian manifods are Jordan. ...

Added: April 3, 2018

Popov V., Underlying varieties and group structures / Cornell University. Series math "arxiv.org". 2021. No. 2105.12861.

Starting with exploration of the possibility to present the underlying variety of an affine algebraic group in the form of a product of some algebraic varieties, we then explore the naturally arising problem as to what extent the group variety of an algebraic group determines its group structure. ...

Added: May 28, 2021

Chepyzhov V. V., Bekmaganbetov K. A., Chechkin G. A., Applicable Analysis 2019 Vol. 98 No. 1-2 P. 256-271

We consider reaction–diffusion systems with random rapidly oscillating coefficient. We do not assume any Lipschitz condition for the nonlinear function in the system, so, the uniqueness theorem for the corresponding initial-value problem may not hold for the considered reaction–diffusion system. Under the assumption that the random function is ergodic and statistically homogeneous in space variables we prove that the trajectory attractors ...

Added: November 11, 2020

Shirokov D., Advances in Applied Clifford Algebras 2010 Vol. 20 No. 2 P. 411-425

In this paper we present new formulas, which represent commutators and anticommutators of Clifford algebra elements as sums of elements of different ranks. Using these formulas we consider subalgebras of Lie algebras of pseudo-unitary groups. Our main techniques are Clifford algebras. We have found 12 types of subalgebras of Lie algebras of pseudo-unitary groups. ...

Added: June 16, 2015

Romanov A., Известия РАН. Серия математическая 2006 Т. 70 № 5 С. 163-178

Для эволюционных уравнений параболического типа c гильбертовым фазовым пространством E рассмотрена проблема эффективной (с липшицевой оценкой) конечной параметризации множеств K в E функционалами из E*, или, в иных терминах, проблема линейного липшицева вложения K в конечномерное евклидово пространство. Если K - глобальный аттрактор уравнения, то такого рода параметризация оказывается равносильной конечномерности динамики на K. Получен ряд признаков параметризации (в различных метриках) ...

Added: December 6, 2012

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., 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

Kharchev S. M., Khoroshkin S. M., Advances in Mathematics 2020 Vol. 375 No. 107368 P. 1-56

We obtain certain Mellin-Barnes integrals which present Whittaker wave functions related to classical real split forms of simple complex Lie groups ...

Added: October 18, 2020

Zhukova N., Journal of Geometry and Physics 2018 Vol. 132 P. 146-154

We present a new method of investigation of G-structures on orbifolds.
This method is founded on the consideration of a G-structure on an
n-dimensional orbifold as the corresponding transversal
structure of an associated foliation. Using this method we prove the
existence and the uniqueness of a finite dimensional Lie group structures
on the full automorphism group of an elliptic G-structure ...

Added: April 4, 2017

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

Zhukova N., 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

А.Ю. Долгоносова .., Zhukova N., 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

Zhukova N., 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., 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

Popov V., Mathematical notes 2018 Vol. 103 No. 5 P. 811-819

We prove that the family of all connected n-dimensional real Lie groups is uniformly Jordan for every n. This
implies that all algebraic (not necessarily affine) groups over fields of characteristic zero and some
transformation groups of complex spaces and Riemannian manifolds are Jordan. ...

Added: April 13, 2018

Kazakov A., Gonchenko S. V., Turaev D. V. et al., Physica D: Nonlinear Phenomena 2017 Vol. 350 P. 45-57

A one-parameter family of time-reversible systems on three-dimensional torus is considered. It is shown that the dynamics is not conservative, namely the attractor and repeller intersect but not coincide. We explain this as the manifestation of the so-called mixed dynamics phenomenon which corresponds to a persistent intersection of the closure of the stable periodic orbits ...

Added: October 13, 2017

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