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Topological classification of Morse-Smale diffeomorphisms without heteroclinic curves on 3-manifolds
Ergodic Theory and Dynamical Systems. 2019. Vol. 39. No. 9. P. 2403-2432.
We show that, up to topological conjugation,} the equivalence class of a Morse-Smale diffeomorphism without heteroclinic curves on a $3$-manifold is completely defined by an embedding of two-dimensional stable and unstable heteroclinic laminations to a characteristic space.
Publication based on the results of:
Pochinka O., Grines V., Mathematical notes 2013 Vol. 94 No. 6 P. 862-875
The results obtained in this paper are related to the Palis–Pugh problem on the existence of an arc withfinitely or countably many bifurcations which joins two Morse–Smale systems on a closed smooth manifoldMn . Newhouse and Peixoto showed that such an arc joining flows exists for any nand, moreover, it is simple. However, there exist ...
Added: September 11, 2014
Pochinka O., Morozov A., / Cornell University. Series arXiv "math". 2019.
In the present paper we consider preserving orientation Morse-Smale diffeomorphisms on surfaces. Using the methods of factorization and linearizing neighborhoods we prove that such diffeomorphisms have a finite number of orientable heteroclinic orbits. ...
Added: October 16, 2019
Bonatti C., Grines V., Pochinka O., Duke Mathematical Journal 2019 Vol. 168 No. 13 P. 2507-2558
Topological classification of even the simplest Morse-Smale diffeomorphisms on 3-manifolds does not fit into the concept of singling out a skeleton consisting of stable and unstable manifolds of periodic orbits. The reason for this lies primarily in the possible ``wild'' behaviour of separatrices of saddle points. Another difference between Morse-Smale diffeomorphisms in dimension 3 from ...
Added: October 23, 2017
Pochinka O., Grines V., Van Strien S., / Cornell University Library. 2017.
In this paper we give a complete topological classification of orientation preserving Morse-Smale diffeomorphisms on orientable closed surfaces. For MS diffeomorphisms with relatively simple behaviour it was known that such a classification
can be given through a directed graph, a three-colour directed graph or by a certain topological object, called a scheme. Here ...
Added: December 7, 2017
Grines V., Gurevich E., Pochinka O., Russian Mathematical Surveys 2017 Vol. 71 No. 6 P. 1146-1148
In the paper a Palis problem on finding sufficient conditions on embedding of Morse-Smale diffeomorphisms in topological flow is discussed. ...
Added: May 17, 2017
Pochinka O., Nozdrinova E., Dolgonosova A., Динамические системы 2017 Vol. 7(35) No. 2 P. 103-111
В настоящей работе рассматриваются диффеоморфизмы Морса-Смейла, заданные на неодносвязном замкнутом многообразии M^n,n>3. Для таких систем вводится понятие тривиальной (нетривиальной) связанности их периодических орбит. Устанавливается, что изотопные тривиальные и нетривиальные диффеоморфизмы не могут быть соединены дугой с бифуркациями коразмерности один. Построены примеры таких каскадов Морса-Смейла на многообразии S^(n-1)xS^1. ...
Added: November 16, 2017
Grines V., Gurevich E., Pochinka O. et al., Nonlinearity 2020 Vol. 33 No. 12 P. 7088-7113
We consider the class G(S^n) of orientation preserving Morse–Smale diffeomorphisms of the sphere S^n of dimension n > 3, assuming that invariant manifolds of different saddle periodic points have no intersection. For any diffeomorphism f ∈ G(S^n), we define a coloured graph Γ_f that describes a mutual arrangement of invariant manifolds of saddle periodic points of the diffeomorphism f. We enrich the graph Γ_f by an ...
Added: November 9, 2020
Pochinka O., Morozov A., Journal of Dynamical and Control Systems 2020 Vol. 26 No. 4 P. 629-639
In the present paper we consider preserving orientation Morse-Smale diffeomorphisms on surfaces. Using the methods of factorization and linearizing neighborhoods we prove that such diffeomorphisms have a finite number of orientable heteroclinic orbits. ...
Added: October 18, 2019
Pochinka O., Таланова Е. А., Shubin D., / Cornell University. Series arXiv "math". 2022.
Lens spaces are the only 3-manifolds that admit gradient-like flows with four fixed points. This is an immediate corollary of Morse inequality and of the Morse function with four critical points existence. A similar question for gradient-like diffeomorphisms is open. Solution can be approached by describing a complete topological conjugacy invariant of the class of ...
Added: September 13, 2022
Nozdrinov A., Починка А. И., Журнал Средневолжского математического общества 2023 Т. 25 № 1 С. 531-541
In this paper we consider gradient-like Morse-Smale diffeomorphisms defined on the three-dimensional sphere S 3 . For such diffeomorphisms, a complete invariant of topological conjugacy was obtained in the works of C. Bonatti, V. Grines, V. Medvedev, E. Pecu. It is an equivalence class of a set of homotopically non-trivially embedded tori and Klein bottles ...
Added: March 22, 2023
Pochinka O., Grines V., Gurevich E. et al., Математический сборник 2012 Т. 203 № 12 С. 81-104
В настоящей работе для разнообразий размерности 3 решена проблема Дж. Палиса о нахождении необходимых и достаточных условий включения каскада Иорса-Смейла в топологический поток. Кроме того, в работе выделен класс диффеоморфизмов, включающихся в топологический поток, для которых полным топологическим вариантом является граф, аналогичный схеме Е.А. Андроновой, А.Г. Майера и графу М. Пейкшото ...
Added: March 25, 2014
Morozov A., Russian Journal of Nonlinear Dynamics 2021 Vol. 17 No. 4 P. 465-473
According to the Nielsen-Thurston classification, the set of homotopy classes of orientation-preserving homeomorphisms of orientable surfaces is split into four disjoint subsets. Each subset consists of homotopy classes of homeomorphisms of one of the following types:
T1) periodic homeomorphism;
T_2) reducible non-periodic
homeomorphism of algebraically finite order;
T_3) a reducible homeomorphism that is not a homeomorphism of algebraically finite order;
T_4) pseudo-Anosov homeomorphism.
It is ...
Added: December 14, 2021
Grines V., Gurevich E., Pochinka O., Успехи математических наук 2016 Т. 71 № 6 С. 163-164
In a paper a solution of Palis problem on sufficient conditions of embedding of a Morse-Smale diffeomorphism in a topological flow is discussed. ...
Added: November 16, 2016
Динамически упорядоченная энергетическая функция для диффеоморфизмов Морса–Смейла на 3-многообразиях
Pochinka O., Лауденбах Ф., Grines V., Труды Математического института им. В.А. Стеклова РАН 2012 Т. 278 № 0,5 С. 34-48
Для произвольного диффеоморфизма Морса-Смейла трехмерного многообразия вводится понятие динамически упорядоченной функции Морса-Ляпунова, свойства которой тесно связаны с динамикой диффеоморфизма. Устанавливается, что необходимые и достаточные условия существования энергетической функции с такими свойствами определяются типом вложения одномерных аттракторов (репеллеров), каждый из которых является объединением нульмерных и одномерных неустойчивых (устойчивых) многообразий периодических орбит диффеоморфизма ...
Added: March 25, 2014
Zhuzhoma E. V., Medvedev V., Journal of Dynamical and Control Systems 2012 Vol. 18 No. 1 P. 21-36
We prove that simplest Morse-Smale systems can have locally flat and wildly embedded separatrices of saddle periodic point. ...
Added: October 17, 2014
Бонатти Х., Grines V., Pochinka O., Труды Математического института им. В.А. Стеклова РАН 2017 Т. 297 С. 46-61
В каждом классе топологической сопряженности по абстрактной схеме реализуется трехмерный сохраняющий ориентацию диффеоморфизм Морса–Смейла. ...
Added: September 5, 2017
Grines V., Gurevich E., Pochinka O. et al., Математические заметки 2012 Т. 91 № 5 С. 791-794
В работе получены необходимые и достаточные условия включения в топологический поток каскадов Морса-Смейла на 3-многообразиях. Условия являются обощением условий Палиса для каскадов на поверхностях. ...
Added: March 25, 2014
Zhuzhoma E. V., Medvedev V., Математические заметки 2012 Т. 92 № 4 С. 541-558
For Morse-Smale diffeomorphisms with three fixed points, one proves that the closure of separatrices are flat spheres provided the dimension of manifold is not less than six, and can be wildly embedded spheres provided the dimension of manifold is four. ...
Added: October 15, 2014
Pochinka O., Гринес В. З., Успехи математических наук 2013 Т. 68 № 1 (409) С. 129-188
Исследования связаны с каскадами Морса-Смейла на ориентируемых 3-многообразиях и включают в себя их полную топологическую классификацию, установление взаимосвязи их динамики с топологией объемлющего многообразия, критерий включения в топологический поток, а также необходимые и достаточные условия существования для таких каскадов энергетической функции. ...
Added: March 25, 2014
Shmukler V., Pochinka O., Таврический вестник информатики и математики 2021 Т. 50 № 1 С. 101-114
In this paper, we consider the class G of orientation-preserving Morse-Smale diffeomorphisms defined on a closed 3-manifold whose non-wandering set consists of exactly four pairwise distinct Morse indices. It is known that the two-dimensional saddle separatrices of any such diffeomorphism always intersect and their intersection necessarily contains non-compact heteroclinic curves, but may also contain compact ...
Added: September 24, 2021
О. В. Починка, Е. А. Таланова, Д. Д. Шубин, Математический сборник 2023 Т. 214 № 8 С. 94-107
It is known that the topological conjugacy of gradient-like 3-difepheromorphisms with a single saddle point is completely determined by the equivalence of Hopf knots on the manifold of S^2 × S^1, which are the projections of a one-dimensional saddle separatics in the basin of node point, and the ambient manifold for all such diffeomorphisms is ...
Added: July 27, 2023
Morozov A., Pochinka O., Журнал Средневолжского математического общества 2020 Т. 22 № 1 С. 71-80
In the present paper, we consider the class of orientation-preserving Morse-Smale diffeomorphisms f defined on an orientable surface M2. The work of A. A. Beznezhennykh and V. Z. Grines showed that such diffeomorphisms have a finite number of heteroclinic orbits. In addition, the classification problem of the diffeomorphisms under consideration is reduced to the problem ...
Added: June 22, 2020
191574970, Functional Analysis and Its Applications 2006 Vol. 40 No. 2 P. 81-90
It is well known that every module M over the algebra ℒ(X) of operators on a finite-dimensional space X can be represented as the tensor product of X by some vector space E, M ≅ = E ⊗ X. We generalize this assertion to the case of topological modules by proving that if X is a stereotype space with the stereotype approximation property, then for each stereotype module M over the ...
Added: September 23, 2016
Losev A. S., Slizovskiy S., JETP Letters 2010 Vol. 91 P. 620-624
Added: February 27, 2013