Vyacheslav Grines
ul. Bolshaya Pecherskaya 25/12, Nizhnii Novgorod, 603155, Russia
National Research University Higher School of Economics
Professor: HSE Campus in Nizhny Novgorod, Faculty of Informatics,
Mathematics, and Computer Science (HSE Nizhny Novgorod), Department of Funda-
mental Mathematics;
Chief Research Fellow: HSE Campus in Nizhny Novgorod / Faculty of Informatics,
Mathematics, and Computer Science (HSE Nizhny Novgorod), International Laboratory
of Dynamical Systems and Applications
Born: December 13, 1946 in Isyaslavl', Ukraina.
Positions held:
2015-Present: Professor: HSE Campus in Nizhny Novgorod,
Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod),
Department of Fundamental Mathematics;
Chief Research Fellow: HSE Campus in Nizhny Novgorod, Faculty of Informatics,
Mathematics, and Computer Science (HSE Nizhny Novgorod), International Laboratory
of Dynamical Systems and Applications;
2013-2015: Professor of department of numerical and functional analysis, Lobachevskii
State University, Nizhnii Novgorod;
1977-2013: Professor of Mathematics, Head of department of mathematics of Nizhny
Novgorod State Agriculture Academy;
1969-1977: Researcher, Res.Inst. of Appl. Math.&Cybernetics, State University,
N.Novgorod
Scientific degrees:
1976: candidate of physical and mathematical sciences.
1998: doctor of physical and mathematical sciences.
Area of expertise:
Dynamical Systems and Foliations on Manifolds.
Publications:
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Baranov D. A., Grines V. Z., Pochinka O. V., Chilina E. E.
On a Classification of Periodic Maps on the 2-Torus
2023, Vol. 19, no. 1, pp. 91-110
Abstract
In this paper, following J. Nielsen, we introduce a complete characteristic of orientationpreserving
periodic maps on the two-dimensional torus. All admissible complete characteristics
were found and realized. In particular, each of the classes of orientation-preserving periodic
homeomorphisms on the 2-torus that are nonhomotopic to the identity is realized by an algebraic
automorphism. Moreover, it is shown that the number of such classes is finite. According to
V. Z. Grines and A.Bezdenezhnykh, any gradient-like orientation-preserving diffeomorphism of
an orientable surface is represented as a superposition of the time-1 map of a gradient-like flow
and some periodic homeomorphism. Thus, the results of this work are directly related to the
complete topological classification of gradient-like diffeomorphisms on surfaces.
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Grines V. Z., Zhuzhoma E. V.
Cantor Type Basic Sets of Surface $A$-endomorphisms
2021, Vol. 17, no. 3, pp. 335-345
Abstract
The paper is devoted to an investigation of the genus of an orientable closed surface $M^2$
which admits $A$-endomorphisms whose nonwandering set contains a one-dimensional strictly
invariant contracting repeller $\Lambda_r^{}$ with a uniquely defined unstable bundle and with
an admissible boundary of finite type. First, we prove that, if $M^2$ is a torus or a
sphere, then $M^2$ admits such an endomorphism. We also show that, if $ \Omega$ is a basic set with a uniquely defined unstable bundle of the endomorphism $f\colon M^2\to M^2$ of a closed orientable surface $M^2$ and $f$ is not a diffeomorphism, then $ \Omega$ cannot be a Cantor type expanding attractor. At last, we prove that, if $f\colon M^2\to M^2$ is an $A$-endomorphism whose nonwandering set consists of a finite number of isolated periodic sink orbits and a one-dimensional strictly invariant contracting repeller of Cantor type $\Omega_r^{}$ with a uniquely defined unstable bundle and such that the lamination consisting of stable manifolds of $\Omega_r^{}$ is regular, then $M^2$ is a two-dimensional torus $\mathbb{T}^2$ or a two-dimensional sphere $\mathbb{S}^2$.
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Grines V. Z., Kruglov E. V., Pochinka O. V.
The Topological Classification of Diffeomorphisms of the Two-Dimensional Torus with an Orientable Attractor
2020, Vol. 16, no. 4, pp. 595-606
Abstract
This paper is devoted to the topological classification of structurally stable diffeomorphisms
of the two-dimensional torus whose nonwandering set consists of an orientable one-dimensional
attractor and finitely many 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
pairwise nonintersecting arcs. The classification of one-dimensional basis sets on surfaces has
been exhaustively obtained in papers by V. Grines. He also obtained a classification of some
classes of structurally stable diffeomorphisms of surfaces using combined algebra-geometric invariants.
In this paper, we distinguish a class of diffeomorphisms that admit purely algebraic
differentiating invariants.
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Grines V. Z., Kurenkov E. D.
On hyperbolic attractors and repellers of endomorphisms
2017, Vol. 13, No. 4, pp. 557–571
Abstract
It is well known that the topological classification of dynamical systems with hyperbolic dynamics is significantly defined by dynamics on a nonwandering set. F. Przytycki generalized axiom $A$ for smooth endomorphisms that was previously introduced by S. Smale for diffeomorphisms, and proved the spectral decomposition theorem which claims that the nonwandering set of an $A$-endomorphism is a union of a finite number of basic sets. In the present paper the criterion for a basic set of an $A$-endomorphism to be an attractor is given. Moreover, dynamics on basic sets of codimension one is studied. It is shown that if an attractor is a topological submanifold of codimension one of type $(n − 1, 1)$, then it is smoothly embedded in the ambient manifold, and the restriction of the endomorphism to this basic set is an expanding endomorphism. If a basic set of type $(n, 0)$ is a topological submanifold of codimension one, then it is a repeller, and the restriction of the endomorphism to this basic set is also an expanding endomorphism. |
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Grines V. Z., Gurevich E. Y., Zhuzhoma E. V., Zinina S. K.
Heteroclinic curves of Morse – Smale cascades and separators in magnetic field of plasma
2014, Vol. 10, No. 4, pp. 427-438
Abstract
We obtain properties of three-dimensional phase space and dynamics of Morse–Smale diffeomorphism that led to existence of at least one heteroclinical curve in non-wandering set of the diffeomorphism. We apply this result to solve a problem of existence of separators in magnetic field of plasma.
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Grines V. Z., Levchenko Y. A., Pochinka O. V.
On topological classification of diffeomorphisms on 3-manifolds with two-dimensional surface attractors and repellers
2014, Vol. 10, No. 1, pp. 17-33
Abstract
We consider a class of diffeomorphisms on 3-manifolds which satisfy S. Smale’s axiom A such that their nonwandering set consists of two-dimensional surface basic sets. Interrelation between dynamics of such diffeomorphism and topology of the ambient manifold is studied. Also we establish that each considered diffeomorphism is Ω-conjugated with a model diffeomorphism of mapping torus. Under certain assumptions on asymptotic properties of two-dimensional invariant manifolds of points from the basic sets, we obtain necessary and sufficient conditions of topological conjugacy of structurally stable diffeomorphisms from the considered class.
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