Vyacheslav Grines
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:
2015Present: 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;
20132015: Professor of department of numerical and functional analysis, Lobachevskii
State University, Nizhnii Novgorod;
19772013: Professor of Mathematics, Head of department of mathematics of Nizhny
Novgorod State Agriculture Academy;
19691977: 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:
Baranov D. A., Grines V. Z., Pochinka O. V., Chilina E. E.
Abstract
In this paper, following J. Nielsen, we introduce a complete characteristic of orientationpreserving
periodic maps on the twodimensional torus. All admissible complete characteristics
were found and realized. In particular, each of the classes of orientationpreserving periodic
homeomorphisms on the 2torus 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 gradientlike orientationpreserving diffeomorphism of
an orientable surface is represented as a superposition of the time1 map of a gradientlike flow
and some periodic homeomorphism. Thus, the results of this work are directly related to the
complete topological classification of gradientlike diffeomorphisms on surfaces.

Grines V. Z., Zhuzhoma E. V.
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 onedimensional 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 onedimensional 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 twodimensional torus $\mathbb{T}^2$ or a twodimensional sphere $\mathbb{S}^2$.

Grines V. Z., Kruglov E. V., Pochinka O. V.
Abstract
This paper is devoted to the topological classification of structurally stable diffeomorphisms
of the twodimensional torus whose nonwandering set consists of an orientable onedimensional
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 onedimensional 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 algebrageometric invariants.
In this paper, we distinguish a class of diffeomorphisms that admit purely algebraic
differentiating invariants.

Grines V. Z., Kurenkov E. D.
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. 
Grines V. Z., Gurevich E. Y., Zhuzhoma E. V., Zinina S. K.
Abstract
We obtain properties of threedimensional phase space and dynamics of Morse–Smale diffeomorphism that led to existence of at least one heteroclinical curve in nonwandering set of the diffeomorphism. We apply this result to solve a problem of existence of separators in magnetic field of plasma.

Grines V. Z., Levchenko Y. A., Pochinka O. V.
Abstract
We consider a class of diffeomorphisms on 3manifolds which satisfy S. Smale’s axiom A such that their nonwandering set consists of twodimensional 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 Ωconjugated with a model diffeomorphism of mapping torus. Under certain assumptions on asymptotic properties of twodimensional 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.
