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## The topological classification of structural stable 3-diffeomorphisms with two-dimensional basic sets

Nonlinearity. 2015. Vol. 28. P. 4081-4102.

In this paper we consider a class of structurally stable diffeomorphisms with two-dimensional basic sets given on a closed 3-manifold. We prove that each such diffeomorphism is a locally direct product of a hyperbolic automorphism of the 2-torus and a rough diffeomorphism of the circle. We find algebraic criteria for topological conjugacy of the systems.

Publication based on the results of:

Grines V., Gurevich E., Pochinka O., Современная математика. Фундаментальные направления 2020 Т. 66 № 2 С. 160-181

This review presents the results of recent years on solving of the J. Palis's problem on finding necessary and sufficient conditions for the embedding of Morse – Smale cascades in topological flows. To date, the problem has been solved by Palis for Morse-Smale diffeomophisms given on manifolds of dimension two. The result for the circle ...

Added: June 4, 2020

Gurevich E., Смирнова А. С., Динамические системы 2018 Т. 2 № 15 С. 159-172

We consider a class $G$ of Morse-Smale diffeomorphisms on the sphere $S^n$ of dimension $n\geq 4$ such that invariant manifolds of different saddle periodic points of any diffeomorphisms from $G$ have no intersection. Dynamics of an arbitrary diffeomorphism $f\in G$ can be represented as ``sink-source'' dynamics where the ``sink'' $A_f$ (the ``source'' $R_f$) is the ...

Added: November 2, 2018

Gurevich E., Труды Средневолжского математического общества 2015 Т. 17 № 3 С. 120-126

We define a class of gradient-like diffeomorphisms that can be presented as local products of diffeomorphisms on the circle and on a surface, provide their topological classification and specify topology of the ambient manifold. ...

Added: December 4, 2015

Grines V., Gurevich E., Zhuzhoma E. V. et al., Успехи математических наук 2019 Т. 74 № 1 С. 41-116

The review is devoted to the presentation of results, including recently obtained by the authors, on the topological classification of Morse-Smale systems and the topology of ambient manifolds. ...

Added: November 20, 2018

Ilyashenko Y., Kudryashov Y., Schurov I., / Cornell University. Series math "arxiv.org". 2015. No. 1506.06797.

This is the first part of a two parts paper dedicated to global bifurcations in the plane. In this part we construct an open set of three parameter families whose topological classification has a numerical invariant that may take an arbitrary positive value. In the second part we construct an open set of six parameter ...

Added: June 24, 2015

Grines V., Gurevich E., Pochinka O., Математические заметки 2019 Т. 105 № 1 С. 136-141

We provide a definition of a two-colored graph of a Morse-Smale diffeomorphism without heteroclinical intersection defined on the sphere $S^n$, $n\geq 4$ and prove that this graph is the complete topological invariant for such diffeomorphisms. ...

Added: October 13, 2018

Pochinka O., Nozdrinova E., Kolobianina A., Журнал Средневолжского математического общества 2018 Т. 20 № 4 С. 408-418

Mathematician A.G. Mayer from Nizhny Novgorod was the first who classify rough diffeomorphisms on a circle. This was one of the pioneering works on the topological classification of dynamic systems. However, the classification results in the work of A.G. Mayer obviously did not stand out, they were part of the proof of the roughness and ...

Added: October 4, 2018

Gurevich E., Malyshev D., Журнал Средневолжского математического общества 2016 Т. 18 № 4 С. 30-33

We consider a class $G$ of orientation preserving Morse-Smale diffeomorphisms without heteroclinical intersection defined on the sphere $S^{n}$ of dimension $n>3$. We put a colored graph $\Gamma_f$, endowed by a automorphism $P_f$ into the correspondence for every diffeomorphism $f\in G$ and give a definition of an isomorphism of such graphs. There is stated that there ...

Added: November 16, 2016

Kruglov V., Malyshev D., Pochinka O. et al., Discrete and Continuous Dynamical Systems 2020

In this paper, we study gradient-like flows without heteroclinic intersections on n-sphere up to topological conjugacy. We prove that such a flow is completely defined by a bi-colour tree corresponding to a skeleton formed by co-dimension one separatrices. Moreover, we show that such a tree is a complete invariant for these flows with respect to ...

Added: October 17, 2019

Pochinka O., Shubin D., / Cornell University. Series math "arxiv.org". 2022.

In the present paper, non-singular Morse-Smale flows on closed orientable 3-manifolds under the assumption that among the periodic orbits of the flow there is only one saddle one and it is twisted are considered. An exhaustive description of the topology of such manifolds is obtained. Namely, it has been established that any manifold admitting such ...

Added: January 30, 2023

Polotovskiy G., Борисов И. М., Итоги науки и техники. Современная математика и ее приложения. Тематические обзоры 2020 Т. 176 С. 3-18

The problem of topological classification of locations in the real projective plane of the union of nonsingular curves of degrees 2 and 6 is considered under some conditions of maximality and general position. After listing the permissible topological models of such locations to be investigated using the Orevkov method, based on the theory of braides ...

Added: October 25, 2019

Pochinka O., Shubin D., / Cornell University. Серия math "arxiv.org". 2021.

In the present paper the exhaustive topological classification of nonsingular Morse-Smale flows of n-manifolds with two limit cycles is presented. Hyperbolicity of periodic orbits implies that among them one is attracting and another is repelling. Due to Poincare-Hopf theorem Euler characteristic of closed manifold Mn which admits the considered flows is equal to zero. Only torus and Klein ...

Added: December 3, 2021

Kurenkov E., Gurevich E., Труды Средневолжского математического общества 2015 Т. 17 № 2 С. 15-26

We introduce the definition of consistent equivalence of energy Morse-Bott functions for Morse-Smale flows on surfaces and state that consistent equivalence of that functions is necessary and sufficient condition for such flows. ...

Added: October 12, 2015

Grines V., Pochinka O., Kruglov E. V., Russian Journal of Nonlinear Dynamics 2020 Vol. 16 No. 4 P. 595-606

This paper is devoted to the topological classification of structurally stable diffeomorphisms of the two-dimensional torus whose non-wandering set consists of an orientable one-dimensional attractor and finitely many of isolated source and saddle periodic points, under the assumption that the closure of the union of the stable manifolds of isolated periodic points consists of simple ...

Added: December 15, 2020

Gurevich E., Павлова Д. А., Журнал Средневолжского математического общества 2018 Т. 20 № 4 С. 378-383

We study a structure of four-dimensional phase space decomposition on trajectories of Morse-Smale flows admitting heteroclinical intersections. We consider a class $G(S^4)$ of Morse-Smale flows on the sphere $S^4$ such that for any flow $f\in G(S^4)$ its non-wandering set consists of exactly four equilibria: source, sink and two saddles. Wandering set of such flows ...

Added: November 11, 2018

Grines V., Gurevich E., Kurenkov E., Математические заметки 2020 Т. 107 № 1 С. 145-148

In the paper the topological classification of gradient-like flows on mapping tori is obtained. Such flows naturally appear in the modelling of processes with at least on cyclic coordinate. ...

Added: October 17, 2019

Grines V., Левченко Ю. А., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2012 Т. 447 № 2 С. 127-129

The paper is devoted to topological classifiication of cascades on 3-manifolds whose nonwandering set consists of surface 2-dimensional basic sets. ...

Added: February 25, 2015

Grines V., Gurevich E., Medvedev V., Труды Математического института им. В.А. Стеклова РАН 2020 Т. 310 С. 119-134

В работе рассматривается класс G(S^n) сохраняющих ориентацию диффеоморфизмов Морса-Смейла, заданных на сфере S^n размерности n≥4 в предположении, что инвариантные многообразия различных седловых периодических точек не пересекаются. Для диффеоморфизмов из этого класса описан алгоритм реализации всех классов топологической сопряженности. ...

Added: June 4, 2020

Kruglov V., Malyshev D., Pochinka O. et al., Regular and Chaotic Dynamics 2020 Vol. 25 No. 6 P. 716-728

In this paper, we study gradient-like flows without heteroclinic intersections on n-sphere up to topological conjugacy. We prove that such a flow is completely defined by a bi-colour tree corresponding to a skeleton formed by co-dimension one separatrices. Moreover, we show that such a tree is a complete invariant for these
flows with respect to the ...

Added: November 15, 2020

Grines V., Zhuzhoma E. V., Medvedev V. et al., Siberian Advances in Mathematics 2018 Т. 21 № 2 С. 163-180

In this paper, we study the relationship between the structure of the set of equilibrium states of a gradient-like flow and the topology of a carrier manifold of dimension 4 and higher. We introduce a class of manifolds admitting a generalized Heegaard decomposition. It is established that if a non-wandering set of a gradient-like flow ...

Added: May 27, 2018

Pochinka O., Shubin D., Applied Mathematics and Nonlinear Sciences 2020 Vol. 5 No. 2 P. 261-266

In the present paper we construct an example of 4-dimensional flows on $S^3\times S^1$ whose saddle periodic orbit has a wildly embedded 2-dimensional unstable manifold. We prove that such a property has every suspension under a non-trivial Pixton's diffeomorphism. Moreover we give a complete topological classification of these suspensions. ...

Added: October 14, 2019

Grines V., Malyshev D., Pochinka O. et al., Regular and Chaotic Dynamics 2016 Vol. 21 No. 2 P. 189-203

It is well known that the topological classification of structurally stable flows on surfaces as well as the topological classification of some multidimensional gradient-like systems can be reduced to a combinatorial problem of distinguishing graphs up to isomorphism. The isomorphism problem of general graphs obviously can be solved by a standard enumeration
algorithm. However, an efficient ...

Added: April 5, 2016

Grines V., Mints D., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2022 Т. 505 С. 66-70

We consider regular Denjoy type homeomorphisms of the two-dimensional torus which are the most natural generalization of Denjoy homeomorphisms of the circle. In particular, they arise as Poincaré maps induced on global cross sections by leaves of one-dimensional orientable unstable foliations of some partially hyperbolic diffeomorphisms of closed three-dimensional manifolds. The nonwandering set of each ...

Added: October 21, 2022

Morozov A., Pochinka O., Moscow Mathematical Journal 2023 Vol. 23 No. 4 P. 571-590

In this paper, we consider orientation-preserving Morse-Smale diffeomorphisms on orientable closed surfaces. Such diffeomorphisms can have infinitely many heteroclinic orbits, which makes their topological classification very difficult. In fact, even in the case of a finite number of heteroclinic orbits, there are no exhaustive classification results. The main problem is that for all currently known ...

Added: November 29, 2023