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О классификации полных аффинных слоений относительно сильной трансверсальной эквивалентности
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 is reduced to the classification up to conjugacy of countable subgroups of the affine group $Aff(A^q)$. It is shown that each equivalence class contains a two-dimensional suspended foliation on the manifold, which is an Elenberg--MacLane space of type $K(\pi,1)$.
In book
Саранск: Средневолжское математическое общество (СВМО), 2017.
Dedaev R., Zhukova N., Russian Journal of Nonlinear Dynamics 2025 Vol. 21 No. 1 P. 85–102
In this work, by a dynamical system we mean a pair (S, X), where S is either a pseudogroup
of local diffeomorphisms, or a transformation group, or a smooth foliation of the manifold X.
The groups of transformations can be both discrete and nondiscrete. We define the concepts of
attractor and global attractor of the dynamical system (S, ...
Added: March 5, 2025
Zhukova N., Sheina K., Известия высших учебных заведений. Прикладная нелинейная динамика 2024 Т. 32 № 6 С. 897–907
The purpose of the work is to study the groups of basic automorphisms of chaotic Cartan foliations with Ehresmann
connection. Cartan foliations form a category where automorphisms preserve not only the foliation, but also its transverse Cartan geometry. The group of basic automorphisms of a foliation is the quotient group of the group of all automorphisms ...
Added: November 11, 2024
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
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
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
N. I. Zhukova, G. S. Levin, N. S. Tonysheva, 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 ...
Added: September 26, 2023
Е. И. Яковлев, В кн.: Труды Математического центра имени Н.И. Лобачевского. Том 62.: КФУ, 2022. С. 125–127.
Исследуются накрытия в категории расслоений Серра. Под накрывающим отображением одного расслоения на другое понимается морфизм указанной категории, состоящий из накрывающих отображений тотальных пространств и баз. Построены инварианты таких накрытий и доказана теорема существования накрытия с заданным инвариантом. Показано, что из локальной тривиальности базового расслоения следует аналогичное свойство для накрывающего расслоения. ...
Added: October 24, 2022
Н.И. Жукова, Г. С. Левин, Тонышева Н. С., Современная математика. Фундаментальные направления 2022 Т. 68 № 3 С. 424–450
Мы называем слоение (M, F) на топологическом многообразии M хаотическим, если оно топологически транзитивно и объединение всех замкнутых слоев всюду плотно в M. При этом компактность слоеного многообразия не предполагается. Исследуемые нами топологические слоения можно рассматривать как многомерные обобщения хаотических динамических систем в смысле Дивани. Для топологических слоений (M, F), накрытых расслоениями, мы доказываем, что ...
Added: September 10, 2022
Жукова Н.И., Г.С. Левин, Н.С. Тонышева, Известия высших учебных заведений. Математика 2022 № 8 С. 81–86
Мы называем слоение $(M, F)$ на топологическом многообразии $M$ хаотическим, если оно топологически транзитивно и объединение всех замкнутых слоев всюду плотно в $M$. Исследуемые нами хаотические топологические слоения произвольной коразмерности на $n$-мерных многообразиях можно рассматривать
как многомерные обобщения хаотических динамических систем в смысле Дивани. Для топологических слоений $(M, F)$, накрытых расслоениями, мы доказываем, что существование хаоса ...
Added: September 5, 2022
Yakovlev E., Russian Mathematics (USA) 2022 № 3 С. 71–84
The category of coverings of a given Serre bundle is investigated. By a covering mapping
of one such bundle onto another we mean a morphism in the category of bundles consisting of
covering mappings of total spaces and bases. An invariant of a covering from the indicated category
is constructed, which is the conjugacy class of some subsequence ...
Added: August 10, 2022
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
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
Yakovlev E., В кн.: Материалы Международной конференции по алгебре, анализу и геометрии 2021.: Каз.: Издательство Академии наук Республики Татарстан, 2021. С. 418–421.
Added: September 21, 2021
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