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## Knot as a complete invariant of a Morse-Smale 3-diffeomorphism with four fixed points

math.
arXiv.
Cornell University
,
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 considered diffeomorphisms and constructing of representative diffeomorphism for every conjugacy class by the abstract invariant. Ch. Bonnati and V. Z. Grines proved that the topological conjugacy class of Morse-Smale flows with unique saddle is defined by the equivalence class of the Hopf knot in S^2×S^1 which is projection of one-dimensional saddle separatrice and used the mentioned approach to prove that the ambient manifold of a diffeomorphism of this class is the three-dimensional sphere. In the present paper similar result is obtained for the gradient-like diffeomorphisms with exactly two saddle points and the unique heteroclinic curve.

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

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

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

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

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

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

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

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

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

Gurevich E., Chernov A., Ivanov A., Динамические системы 2020 Vol. 10 No. 2 P. 129-138

Manifolds admitting a Morse function with three critical points are called projective-like, by analogy with the projective plane. Eells and Kuiper showed that the dimension n of such manifolds takes on the values 2, 4, 8, and 16, and the critical points of the Morse function have indices 0, n / 2, and n. Zhuzhoma ...

Added: November 16, 2020

Grines V., Gurevich E., Zhuzhoma E. V. et al., Siberian Advances in Mathematics 2019 Vol. 29 No. 2 P. 116-127

We study relations between the structure of the set of equilibrium points of a gradient-like flows
and the topology of the support manifold of dimension 4 and higher. We introduce a class
of manifolds that admit a generalized Heegaard splitting. We consider gradient-like
flows such that the non-wandering set consists of exactly μ node and ν
saddle equilibrium points of indices equal to either 1 or n − 1. We show ...

Added: May 29, 2019

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

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

Akhmet’ev P., Medvedev T. V., Pochinka O., Qualitative Theory of Dynamical Systems 2021 Vol. 20 Article 76

For a wide class of dynamical systems known as Pixton diffeomorphisms the topological conjugacy class is completely defined by the Hopf knot equivalence class, i.e. the knot whose equivalence class under homotopy of the loops is a generator of the fundamental group π1(S2×S1)π1(S2×S1). Moreover, any Hopf knot can be realized by a Pixton diffeomorphism. Nevertheless, the ...

Added: August 24, 2021

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

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

Grines V., Gurevich E., Pochinka O., Moscow Mathematical Journal 2019 Vol. 19 No. 4 P. 739-760

J.~Palis found necessary conditions for a Morse-Smale diffeomorphism on a closed $n$-dimensional manifold $M^n$ to embed into a topological flow and proved that these conditions are also sufficient for $n=2$. For the case $n=3$ a possibility of wild embedding of closures of separatrices of saddles is an additional obstacle for Morse-Smale cascades to embed into ...

Added: October 17, 2019

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., 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., Shubin D., Nonsingular Morse-Smale flows of n-manifolds with attractor-repeller dynamics / 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

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., Gurevich E., Kevlia S. S., Lobachevskii Journal of Mathematics 2021 Vol. 42 No. 5 P. 901-910

We consider a class of gradient-like flows on three-dimensional closedmanifolds whose
attractors and repellers belongs to a finite union of embedded surfaces and find conditions when the
ambient manifold is Seifert. ...

Added: April 28, 2021

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

Grines V., Zhuzhoma E. V., Pochinka O., Journal of Mathematical Sciences 2019 Vol. 239 No. 5 P. 549-581

We revier the results describing the connection between the global dynamics of Morse-Smale systems on closed manifolds and the topology of carrier manifolds. Also we consider the rezults related to topological classification of Morse-Smale systems. ...

Added: May 18, 2019